Finite Group Databasebeta

← #17005205

Subgroups — #17005205

Order 4352.

748 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538782C211yesmaximal, characteristic
2#14537523C211yesmaximal, characteristic, complemented
3D4×#63218C211yesmaximal, characteristic, complemented
4#14527670C211yesmaximal, characteristic, complemented
5#14527666C211yesmaximal, characteristic, complemented
6#14527614C211yesmaximal, characteristic, complemented
7#14527613C211yesmaximal, characteristic, complemented
8#14527613C211yesmaximal, characteristic, complemented
9#14527612C211yesmaximal, characteristic, complemented
10#14527576C211yesmaximal, characteristic, complemented
11#14527557C211yesmaximal, characteristic, complemented
12#14527233C211yesmaximal, characteristic, complemented
13D17×D4⋊C8C211yesmaximal, characteristic, complemented
14C8×#63218C211yesmaximal, characteristic, complemented
15C8×D8×C17C211yesmaximal, characteristic, complemented
16C2×#10587587—21no—
17C22×#63218—21no—
18#11763356C2211yescharacteristic
19C4×D4×D17C2211yescharacteristic
20#11763124C411yescharacteristic
21#11763121C411yescharacteristic
22#11763088C411yescharacteristic
23#11763087C411yescharacteristic
24#11763003C411yescharacteristic
25#11763002C411yescharacteristic
26#11762944C411yescharacteristic
27#11762941C411yescharacteristic
28C4×D136C2211yescharacteristic
29C4×#63228C2211yescharacteristic
30C4×#63228C2211yescharacteristic
31#11762888C2211yescharacteristic
32#11762888C2211yescharacteristic, complemented
33#11762887C2211yescharacteristic, complemented
34#11762886C2211yescharacteristic
35#11762883—21no—
36#11762880—21no—
37#11762880—21no—
38#11762877—21no—
39D4×#3622C2211yescharacteristic
40#11762874C2211yescharacteristic, complemented
41#11762860C2211yescharacteristic
42#11762831C2211yescharacteristic
43C4⋊C8×D17C2211yescharacteristic
44#11762774—21no—
45#11762771—21no—
46C4×#63218C2211yescharacteristic
47C4×C8×D17C2211yescharacteristic
48#11762562C2211yescharacteristic, complemented
49#11762561C2211yescharacteristic, complemented
50#11762561C2211yescharacteristic, complemented
51#11762538C2211yescharacteristic
52#11762538C2211yescharacteristic, complemented
53D4⋊#3622C2211yescharacteristic
54D4⋊#3622C2211yescharacteristic, complemented
55#11762536C2211yescharacteristic
56#11762536C2211yescharacteristic, complemented
57#11762531C2211yescharacteristic, complemented
58#11762530C2211yescharacteristic, complemented
59C8×#3622C2211yescharacteristic, complemented
60C4×D8×C17C2211yescharacteristic
61C8×D4×C17C2211yescharacteristic
62C8×D4×C17C2211yescharacteristic, complemented
63C17×#335C2211yescharacteristic, complemented
64C17×D4⋊C8C2211yescharacteristic
65C17×D4⋊C8C2211yescharacteristic, complemented
66C82×C17C2211yescharacteristic, complemented
67C2×D4×D17C2×C411yescharacteristic
68C2×#63252—21no—
69C22×C4×D17—21no—
70C2×D8×C17C2×C411yescharacteristic
71C22×C8×C17—21no—
72C22×C8×C17—21no—
73C4×D4×C17C2311yescharacteristic
74C4×D4×C17C2311yescharacteristic
75#10587626C2×C411yescharacteristic
76#10587624C2×C411yescharacteristic
77D4×Dic17C2×C411yescharacteristic
78C2×#63228C2×C411yescharacteristic
79C2×#63228C2×C411yescharacteristic
80C4×C22⋊D17—21no—
