Finite Group Databasebeta

← #11166395

Subgroups — #11166395

Order 768.

2054 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#555C211yesmaximal, characteristic
2C2×#5519C211yesmaximal, characteristic, complemented
3#83241C212yesmaximal, complemented
4#82992C212yesmaximal, complemented
5#82985C214yesmaximal, complemented
6#82978C212yesmaximal, complemented
7#82685C214yesmaximal, complemented
8#82678C211yesmaximal, characteristic, complemented
9#82665C211yesmaximal, characteristic, complemented
10#82661C212yesmaximal, complemented
11#82373C211yesmaximal, characteristic, complemented
12#82362C212yesmaximal, complemented
13#82346C211yesmaximal, characteristic, complemented
14#82262C212yesmaximal, complemented
15#82261C212yesmaximal, complemented
16#82248C212yesmaximal, complemented
17#82239C211yesmaximal, characteristic, complemented
18#42965—31nomaximal, Sylow
19C3×#555—28no—
20C2×C3×C4⋊Q8C2211yescharacteristic
21C2×C3×#124C2212yes—
22C2×C3×#124C2212yes—
23C2×C3×#121C2211yescharacteristic
24C2×C3×#121C2211yescharacteristic
25C2×C3×#121C2212yes—
26C2×C3×C4×D4C2211yescharacteristic
27C2×C3×C4×D4C2211yescharacteristic
28C2×C3×#117C2211yescharacteristic
29C2×C3×#117C2212yes—
30C2×C3×#117C2211yescharacteristic
31C2×#935C2212yes—
32C2×#926C2211yescharacteristic
33C22×C4⋊Dic3C2211yescharacteristic
34C2×#924C2214yes—
35#5519—28no—
36C2×#890C2212yes—
37C2×#888C2212yes—
38C2×#878C2214yes—
39C2×#870C2212yes—
40C2×#869C2212yes—
41#5230C2214yes—
42#5229C2214yes—
43#5222C2214yes—
44#5222C2214yes—
45#5222C2212yes—
46#5221C2214yes—
47#5221C2212yes—
48#5221C2212yes—
49#5220C2214yes—
50#5212C2211yescharacteristic
51#5212C2212yes—
52#5210C2214yes—
53C4×#217C2211yescharacteristic
54C4×#217C2212yes—
55#4986C2211yescharacteristic
56#4984C2212yes—
57#4981C2214yes—
58#4981C2212yes—
59#4981C2211yescharacteristic
60#4977C2212yes—
61#4976C2214yes—
62Dic3×C4⋊C4C2212yes—
63#4972C2212yes—
64#4953C2211yescharacteristic
65#4953C2212yes—
66#4953C2212yes—
67#4953C2212yes—
68#4953C2214yes—
69#4953C2212yes—
70#4952C2212yes—
71#4951C2214yes—
72#4949C2211yescharacteristic
73#4949C2212yes—
74#4948C2214yes—
75#4948C2214yes—
76#4947C2214yes—
77#4947C2214yes—
78#4945C2214yes—
79#4943C2212yes—
80#4936C2212yes—
81#4933C2214yes—
82#4932C2212yes—
83C4×Dic3⋊C4C2212yes—
84#4929C2214yes—
85#4928C2212yes—
86C2×#555D311yescharacteristic
87C2×#544—31no—
88#2820—32no—
89#2700—34no—
90#2681—34no—
91#2640—32no—
92#2624—31no—
93#2621—32no—
94#2611—32no—
