Finite Group Databasebeta

← #10701599

Subgroups — #10701599

Order 768.

916 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83768C211yesmaximal, characteristic
2#83747C211yesmaximal, characteristic, complemented
3C2×#5006C211yesmaximal, characteristic
4C2×#5005C211yesmaximal, characteristic
5C2×C3×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
6#71592C211yesmaximal, characteristic, complemented
7#71574C211yesmaximal, characteristic, complemented
8#10951—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C2×#926C2211yescharacteristic
15C22×#208C2211yescharacteristic
16#5133—22no—
17#5131C412yescomplemented
18#5122—22no—
19#5120C412yescomplemented
20Dic3×M16—22no—
21#5113—22no—
22#5101—22no—
23#5006—22no—
24#5006C414yescomplemented
25#5005—22no—
26#5005—22no—
27C2×#806—22no—
28C2×#805C414yescomplemented
29C3×H(3,ℤ/4)—22no—
30#4532—22no—
31#4526—22no—
32C2×#421—31no—
33#2896—31no—
34C2×#410—31no—
35#2873—31no—
36#1740—31no—
37#1733—31no—
38C2×H(3,ℤ/4)D311yescharacteristic, complemented
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×M16C2×C412yes—
61C2×C3×M16D412yes—
62C22×C3×C8D412yes—
63C3×#117—21no—
64C3×#117—21no—
65C3×#117—21no—
66C3×#117—21no—
67C3×#117—21no—
68C3×#117—21no—
69C3×#117—21no—
70C3×#117C2×C412yes—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C42—21no—
76C2×C3×C42C2×C412yes—
77C2×#217—21no—
78C2×#217D412yes—
79#926—21no—
80#926—21no—
81#926—21no—
82#926—21no—
83#926—21no—
84#926C2×C412yes—
85#926C2×C412yes—
86#926D412yes—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4×Dic3—21no—
90C2×C4×Dic3—21no—
91C2×#208—21no—
92C2×#208—21no—
93C2×#208—21no—
94C2×#208C2×C412yes—
95C2×#208D412yes—
96C2×#208D412yes—
97C22×C3⋊C8—21no—
98#821—24no—
99#816—24no—
100#815—24no—
101C8×Dic3—24no—
102#806—28no—
103#805—28no—
104C22×#41—31no—
105C22×M16—31no—
106C22×M16C2×D311yescharacteristic
107C2×#117—31no—
108C2×#117—31no—
109C2×#117C2×D311yescharacteristic
110C2×#117C2×D311yescharacteristic
111#421—62no—
112#421—62no—
113C2×#104—34no—
114#419—32no—
115#418—32no—
116#417—62no—
117#416—62no—
118#410—34no—
119#410—62no—
120C2×#99—62no—
121#405—62no—
122#405—62no—
123C4×M16—62no—
124#342—62no—
125#339—62no—
126H(3,ℤ/4)—22no—
127C24×C3C2×D411yescharacteristic
128C23×D3—21no—
129C22×C3×C4—22no—
130C22×C3×C4—41no—
131C22×C3×C4C22×C411yescharacteristic
132C22×C3×C4C22×C411yescharacteristic
133C22×C3×C4C22×C411yescharacteristic
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
144C22×C3×C4#4111yescharacteristic
145C2×#67—22no—
146C2×#67—22no—
147C2×#67—41no—
148C2×#67—41no—
149C2×#67—41no—
150C2×#67—41no—
151C22×Dic3—21no—
152C22×Dic3—22no—
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×D12C4212yes—
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×Dic6C4212yes—
188C3×M16—22no—
189C3×M16—22no—
190C3×M16—22no—
191C3×M16—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8C22⋊C414yes—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—41no—
197C3×C4⋊C4—41no—
198C3×C4⋊C4—41no—
199C3×C22⋊C4—22no—
200C3×C22⋊C4—41no—
201C3×C22⋊C4—41no—
202C3×C22⋊C4—41no—
203C3×C42—22no—
204C3×C42—22no—
205C3×C42—22no—
206C3×C42—41no—
207#217—22no—
208#217—22no—
209#217—41no—
210#217C22⋊C414yes—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3—41no—
215C4×Dic3—22no—
216C4×Dic3—22no—
217C4×Dic3—22no—
218C4×Dic3—41no—
219#208—22no—
220#208—22no—
221#208—22no—
222#208C22⋊C414yes—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
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—32no—
248C2×M16—32no—
249C2×M16—32no—
250C2×M16—61no—
251C2×M16—61no—
252C2×M16—61no—
253C2×M16C4×D312yes—
254C2×M16#6712yes—
255C22×C8—61no—
256C22×C8D1212yes—
257#117—21no—
258#117—21no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—32no—
265#117—32no—
