Finite Group Databasebeta

← #10701587

Subgroups — #10701587

Order 768.

904 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5139C211yesmaximal, characteristic
2#83762C211yesmaximal, characteristic
3#83747C211yesmaximal, characteristic, complemented
4#82213C211yesmaximal, characteristic
5C2×C3×#344C211yesmaximal, characteristic, complemented
6C2×C3⋊#338C211yesmaximal, characteristic, complemented
7C2×#4531C211yesmaximal, characteristic, complemented
8#10950—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C22×#208C2211yescharacteristic
15C22×#208C2211yescharacteristic
16#5139—22no—
17#5139C414yescomplemented
18C2×#825—22no—
19#5129C414yescomplemented
20#5126—22no—
21#5122—22no—
22Dic3×M16—22no—
23#5110—22no—
24#5101—22no—
25#5004C412yescomplemented
26#5002—22no—
27#5001C412yescomplemented
28#4966—22no—
29C3×#344—22no—
30C3⋊#338—22no—
31#4531—22no—
32C2×#421—31no—
33#2896—31no—
34#2879—31no—
35#2873—31no—
36C2×#344D311yescharacteristic, complemented
37C2×#338—31no—
38C2×H(3,ℤ/4)—31no—
39C23×C3×C4C2311yescharacteristic
40C22×#67—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×#235C2×C412yes—
53C22×D12C2×C411yescharacteristic
54C22×C4×D3—21no—
55C22×Dic6C2×C411yescharacteristic
56C2×C3×M16—21no—
57C2×C3×M16—21no—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16—21no—
63C2×C3×M16—21no—
64C2×C3×M16—21no—
65C2×C3×M16C2×C412yes—
66C2×C3×M16C2×C412yes—
67C2×C3×M16D412yes—
68C22×C3×C8—21no—
69C22×C3×C8D412yes—
70C3×#117—21no—
71C3×#117—21no—
72C3×#117—21no—
73C3×#117C2×C412yes—
74C2×C3×C4⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C42—21no—
77C2×#217—21no—
78#926—21no—
79#926—21no—
80#926C2×C412yes—
81#926D412yes—
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⋊C8C2×C412yes—
97C22×C3⋊C8D412yes—
98#825—28no—
99(C2×D3)⋊C8—28no—
100#815—24no—
101C8×Dic3—24no—
102#810—24no—
103#809—24no—
104C22×#41—31no—
105C22×M16—31no—
106C22×M16—31no—
107C22×M16C2×D311yescharacteristic
108C22×M16C2×D311yescharacteristic
109C2×#117—31no—
110C2×#117C2×D311yescharacteristic
111#421—34no—
112#421—62no—
113C2×#104—62no—
114#419—32no—
115#418—32no—
116#417—62no—
117#416—62no—
118#408—34no—
119#408—62no—
120#407—62no—
121#405—62no—
122#405—62no—
123C4×M16—62no—
124#344—22no—
125#338—62no—
126H(3,ℤ/4)—62no—
127C24×C3#4111yescharacteristic
128C23×D3—21no—
129C22×C3×C4—41no—
130C22×C3×C4C22×C411yescharacteristic
131C22×C3×C4C22×C411yescharacteristic
132C22×C3×C4C22×C411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×C3×C4#4111yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C22×C3×C4#4111yescharacteristic
144C2×#67—22no—
145C2×#67—22no—
146C2×#67—41no—
147C2×#67—41no—
148C2×#67—41no—
149C2×#67—41no—
150C22×Dic3—21no—
151C22×Dic3—22no—
152#235—22no—
153#235—22no—
154#235—22no—
155#235—22no—
156#235—22no—
157#235—22no—
158#235—22no—
159#235—22no—
160#235—41no—
161#235—41no—
162#235—41no—
163#235—41no—
164#235—41no—
165#235—41no—
166#235—41no—
167#235—41no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12C4212yes—
174C2×C4×D3—22no—
175C2×C4×D3—22no—
176C2×C4×D3—41no—