95#2550—32no—
96#2504—31no—
97#2490—32no—
98#2486—31no—
99#2452—32no—
100#2439—32no—
101#2395—31no—
102#2346—31no—
103C22×C3×Q8C2312yes—
104C22×C3×D4C2311yescharacteristic
105C22×C3×D4C2311yescharacteristic
106C22×C3×D4C2312yes—
107C22×C3×D4C2312yes—
108C23×C3×C4C2311yescharacteristic
109C23×C3×C4C2312yes—
110C23×C3×C4C2311yescharacteristic
111C23×Dic3C2312yes—
112C22×Dic6C2314yes—
113C3×C4⋊Q8—24no—
114C3×#124—28no—
115C3×#124—28no—
116C3×#121—24no—
117C3×#121—24no—
118C3×#121—28no—
119C3×C4×D4—24no—
120C3×C4×D4—24no—
121C3×#117—24no—
122C3×#117—28no—
123C3×#117D418yes—
124C2×C3×C4⋊C4C2311yescharacteristic
125C2×C3×C4⋊C4C2312yes—
126C2×C3×C4⋊C4C2311yescharacteristic
127C2×C3×C4⋊C4C2311yescharacteristic
128C2×C3×C4⋊C4C2311yescharacteristic
129C2×C3×C22⋊C4C2311yescharacteristic
130C2×C3×C22⋊C4C2312yes—
131C2×C3×C22⋊C4C2312yes—
132C2×C3×C22⋊C4C2312yes—
133C2×C3×C22⋊C4C2312yes—
134C2×C3×C22⋊C4C2311yescharacteristic
135C2×C3×C22⋊C4C2312yes—
136C2×C3×C42C2311yescharacteristic
137C2×C3×C42C2312yes—
138C2×C3×C42C2312yes—
139C2×#217C2312yes—
140C2×#217C2312yes—
141C2×#217C2314yes—
142C2×#217C2314yes—
143C2×#217C2312yes—
144C2×#217C2314yes—
145C2×#217C2314yes—
146C2×#217C2312yes—
147C2×#217C2314yes—
148C2×#217C2312yes—
149#935—44no—
150#926—24no—
151C2×C4⋊Dic3—24no—
152C2×C4⋊Dic3C2312yes—
153C2×C4⋊Dic3C2312yes—
154C2×C4⋊Dic3C2311yescharacteristic
155C2×C4⋊Dic3C2311yescharacteristic
156C2×C4⋊Dic3C2312yes—
157C2×C4⋊Dic3C2314yes—
158C2×C4⋊Dic3C2314yes—
159#924—48no—
160C2×Dic3⋊C4C2314yes—
161C2×Dic3⋊C4C2312yes—
162C2×Dic3⋊C4C2314yes—
163C2×Dic3⋊C4C2314yes—
164C2×Dic3⋊C4C2312yes—
165C2×Dic3⋊C4C2314yes—
166C2×C4×Dic3C2312yes—
167C2×C4×Dic3C2312yes—
168C2×C4×Dic3C2314yes—
169#890—28no—
170#888—28no—
171#878—216no—
172#870—28no—
173#869—28no—
174C3⋊#95C2314yes—
175C3⋊#95C2312yes—
176C3⋊#95C2314yes—
177C3⋊#95C2314yes—
178C3⋊#95C2314yes—
179C3⋊#95C2312yes—
180C3⋊#95C2312yes—
181C3⋊#95C2311yescharacteristic
182C3⋊#95C2314yes—
183C3⋊#95C2312yes—
184C3⋊#95C2312yes—
185C3⋊#95C2314yes—
186C3⋊#95C2311yescharacteristic
187C3⋊#95C2314yes—
188#555—28no—
189#544—68no—
190C2×C4⋊Q8—32no—
191C2×C4⋊Q8—32no—
192C2×C4⋊Q8C2×D311yescharacteristic
193C2×#125—32no—
194C2×#125—32no—
195C2×#124C2×D312yes—
196C2×#124C2×D312yes—
197C2×#123—32no—
198C2×C22⋊Q8—34no—
199C2×C22⋊Q8—34no—
200C2×#121C2×D311yescharacteristic
201C2×#121C2×D312yes—
202C2×#121C2×D311yescharacteristic
203C2×C4×D4C2×D311yescharacteristic
204C2×C4×D4C2×D311yescharacteristic
