Finite Group Databasebeta

← #10801788

Subgroups — #10801788

Order 768.

954 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82488C211yesmaximal, characteristic
2C2×#5030C211yesmaximal, characteristic, complemented
3C2×#4813C211yesmaximal, characteristic
4C2×#4689C211yesmaximal, characteristic
5#73762C211yesmaximal, characteristic
6C3×C4×#103C211yesmaximal, characteristic, complemented
7#70271C211yesmaximal, characteristic
8#70259C211yesmaximal, characteristic, complemented
9#70253C211yesmaximal, characteristic, complemented
10C4×#816C211yesmaximal, characteristic, complemented
11#69481C211yesmaximal, characteristic, complemented
12#68884C211yesmaximal, characteristic
13#68868C211yesmaximal, characteristic
14#68860C211yesmaximal, characteristic, complemented
15#68850C211yesmaximal, characteristic, complemented
16#21530—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C22×#216C2211yescharacteristic
20C22×ΓL(1,𝔽25)C2211yescharacteristic
21C22×D3×C8C2211yescharacteristic
22C2×#897C2211yescharacteristic
23C2×#872C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28C3×C4×C4⋊C4C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#815C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35#5030—24no—
36#4990—21no—
37#4987C2211yescharacteristic
38#4987C2211yescharacteristic
39#4984C2211yescharacteristic
40#4976—21no—
41#4973C2211yescharacteristic
42#4972C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4940—21no—
48#4940—21no—
49C4×C4⋊Dic3C2211yescharacteristic
50#4933—21no—
51#4928—21no—
52C2×#803C2211yescharacteristic
53#4813C418yescomplemented
54#4689—24no—
55#4670—21no—
56#4653—21no—
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×#467—31no—
64C2×#441—31no—
65C2×#405—31no—
66#2323—31no—
67#2323—31no—
68#1971—31no—
69#1943—31no—
70#1933—31no—
71#1933—31no—
72#1894—31no—
73#1894—31no—
74#1807—31no—
75#1776—31no—
76C4×#103—31no—
77C4×#103D311yescharacteristic, complemented
78C22×C3×Q8C2311yescharacteristic
79C22×C4×D3C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42—21no—
91C2×C3×C42C2311yescharacteristic
92#946—22no—
93#946C2×C414yes—
94C2×#216—22no—
95C2×#216—22no—
96C2×#216C2×C414yes—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4D411yescharacteristic
102C2×(C2×D3)⋊C4D411yescharacteristic
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4D411yescharacteristic
112C2×Dic3⋊C4D411yescharacteristic
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C2×ΓL(1,𝔽25)—22no—
119C2×ΓL(1,𝔽25)—22no—
120C2×ΓL(1,𝔽25)—22no—
121C2×D3×C8—22no—
122C2×D3×C8—22no—
123C2×D3×C8C2×C414yes—
124#897—22no—
125#897C2×C414yes—
126#872—22no—
127#872—22no—
128C4×Dic6—22no—
129C4×Dic6—22no—
130C3×#103—22no—
131C3×#103C2×C414yes—
132C3×C4×C8—22no—
133C3×#95—21no—
134#835—22no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144#816—22no—
145#816C2×C414yes—
146#815—22no—
147C8×Dic3—22no—
148#810—22no—
149#808—22no—
150#808—22no—
151#807C2×C414yes—
152#803D414yes—
153C22×Q16—31no—
154C22×M16—31no—
155C23×C8—31no—
156C2×C22⋊Q8—31no—
157C2×C22⋊Q8—31no—
158C2×C4×Q8—31no—
159C2×C4×Q8C2×D311yescharacteristic
160C2×#117—31no—
161#467—38no—
162#441—64no—
163C2×#107—31no—
164C2×#106—31no—
165C2×#106—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103C2×D311yescharacteristic
171C2×#103C2×D311yescharacteristic
172#405—64no—
173C2×#97—31no—
174C2×#97—31no—
175C2×C4×C8—31no—
176C2×C4×C8C2×D311yescharacteristic
177#391—31no—
178#391—31no—
179#389—31no—
180#389—31no—
181#389—61no—
182#389—61no—
183#388—61no—
184#388—61no—
185#388—61no—
186#388—61no—
187#387—61no—
188#387—61no—
189#380—31no—
190#380—31no—
191C4×C4⋊C4—31no—
