Finite Group Databasebeta

← #10835924

Subgroups — #10835924

Order 768.

856 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5185C211yesmaximal, characteristic, complemented
2#82821C211yesmaximal, characteristic
3#82744C211yesmaximal, characteristic, complemented
4C2×#4720C211yesmaximal, characteristic, complemented
5C3×#2001C211yesmaximal, characteristic, complemented
6#72066C211yesmaximal, characteristic, complemented
7#71699C211yesmaximal, characteristic
8#71695C211yesmaximal, characteristic, complemented
9#70260C211yesmaximal, characteristic, complemented
10#70255C211yesmaximal, characteristic, complemented
11#70230C211yesmaximal, characteristic, complemented
12#70221C211yesmaximal, characteristic, complemented
13#69934C211yesmaximal, characteristic
14#68853C211yesmaximal, characteristic, complemented
15#68795C211yesmaximal, characteristic, complemented
16#24665—31nomaximal, Sylow
17C22×C3×Q16C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×#947C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C22×#215C2211yescharacteristic
23C2×#930C2211yescharacteristic
24C2×#926C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28#5229C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33#5212C2211yescharacteristic
34C2×#830C2211yescharacteristic
35#5185—24no—
36C2×#821C2211yescharacteristic
37C2×#821C2211yescharacteristic
38C2×#818C2211yescharacteristic
39C2×#815C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41C2×Dic3⋊C8C2211yescharacteristic
42C2×C8×Dic3C2211yescharacteristic
43#4994—21no—
44#4994—21no—
45#4988C2211yescharacteristic
46#4984—21no—
47#4984C2211yescharacteristic
48#4980—21no—
49#4975C2211yescharacteristic
50Dic3×C4⋊C4C2211yescharacteristic
51C2×#809C2211yescharacteristic
52C2×#807C2211yescharacteristic
53#4959—21no—
54#4959—21no—
55#4951—21no—
56#4951—21no—
57#4720—24no—
58C3×#340C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#464—31no—
63C2×#433—31no—
64#2620—31no—
65#2323—31no—
66#2036—31no—
67#2033—31no—
68#2001D311yescharacteristic, complemented
69#2000—31no—
70#1983—31no—
71#1973—31no—
72#1969—31no—
73#1947—31no—
74#1935—31no—
75#1934—31no—
76#1803—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×Q8C2311yescharacteristic
79C23×C3×C4C2311yescharacteristic
80C22×#67—21no—
81C22×D12C2311yescharacteristic
82C2×C3×Q16—22no—
83C2×C3×Q16—22no—
84C2×C3×Q16—41no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—41no—
90C3×C22⋊Q8—41no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×#217—21no—
98C2×#217D411yescharacteristic
99C2×#217D411yescharacteristic
100#947—22no—
101#947—22no—
102Q8×Dic3—22no—
103Q8×Dic3—22no—
104C2×#215—22no—
105C2×#215—22no—
106C2×#215—41no—
107#930—41no—
108#930—41no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×(C2×D3)⋊C4—21no—
114#926—22no—
115#926—22no—
116C2×C4⋊Dic3—21no—
117C2×C4⋊Dic3—21no—
118C2×C4⋊Dic3D411yescharacteristic
119C2×C4⋊Dic3D411yescharacteristic
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4⋊Dic3C2311yescharacteristic
122C2×C4⋊Dic3C2311yescharacteristic
123C2×Dic3⋊C4—21no—
124C2×Dic3⋊C4—21no—
125C2×C4×Dic3—21no—
126C2×C4×Dic3—21no—
127C2×C4×Dic3C2311yescharacteristic
128C2×C4×Dic3C2311yescharacteristic
129C22×C3⋊C8C2311yescharacteristic
130C22×C3⋊C8C2311yescharacteristic
131C3×#103—22no—
132C3×#103—41no—
133C3×C22⋊C8—22no—
134#835—22no—
135#835—41no—
136#835—41no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142C3⋊#95—21no—
143C3⋊#95—21no—
144#830—22no—
145#821—22no—
146#821—41no—
147#818—41no—
148#815—22no—
149Dic3⋊C8—22no—
150Dic3⋊C8D414yescomplemented
151C8×Dic3—22no—
152#809—22no—
153#807—41no—
154C22×Q16C2×D311yescharacteristic
155C22×SD16—31no—
156C2×#124—31no—
157C2×C22⋊Q8C2×D311yescharacteristic
158C2×C22⋊Q8C2×D311yescharacteristic
159C2×#121—31no—
160C2×C4×Q8—31no—
161C2×#117—31no—
162#464—64no—
163#433—64no—
164C2×#107—31no—
165C2×#107—31no—
166C2×C4⋊C8—31no—
167C2×C4⋊C8—31no—
168C2×#103—31no—
169C2×#103—31no—
170C2×#103—31no—
171C2×#103C2×D311yescharacteristic
172C2×#103C2×D311yescharacteristic
173C2×#102—31no—
174C2×#102—31no—
175C2×#102—31no—
176C2×C22⋊C8—31no—
177C2×C22⋊C8C2×D311yescharacteristic
178C2×#97—31no—
179C2×C4×C8—31no—
180#396—61no—
181#396—61no—
