Finite Group Databasebeta

← #10703030

Subgroups — #10703030

Order 768.

894 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82228C211yesmaximal, characteristic
2C2×#5005C211yesmaximal, characteristic
3#82213C211yesmaximal, characteristic
4C3×#1740C211yesmaximal, characteristic, complemented
5C2×#4533C211yesmaximal, characteristic, complemented
6C2×#4532C211yesmaximal, characteristic, complemented
7C2×#4532C211yesmaximal, characteristic, complemented
8#11193—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C2×C3×#117C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C22×C4×Dic3C2211yescharacteristic
14C22×#208C2211yescharacteristic
15C22×#208C2211yescharacteristic
16#5005—22no—
17#5005—22no—
18#5004—21no—
19#5002—22no—
20#5001—21no—
21#4988—21no—
22#4973—21no—
23#4966C414yescomplemented
24#4958—22no—
25C4×(C2×D3)⋊C4—22no—
26C2×#805—22no—
27C3×#342—22no—
28#4533—22no—
29#4532—22no—
30#4532—22no—
31C2×#421—31no—
32#2896—31no—
33#2297—31no—
34C2×#344—31no—
35#1740D311yescharacteristic, complemented
36C2×#339—31no—
37C2×#339—31no—
38C23×C3×C4C2311yescharacteristic
39C22×#67—21no—
40C23×Dic3—21no—
41C2×#235—21no—
42C2×#235—21no—
43C2×#235—21no—
44C2×#235—21no—
45C2×#235—21no—
46C2×#235—21no—
47C2×#235—21no—
48C2×#235—21no—
49C2×#235—21no—
50C2×#235—21no—
51C2×#235C2×C412yes—
52C2×#235D412yes—
53C22×D12D411yescharacteristic
54C22×C4×D3—21no—
55C22×Dic6D411yescharacteristic
56C3×#117—21no—
57C3×#117—21no—
58C3×#117—21no—
59C3×#117—21no—
60C3×#117—21no—
61C3×#117—21no—
62C3×#117—21no—
63C3×#117—21no—
64C3×#117—21no—
65C3×#117—21no—
66C3×#117—21no—
67C3×#117D412yes—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2×C412yes—
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4D412yes—
74C2×C3×C42—21no—
75C2×C3×C42—21no—
76C2×C3×C42—21no—
77C2×(C2×D3)⋊C4—24no—
78C2×C4×Dic3—21no—
79C2×C4×Dic3—21no—
80C2×C4×Dic3—21no—
81C2×C4×Dic3—21no—
82C2×C4×Dic3—21no—
83C2×C4×Dic3D412yes—
84C2×#208—21no—
85C2×#208—21no—
86C2×#208—21no—
87C2×#208—21no—
88C2×#208—21no—
89C2×#208—21no—
90C2×#208—21no—
91C2×#208—21no—
92C2×#208—21no—
93C2×#208—21no—
94C2×#208C2×C412yes—
95C2×#208D412yes—
96C22×C3⋊C8—21no—
97C22×C3⋊C8—21no—
98C3⋊#95—24no—
99#810—24no—
100#809—24no—
101#805—28no—
102C22×#41—31no—
103C22×M16—31no—
104C22×M16—31no—
105C2×#117C2×D311yescharacteristic
106C2×#117C2×D311yescharacteristic
107C2×#117C2×D311yescharacteristic
108C22×C42—31no—
109#421—62no—
110#421—62no—
111C2×#104—62no—
112#419—61no—
113#418—61no—
114#417—34no—
115#416—62no—
116#391—61no—
117#390—61no—
118#386—62no—
119C4×C22⋊C4—62no—
120#344—62no—
121#342—22no—
122#339—62no—
123#339—62no—
124C24×C3C2×D411yescharacteristic
125C23×D3—21no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—41no—
129C22×C3×C4C22×C411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×C3×C4#4111yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C2×#67—22no—
144C2×#67—22no—
145C2×#67—41no—
146C2×#67—41no—
147C2×#67—41no—
148C2×#67—41no—
149C22×Dic3—21no—
150C22×Dic3—22no—
151C22×Dic3—44no—
152C22×Dic3—41no—
153C22×Dic3—41no—
154#235—22no—
155#235—22no—
156#235—22no—
157#235—22no—
158#235—22no—
