Finite Group Databasebeta

← #30883265

Subgroups — #30883265

Order 17152.

1148 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22573614C211yesmaximal, characteristic, complemented
2C2×#16863413C211yesmaximal, characteristic
3C2×#16863413C211yesmaximal, characteristic
4#22561383C218yesmaximal, complemented
5C2×#16863166C211yesmaximal, characteristic, complemented
6#22559918C211yesmaximal, characteristic, complemented
7C2×#16863152C211yesmaximal, characteristic, complemented
8C2×#410×C67C211yesmaximal, characteristic
9C22×#11761612C2211yescharacteristic
10C22×#11761612C2211yescharacteristic
11#16864027—22no—
12#16864024—22no—
13#16864024—22no—
14#16864022—22no—
15#16863953—22no—
16#16863949—22no—
17#16863949C2214yescomplemented
18C2×#14524002C2211yescharacteristic
19C2×#14524002C2211yescharacteristic
20#16863413—22no—
21#16863413—22no—
22#16863413—22no—
23#16863413C2214yes—
24C2×#14523970—22no—
25C2×#14523970C2214yes—
26#16863166C2214yescomplemented
27#16863155C2214yescomplemented
28#16863152C2214yescomplemented
29C22×#41×C67C2211yescharacteristic
30C2×#117×C67C2211yescharacteristic
31C2×#117×C67C2211yescharacteristic
32#410×C67—22no—
33#410×C67C2214yes—
34C2×#99×C67—22no—
35C2×#11761612—21no—
36C2×#11761612—21no—
37C2×#11761612—21no—
38C2×#11761612—21no—
39C2×#11761612—21no—
40C2×#11761612—21no—
41C2×#11761612—21no—
42C2×#11761612—21no—
43C2×#11761612—21no—
44C2×#11761612—21no—
45C2×#11761612C2312yes—
46C2×#11761612C2312yes—
47C22×C67⋊C8—21no—
48C22×C67⋊C8—21no—
49#14524043—24no—
50#14524042—24no—
51#14524042—24no—
52#14524035—24no—
53#14524034—24no—
54#14524034—24no—
55#14524022—28no—
56#14524020—28no—
57#14524020—28no—
58C4×Dic134—28no—
59#14524002—21no—
60#14524002—21no—
61#14524002—21no—
62#14524002—21no—
63#14524002D412yes—
64#14524002D412yes—
65#14524002D412yes—
66#14524002C2312yes—
67C2×C4⋊Dic67—21no—
68C2×C4⋊Dic67—21no—
69C2×C22⋊Dic67—21no—
70C2×C22⋊Dic67—21no—
71C2×C4×Dic67D412yes—
72C2×C4×Dic67C2312yes—
73#14523970—28no—
74#14523970—28no—
75C2×#41×C67—21no—
76C2×#41×C67—21no—
77C2×#41×C67—21no—
78C2×#41×C67—21no—
79C2×#41×C67—21no—
80C2×#41×C67—21no—
81C2×#41×C67—21no—
82C2×#41×C67—21no—
83C2×#41×C67—21no—
84C2×#41×C67—21no—
85C2×#41×C67Q812yescomplemented
86C2×#41×C67C2312yes—
87C22×Q8×C67Q811yescharacteristic, complemented
88C22×D4×C67—21no—
89C22×D4×C67Q811yescharacteristic, complemented
90C23×C4×C67—21no—
91C23×C4×C67C2311yescharacteristic
92#117×C67—21no—
93#117×C67—21no—
94#117×C67—21no—
95#117×C67—21no—
96#117×C67—21no—
97#117×C67D412yes—
98#117×C67C2312yes—
99#117×C67C2312yes—
100C2×C4⋊C4×C67—21no—
101C2×C4⋊C4×C67—21no—
102C2×C22⋊C4×C67—21no—
103C2×C22⋊C4×C67D412yes—
104C2×C42×C67—21no—
105C2×C42×C67—21no—
106#99×C67—28no—
107C2×Dic134—28no—
108C22×Dic67—22no—
109C22×Dic67—22no—
110#11761612—22no—
111#11761612—22no—
112#11761612—22no—
113#11761612—22no—
114#11761612—41no—
115#11761612—41no—
116#11761612—41no—
117#11761612—41no—
118C2×C67⋊C8—22no—
119C2×C67⋊C8—22no—
120C2×C67⋊C8—41no—
121C2×C67⋊C8—41no—
