Finite Group Databasebeta

← #10901356

Subgroups — #10901356

Order 768.

1068 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5622C211yesmaximal, characteristic
2C2×#4854C211yesmaximal, characteristic, complemented
3C2×#4791C211yesmaximal, characteristic, complemented
4#74001C211yesmaximal, characteristic
5C3×#2035C211yesmaximal, characteristic
6#71694C211yesmaximal, characteristic
7#70565C211yesmaximal, characteristic, complemented
8#70284C211yesmaximal, characteristic, complemented
9#70246C211yesmaximal, characteristic, complemented
10#70221C211yesmaximal, characteristic, complemented
11#69936C211yesmaximal, characteristic
12#69908C211yesmaximal, characteristic
13#69526C211yesmaximal, characteristic
14#68904C211yesmaximal, characteristic, complemented
15#68827C211yesmaximal, characteristic, complemented
16#29688—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×#944C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C22×C4×Dic3C2211yescharacteristic
22#5622—24no—
23#5622—24no—
24C2×#889—21no—
25C2×#888C2211yescharacteristic
26C2×#888C2211yescharacteristic
27C2×#887—21no—
28C2×#887C2211yescharacteristic
29C2×#878—21no—
30C2×#878—21no—
31C2×#877—21no—
32C2×C3×#106C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×#103C2211yescharacteristic
35C2×C3×C22⋊C8C2211yescharacteristic
36C3×#380C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×#835C2211yescharacteristic
39#5212C2211yescharacteristic
40C2×#830C2211yescharacteristic
41C2×#817C2211yescharacteristic
42C2×#816C2211yescharacteristic
43C2×#816C2211yescharacteristic
44C2×Dic3⋊C8C2211yescharacteristic
45C2×Dic3⋊C8C2211yescharacteristic
46#4984C2211yescharacteristic
47#4975C2211yescharacteristic
48#4972C2211yescharacteristic
49C2×#810C2211yescharacteristic
50C2×#810C2211yescharacteristic
51C2×#808C2211yescharacteristic
52C2×#808C2211yescharacteristic
53#4942—21no—
54#4854—24no—
55#4791—24no—
56#4647—21no—
57#4646—21no—
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4529C2211yescharacteristic
64C2×#533—31no—
65C2×#475—31no—
66C2×#475—31no—
67#2857—31no—
68#2058—31no—
69#2058—31no—
70#2043—31no—
71#2043—31no—
72#2035—31no—
73#2035—31no—
74#2035—31no—
75#2035D311yescharacteristic
76#1803—31no—
77#1803—31no—
78#1797—31no—
79C22×C3×Q8C2311yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C23×Dic3D411yescharacteristic
82C23×Dic3D411yescharacteristic
83C22×Dic6—21no—
84C22×Dic6—21no—
85C22×Dic6C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×#217—21no—
97C2×#217—21no—
98Q8×Dic3—22no—
99Q8×Dic3D414yes—
100#944—22no—
101#944—22no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107#924—22no—
108#924—22no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3—22no—
121C2×C4×Dic3—22no—
122C2×C4×Dic3D411yescharacteristic
123C2×C4×Dic3D411yescharacteristic
124C2×C4×Dic3C2311yescharacteristic
125C2×C4×Dic3C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128#889—28no—
129#888—22no—
130#888—22no—
131#888—22no—
132#888—22no—
133#887—28no—
134#887—22no—
135#887D414yes—
136#878—28no—
137#878—44no—
138#877—28no—
139C3×#106—22no—
