Finite Group Databasebeta

← #17097392

Subgroups — #17097392

Order 4352.

900 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763096C211yesmaximal, characteristic
2C2×C4×#63230C211yesmaximal, characteristic
3C4⋊C4×D68C211yesmaximal, characteristic
4#14530327C211yesmaximal, characteristic, complemented
5#14530260C211yesmaximal, characteristic, complemented
6#14530165C211yesmaximal, characteristic, complemented
7#14530114C211yesmaximal, characteristic, complemented
8#14529509C211yesmaximal, characteristic, complemented
9#14529343C211yesmaximal, characteristic, complemented
10#14529121C211yesmaximal, characteristic
11#14529121C211yesmaximal, characteristic
12#14529052C211yesmaximal, characteristic, complemented
13#14529044C211yesmaximal, characteristic
14#14529044C211yesmaximal, characteristic
15C17×#2451C211yesmaximal, characteristic, complemented
16C22×#63230C2211yescharacteristic
17C2×C4×D68C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19C2×C4×D68C2211yescharacteristic
20C2×C4⋊C4×D17—21no—
21#11763096—24no—
22C4×#63230—24no—
23C2×#10587479C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×#10587479C2211yescharacteristic
27C2×#10587479C2211yescharacteristic
28C2×C4⋊#3622C2211yescharacteristic
29C2×#10587506C2211yescharacteristic
30C2×#10587506C2211yescharacteristic
31C2×#10587506C2211yescharacteristic
32C2×#10587480C2211yescharacteristic
33C2×#10587480C2211yescharacteristic
34C2×#10587480C2211yescharacteristic
35C2×C4×#3622C2211yescharacteristic
36#11762707C2211yescharacteristic
37#11762707C2211yescharacteristic
38#11762683—21no—
39#11762683—21no—
40#11762679C2211yescharacteristic
41#11762666—21no—
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C2×C17×C4⋊Q8C2211yescharacteristic
51C2×C4×Q8×C17C2211yescharacteristic
52C17×#398—21no—
53C17×#395—21no—
54C17×#391—21no—
55C17×#391C2211yescharacteristic
56C17×#391C2211yescharacteristic
57C17×#389C2211yescharacteristic
58C17×#389C2211yescharacteristic
59C17×#387—21no—
60C4×C4⋊C4×C17C2211yescharacteristic
61C22×Q8×C17—21no—
62C22×Q8×C17C2311yescharacteristic
63C22×D68C2311yescharacteristic
64C22×C4×D17—21no—
65C22×C4×D17—21no—
66C22×C4×D17—21no—
67C17×C4⋊Q8—22no—
68C17×C4⋊Q8—41no—
69C4×Q8×C17—22no—
70C4×Q8×C17—41no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17D411yescharacteristic
76C2×C4⋊C4×C17D411yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C42×C17—21no—
81C2×C42×C17C2311yescharacteristic
82C2×C42×C17C2311yescharacteristic
83C2×C42×C17C2311yescharacteristic
84C2×#63230—22no—
85C2×#63230—22no—
86C2×#63230Q814yescomplemented
87C2×#63227—21no—
88C2×#63227—21no—
89C2×#63227—21no—
90C2×C4⋊Dic17—21no—
91C2×C4⋊Dic17—21no—
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×#63225—21no—
96C2×#63225—21no—
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101C4⋊C4×D17—44no—
102C4×D68—22no—
103C4×D68—22no—
104C4×D68—22no—
105C4×D68—22no—
106C4×D68—41no—
107C4×D68—41no—
108C17×#95—21no—
109C17×#95—21no—
110C17×#95—21no—
111C17×#95—21no—
112C17×#95—21no—
113C17×#95—21no—
114C17×#95D411yescharacteristic
115C17×#95D411yescharacteristic
116#10587506—22no—
117#10587506—22no—
118#10587506—41no—
119#10587502—21no—
120#10587502—21no—
121#10587480—22no—
122#10587480—22no—
123#10587480D414yes—
124#10587479—22no—