177C2×C4×D3—41no—
178C2×C4×D3—41no—
179C2×C4×D3—41no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6C4212yes—
186C3×M16—22no—
187C3×M16—22no—
188C3×M16—22no—
189C3×M16—22no—
190C3×M16—22no—
191C3×M16—22no—
192C3×M16—41no—
193C3×M16—41no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—41no—
198C2×C3×C8—41no—
199C2×C3×C8C22⋊C414yes—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—41no—
202C3×C22⋊C4—22no—
203C3×C22⋊C4—41no—
204C3×C42—22no—
205C3×C42—41no—
206#217—22no—
207#217—22no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210C4×Dic3—22no—
211C4×Dic3C22⋊C414yes—
212#208—22no—
213#208—22no—
214#208—22no—
215#208—22no—
216#208—41no—
217#208—41no—
218#208—41no—
219#208—41no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8C22⋊C414yes—
226C2×#41—32no—
227C2×#41—32no—
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—
237C2×#41—61no—
238C22×Q8—31no—
239C22×D4—31no—
240C22×D4—61no—
241C23×C4—61no—
242C23×C4C22×D311yescharacteristic
243C2×M16—21no—
244C2×M16—21no—
245C2×M16—21no—
246C2×M16—21no—
247C2×M16—21no—
248C2×M16—21no—
249C2×M16—21no—
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—32no—
253C2×M16—32no—
254C2×M16—61no—
255C2×M16—61no—
256C2×M16—61no—
257C2×M16—61no—
258C2×M16—61no—
259C2×M16—61no—
260C2×M16—61no—
261C2×M16—61no—
262C2×M16—61no—
263C2×M16—61no—
264C2×M16C4×D312yes—
265C2×M16C4×D312yes—
266C2×M16#6712yes—
267C22×C8—21no—
268C22×C8—32no—
269C22×C8—32no—
270C22×C8D1212yes—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—32no—
275#117—32no—
276#117—61no—
277#117—61no—
278#117C4×D312yes—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—61no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—61no—
283C2×C42—21no—
284C2×C42—32no—
285#104—68no—
286#103—64no—
287#102—64no—
288C22⋊C8—68no—
289#97—64no—
290C4×C8—64no—
291C23×C3—81no—
292C23×C3#11711yescharacteristic
293C23×C3C4×D411yescharacteristic
294C23×C3C4×D411yescharacteristic
295C23×C3#12111yescharacteristic
296C23×C3#12311yescharacteristic
297C23×C3#12411yescharacteristic
298C23×C3#12611yescharacteristic
299C22×D3—22no—
300C22×D3—41no—
301C22×D3—41no—
302C22×D3—82no—
303C2×C3×C4—42no—
304C2×C3×C4—82no—
305C2×C3×C4C2×C4211yescharacteristic
306C2×C3×C4C2×C22⋊C411yescharacteristic
307C2×C3×C4C2×C22⋊C411yescharacteristic
308C2×C3×C4C2×C22⋊C411yescharacteristic
309C2×C3×C4#11711yescharacteristic
310C2×C3×C4#11711yescharacteristic
311C2×C3×C4C4×D411yescharacteristic
312C2×C3×C4C4×D411yescharacteristic
313C2×C3×C4C4×D411yescharacteristic
314C2×C3×C4C4×D411yescharacteristic
315C2×C3×C4C4×D411yescharacteristic
316C2×C3×C4C4×D411yescharacteristic
317C2×C3×C4C4×D411yescharacteristic
318C2×C3×C4C4×D411yescharacteristic
319C2×C3×C4C4×D411yescharacteristic
320C2×C3×C4C4×D411yescharacteristic
321C2×C3×C4#12011yescharacteristic
322C2×C3×C4#12111yescharacteristic
323C2×C3×C4#12111yescharacteristic
324C2×C3×C4#12111yescharacteristic
325C2×C3×C4#12111yescharacteristic
326C2×C3×C4#12111yescharacteristic
327C2×C3×C4#12311yescharacteristic
328C2×C3×C4#12311yescharacteristic