159#235—22no—
160#235—22no—
161#235—22no—
162#235—41no—
163#235—41no—
164#235—41no—
165#235—41no—
166#235—41no—
167#235—41no—
168#235—41no—
169#235—41no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12C22⋊C412yes—
176C2×C4×D3—22no—
177C2×C4×D3—22no—
178C2×C4×D3—41no—
179C2×C4×D3—41no—
180C2×C4×D3—41no—
181C2×C4×D3—41no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6C22⋊C412yes—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—41no—
192C3×C4⋊C4—41no—
193C3×C4⋊C4—41no—
194C3×C22⋊C4—22no—
195C3×C22⋊C4—22no—
196C3×C22⋊C4—22no—
197C3×C22⋊C4—41no—
198C3×C22⋊C4—41no—
199C3×C22⋊C4—41no—
200C3×C42—22no—
201C3×C42—41no—
202C3×C42—41no—
203C3×C42—41no—
204C3×C42—41no—
205C3×C42—41no—
206(C2×D3)⋊C4—48no—
207C4×Dic3—22no—
208C4×Dic3—22no—
209C4×Dic3—41no—
210C4×Dic3—41no—
211#208—22no—
212#208—22no—
213#208—22no—
214#208—22no—
215#208—41no—
216#208—41no—
217#208—41no—
218#208—41no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—41no—
222C2×C3⋊C8—41no—
223C2×C3⋊C8—41no—
224C2×C3⋊C8—41no—
225C2×#41—32no—
226C2×#41—32no—
227C2×#41—61no—
228C2×#41—61no—
229C2×#41—61no—
230C2×#41—61no—
231C2×#41—61no—
232C2×#41—61no—
233C2×#41—61no—
234C2×#41—61no—
235C2×#41—61no—
236C2×#41—61no—
237C22×Q8—31no—
238C22×D4—31no—
239C22×D4—61no—
240C23×C4—61no—
241C23×C4—61no—
242C23×C4C22×D311yescharacteristic
243C2×M16—32no—
244C2×M16—32no—
245C2×M16—61no—
246C2×M16—61no—
247C2×M16—61no—
248C2×M16—61no—
249C2×M16—61no—
250C2×M16—61no—
251C2×M16—61no—
252C2×M16—61no—
253C2×M16—61no—
254C2×M16—61no—
255C22×C8—61no—
256C22×C8—61no—
257#117—21no—
258#117—21no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117D1212yes—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4C4×D312yes—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—64no—
275C2×C22⋊C4#6712yes—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—32no—
280C2×C42—61no—
281C2×C42—61no—
282C2×C42—61no—
283C2×C42—61no—
284C2×C42—61no—
285#104—68no—
286#103—64no—
287#102—64no—
288#95—64no—
289C23×C3—42no—
290C23×C3C4×D411yescharacteristic
291C23×C3#12011yescharacteristic
292C23×C3#12111yescharacteristic
293C23×C3#12111yescharacteristic
294C23×C3#12111yescharacteristic
295C23×C3#12311yescharacteristic
296C23×C3#12411yescharacteristic
297C22×D3—22no—
298C22×D3—44no—
299C22×D3—41no—
300C22×D3—41no—
301C2×C3×C4—44no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—82no—
306C2×C3×C4—82no—
307C2×C3×C4C2×C22⋊C411yescharacteristic
308C2×C3×C4#11711yescharacteristic
309C2×C3×C4#11711yescharacteristic
310C2×C3×C4C4×D411yescharacteristic
311C2×C3×C4C4×D411yescharacteristic
312C2×C3×C4C4×D411yescharacteristic
313C2×C3×C4#12011yescharacteristic
314C2×C3×C4#12111yescharacteristic
315C2×C3×C4#12111yescharacteristic
316C2×C3×C4#12111yescharacteristic
317C2×C3×C4#12111yescharacteristic
318C2×C3×C4#12111yescharacteristic
319C2×C3×C4#12311yescharacteristic
320C2×C3×C4#12311yescharacteristic
321C2×C3×C4#12311yescharacteristic
322C2×C3×C4#12311yescharacteristic
323C2×C3×C4#12311yescharacteristic
324C2×C3×C4#12311yescharacteristic
325C2×C3×C4#12311yescharacteristic
326C2×C3×C4#12311yescharacteristic
