Finite Group Databasebeta

← #10702236

Subgroups — #10702236

Order 768.

870 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5139C211yesmaximal, characteristic
2#83768C211yesmaximal, characteristic
3#83750C211yesmaximal, characteristic, complemented
4C2×#5110C211yesmaximal, characteristic
5C2×C3×#341C211yesmaximal, characteristic, complemented
6C2×#4559C211yesmaximal, characteristic, complemented
7#71969C211yesmaximal, characteristic, complemented
8#11065—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C23×C3×C8C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C2×#926C2211yescharacteristic
15C22×#208C2211yescharacteristic
16#5139—22no—
17#5139—22no—
18C2×#825—22no—
19#5133—22no—
20#5131C412yescomplemented
21#5120C412yescomplemented
22#5119—22no—
23#5118—22no—
24#5113—22no—
25#5110—22no—
26#5110—22no—
27C2×(C2×D3)⋊C8—22no—
28#5102—22no—
29C3×#341—22no—
30#4559—22no—
31#4551—22no—
32#2905—31no—
33C2×#421—31no—
34#2896—31no—
35C2×#408—31no—
36C2×#344—31no—
37C2×#341D311yescharacteristic, complemented
38#1735—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×#235D412yes—
52C2×#235Q812yes—
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×M16—21no—
66C2×C3×M16C2×C412yes—
67C2×C3×M16D412yes—
68C22×C3×C8—21no—
69C22×C3×C8—21no—
70C22×C3×C8—21no—
71C22×C3×C8—21no—
72C22×C3×C8—21no—
73C22×C3×C8—21no—
74C22×C3×C8—21no—
75C22×C3×C8—21no—
76C22×C3×C8D412yes—
77C2×#217—21no—
78C2×#217—21no—
79#926—21no—
80#926—21no—
81#926—21no—
82#926—21no—
83#926—21no—
84#926C2×C412yes—
85#926D412yes—
86#926D412yes—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3D412yes—
89C2×C4×Dic3—21no—
90C2×C4×Dic3—21no—
91C2×#208—21no—
92C2×#208—21no—
93C2×#208—21no—
94C2×#208—21no—
95C2×#208—21no—
96C2×#208—21no—
97C22×C3⋊C8D412yes—
98#825—28no—
99#821—24no—
100(C2×D3)⋊C8—28no—
101#816—24no—
102Dic3⋊C8—28no—
103C22×#41—31no—
104C22×M16—31no—
105C22×M16C2×D311yescharacteristic
106C22×M16C2×D311yescharacteristic
107C23×C8C2×D311yescharacteristic
108C2×#117—31no—
109C2×#117—31no—
110#424—62no—
111#424—62no—
112#423—62no—
113#421—62no—
114#421—62no—
115C2×#104—62no—
116#419—32no—
117#418—32no—
118#417—62no—
119#416—62no—
120#408—62no—
121#408—62no—
122C2×C22⋊C8—62no—
123#344—62no—
124#341—22no—
125#340—62no—
126C24×C3#4111yescharacteristic
127C23×D3—21no—
128C22×C3×C4C22×C411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×Q811yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×C3×C4#4111yescharacteristic
142C2×#67—22no—
143C2×#67—22no—
144C2×#67—41no—
145C2×#67—41no—
146C2×#67—41no—
147C2×#67—41no—
148C22×Dic3—21no—
149C22×Dic3—22no—
150C22×Dic3—41no—
151#235—22no—
152#235—22no—
153#235—22no—
154#235—22no—
155#235—22no—
156#235—22no—
157#235—22no—
158#235—22no—
159#235—41no—
160#235—41no—
161#235—41no—
162#235—41no—
163#235—41no—
164#235—41no—
165#235—41no—
166#235—41no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12C4⋊C412yes—
173C2×C4×D3—22no—
174C2×C4×D3—22no—
175C2×C4×D3—41no—
176C2×C4×D3—41no—
177C2×C4×D3—41no—
178C2×C4×D3—41no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6C4⋊C412yes—
185C3×M16—22no—
186C3×M16—22no—
187C3×M16—22no—
188C3×M16—22no—
189C3×M16—41no—
190C3×M16—41no—
191C3×M16—41no—
192C3×M16—41no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8—41no—
199C2×C3×C8—41no—
200C2×C3×C8—41no—
201C2×C3×C8—41no—
202C2×C3×C8—41no—
203C2×C3×C8—41no—
204C2×C3×C8—41no—
205C2×C3×C8—41no—
206#217—22no—
207#217—22no—
208#217—22no—
209#217—41no—
210C4⋊Dic3—22no—
211C4⋊Dic3—22no—
212C4⋊Dic3—41no—
213C4⋊Dic3C22⋊C414yes—
214C4×Dic3—22no—
215C4×Dic3—22no—
216C4×Dic3—22no—
217C4×Dic3—41no—
218#208—41no—
219#208—41no—
220#208—41no—
221#208—41no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—22no—
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×C4C22×D311yescharacteristic
