Finite Group Databasebeta

← #10835931

Subgroups — #10835931

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
1C3×#2323C211yesmaximal, characteristic, complemented
2C2×#5028C211yesmaximal, characteristic, complemented
3#82287C211yesmaximal, characteristic
4C2×#5000C211yesmaximal, characteristic, complemented
5#71700C211yesmaximal, characteristic, complemented
6#71699C211yesmaximal, characteristic
7#71626C211yesmaximal, characteristic, complemented
8#69679C211yesmaximal, characteristic, complemented
9#69677C211yesmaximal, characteristic, complemented
10#69616C211yesmaximal, characteristic, complemented
11#69615C211yesmaximal, characteristic, complemented
12#69527C211yesmaximal, characteristic, complemented
13#69523C211yesmaximal, characteristic
14#69488C211yesmaximal, characteristic, complemented
15#69466C211yesmaximal, characteristic, complemented
16#24665—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C22×#215C2211yescharacteristic
22C2×#930C2211yescharacteristic
23C2×#924C2211yescharacteristic
24C2×#875C2211yescharacteristic
25C3×#391C2211yescharacteristic
26C3×#389—21no—
27C3×#389C2211yescharacteristic
28C3×#388—21no—
29C3×#388—21no—
30C3×#387—21no—
31C3×#380C2211yescharacteristic
32C3×C4×C4⋊C4C2211yescharacteristic
33C2×#835C2211yescharacteristic
34C2×#835C2211yescharacteristic
35C2×#830C2211yescharacteristic
36C2×#830C2211yescharacteristic
37#5028—24no—
38#5000—24no—
39#4994—21no—
40#4994—21no—
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#809C2211yescharacteristic
45C2×#809C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#807C2211yescharacteristic
48C2×#807C2211yescharacteristic
49#4961—21no—
50#4961—21no—
51#4939C2211yescharacteristic
52#4935C2211yescharacteristic
53#4929C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C2×C4⋊(C3⋊C8)C2211yescharacteristic
56C2×#803C2211yescharacteristic
57C2×C4×C3⋊C8C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#464—31no—
63C2×#433—31no—
64#2620—31no—
65#2323D311yescharacteristic, complemented
66#2036—31no—
67#2033—31no—
68#2001—31no—
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
78C23×C3×C4C2311yescharacteristic
79C22×#67—21no—
80C22×D12C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8—22no—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C3×#117—22no—
87C3×#117—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4D411yescharacteristic
99C2×C3×C22⋊C4D411yescharacteristic
100C2×C3×C42—21no—
101C2×C3×C42—21no—
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×#217—21no—
105C2×#216—22no—
106C2×#216—22no—
107C2×#216—41no—
108C2×#215—22no—
109C2×#215—22no—
110C2×#215—41no—
111#930—41no—
112#930—41no—
113C2×(C2×D3)⋊C4—21no—
114C2×(C2×D3)⋊C4—21no—
115C2×(C2×D3)⋊C4—21no—
116C2×(C2×D3)⋊C4—21no—
117C2×(C2×D3)⋊C4—21no—
118C2×C4⋊Dic3—21no—
119C2×C4⋊Dic3—21no—
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4⋊Dic3C2311yescharacteristic
122#924—41no—
123#924—41no—
124C2×Dic3⋊C4—21no—
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128C22×C3⋊C8C2311yescharacteristic
129#875—22no—
130#875—22no—
131C3×#95—21no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136#835—41no—
137#835—41no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140#830—22no—
141#830—22no—
142#810—22no—
143#810—22no—
144#810—41no—
145#809—22no—
146#809—22no—
147#809—41no—
148#807—41no—
149#807—41no—
150C4⋊(C3⋊C8)—22no—
151C4⋊(C3⋊C8)D414yescomplemented
152#803—22no—
153C4×C3⋊C8—22no—
154C22×Q16—31no—
155C22×SD16—31no—
156C2×#124—31no—
157C2×C22⋊Q8—31no—