329C2×C3×C4#12411yescharacteristic
330C2×C3×C4#12411yescharacteristic
331C2×C3×C4#12611yescharacteristic
332C2×C3×C4#12711yescharacteristic
333#67—42no—
334#67—42no—
335#67—42no—
336#67—42no—
337C2×Dic3—22no—
338C2×Dic3—24no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—82no—
342D12—22no—
343D12—22no—
344D12—41no—
345D12—41no—
346C4×D3—42no—
347C4×D3—42no—
348C4×D3—42no—
349C4×D3—42no—
350Dic6—22no—
351Dic6—22no—
352Dic6—41no—
353Dic6—41no—
354C3×C8—24no—
355C3×C8—42no—
356C3⋊C8—24no—
357C3⋊C8—42no—
358C24—61no—
359C24#23511yescharacteristic
360#41—62no—
361#41—62no—
362#41—62no—
363#41—62no—
364#41—62no—
365#41—62no—
366#41—62no—
367#41—62no—
368#41—121no—
369#41—121no—
370#41—121no—
371#41—121no—
372#41—121no—
373#41—121no—
374#41—121no—
375#41—121no—
376C2×Q8—32no—
377C2×Q8—61no—
378C2×Q8—61no—
379C2×Q8—61no—
380C2×Q8—61no—
381C2×Q8—61no—
382C2×D4—32no—
383C2×D4—62no—
384C2×D4—62no—
385C2×D4—61no—
386C2×D4—61no—
387C2×D4—61no—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4—121no—
391C2×D4—121no—
392C2×D4—121no—
393C2×D4—121no—
394C22×C4—41no—
395C22×C4—61no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—121no—
400C22×C4—121no—
401C22×C4—121no—
402C22×C4—121no—
403C22×C4C2×C4×D311yescharacteristic
404C22×C4C2×C4×D311yescharacteristic
405C22×C4C2×C4×D311yescharacteristic
406C22×C4C2×D1211yescharacteristic
407C22×C4#23511yescharacteristic
408C22×C4D3×D411yescharacteristic
409C22×C4D3×D411yescharacteristic
410C22×C4D3×D411yescharacteristic
411C22×C4D3×D411yescharacteristic
412C22×C4#23711yescharacteristic
413C22×C4#23711yescharacteristic
414C22×C4#23911yescharacteristic
415C22×C4#23911yescharacteristic
416C22×C4C2×#6711yescharacteristic
417M16—22no—
418M16—22no—
419M16—22no—
420M16—22no—
421M16—22no—
422M16—22no—
423M16—41no—
424M16—41no—
425M16—62no—
426M16—62no—
427M16—62no—
428M16—62no—
429M16—121no—
430M16—121no—
431M16—121no—
432M16—121no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—34no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—62no—
440C2×C8—62no—
441C2×C8—62no—
442C2×C8—62no—
443C2×C8—62no—
444C2×C8(C2×D3)⋊C414yes—
445C4⋊C4—22no—
446C4⋊C4—41no—
447C4⋊C4—62no—
448C4⋊C4—62no—
449C22⋊C4—22no—
450C22⋊C4—41no—
451C22⋊C4—62no—
452C22⋊C4—62no—
453C42—22no—
454C42—34no—
455C42—41no—
456C42—62no—
457C22×C3—41no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3C4×C22⋊C411yescharacteristic
465C22×C3#38611yescharacteristic
466C22×C3#38811yescharacteristic
467C22×C3#38811yescharacteristic
468C22×C3#38811yescharacteristic
469C22×C3#38811yescharacteristic
470C22×C3#39011yescharacteristic
471C2×D3—44no—
472C2×D3—42no—
473C2×D3—82no—
474C2×D3—82no—
475C3×C4—82no—
476C3×C4C4×C22⋊C411yescharacteristic
477C3×C4C4×C22⋊C411yescharacteristic
478C3×C4#38611yescharacteristic
479C3×C4#38611yescharacteristic
480C3×C4#38811yescharacteristic
481C3×C4#38811yescharacteristic
482C3×C4#39011yescharacteristic
483C3×C4#39011yescharacteristic
484Dic3—44no—
485Dic3—42no—
486C23—62no—
487C23—81no—
488C23—121no—
489C23—121no—
490C23—242no—
491C23#87211yescharacteristic