242C2×M16—21no—
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—61no—
253C2×M16—61no—
254C2×M16—61no—
255C2×M16—61no—
256C2×M16—61no—
257C2×M16—61no—
258C2×M16C4×D312yes—
259C2×M16D1212yes—
260C22×C8—21no—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8—21no—
268C22×C8—32no—
269C22×C8#6712yes—
270#117—32no—
271#117—32no—
272#117—32no—
273#117—61no—
274#117—61no—
275#117—61no—
276#117—61no—
277#117—61no—
278C2×C4⋊C4—32no—
279C2×C4⋊C4—61no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282C2×C42—61no—
283C2×C42—61no—
284C4⋊C8—68no—
285#104—68no—
286#103—64no—
287#102—64no—
288C22⋊C8—68no—
289C23×C3—42no—
290C23×C3C4×D411yescharacteristic
291C23×C3#12111yescharacteristic
292C23×C3C22⋊Q811yescharacteristic
293C23×C3#12311yescharacteristic
294C23×C3#12311yescharacteristic
295C23×C3#12311yescharacteristic
296C23×C3#12611yescharacteristic
297C22×D3—22no—
298C22×D3—44no—
299C22×D3—41no—
300C22×D3—41no—
301C2×C3×C4C2×C22⋊C411yescharacteristic
302C2×C3×C4C2×C4⋊C411yescharacteristic
303C2×C3×C4#11711yescharacteristic
304C2×C3×C4C4×D411yescharacteristic
305C2×C3×C4C4×D411yescharacteristic
306C2×C3×C4C4×D411yescharacteristic
307C2×C3×C4#12011yescharacteristic
308C2×C3×C4#12011yescharacteristic
309C2×C3×C4#12011yescharacteristic
310C2×C3×C4#12111yescharacteristic
311C2×C3×C4#12111yescharacteristic
312C2×C3×C4#12111yescharacteristic
313C2×C3×C4#12111yescharacteristic
314C2×C3×C4#12111yescharacteristic
315C2×C3×C4#12111yescharacteristic
316C2×C3×C4#12111yescharacteristic
317C2×C3×C4C22⋊Q811yescharacteristic
318C2×C3×C4C22⋊Q811yescharacteristic
319C2×C3×C4C22⋊Q811yescharacteristic
320C2×C3×C4C22⋊Q811yescharacteristic
321C2×C3×C4C22⋊Q811yescharacteristic
322C2×C3×C4#12311yescharacteristic
323C2×C3×C4#12311yescharacteristic
324C2×C3×C4#12411yescharacteristic
325C2×C3×C4#12411yescharacteristic
326C2×C3×C4#12411yescharacteristic
327C2×C3×C4#12611yescharacteristic
328C2×C3×C4#12711yescharacteristic
329#67—42no—
330#67—42no—
331#67—42no—
332#67—42no—
333C2×Dic3—22no—
334C2×Dic3—24no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—42no—
338C2×Dic3—82no—
339C2×Dic3—82no—
340D12—22no—
341D12—22no—
342D12—41no—
343D12—41no—
344C4×D3—42no—
345C4×D3—42no—
346C4×D3—42no—
347C4×D3—42no—
348Dic6—22no—
349Dic6—22no—
350Dic6—41no—
351Dic6—41no—
352C3×C8—42no—
353C3×C8—42no—
354C3×C8—42no—
355C3⋊C8—42no—
356C24—61no—
357C24#23911yescharacteristic
358#41—62no—
359#41—62no—
360#41—62no—
361#41—62no—
362#41—62no—
363#41—62no—
364#41—62no—
365#41—62no—
366#41—121no—
367#41—121no—
368#41—121no—
369#41—121no—
370#41—121no—
371#41—121no—
372#41—121no—
373#41—121no—
374C2×Q8—32no—
375C2×Q8—61no—
376C2×Q8—61no—
377C2×Q8—61no—
378C2×Q8—61no—
379C2×Q8—61no—
380C2×D4—32no—
381C2×D4—62no—
382C2×D4—62no—
383C2×D4—61no—
384C2×D4—61no—
385C2×D4—61no—
386C2×D4—61no—
387C2×D4—61no—
388C2×D4—121no—
389C2×D4—121no—
390C2×D4—121no—
391C2×D4—121no—
392C22×C4—61no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—121no—
397C22×C4—121no—
398C22×C4—121no—
399C22×C4—121no—
400C22×C4—121no—
401C22×C4C2×C4×D311yescharacteristic
402C22×C4C2×D1211yescharacteristic
403C22×C4#23511yescharacteristic
404C22×C4#23511yescharacteristic
405C22×C4#23511yescharacteristic
406C22×C4#23511yescharacteristic
407C22×C4D3×D411yescharacteristic
408C22×C4D3×D411yescharacteristic
409C22×C4D3×D411yescharacteristic
410C22×C4D3×D411yescharacteristic
411C22×C4D3×D411yescharacteristic
412C22×C4#23711yescharacteristic
413C22×C4D3×Q811yescharacteristic
414C22×C4C2×#6711yescharacteristic
415M16—22no—
416M16—22no—
417M16—22no—
418M16—22no—
419M16—41no—
420M16—41no—
421M16—41no—
422M16—41no—
423M16—121no—
424M16—121no—
425M16—121no—
426M16—121no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—62no—
441C2×C8—62no—
442C2×C8—62no—
443C4⋊C4—34no—
444C4⋊C4—62no—
445C4⋊C4—62no—