140C3×#103—22no—
141C3×#103—41no—
142C3×C22⋊C8—41no—
143C3×#95—21no—
144#835—22no—
145#835—41no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149#830—41no—
150#817—22no—
151#816—22no—
152#816—41no—
153Dic3⋊C8—22no—
154Dic3⋊C8—22no—
155#810—22no—
156#810—41no—
157#808—22no—
158#808—22no—
159#533—64no—
160#533—64no—
161C2×C4⋊Q8—31no—
162C2×C4⋊Q8—31no—
163C2×C4⋊Q8—31no—
164C2×#125—61no—
165C2×C22⋊Q8—31no—
166C2×C22⋊Q8—61no—
167C2×C22⋊Q8—61no—
168C2×C22⋊Q8C2×D311yescharacteristic
169C2×C4×Q8—31no—
170C2×C4×Q8—31no—
171C2×C4×Q8—61no—
172C2×#117—61no—
173C22×C42—31no—
174#475—64no—
175#475—64no—
176C2×#106—31no—
177C2×#106—31no—
178C2×#106—31no—
179C2×#106C2×D311yescharacteristic
180C2×C4⋊C8—31no—
181C2×C4⋊C8—31no—
182C2×#103—31no—
183C2×#103—31no—
184C2×#103—31no—
185C2×#103—31no—
186C2×#103—31no—
187C2×#103—31no—
188C2×#103C2×D311yescharacteristic
189C2×#103C2×D311yescharacteristic
190C2×C22⋊C8—31no—
191C2×C22⋊C8C2×D311yescharacteristic
192#391—61no—
193#389—31no—
194#389—31no—
195#389—61no—
196#384—31no—
197#380—31no—
198#380—61no—
199#380C2×D311yescharacteristic
200#340—31no—
201#340—31no—
202#340—31no—
203#340C2×D311yescharacteristic
204#336—31no—
205#336—31no—
206C24×C3C2×D411yescharacteristic
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8C2×D412yes—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2411yescharacteristic
225C22×Dic3—22no—
226C22×Dic3—22no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C22×Dic3—22no—
230C22×Dic3—22no—
231C22×Dic3—21no—
232C22×Dic3—21no—
233C22×Dic3—41no—
234C22×Dic3SD1614yes—
235C22×Dic3C2×D411yescharacteristic
236C22×Dic3C2×D411yescharacteristic
237C22×Dic3C2×D411yescharacteristic
238C22×Dic3C2×D411yescharacteristic
239C2×Dic6—21no—
240C2×Dic6—21no—
241C2×Dic6—21no—
242C2×Dic6—21no—
243C2×Dic6—24no—
244C2×Dic6—21no—
245C2×Dic6—42no—
246C2×Dic6—42no—
247C2×Dic6—42no—
248C2×Dic6—42no—
249C2×Dic6—42no—
250C2×Dic6C2×D412yes—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C3×C4⋊C4—24no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—21no—
263C3×C4⋊C4C2×D412yes—
264C3×C22⋊C4—24no—
265C3×C22⋊C4—42no—
266#217—24no—
267#217—42no—
268C4⋊Dic3—21no—
269C4⋊Dic3—24no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—42no—
275C4⋊Dic3C2×D412yes—
276Dic3⋊C4—24no—
277Dic3⋊C4—24no—
278Dic3⋊C4—24no—
279Dic3⋊C4—24no—
280Dic3⋊C4—24no—
281Dic3⋊C4—42no—
282Dic3⋊C4—42no—
283Dic3⋊C4—42no—
284Dic3⋊C4—42no—
285C4×Dic3—21no—
286C4×Dic3—21no—
287C4×Dic3—24no—
288C4×Dic3—24no—
289C4×Dic3—22no—
290C4×Dic3—22no—
291C4×Dic3C2×D412yes—
292C4×Dic3C2×D412yes—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
299C22×Q8—31no—
300C22×Q8—61no—
301C22×Q8—61no—
302C22×Q8C22×D311yescharacteristic
303C23×C4—31no—
304C23×C4—31no—
305C23×C4C22×D311yescharacteristic
306C22×C8—31no—
307C22×C8—31no—
308C22×C8C22×D311yescharacteristic
309C22×C8C22×D311yescharacteristic
310C4⋊Q8—62no—
311C4⋊Q8—62no—
312C4⋊Q8—62no—
313C4⋊Q8—62no—
314C4⋊Q8—62no—
315C4⋊Q8—62no—
316#125—68no—
317C22⋊Q8—22no—
318C22⋊Q8—22no—
319C22⋊Q8—68no—
320C22⋊Q8—62no—
321C22⋊Q8—62no—
322C22⋊Q8—124no—
323C4×Q8—34no—
