Finite Group Databasebeta

← #17005290

Subgroups — #17005290

Order 4352.

740 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#14527699C211yesmaximal, characteristic, complemented
5#14527695C211yesmaximal, characteristic, complemented
6#14527649C211yesmaximal, characteristic, complemented
7#14527648C211yesmaximal, characteristic, complemented
8#14527648C211yesmaximal, characteristic, complemented
9#14527647C211yesmaximal, characteristic, complemented
10#14527605C211yesmaximal, characteristic, complemented
11#14527553C211yesmaximal, characteristic, complemented
12#14527233C211yesmaximal, characteristic, complemented
13D17×D4⋊C8C211yesmaximal, characteristic, complemented
14#14527072C211yesmaximal, characteristic, complemented
15C17×#1580C211yesmaximal, 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#11762560C2211yescharacteristic, complemented
49#11762559C2211yescharacteristic, complemented
50#11762559C2211yescharacteristic, complemented
51#11762538C2211yescharacteristic
52#11762538C2211yescharacteristic, complemented
53D4⋊#3622C2211yescharacteristic
54D4⋊#3622C2211yescharacteristic, complemented
55#11762536C2211yescharacteristic
56#11762536C2211yescharacteristic, complemented
57#11762531C2211yescharacteristic, complemented
58#11762531C2211yescharacteristic, complemented
59#11762531C2211yescharacteristic, complemented
60C4×D8×C17C2211yescharacteristic
61C8×D4×C17C2211yescharacteristic
62C8×D4×C17C2211yescharacteristic, complemented
63C17×#334C2211yescharacteristic, complemented
64C17×D4⋊C8C2211yescharacteristic
65C17×D4⋊C8C2211yescharacteristic, complemented
66C17×#322C2211yescharacteristic, 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—22no—
215#63218—42no—
216#63218—42no—
217#63218—42no—
218#63218—42no—
219C8×D17—42no—
220C8×D17—42no—
221C8×D17—42no—
222#18001—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×D17AGL(1,ℤ/8)12yes—
269C4×D17AGL(1,ℤ/8)12yes—
270Dic34—42no—
271C8×C17—21no—
272C8×C17—21no—
273C8×C17—21no—
274C8×C17—21no—
275C8×C17—41no—
276C8×C17—41no—
277C8×C17—41no—
278#3622—22no—
279#3622—21no—
280#3622—21no—
281#3622—21no—
282#3622—21no—
283#3622—41no—
284#3622—41no—
285#3100—171no—
286#2935—171no—
287D4×M16—171no—
288#1594—171no—
289#1594—171no—
290#1586—171no—
291#1586—171no—
292#1586—171no—
293#1586—171no—
294#1580—171no—
295#1580D1711yescharacteristic, complemented
296#1570—171no—
297#1492—171no—
298C2×D4⋊C8—171no—
299#1451—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×D17#44212yes—
317C4×C17—21no—
318C4×C17—41no—
319C4×C17—41no—
320C4×C17C2×#13111yescharacteristic
321C4×C17#56811yescharacteristic
322C4×C17C2×AGL(1,ℤ/8)11yescharacteristic
323C4×C17C2×AGL(1,ℤ/8)11yescharacteristic
324Dic17—21no—
325Dic17—21no—
326Dic17—41no—
327Dic17—41no—
328Dic17#44212yes—
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#334—171no—
365#334—171no—
366#334—171no—
367#334C2×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—
378#322—171no—
379#322C2×D1711yescharacteristic, complemented
380C2×C17—42no—
381C2×C17—81no—
382C2×C17#293311yescharacteristic
383C2×C17C2×#44211yescharacteristic
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#1800111yescharacteristic, 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—342no—
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—21no—
593C8—21no—
594C8—21no—
595C8—21no—
596C8—41no—
597C8—41no—
598C8—41no—
599C8—342no—
600C8—341no—
601C8—341no—
602C8—341no—
603C8—341no—
604C8—681no—
605C8—681no—
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
633C4AGL(1,ℤ/8)×D1711yescharacteristic
634C4AGL(1,ℤ/8)×D1711yescharacteristic
635C2—42no—
636C2—81no—
637C2—342no—
638C2—1361no—
639C2—1361no—
640C2#1453750411yescharacteristic
641C2D17×#44211yescharacteristic
642C2#1453769211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#1580C2
socleC22×C17C2×C4×D4
prime radical#14538782C2
cosocleC42×C17C24
hypercenter#1580D17
hypocenterC17#18001
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1580

Complements

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

NH ≅ G/Ntype
2. #14537523C2semidirect
3. D4×#63218C2semidirect
4. #14527699C2semidirect
5. #14527695C2semidirect
6. #14527649C2semidirect
7. #14527648C2semidirect
8. #14527648C2semidirect
9. #14527647C2semidirect
10. #14527605C2semidirect
11. #14527553C2semidirect
12. #14527233C2semidirect
13. D17×D4⋊C8C2semidirect
14. #14527072C2semidirect
15. C17×#1580C2semidirect
32. #11762888C22semidirect
33. #11762887C22semidirect
40. #11762874C22semidirect
48. #11762560C22semidirect
49. #11762559C22semidirect
50. #11762559C22semidirect
52. #11762538C22semidirect
54. D4⋊#3622C22semidirect
56. #11762536C22semidirect
57. #11762531C22semidirect
58. #11762531C22semidirect
59. #11762531C22semidirect
62. C8×D4×C17C22semidirect
63. C17×#334C22semidirect
65. C17×D4⋊C8C22semidirect
66. C17×#322C22semidirect
234. D4×C17#131semidirect
295. #1580D17semidirect
356. C8×D4C2×D17semidirect
367. #334C2×D17semidirect
375. D4⋊C8C2×D17semidirect
379. #322C2×D17semidirect
467. C17#18001semidirect
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
#145276992
#145276952
#145276492
#145276482
#145276472
#145276052
#145275532
#145272332
D17×D4⋊C82
#145270722
C17×#15802
#1800117