Finite Group Databasebeta

← #10845537

Subgroups — #10845537

Order 768.

840 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82777C211yesmaximal, characteristic
2C2×#4791C211yesmaximal, characteristic, complemented
3C2×#4730C211yesmaximal, characteristic
4#73942C211yesmaximal, characteristic
5C3×#2036C211yesmaximal, characteristic, complemented
6#71696C211yesmaximal, characteristic, complemented
7#70631C211yesmaximal, characteristic, complemented
8#70616C211yesmaximal, characteristic, complemented
9#70274C211yesmaximal, characteristic, complemented
10#70271C211yesmaximal, characteristic, complemented
11#70220C211yesmaximal, characteristic, complemented
12#69940C211yesmaximal, characteristic, complemented
13#69511C211yesmaximal, characteristic, complemented
14#68891C211yesmaximal, characteristic, complemented
15#68806C211yesmaximal, characteristic, complemented
16#25356—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×#946C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×C4×#67C2211yescharacteristic
21C23×C3⋊C8C2211yescharacteristic
22C22×ΓL(1,𝔽25)C2211yescharacteristic
23C2×#888C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C3×#380C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31#5212C2211yescharacteristic
32#5212C2211yescharacteristic
33C2×(C2×D3)⋊C8C2211yescharacteristic
34C2×#817C2211yescharacteristic
35C2×#815C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37#4990—21no—
38#4987C2211yescharacteristic
39#4987C2211yescharacteristic
40#4987C2211yescharacteristic
41#4982—21no—
42#4977C2211yescharacteristic
43Dic3×C4⋊C4C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#808C2211yescharacteristic
46#4961—21no—
47#4957—21no—
48#4947—21no—
49#4945—21no—
50#4791—24no—
51#4730—24no—
52#4673—21no—
53#4650—21no—
54C3×#340C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#475—31no—
63C2×#435—31no—
64#2563—31no—
65#2536—31no—
66#2077—31no—
67#2077—31no—
68#2036D311yescharacteristic, complemented
69#2035—31no—
70#1993—31no—
71#1949—31no—
72#1894—31no—
73#1892—31no—
74#1809—31no—
75#1807—31no—
76#1763—31no—
77C22×C3×Q8C2311yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C22×#67D411yescharacteristic
80C22×C4×D3C2311yescharacteristic
81C22×Dic6—21no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—41no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×#217—21no—
93C2×#217—21no—
94C2×#217D411yescharacteristic
95#946—22no—
96#946—41no—
97Q8×Dic3—22no—
98Q8×Dic3—22no—
99C4×#67—22no—
100C4×#67—22no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4D411yescharacteristic
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4D411yescharacteristic
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8—22no—
118C22×C3⋊C8—22no—
119C22×C3⋊C8—41no—
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C2×ΓL(1,𝔽25)—22no—
123C2×ΓL(1,𝔽25)—22no—
124C2×ΓL(1,𝔽25)—41no—
125#888—22no—
126#888—22no—
127C3×#107—41no—
128C3×#103—22no—
129C3×#103—22no—
130C3×C22⋊C8—22no—
131C3×#95—21no—
132#835—22no—
133#835—22no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142(C2×D3)⋊C8—22no—
143#817—22no—
144#815—22no—
145Dic3⋊C8D414yes—
146#808—22no—
147#808—41no—
148C22×M16—31no—
149C23×C8—31no—
150C2×C4⋊Q8—31no—
151C2×C22⋊Q8—31no—
152C2×C22⋊Q8C2×D311yescharacteristic
153C2×C4×Q8—31no—
154C2×C4×D4—31no—
155#475—64no—
156#435—64no—
157C2×#107C2×D311yescharacteristic
158C2×#106—31no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#103C2×D311yescharacteristic
166C2×C22⋊C8—31no—
167C2×C22⋊C8C2×D311yescharacteristic
168C2×#97—31no—
169#398—61no—
170#394—61no—
171#393—61no—
172#393—61no—
173#390—61no—
174#388—61no—
175#388—61no—
176#382—31no—
177#382—61no—