192C4×C4⋊C4C2×D311yescharacteristic
193#340—31no—
194#340—31no—
195#340—31no—
196#340—31no—
197#340C2×D311yescharacteristic
198#340C2×D311yescharacteristic
199C23×D3C2×D411yescharacteristic
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8C22×C412yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3C22×C412yes—
225C2×C4×D3#4112yes—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6C22×C412yes—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8C22×C412yes—
237C2×C3×C8#4112yes—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—42no—
244C3×C4⋊C4C22×C412yes—
245C3×C42—42no—
246C3×C42—81no—
247C3×C42C2×D412yes—
248C3×C42#4112yes—
249#216—28no—
250(C2×D3)⋊C4—24no—
251(C2×D3)⋊C4—24no—
252(C2×D3)⋊C4—22no—
253(C2×D3)⋊C4—22no—
254(C2×D3)⋊C4—81no—
255(C2×D3)⋊C4—81no—
256C4⋊Dic3—21no—
257C4⋊Dic3—24no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—81no—
263C4⋊Dic3C22×C412yes—
264Dic3⋊C4—22no—
265Dic3⋊C4—22no—
266Dic3⋊C4—24no—
267Dic3⋊C4—42no—
268Dic3⋊C4—81no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—42no—
272C4×Dic3—81no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8C22×C412yes—
276C2×C3⋊C8C2×D412yes—
277C2×C3⋊C8C2×D412yes—
278C2×C3⋊C8#4112yes—
279ΓL(1,𝔽25)—28no—
280D3×C8—28no—
281C22×Q8—31no—
282C22×Q8C22×D311yescharacteristic
283C23×C4—31no—
284C2×Q16—34no—
285C2×Q16—62no—
286C2×Q16—62no—
287C2×M16—62no—
288C2×M16—62no—
289C2×M16—62no—
290C22×C8—31no—
291C22×C8—31no—
292C22×C8—34no—
293C22×C8—62no—
294C22×C8—62no—
295C22×C8C22×D311yescharacteristic
296C22×C8C22×D311yescharacteristic
297C22⋊Q8—34no—
298C22⋊Q8—34no—
299C22⋊Q8—62no—
300C22⋊Q8—62no—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×Q8—62no—
304C4×Q8—62no—
305#117—62no—
306#117—62no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4—61no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C4⋊C4C22×D311yescharacteristic
321C2×C22⋊C4—31no—
322C2×C22⋊C4—31no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C22⋊C4—61no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—31no—
330C2×C42—61no—
331C2×C42—61no—
332C2×C42C22×D311yescharacteristic
333#107—34no—
334#106—62no—
335#106—62no—
336#103—22no—
337#103—34no—
338#103—62no—
339#103—62no—
340#103—62no—
341#103C4×D314yes—
342#97—34no—
343#97—62no—
344C4×C8—22no—
345C4×C8—62no—
346#95—21no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3C22×D411yescharacteristic
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—21no—
361C22×D3—21no—
362C22×D3—21no—
363C22×D3C4×D412yes—
364C3×Q8—22no—
365C3×Q8—22no—
366C3×Q8—41no—
367C3×Q8—41no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4#12112yes—
385C2×C3×C4#12112yes—
386C2×C3×C4C23×C411yescharacteristic
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3C4×D412yes—
413C4×D3—22no—
414C4×D3—22no—
415C4×D3—22no—
416C4×D3—22no—
417Dic6—22no—
418Dic6—22no—
419Dic6—41no—
420Dic6—41no—
421C3×C8—22no—
422C3×C8—22no—
423C3⋊C8—22no—
424C3⋊C8C4×D414yes—
425C24—31no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—32no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8C2×C4×D312yes—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—31no—
443C22×C4—32no—
444C22×C4—32no—
445C22×C4—31no—
446C22×C4—31no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4D3×D411yescharacteristic
456C22×C4D3×D411yescharacteristic
457C22×C4C23×D311yescharacteristic
458Q16—68no—
459M16—68no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—32no—
465C2×C8—32no—
466C2×C8—32no—
467C2×C8—32no—
468C2×C8—68no—
469C2×C8—61no—
470C2×C8—61no—
471C2×C8C2×C4×D312yes—
472C2×C8#23512yes—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—32no—
479C4⋊C4—42no—
480C4⋊C4—64no—
