Finite Group Databasebeta

← #10702259

Subgroups — #10702259

Order 768.

934 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5129C211yesmaximal, characteristic
2C2×#5002C211yesmaximal, characteristic
3C2×#4999C211yesmaximal, characteristic, complemented
4C22×#805C211yesmaximal, characteristic
5C2×C3×#339C211yesmaximal, characteristic, complemented
6C2×#4558C211yesmaximal, characteristic, complemented
7C2×#4524C211yesmaximal, characteristic, complemented
8#11068—31nomaximal, Sylow
9C22×#235C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C22×C3×C42C2211yescharacteristic
13C22×#208C2211yescharacteristic
14C23×C3⋊C8C2211yescharacteristic
15C23×C3⋊C8C2211yescharacteristic
16#5129—22no—
17#5129—22no—
18C2×(C2×D3)⋊C8—22no—
19#5002—22no—
20#5002—22no—
21#4999—22no—
22#4999—22no—
23C2×#810C412yescomplemented
24C2×#809C412yescomplemented
25C2×#805—22no—
26C2×#805—22no—
27C2×#805—22no—
28C2×C4⋊(C3⋊C8)—22no—
29C3×#339—22no—
30#4558—22no—
31#4524—22no—
32C2×#424—31no—
33C22×#104—31no—
34C2×#416—31no—
35C2×#408—31no—
36C2×#341—31no—
37C2×#339D311yescharacteristic, complemented
38C2×#336—31no—
39C23×C3×C4—21no—
40C23×C3×C4C2311yescharacteristic
41C22×#67—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×#235—21no—
52C2×#235D412yes—
53C2×#235Q812yes—
54C22×D12C2×C411yescharacteristic
55C22×C4×D3—21no—
56C22×Dic6C2×C411yescharacteristic
57C2×C3×M16—21no—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16D412yes—
63C22×C3×C8—21no—
64C3×#117—21no—
65C3×#117—21no—
66C3×#117D412yes—
67C3×#117D412yes—
68C2×C3×C4⋊C4C2×C412yes—
69C2×C3×C22⋊C4—21no—
70C2×C3×C42—21no—
71C2×C3×C42—21no—
72C2×C3×C42—21no—
73C2×C3×C42—21no—
74C2×C3×C42—21no—
75C2×C3×C42—21no—
76C2×C3×C42D412yes—
77C2×#208—21no—
78C2×#208—21no—
79C2×#208—21no—
80C2×#208—21no—
81C2×#208—21no—
82C2×#208D412yes—
83C22×C3⋊C8—21no—
84C22×C3⋊C8—21no—
85C22×C3⋊C8—21no—
86C22×C3⋊C8—21no—
87C22×C3⋊C8—21no—
88C22×C3⋊C8—21no—
89C22×C3⋊C8—21no—
90C22×C3⋊C8—21no—
91C22×C3⋊C8—21no—
92C22×C3⋊C8—21no—
93C22×C3⋊C8—21no—
94C22×C3⋊C8—21no—
95C22×C3⋊C8—21no—
96C22×C3⋊C8C2×C412yes—
97C22×C3⋊C8D412yes—
98(C2×D3)⋊C8—28no—
99#810—24no—
100#809—24no—
101#805—28no—
102C4⋊(C3⋊C8)—28no—
103C22×#41—31no—
104C22×M16—31no—
105C22×M16C2×D311yescharacteristic
106C23×C8—31no—
107C23×C8—31no—
108C2×#117C2×D311yescharacteristic
109C22×C42C2×D311yescharacteristic
110#424—62no—
111#424—62no—
112C2×C4⋊C8—62no—
113C2×#104—62no—
114C2×#104—62no—
115C2×#104—62no—
116#416—62no—
117#416—62no—
118C2×#103—32no—
119C2×#102—32no—
120#408—62no—
121#408—62no—
122C2×C22⋊C8—62no—
123#341—62no—
124#339—22no—
125#336—62no—
126C24×C3#4111yescharacteristic
127C23×D3—21no—
128C22×C3×C4—24no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—42no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4C22×C411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×Q811yescharacteristic
143C22×C3×C4#4111yescharacteristic
144C22×C3×C4#4111yescharacteristic
145C22×C3×C4#4111yescharacteristic
146C22×C3×C4#4111yescharacteristic
147C22×C3×C4#4111yescharacteristic
148C2×#67—22no—
149C2×#67—22no—
150C2×#67—41no—
151C2×#67—41no—
152C2×#67—41no—
153C2×#67—41no—
154C22×Dic3—21no—