158C2×C22⋊Q8C2×D311yescharacteristic
159C2×#121—31no—
160C2×C4×Q8C2×D311yescharacteristic
161C2×#117C2×D311yescharacteristic
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×#103—31no—
172C2×#103—31no—
173C2×#102—31no—
174C2×#102—31no—
175C2×#102—31no—
176C2×C22⋊C8—31no—
177C2×C22⋊C8—31no—
178C2×#97—31no—
179C2×C4×C8—31no—
180#396—61no—
181#396—61no—
182#394—61no—
183#394—61no—
184#391—31no—
185#391C2×D311yescharacteristic
186#390—31no—
187#389—21no—
188#389C2×D311yescharacteristic
189#388—21no—
190#388—21no—
191#387—21no—
192#384—31no—
193#380C2×D311yescharacteristic
194C4×C4⋊C4C2×D311yescharacteristic
195#340—31no—
196#340—31no—
197#340—31no—
198#340—31no—
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#4112yes—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2411yescharacteristic
223C2×#67—42no—
224C2×#67—81no—
225C2×#67—81no—
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—21no—
241C3×C4⋊C4—24no—
242C3×C4⋊C4—22no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—42no—
250C3×C4⋊C4—81no—
251C3×C4⋊C4#4112yes—
252C3×C22⋊C4—24no—
253C3×C22⋊C4—22no—
254C3×C22⋊C4—22no—
255C3×C22⋊C4—81no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42—42no—
259C3×C42—81no—
260C3×C42C2×D412yes—
261C3×C42#4112yes—
262#217—42no—
263#216—44no—
264#215—44no—
265(C2×D3)⋊C4—42no—
266(C2×D3)⋊C4—42no—
267(C2×D3)⋊C4—42no—
268(C2×D3)⋊C4—81no—
269(C2×D3)⋊C4—81no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—81no—
275C4⋊Dic3—81no—
276Dic3⋊C4—42no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8C2×D412yes—
288C2×C3⋊C8C2×D412yes—
289C22×Q8—31no—
290C22×Q8C22×D311yescharacteristic
291C22×D4—31no—
292C22×D4—61no—
293C23×C4C22×D311yescharacteristic
294C2×Q16—62no—
295C2×Q16—62no—
296C2×Q16—121no—
297C2×SD16—62no—
298C2×SD16—62no—
299C2×SD16—121no—
300C22×C8—31no—
301C22×C8—31no—
302C22×C8—31no—
303C22×C8—31no—
304#124—62no—
305#124—62no—
306C22⋊Q8—22no—
307C22⋊Q8—22no—
308C22⋊Q8—121no—
309C22⋊Q8—121no—
310#121—121no—
311#121—121no—
312C4×Q8—22no—
313C4×Q8—22no—
314#117—22no—
315#117—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4#6711yescharacteristic
325C2×C4⋊C4#6711yescharacteristic
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C4⋊C4C22×D311yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4#6711yescharacteristic
338C2×C22⋊C4#6711yescharacteristic
339C2×C42—21no—
340C2×C42—21no—
341C2×C42C22×D311yescharacteristic
342C2×C42C22×D311yescharacteristic
343#107—121no—
344#107—121no—
345C4⋊C8—34no—
346C4⋊C8—62no—
347#103—62no—
348#103—62no—
349#103—121no—
350#103—121no—
351#103—121no—
352#102—62no—
353#102—62no—
354#102—121no—
355C22⋊C8—62no—
356C22⋊C8—62no—
357#97—62no—
358C4×C8—62no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
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—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—21no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
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×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4#12112yes—
415C2×C3×C4#12112yes—
416C2×C3×C4C22×D411yescharacteristic
417C2×C3×C4C2×#4111yescharacteristic
418C2×C3×C4C2×#4111yescharacteristic
419C2×C3×C4C2×#4111yescharacteristic
420C2×C3×C4H(5,𝔽2)11yescharacteristic
421C2×C3×C4H(5,𝔽2)11yescharacteristic
422#67—84no—
423C2×Dic3—41no—
424C2×Dic3—41no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432D12—41no—
433D12—41no—
434D12—41no—
435D12—41no—
436Dic6—41no—
437Dic6—41no—
438Dic6—41no—
439Dic6—41no—
440C3⋊C8—22no—
441C3⋊C8—22no—
442C3⋊C8—41no—
443C3⋊C8—41no—
444C24—61no—