81C4×C22⋊D17—21no—
82C2×#63223—21no—
83C22×#3622—21no—
84C2×D136C2×C411yescharacteristic
85#10587587—44no—
86C2×#63218—21no—
87C2×#63218—21no—
88C2×#63218—21no—
89C2×#63218—21no—
90C2×#63218—44no—
91C2×#63218D411yescharacteristic
92C2×#63218D411yescharacteristic
93C2×C8×D17—21no—
94C2×C8×D17—21no—
95C2×C8×D17—21no—
96#10587581C2×C411yescharacteristic
97#10587579C2×C411yescharacteristic
98C4⋊C4×D17C2×C411yescharacteristic
99#10587574C2×C411yescharacteristic
100#10587571—21no—
101#10587570—21no—
102#10587567—21no—
103C22⋊C4×D17—21no—
104C4×D68C2311yescharacteristic
105C4×D68C2311yescharacteristic
106C42×D17C2311yescharacteristic
107C17×#107C2×C411yescharacteristic
108C17×C4⋊C8C2311yescharacteristic
109C17×C4⋊C8C2311yescharacteristic
110C17×#102C2×C411yescharacteristic
111C17×#102C2×C411yescharacteristic
112C17×C22⋊C8—21no—
113C17×C22⋊C8—21no—
114C4×C8×C17C2311yescharacteristic
115C4×C8×C17C2311yescharacteristic
116C4×C8×C17C2311yescharacteristic
117#10587503C2×C411yescharacteristic
118#10587503C2×C411yescharacteristic
119C22⋊#3622—21no—
120C22⋊#3622—21no—
121#10587492C2×C411yescharacteristic
122#10587492C2×C411yescharacteristic
123#10587491—21no—
124#10587491—21no—
125#10587491—21no—
126#10587491—21no—
127#10587491C2×C411yescharacteristic
128#10587491C2×C411yescharacteristic
129#10587489C2×C411yescharacteristic
130#10587486D411yescharacteristic
131#10587486D411yescharacteristic
132#10587485—21no—
133#10587485—21no—
134#10587485C2×C411yescharacteristic
135#10587485C2×C411yescharacteristic
136C8×Dic17—21no—
137#10587480C2×C411yescharacteristic
138#10587480C2×C411yescharacteristic
139#10587478C2×C411yescharacteristic
140#10587478C2×C411yescharacteristic
141C4⋊#3622C2311yescharacteristic
142C4⋊#3622C2311yescharacteristic
143C4×#3622C2311yescharacteristic
144C4×#3622C2311yescharacteristic
145C4×#3622C2311yescharacteristic
146C23×D17—21no—
147C2×D4×C17C22×C411yescharacteristic
148C2×D4×C17C22×C411yescharacteristic
149C22×C4×C17—21no—
150C22×C4×C17—21no—
151C2×C22⋊D17—21no—
152C2×C22⋊D17—21no—
153C2×C22⋊D17—21no—
154C2×C22⋊D17—21no—
155C22×Dic17—21no—
156D4×D17—22no—
157D4×D17—41no—
158#63252—42no—
159#63252—42no—
160C2×D68—21no—
161C2×D68C22×C411yescharacteristic
162C2×D68C22×C411yescharacteristic
163C2×C4×D17—21no—
164C2×C4×D17—21no—
165C2×C4×D17—21no—
166C2×C4×D17—21no—
167C2×C4×D17—42no—
168C2×C4×D17—42no—
169C2×C4×D17C22×C411yescharacteristic
170C2×C4×D17C2×D411yescharacteristic
171C2×C4×D17#4111yescharacteristic
172C2×Dic34—21no—
173D8×C17—22no—
174C2×C8×C17—21no—
175C2×C8×C17—21no—
176C2×C8×C17—21no—
177C2×C8×C17—42no—
178C2×C8×C17—42no—
179C2×C8×C17C22×C411yescharacteristic
180C2×C8×C17C22×C411yescharacteristic
181C2×C8×C17C2×D411yescharacteristic
182C2×C8×C17C2×D411yescharacteristic
183C4⋊C4×C17C22×C411yescharacteristic
184C4⋊C4×C17C22×C411yescharacteristic
185C22⋊C4×C17—21no—
186C22⋊C4×C17—21no—
187C42×C17C2411yescharacteristic
188C22⋊Dic17—21no—
189C22⋊Dic17—21no—
190#63228—22no—
191#63228—41no—
192#63227—21no—
193#63227—21no—
194#63227—21no—
195#63227—21no—
196C4⋊Dic17C22×C411yescharacteristic
197C4⋊Dic17C22×C411yescharacteristic
198#63225—21no—
199#63225—21no—
200C4×Dic17C22×C411yescharacteristic