205C2×#117—31no—
206C2×#117C2×D312yes—
207C2×#117C2×D311yescharacteristic
208C2×#117C2×D311yescharacteristic
209C22×C4⋊C4—31no—
210#400—31no—
211#399—31no—
212#399—32no—
213#398—32no—
214#398—31no—
215#398—34no—
216#397—32no—
217#397—32no—
218#397—32no—
219#397—32no—
220#397—34no—
221#397—31no—
222#397—32no—
223#396—34no—
224#396—34no—
225#395—34no—
226#394—34no—
227#394—34no—
228#394—32no—
229#393—34no—
230#393—34no—
231#391—34no—
232#391—34no—
233#390—32no—
234#390—34no—
235#390—32no—
236#389—32no—
237#389—34no—
238#389—32no—
239#389—32no—
240#388—34no—
241#388—34no—
242#387—34no—
243#386—34no—
244#385—34no—
245#382—32no—
246#382—32no—
247#381—32no—
248#380—31no—
249#380—32no—
250C4×C4⋊C4—32no—
251C4×C4⋊C4—32no—
252C4×C22⋊C4—31no—
253C4×C22⋊C4—32no—
254#376—32no—
255C24×C3C2412yes—
256C24×C3C2411yescharacteristic
257C24×C3C2411yescharacteristic
258C2×C3×Q8—24no—
259C2×C3×Q8—44no—
260C2×C3×D4—22no—
261C2×C3×D4—24no—
262C2×C3×D4—22no—
263C2×C3×D4—24no—
264C2×C3×D4—22no—
265C2×C3×D4—24no—
266C2×C3×D4—41no—
267C2×C3×D4—44no—
268C2×C3×D4—44no—
269C22×C3×C4—24no—
270C22×C3×C4—24no—
271C22×C3×C4—22no—
272C22×C3×C4—24no—
273C22×C3×C4—24no—
274C22×C3×C4—42no—
275C22×C3×C4C2×D414yes—
276C22×C3×C4C2411yescharacteristic
277C22×C3×C4C2411yescharacteristic
278C22×C3×C4C2412yes—
279C22×C3×C4C2411yescharacteristic
280C22×C3×C4C2412yes—
281C22×C3×C4C2412yes—
282C22×C3×C4C2411yescharacteristic
283C22×C3×C4C2411yescharacteristic
284C22×Dic3—24no—
285C22×Dic3—42no—
286C22×Dic3—42no—
287C22×Dic3C2412yes—
288C22×Dic3C2412yes—
289C22×Dic3C2414yes—
290C22×Dic3C2412yes—
291C22×Dic3C2414yes—
292C22×Dic3C2412yes—
293C2×Dic6—28no—
294C2×Dic6—44no—
295C2×Dic6—44no—
296C3×C4⋊C4—22no—
297C3×C4⋊C4—24no—
298C3×C4⋊C4—22no—
299C3×C4⋊C4C2×D414yes—
300C3×C4⋊C4C2×D414yes—
301C3×C22⋊C4—22no—
302C3×C22⋊C4—24no—
303C3×C22⋊C4—24no—
304C3×C22⋊C4—24no—
305C3×C22⋊C4—24no—
306C3×C22⋊C4—22no—
307C3×C22⋊C4C2×D418yes—
308C3×C42—24no—
309C3×C42—22no—
310C3×C42C2×D418yes—
311#217—24no—
312#217—44no—
313#217—44no—
314#217—42no—
315#217—44no—
316#217—42no—
317#217—44no—
318#217—42no—
319#217—42no—
320#217—44no—
321C4⋊Dic3—22no—
322C4⋊Dic3—24no—
323C4⋊Dic3—28no—
324C4⋊Dic3—22no—
325C4⋊Dic3—24no—
326C4⋊Dic3—24no—
327C4⋊Dic3—44no—
328Dic3⋊C4—28no—
329Dic3⋊C4—28no—
330Dic3⋊C4—24no—
331Dic3⋊C4—28no—
332Dic3⋊C4—44no—
333Dic3⋊C4—42no—
334C4×Dic3—24no—
335C4×Dic3—24no—
336C4×Dic3—44no—
337C22×Q8—34no—
338C22×Q8C22×D312yes—
339C22×D4C22×D311yescharacteristic