155#235—22no—
156#235—22no—
157#235—22no—
158#235—22no—
159#235—22no—
160#235—22no—
161#235—22no—
162#235—22no—
163#235—41no—
164#235—41no—
165#235—41no—
166#235—41no—
167#235—41no—
168#235—41no—
169#235—41no—
170#235—41no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12C4⋊C412yes—
177C2×C4×D3—22no—
178C2×C4×D3—22no—
179C2×C4×D3—41no—
180C2×C4×D3—41no—
181C2×C4×D3—41no—
182C2×C4×D3—41no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6C4⋊C412yes—
189C3×M16—22no—
190C3×M16—22no—
191C3×M16—41no—
192C3×M16—41no—
193C2×C3×C8—22no—
194C2×C3×C8—41no—
195C2×C3×C8—41no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4C22⋊C414yes—
198C3×C22⋊C4—22no—
199C3×C22⋊C4—22no—
200C3×C42—22no—
201C3×C42—22no—
202C3×C42—22no—
203C3×C42—22no—
204C3×C42—41no—
205C3×C42—41no—
206#208—22no—
207#208—22no—
208#208—41no—
209#208—41no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—41no—
218C2×C3⋊C8—41no—
219C2×C3⋊C8—41no—
220C2×C3⋊C8—41no—
221C2×C3⋊C8—41no—
222C2×C3⋊C8—41no—
223C2×C3⋊C8—41no—
224C2×C3⋊C8—41no—
225C2×C3⋊C8—41no—
226C2×C3⋊C8—41no—
227C2×#41—32no—
228C2×#41—32no—
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—
238C2×#41—61no—
239C22×Q8—31no—
240C22×D4—31no—
241C22×D4—61no—
242C23×C4—21no—
243C23×C4—61no—
244C23×C4C22×D311yescharacteristic
245C2×M16—21no—
246C2×M16—21no—
247C2×M16—21no—
248C2×M16—21no—
249C2×M16—21no—
250C2×M16—32no—
251C2×M16—61no—
252C2×M16—61no—
253C2×M16—61no—
254C2×M16—61no—
255C2×M16—61no—
256C2×M16D1212yes—
257C22×C8—21no—
258C22×C8—32no—
259C22×C8—32no—
260C22×C8—61no—
261C22×C8—61no—
262C22×C8—61no—
263C22×C8—61no—
264C22×C8—61no—
265C22×C8—61no—
266C22×C8—61no—
267C22×C8—61no—
268C22×C8—61no—
269C22×C8—61no—
270C22×C8—61no—
271C22×C8—61no—
272C22×C8—61no—
273#117—21no—
274#117—21no—
275#117D1212yes—
276#117#6712yes—
277C2×C4⋊C4C4×D312yes—
278C2×C22⋊C4—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42D1212yes—
286C4⋊C8—68no—
287#104—68no—
288#103—64no—
289#102—64no—
290C22⋊C8—68no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3C4×D411yescharacteristic
294C23×C3#12111yescharacteristic
295C23×C3C22⋊Q811yescharacteristic
296C23×C3#12311yescharacteristic
297C23×C3#12311yescharacteristic
298C23×C3#12311yescharacteristic
299C23×C3#12611yescharacteristic
300C22×D3—22no—
301C22×D3—41no—
302C22×D3—41no—
303C22×D3—42no—
304C22×D3—42no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—44no—
308C2×C3×C4—44no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4C2×C22⋊C411yescharacteristic
314C2×C3×C4C2×C4⋊C411yescharacteristic
315C2×C3×C4#11711yescharacteristic
316C2×C3×C4C4×D411yescharacteristic
317C2×C3×C4C4×D411yescharacteristic
318C2×C3×C4C4×D411yescharacteristic
319C2×C3×C4#12011yescharacteristic
320C2×C3×C4#12011yescharacteristic
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#12111yescharacteristic
328C2×C3×C4#12111yescharacteristic
329C2×C3×C4C22⋊Q811yescharacteristic
330C2×C3×C4C22⋊Q811yescharacteristic
331C2×C3×C4C22⋊Q811yescharacteristic
332C2×C3×C4C22⋊Q811yescharacteristic
333C2×C3×C4C22⋊Q811yescharacteristic
334C2×C3×C4#12311yescharacteristic
335C2×C3×C4#12311yescharacteristic
336C2×C3×C4#12411yescharacteristic