266#117—32no—
267#117—61no—
268#117—61no—
269#117—61no—
270#117—61no—
271#117—61no—
272#117C4×D312yes—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—32no—
280C2×C22⋊C4—61no—
281C2×C42—21no—
282C2×C42—61no—
283C2×C42—61no—
284C2×C42C4×D312yes—
285#104—68no—
286#103—64no—
287#102—64no—
288#99—68no—
289#97—64no—
290C4×C8—64no—
291C23×C3—81no—
292C23×C3C2×C22⋊C411yescharacteristic
293C23×C3C4×D411yescharacteristic
294C23×C3C4×D411yescharacteristic
295C23×C3#12111yescharacteristic
296C23×C3#12111yescharacteristic
297C23×C3#12411yescharacteristic
298C23×C3#12711yescharacteristic
299C22×D3—22no—
300C22×D3—41no—
301C22×D3—41no—
302C22×D3—82no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—82no—
306C2×C3×C4—82no—
307C2×C3×C4C2×C4211yescharacteristic
308C2×C3×C4C2×C22⋊C411yescharacteristic
309C2×C3×C4C2×C22⋊C411yescharacteristic
310C2×C3×C4#11711yescharacteristic
311C2×C3×C4#11711yescharacteristic
312C2×C3×C4#11711yescharacteristic
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×C4C4×D411yescharacteristic
322C2×C3×C4C4×D411yescharacteristic
323C2×C3×C4#12011yescharacteristic
324C2×C3×C4#12111yescharacteristic
325C2×C3×C4#12111yescharacteristic
326C2×C3×C4#12111yescharacteristic
327C2×C3×C4#12111yescharacteristic
328C2×C3×C4#12311yescharacteristic
329C2×C3×C4#12311yescharacteristic
330C2×C3×C4#12311yescharacteristic
331C2×C3×C4#12411yescharacteristic
332C2×C3×C4#12411yescharacteristic
333C2×C3×C4#12611yescharacteristic
334C2×C3×C4#12611yescharacteristic
335#67—42no—
336#67—42no—
337#67—42no—
338#67—42no—
339C2×Dic3—22no—
340C2×Dic3—24no—
341C2×Dic3—41no—
342C2×Dic3—41no—
343C2×Dic3—42no—
344C2×Dic3—82no—
345C2×Dic3—82no—
346D12—22no—
347D12—22no—
348D12—41no—
349D12—41no—
350C4×D3—42no—
351C4×D3—42no—
352C4×D3—42no—
353C4×D3—42no—
354Dic6—22no—
355Dic6—22no—
356Dic6—41no—
357Dic6—41no—
358C3×C8—24no—
359C3⋊C8—24no—
360C24—61no—
361C24D3×D411yescharacteristic
362#41—62no—
363#41—62no—
364#41—62no—
365#41—62no—
366#41—62no—
367#41—62no—
368#41—62no—
369#41—62no—
370#41—121no—
371#41—121no—
372#41—121no—
373#41—121no—
374#41—121no—
375#41—121no—
376#41—121no—
377#41—121no—
378C2×Q8—32no—
379C2×Q8—61no—
380C2×Q8—61no—
381C2×Q8—61no—
382C2×Q8—61no—
383C2×Q8—61no—
384C2×D4—32no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4—61no—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4—61no—
391C2×D4—61no—
392C2×D4—121no—
393C2×D4—121no—
394C2×D4—121no—
395C2×D4—121no—
396C22×C4—22no—
397C22×C4—41no—
398C22×C4—61no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—121no—
403C22×C4—121no—
404C22×C4—121no—
405C22×C4—121no—
406C22×C4—121no—
407C22×C4C2×C4×D311yescharacteristic
408C22×C4C2×C4×D311yescharacteristic
409C22×C4C2×C4×D311yescharacteristic
410C22×C4C2×D1211yescharacteristic
411C22×C4#23511yescharacteristic
412C22×C4#23511yescharacteristic
413C22×C4D3×D411yescharacteristic
414C22×C4D3×D411yescharacteristic
415C22×C4D3×D411yescharacteristic
416C22×C4#23711yescharacteristic
417C22×C4#23711yescharacteristic
418C22×C4#23911yescharacteristic
419C22×C4#23911yescharacteristic
420C22×C4C2×#6711yescharacteristic
421M16—22no—
422M16—22no—
423M16—22no—
424M16—22no—
425M16—34no—
426M16—62no—
427M16—62no—
428M16—62no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—62no—
432C2×C8—62no—
433C2×C8—62no—
434C2×C8(C2×D3)⋊C414yes—
435C4⋊C4—22no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—62no—
440C4⋊C4—62no—
441C4⋊C4—62no—
442C4⋊C4—121no—
443C22⋊C4—22no—
444C22⋊C4—34no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—62no—
449C22⋊C4—62no—
450C22⋊C4—121no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—41no—
455C42—62no—
456C42—62no—
457C42—62no—
458C42—121no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—41no—
465C22×C3—41no—