122C2×C67⋊C8—41no—
123C2×C67⋊C8—41no—
124C4⋊Dic67—28no—
125C4⋊Dic67—22no—
126C4⋊Dic67—22no—
127C4⋊Dic67—22no—
128C4⋊Dic67—22no—
129Dic67⋊C4—28no—
130Dic67⋊C4—28no—
131C22⋊Dic67—22no—
132C22⋊Dic67—22no—
133C22⋊Dic67—22no—
134C22⋊Dic67—22no—
135C4×Dic67—22no—
136C4×Dic67—22no—
137C4×Dic67C2×D414yes—
138C4×Dic67C2×D414yes—
139C24×C67—21no—
140C24×C67C2×D411yescharacteristic
141#41×C67—22no—
142#41×C67—22no—
143#41×C67—22no—
144#41×C67—22no—
145#41×C67—22no—
146#41×C67—22no—
147#41×C67—22no—
148#41×C67—22no—
149#41×C67—41no—
150#41×C67—41no—
151#41×C67—41no—
152#41×C67—41no—
153#41×C67—41no—
154#41×C67—41no—
155#41×C67—41no—
156#41×C67—41no—
157C2×Q8×C67—21no—
158C2×Q8×C67—21no—
159C2×Q8×C67—21no—
160C2×Q8×C67—21no—
161C2×Q8×C67—21no—
162C2×Q8×C67C2×Q812yes—
163C2×D4×C67—22no—
164C2×D4×C67—22no—
165C2×D4×C67—21no—
166C2×D4×C67—21no—
167C2×D4×C67—21no—
168C2×D4×C67—21no—
169C2×D4×C67—21no—
170C2×D4×C67—41no—
171C2×D4×C67—41no—
172C2×D4×C67—41no—
173C2×D4×C67—41no—
174C2×D4×C67C2×Q812yes—
175C22×C4×C67—21no—
176C22×C4×C67—22no—
177C22×C4×C67—22no—
178C22×C4×C67—22no—
179C22×C4×C67—41no—
180C22×C4×C67—41no—
181C22×C4×C67—41no—
182C22×C4×C67—41no—
183C22×C4×C67—41no—
184C22×C4×C67C2×D411yescharacteristic
185C22×C4×C67C2×D411yescharacteristic
186C22×C4×C67C2×D411yescharacteristic
187C22×C4×C67C2×D411yescharacteristic
188C22×C4×C67C2×D411yescharacteristic
189C22×C4×C67C2×Q811yescharacteristic
190C22×C4×C67C2×Q811yescharacteristic
191C22×C4×C67#4111yescharacteristic
192C22×C4×C67#4111yescharacteristic
193C22×C4×C67#4111yescharacteristic
194C22×C4×C67#4111yescharacteristic
195C22×C4×C67#4111yescharacteristic
196C22×C4×C67#4111yescharacteristic
197C22×C4×C67C2411yescharacteristic
198C4⋊C4×C67—22no—
199C4⋊C4×C67—22no—
200C4⋊C4×C67—22no—
201C4⋊C4×C67—41no—
202C22⋊C4×C67—22no—
203C22⋊C4×C67—22no—
204C22⋊C4×C67—41no—
205C22⋊C4×C67C2×D414yes—
206C42×C67—22no—
207C42×C67—22no—
208C42×C67—22no—
209C42×C67—41no—
210C23×C67—22no—
211C23×C67—41no—
212C23×C67—41no—
213C23×C67—81no—
214C23×C67—82no—
215C23×C67#12011yescharacteristic
216C23×C67#12011yescharacteristic
217C23×C67C22⋊Q811yescharacteristic
218C23×C67C22⋊Q811yescharacteristic
219C23×C67#12411yescharacteristic
220C23×C67#12411yescharacteristic
221C23×C67C22×D411yescharacteristic
222Q8×C67—22no—
223Q8×C67—22no—
224Q8×C67—41no—
225Q8×C67—41no—
226D4×C67—22no—
227D4×C67—22no—
228D4×C67—42no—
229D4×C67—42no—
230D4×C67—42no—
231D4×C67—42no—
232D4×C67—41no—
233D4×C67—41no—
234C2×C4×C67—24no—
235C2×C4×C67—22no—
236C2×C4×C67—41no—
237C2×C4×C67—41no—
238C2×C4×C67—42no—
239C2×C4×C67—42no—
240C2×C4×C67—42no—
241C2×C4×C67—42no—
242C2×C4×C67—42no—
243C2×C4×C67—82no—
244C2×C4×C67—82no—
245C2×C4×C67#12011yescharacteristic
246C2×C4×C67#12011yescharacteristic
247C2×C4×C67C22⋊Q811yescharacteristic
248C2×C4×C67C22⋊Q811yescharacteristic
249C2×C4×C67C22⋊Q811yescharacteristic
250C2×C4×C67C22⋊Q811yescharacteristic
251C2×C4×C67C22⋊Q811yescharacteristic
252C2×C4×C67C22⋊Q811yescharacteristic
253C2×C4×C67C22⋊Q811yescharacteristic
