Finite Group Databasebeta

← C2×#69032 #10794848

Subgroups — C2×#69032 #10794848

Order 768.

1330 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#907C211yesmaximal, characteristic
2C2×#5139C211yesmaximal, characteristic
3C22×#822C211yesmaximal, characteristic, complemented
4C2×#5005C211yesmaximal, characteristic
5C2×#4998C211yesmaximal, characteristic, complemented
6C2×C3×#344C211yesmaximal, characteristic, complemented
7C2×#4559C211yesmaximal, characteristic, complemented
8#69032C218yesmaximal, direct product factor
9C2×#1814—31nomaximal, Sylow
10C22×#235C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C22×C3×M16C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C2×#926C2211yescharacteristic
15C22×#208C2211yescharacteristic
16C23×C3⋊C8C2211yescharacteristic
17C2×#907—21no—
18C2×#907—21no—
19C2×#907—22no—
20C2×#907—22no—
21C2×#907C412yescomplemented
22C2×#907C412yescomplemented
23C2×#907C414yescomplemented
24C2×#907C2214yes—
25C22×ΓL(1,𝔽25)—21no—
26C22×D3×C8—21no—
27#5139—22no—
28#5139C2214yes—
29C2×#825—22no—
30C2×#822—21no—
31C2×#822—21no—
32C2×#822—22no—
33C2×#822C2214yescomplemented
34#5005—22no—
35#5005C2214yes—
36#4998—22no—
37#4998C2214yescomplemented
38C2×#808—21no—
39C2×#807—21no—
40C2×#805—22no—
41C3×#344C2214yescomplemented
42#4559C2214yescomplemented
43C22×#131—31no—
44C22×#108—31no—
45C2×#427—31no—
46C2×#421—31no—
47C2×#421—31no—
48#1814—38no—
49C2×#344—31no—
50C2×#344D311yescharacteristic, complemented
51C23×C3×C4C2311yescharacteristic
52C22×#67—21no—
53C2×#235—21no—
54C2×#235—21no—
55C2×#235—21no—
56C2×#235—21no—
57C2×#235—21no—
58C2×#235—21no—
59C2×#235C2×C412yes—
60C2×#235C2×C412yes—
61C2×#235D412yes—
62C2×#235D412yes—
63C2×#235Q812yes—
64C2×#235C2312yes—
65C22×D12Q811yescharacteristic
66C22×C4×D3—21no—
67C22×Dic6Q811yescharacteristic
68C2×C3×M16—21no—
69C2×C3×M16—21no—
70C2×C3×M16—21no—
71C2×C3×M16—21no—
72C2×C3×M16—21no—
73C2×C3×M16—21no—
74C2×C3×M16—21no—
75C2×C3×M16—21no—
76C2×C3×M16C2×C412yes—
77C2×C3×M16C2×C412yes—
78C2×C3×M16C2312yes—
79C2×C3×M16C2312yes—
80C22×C3×C8—21no—
81C22×C3×C8—21no—
82C3×#117—21no—
83C3×#117—21no—
84C3×#117—21no—
85C3×#117C2312yes—
86C2×C3×C4⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C42—21no—
89C2×#217—21no—
90#926—21no—
91#926—21no—
92#926—21no—
93#926C2312yes—
94C2×C4⋊Dic3—21no—
95C2×C4×Dic3—21no—
96C2×#208—21no—
97C2×#208—21no—
98C2×#208—21no—
99C2×#208—21no—
100C2×#208—21no—
101C2×#208C2312yes—
102C22×C3⋊C8—21no—
103C22×C3⋊C8C2×C411yescharacteristic
104C22×C3⋊C8C2×C411yescharacteristic
105C22×C3⋊C8C2×C412yes—
106C22×C3⋊C8D411yescharacteristic
107C22×C3⋊C8D411yescharacteristic
108C22×C3⋊C8D412yes—
109C22×C3⋊C8D411yescharacteristic
110C22×C3⋊C8D411yescharacteristic
111C22×C3⋊C8D412yes—
112C22×C3⋊C8C2312yes—
113#907—24no—
114#907—24no—
115#907—24no—
116#907—24no—
117#907—24no—
118#907—24no—
119#907C2×C418yescomplemented
120#907C2×C418yescomplemented
121C2×ΓL(1,𝔽25)—22no—
122C2×ΓL(1,𝔽25)—22no—
123C2×ΓL(1,𝔽25)—24no—
124C2×ΓL(1,𝔽25)—24no—
125C2×D3×C8—22no—
126C2×D3×C8—22no—
127C2×D3×C8—24no—
128C2×D3×C8—24no—
129#825—28no—