337C2×C3×C4#12411yescharacteristic
338C2×C3×C4#12411yescharacteristic
339C2×C3×C4#12611yescharacteristic
340C2×C3×C4#12711yescharacteristic
341#67—42no—
342#67—42no—
343#67—42no—
344#67—42no—
345C2×Dic3—22no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348D12—22no—
349D12—22no—
350D12—41no—
351D12—41no—
352C4×D3—42no—
353C4×D3—42no—
354C4×D3—42no—
355C4×D3—42no—
356Dic6—22no—
357Dic6—22no—
358Dic6—41no—
359Dic6—41no—
360C3×C8—42no—
361C3⋊C8—42no—
362C3⋊C8—42no—
363C3⋊C8—42no—
364C24—61no—
365C24#23711yescharacteristic
366#41—62no—
367#41—62no—
368#41—62no—
369#41—62no—
370#41—62no—
371#41—62no—
372#41—62no—
373#41—62no—
374#41—121no—
375#41—121no—
376#41—121no—
377#41—121no—
378#41—121no—
379#41—121no—
380#41—121no—
381#41—121no—
382C2×Q8—32no—
383C2×Q8—61no—
384C2×Q8—61no—
385C2×Q8—61no—
386C2×Q8—61no—
387C2×Q8—61no—
388C2×D4—32no—
389C2×D4—62no—
390C2×D4—62no—
391C2×D4—61no—
392C2×D4—61no—
393C2×D4—61no—
394C2×D4—61no—
395C2×D4—61no—
396C2×D4—121no—
397C2×D4—121no—
398C2×D4—121no—
399C2×D4—121no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—42no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—61no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—121no—
410C22×C4—121no—
411C22×C4—121no—
412C22×C4—121no—
413C22×C4C2×C4×D311yescharacteristic
414C22×C4C2×D1211yescharacteristic
415C22×C4C2×D1211yescharacteristic
416C22×C4C2×D1211yescharacteristic
417C22×C4#23511yescharacteristic
418C22×C4#23511yescharacteristic
419C22×C4D3×D411yescharacteristic
420C22×C4D3×D411yescharacteristic
421C22×C4D3×D411yescharacteristic
422C22×C4#23711yescharacteristic
423C22×C4#23711yescharacteristic
424C22×C4D3×Q811yescharacteristic
425C22×C4#23911yescharacteristic
426C22×C4C2×#6711yescharacteristic
427M16—22no—
428M16—22no—
429M16—41no—
430M16—41no—
431M16—62no—
432M16—62no—
433M16—121no—
434M16—121no—
435C2×C8—22no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—62no—
439C2×C8—62no—
440C2×C8—62no—
441C2×C8—62no—
442C2×C8—62no—
443C2×C8—62no—
444C2×C8—62no—
445C2×C8—121no—
446C2×C8—121no—
447C2×C8—121no—
448C2×C8—121no—
449C2×C8—121no—
450C2×C8—121no—
451C2×C8—121no—
452C2×C8—121no—
453C2×C8—121no—
454C2×C8—121no—
455C4⋊C4—22no—
456C4⋊C4(C2×D3)⋊C414yes—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—41no—
464C42—41no—
465C22×C3—22no—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3#38511yescharacteristic
474C22×C3#38611yescharacteristic
475C22×C3#38811yescharacteristic
476C22×C3#39411yescharacteristic
477C22×C3#39611yescharacteristic
478C22×C3#39711yescharacteristic
479C22×C3#39911yescharacteristic
480C22×C3#44312yes—
481C22×C3#44412yes—
482C2×D3—44no—
483C2×D3—42no—
484C2×D3—82no—
485C2×D3—81no—
486C2×D3—81no—
487C3×C4—44no—
488C3×C4—44no—
489C3×C4#38011yescharacteristic
490C3×C4#38511yescharacteristic
491C3×C4#38811yescharacteristic
492C3×C4#39011yescharacteristic
493C3×C4#39211yescharacteristic
494C3×C4#39311yescharacteristic
495C3×C4#39411yescharacteristic
496C3×C4#39611yescharacteristic
497Dic3—42no—
498C23—22no—
499C23—22no—
500C23—62no—
501C23—121no—
502C23—121no—
503C23—122no—
504C23—122no—
505C23#67⋊C411yescharacteristic
506C23#88311yescharacteristic
