Finite Group Databasebeta

← #10852335

Subgroups — #10852335

Order 768.

986 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4847C211yesmaximal, characteristic, complemented
2C2×#4804C211yesmaximal, characteristic
3#76613C211yesmaximal, characteristic
4#76224C211yesmaximal, characteristic, complemented
5C3×#2068C211yesmaximal, characteristic, complemented
6#70638C211yesmaximal, characteristic, complemented
7#70630C211yesmaximal, characteristic, complemented
8#70477C211yesmaximal, characteristic, complemented
9#70267C211yesmaximal, characteristic
10#70249C211yesmaximal, characteristic, complemented
11#70247C211yesmaximal, characteristic, complemented
12#69669C211yesmaximal, characteristic, complemented
13#69625C211yesmaximal, characteristic, complemented
14#68892C211yesmaximal, characteristic, complemented
15#68876C211yesmaximal, characteristic, complemented
16#25956—31nomaximal, Sylow
17C22×#239C2211yescharacteristic
18C22×D3×C8C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×#870C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#389C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×(C2×D3)⋊C8C2211yescharacteristic
35C2×#818C2211yescharacteristic
36C2×#817C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38#4993—21no—
39#4987C2211yescharacteristic
40#4987C2211yescharacteristic
41#4976—21no—
42#4975C2211yescharacteristic
43#4972C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C4×(C2×D3)⋊C4C2211yescharacteristic
48#4847—24no—
49#4804—24no—
50#4676—21no—
51#4665—21no—
52#4665—21no—
53#4664—21no—
54D3×#95—21no—
55#4652—21no—
56C3×#340C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#478—31no—
63C2×#416—31no—
64#2441—31no—
65#2339—31no—
66#2095—31no—
67#2068D311yescharacteristic, complemented
68#2066—31no—
69#2026—31no—
70#1862—31no—
71#1862—31no—
72#1854—31no—
73#1850—31no—
74#1808—31no—
75#1802—31no—
76#1788—31no—
77C22×C3×Q8C2311yescharacteristic
78C2×#239—22no—
79C2×#239—22no—
80C2×#239—22no—
81C2×#239—41no—
82C2×#239—41no—
83C2×#239Q814yes—
84C22×D12—21no—
85C22×D12C2311yescharacteristic
86C22×C4×D3—21no—
87C22×C4×D3—21no—
88C22×C4×D3—21no—
89C22×C4×D3C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4D411yescharacteristic
103C2×(C2×D3)⋊C4D411yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C2×D3×C8—22no—
117C2×D3×C8—22no—
118C2×D3×C8—41no—
119#893—22no—
120#893—41no—
121#892—22no—
122#892—41no—
123C4×D12—22no—
124C4×D12—41no—
125#870—22no—
126#870—41no—
127C3×#107—41no—
128C3×C4⋊C8—22no—
129C3×#103—22no—
130C3×#103—41no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134#835—22no—
135#835—41no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95D411yescharacteristic
142C3⋊#95D411yescharacteristic
143#821—22no—
144#821D414yes—
145(C2×D3)⋊C8—22no—
146#818—22no—
147#817—22no—
148Dic3⋊C8—41no—
149#809—22no—
150#809—22no—
151#807—41no—
152C22×#41—31no—
153C23×C8—31no—
154C2×#125—31no—
155C2×C4×D4—31no—
156C2×C4×D4—31no—
157C2×#117—31no—
158#478—64no—
159C2×#107—31no—
160C2×#107—31no—
161C2×#107C2×D311yescharacteristic
162C2×#106—31no—
163C2×C4⋊C8—31no—
164C2×C4⋊C8C2×D311yescharacteristic
165#416—64no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103C2×D311yescharacteristic
169C2×#103C2×D311yescharacteristic
170C2×#102—31no—
171C2×#102—31no—
172C2×#102—31no—
173C2×#102—31no—
174C2×C22⋊C8—31no—
175#399—61no—
176#396—61no—
177#391—31no—
178#391C2×D311yescharacteristic
179#389—31no—
180#389—61no—
181#389C2×D311yescharacteristic
182#387—61no—
183#386—61no—
184#386—61no—
185#384—31no—
186#381—61no—
187#380—31no—
188#380—31no—
189C4×C22⋊C4—31no—
190C2×#95—61no—
191#340—31no—
192#340—31no—
193#340—31no—
194#340C2×D311yescharacteristic
195#336—31no—
196#336—31no—
197C23×D3—21no—
198C23×D3C2×D411yescharacteristic
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8C2×Q812yes—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218#239—28no—
219#239—44no—
220#239—44no—
221#239—44no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—24no—
226C2×D12—42no—
227C2×D12—42no—
228C2×D12C2×D412yes—