130#822—24no—
131#822—24no—
132#808—24no—
133#807—24no—
134#805—28no—
135C22×#41—31no—
136C2×#131—32no—
137C2×#131—32no—
138C2×#131—34no—
139C2×#131—34no—
140C2×#131—61no—
141C2×#131—61no—
142C2×#131—62no—
143C2×#131—62no—
144C22×M16—31no—
145C22×M16—61no—
146C22×M16C2×D311yescharacteristic
147C22×M16C2×D311yescharacteristic
148C23×C8—31no—
149C23×C8—61no—
150C2×#117—31no—
151C2×#117C2×D311yescharacteristic
152C2×#108—34no—
153C2×#108—61no—
154C2×#108—61no—
155C2×#108—62no—
156#427—34no—
157#427—62no—
158C2×#107—61no—
159C2×#106—61no—
160#421—34no—
161#421—34no—
162#421—62no—
163#421—62no—
164C2×#104—62no—
165C2×#104—62no—
166#344—34no—
167#344C2×D314yescomplemented
168C24×C3#4111yescharacteristic
169C23×D3—21no—
170C22×C3×C4—41no—
171C22×C3×C4C22×C411yescharacteristic
172C22×C3×C4C22×C411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×Q811yescharacteristic
180C22×C3×C4C2×Q811yescharacteristic
181C22×C3×C4#4111yescharacteristic
182C22×C3×C4#4111yescharacteristic
183C22×C3×C4#4111yescharacteristic
184C22×C3×C4C2411yescharacteristic
185C2×#67—22no—
186C2×#67—22no—
187C2×#67—22no—
188C2×#67—22no—
189C2×#67—22no—
190C2×#67—22no—
191C22×Dic3—21no—
192C22×Dic3—41no—
193#235—22no—
194#235—22no—
195#235—22no—
196#235—22no—
197#235—22no—
198#235—22no—
199#235—22no—
200#235—22no—
201#235—22no—
202#235—22no—
203#235—22no—
204#235—22no—
205#235C4⋊C414yes—
206#235C4⋊C414yes—
207#235C22×C414yes—
208#235C2×D414yes—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12C4⋊C412yes—
213C2×D12C4⋊C412yes—
214C2×D12C2×Q812yes—
215C2×C4×D3—22no—
216C2×C4×D3—22no—
217C2×C4×D3—22no—
218C2×C4×D3—22no—
219C2×C4×D3—22no—
220C2×C4×D3—22no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6C4⋊C412yes—
225C2×Dic6C4⋊C412yes—
226C2×Dic6C2×Q812yes—
227C3×M16—22no—
228C3×M16—22no—
229C3×M16—22no—
230C3×M16—22no—
231C3×M16—22no—
232C3×M16—41no—
233C3×M16—41no—
234C3×M16C22×C414yes—
235C2×C3×C8—22no—
236C2×C3×C8—22no—
237C2×C3×C8—22no—
238C2×C3×C8—22no—
239C2×C3×C8—41no—
240C2×C3×C8—41no—
241C3×C4⋊C4—22no—
242C3×C4⋊C4—41no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—41no—
245C3×C42—22no—
246C3×C42—41no—
247#217—22no—
248#217—41no—
249C4⋊Dic3—22no—
250C4⋊Dic3—41no—
251C4×Dic3—22no—
252C4×Dic3—41no—
253#208—22no—
254#208—22no—
255#208—41no—
256#208—41no—
257C2×C3⋊C8—22no—
258C2×C3⋊C8—41no—
259C2×C3⋊C8—41no—
260C2×C3⋊C8C22⋊C411yescharacteristic
261C2×C3⋊C8C22⋊C411yescharacteristic
262C2×C3⋊C8C22⋊C412yes—
263C2×C3⋊C8C22⋊C411yescharacteristic
264C2×C3⋊C8C22⋊C411yescharacteristic
265C2×C3⋊C8C22⋊C412yes—
266C2×C3⋊C8C22⋊C412yes—
267C2×C3⋊C8C22⋊C412yes—
268C2×C3⋊C8C22⋊C412yes—
269C2×C3⋊C8C22⋊C412yes—
270C2×C3⋊C8C22×C412yes—
271C2×C3⋊C8C22×C412yes—
272C2×C3⋊C8C2×D412yes—
273C2×C3⋊C8C2×D412yes—
274C2×C3⋊C8C2×D412yes—
275C2×C3⋊C8C2×D412yes—
276ΓL(1,𝔽25)—28no—
277ΓL(1,𝔽25)—28no—
278D3×C8—28no—
279D3×C8—28no—
280C2×#41—32no—
281C2×#41—32no—
282C2×#41—32no—
283C2×#41—32no—
284C2×#41—32no—
285C2×#41—32no—
286C2×#41—61no—
287C2×#41—61no—
288C2×#41—61no—
289C2×#41—61no—
290C2×#41—61no—
291C2×#41—61no—
292C22×Q8—31no—
293C22×D4—31no—