201C4×Dic17C2×D411yescharacteristic
202C4×Dic17#4111yescharacteristic
203#63223—42no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—42no—
208C2×#3622C22×C411yescharacteristic
209C2×#3622C22×C411yescharacteristic
210C2×#3622C2×D411yescharacteristic
211C2×#3622C2×D411yescharacteristic
212D136—41no—
213#63218—22no—
214#63218—42no—
215#63218—42no—
216#63218—42no—
217#63218—42no—
218#63218D814yescomplemented
219C8×D17—42no—
220C8×D17—42no—
221C8×D17—42no—
222#17994—171nomaximal, Sylow
223C23×C17—21no—
224C23×C17—21no—
225C22×D17—21no—
226C22×D17—21no—
227C22×D17—42no—
228C22×D17—42no—
229C22×D17—81no—
230C22×D17C4×D411yescharacteristic
231D4×C17—21no—
232D4×C17—21no—
233D4×C17—21no—
234D4×C17#13112yescomplemented
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17C23×C411yescharacteristic
242C2×C4×C17C22×D411yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C22⋊D17—42no—
245C22⋊D17—42no—
246C22⋊D17—42no—
247C22⋊D17—81no—
248C2×Dic17—21no—
249C2×Dic17—21no—
250C2×Dic17—42no—
251C2×Dic17C4×D411yescharacteristic
252C2×Dic17C4×D411yescharacteristic
253C2×Dic17C4×D411yescharacteristic
254D68—21no—
255D68—21no—
256D68—21no—
257D68—21no—
258D68—42no—
259C4×D17—21no—
260C4×D17—21no—
261C4×D17—41no—
262C4×D17—42no—
263C4×D17—41no—
264C4×D17—41no—
265C4×D17—41no—
266C4×D17—41no—
267C4×D17—42no—
268C4×D17C2×D812yes—
269C4×D17#13512yes—
270Dic34—42no—
271C8×C17—22no—
272C8×C17—21no—
273C8×C17—21no—
274C8×C17—21no—
275C8×C17—41no—
276C8×C17—41no—
277C8×C17C2×D812yes—
278#3622—21no—
279#3622—21no—
280#3622—21no—
281#3622—41no—
282#3622—41no—
283#3622—41no—
284#3622C2×D812yes—
285#3100—171no—
286#2935—171no—
287D4×M16—171no—
288#1589—171no—
289#1589—171no—
290#1581—171no—
291#1581—171no—
292#1581—171no—
293#1581—171no—
294C8×D8—171no—
295C8×D8D1711yescharacteristic, complemented
296#1571—171no—
297#1492—171no—
298C2×D4⋊C8—171no—
299C8×M16—171no—
300C22×C17—22no—
301C22×C17—41no—
302C22×C17—41no—
303C22×C17—41no—
304C22×C17—41no—
305C22×C17—41no—
306C22×C17C2×C4×D411yescharacteristic
307C2×D17—21no—
308C2×D17—21no—
309C2×D17—41no—
310C2×D17—41no—
311C2×D17—41no—
312C2×D17—41no—
313C2×D17—41no—
314C2×D17—41no—
315C2×D17—84no—
316C2×D17C4×D812yes—
317C4×C17—21no—
318C4×C17—41no—
319C4×C17—41no—
320C4×C17C2×#13111yescharacteristic
321C4×C17#56811yescharacteristic
322C4×C17C22×D811yescharacteristic
323C4×C17C2×#13511yescharacteristic
324Dic17—21no—
325Dic17—21no—
326Dic17—41no—
327Dic17—41no—
328Dic17C4×D812yes—
329C2×#131—341no—
330C22×M16—341no—
331#532—171no—
332C2×C4×D4—171no—
333#480—171no—
334#480—171no—
335#474—171no—
336#474—171no—
337#459—171no—
338#459—171no—
339#447—171no—
340#447—171no—
341C4×D8—171no—
342C4×D8—171no—
343C4×D8—171no—
344C4×D8C2×D1711yescharacteristic
345#436—171no—
346#436—171no—
347#436—171no—
348#436—171no—
349#435—341no—
350#435—341no—
351#435—341no—
352#435—341no—
353C8×D4—171no—
354C8×D4—171no—
355C8×D4C2×D1711yescharacteristic
356C8×D4C2×D1711yescharacteristic, complemented
357#431—171no—
358#423—171no—
359C2×C4⋊C8—171no—
360#408—341no—
361#407—341no—
362C4×M16—171no—
363C2×C4×C8—171no—
364#335—171no—
365#335—171no—
366#335—171no—
367#335C2×D1711yescharacteristic, complemented
368D4⋊C8—171no—
369D4⋊C8—171no—