182#394—61no—
183#394—61no—
184#391—31no—
185#391—31no—
186#390—31no—
187#389—31no—
188#389—61no—
189#388—61no—
190#388—61no—
191#387—61no—
192#384—31no—
193#380—31no—
194C4×C4⋊C4—31no—
195#340—31no—
196#340—31no—
197#340—31no—
198#340C2×D311yescharacteristic
199C24×C3C2×D411yescharacteristic
200C23×D3—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8#4112yes—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2411yescharacteristic
225C2×#67—42no—
226C2×#67—81no—
227C2×#67—81no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3C2×D411yescharacteristic
232C22×Dic3C2×D411yescharacteristic
233C22×Dic3C2×D411yescharacteristic
234C22×Dic3C2×D411yescharacteristic
235C2×D12—21no—
236C2×D12—21no—
237C2×D12—21no—
238C2×D12—21no—
239C2×D12—21no—
240C2×D12—21no—
241C3×Q16—44no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8C2×D412yes—
248C3×C4⋊C4—24no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—42no—
253C3×C4⋊C4#4112yes—
254C3×C22⋊C4—24no—
255C3×C22⋊C4—42no—
256#217—22no—
257#217—22no—
258#217—81no—
259#215—44no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4—81no—
264(C2×D3)⋊C4—81no—
265C4⋊Dic3—22no—
266C4⋊Dic3—22no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—42no—
274C4⋊Dic3—81no—
275Dic3⋊C4—81no—
276Dic3⋊C4—81no—
277C4×Dic3—21no—
278C4×Dic3—21no—
279C4×Dic3—42no—
280C4×Dic3—81no—
281C4×Dic3C2×D412yes—
282C4×Dic3#4112yes—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8C2×D412yes—
289C22×Q8C22×D311yescharacteristic
290C22×Q8C22×D311yescharacteristic
291C22×D4—31no—
292C22×D4—61no—
293C23×C4C22×D311yescharacteristic
294C2×Q16—22no—
295C2×Q16—22no—
296C2×Q16—41no—
297C2×SD16—62no—
298C2×SD16—62no—
299C2×SD16—121no—
300C22×C8—31no—
301C22×C8—31no—
302C22×C8C22×D311yescharacteristic
303C22×C8C22×D311yescharacteristic
304#124—62no—
305#124—62no—
306C22⋊Q8—22no—
307C22⋊Q8—22no—
308C22⋊Q8—41no—
309C22⋊Q8—41no—
310#121—121no—
311#121—121no—
312C4×Q8—62no—
313C4×Q8—62no—
314#117—62no—
315#117—62no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4—61no—
326C2×C4⋊C4—61no—
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C4⋊C4C22×D311yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—31no—
332C2×C22⋊C4—31no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C22⋊C4—61no—
339C2×C42—31no—
340C2×C42—31no—
341C2×C42—61no—
342C2×C42—61no—
343#107—121no—
344#107—121no—
345C4⋊C8—34no—
346C4⋊C8—62no—
347#103—22no—
348#103—41no—
349#103—62no—
350#103—121no—
351#103—121no—
352#102—62no—
353#102—62no—
354#102—121no—
355C22⋊C8—22no—
356C22⋊C8—62no—
357#97—62no—
358C4×C8—62no—
359#95—61no—
360#95—61no—
361#95—61no—
362#95—61no—
363#95—61no—
364#95—61no—
365#95—61no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3—21no—
372C23×C3—21no—
373C23×C3C22×D411yescharacteristic
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—41no—
381C3×Q8—22no—
382C3×Q8—22no—
383C3×Q8—41no—
384C3×Q8—41no—
385C3×Q8—41no—
386C3×Q8—41no—
387C3×Q8—41no—
388C3×Q8—41no—
389C2×C3×C4—21no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
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×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4C22×D411yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4C2×#4111yescharacteristic
411C2×C3×C4C2×#4111yescharacteristic
412C2×C3×C4H(5,𝔽2)11yescharacteristic
413C2×C3×C4H(5,𝔽2)11yescharacteristic
414#67—84no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—21no—
424C2×Dic3—21no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3#12112yes—
435C2×Dic3#12112yes—
436D12—41no—
437D12—41no—
438D12—41no—
439D12—41no—
440C3×C8—22no—
441C3×C8—41no—
442C3⋊C8—22no—
443C3⋊C8—41no—
444C24—61no—
445C24D3×D411yescharacteristic
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8#23712yes—
458C2×D4—61no—
459C2×D4—61no—
460C2×D4—61no—
461C2×D4—61no—
462C2×D4—61no—
463C2×D4—61no—
464C2×D4—122no—
465C2×D4—241no—
466C2×D4—241no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—31no—
478C22×C4—31no—
479C22×C4—31no—
480C22×C4—31no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4D3×D411yescharacteristic
485C22×C4C23×D311yescharacteristic
486Q16—44no—
487SD16—124no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—32no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8D3×D412yes—
