Finite Group Databasebeta

← #10845538

Subgroups — #10845538

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#83260C211yesmaximal, characteristic
2C2×#5094C211yesmaximal, characteristic, complemented
3C2×#4706C211yesmaximal, characteristic
4#74240C211yesmaximal, characteristic
5C3×#1993C211yesmaximal, characteristic, complemented
6#69813C211yesmaximal, characteristic, complemented
7#69540C211yesmaximal, characteristic, complemented
8#68891C211yesmaximal, characteristic, complemented
9#68884C211yesmaximal, characteristic, complemented
10#68882C211yesmaximal, characteristic, complemented
11#68870C211yesmaximal, characteristic, complemented
12#68868C211yesmaximal, characteristic, complemented
13#68833C211yesmaximal, characteristic, complemented
14#68825C211yesmaximal, characteristic, complemented
15#68824C211yesmaximal, characteristic, complemented
16#25356—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C22×ΓL(1,𝔽25)C2211yescharacteristic
19C22×D3×C8C2211yescharacteristic
20C2×#897C2211yescharacteristic
21C2×#897C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×#97C2211yescharacteristic
25C3×#382C2211yescharacteristic
26#5231—21no—
27#5231—21no—
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32#5094—24no—
33#4988—21no—
34#4987C2211yescharacteristic
35#4987C2211yescharacteristic
36#4987C2211yescharacteristic
37#4987C2211yescharacteristic
38#4987C2211yescharacteristic
39#4987C2211yescharacteristic
40#4977—21no—
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C4×C4⋊Dic3C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49#4706—24no—
50#4672—21no—
51#4670—21no—
52#4670—21no—
53#4661—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#475—31no—
63C2×#435—31no—
64#2563—31no—
65#2536—31no—
66#2077—31no—
67#2077—31no—
68#2036—31no—
69#2035—31no—
70#1993D311yescharacteristic, complemented
71#1949—31no—
72#1894—31no—
73#1892—31no—
74#1809—31no—
75#1807—31no—
76#1763—31no—
77C22×C3×Q8—21no—
78C22×D12D411yescharacteristic
79C22×C4×D3C2311yescharacteristic
80C22×C4×D3C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C4⋊Q8—22no—
85C3×C4⋊Q8—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4D411yescharacteristic
101C2×(C2×D3)⋊C4D411yescharacteristic
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3D411yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C2×ΓL(1,𝔽25)—22no—
114C2×ΓL(1,𝔽25)—22no—
115C2×ΓL(1,𝔽25)—41no—
116C2×D3×C8—22no—
117C2×D3×C8—22no—
118C2×D3×C8—41no—
119#897—22no—
120#897—22no—
121#897—41no—
122#897—41no—
123C4×D12—22no—
124C4×D12—22no—
125C4×Dic6—22no—
126C4×Dic6—22no—
127C3×#97—22no—
128C3×#95—21no—
129C3×#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137(C2×D3)⋊C8—22no—
138(C2×D3)⋊C8—22no—
139#816—22no—
140#816—22no—
141#810—22no—
142#810—22no—
143#808—22no—
144#808—22no—
145#808—41no—
146#807—41no—
147C4⋊(C3⋊C8)D414yes—
148C22×M16—31no—
149C23×C8—31no—
150C2×C4⋊Q8C2×D311yescharacteristic
151C2×C22⋊Q8—31no—
152C2×C22⋊Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×D4—31no—
155#475—64no—
156#435—64no—
157C2×#107—31no—
158C2×#106—31no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×C22⋊C8—31no—
167C2×C22⋊C8—31no—
168C2×#97C2×D311yescharacteristic
169#398—61no—
170#394—61no—
171#393—61no—
172#393—61no—
173#390—61no—
174#388—61no—
175#388—61no—
176#382—61no—
177#382C2×D311yescharacteristic
178#380—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183#380—31no—
184C4×C4⋊C4—31no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340C2×D311yescharacteristic
190#340C2×D311yescharacteristic
191#340C2×D311yescharacteristic
192#340C2×D311yescharacteristic