125#10587479—22no—
126#10587479—22no—
127#10587479—22no—
128#10587479—41no—
129C4⋊#3622—22no—
130C4×#3622—22no—
131C23×D17—21no—
132C2×Q8×C17—21no—
133C2×Q8×C17—21no—
134C2×Q8×C17—21no—
135C2×Q8×C17—21no—
136C2×Q8×C17—21no—
137C2×Q8×C17—42no—
138C2×Q8×C17—42no—
139C2×Q8×C17—81no—
140C2×Q8×C17C2×Q812yes—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—21no—
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2411yescharacteristic
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C2×D68—21no—
156C2×D68—21no—
157C2×D68—21no—
158C2×D68C2×D412yes—
159C2×D68C2×Q812yes—
160C2×D68#4112yes—
161C2×C4×D17—24no—
162C2×C4×D17—42no—
163C2×C4×D17—42no—
164C2×C4×D17—42no—
165C2×C4×D17—42no—
166C2×C4×D17—42no—
167C2×C4×D17—42no—
168C2×C4×D17—81no—
169C2×C4×D17—81no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17—21no—
174C4⋊C4×C17—21no—
175C4⋊C4×C17—41no—
176C4⋊C4×C17—42no—
177C4⋊C4×C17—42no—
178C4⋊C4×C17—81no—
179C4⋊C4×C17—81no—
180C4⋊C4×C17SD1614yescomplemented
181C4⋊C4×C17C2×D412yes—
182C42×C17—21no—
183C42×C17—21no—
184C42×C17—21no—
185C42×C17—21no—
186C42×C17—21no—
187C42×C17—21no—
188C42×C17—42no—
189#63230—28no—
190#63227—24no—
191#63227—42no—
192#63227—42no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—42no—
199C4⋊Dic17—81no—
200C4⋊Dic17#4112yes—
201#63225—42no—
202#63225—42no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622—21no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622C2×D412yes—
214C2×#3622C2×Q812yes—
215#26376—171nomaximal, Sylow
216C23×C17C22×D411yescharacteristic
217C22×D17—22no—
218C22×D17—22no—
219C22×D17—22no—
220C22×D17—41no—
221C22×D17—41no—
222C22×D17—41no—
223C22×D17—41no—
224Q8×C17—22no—
225Q8×C17—22no—
226Q8×C17—41no—
227Q8×C17—41no—
228Q8×C17—84no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—22no—
239C2×C4×C17—22no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17C2×SD1612yes—
257C2×C4×C17C2×SD1612yes—
258C2×C4×C17C2×SD1612yes—
259C2×C4×C17C22×D411yescharacteristic
260C2×C4×C17C22×Q811yescharacteristic
261C2×C4×C17C2×#4111yescharacteristic
262C2×C4×C17C2×#4111yescharacteristic
263C2×C4×C17H(5,𝔽2)11yescharacteristic
264C2×C4×C172−1+411yescharacteristic
265C2×Dic17—22no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274D68—22no—
275D68—22no—
276D68—22no—
277D68C22⋊Q814yes—
278C4×D17—48no—
279C4×D17—84no—
280C4×D17—84no—
281#3622—22no—
282#3622—22no—
283#3622—41no—
284#3622—41no—
285C2×#476—171no—
286C2×C4×SD16—171no—
287#2451D1711yescharacteristic, complemented
288D4×C4⋊C4—171no—
289#2096—171no—
290#2084—171no—
291#2066—171no—
292#2058—171no—
293#1941—171no—
294#1916—171no—
295#1877—171no—
296#1877—171no—
297#1865—171no—
298#1864—171no—
299#1864—171no—
300C22×C17#52511yescharacteristic
301C22×C17D4211yescharacteristic
302C22×C17D4×Q811yescharacteristic
303C22×C17C22×SD1611yescharacteristic
304C22×C17#57511yescharacteristic
305C22×C17#57711yescharacteristic
306C22×C17#57711yescharacteristic
307C2×D17—24no—
308C2×D17—42no—
309C2×D17—42no—
310C2×D17—42no—
311C2×D17—42no—
312C2×D17—42no—
313C2×D17—42no—
314C4×C17—22no—
315C4×C17—22no—
316C4×C17—22no—
317C4×C17—41no—
318C4×C17—42no—
319C4×C17—41no—
320C4×C17—41no—
321C4×C17—42no—
322C4×C17—81no—
323C4×C17—81no—
324C4×C17C2×C22⋊Q811yescharacteristic
325C4×C17#53911yescharacteristic