229C2×D12C2×Q812yes—
230C2×D12#4112yes—
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—24no—
236C2×C4×D3—42no—
237C2×C4×D3—42no—
238C2×C4×D3—42no—
239C2×C4×D3—42no—
240C2×C4×D3—42no—
241C2×C4×D3—42no—
242C2×C4×D3—81no—
243C2×C4×D3—81no—
244C2×C4×D3C2×Q812yes—
245C2×C4×D3#4112yes—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8C2×D412yes—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—81no—
259C3×C42—21no—
260C3×C42—21no—
261(C2×D3)⋊C4—22no—
262(C2×D3)⋊C4—41no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—42no—
266(C2×D3)⋊C4—81no—
267(C2×D3)⋊C4—81no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3C2×D412yes—
274Dic3⋊C4—42no—
275Dic3⋊C4—42no—
276Dic3⋊C4—42no—
277Dic3⋊C4—42no—
278Dic3⋊C4—81no—
279C4×Dic3—21no—
280C4×Dic3—21no—
281C4×Dic3—42no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288D3×C8—44no—
289C2×#41—34no—
290C2×#41—62no—
291C2×#41—62no—
292C2×#41—62no—
293C2×#41—121no—
294C2×#41—121no—
295C22×Q8C22×D311yescharacteristic
296C22×D4—31no—
297C22×D4—61no—
298C23×C4—31no—
299C23×C4—61no—
300C23×C4—61no—
301C23×C4—61no—
302C22×C8—31no—
303C22×C8—31no—
304C22×C8—62no—
305C22×C8—62no—
306C22×C8—121no—
307C22×C8C22×D311yescharacteristic
308C22×C8C22×D311yescharacteristic
309#125—62no—
310#125—121no—
311C4×D4—62no—
312C4×D4—62no—
313C4×D4—121no—
314C4×D4—121no—
315#117—62no—
316#117—121no—
317C2×C4⋊C4—21no—
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⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C4⋊C4C22×D311yescharacteristic
329C2×C22⋊C4—31no—
330C2×C22⋊C4—31no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C42—31no—
337C2×C42—61no—
338C2×C42C22×D311yescharacteristic
339#107—41no—
340#107—62no—
341#107—121no—
342#106—62no—
343C4⋊C8—22no—
344C4⋊C8—121no—
345#103—22no—
346#103—41no—
347#103—62no—
348#103—121no—
349#102—34no—
350#102—62no—
351#102—62no—
352#102—62no—
353C22⋊C8—62no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—31no—
358#95—31no—
359#95—61no—
360#95—61no—
361#95—61no—
362#95—61no—
363#95—61no—
364C23×C3C22×D411yescharacteristic
365C22×D3—22no—
366C22×D3—21no—
367C22×D3—22no—
368C22×D3—22no—
369C22×D3—21no—
370C22×D3—21no—
371C22×D3—21no—
372C22×D3—21no—
373C22×D3—21no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3C22⋊Q812yes—
379C3×Q8—22no—
380C3×Q8—22no—
381C3×Q8—41no—
382C3×Q8—41no—
383C2×C3×C4—22no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
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—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C22×Q811yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4H(5,𝔽2)11yescharacteristic
406C2×C3×C4H(5,𝔽2)11yescharacteristic
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—22no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3C22⋊Q812yes—
424C2×Dic3#13512yes—
425D12—22no—
426D12—22no—
427D12—22no—
428D12—48no—
429D12C22⋊Q814yes—
430C4×D3—22no—
431C4×D3—22no—
432C4×D3—22no—
433C4×D3—22no—
434C4×D3—48no—
435C4×D3—84no—
436C4×D3—84no—
437C3×C8—22no—
438C3×C8—41no—
439C3⋊C8—41no—
440C3⋊C8—41no—
441C24—31no—
442C24—61no—
443#41—68no—
444#41—124no—
445#41—124no—
446#41—124no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8D3×Q812yes—
453C2×D4—32no—
454C2×D4—32no—
455C2×D4—32no—
456C2×D4—64no—
457C2×D4—61no—
458C2×D4—61no—
459C2×D4—61no—
460C2×D4—122no—
461C2×D4—122no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—31no—
467C22×C4—31no—
468C22×C4—32no—
469C22×C4—32no—
470C22×C4—31no—
471C22×C4—64no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—122no—
480C22×C4—122no—
481C22×C4—122no—
482C22×C4—122no—
483C22×C4—122no—
484C22×C4—122no—
485C22×C4—241no—
486C22×C4—241no—
487C22×C4D3×D411yescharacteristic
488C22×C4D3×D411yescharacteristic
489C22×C4C23×D311yescharacteristic
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8—61no—
500C2×C8—61no—
501C2×C8—124no—
502C2×C8D3×D412yes—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—32no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—81no—
516C4⋊C4—122no—
517C4⋊C4—122no—
518C4⋊C4—122no—
519C4⋊C4—122no—
520C4⋊C4—241no—