193C23×D3C2×D411yescharacteristic
194C23×D3C2×D411yescharacteristic
195C2×C3×Q8—42no—
196C2×C3×Q8—81no—
197C2×C3×Q8—81no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C2×D12—22no—
212C2×D12—22no—
213C2×D12—41no—
214C2×C4×D3—21no—
215C2×C4×D3—21no—
216C2×C4×D3—21no—
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—21no—
220C2×C4×D3—21no—
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6#4112yes—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
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—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—42no—
247C3×C42C2×D412yes—
248C3×C42#4112yes—
249(C2×D3)⋊C4—22no—
250(C2×D3)⋊C4—22no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—42no—
253(C2×D3)⋊C4—42no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—81no—
256(C2×D3)⋊C4—81no—
257(C2×D3)⋊C4—81no—
258(C2×D3)⋊C4—81no—
259C4⋊Dic3—22no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—42no—
267C4⋊Dic3—42no—
268Dic3⋊C4—42no—
269C4×Dic3—42no—
270C4×Dic3—42no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8C2×D412yes—
276C2×C3⋊C8C2×D412yes—
277ΓL(1,𝔽25)—44no—
278D3×C8—44no—
279C22×Q8—21no—
280C22×Q8—31no—
281C22×D4—31no—
282C23×C4—31no—
283C23×C4—31no—
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—22no—
295C4⋊Q8—22no—
296C22⋊Q8—62no—
297C22⋊Q8—62no—
298C22⋊Q8—121no—
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—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4—61no—
312C2×C4⋊C4C22×D311yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—31no—
317C2×C22⋊C4—31no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
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—61no—
327C2×C42—61no—
328C2×C42C22×D311yescharacteristic
329#107—121no—
330#106—62no—
331#106—62no—
332#106—121no—
333C4⋊C8—34no—
334#103—62no—
335#103—62no—
336#103—62no—
337#103—62no—
338C22⋊C8—62no—
339C22⋊C8—62no—
340#97—22no—
341#95—21no—
342#95—21no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350C23×C3C22×D411yescharacteristic
351C22×D3—21no—
352C22×D3—21no—
353C22×D3—21no—
354C22×D3—21no—
355C22×D3—21no—
356C22×D3—21no—
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—21no—
361C22×D3—21no—
362C22×D3—21no—
363C22×D3—21no—
364C22×D3#13712yes—
365C3×Q8—84no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
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#12112yes—
383C2×C3×C4#12112yes—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4H(5,𝔽2)11yescharacteristic
389C2×C3×C42−1+411yescharacteristic
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—22no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408D12—44no—
409C4×D3—41no—
410C4×D3—41no—
411C4×D3—41no—
412C4×D3—41no—
413C4×D3—41no—
414C4×D3—41no—
415C4×D3—41no—
416C4×D3—41no—
417Dic6—22no—
418Dic6—22no—
419Dic6—41no—
420Dic6—41no—
421C3×C8—22no—
422C3×C8—41no—
423C3⋊C8—22no—
424C3⋊C8—22no—
425C24—31no—
426C24—31no—
427C2×Q8—32no—
428C2×Q8—42no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—81no—
435C2×Q8—81no—
436C2×D4—62no—
437C2×D4—62no—
438C2×D4—121no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—61no—
446C22×C4—61no—
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×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4D3×D411yescharacteristic
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×C8#23912yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—62no—
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—62no—
498C22⋊C4—62no—
499C22⋊C4—122no—
500C22⋊C4—122no—
501C22⋊C4—122no—
502C22⋊C4—122no—
503C22⋊C4—241no—
504C22⋊C4—241no—
505C22⋊C4—241no—
506C22⋊C4—241no—