254C2×C4×C67C22⋊Q811yescharacteristic
255C2×C4×C67C22⋊Q811yescharacteristic
256C2×C4×C67C22⋊Q811yescharacteristic
257C2×C4×C67#12411yescharacteristic
258C2×C4×C67#12411yescharacteristic
259C2×C4×C67#12411yescharacteristic
260C2×C4×C67#12411yescharacteristic
261C2×C4×C67#12411yescharacteristic
262C2×C4×C67#12411yescharacteristic
263C2×C4×C67#12411yescharacteristic
264C2×C4×C67#12411yescharacteristic
265C2×C4×C67#12411yescharacteristic
266C2×C4×C67#12411yescharacteristic
267C2×C4×C67C22×D411yescharacteristic
268C2×C4×C67C22×D411yescharacteristic
269C2×C4×C67C22×Q811yescharacteristic
270C2×C4×C67C2×#4111yescharacteristic
271C2×C4×C67C2×#4111yescharacteristic
272C2×C4×C67C2×#4111yescharacteristic
273C2×Dic67—24no—
274C2×Dic67—24no—
275C2×Dic67—82no—
276C2×Dic67—82no—
277Dic134—88no—
278C67⋊C8—42no—
279C67⋊C8—42no—
280C22×C67—41no—
281C22×C67—41no—
282C22×C67—41no—
283C22×C67—41no—
284C22×C67—41no—
285C22×C67—41no—
286C22×C67—41no—
287C22×C67—42no—
288C22×C67—82no—
289C22×C67—82no—
290C22×C67—82no—
291C22×C67#39311yescharacteristic
292C22×C67#39311yescharacteristic
293C22×C67#39311yescharacteristic
294C22×C67#39311yescharacteristic
295C22×C67C2×#12011yescharacteristic
296C22×C67C2×C22⋊Q811yescharacteristic
297C22×C67C2×#12411yescharacteristic
298C4×C67—44no—
299C4×C67—42no—
300C4×C67—82no—
301C4×C67#39311yescharacteristic
302C4×C67#39311yescharacteristic
303C4×C67#39311yescharacteristic
304C4×C67#39311yescharacteristic
305C4×C67C2×C22⋊Q811yescharacteristic
306C4×C67C2×C22⋊Q811yescharacteristic
307C4×C67C2×#12411yescharacteristic
308C4×C67C2×#12411yescharacteristic
309Dic67—44no—
310Dic67—44no—
311#23472—671nomaximal, Sylow
312C2×C67—21no—
313C2×C67—21no—
314C2×C67—21no—
315C2×C67—21no—
316C2×C67—21no—
317C2×C67—21no—
318C2×C67—21no—
319C2×C67—82no—
320C2×C67C2×#39311yescharacteristic
321#3458—671no—
322C2×#421—671no—
323C2×#421—671no—
324C2×#410D6711yescharacteristic
325#2031—678no—
326C2×#344—671no—
327#1733—671no—
328C2×H(3,ℤ/4)—671no—
329C67#2347211yescharacteristic, complemented, Sylow
330C22×#41C2×D6711yescharacteristic
331C22×M16—671no—
332C22×M16—671no—
333#544—1342no—
334#544—1342no—
335#544—1342no—
336#543—1342no—
337#533—674no—
338#533—1342no—
339#533—1342no—
340C2×#117—671no—
341C2×#117—671no—
342C2×#117C2×D6711yescharacteristic
343C2×#117C2×D6711yescharacteristic
344#421—674no—
345#421—1342no—
346#421—1342no—
347#421—1342no—
348C2×#104—674no—
349C2×#104—1342no—
350#410—22no—
351#410C2×D6714yes—
352C2×#99—22no—
353#344—674no—
354#339—674no—
355H(3,ℤ/4)—674no—
356C2×#41—21no—
357C2×#41—21no—
358C2×#41—21no—
359C2×#41—21no—
360C2×#41—21no—
361C2×#41—21no—
362C2×#41—21no—
363C2×#41—21no—
364C2×#41—21no—
365C2×#41—21no—
366C2×#41Dic13412yescomplemented
367C2×#41C22×D6712yes—
368C22×Q8Dic13411yescharacteristic, complemented
369C22×D4—21no—
370C22×D4Dic13411yescharacteristic, complemented
371C23×C4—21no—
372C23×C4C22×D6711yescharacteristic
373C2×M16—672no—
374C2×M16—672no—
375C2×M16—1341no—
376C2×M16—1341no—