178#380—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183#380C2×D311yescharacteristic
184C4×C4⋊C4—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340C2×D311yescharacteristic
193C24×C3C2×D411yescharacteristic
194C23×D3C2×D411yescharacteristic
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8#4112yes—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C2×#67—22no—
214C2×#67—22no—
215C2×#67—41no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—21no—
229C2×Dic6—42no—
230C2×Dic6—81no—
231C2×Dic6—81no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8C2×D412yes—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—42no—
245C3×C22⋊C4—42no—
246C3×C22⋊C4—81no—
247#217—22no—
248#217—42no—
249#217—81no—
250(C2×D3)⋊C4—22no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—81no—
254(C2×D3)⋊C4—81no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—42no—
264Dic3⋊C4—22no—
265Dic3⋊C4—42no—
266Dic3⋊C4—42no—
267C4×Dic3—42no—
268C4×Dic3—42no—
269C4×Dic3C2×D412yes—
270C4×Dic3#4112yes—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—44no—
276C2×C3⋊C8C2×D412yes—
277C2×C3⋊C8#4112yes—
278ΓL(1,𝔽25)—44no—
279C22×Q8—61no—
280C22×Q8C22×D311yescharacteristic
281C22×D4—31no—
282C23×C4—31no—
283C23×C4C22×D311yescharacteristic
284C2×M16—62no—
285C2×M16—62no—
286C2×M16—121no—
287C22×C8—31no—
288C22×C8—31no—
289C22×C8—62no—
290C22×C8—62no—
291C22×C8—121no—
292C22×C8C22×D311yescharacteristic
293C22×C8C22×D311yescharacteristic
294C4⋊Q8—62no—
295C4⋊Q8—62no—
296C22⋊Q8—22no—
297C22⋊Q8—41no—
298C22⋊Q8—62no—
299C22⋊Q8—121no—
300C4×Q8—62no—
301C4×Q8—62no—
302C4×D4—62no—
303C4×D4—62no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—31no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—31no—
319C2×C22⋊C4—31no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C42—31no—
327C2×C42—61no—
328C2×C42—61no—
329#107—41no—
330#106—62no—
331#106—62no—
332#106—121no—
333C4⋊C8—34no—
334#103—22no—
335#103—22no—
336#103—62no—
337#103—62no—
338C22⋊C8—22no—
339C22⋊C8—62no—
340#97—62no—
341#95—21no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3#13712yes—
357C23×C3C22×D411yescharacteristic
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—21no—
361C22×D3—21no—
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C3×Q8—22no—
366C3×Q8—22no—
367C3×Q8—41no—
368C3×Q8—41no—
369C2×C3×C4—21no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—41no—
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—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4H(5,𝔽2)11yescharacteristic
393C2×C3×C42−1+411yescharacteristic
394#67—44no—
395C2×Dic3—21no—
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×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3#12112yes—
415C2×Dic3#12112yes—
416C4×D3—41no—
417C4×D3—41no—
418C4×D3—41no—
419C4×D3—41no—
420Dic6—84no—
421C3×C8—22no—
422C3×C8—41no—
423C3⋊C8—22no—
424C3⋊C8—22no—
425C24—31no—
426C24D3×D411yescharacteristic
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—122no—
433C2×Q8—241no—
434C2×Q8—241no—
435C2×Q8#23712yes—
436C2×D4—62no—
437C2×D4—62no—
438C2×D4—121no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—31no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4D3×D411yescharacteristic
463C22×C4C23×D311yescharacteristic
464M16—124no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—32no—
471C2×C8—32no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C2×C8—124no—
477C2×C8D3×D412yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4—62no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—61no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—122no—
495C4⋊C4—122no—
496C4⋊C4—122no—
497C22⋊C4—42no—