294C22×D4—61no—
295C23×C4—61no—
296C23×C4C22×D311yescharacteristic
297#131—38no—
298#131—38no—
299#131—64no—
300#131—64no—
301#131—64no—
302#131—64no—
303#131—64no—
304#131—64no—
305C2×M16—21no—
306C2×M16—21no—
307C2×M16—21no—
308C2×M16—21no—
309C2×M16—21no—
310C2×M16—21no—
311C2×M16—21no—
312C2×M16—21no—
313C2×M16—32no—
314C2×M16—62no—
315C2×M16—62no—
316C2×M16—64no—
317C2×M16—64no—
318C2×M16—61no—
319C2×M16—61no—
320C2×M16—61no—
321C2×M16—61no—
322C2×M16—61no—
323C2×M16C4×D312yes—
324C2×M16C4×D312yes—
325C2×M16C22×D312yes—
326C2×M16C22×D312yes—
327C22×C8—21no—
328C22×C8—21no—
329C22×C8—31no—
330C22×C8—31no—
331C22×C8—32no—
332C22×C8—31no—
333C22×C8—31no—
334C22×C8—31no—
335C22×C8—31no—
336C22×C8—32no—
337C22×C8—32no—
338C22×C8—32no—
339C22×C8—62no—
340C22×C8—62no—
341C22×C8—64no—
342C22×C8—64no—
343C22×C8—61no—
344#117—21no—
345#117—21no—
346#117—21no—
347#117—32no—
348#117—61no—
349#117—61no—
350#117—61no—
351#117C22×D312yes—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—61no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—61no—
356C2×C42—21no—
357C2×C42—61no—
358#108—64no—
359#108—64no—
360#107—64no—
361#106—64no—
362#104—68no—
363#104—68no—
364C23×C3—42no—
365C23×C3#11711yescharacteristic
366C23×C3#11711yescharacteristic
367C23×C3#12111yescharacteristic
368C23×C3#12111yescharacteristic
369C23×C3C22⋊Q811yescharacteristic
370C23×C3C22⋊Q811yescharacteristic
371C23×C3C2×#4111yescharacteristic
372C22×D3—22no—
373C22×D3—22no—
374C22×D3—22no—
375C22×D3—44no—
376C2×C3×C4—42no—
377C2×C3×C4—82no—
378C2×C3×C4C2×C22⋊C411yescharacteristic
379C2×C3×C4C2×C22⋊C411yescharacteristic
380C2×C3×C4C2×C22⋊C411yescharacteristic
381C2×C3×C4C2×C22⋊C411yescharacteristic
382C2×C3×C4C2×C4⋊C411yescharacteristic
383C2×C3×C4C2×C4⋊C411yescharacteristic
384C2×C3×C4C2×C4⋊C411yescharacteristic
385C2×C3×C4C2×C4⋊C411yescharacteristic
386C2×C3×C4#11711yescharacteristic
387C2×C3×C4#11711yescharacteristic
388C2×C3×C4#12111yescharacteristic
389C2×C3×C4#12111yescharacteristic
390C2×C3×C4#12111yescharacteristic
391C2×C3×C4#12111yescharacteristic
392C2×C3×C4#12111yescharacteristic
393C2×C3×C4#12111yescharacteristic
394C2×C3×C4C22⋊Q811yescharacteristic
395C2×C3×C4C22⋊Q811yescharacteristic
396C2×C3×C4C22⋊Q811yescharacteristic
397C2×C3×C4C22⋊Q811yescharacteristic
398C2×C3×C4C22⋊Q811yescharacteristic
399C2×C3×C4C22⋊Q811yescharacteristic
400C2×C3×C4C23×C411yescharacteristic
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C22×D411yescharacteristic
404C2×C3×C4C22×Q811yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406#67—24no—
407#67—24no—
408#67—24no—
409#67—24no—
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—22no—
413C2×Dic3—42no—
414C2×Dic3—82no—
415D12—22no—
416D12—22no—
417D12—22no—
418D12C2×C4⋊C414yes—
419C4×D3—24no—
420C4×D3—24no—
421C4×D3—24no—
422C4×D3—24no—
423Dic6—22no—
424Dic6—22no—
425Dic6—22no—
426Dic6C2×C4⋊C414yes—
427C3×C8—24no—
428C3×C8—42no—
429C3⋊C8—42no—
430C3⋊C8C2×C22⋊C412yes—
431C3⋊C8C2×C22⋊C412yes—
432C3⋊C8C2×C22⋊C412yes—
433C3⋊C8C2×C22⋊C412yes—
434C24—61no—
435C24#23511yescharacteristic
436#41—34no—