377C2×M16—1341no—
378C2×M16—1341no—
379C2×M16—1341no—
380C2×M16—1341no—
381C2×M16—1341no—
382C2×M16—1341no—
383C2×M16—1341no—
384C2×M16—1341no—
385C22×C8—1341no—
386C22×C8—1341no—
387C4⋊Q8—1344no—
388C4⋊Q8—1344no—
389C4⋊Q8—1344no—
390#125—1344no—
391#125—1344no—
392#125—1344no—
393C22⋊Q8—1348no—
394C22⋊Q8—1348no—
395C22⋊Q8—1348no—
396C4×Q8—1348no—
397#117—21no—
398#117—21no—
399#117—21no—
400#117—21no—
401#117—21no—
402#117—672no—
403#117—672no—
404#117—672no—
405#117—672no—
406#117—1341no—
407#117—1341no—
408#117—1341no—
409#117—1341no—
410#117C22⋊D6712yes—
411#117C22×D6712yes—
412#117C22×D6712yes—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—1341no—
416C2×C4⋊C4—1341no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—1341no—
419C2×C22⋊C4—1341no—
420C2×C22⋊C4C22⋊D6712yes—
421C2×C42—21no—
422C2×C42—21no—
423C2×C42—672no—
424C2×C42—672no—
425#104—1348no—
426#104—1348no—
427#99—28no—
428C24—21no—
429C24C2×C22⋊D6711yescharacteristic
430#41—22no—
431#41—22no—
432#41—22no—
433#41—22no—
434#41—22no—
435#41—22no—
436#41—22no—
437#41—22no—
438#41—41no—
439#41—41no—
440#41—41no—
441#41—41no—
442#41—41no—
443#41—41no—
444#41—41no—
445#41—41no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—1348no—
452C2×Q8C2×Dic13412yes—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4C2×Dic13412yes—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—1342no—
475C22×C4—1342no—
476C22×C4D4×D6711yescharacteristic
477C22×C4D4×D6711yescharacteristic
478C22×C4D4×D6711yescharacteristic
479C22×C4D4×D6711yescharacteristic
480C22×C4C2×C22⋊D6711yescharacteristic
481C22×C4C2×Dic13411yescharacteristic
482C22×C4C2×Dic13411yescharacteristic
483C22×C4#1176162711yescharacteristic
484C22×C4#1176162711yescharacteristic
485C22×C4#1176162711yescharacteristic
486C22×C4#1176162711yescharacteristic
487C22×C4#1176162811yescharacteristic
488C22×C4#1176162811yescharacteristic
489C22×C4C23×D6711yescharacteristic
490M16—2681no—
491M16—2681no—
492M16—2681no—
493M16—2681no—
494M16—1342no—
495M16—1342no—
496M16—1342no—
497M16—1342no—
498C2×C8—2681no—
499C2×C8—2681no—
500C2×C8—2681no—
501C2×C8—2681no—
502C2×C8—1342no—
503C2×C8—1342no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—41no—
508C4⋊C4—1348no—
509C4⋊C4—1348no—
510C4⋊C4—1348no—
511C4⋊C4—1342no—
512C4⋊C4—1342no—
513C4⋊C4—1342no—
514C4⋊C4—1342no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—41no—
518C22⋊C4—1342no—
519C22⋊C4—1342no—
520C22⋊C4—1342no—
521C22⋊C4—1342no—
522C22⋊C4C2×C22⋊D6714yes—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—41no—
527C42—674no—
528C42—674no—
529C42—1342no—
530C42—1342no—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—81no—
535C23—82no—
536C23#1452401111yescharacteristic
537C23#1452401111yescharacteristic
538C23#1452402211yescharacteristic
539C23#1452402211yescharacteristic
540C23#1452403011yescharacteristic
541C23#1452403011yescharacteristic
542C23C22×C22⋊D6711yescharacteristic
543Q8—22no—
544Q8—22no—
545Q8—41no—
546Q8—41no—
547Q8—5368no—
548D4—22no—