340C22×D4C22×D311yescharacteristic
341C22×D4C22×D312yes—
342C22×D4C22×D312yes—
343C23×C4—32no—
344C23×C4C22×D311yescharacteristic
345C23×C4C22×D312yes—
346C23×C4C22×D311yescharacteristic
347C4⋊Q8—24no—
348C4⋊Q8—68no—
349C4⋊Q8—68no—
350#125—68no—
351#125—68no—
352#124—28no—
353#124—28no—
354#123—124no—
355C22⋊Q8—616no—
356C22⋊Q8—128no—
357#121—24no—
358#121—24no—
359#121—28no—
360C4×D4—24no—
361C4×D4—24no—
362#117—24no—
363#117—28no—
364#117—64no—
365#117#6718yes—
366C2×C4⋊C4—32no—
367C2×C4⋊C4—32no—
368C2×C4⋊C4—31no—
369C2×C4⋊C4—31no—
370C2×C4⋊C4—32no—
371C2×C4⋊C4—34no—
372C2×C4⋊C4—34no—
373C2×C4⋊C4—34no—
374C2×C4⋊C4—34no—
375C2×C4⋊C4—32no—
376C2×C4⋊C4—34no—
377C2×C4⋊C4—32no—
378C2×C4⋊C4—34no—
379C2×C4⋊C4—64no—
380C2×C4⋊C4C22×D311yescharacteristic
381C2×C4⋊C4C22×D311yescharacteristic
382C2×C4⋊C4C22×D312yes—
383C2×C4⋊C4C22×D311yescharacteristic
384C2×C4⋊C4C22×D311yescharacteristic
385C2×C22⋊C4—32no—
386C2×C22⋊C4—34no—
387C2×C22⋊C4—34no—
388C2×C22⋊C4—32no—
389C2×C22⋊C4—34no—
390C2×C22⋊C4—32no—
391C2×C22⋊C4—32no—
392C2×C22⋊C4—32no—
393C2×C22⋊C4—34no—
394C2×C22⋊C4—34no—
395C2×C22⋊C4C22×D312yes—
396C2×C22⋊C4C22×D312yes—
397C2×C22⋊C4C22×D312yes—
398C2×C22⋊C4C22×D311yescharacteristic
399C2×C22⋊C4C22×D312yes—
400C2×C22⋊C4C22×D311yescharacteristic
401C2×C22⋊C4C22×D312yes—
402C2×C42—32no—
403C2×C42—32no—
404C2×C42—34no—
405C2×C42C22×D312yes—
406C2×C42C22×D311yescharacteristic
407C2×C42C22×D312yes—
408#95—31no—
409#95—32no—
410#95—34no—
411#95—31no—
412#95—32no—
413#95—32no—
414#95—34no—
415#95—32no—
416#95—32no—
417#95—34no—
418#95—34no—
419#95—34no—
420#95—34no—
421#95—34no—
422C23×C3—21no—
423C23×C3—22no—
424C23×C3—22no—
425C23×C3—21no—
426C23×C3—21no—
427C23×C3—22no—
428C23×C3—22no—
429C23×C3—21no—
430C23×C3—22no—
431C23×C3—24no—
432C23×C3—22no—
433C23×C3—21no—
434C23×C3—22no—
435C23×C3—24no—
436C23×C3C22×D412yes—
437C23×C3C2511yescharacteristic
438C3×Q8—84no—
439C3×D4—44no—
440C3×D4—44no—
441C3×D4—84no—
442C3×D4—84no—
443C2×C3×C4—21no—
444C2×C3×C4—22no—
445C2×C3×C4—22no—
446C2×C3×C4—24no—
447C2×C3×C4—22no—
448C2×C3×C4—21no—
449C2×C3×C4—28no—
450C2×C3×C4—21no—
451C2×C3×C4—21no—
452C2×C3×C4—22no—
453C2×C3×C4—22no—
454C2×C3×C4—24no—
455C2×C3×C4—22no—
456C2×C3×C4—48no—
457C2×C3×C4—44no—
458C2×C3×C4C22×D414yes—
459C2×C3×C4C22×D414yes—
460C2×C3×C4C22×D412yes—
461C2×C3×C4C22×D412yes—
462C2×C3×C4H(5,𝔽2)12yes—
463C2×C3×C42−1+414yes—
464C2×Dic3—22no—
465C2×Dic3—22no—
466C2×Dic3—24no—
467C2×Dic3—22no—
