Finite Group Databasebeta

← #10801955

Subgroups — #10801955

Order 768.

1010 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
1#82488C211yesmaximal, characteristic, complemented
2C2×#5026C211yesmaximal, characteristic, complemented
3C2×#4812C211yesmaximal, characteristic
4C2×#4689C211yesmaximal, characteristic, complemented
5#73763C211yesmaximal, characteristic
6C3×C4×#103C211yesmaximal, characteristic, complemented
7#70270C211yesmaximal, characteristic, complemented
8#70257C211yesmaximal, characteristic, complemented
9#70255C211yesmaximal, characteristic, complemented
10C4×#821C211yesmaximal, characteristic, complemented
11#69481C211yesmaximal, characteristic, complemented
12#68883C211yesmaximal, characteristic
13#68870C211yesmaximal, characteristic, complemented
14#68859C211yesmaximal, characteristic, complemented
15#68856C211yesmaximal, characteristic, complemented
16#21543—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C22×#215C2211yescharacteristic
20C22×ΓL(1,𝔽25)C2211yescharacteristic
21C22×D3×C8C2211yescharacteristic
22C2×#895C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×#872C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28C3×C4×C4⋊C4C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#815C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35#5026—24no—
36#4990—21no—
37#4988C2211yescharacteristic
38#4987C2211yescharacteristic
39#4987C2211yescharacteristic
40#4984C2211yescharacteristic
41#4976—21no—
42#4972C2211yescharacteristic
43C2×#809C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4940—21no—
48#4940—21no—
49#4939—21no—
50C4×C4⋊Dic3C2211yescharacteristic
51#4933—21no—
52C2×#803C2211yescharacteristic
53#4812C418yescomplemented
54#4689—24no—
55#4668—21no—
56#4665—21no—
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4551C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63C2×#468—31no—
64C2×#440—31no—
65C2×#405—31no—
66#2324—31no—
67#2323—31no—
68#1968—31no—
69#1942—31no—
70#1936—31no—
71#1935—31no—
72#1893—31no—
73#1892—31no—
74#1809—31no—
75#1776—31no—
76C4×#103D311yescharacteristic, complemented
77C4×#102—31no—
78C22×C3×Q8C2311yescharacteristic
79C22×D12—21no—
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×C4×D3C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C42—21no—
92C2×C3×C42—21no—
93C2×C3×C42C2311yescharacteristic
94#946—22no—
95#946C2×C414yescomplemented
96C2×#215—22no—
97C2×#215—22no—
98C2×#215C2×C414yes—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4D411yescharacteristic
107C2×(C2×D3)⋊C4D411yescharacteristic
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4D411yescharacteristic
115C2×Dic3⋊C4D411yescharacteristic
116C2×C4×Dic3—21no—
117C2×C4×Dic3C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C2×ΓL(1,𝔽25)—22no—
121C2×ΓL(1,𝔽25)—22no—
122C2×ΓL(1,𝔽25)C2×C414yescomplemented
123C2×D3×C8—22no—
124C2×D3×C8—22no—
125C2×D3×C8—22no—
126#895—22no—
127#895C2×C414yes—
128C4×D12—22no—
129C4×D12—22no—
130#872—22no—
131#872—22no—
132C3×#103—22no—
133C3×#103C2×C414yescomplemented
134C3×C4×C8—22no—
135C3×#95—21no—
136#835—22no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144#821—22no—
145#821C2×C414yes—
146#815—22no—
147C8×Dic3—22no—
148#809—22no—
149#808—22no—
150#807—22no—
151#807C2×C414yescomplemented
152#803D414yescomplemented
153C22×SD16—31no—
154C22×M16—31no—
155C23×C8—31no—
156C2×C22⋊Q8—31no—
157C2×#121—31no—
158C2×C4×Q8C2×D311yescharacteristic
159C2×C4×D4—31no—
160C2×#117—31no—
161#468—38no—
162#440—64no—
163C2×#107—31no—
164C2×#107—31no—
165C2×#106—31no—
166C2×#103—31no—
167C2×#103C2×D311yescharacteristic
168C2×#103C2×D311yescharacteristic
169C2×#102—31no—
170C2×#102—31no—
171C2×#102—31no—
172#405—64no—
173C2×#97—31no—
174C2×#97—31no—
175C2×C4×C8—31no—
176C2×C4×C8C2×D311yescharacteristic
177#391—31no—
178#390—31no—
179#390—61no—
180#389—31no—
181#389—31no—
182#389—61no—
183#388—61no—
184#388—61no—
185#388—61no—
186#387—61no—
187#386—61no—
188#385—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×D3—21no—
