Finite Group Databasebeta

← #10852257

Subgroups — #10852257

Order 768.

966 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82457C211yesmaximal, characteristic
2C2×#4804C211yesmaximal, characteristic
3C2×#4791C211yesmaximal, characteristic, complemented
4#75997C211yesmaximal, characteristic, complemented
5C3×#1865C211yesmaximal, characteristic, complemented
6#71627C211yesmaximal, characteristic, complemented
7#70585C211yesmaximal, characteristic, complemented
8#70577C211yesmaximal, characteristic, complemented
9#70267C211yesmaximal, characteristic
10#70236C211yesmaximal, characteristic, complemented
11#69629C211yesmaximal, characteristic, complemented
12#68898C211yesmaximal, characteristic, complemented
13#68877C211yesmaximal, characteristic, complemented
14#68680C211yesmaximal, characteristic, complemented
15#68666C211yesmaximal, characteristic, complemented
16#25951—31nomaximal, Sylow
17C22×#239C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C22×D3×C8C2211yescharacteristic
21C2×#892C2211yescharacteristic
22C2×#888C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×#818C2211yescharacteristic
33C2×#817C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35#4994—21no—
36#4988—21no—
37#4988C2211yescharacteristic
38#4987—21no—
39#4987C2211yescharacteristic
40#4987C2211yescharacteristic
41#4984C2211yescharacteristic
42#4973—21no—
43C2×#809C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#808C2211yescharacteristic
46#4939—21no—
47#4939C2211yescharacteristic
48C4×(C2×D3)⋊C4—21no—
49C4×(C2×D3)⋊C4C2211yescharacteristic
50#4936C2211yescharacteristic
51C2×C4⋊(C3⋊C8)C2211yescharacteristic
52#4804—24no—
53#4791—24no—
54#4673—21no—
55#4653—21no—
56C3×#340C2211yescharacteristic
57C3×#336C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#475—31no—
63C2×#416—31no—
64#2439—31no—
65#2341—31no—
66#2092—31no—
67#2065—31no—
68#2065—31no—
69#2023—31no—
70#1865D311yescharacteristic, complemented
71#1863—31no—
72#1854—31no—
73#1850—31no—
74#1807—31no—
75#1805—31no—
76#1788—31no—
77C22×C3×Q8C2311yescharacteristic
78C2×#239—22no—
79C2×#239—22no—
80C2×#239—22no—
81C2×#239—41no—
82C2×#239—41no—
83C2×#239Q814yescomplemented
84C22×D12—21no—
85C22×D12C2311yescharacteristic
86C22×C4×D3—21no—
87C22×C4×D3C2311yescharacteristic
88C22×Dic6—21no—
89C22×C3×C8C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C3×C4×Q8—22no—
92C3×C4×Q8—41no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C42—21no—
99C2×C3×C42C2311yescharacteristic
100Q8×Dic3—22no—
101Q8×Dic3—41no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4D411yescharacteristic
110C2×(C2×D3)⋊C4D411yescharacteristic
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×Dic3⋊C4—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C2×D3×C8—22no—
123C2×D3×C8—22no—
124C2×D3×C8—41no—
125#892—22no—
126#892—41no—
127#888—22no—
128#888—41no—
129C3×#106—41no—
130C3×#103—22no—
131C3×#103D414yes—
132C3×#95—21no—
133#835—22no—
134#835—22no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95D411yescharacteristic
141C3⋊#95D411yescharacteristic
142#821—22no—
143#821—41no—
144(C2×D3)⋊C8—22no—
145#818—41no—
146#817—22no—
147Dic3⋊C8—22no—
148#809—22no—
149#809—41no—
150#808—22no—
151C4⋊(C3⋊C8)—41no—
152C22×#41—31no—
153C23×C8—31no—
154C2×C4⋊Q8—31no—
155C2×C4×Q8—31no—
156C2×C4×Q8C2×D311yescharacteristic
157C2×#117—31no—
158#475—64no—
159C2×#107—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×#106C2×D311yescharacteristic
163C2×C4⋊C8—31no—
164C2×C4⋊C8—31no—
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#396—61no—
176#393—61no—
177#391—61no—
178#390—31no—
179#390—31no—
180#390—61no—
181#390—61no—
182#389—31no—
183#389C2×D311yescharacteristic
184#387—61no—
185#382—31no—
186#380—31no—
187#380—31no—
188#380—61no—
189C4×C22⋊C4—31no—