370D4⋊C8—171no—
371D4⋊C8—171no—
372D4⋊C8—171no—
373D4⋊C8—171no—
374D4⋊C8C2×D1711yescharacteristic
375D4⋊C8C2×D1711yescharacteristic, complemented
376#322—171no—
377#322—171no—
378C82—171no—
379C82C2×D1711yescharacteristic, complemented
380C2×C17—42no—
381C2×C17—81no—
382C2×C17#293311yescharacteristic
383C2×C17C2×C4×D811yescharacteristic
384C2×C17#295611yescharacteristic
385D17—22no—
386D17—81no—
387D17—81no—
388C2×#41—341no—
389C22×D4—171no—
390C23×C4—341no—
391C2×D8—171no—
392C2×D8—171no—
393C2×D8—171no—
394C2×D8C4×D1711yescharacteristic
395#131—684no—
396C2×M16—171no—
397C2×M16—171no—
398C2×M16—341no—
399C2×M16—341no—
400C2×M16—341no—
401C2×M16—341no—
402C2×M16—341no—
403C2×M16—684no—
404C22×C8—21no—
405C22×C8—21no—
406C22×C8—341no—
407C22×C8—341no—
408C22×C8—341no—
409C22×C8—341no—
410C4⋊Q8—171no—
411#127—171no—
412#124—341no—
413#121—171no—
414#121—171no—
415#121—341no—
416C4×D4—171no—
417C4×D4—171no—
418C4×D4—171no—
419C4×D4—171no—
420C4×D4—341no—
421C4×D4—341no—
422C4×D4—341no—
423C4×D4C22×D1711yescharacteristic
424C4×D4C22×D1711yescharacteristic
425C2×C4⋊C4—171no—
426C2×C22⋊C4—341no—
427C2×C42—171no—
428#107—171no—
429#107—171no—
430#107—171no—
431#107C4×D1711yescharacteristic
432C4⋊C8—171no—
433C4⋊C8—171no—
434C4⋊C8—171no—
435C4⋊C8—171no—
436C4⋊C8—341no—
437C4⋊C8—341no—
438C4⋊C8C22×D1711yescharacteristic
439C4⋊C8C22×D1711yescharacteristic
440#102—171no—
441#102—171no—
442#102—171no—
443#102—171no—
444#102—171no—
445#102—171no—
446#102C4×D1711yescharacteristic
447#102C4×D1711yescharacteristic
448C22⋊C8—21no—
449C22⋊C8—21no—
450C22⋊C8—171no—
451C22⋊C8—171no—
452C22⋊C8—341no—
453C22⋊C8—341no—
454C22⋊C8—341no—
455C22⋊C8—341no—
456C22⋊C8—341no—
457C22⋊C8—341no—
458#97—171no—
459#97—171no—
460C4×C8—171no—
461C4×C8—171no—
462C4×C8—171no—
463C4×C8—341no—
464C4×C8C22×D1711yescharacteristic
465C4×C8C22×D1711yescharacteristic
466C4×C8C22×D1711yescharacteristic
467C17#1799411yescharacteristic, complemented, Sylow
468C24—341no—
469#41—682no—
470#41—682no—
471C2×Q8—341no—
472C2×D4—171no—
473C2×D4—171no—
474C2×D4—341no—
475C2×D4—341no—
476C2×D4—341no—
477C2×D4—341no—
478C2×D4—342no—
479C2×D4—341no—
480C2×D4—681no—
481C2×D4C2×C4×D1711yescharacteristic
482C2×D4C2×C4×D1711yescharacteristic
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—171no—
486C22×C4—171no—
487C22×C4—171no—
488C22×C4—341no—
489C22×C4—341no—
490C22×C4—341no—
491C22×C4—341no—
492C22×C4—341no—
493C22×C4—682no—
494C22×C4—682no—
495D8—22no—
496D8—342no—
497D8—681no—
498D8—681no—
499M16—174no—
500M16—342no—
501M16—682no—
502M16—682no—
503M16—682no—
504M16—682no—
505M16—682no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—42no—
510C2×C8—42no—
511C2×C8—171no—
512C2×C8—171no—
513C2×C8—171no—
514C2×C8—171no—
515C2×C8—341no—
516C2×C8—341no—
517C2×C8—341no—
518C2×C8—682no—
519C2×C8—682no—
520C2×C8—682no—
521C2×C8—682no—
522C2×C8C2×C4×D1711yescharacteristic
523C2×C8C2×C4×D1711yescharacteristic
524C2×C8D4×D1711yescharacteristic
525C2×C8D4×D1711yescharacteristic
526C4⋊C4—171no—
527C4⋊C4—171no—
528C4⋊C4—341no—
529C4⋊C4—341no—
530C4⋊C4C2×C4×D1711yescharacteristic
531C4⋊C4C2×C4×D1711yescharacteristic
532C22⋊C4—21no—
533C22⋊C4—21no—
534C22⋊C4—341no—