507C42—122no—
508C42—122no—
509C42#23512yes—
510C42D3×D412yes—
511C22×C3C2×#12111yescharacteristic
512C22×C3#54611yescharacteristic
513C22×C3#54711yescharacteristic
514C22×C3C2×#13711yescharacteristic
515C22×C3#57511yescharacteristic
516C22×C3#57711yescharacteristic
517C22×C3#57711yescharacteristic
518C2×D3—22no—
519C2×D3—22no—
520C2×D3—22no—
521C2×D3—41no—
522C2×D3—41no—
523C2×D3—41no—
524C2×D3—41no—
525C2×D3—41no—
526C2×D3—41no—
527C2×D3—41no—
528C2×D3—41no—
529C2×D3—41no—
530C2×D3—41no—
531C2×D3—41no—
532C3×C4—22no—
533C3×C4—22no—
534C3×C4—81no—
535C3×C4—81no—
536C3×C4—81no—
537C3×C4—81no—
538C3×C4#52911yescharacteristic
539C3×C4#54211yescharacteristic
540C3×C4#54611yescharacteristic
541C3×C4#54811yescharacteristic
542Dic3—42no—
543Dic3—41no—
544Dic3—41no—
545Dic3—81no—
546Dic3—81no—
547Dic3—81no—
548C23—32no—
549C23—61no—
550C23—61no—
551C23—61no—
552C23—61no—
553C23—61no—
554C23—61no—
555C23—61no—
556C23—61no—
557C23—61no—
558C23—61no—
559C23—61no—
560C23—61no—
561C23—61no—
562C23C2×D3×D411yescharacteristic
563Q8—62no—
564Q8—62no—
565Q8—84no—
566Q8—121no—
567Q8—121no—
568D4—124no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—41no—
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—62no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
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#88412yes—
612C2×C4#88412yes—
613C2×C4C2×#23511yescharacteristic
614C2×C4C2×D3×D411yescharacteristic
615C2×C4C2×#23911yescharacteristic
616C2×C4#100711yescharacteristic
617C2×C4D3×#4111yescharacteristic
618C2×C4#100911yescharacteristic
619C8—22no—
620C8—41no—
621C8—62no—
622C8—62no—
623C2×C3#242711yescharacteristic
624C2×C3#305611yescharacteristic
625C2×C3#305711yescharacteristic
626C2×C3#331611yescharacteristic
627C2×C3#332111yescharacteristic
628C2×C3#334411yescharacteristic
629C2×C3#334811yescharacteristic
630D3—42no—
631D3—81no—
632C22—62no—
633C22—62no—
634C22—62no—
635C22—121no—
636C22—121no—
637C22—121no—
638C22—121no—
639C22—121no—
640C22—121no—
641C22—121no—
642C22—121no—
643C22—121no—
644C22—121no—
645C22—121no—
646C22C2×#88411yescharacteristic
647C22#567611yescharacteristic
648C22#572611yescharacteristic
649C22#577511yescharacteristic
650C22#577511yescharacteristic
651C22D3×#13711yescharacteristic
652C22#577711yescharacteristic
653C4—22no—
654C4—22no—
655C4—81no—
656C4—81no—
657C4—81no—
658C4—81no—
659C4—121no—
660C4—121no—
661C4—122no—
662C4—241no—
663C4—241no—
664C4—241no—
665C4#558011yescharacteristic
666C4#558211yescharacteristic
667C4#563011yescharacteristic
668C4#563311yescharacteristic
669C3#2535611yescharacteristic, complemented, Sylow
670C2—122no—
671C2—241no—
672C2#7662411yescharacteristic
673C2#7893111yescharacteristic
674C2#7896211yescharacteristic
675C2#8170111yescharacteristic
676C2#8170311yescharacteristic
677C2#8199111yescharacteristic
678C2#8199511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1993C2
socleC23×C3C22×D4
prime radicalC22×D12D4
hypercenter#1993D3
hypocenterC3#25356
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
2. C2×#5094C2semidirect
5. C3×#1993C2semidirect
6. #69813C2semidirect
7. #69540C2semidirect
8. #68891C2semidirect
9. #68884C2semidirect
10. #68882C2semidirect
11. #68870C2semidirect
12. #68868C2semidirect
13. #68833C2semidirect
14. #68825C2semidirect
15. #68824C2semidirect
70. #1993D3semidirect
669. C3#25356semidirect

Maximal subgroups

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

subgroupindex
#832602
C2×#50942
C2×#47062
#742402
C3×#19932
#698132
#695402
#688912
#688842
#688822
#688702
#688682
#688332
#688252
#688242
#253563