190C4×C22⋊C4—61no—
191#340—31no—
192#340—31no—
193#340—31no—
194#340C2×D311yescharacteristic
195#336—31no—
196#336C2×D311yescharacteristic
197C23×D3—21no—
198C23×D3C2×D411yescharacteristic
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8C2×D412yes—
203C2×C3×Q8C2×Q812yes—
204C2×C3×Q8#4112yes—
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—21no—
226C2×D12—24no—
227C2×D12—21no—
228C2×D12—42no—
229C2×D12—42no—
230C2×D12C2×Q812yes—
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×D3C2×Q812yes—
239C2×C4×D3#4112yes—
240C2×Dic6—42no—
241C2×Dic6—42no—
242C2×Dic6—81no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8C2×D412yes—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—42no—
255C3×C4⋊C4—81no—
256C3×C4⋊C4C2×D412yes—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42—42no—
260(C2×D3)⋊C4—22no—
261(C2×D3)⋊C4—41no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—81no—
266(C2×D3)⋊C4—81no—
267(C2×D3)⋊C4—81no—
268(C2×D3)⋊C4—81no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—42no—
276C4⋊Dic3—42no—
277Dic3⋊C4—42no—
278C4×Dic3—21no—
279C4×Dic3—21no—
280C4×Dic3—42no—
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×Q8—61no—
296C22×Q8C22×D311yescharacteristic
297C22×D4—31no—
298C22×D4—61no—
299C23×C4—31no—
300C23×C4—61no—
301C22×C8—31no—
302C22×C8—31no—
303C22×C8—62no—
304C22×C8—62no—
305C22×C8—121no—
306C22×C8C22×D311yescharacteristic
307C22×C8C22×D311yescharacteristic
308C4⋊Q8—62no—
309C4⋊Q8—121no—
310C4×Q8—22no—
311C4×Q8—41no—
312C4×Q8—62no—
313C4×Q8—121no—
314#117—62no—
315#117—121no—
316C2×C4⋊C4—21no—
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⋊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—31no—
338C2×C42—61no—
339C2×C42—61no—
340C2×C42C22×D311yescharacteristic
341#107—121no—
342#106—41no—
343#106—62no—
344#106—62no—
345C4⋊C8—62no—
346C4⋊C8—121no—
347#103—22no—
348#103—62no—
349#103—62no—
350#103D1214yes—
351#102—62no—
352#102—62no—
353#102—121no—
354#102—121no—
355C22⋊C8—62no—
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—21no—
366C22×D3—21no—
367C22×D3—21no—
368C22×D3—21no—
369C22×D3—21no—
370C22×D3—21no—
371C22×D3—22no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3C22⋊Q812yes—
379C3×Q8—22no—
380C3×Q8—22no—
381C3×Q8—22no—
382C3×Q8C22⋊Q814yes—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
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—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×Dic3—41no—
423C2×Dic3C22⋊Q812yes—
424C2×Dic3#13512yes—
425D12—22no—
426D12—22no—
427D12—48no—
428D12—41no—
429D12—41no—
430C4×D3—22no—
431C4×D3—22no—
432C4×D3—22no—
433C4×D3—22no—
434C4×D3—48no—
435Dic6—84no—
436C3×C8—22no—
437C3×C8—41no—
438C3⋊C8—41no—
439C3⋊C8—41no—
440C24—31no—
441C24—61no—
442#41—68no—
443#41—124no—
444#41—124no—
445#41—124no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—122no—
450C2×Q8—122no—
451C2×Q8—241no—
452C2×Q8C2×D1212yes—
453C2×Q8#23712yes—
454C2×Q8D3×Q812yes—
455C2×D4—32no—
456C2×D4—64no—
457C2×D4—61no—
458C2×D4—61no—
459C2×D4—61no—
460C2×D4—61no—
461C2×D4—61no—
462C2×D4—122no—
463C2×D4—122no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—31no—
469C22×C4—31no—
470C22×C4—32no—
471C22×C4—32no—
472C22×C4—31no—
473C22×C4—64no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—122no—
482C22×C4—122no—
483C22×C4D3×D411yescharacteristic
484C22×C4D3×D411yescharacteristic
485C22×C4C23×D311yescharacteristic
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—124no—
498C2×C8C2×D1212yes—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—42no—
505C4⋊C4—61no—
506C4⋊C4—61no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—81no—
512C4⋊C4—122no—
513C4⋊C4—122no—
514C4⋊C4—122no—
515C4⋊C4C2×D1212yes—
516C22⋊C4—62no—
517C22⋊C4—121no—
518C22⋊C4—122no—
519C22⋊C4—122no—