437#41—34no—
438#41—34no—
439#41—34no—
440#41—62no—
441#41—62no—
442#41—62no—
443#41—62no—
444#41—62no—
445#41—62no—
446#41—62no—
447#41—62no—
448#41—62no—
449#41—62no—
450#41—62no—
451#41—62no—
452C2×Q8—32no—
453C2×Q8—32no—
454C2×Q8—32no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—61no—
458C2×D4—32no—
459C2×D4—32no—
460C2×D4—32no—
461C2×D4—62no—
462C2×D4—62no—
463C2×D4—62no—
464C2×D4—62no—
465C2×D4—62no—
466C2×D4—62no—
467C2×D4—61no—
468C2×D4—61no—
469C2×D4—61no—
470C22×C4—41no—
471C22×C4—61no—
472C22×C4—62no—
473C22×C4—62no—
474C22×C4—62no—
475C22×C4—62no—
476C22×C4—62no—
477C22×C4—62no—
478C22×C4—121no—
479C22×C4C2×C4×D311yescharacteristic
480C22×C4C2×C4×D311yescharacteristic
481C22×C4#23511yescharacteristic
482C22×C4#23511yescharacteristic
483C22×C4#23511yescharacteristic
484C22×C4D3×D411yescharacteristic
485C22×C4D3×D411yescharacteristic
486C22×C4D3×D411yescharacteristic
487C22×C4D3×D411yescharacteristic
488C22×C4D3×D411yescharacteristic
489C22×C4D3×D411yescharacteristic
490C22×C4D3×Q811yescharacteristic
491C22×C4D3×Q811yescharacteristic
492C22×C4C23×D311yescharacteristic
493M16—22no—
494M16—22no—
495M16—22no—
496M16—22no—
497M16—22no—
498M16—41no—
499M16—41no—
500M16—68no—
501M16—68no—
502M16—62no—
503M16—62no—
504M16—121no—
505M16—121no—
506M16C2×C4×D314yes—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—31no—
512C2×C8—31no—
513C2×C8—32no—
514C2×C8—31no—
515C2×C8—31no—
516C2×C8—32no—
517C2×C8—32no—
518C2×C8—32no—
519C2×C8—32no—
520C2×C8—32no—
521C2×C8—32no—
522C2×C8—32no—
523C2×C8—32no—
524C2×C8—32no—
525C2×C8—32no—
526C2×C8—32no—
527C2×C8—41no—
528C2×C8—41no—
529C2×C8—68no—
530C2×C8—68no—
531C2×C8—62no—
532C2×C8—121no—
533C2×C8—121no—
534C4⋊C4—22no—
535C4⋊C4—41no—
536C4⋊C4—62no—
537C4⋊C4—121no—
538C22⋊C4—22no—
539C22⋊C4—41no—
540C22⋊C4—62no—
541C22⋊C4—121no—
542C42—22no—
543C42—41no—
544C42—62no—
545C42—121no—
546C22×C3—22no—
547C22×C3—22no—
548C22×C3—22no—
549C22×C3—41no—
550C22×C3—41no—
551C22×C3—41no—
552C22×C3—41no—
553C22×C3#38011yescharacteristic
554C22×C3#38011yescharacteristic
555C22×C3#38011yescharacteristic
556C22×C3#38011yescharacteristic
557C22×C3C2×#11711yescharacteristic
558C22×C3C2×#12111yescharacteristic
559C22×C3C2×C22⋊Q811yescharacteristic
560C2×D3—24no—
561C2×D3—44no—
562C2×D3—44no—
563C2×D3—44no—
564C3×C4—82no—
565C3×C4#38011yescharacteristic
566C3×C4#38011yescharacteristic
567C3×C4#38011yescharacteristic
568C3×C4#38011yescharacteristic
569C3×C4C22×C22⋊C411yescharacteristic
570C3×C4C22×C4⋊C411yescharacteristic
571C3×C4C2×#12111yescharacteristic
572C3×C4C2×C22⋊Q811yescharacteristic
573Dic3—24no—
574Dic3—82no—
575C23—42no—
576C23—62no—
577C23—62no—
578C23—62no—
579C23—124no—
580C23#87211yescharacteristic
581C23#87211yescharacteristic
582C23#88411yescharacteristic
583C23#88411yescharacteristic
584C23#89611yescharacteristic
585C23#89611yescharacteristic
586C23C2×#23511yescharacteristic
587Q8—34no—
588Q8—62no—
589Q8—62no—
590Q8—62no—
591D4—34no—
592D4—64no—
593D4—64no—
594D4—64no—
595D4—64no—
596D4—62no—
597D4—62no—
598D4—62no—