521C22⋊C4—62no—
522C22⋊C4—121no—
523C22⋊C4—122no—
524C22⋊C4—122no—
525C22⋊C4—122no—
526C22⋊C4—241no—
527C22⋊C4—241no—
528C42—21no—
529C42—21no—
530C42—61no—
531C42—61no—
532C42—122no—
533C22×C3C2×C22⋊Q811yescharacteristic
534C22×C3D4211yescharacteristic
535C22×C3#54611yescharacteristic
536C22×C3C2×#13511yescharacteristic
537C22×C3#57511yescharacteristic
538C22×C3#57611yescharacteristic
539C22×C3#57811yescharacteristic
540C2×D3—22no—
541C2×D3—22no—
542C2×D3—22no—
543C2×D3—24no—
544C2×D3—42no—
545C2×D3—42no—
546C2×D3—42no—
547C2×D3—41no—
548C2×D3—41no—
549C2×D3—42no—
550C2×D3—42no—
551C2×D3—41no—
552C2×D3—41no—
553C2×D3—42no—
554C3×C4—42no—
555C3×C4—41no—
556C3×C4—41no—
557C3×C4—81no—
558C3×C4—81no—
559C3×C4—81no—
560C3×C4C2×C22⋊Q811yescharacteristic
561C3×C4#54311yescharacteristic
562C3×C4#54611yescharacteristic
563C3×C4#55111yescharacteristic
564Dic3—22no—
565Dic3—22no—
566Dic3—42no—
567Dic3—41no—
568Dic3—81no—
569Dic3—81no—
570C23—32no—
571C23—61no—
572C23—61no—
573C23—61no—
574C23—61no—
575C23—61no—
576C23—62no—
577C23—62no—
578C23—61no—
579C23—62no—
580C23—121no—
581C23—121no—
582C23—121no—
583C23—121no—
584C23C2×D3×D411yescharacteristic
585Q8—22no—
586Q8—22no—
587Q8—41no—
588Q8—41no—
589D4—34no—
590D4—62no—
591D4—62no—
592D4—62no—
593D4—128no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—32no—
602C2×C4—32no—
603C2×C4—41no—
604C2×C4—41no—
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—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—61no—
618C2×C4—62no—
619C2×C4—62no—
620C2×C4—62no—
621C2×C4—62no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—62no—
625C2×C4—61no—
626C2×C4—128no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—244no—
636C2×C4—244no—
637C2×C4C2×D3×D411yescharacteristic
638C2×C4#100411yescharacteristic
639C2×C4C2×D3×Q811yescharacteristic
640C2×C4D3×#4111yescharacteristic
641C2×C4D3×#4111yescharacteristic
642C2×C4#100911yescharacteristic
643C8—22no—
644C8—41no—
645C8—121no—
646C8—121no—
647C2×C3#243711yescharacteristic
648C2×C3#307611yescharacteristic
649C2×C3#308311yescharacteristic
650C2×C3#326811yescharacteristic
651C2×C3#328311yescharacteristic
652C2×C3#329711yescharacteristic
653C2×C3#331511yescharacteristic
654D3—42no—
655D3—44no—
656C22—62no—
657C22—62no—
658C22—64no—
659C22—62no—
660C22—122no—
661C22—121no—
662C22—122no—
663C22—121no—
664C22—122no—
665C22—122no—
666C22—121no—
667C22—122no—
668C22—121no—
669C22—122no—
670C22#560911yescharacteristic
671C22#561211yescharacteristic
672C22D3×C22⋊Q811yescharacteristic
673C22#576611yescharacteristic
674C22D3×#13511yescharacteristic
675C22#577411yescharacteristic
676C22#577711yescharacteristic
677C4—42no—
678C4—41no—
679C4—41no—
680C4—62no—
681C4—62no—
682C4—81no—
683C4—81no—
684C4—81no—
685C4—121no—
686C4—122no—
687C4—241no—
688C4—241no—
689C4#561011yescharacteristic
690C4D3×C22⋊Q811yescharacteristic
691C4#563411yescharacteristic
692C4#570511yescharacteristic
693C3#2595611yescharacteristic, complemented, Sylow
694C2—122no—
695C2—124no—
696C2#7741811yescharacteristic
697C2#8049411yescharacteristic
698C2#8055911yescharacteristic
699C2#8077411yescharacteristic
700C2#8083611yescharacteristic
701C2#8111111yescharacteristic
702C2#8122911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2068C2
socleC23×C3C22×D4
prime radicalC22×#239C22
hypercenter#2068D3
hypocenterC3#25956
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2068

Complements

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

NH ≅ G/Ntype
1. C2×#4847C2semidirect
4. #76224C2semidirect
5. C3×#2068C2semidirect
6. #70638C2semidirect
7. #70630C2semidirect
8. #70477C2semidirect
10. #70249C2semidirect
11. #70247C2semidirect
12. #69669C2semidirect
13. #69625C2semidirect
14. #68892C2semidirect
15. #68876C2semidirect
67. #2068D3semidirect
693. C3#25956semidirect

Maximal subgroups

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

subgroupindex
C2×#48472
C2×#48042
#766132
#762242
C3×#20682
#706382
#706302
#704772
#702672
#702492
#702472
#696692
#696252
#688922
#688762
#259563