326C4×C17#54711yescharacteristic
327C4×C17#55411yescharacteristic
328Dic17—42no—
329Dic17—81no—
330Dic17—81no—
331C22×SD16—171no—
332C2×C4⋊Q8C2×D1711yescharacteristic
333C2×C4×Q8C2×D1711yescharacteristic
334C2×C4×D4—171no—
335C2×C4×D4—171no—
336C2×C4×D4—171no—
337C22×C4⋊C4—341no—
338#476—344no—
339C4×SD16—344no—
340C2×#106—171no—
341C2×#106—171no—
342C2×#106—171no—
343C2×#106—171no—
344C2×#106—171no—
345C2×C4⋊C8—171no—
346C2×#103—171no—
347C2×#103—171no—
348C2×#103—171no—
349C2×#102—171no—
350C2×#102—171no—
351C2×#102—171no—
352C2×C4×C8—171no—
353#398—21no—
354#395—21no—
355#391—21no—
356#391C2×D1711yescharacteristic
357#391C2×D1711yescharacteristic
358#389—171no—
359#389—171no—
360#389C2×D1711yescharacteristic
361#389C2×D1711yescharacteristic
362#387—21no—
363#385—341no—
364#385—341no—
365#384—171no—
366#380—341no—
367C4×C4⋊C4C2×D1711yescharacteristic
368#340—171no—
369#340—171no—
370#340—171no—
371#340—171no—
372#336—171no—
373#336—171no—
374#336—171no—
375#336—171no—
376C2×C17#244911yescharacteristic
377C2×C17#310111yescharacteristic
378C2×C17#312511yescharacteristic
379C2×C17#326711yescharacteristic
380C2×C17D4×SD1611yescharacteristic
381C2×C17Q8×SD1611yescharacteristic
382C2×C17#338811yescharacteristic
383D17—44no—
384C22×Q8—21no—
385C22×Q8C22×D1711yescharacteristic
386C22×D4—171no—
387C23×C4—341no—
388C23×C4—341no—
389C23×C4—341no—
390C2×SD16—174no—
391C2×SD16—342no—
392C2×SD16—342no—
393C22×C8—171no—
394C22×C8—171no—
395C22×C8—171no—
396C22×C8—171no—
397C4⋊Q8—22no—
398C4⋊Q8—41no—
399C4×Q8—22no—
400C4×Q8—41no—
401C4×D4—342no—
402C4×D4—342no—
403C4×D4—342no—
404C4×D4—342no—
405C4×D4—681no—
406C4×D4—681no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—171no—
412C2×C4⋊C4—171no—
413C2×C4⋊C4—171no—
414C2×C4⋊C4—341no—
415C2×C4⋊C4—341no—
416C2×C4⋊C4—341no—
417C2×C4⋊C4—341no—
418C2×C4⋊C4—684no—
419C2×C4⋊C4C22⋊D1711yescharacteristic
420C2×C4⋊C4C22⋊D1711yescharacteristic
421C2×C4⋊C4C22×D1711yescharacteristic
422C2×C4⋊C4C22×D1711yescharacteristic
423C2×C4⋊C4C22×D1711yescharacteristic
424C2×C22⋊C4—341no—
425C2×C22⋊C4—341no—
426C2×C22⋊C4—341no—
427C2×C42—21no—
428C2×C42C22×D1711yescharacteristic
429C2×C42C22×D1711yescharacteristic
430C2×C42C22×D1711yescharacteristic
431#106—342no—
432#106—342no—
433#106—342no—
434#106—342no—
435#106—681no—
436C4⋊C8—342no—
437#103—342no—
438#103—342no—
439#103—681no—
440#102—174no—
441#102—342no—
442#102—342no—
443C4×C8—342no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—341no—
451#95—341no—
452#95C22⋊D1711yescharacteristic
453#95C22⋊D1711yescharacteristic
454C17#2637611yescharacteristic, complemented, Sylow
455C24—341no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—81no—
464C2×Q8Q8×D1712yes—
465C2×D4—172no—
466C2×D4—172no—
467C2×D4—172no—
468C2×D4—341no—
469C2×D4—341no—
470C2×D4—341no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—344no—
476C22×C4—341no—
477C22×C4—341no—
478C22×C4—341no—
479C22×C4—682no—
480C22×C4—682no—
481C22×C4—682no—
482C22×C4—682no—
483C22×C4—682no—
484C22×C4—682no—
485C22×C4—1361no—
486C22×C4—1361no—
487C22×C4C2×C22⋊D1711yescharacteristic
488C22×C4C2×C22⋊D1711yescharacteristic
489C22×C4C2×C22⋊D1711yescharacteristic
490C22×C4C2×C22⋊D1711yescharacteristic
491C22×C4C2×C22⋊D1711yescharacteristic
492C22×C4C2×C22⋊D1711yescharacteristic