468C2×Dic3—24no—
469C2×Dic3—22no—
470C2×Dic3—24no—
471C2×Dic3—22no—
472C2×Dic3—22no—
473C2×Dic3—24no—
474C2×Dic3—22no—
475C2×Dic3—24no—
476C2×Dic3—22no—
477C2×Dic3—22no—
478C2×Dic3—22no—
479C2×Dic3—22no—
480C2×Dic3—22no—
481C2×Dic3—24no—
482C2×Dic3—84no—
483Dic6—88no—
484C24C23×D312yes—
485C24C23×D311yescharacteristic
486C24C23×D311yescharacteristic
487C2×Q8—24no—
488C2×Q8—44no—
489C2×Q8—68no—
490C2×Q8—124no—
491C2×Q8—124no—
492C2×D4—22no—
493C2×D4—24no—
494C2×D4—22no—
495C2×D4—24no—
496C2×D4—22no—
497C2×D4—24no—
498C2×D4—41no—
499C2×D4—44no—
500C2×D4—44no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—24no—
504C22×C4—24no—
505C22×C4—22no—
506C22×C4—32no—
507C22×C4—34no—
508C22×C4—32no—
509C22×C4—32no—
510C22×C4—34no—
511C22×C4—32no—
512C22×C4—42no—
513C22×C4—64no—
514C22×C4—122no—
515C22×C4—122no—
516C22×C4C2×#6714yes—
517C22×C4C23×D311yescharacteristic
518C22×C4C23×D311yescharacteristic
519C22×C4C23×D312yes—
520C22×C4C23×D311yescharacteristic
521C22×C4C23×D311yescharacteristic
522C22×C4C23×D312yes—
523C22×C4C23×D312yes—
524C22×C4C23×D311yescharacteristic
525C4⋊C4—22no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—64no—
529C4⋊C4—68no—
530C4⋊C4—64no—
531C4⋊C4—64no—
532C4⋊C4—64no—
533C4⋊C4—62no—
534C4⋊C4—62no—
535C4⋊C4—68no—
536C4⋊C4—68no—
537C4⋊C4—68no—
538C4⋊C4—124no—
539C4⋊C4—124no—
540C4⋊C4—122no—
541C4⋊C4C2×#6714yes—
542C4⋊C4C2×#6714yes—
543C22⋊C4—22no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4—22no—
549C22⋊C4—64no—
550C22⋊C4—124no—
551C22⋊C4—124no—
552C22⋊C4—122no—
553C22⋊C4—124no—
554C22⋊C4—124no—
555C22⋊C4—122no—
556C22⋊C4—122no—
557C22⋊C4—124no—
558C22⋊C4—122no—
559C22⋊C4C2×#6718yes—
560C42—24no—
561C42—22no—
562C42—64no—
563C42—64no—
564C42—124no—
565C42C2×#6718yes—
566C22×C3—22no—
567C22×C3—22no—
568C22×C3—22no—
569C22×C3—42no—
570C22×C3—42no—
571C22×C3—41no—
572C22×C3—42no—
573C22×C3—42no—
574C22×C3—41no—
575C22×C3—41no—
576C22×C3—42no—
577C22×C3—44no—
578C22×C3—42no—
579C22×C3—42no—
580C22×C3—44no—
581C22×C3C23×D411yescharacteristic
582C22×C3C2×H(5,𝔽2)11yescharacteristic
583C22×C3C2×2−1+412yes—
584C22×C3#58511yescharacteristic
585C22×C3#58512yes—
586C3×C4—22no—
587C3×C4—24no—
588C3×C4—22no—
589C3×C4—24no—
590C3×C4—41no—
591C3×C4—42no—
592C3×C4—41no—
593C3×C4#53614yes—
594Dic3—42no—
595Dic3—44no—
596Dic3—42no—
597Dic3—42no—
598Dic3—44no—
599Dic3—42no—
600C23—21no—
601C23—22no—
602C23—21no—
603C23—22no—
604C23—22no—
605C23—21no—
606C23—22no—
607C23—24no—
608C23—22no—