445C24C2×#6711yescharacteristic
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—61no—
452C2×Q8—61no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8#23512yes—
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—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4C2×#6711yescharacteristic
481C22×C4C2×#6711yescharacteristic
482C22×C4C2×#6711yescharacteristic
483C22×C4C2×#6711yescharacteristic
484C22×C4C2×#6711yescharacteristic
485C22×C4C23×D311yescharacteristic
486Q16—124no—
487SD16—124no—
488C2×C8—32no—
489C2×C8—32no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8—61no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—42no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—81no—
514C4⋊C4—122no—
515C4⋊C4—241no—
516C4⋊C4—241no—
517C4⋊C4#23512yes—
518C22⋊C4—24no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—81no—
522C22⋊C4—122no—
523C22⋊C4—122no—
524C22⋊C4—122no—
525C22⋊C4—122no—
526C22⋊C4—241no—
527C22⋊C4—241no—
528C42—21no—
529C42—21no—
530C42—42no—
531C42—81no—
532C42D3×D412yes—
533C42#23712yes—
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—22no—
556C3×C4—22no—
557C3×C4—42no—
558C3×C4—41no—
559C3×C4—42no—
560C3×C4—41no—
561C3×C4—41no—
562C3×C4—81no—
563C3×C4—81no—
564C3×C4#54611yescharacteristic
565C3×C4#55011yescharacteristic
566C3×C4#55311yescharacteristic
567C3×C4#55511yescharacteristic
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—
585C23C22×#6711yescharacteristic
586Q8—22no—
587Q8—22no—
588Q8—41no—
589Q8—41no—
590Q8—121no—
591Q8—121no—
592Q8—121no—
593Q8—121no—
594D4—121no—
595D4—121no—
596D4—121no—
597D4—121no—
598D4—244no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4#93912yes—
638C2×C4#93912yes—
639C2×C4C2×#23511yescharacteristic
640C2×C4C2×#23511yescharacteristic
641C2×C4C2×D3×D411yescharacteristic
642C2×C4C2×#23711yescharacteristic
643C2×C4#100911yescharacteristic
644C2×C4#100911yescharacteristic
645C8—62no—
646C8—62no—
647C8—121no—
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#579111yescharacteristic
673C22C2×#93911yescharacteristic
674C22#583011yescharacteristic
675C22#583511yescharacteristic
676C22#583711yescharacteristic
677C22#583811yescharacteristic
678C22#583811yescharacteristic
679C4—22no—
680C4—22no—
681C4—42no—
682C4—41no—
683C4—42no—
684C4—41no—
685C4—41no—
686C4—81no—
687C4—81no—
688C4—241no—
689C4—241no—
690C4—241no—
691C4#555011yescharacteristic
692C4#555311yescharacteristic
693C4#556311yescharacteristic
694C4#556511yescharacteristic
695C3#2466511yescharacteristic, complemented, Sylow
696C2—81no—
697C2—481no—
698C2#8235011yescharacteristic
699C2#8255211yescharacteristic
700C2#8256911yescharacteristic
701C2#8340011yescharacteristic
702C2#8342111yescharacteristic
703C2#8345211yescharacteristic
704C2#8346311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2323C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#2323D3
hypocenterC3#24665
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
1. C3×#2323C2semidirect
2. C2×#5028C2semidirect
4. C2×#5000C2semidirect
5. #71700C2semidirect
7. #71626C2semidirect
8. #69679C2semidirect
9. #69677C2semidirect
10. #69616C2semidirect
11. #69615C2semidirect
12. #69527C2semidirect
14. #69488C2semidirect
15. #69466C2semidirect
65. #2323D3semidirect
151. C4⋊(C3⋊C8)D4semidirect
695. C3#24665semidirect

Maximal subgroups

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

subgroupindex
C3×#23232
C2×#50282
#822872
C2×#50002
#717002
#716992
#716262
#696792
#696772
#696162
#696152
#695272
#695232
#694882
#694662
#246653