466C22×C3C4×C22⋊C411yescharacteristic
467C22×C3#38611yescharacteristic
468C22×C3#38811yescharacteristic
469C22×C3#38811yescharacteristic
470C22×C3#39011yescharacteristic
471C22×C3#39011yescharacteristic
472C22×C3#39011yescharacteristic
473C2×D3—42no—
474C2×D3—82no—
475C2×D3—82no—
476C2×D3—82no—
477C3×C4—82no—
478C3×C4—82no—
479C3×C4C4×C22⋊C411yescharacteristic
480C3×C4C4×C22⋊C411yescharacteristic
481C3×C4#38611yescharacteristic
482C3×C4#38611yescharacteristic
483C3×C4#38811yescharacteristic
484C3×C4#38811yescharacteristic
485C3×C4#38811yescharacteristic
486C3×C4#38811yescharacteristic
487Dic3—44no—
488Dic3—42no—
489Dic3—82no—
490C23—62no—
491C23—81no—
492C23—121no—
493C23—121no—
494C23—242no—
495C23D3×C22⋊C411yescharacteristic
496C23#67⋊C411yescharacteristic
497C23#88411yescharacteristic
498C23#88511yescharacteristic
499C23#89311yescharacteristic
500C23#89511yescharacteristic
501C23#94011yescharacteristic
502Q8—62no—
503Q8—62no—
504Q8—121no—
505Q8—121no—
506D4—62no—
507D4—62no—
508D4—122no—
509D4—122no—
510D4—122no—
511D4—122no—
512D4—121no—
513D4—121no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—64no—
517C2×C4—62no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4—242no—
528C2×C4—242no—
529C2×C4C42×D311yescharacteristic
530C2×C4#87211yescharacteristic
531C2×C4C4×D1211yescharacteristic
532C2×C4C4×D1211yescharacteristic
533C2×C4D3×C22⋊C411yescharacteristic
534C2×C4#67⋊C411yescharacteristic
535C2×C4#67⋊C411yescharacteristic
536C2×C4#67⋊C411yescharacteristic
537C2×C4#88211yescharacteristic
538C2×C4#88411yescharacteristic
539C2×C4#88511yescharacteristic
540C2×C4#88611yescharacteristic
541C2×C4#89211yescharacteristic
542C2×C4#89211yescharacteristic
543C2×C4#89311yescharacteristic
544C2×C4#89311yescharacteristic
545C2×C4#89311yescharacteristic
546C2×C4#89411yescharacteristic
547C2×C4#89411yescharacteristic
548C2×C4#89511yescharacteristic
549C2×C4#89811yescharacteristic
550C2×C4#89811yescharacteristic
551C2×C4C2×(C2×D3)⋊C411yescharacteristic
552C2×C4C4×#6711yescharacteristic
553C2×C4C4×#6711yescharacteristic
554C2×C4#93911yescharacteristic
555C2×C4#93911yescharacteristic
556C2×C4#94711yescharacteristic
557C8—24no—
558C8—64no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3#143711yescharacteristic
567D3—82no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—122no—
576C22—242no—
577C22—242no—
578C22—242no—
579C22#466511yescharacteristic
580C22#466611yescharacteristic
581C22#466911yescharacteristic
582C22#467011yescharacteristic
583C22#495711yescharacteristic
584C22#495711yescharacteristic
585C22#498811yescharacteristic
586C4—82no—
587C4—82no—
588C4—124no—
589C4—122no—
590C4—242no—
591C4#466511yescharacteristic
592C4#466611yescharacteristic
593C4#466911yescharacteristic
594C4#467011yescharacteristic
595C4C4×(C2×D3)⋊C411yescharacteristic
596C4#495811yescharacteristic
597C4#499011yescharacteristic
598C4#499011yescharacteristic
599C3#1095111yescharacteristic, complemented, Sylow
600C2—21no—
601C2—21no—
602C2—21no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—242no—
608C2#6940811yescharacteristic
subgroupG/H
centerC4#4665
commutatorC2×C3×C4C2×C42
FrattiniC23×C4C22×D3
FittingC2×C3×H(3,ℤ/4)C2
socleC2×C3#1437
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenterC2×H(3,ℤ/4)D3
hypocenterC3#10951
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
2. #83747C2semidirect
5. C2×C3×H(3,ℤ/4)C2semidirect
6. #71592C2semidirect
7. #71574C2semidirect
17. #5131C4semidirect
19. #5120C4semidirect
24. #5006C4semidirect
28. C2×#805C4semidirect
38. C2×H(3,ℤ/4)D3semidirect
599. C3#10951semidirect

Maximal subgroups

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

subgroupindex
#837682
#837472
C2×#50062
C2×#50052
C2×C3×H(3,ℤ/4)2
#715922
#715742
#109513