327C2×C3×C4#12311yescharacteristic
328C2×C3×C4#12411yescharacteristic
329C2×C3×C4#12411yescharacteristic
330C2×C3×C4#12411yescharacteristic
331C2×C3×C4#12611yescharacteristic
332C2×C3×C4#12611yescharacteristic
333C2×C3×C4#12611yescharacteristic
334C2×C3×C4#12611yescharacteristic
335#67—42no—
336#67—42no—
337#67—42no—
338#67—42no—
339C2×Dic3—24no—
340C2×Dic3—22no—
341C2×Dic3—42no—
342C2×Dic3—44no—
343C2×Dic3—44no—
344C2×Dic3—42no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347D12—22no—
348D12—22no—
349D12—41no—
350D12—41no—
351C4×D3—42no—
352C4×D3—42no—
353C4×D3—42no—
354C4×D3—42no—
355Dic6—22no—
356Dic6—22no—
357Dic6—41no—
358Dic6—41no—
359C3⋊C8—42no—
360C3⋊C8—42no—
361C24—61no—
362C24C2×#6711yescharacteristic
363#41—62no—
364#41—62no—
365#41—62no—
366#41—62no—
367#41—62no—
368#41—62no—
369#41—62no—
370#41—62no—
371#41—121no—
372#41—121no—
373#41—121no—
374#41—121no—
375#41—121no—
376#41—121no—
377#41—121no—
378#41—121no—
379C2×Q8—32no—
380C2×Q8—61no—
381C2×Q8—61no—
382C2×Q8—61no—
383C2×Q8—61no—
384C2×Q8—61no—
385C2×D4—32no—
386C2×D4—62no—
387C2×D4—62no—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4—61no—
391C2×D4—61no—
392C2×D4—61no—
393C2×D4—121no—
394C2×D4—121no—
395C2×D4—121no—
396C2×D4—121no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—41no—
400C22×C4—61no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—124no—
405C22×C4—121no—
406C22×C4—121no—
407C22×C4—121no—
408C22×C4—121no—
409C22×C4—121no—
410C22×C4—121no—
411C22×C4C2×C4×D311yescharacteristic
412C22×C4C2×D1211yescharacteristic
413C22×C4#23511yescharacteristic
414C22×C4#23511yescharacteristic
415C22×C4#23511yescharacteristic
416C22×C4#23511yescharacteristic
417C22×C4D3×D411yescharacteristic
418C22×C4D3×D411yescharacteristic
419C22×C4D3×D411yescharacteristic
420C22×C4D3×D411yescharacteristic
421C22×C4#23711yescharacteristic
422C22×C4#23711yescharacteristic
423C22×C4#23911yescharacteristic
424C22×C4#23911yescharacteristic
425M16—62no—
426M16—62no—
427M16—62no—
428M16—62no—
429M16—121no—
430M16—121no—
431M16—121no—
432M16—121no—
433C2×C8—62no—
434C2×C8—62no—
435C2×C8—121no—
436C2×C8—121no—
437C2×C8—121no—
438C2×C8—121no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—128no—
452C42—22no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—62no—
459C42—62no—
460C42—121no—
461C42—121no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—41no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3#38611yescharacteristic
470C22×C3#38811yescharacteristic
471C22×C3#38811yescharacteristic
472C22×C3#39411yescharacteristic
473C22×C3#39411yescharacteristic
474C22×C3#39411yescharacteristic
475C22×C3#39711yescharacteristic
476C2×D3—44no—
477C2×D3—42no—
478C2×D3—82no—
479C2×D3—82no—
480C3×C4—82no—
481C3×C4—82no—
482C3×C4—82no—
483C3×C4#38111yescharacteristic
484C3×C4#38611yescharacteristic
485C3×C4#38811yescharacteristic
486C3×C4#38811yescharacteristic
487C3×C4#39411yescharacteristic
488C3×C4#39711yescharacteristic
489C3×C4#39711yescharacteristic
490C3×C4#39711yescharacteristic
491Dic3—44no—
492Dic3—42no—
493C23—42no—
494C23—62no—
495C23—124no—
496C23—121no—
497C23—121no—