535C22⋊C4—341no—
536C22⋊C4—341no—
537C22⋊C4—341no—
538C22⋊C4—341no—
539C22⋊C4—341no—
540C42—171no—
541C42—171no—
542C42—171no—
543C42C23×D1711yescharacteristic
544C23—21no—
545C23—21no—
546C23—171no—
547C23—341no—
548C23—341no—
549C23—682no—
550C23—682no—
551C23—1361no—
552Q8—682no—
553D4—21no—
554D4—21no—
555D4—21no—
556D4—341no—
557D4—341no—
558D4—341no—
559D4—341no—
560D4—682no—
561D4—682no—
562D4—682no—
563D4—682no—
564D4—1361no—
565D4#1058758712yescomplemented
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—171no—
573C2×C4—172no—
574C2×C4—172no—
575C2×C4—171no—
576C2×C4—171no—
577C2×C4—341no—
578C2×C4—341no—
579C2×C4—341no—
580C2×C4—341no—
581C2×C4—681no—
582C2×C4—682no—
583C2×C4—682no—
584C2×C4—682no—
585C2×C4—681no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4C22×C4×D1711yescharacteristic
590C2×C4C2×D4×D1711yescharacteristic
591C2×C4#41×D1711yescharacteristic
592C8—22no—
593C8—21no—
594C8—21no—
595C8—21no—
596C8—41no—
597C8—41no—
598C8—172no—
599C8—341no—
600C8—341no—
601C8—341no—
602C8—681no—
603C8—681no—
604C8—681no—
605C8D8×D1712yes—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—172no—
613C22—341no—
614C22—341no—
615C22—681no—
616C22—681no—
617C22—681no—
618C22—681no—
619C22—681no—
620C22—681no—
621C22—1364no—
622C22C4×D4×D1711yescharacteristic
623C4—21no—
624C4—41no—
625C4—41no—
626C4—172no—
627C4—341no—
628C4—341no—
629C4—681no—
630C4—681no—
631C4C2×#1058758711yescharacteristic
632C4#1176357411yescharacteristic
633C4C2×D8×D1711yescharacteristic
634C4#135×D1711yescharacteristic
635C2—42no—
636C2—81no—
637C2—342no—
638C2—1361no—
639C2—1361no—
640C2#1453750411yescharacteristic
641C2C4×D8×D1711yescharacteristic
642C2#1453768911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC8×D8×C17C2
socleC22×C17C2×C4×D4
prime radical#14538782C2
cosocleC42×C17C24
hypercenterC8×D8D17
hypocenterC17#17994
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×D8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #14537523C2semidirect
3. D4×#63218C2semidirect
4. #14527670C2semidirect
5. #14527666C2semidirect
6. #14527614C2semidirect
7. #14527613C2semidirect
8. #14527613C2semidirect
9. #14527612C2semidirect
10. #14527576C2semidirect
11. #14527557C2semidirect
12. #14527233C2semidirect
13. D17×D4⋊C8C2semidirect
14. C8×#63218C2semidirect
15. C8×D8×C17C2semidirect
32. #11762888C22semidirect
33. #11762887C22semidirect
40. #11762874C22semidirect
48. #11762562C22semidirect
49. #11762561C22semidirect
50. #11762561C22semidirect
52. #11762538C22semidirect
54. D4⋊#3622C22semidirect
56. #11762536C22semidirect
57. #11762531C22semidirect
58. #11762530C22semidirect
59. C8×#3622C22semidirect
62. C8×D4×C17C22semidirect
63. C17×#335C22semidirect
65. C17×D4⋊C8C22semidirect
66. C82×C17C22semidirect
218. #63218D8semidirect
234. D4×C17#131semidirect
295. C8×D8D17semidirect
356. C8×D4C2×D17semidirect
367. #335C2×D17semidirect
375. D4⋊C8C2×D17semidirect
379. C82C2×D17semidirect
467. C17#17994semidirect
565. D4#10587587semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#145387822
#145375232
D4×#632182
#145276702
#145276662
#145276142
#145276132
#145276122
#145275762
#145275572
#145272332
D17×D4⋊C82
C8×#632182
C8×D8×C172
#1799417