507C23#88611yescharacteristic
508C23#88611yescharacteristic
509C23#89711yescharacteristic
510C23#89811yescharacteristic
511C23#93911yescharacteristic
512Q8—62no—
513Q8—62no—
514Q8—121no—
515Q8—121no—
516D4—62no—
517D4—62no—
518D4—122no—
519D4—122no—
520D4—122no—
521D4—122no—
522D4—121no—
523D4—121no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—62no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—122no—
536C2×C4—122no—
537C2×C4—122no—
538C2×C4—122no—
539C2×C4C4×D1211yescharacteristic
540C2×C4C4×D1211yescharacteristic
541C2×C4#87411yescharacteristic
542C2×C4#87511yescharacteristic
543C2×C4#67⋊C411yescharacteristic
544C2×C4#88211yescharacteristic
545C2×C4#88211yescharacteristic
546C2×C4#88211yescharacteristic
547C2×C4#88311yescharacteristic
548C2×C4#88511yescharacteristic
549C2×C4#88511yescharacteristic
550C2×C4#88611yescharacteristic
551C2×C4D3×C4⋊C411yescharacteristic
552C2×C4#89211yescharacteristic
553C2×C4#89511yescharacteristic
554C2×C4#89511yescharacteristic
555C2×C4#89511yescharacteristic
556C2×C4#89611yescharacteristic
557C2×C4#89611yescharacteristic
558C2×C4#89711yescharacteristic
559C2×C4#89711yescharacteristic
560C2×C4#89811yescharacteristic
561C2×C4C2×(C2×D3)⋊C411yescharacteristic
562C2×C4#93011yescharacteristic
563C2×C4#93011yescharacteristic
564C2×C4#93811yescharacteristic
565C2×C4#93911yescharacteristic
566C2×C4#94611yescharacteristic
567C8—42no—
568C8—122no—
569C8—122no—
570C8—122no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3#143911yescharacteristic
576C2×C3#186612yes—
577C2×C3#187612yes—
578C2×C3#193711yescharacteristic
579C2×C3#193811yescharacteristic
580D3—82no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—122no—
590C22—124no—
591C22—241no—
592C22—241no—
593C22—242no—
594C22#466811yescharacteristic
595C22#467011yescharacteristic
596C22#467411yescharacteristic
597C22#467711yescharacteristic
598C22#482312yes—
599C22#482412yes—
600C22#495811yescharacteristic
601C22#496111yescharacteristic
602C22#499411yescharacteristic
603C4—44no—
604C4—44no—
605C4—122no—
606C4#466811yescharacteristic
607C4#467011yescharacteristic
608C4#467111yescharacteristic
609C4#467311yescharacteristic
610C4#493911yescharacteristic
611C4#496111yescharacteristic
612C4#498711yescharacteristic
613C4#499411yescharacteristic
614C3#1106811yescharacteristic, complemented, Sylow
615C2—21no—
616C2—21no—
617C2—21no—
618C2—21no—
619C2—242no—
620C2#6874511yescharacteristic
621C2#6874611yescharacteristic
622C2#6877412yes—
623C2#6942711yescharacteristic
624C2#7037712yes—
subgroupG/H
centerC22×C4D3×D4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#339C2
socleC23×C3C4×D4
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenterC2×#339D3
hypocenterC3#11068
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
3. C2×#4999C2semidirect
5. C2×C3×#339C2semidirect
6. C2×#4558C2semidirect
7. C2×#4524C2semidirect
23. C2×#810C4semidirect
24. C2×#809C4semidirect
37. C2×#339D3semidirect
614. C3#11068semidirect

Maximal subgroups

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

subgroupindex
C2×#51292
C2×#50022
C2×#49992
C22×#8052
C2×C3×#3392
C2×#45582
C2×#45242
#110683