492C23C4×D1211yescharacteristic
493C23#88411yescharacteristic
494C23#88511yescharacteristic
495C23#89411yescharacteristic
496C23#89811yescharacteristic
497C23C4×#6711yescharacteristic
498Q8—62no—
499Q8—62no—
500Q8—121no—
501Q8—121no—
502D4—62no—
503D4—62no—
504D4—122no—
505D4—122no—
506D4—122no—
507D4—122no—
508D4—121no—
509D4—121no—
510C2×C4—42no—
511C2×C4—64no—
512C2×C4—62no—
513C2×C4—82no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—242no—
521C2×C4C42×D311yescharacteristic
522C2×C4C4×D1211yescharacteristic
523C2×C4D3×C22⋊C411yescharacteristic
524C2×C4D3×C22⋊C411yescharacteristic
525C2×C4#67⋊C411yescharacteristic
526C2×C4#67⋊C411yescharacteristic
527C2×C4#67⋊C411yescharacteristic
528C2×C4#67⋊C411yescharacteristic
529C2×C4#88211yescharacteristic
530C2×C4#88411yescharacteristic
531C2×C4#88511yescharacteristic
532C2×C4#88611yescharacteristic
533C2×C4#89211yescharacteristic
534C2×C4#89211yescharacteristic
535C2×C4#89311yescharacteristic
536C2×C4#89311yescharacteristic
537C2×C4#89311yescharacteristic
538C2×C4#89311yescharacteristic
539C2×C4#89411yescharacteristic
540C2×C4#89511yescharacteristic
541C2×C4#89511yescharacteristic
542C2×C4#89811yescharacteristic
543C2×C4C2×(C2×D3)⋊C411yescharacteristic
544C2×C4C4×#6711yescharacteristic
545C2×C4#93911yescharacteristic
546C2×C4#93911yescharacteristic
547C2×C4#94011yescharacteristic
548C2×C4#94711yescharacteristic
549C8—24no—
550C8—42no—
551C8—64no—
552C8—122no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3#143711yescharacteristic
561D3—82no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—124no—
570C22—122no—
571C22—242no—
572C22—242no—
573C22#466511yescharacteristic
574C22#466911yescharacteristic
575C22#466911yescharacteristic
576C22#467011yescharacteristic
577C22C4×(C2×D3)⋊C411yescharacteristic
578C22#495711yescharacteristic
579C22#499011yescharacteristic
580C4—82no—
581C4—124no—
582C4—122no—
583C4#466511yescharacteristic
584C4#466611yescharacteristic
585C4#466611yescharacteristic
586C4#467011yescharacteristic
587C4#495711yescharacteristic
588C4#495811yescharacteristic
589C4#498811yescharacteristic
590C4#499011yescharacteristic
591C3#1095011yescharacteristic, complemented, Sylow
592C2—21no—
593C2—21no—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—21no—
599C2—242no—
600C2#6940811yescharacteristic
subgroupG/H
centerC4#4958
commutatorC2×C3×C4C2×C42
FrattiniC23×C4C22×D3
FittingC2×C3×#344C2
socleC2×C3#1437
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenterC2×#344D3
hypocenterC3#10950
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#344

Complements

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

NH ≅ G/Ntype
3. #83747C2semidirect
5. C2×C3×#344C2semidirect
6. C2×C3⋊#338C2semidirect
7. C2×#4531C2semidirect
17. #5139C4semidirect
19. #5129C4semidirect
25. #5004C4semidirect
27. #5001C4semidirect
36. C2×#344D3semidirect
591. C3#10950semidirect

Maximal subgroups

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

subgroupindex
C2×#51392
#837622
#837472
#822132
C2×C3×#3442
C2×C3⋊#3382
C2×#45312
#109503