520C22⋊C4—122no—
521C22⋊C4—241no—
522C22⋊C4—241no—
523C22⋊C4—241no—
524C22⋊C4—241no—
525C42—21no—
526C42—21no—
527C42—42no—
528C42—61no—
529C42—61no—
530C42—122no—
531C42—122no—
532C22×C3C2×C22⋊Q811yescharacteristic
533C22×C3D4211yescharacteristic
534C22×C3#54611yescharacteristic
535C22×C3C2×#13511yescharacteristic
536C22×C3#57511yescharacteristic
537C22×C3#57711yescharacteristic
538C22×C3#57711yescharacteristic
539C2×D3—22no—
540C2×D3—22no—
541C2×D3—22no—
542C2×D3—41no—
543C2×D3—41no—
544C2×D3—41no—
545C2×D3—41no—
546C2×D3—42no—
547C2×D3—42no—
548C2×D3—42no—
549C2×D3—81no—
550C2×D3—81no—
551C2×D3—81no—
552C2×D3—81no—
553C3×C4—24no—
554C3×C4—42no—
555C3×C4—41no—
556C3×C4—41no—
557C3×C4—81no—
558C3×C4—81no—
559C3×C4C2×C22⋊Q811yescharacteristic
560C3×C4#54311yescharacteristic
561C3×C4#54611yescharacteristic
562C3×C4#55111yescharacteristic
563Dic3—22no—
564Dic3—22no—
565Dic3—41no—
566Dic3—81no—
567Dic3—81no—
568Dic3—81no—
569C23—32no—
570C23—61no—
571C23—61no—
572C23—61no—
573C23—61no—
574C23—61no—
575C23—61no—
576C23—62no—
577C23—121no—
578C23—121no—
579C23—121no—
580C23—121no—
581C23—121no—
582C23—121no—
583C23C2×D3×D411yescharacteristic
584Q8—22no—
585Q8—22no—
586Q8—22no—
587Q8—244no—
588Q8#89714yes—
589D4—62no—
590D4—62no—
591D4—128no—
592D4—121no—
593D4—121no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—32no—
605C2×C4—32no—
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—61no—
625C2×C4—128no—
626C2×C4—121no—
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×C4C22×D1211yescharacteristic
636C2×C4C2×#23711yescharacteristic
637C2×C4#100411yescharacteristic
638C2×C4C2×D3×Q811yescharacteristic
639C2×C4D3×#4111yescharacteristic
640C2×C4#100911yescharacteristic
641C8—22no—
642C8—41no—
643C8—121no—
644C8—121no—
645C2×C3#243711yescharacteristic
646C2×C3#307611yescharacteristic
647C2×C3#308411yescharacteristic
648C2×C3#327111yescharacteristic
649C2×C3#328211yescharacteristic
650C2×C3#330411yescharacteristic
651C2×C3#332111yescharacteristic
652D3—42no—
653D3—82no—
654C22—62no—
655C22—62no—
656C22—62no—
657C22—121no—
658C22—121no—
659C22—121no—
660C22—121no—
661C22—122no—
662C22—122no—
663C22—122no—
664C22—241no—
665C22—241no—
666C22—241no—
667C22—241no—
668C22D4×D1211yescharacteristic
669C22#559111yescharacteristic
670C22D3×C22⋊Q811yescharacteristic
671C22#576211yescharacteristic
672C22D3×#13511yescharacteristic
673C22#577711yescharacteristic
674C22#577811yescharacteristic
675C4—24no—
676C4—42no—
677C4—41no—
678C4—41no—
679C4—62no—
680C4—62no—
681C4—81no—
682C4—81no—
683C4—121no—
684C4—241no—
685C4—241no—
686C4—241no—
687C4C2×#89711yescharacteristic
688C4#555411yescharacteristic
689C4#563411yescharacteristic
690C4#565811yescharacteristic
691C3#2595111yescharacteristic, complemented, Sylow
692C2—122no—
693C2—242no—
694C2#7441311yescharacteristic
695C2#7967411yescharacteristic
696C2#7984011yescharacteristic
697C2#8005411yescharacteristic
698C2#8024911yescharacteristic
699C2#8101411yescharacteristic
700C2#8122911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1865C2
socleC23×C3C22×D4
prime radicalC22×#239C22
hypercenter#1865D3
hypocenterC3#25951
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
3. C2×#4791C2semidirect
4. #75997C2semidirect
5. C3×#1865C2semidirect
6. #71627C2semidirect
7. #70585C2semidirect
8. #70577C2semidirect
10. #70236C2semidirect
11. #69629C2semidirect
12. #68898C2semidirect
13. #68877C2semidirect
14. #68680C2semidirect
15. #68666C2semidirect
70. #1865D3semidirect
83. C2×#239Q8semidirect
691. C3#25951semidirect

Maximal subgroups

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

subgroupindex
#824572
C2×#48042
C2×#47912
#759972
C3×#18652
#716272
#705852
#705772
#702672
#702362
#696292
#688982
#688772
#686802
#686662
#259513