324C4×Q8—34no—
325C4×Q8—62no—
326C4×Q8—62no—
327C4×Q8—68no—
328#117—68no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4—61no—
344C2×C4⋊C4C22×D311yescharacteristic
345C2×C4⋊C4C22×D311yescharacteristic
346C2×C4⋊C4C22×D311yescharacteristic
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—61no—
350C2×C22⋊C4—61no—
351C2×C42—31no—
352C2×C42—31no—
353C2×C42—31no—
354C2×C42—31no—
355C2×C42—62no—
356C2×C42—62no—
357C2×C42—61no—
358C2×C42—61no—
359#106—22no—
360#106—62no—
361#106—62no—
362#106—62no—
363C4⋊C8—62no—
364C4⋊C8—62no—
365#103—22no—
366#103—41no—
367#103—62no—
368#103—62no—
369#103—62no—
370#103—121no—
371#103—121no—
372#103—121no—
373C22⋊C8—41no—
374C22⋊C8—121no—
375#95—21no—
376#95—61no—
377#95—61no—
378#95—61no—
379C23×C3—21no—
380C23×C3—21no—
381C23×C3—21no—
382C23×C3—21no—
383C23×C3—21no—
384C23×C3—21no—
385C23×C3C2×SD1612yes—
386C23×C3C22×D411yescharacteristic
387C3×Q8—22no—
388C3×Q8—22no—
389C3×Q8—41no—
390C3×Q8—41no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4C22×D411yescharacteristic
411C2×C3×C4C22×D411yescharacteristic
412C2×C3×C4H(5,𝔽2)11yescharacteristic
413C2×C3×C42−1+411yescharacteristic
414C2×C3×C42−1+411yescharacteristic
415C2×C3×C42−1+411yescharacteristic
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—28no—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—44no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3#12012yes—
437C2×Dic3#12012yes—
438C2×Dic3C2×SD1612yes—
439C2×Dic3C2×SD1612yes—
440C2×Dic3#13712yes—
441C2×Dic3#13712yes—
442Dic6—22no—
443Dic6—22no—
444Dic6—48no—
445Dic6—41no—
446Dic6—41no—
447Dic6—84no—
448C3×C8—41no—
449C3×C8—41no—
450C3⋊C8—41no—
451C3⋊C8—41no—
452C24D3×D411yescharacteristic
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—32no—
459C2×Q8—64no—
460C2×Q8—61no—
461C2×Q8—61no—
462C2×Q8—61no—
463C2×Q8—61no—
464C2×Q8—61no—
465C2×Q8—122no—
466C2×Q8—122no—
467C2×Q8—122no—
468C2×Q8—122no—
469C2×Q8—122no—
470C2×Q8D3×D412yes—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—31no—
482C22×C4—34no—
483C22×C4—31no—
484C22×C4—31no—
485C22×C4—31no—
486C22×C4—62no—
487C22×C4—62no—
488C22×C4—62no—
489C22×C4—62no—
490C22×C4—61no—
491C22×C4—61no—
492C22×C4—61no—
493C22×C4—61no—
494C22×C4—121no—
495C22×C4D3×D411yescharacteristic
496C22×C4C23×D311yescharacteristic
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—61no—
504C2×C8—61no—
505C2×C8—61no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C4⋊C4—24no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—32no—
516C4⋊C4—64no—
517C4⋊C4—64no—
518C4⋊C4—64no—
519C4⋊C4—64no—
520C4⋊C4—64no—
521C4⋊C4—64no—
522C4⋊C4—61no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4—122no—
528C4⋊C4—122no—
529C4⋊C4—122no—
530C4⋊C4—122no—
531C4⋊C4—122no—
532C4⋊C4D3×D412yes—
533C22⋊C4—24no—
534C22⋊C4—42no—
535C22⋊C4—64no—
536C22⋊C4—122no—
537C42—32no—
538C42—32no—
539C42—62no—
540C42—62no—
541C42—64no—
542C42—64no—
543C42—61no—
544C42—61no—
545C22×C3—22no—
546C22×C3—22no—
547C22×C3—22no—
548C22×C3—41no—
549C22×C3—41no—
550C22×C3—41no—
551C22×C3—41no—
552C22×C3C2×#12011yescharacteristic