481C4⋊C4—64no—
482C4⋊C4—62no—
483C4⋊C4—62no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—122no—
490C4⋊C4—241no—
491C4⋊C4—241no—
492C4⋊C4C2×C4×D312yes—
493C22⋊C4—64no—
494C22⋊C4—64no—
495C22⋊C4—62no—
496C22⋊C4—62no—
497C22⋊C4—241no—
498C22⋊C4—241no—
499C42—42no—
500C42—61no—
501C42—61no—
502C42—81no—
503C42—122no—
504C42—241no—
505C42#23512yes—
506C42D3×D412yes—
507C22×C3C2×C4×D411yescharacteristic
508C22×C3C2×#12111yescharacteristic
509C22×C3#53211yescharacteristic
510C22×C3#57711yescharacteristic
511C22×C3#57711yescharacteristic
512C22×C3#57811yescharacteristic
513C22×C3#57811yescharacteristic
514C2×D3—22no—
515C2×D3—22no—
516C2×D3—22no—
517C2×D3—41no—
518C2×D3—41no—
519C2×D3—41no—
520C2×D3—41no—
521C3×C4—22no—
522C3×C4—22no—
523C3×C4—42no—
524C3×C4—42no—
525C3×C4—81no—
526C3×C4—81no—
527C3×C4C2×C4×D411yescharacteristic
528C3×C4C4×#4111yescharacteristic
529C3×C4#52511yescharacteristic
530C3×C4#52911yescharacteristic
531Dic3—22no—
532Dic3—42no—
533Dic3—41no—
534Dic3—41no—
535Dic3—42no—
536Dic3—81no—
537Dic3—81no—
538C23—32no—
539C23—61no—
540C23—61no—
541C23—61no—
542C23—61no—
543C23—61no—
544C23—61no—
545C23C2×D3×D411yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—62no—
551Q8—62no—
552Q8—121no—
553Q8—121no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—32no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—62no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—62no—
584C2×C4—62no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4#88412yes—
596C2×C4#88412yes—
597C2×C4C22×C4×D311yescharacteristic
598C2×C4C2×#23511yescharacteristic
599C2×C4C2×#23511yescharacteristic
600C2×C4C2×D3×D411yescharacteristic
601C2×C4D3×#4111yescharacteristic
602C2×C4D3×#4111yescharacteristic
603C8—22no—
604C8—22no—
605C8—34no—
606C8—62no—
607C2×C3C4×#12111yescharacteristic
608C2×C3#295011yescharacteristic
609C2×C3#295211yescharacteristic
610C2×C3#306311yescharacteristic
611C2×C3#306311yescharacteristic
612C2×C3#317811yescharacteristic
613C2×C3#317811yescharacteristic
614D3—42no—
615C22—62no—
616C22—62no—
617C22—62no—
618C22—121no—
619C22—121no—
620C22—121no—
621C22—121no—
622C22C2×#88411yescharacteristic
623C22C4×D3×D411yescharacteristic
624C22#554811yescharacteristic
625C22#576211yescharacteristic
626C22#576611yescharacteristic
627C22#577011yescharacteristic
628C22#577111yescharacteristic
629C4—22no—
630C4—22no—
631C4—42no—
632C4—42no—
633C4—62no—
634C4—81no—
635C4—81no—
636C4—122no—
637C4—122no—
638C4—121no—
639C4—121no—
640C4—241no—
641C4—241no—
642C4C4×#23511yescharacteristic
643C4#552811yescharacteristic
644C4C4×D3×D411yescharacteristic
645C4#554711yescharacteristic
646C3#2153011yescharacteristic, complemented, Sylow
647C2—122no—
648C2C4×#88411yescharacteristic
649C2#7880011yescharacteristic
650C2#7884111yescharacteristic
651C2#7965011yescharacteristic
652C2#7966811yescharacteristic
653C2#7972011yescharacteristic
654C2#7972911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×C4×#103C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
cosocleC22×C3×C4C24
hypercenterC4×#103D3
hypocenterC3#21530
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
2. C2×#5030C2semidirect
6. C3×C4×#103C2semidirect
8. #70259C2semidirect
9. #70253C2semidirect
10. C4×#816C2semidirect
11. #69481C2semidirect
14. #68860C2semidirect
15. #68850C2semidirect
53. #4813C4semidirect
77. C4×#103D3semidirect
646. C3#21530semidirect

Maximal subgroups

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

subgroupindex
#824882
C2×#50302
C2×#48132
C2×#46892
#737622
C3×C4×#1032
#702712
#702592
#702532
C4×#8162
#694812
#688842
#688682
#688602
#688502
#215303