200C23×D3C2×D411yescharacteristic
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8C22×C412yes—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—24no—
226C2×D12—81no—
227C2×D12—81no—
228C2×D12C22×C412yes—
229C2×C4×D3—21no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C4×D3—42no—
234C2×C4×D3—81no—
235C2×C4×D3—81no—
236C2×C4×D3C22×C412yes—
237C2×C4×D3#4112yes—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8C22×C412yes—
243C2×C3×C8#4112yes—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—42no—
250C3×C4⋊C4C22×C412yes—
251C3×C42—42no—
252C3×C42—81no—
253C3×C42C2×D412yes—
254C3×C42#4112yes—
255#215—28no—
256(C2×D3)⋊C4—24no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—22no—
259(C2×D3)⋊C4—24no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—81no—
262(C2×D3)⋊C4—81no—
263(C2×D3)⋊C4—81no—
264(C2×D3)⋊C4—81no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—81no—
271C4⋊Dic3C22×C412yes—
272Dic3⋊C4—22no—
273Dic3⋊C4—22no—
274Dic3⋊C4—24no—
275Dic3⋊C4—81no—
276C4×Dic3—21no—
277C4×Dic3—21no—
278C4×Dic3—81no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8C22×C412yes—
282C2×C3⋊C8C2×D412yes—
283C2×C3⋊C8C2×D412yes—
284C2×C3⋊C8#4112yes—
285ΓL(1,𝔽25)—28no—
286D3×C8—28no—
287C22×Q8C22×D311yescharacteristic
288C22×D4—31no—
289C22×D4—61no—
290C23×C4—31no—
291C23×C4—61no—
292C2×SD16—34no—
293C2×SD16—62no—
294C2×SD16—62no—
295C2×M16—34no—
296C2×M16—62no—
297C2×M16—62no—
298C22×C8—31no—
299C22×C8—31no—
300C22×C8—62no—
301C22×C8—62no—
302C22×C8—62no—
303C22×C8C22×D311yescharacteristic
304C22×C8C22×D311yescharacteristic
305C22⋊Q8—34no—
306C22⋊Q8—62no—
307#121—34no—
308#121—62no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×D4—62no—
312C4×D4—62no—
313#117—62no—
314#117—62no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C22⋊C4—31no—
328C2×C22⋊C4—31no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—31no—
339C2×C42—61no—
340C2×C42C22×D311yescharacteristic
341#107—34no—
342#107—62no—
343#106—62no—
344#103—22no—
345#103—62no—
346#103C4×D314yescomplemented
347#102—34no—
348#102—62no—
349#102—62no—
350#97—34no—
351#97—62no—
352C4×C8—22no—
353C4×C8—62no—
354#95—21no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360#95—61no—
361#95—61no—
362C23×C3C22×D411yescharacteristic
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—21no—
367C22×D3—21no—
368C22×D3—21no—
369C22×D3—22no—
370C22×D3—41no—
371C22×D3—41no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3C4×D412yes—
377C3×Q8—22no—
378C3×Q8—22no—
379C3×Q8—41no—
380C3×Q8—41no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4#12112yes—
398C2×C3×C4#12112yes—
399C2×C3×C4C23×C411yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—22no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3C4×D412yes—
423D12—22no—
424D12—22no—
425D12—41no—
426D12—41no—
427D12—84no—
428C4×D3—22no—
429C4×D3—22no—
430C4×D3—22no—
431C4×D3—22no—
432C4×D3—84no—
433C3×C8—22no—
434C3×C8—22no—
435C3⋊C8—22no—
436C3⋊C8C4×D414yes—
437C24—31no—
438C24—61no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C2×C4×D312yes—
445C2×D4—32no—
446C2×D4—64no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—61no—
451C2×D4—61no—
452C2×D4—241no—
453C2×D4—241no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—31no—
459C22×C4—32no—
460C22×C4—32no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—122no—
471C22×C4—241no—
472C22×C4—241no—
473C22×C4D3×D411yescharacteristic
474C22×C4D3×D411yescharacteristic
475C22×C4C23×D311yescharacteristic
476SD16—68no—
477M16—68no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—32no—
483C2×C8—32no—
484C2×C8—32no—
485C2×C8—32no—
486C2×C8—68no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8C2×C4×D312yes—
490C2×C8#23512yes—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—32no—
497C4⋊C4—42no—
498C4⋊C4—64no—
499C4⋊C4—62no—
500C4⋊C4—62no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—61no—
504C4⋊C4—61no—
505C4⋊C4—61no—
506C4⋊C4—241no—
507C4⋊C4—241no—
508C4⋊C4C2×C4×D312yes—
509C22⋊C4—64no—