553C22×C3#53611yescharacteristic
554C22×C3#53711yescharacteristic
555C22×C3C22×SD1611yescharacteristic
556C22×C3C2×#13711yescharacteristic
557C22×C3C2×#13711yescharacteristic
558C22×C3#57511yescharacteristic
559C3×C4—41no—
560C3×C4—42no—
561C3×C4—42no—
562C3×C4—81no—
563C3×C4—81no—
564C3×C4#53611yescharacteristic
565C3×C4#53611yescharacteristic
566C3×C4#53711yescharacteristic
567C3×C4#53711yescharacteristic
568Dic3—22no—
569Dic3—22no—
570Dic3—22no—
571Dic3—42no—
572Dic3—42no—
573Dic3—81no—
574Dic3—81no—
575Dic3#44814yes—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23D3×SD1612yes—
583C23C2×D3×D411yescharacteristic
584Q8—22no—
585Q8—22no—
586Q8—41no—
587Q8—41no—
588Q8—62no—
589Q8—62no—
590Q8—128no—
591Q8—121no—
592Q8—121no—
593Q8—244no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—32no—
600C2×C4—32no—
601C2×C4—32no—
602C2×C4—32no—
603C2×C4—32no—
604C2×C4—32no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—68no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—62no—
627C2×C4—62no—
628C2×C4—124no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4C2×D3×D411yescharacteristic
640C2×C4C2×D3×D411yescharacteristic
641C2×C4#100411yescharacteristic
642C2×C4#101011yescharacteristic
643C2×C4#101011yescharacteristic
644C2×C4#101011yescharacteristic
645C8—41no—
646C8—41no—
647C8—121no—
648C8—121no—
649C2×C3—42no—
650C2×C3#261611yescharacteristic
651C2×C3C2×#44811yescharacteristic
652C2×C3#300711yescharacteristic
653C2×C3#322911yescharacteristic
654C2×C3#323711yescharacteristic
655C2×C3#324611yescharacteristic
656C2×C3#325511yescharacteristic
657C22—22no—
658C22—22no—
659C22—22no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22D3×#12011yescharacteristic
665C22#559711yescharacteristic
666C22#564811yescharacteristic
667C22C2×D3×SD1611yescharacteristic
668C22#576811yescharacteristic
669C22D3×#13711yescharacteristic
670C22D3×#13711yescharacteristic
671C4—34no—
672C4—42no—
673C4—41no—
674C4—42no—
675C4—62no—
676C4—62no—
677C4—62no—
678C4—81no—
679C4—81no—
680C4—122no—
681C4—122no—
682C4—241no—
683C4—241no—
684C4#559711yescharacteristic
685C4#560811yescharacteristic
686C4#564811yescharacteristic
687C4#567511yescharacteristic
688C3#2968811yescharacteristic, complemented, Sylow
689C2—42no—
690C2#7697011yescharacteristic
691C2D3×#44811yescharacteristic
692C2#8022811yescharacteristic
693C2#8064011yescharacteristic
694C2#8089511yescharacteristic
695C2#8141811yescharacteristic
696C2#8150211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2035C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2035D3
hypocenterC3#29688
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
2. C2×#4854C2semidirect
3. C2×#4791C2semidirect
7. #70565C2semidirect
8. #70284C2semidirect
9. #70246C2semidirect
10. #70221C2semidirect
14. #68904C2semidirect
15. #68827C2semidirect
688. C3#29688semidirect

Maximal subgroups

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

subgroupindex
C2×#56222
C2×#48542
C2×#47912
#740012
C3×#20352
#716942
#705652
#702842
#702462
#702212
#699362
#699082
#695262
#689042
#688272
#296883