549D4—22no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—41no—
555D4—41no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—5362no—
568C2×C4—5362no—
569C2×C4—1344no—
570C2×C4—1344no—
571C2×C4#1452401111yescharacteristic
572C2×C4#1452401111yescharacteristic
573C2×C4#1452402011yescharacteristic
574C2×C4#1452402011yescharacteristic
575C2×C4#1452402011yescharacteristic
576C2×C4#1452402011yescharacteristic
577C2×C4#1452402011yescharacteristic
578C2×C4#1452402011yescharacteristic
579C2×C4#1452402011yescharacteristic
580C2×C4#1452402011yescharacteristic
581C2×C4#1452402211yescharacteristic
582C2×C4#1452402211yescharacteristic
583C2×C4#1452402911yescharacteristic
584C2×C4#1452402911yescharacteristic
585C2×C4#1452402911yescharacteristic
586C2×C4#1452402911yescharacteristic
587C2×C4#1452402911yescharacteristic
588C2×C4#1452402911yescharacteristic
589C2×C4#1452402911yescharacteristic
590C2×C4#1452402911yescharacteristic
591C2×C4#1452403011yescharacteristic
592C2×C4#1452403011yescharacteristic
593C2×C4C2×D4×D6711yescharacteristic
594C2×C4C2×D4×D6711yescharacteristic
595C2×C4C22×Dic13411yescharacteristic
596C2×C4C2×#1176162711yescharacteristic
597C2×C4C2×#1176162711yescharacteristic
598C2×C4C2×#1176162811yescharacteristic
599C8—2682no—
600C8—2682no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—42no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22#1686330811yescharacteristic
613C22#1686330811yescharacteristic
614C22#1686330811yescharacteristic
615C22#1686330811yescharacteristic
616C22C2×#1452401111yescharacteristic
617C22C2×#1452402211yescharacteristic
618C22C2×#1452403011yescharacteristic
619C4—44no—
620C4—42no—
621C4—82no—
622C4—2684no—
623C4—2684no—
624C4#1686330811yescharacteristic
625C4#1686330811yescharacteristic
626C4#1686330811yescharacteristic
627C4#1686330811yescharacteristic
628C4C2×#1452402011yescharacteristic
629C4C2×#1452402011yescharacteristic
630C4C2×#1452402911yescharacteristic
631C4C2×#1452402911yescharacteristic
632C2—21no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—21no—
637C2—21no—
638C2—21no—
639C2—82no—
640C2C2×#1686330811yescharacteristic
subgroupG/H
centerC4C2×#14524029
commutator, cosocleC22×C4×C67C24
FrattiniC22×C4C23×D67
FittingC2×#410×C67C2
socleC2×C67C2×#393
prime radicalC22×#41×C67C22
hypercenterC2×#410D67
hypocenterC67#23472
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
1. #22573614C2semidirect
4. #22561383C2semidirect
5. C2×#16863166C2semidirect
6. #22559918C2semidirect
7. C2×#16863152C2semidirect
17. #16863949C22semidirect
26. #16863166C22semidirect
27. #16863155C22semidirect
28. #16863152C22semidirect
85. C2×#41×C67Q8semidirect
87. C22×Q8×C67Q8semidirect
89. C22×D4×C67Q8semidirect
329. C67#23472semidirect
366. C2×#41Dic134semidirect
368. C22×Q8Dic134semidirect
370. C22×D4Dic134semidirect

Maximal subgroups

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

subgroupindex
#225736142
C2×#168634132
#225613832
C2×#168631662
#225599182
C2×#168631522
C2×#410×C672
#2347267