510C22⋊C4—64no—
511C22⋊C4—62no—
512C22⋊C4—62no—
513C22⋊C4—122no—
514C22⋊C4—241no—
515C22⋊C4—241no—
516C22⋊C4—241no—
517C22⋊C4—241no—
518C42—42no—
519C42—61no—
520C42—61no—
521C42—81no—
522C42—241no—
523C42#23512yes—
524C42D3×D412yes—
525C22×C3C2×C4×D411yescharacteristic
526C22×C3C2×#12111yescharacteristic
527C22×C3#53211yescharacteristic
528C22×C3#57611yescharacteristic
529C22×C3#57711yescharacteristic
530C22×C3#57711yescharacteristic
531C22×C3#57811yescharacteristic
532C2×D3—22no—
533C2×D3—22no—
534C2×D3—22no—
535C2×D3—41no—
536C2×D3—41no—
537C2×D3—41no—
538C2×D3—41no—
539C2×D3—42no—
540C2×D3—42no—
541C2×D3—42no—
542C2×D3—81no—
543C2×D3—81no—
544C2×D3—81no—
545C2×D3—81no—
546C3×C4—22no—
547C3×C4—22no—
548C3×C4—42no—
549C3×C4—42no—
550C3×C4—81no—
551C3×C4—81no—
552C3×C4C2×C4×D411yescharacteristic
553C3×C4C4×#4111yescharacteristic
554C3×C4#52511yescharacteristic
555C3×C4#52911yescharacteristic
556Dic3—22no—
557Dic3—42no—
558Dic3—41no—
559Dic3—41no—
560Dic3—81no—
561Dic3—81no—
562C23—32no—
563C23—61no—
564C23—61no—
565C23—61no—
566C23—61no—
567C23—61no—
568C23—61no—
569C23—62no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23—121no—
574C23—121no—
575C23—121no—
576C23C2×D3×D411yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—41no—
580Q8—41no—
581D4—62no—
582D4—62no—
583D4—121no—
584D4—121no—
585D4—244no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—32no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—62no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—244no—
625C2×C4#88412yes—
626C2×C4#88412yes—
627C2×C4C22×C4×D311yescharacteristic
628C2×C4C2×#23511yescharacteristic
629C2×C4C2×#23511yescharacteristic
630C2×C4C2×D3×D411yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4D3×#4111yescharacteristic
633C8—22no—
634C8—22no—
635C8—34no—
636C8—62no—
637C2×C3C4×#12111yescharacteristic
638C2×C3#294611yescharacteristic
639C2×C3#294711yescharacteristic
640C2×C3#306211yescharacteristic
641C2×C3#306311yescharacteristic
642C2×C3#317411yescharacteristic
643C2×C3#317511yescharacteristic
644D3—42no—
645D3—82no—
646C22—62no—
647C22—62no—
648C22—62no—
649C22—121no—
650C22—121no—
651C22—121no—
652C22—121no—
653C22—122no—
654C22—122no—
655C22—122no—
656C22—241no—
657C22—241no—
658C22—241no—
659C22—241no—
660C22C2×#88411yescharacteristic
661C22C4×D3×D411yescharacteristic
662C22#554811yescharacteristic
663C22#576211yescharacteristic
664C22#576611yescharacteristic
665C22#576911yescharacteristic
666C22#577011yescharacteristic
667C4—22no—
668C4—22no—
669C4—42no—
670C4—42no—
671C4—62no—
672C4—81no—
673C4—81no—
674C4—122no—
675C4—121no—
676C4—121no—
677C4—241no—
678C4—241no—
679C4C4×#23511yescharacteristic
680C4#552811yescharacteristic
681C4C4×D3×D411yescharacteristic
682C4#554711yescharacteristic
683C3#2154311yescharacteristic, complemented, Sylow
684C2—122no—
685C2—242no—
686C2C4×#88411yescharacteristic
687C2#7880011yescharacteristic
688C2#7884011yescharacteristic
689C2#7964711yescharacteristic
690C2#7966711yescharacteristic
691C2#7971911yescharacteristic
692C2#7972711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×C4×#103C2
socleC23×C3C22×D4
prime radicalC2×#895C22
cosocleC22×C3×C4C24
hypercenterC4×#103D3
hypocenterC3#21543
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
1. #82488C2semidirect
2. C2×#5026C2semidirect
4. C2×#4689C2semidirect
6. C3×C4×#103C2semidirect
7. #70270C2semidirect
8. #70257C2semidirect
9. #70255C2semidirect
10. C4×#821C2semidirect
11. #69481C2semidirect
13. #68870C2semidirect
14. #68859C2semidirect
15. #68856C2semidirect
53. #4812C4semidirect
76. C4×#103D3semidirect
95. #946C2×C4semidirect
122. C2×ΓL(1,𝔽25)C2×C4semidirect
133. C3×#103C2×C4semidirect
151. #807C2×C4semidirect
152. #803D4semidirect
346. #103C4×D3semidirect
683. C3#21543semidirect

Maximal subgroups

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

subgroupindex
#824882
C2×#50262
C2×#48122
C2×#46892
#737632
C3×C4×#1032
#702702
#702572
#702552
C4×#8212
#694812
#688832
#688702
#688592
#688562
#215433