498C23C4×#6711yescharacteristic
499C23#92911yescharacteristic
500C23#93011yescharacteristic
501C23#93711yescharacteristic
502C23#93811yescharacteristic
503C23#93911yescharacteristic
504C23#94711yescharacteristic
505Q8—62no—
506Q8—62no—
507Q8—121no—
508Q8—121no—
509D4—62no—
510D4—62no—
511D4—122no—
512D4—122no—
513D4—122no—
514D4—122no—
515D4—121no—
516D4—121no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—64no—
522C2×C4—62no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—122no—
526C2×C4—124no—
527C2×C4—124no—
528C2×C4—121no—
529C2×C4—122no—
530C2×C4—121no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4#87211yescharacteristic
536C2×C4C4×D1211yescharacteristic
537C2×C4#87511yescharacteristic
538C2×C4#87611yescharacteristic
539C2×C4D3×C22⋊C411yescharacteristic
540C2×C4#67⋊C411yescharacteristic
541C2×C4#88211yescharacteristic
542C2×C4#88311yescharacteristic
543C2×C4#88311yescharacteristic
544C2×C4#88311yescharacteristic
545C2×C4#88411yescharacteristic
546C2×C4#88411yescharacteristic
547C2×C4#88411yescharacteristic
548C2×C4#88511yescharacteristic
549C2×C4#88511yescharacteristic
550C2×C4#88611yescharacteristic
551C2×C4#89211yescharacteristic
552C2×C4#89311yescharacteristic
553C2×C4#89411yescharacteristic
554C2×C4#89411yescharacteristic
555C2×C4#89411yescharacteristic
556C2×C4#89411yescharacteristic
557C2×C4#89411yescharacteristic
558C2×C4#89511yescharacteristic
559C2×C4#89511yescharacteristic
560C2×C4#89811yescharacteristic
561C2×C4#89811yescharacteristic
562C2×C4#89811yescharacteristic
563C8—122no—
564C8—122no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3#144011yescharacteristic
572C2×C3#190612yes—
573D3—82no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—122no—
582C22—124no—
583C22—242no—
584C22—242no—
585C22#494011yescharacteristic
586C22#495611yescharacteristic
587C22#495911yescharacteristic
588C22#496111yescharacteristic
589C22#499011yescharacteristic
590C22#499111yescharacteristic
591C22#499211yescharacteristic
592C4—82no—
593C4—82no—
594C4—82no—
595C4—124no—
596C4—122no—
597C4#466411yescharacteristic
598C4#466511yescharacteristic
599C4#466911yescharacteristic
600C4#467011yescharacteristic
601C4#467211yescharacteristic
602C4#467411yescharacteristic
603C4#467511yescharacteristic
604C4#467511yescharacteristic
605C3#1119311yescharacteristic, complemented, Sylow
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—21no—
611C2—21no—
612C2—242no—
613C2#6942911yescharacteristic
614C2#7005512yes—
subgroupG/H
centerC2×C4#884
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1740C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenter#1740D3
hypocenterC3#11193
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1740

Complements

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

NH ≅ G/Ntype
4. C3×#1740C2semidirect
5. C2×#4533C2semidirect
6. C2×#4532C2semidirect
7. C2×#4532C2semidirect
23. #4966C4semidirect
35. #1740D3semidirect
605. C3#11193semidirect

Maximal subgroups

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

subgroupindex
#822282
C2×#50052
#822132
C3×#17402
C2×#45332
C2×#45322
#111933