446C4⋊C4—121no—
447C22⋊C4—62no—
448C22⋊C4—62no—
449C22⋊C4—62no—
450C22⋊C4—121no—
451C42—62no—
452C42—62no—
453C42—62no—
454C42—121no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3#38511yescharacteristic
463C22×C3#38611yescharacteristic
464C22×C3#38811yescharacteristic
465C22×C3#39411yescharacteristic
466C22×C3#39611yescharacteristic
467C22×C3#39711yescharacteristic
468C22×C3#39911yescharacteristic
469C2×D3—44no—
470C2×D3—42no—
471C2×D3—82no—
472C2×D3—82no—
473C3×C4#38011yescharacteristic
474C3×C4#38511yescharacteristic
475C3×C4#38811yescharacteristic
476C3×C4#39011yescharacteristic
477C3×C4#39211yescharacteristic
478C3×C4#39311yescharacteristic
479C3×C4#39411yescharacteristic
480C3×C4#39611yescharacteristic
481Dic3—44no—
482Dic3—42no—
483Dic3—82no—
484C23—42no—
485C23—62no—
486C23—124no—
487C23—121no—
488C23—121no—
489C23#89311yescharacteristic
490C23#89411yescharacteristic
491C23#89411yescharacteristic
492C23#89411yescharacteristic
493C23#89511yescharacteristic
494C23#89811yescharacteristic
495C23#94611yescharacteristic
496Q8—62no—
497Q8—62no—
498Q8—121no—
499Q8—121no—
500D4—62no—
501D4—62no—
502D4—122no—
503D4—122no—
504D4—122no—
505D4—122no—
506D4—121no—
507D4—121no—
508C2×C4—64no—
509C2×C4—62no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—242no—
518C2×C4—242no—
519C2×C4#87211yescharacteristic
520C2×C4C4×D1211yescharacteristic
521C2×C4#87511yescharacteristic
522C2×C4#87611yescharacteristic
523C2×C4D3×C22⋊C411yescharacteristic
524C2×C4#67⋊C411yescharacteristic
525C2×C4#88211yescharacteristic
526C2×C4#88211yescharacteristic
527C2×C4#88311yescharacteristic
528C2×C4#88411yescharacteristic
529C2×C4#88411yescharacteristic
530C2×C4#88411yescharacteristic
531C2×C4#88411yescharacteristic
532C2×C4#88411yescharacteristic
533C2×C4#88511yescharacteristic
534C2×C4#88511yescharacteristic
535C2×C4D3×C4⋊C411yescharacteristic
536C2×C4#89611yescharacteristic
537C2×C4#89611yescharacteristic
538C2×C4#89611yescharacteristic
539C2×C4#89611yescharacteristic
540C2×C4#89711yescharacteristic
541C2×C4C4×#6711yescharacteristic
542C2×C4#92911yescharacteristic
543C2×C4#93011yescharacteristic
544C2×C4#93711yescharacteristic
545C2×C4#93911yescharacteristic
546C2×C4#94011yescharacteristic
547C8—42no—
548C8—42no—
549C8—42no—
550C8—122no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3#143911yescharacteristic
558C2×C3#188112yes—
559D3—82no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—124no—
568C22—122no—
569C22—242no—
570C22—242no—
571C22#466511yescharacteristic
572C22#466911yescharacteristic
573C22#467211yescharacteristic
574C22#467511yescharacteristic
575C22#498911yescharacteristic
576C22#499311yescharacteristic
577C22#499411yescharacteristic
578C4—122no—
579C4—124no—
580C4—242no—
581C4#466711yescharacteristic
582C4#466811yescharacteristic
583C4#467311yescharacteristic
584C4#467611yescharacteristic
585C4#494011yescharacteristic
586C4#495711yescharacteristic
587C4#495911yescharacteristic
588C4#496011yescharacteristic
589C3#1106511yescharacteristic, complemented, Sylow
590C2—21no—
591C2—21no—
592C2—21no—
593C2—21no—
594C2—21no—
595C2—21no—
596C2—242no—
597C2#6894712yes—
598C2#6942811yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#341C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenterC2×#341D3
hypocenterC3#11065
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#341

Complements

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

NH ≅ G/Ntype
3. #83750C2semidirect
5. C2×C3×#341C2semidirect
6. C2×#4559C2semidirect
7. #71969C2semidirect
20. #5131C4semidirect
21. #5120C4semidirect
37. C2×#341D3semidirect
589. C3#11065semidirect

Maximal subgroups

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

subgroupindex
C2×#51392
#837682
#837502
C2×#51102
C2×C3×#3412
C2×#45592
#719692
#110653