498C22⋊C4—62no—
499C22⋊C4—62no—
500C22⋊C4—81no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—122no—
504C22⋊C4—241no—
505C22⋊C4—241no—
506C22⋊C4—241no—
507C42—32no—
508C42—32no—
509C42—122no—
510C42—122no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3—41no—
518C22×C3C2×#12111yescharacteristic
519C22×C3#54611yescharacteristic
520C22×C3#54711yescharacteristic
521C22×C3C2×#13711yescharacteristic
522C22×C3#57511yescharacteristic
523C22×C3#57711yescharacteristic
524C22×C3#57711yescharacteristic
525C2×D3—41no—
526C2×D3—41no—
527C2×D3—41no—
528C2×D3—41no—
529C2×D3—41no—
530C2×D3—41no—
531C2×D3—41no—
532C3×C4—41no—
533C3×C4—42no—
534C3×C4—81no—
535C3×C4—81no—
536C3×C4—81no—
537C3×C4#52911yescharacteristic
538C3×C4#54211yescharacteristic
539C3×C4#54611yescharacteristic
540C3×C4#54811yescharacteristic
541Dic3—22no—
542Dic3—22no—
543Dic3—41no—
544Dic3—81no—
545Dic3—81no—
546Dic3—81no—
547Dic3—81no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—61no—
555C23—61no—
556C23—61no—
557C23—61no—
558C23—61no—
559C23—61no—
560C23—61no—
561C23#91812yes—
562C23C2×D3×D411yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8—244no—
568D4—124no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—32no—
574C2×C4—32no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—61no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—121no—
610C2×C4—121no—
611C2×C4—121no—
612C2×C4—121no—
613C2×C4C2×D3×D411yescharacteristic
614C2×C4C2×#23711yescharacteristic
615C2×C4#100411yescharacteristic
616C2×C4D3×#4111yescharacteristic
617C2×C4D3×#4111yescharacteristic
618C2×C4#101011yescharacteristic
619C8—22no—
620C8—41no—
621C8—62no—
622C8—62no—
623C2×C3—42no—
624C2×C3#242711yescharacteristic
625C2×C3#305611yescharacteristic
626C2×C3#305711yescharacteristic
627C2×C3#331611yescharacteristic
628C2×C3#332111yescharacteristic
629C2×C3#334411yescharacteristic
630C2×C3#334811yescharacteristic
631D3—81no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—121no—
640C22—121no—
641C22—121no—
642C22—121no—
643C22—121no—
644C22—121no—
645C22—121no—
646C22#559111yescharacteristic
647C22D3×#12111yescharacteristic
648C22#565511yescharacteristic
649C22C2×#91811yescharacteristic
650C22#576811yescharacteristic
651C22#577511yescharacteristic
652C22#577811yescharacteristic
653C4—41no—
654C4—42no—
655C4—62no—
656C4—62no—
657C4—81no—
658C4—81no—
659C4—81no—
660C4—121no—
661C4—241no—
662C4—241no—
663C4—241no—
664C4—241no—
665C4#559811yescharacteristic
666C4#561011yescharacteristic
667C4#564911yescharacteristic
668C4#566911yescharacteristic
669C3#2535611yescharacteristic, complemented, Sylow
670C2—42no—
671C2—241no—
672C2#7690211yescharacteristic
673C2#8011311yescharacteristic
674C2#8019811yescharacteristic
675C2#8075911yescharacteristic
676C2#8080711yescharacteristic
677C2#8146511yescharacteristic
678C2#8153611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2036C2
socleC23×C3C22×D4
prime radicalC22×#67D4
hypercenter#2036D3
hypocenterC3#25356
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2036

Complements

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

NH ≅ G/Ntype
2. C2×#4791C2semidirect
5. C3×#2036C2semidirect
6. #71696C2semidirect
7. #70631C2semidirect
8. #70616C2semidirect
9. #70274C2semidirect
10. #70271C2semidirect
11. #70220C2semidirect
12. #69940C2semidirect
13. #69511C2semidirect
14. #68891C2semidirect
15. #68806C2semidirect
68. #2036D3semidirect
669. C3#25356semidirect

Maximal subgroups

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

subgroupindex
#827772
C2×#47912
C2×#47302
#739422
C3×#20362
#716962
#706312
#706162
#702742
#702712
#702202
#699402
#695112
#688912
#688062
#253563