609C23—21no—
610C23—22no—
611C23—24no—
612C23—21no—
613C23—22no—
614C23C22×#6712yes—
615C23C24×D311yescharacteristic
616Q8—84no—
617Q8—248no—
618D4—44no—
619D4—44no—
620D4—84no—
621D4—84no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—21no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—21no—
628C2×C4—28no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—22no—
635C2×C4—48no—
636C2×C4—44no—
637C2×C4—64no—
638C2×C4—62no—
639C2×C4—64no—
640C2×C4—62no—
641C2×C4—62no—
642C2×C4—62no—
643C2×C4—64no—
644C2×C4—62no—
645C2×C4—64no—
646C2×C4—62no—
647C2×C4—62no—
648C2×C4—64no—
649C2×C4—62no—
650C2×C4—62no—
651C2×C4—62no—
652C2×C4—62no—
653C2×C4—62no—
654C2×C4—64no—
655C2×C4—244no—
656C2×C4#100912yes—
657C2×C4#101014yes—
658C2×C4C22×#6714yes—
659C2×C4C22×#6712yes—
660C2×C4C22×#6714yes—
661C2×C4C22×#6712yes—
662C2×C3—42no—
663C2×C3—81no—
664C2×C3—82no—
665C2×C3C2×#53611yescharacteristic
666C2×C3#348811yescharacteristic
667C2×C3#350011yescharacteristic
668C2×C3#354712yes—
669C2×C3#356512yes—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—44no—
677C22—44no—
678C22—42no—
679C22—41no—
680C22—42no—
681C22—42no—
682C22—41no—
683C22—41no—
684C22—42no—
685C22C2×#100911yescharacteristic
686C22C2×#101012yes—
687C22#596411yescharacteristic
688C22#596612yes—
689C22C23×#6711yescharacteristic
690C4—22no—
691C4—22no—
692C4—24no—
693C4—24no—
694C4—41no—
695C4—42no—
696C4—41no—
697C4—122no—
698C4—124no—
699C4—122no—
700C4—124no—
701C4—122no—
702C4—122no—
703C4#582514yes—
704C3#4296511yescharacteristic, complemented, Sylow
705C2—42no—
706C2—81no—
707C2—82no—
708C2#8697012yes—
709C2#8697712yes—
710C2C2×#582511yescharacteristic
711C2#8727511yescharacteristic
712C2#8728311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
Fitting, prime radicalC2×C3×#555C2
hypercenterC2×#555D3
hypocenterC3#42965
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
2. C2×#5519C2semidirect
3. #83241C2semidirect
4. #82992C2semidirect
5. #82985C2semidirect
6. #82978C2semidirect
7. #82685C2semidirect
8. #82678C2semidirect
9. #82665C2semidirect
10. #82661C2semidirect
11. #82373C2semidirect
12. #82362C2semidirect
13. #82346C2semidirect
14. #82262C2semidirect
15. #82261C2semidirect
16. #82248C2semidirect
17. #82239C2semidirect
704. C3#42965semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5552
C2×#55192
#832412
#829922
#829852
#829782
#826852
#826782
#826652
#826612
#823732
#823622
#823462
#822622
#822612
#822482
#822392
#429653