493C22×C4C23×D1711yescharacteristic
494SD16—348no—
495C2×C8—172no—
496C2×C8—172no—
497C2×C8—341no—
498C2×C8—341no—
499C2×C8—341no—
500C2×C8—341no—
501C2×C8—341no—
502C2×C8—341no—
503C2×C8—341no—
504C2×C8—341no—
505C2×C8—341no—
506C2×C8—341no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—41no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—81no—
516C4⋊C4—81no—
517C4⋊C4—172no—
518C4⋊C4—341no—
519C4⋊C4—341no—
520C4⋊C4—341no—
521C4⋊C4—341no—
522C4⋊C4—341no—
523C4⋊C4—682no—
524C4⋊C4—682no—
525C4⋊C4—682no—
526C4⋊C4—1361no—
527C4⋊C4#6323014yescomplemented
528C4⋊C4D4×D1712yes—
529C22⋊C4—344no—
530C22⋊C4—682no—
531C22⋊C4—682no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—42no—
539C23—342no—
540C23—342no—
541C23—342no—
542C23—681no—
543C23—681no—
544C23—681no—
545C23—681no—
546C23C22×C22⋊D1711yescharacteristic
547Q8—22no—
548Q8—22no—
549Q8—41no—
550Q8—41no—
551Q8—84no—
552D4—174no—
553D4—342no—
554D4—342no—
555D4—342no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—342no—
584C2×C4—688no—
585C2×C4—681no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4—681no—
590C2×C4—681no—
591C2×C4—681no—
592C2×C4—681no—
593C2×C4—1364no—
594C2×C4—1364no—
595C2×C4C2×#6323012yes—
596C2×C4C2×#6323012yes—
597C2×C4C2×#6323012yes—
598C2×C4C2×D4×D1711yescharacteristic
599C2×C4#1058769111yescharacteristic
600C2×C4C2×Q8×D1711yescharacteristic
601C2×C4#1058769411yescharacteristic
602C2×C4#41×D1711yescharacteristic
603C2×C4#41×D1711yescharacteristic
604C8—342no—
605C8—342no—
606C8—681no—
607C8—681no—
608C22—344no—
609C22—682no—
610C22—682no—
611C22—682no—
612C22—682no—
613C22—682no—
614C22—682no—
615C22#1176331611yescharacteristic
616C22D4×C22⋊D1711yescharacteristic
617C22Q8×C22⋊D1711yescharacteristic
618C22C22×#6323011yescharacteristic
619C22#1176360711yescharacteristic
620C22#1176361711yescharacteristic
621C22#1176362011yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—42no—
628C4—41no—
629C4—42no—
630C4—81no—
631C4—81no—
632C4—682no—
633C4—1361no—
634C4—1361no—
635C4C22⋊Q8×D1711yescharacteristic
636C4#1176340511yescharacteristic
637C4#1176346111yescharacteristic
638C4#1176351311yescharacteristic
639C2—684no—
640C2#1453259011yescharacteristic
641C2#1453878811yescharacteristic
642C2#1453905511yescharacteristic
643C2#1454030511yescharacteristic
644C2D4×#6323011yescharacteristic
645C2Q8×#6323011yescharacteristic
646C2#1454172311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2451C2
socleC23×C17C22×D4
prime radicalC22×D68C23
hypercenter#2451D17
hypocenterC17#26376
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2451

Complements

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

NH ≅ G/Ntype
4. #14530327C2semidirect
5. #14530260C2semidirect
6. #14530165C2semidirect
7. #14530114C2semidirect
8. #14529509C2semidirect
9. #14529343C2semidirect
12. #14529052C2semidirect
15. C17×#2451C2semidirect
86. C2×#63230Q8semidirect
180. C4⋊C4×C17SD16semidirect
287. #2451D17semidirect
454. C17#26376semidirect
527. C4⋊C4#63230semidirect

Maximal subgroups

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

subgroupindex
C2×#117630962
C2×C4×#632302
C4⋊C4×D682
#145303272
#145302602
#145301652
#145301142
#145295092
#145293432
#145291212
#145290522
#145290442
C17×#24512
#2637617