500C4⋊C4—24no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—42no—
505C4⋊C4—62no—
506C4⋊C4—62no—
507C4⋊C4—61no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—122no—
514C4⋊C4—241no—
515C4⋊C4—241no—
516C4⋊C4—241no—
517C4⋊C4#23712yes—
518C22⋊C4—24no—
519C22⋊C4—42no—
520C22⋊C4—62no—
521C22⋊C4—62no—
522C22⋊C4—122no—
523C22⋊C4—122no—
524C22⋊C4—122no—
525C22⋊C4—241no—
526C22⋊C4—241no—
527C22⋊C4—241no—
528C42—32no—
529C42—32no—
530C42—61no—
531C42—61no—
532C42—122no—
533C42—241no—
534C22×C3—41no—
535C22×C3—41no—
536C22×C3—41no—
537C22×C3—41no—
538C22×C3—41no—
539C22×C3—41no—
540C22×C3—41no—
541C22×C3C2×#12111yescharacteristic
542C22×C3#52511yescharacteristic
543C22×C3#53511yescharacteristic
544C22×C3#57611yescharacteristic
545C22×C3#57711yescharacteristic
546C22×C3#57811yescharacteristic
547C22×C3#57811yescharacteristic
548C2×D3—81no—
549C2×D3—81no—
550C2×D3—81no—
551C2×D3—81no—
552C2×D3—81no—
553C2×D3—81no—
554C2×D3—81no—
555C3×C4—42no—
556C3×C4—41no—
557C3×C4—42no—
558C3×C4—81no—
559C3×C4—81no—
560C3×C4#54611yescharacteristic
561C3×C4#55011yescharacteristic
562C3×C4#55311yescharacteristic
563C3×C4#55511yescharacteristic
564Dic3—22no—
565Dic3—22no—
566Dic3—41no—
567Dic3—41no—
568Dic3—81no—
569Dic3—81no—
570Dic3—81no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—121no—
579C23—121no—
580C23—121no—
581C23—121no—
582C23—121no—
583C23—121no—
584C23—121no—
585C23C2×D3×D411yescharacteristic
586Q8—22no—
587Q8—22no—
588Q8—41no—
589Q8—41no—
590Q8—41no—
591Q8—41no—
592Q8—41no—
593Q8—41no—
594D4—121no—
595D4—121no—
596D4—121no—
597D4—121no—
598D4—244no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
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—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—61no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—61no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4C2×D3×D411yescharacteristic
640C2×C4C2×#23711yescharacteristic
641C2×C4C2×#23711yescharacteristic
642C2×C4#100411yescharacteristic
643C2×C4#100411yescharacteristic
644C2×C4D3×#4111yescharacteristic
645C8—22no—
646C8—41no—
647C8—62no—
648C8—121no—
649C2×C3—81no—
650C2×C3#241411yescharacteristic
651C2×C3#306111yescharacteristic
652C2×C3#306311yescharacteristic
653C2×C3#312111yescharacteristic
654C2×C3#312411yescharacteristic
655C2×C3#322311yescharacteristic
656C2×C3#322411yescharacteristic
657D3—161no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—241no—
666C22—241no—
667C22—241no—
668C22—241no—
669C22—241no—
670C22—241no—
671C22—241no—
672C22#558611yescharacteristic
673C22#559211yescharacteristic
674C22D3×#12111yescharacteristic
675C22#576611yescharacteristic
676C22#576911yescharacteristic
677C22#577811yescharacteristic
678C22#577911yescharacteristic
679C4—42no—
680C4—41no—
681C4—42no—
682C4—62no—
683C4—62no—
684C4—81no—
685C4—81no—
686C4—121no—
687C4—121no—
688C4—241no—
689C4—241no—
690C4—241no—
691C4#559111yescharacteristic
692C4#560111yescharacteristic
693C4#561611yescharacteristic
694C4#571811yescharacteristic
695C3#2466511yescharacteristic, complemented, Sylow
696C2—81no—
697C2—481no—
698C2#7667011yescharacteristic
699C2#8017611yescharacteristic
700C2#8018011yescharacteristic
701C2#8021811yescharacteristic
702C2#8026311yescharacteristic
703C2#8076111yescharacteristic
704C2#8090711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2001C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#2001D3
hypocenterC3#24665
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2001

Complements

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

NH ≅ G/Ntype
1. C2×#5185C2semidirect
3. #82744C2semidirect
4. C2×#4720C2semidirect
5. C3×#2001C2semidirect
6. #72066C2semidirect
8. #71695C2semidirect
9. #70260C2semidirect
10. #70255C2semidirect
11. #70230C2semidirect
12. #70221C2semidirect
14. #68853C2semidirect
15. #68795C2semidirect
68. #2001D3semidirect
150. Dic3⋊C8D4semidirect
695. C3#24665semidirect

Maximal subgroups

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

subgroupindex
C2×#51852
#828212
#827442
C2×#47202
C3×#20012
#720662
#716992
#716952
#702602
#702552
#702302
#702212
#699342
#688532
#687952
#246653