599C2×C4—42no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—64no—
604C2×C4—64no—
605C2×C4—64no—
606C2×C4—64no—
607C2×C4—82no—
608C2×C4—122no—
609C2×C4—242no—
610C2×C4#87211yescharacteristic
611C2×C4#87211yescharacteristic
612C2×C4D3×C22⋊C411yescharacteristic
613C2×C4D3×C22⋊C411yescharacteristic
614C2×C4D3×C22⋊C411yescharacteristic
615C2×C4D3×C22⋊C411yescharacteristic
616C2×C4#88411yescharacteristic
617C2×C4#88411yescharacteristic
618C2×C4#88411yescharacteristic
619C2×C4#88411yescharacteristic
620C2×C4#88411yescharacteristic
621C2×C4#88411yescharacteristic
622C2×C4D3×C4⋊C411yescharacteristic
623C2×C4D3×C4⋊C411yescharacteristic
624C2×C4D3×C4⋊C411yescharacteristic
625C2×C4D3×C4⋊C411yescharacteristic
626C2×C4#89611yescharacteristic
627C2×C4#89611yescharacteristic
628C2×C4#89611yescharacteristic
629C2×C4#89611yescharacteristic
630C2×C4#89611yescharacteristic
631C2×C4#89611yescharacteristic
632C2×C4C22×C4×D311yescharacteristic
633C2×C4C2×#23511yescharacteristic
634C2×C4C2×D3×D411yescharacteristic
635C2×C4C2×D3×D411yescharacteristic
636C2×C4C2×D3×D411yescharacteristic
637C2×C4C2×D3×Q811yescharacteristic
638C8—24no—
639C8—32no—
640C8—32no—
641C8—32no—
642C8—32no—
643C8—42no—
644C8—122no—
645C2×C3—21no—
646C2×C3—21no—
647C2×C3—21no—
648C2×C3—21no—
649C2×C3—21no—
650C2×C3—21no—
651C2×C3#181412yescomplemented
652C2×C3C2×#38011yescharacteristic
653D3—44no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—64no—
662C22—124no—
663C22—124no—
664C22—124no—
665C22#466711yescharacteristic
666C22#466711yescharacteristic
667C22#466711yescharacteristic
668C22#466711yescharacteristic
669C22C2×#87211yescharacteristic
670C22C2×#88411yescharacteristic
671C22C2×#89611yescharacteristic
672C4—64no—
673C4—82no—
674C4—242no—
675C4#466711yescharacteristic
676C4#466711yescharacteristic
677C4#466711yescharacteristic
678C4#466711yescharacteristic
679C4C2×D3×C22⋊C411yescharacteristic
680C4C2×#88411yescharacteristic
681C4C2×D3×C4⋊C411yescharacteristic
682C4C2×#89611yescharacteristic
683C3C2×#181411yescharacteristic, complemented, Sylow
684C2—21no—
685C2—21no—
686C2—21no—
687C2—21no—
688C2—21no—
689C2—21no—
690C2—124no—
691C2#6903212yesdirect product factor
692C2C2×#466711yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#344C2
socleC22×C3#380
prime radicalC22×#235C22
cosocleC22×C3×C4C24
hypercenterC2×#344D3
hypocenterC3C2×#1814
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#344

Complements

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

NH ≅ G/Ntype
3. C22×#822C2semidirect
5. C2×#4998C2semidirect
6. C2×C3×#344C2semidirect
7. C2×#4559C2semidirect
8. #69032C2direct (N × H)
21. C2×#907C4semidirect
22. C2×#907C4semidirect
23. C2×#907C4semidirect
33. C2×#822C22semidirect
37. #4998C22semidirect
41. C3×#344C22semidirect
42. #4559C22semidirect
50. C2×#344D3semidirect
119. #907C2×C4semidirect
120. #907C2×C4semidirect
167. #344C2×D3semidirect
651. C2×C3#1814semidirect
683. C3C2×#1814semidirect
691. C2#69032direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9072
C2×#51392
C22×#8222
C2×#50052
C2×#49982
C2×C3×#3442
C2×#45592
#690322
C2×#18143