Finite Group Databasebeta

← #11069655

Subgroups — #11069655

Order 768.

2030 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5625C212yesmaximal, complemented
2C3×#2542C211yesmaximal, characteristic
3#83253C211yesmaximal, characteristic, complemented
4#83229C211yesmaximal, characteristic, complemented
5#82786C212yesmaximal, complemented
6#82758C212yesmaximal, complemented
7#82757C212yesmaximal, complemented
8#82379C211yesmaximal, characteristic
9#82319C211yesmaximal, characteristic
10#78070C214yesmaximal, complemented
11#78069C214yesmaximal, complemented
12#76339C212yesmaximal
13#76338C212yesmaximal
14#74337C212yesmaximal, complemented
15#74336C212yesmaximal, complemented
16#73650C211yesmaximal, characteristic, complemented
17C4×#890C211yesmaximal, characteristic, complemented
18#39385—31nomaximal, Sylow
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×C4×D4C2211yescharacteristic
23C2×Q8×Dic3C2212yes—
24C2×#944C2212yes—
25C2×#944C2212yes—
26C2×#926C2212yes—
27C2×#924C2212yes—
28C2×#924C2212yes—
29#5625—216no—
30C2×#890C2214yes—
31C2×#890C2211yescharacteristic
32C2×#889C2214yes—
33C2×#887C2212yes—
34C2×#879C2214yes—
35C2×#879C2214yes—
36C2×#869C2211yescharacteristic
37C3×#399C2214yes—
38C3×#390C2212yes—
39C3×#384C2212yes—
40C3×C4×C4⋊C4C2211yescharacteristic
41#5229C2212yes—
42#5221C2211yescharacteristic
43#5221C2212yes—
44#5221C2211yescharacteristic
45C4×#217C2212yes—
46#4984C2212yes—
47#4984C2211yescharacteristic
48#4984C2211yescharacteristic
49#4984C2211yescharacteristic
50#4983C2214yes—
51#4982C2214yes—
52#4982C2214yes—
53#4982C2214yes—
54#4980C2214yes—
55#4977C2211yescharacteristic
56#4976C2214yes—
57#4975C2212yes—
58#4975C2212yes—
59#4975C2212yes—
60#4975C2212yes—
61#4975C2212yes—
62Dic3×C4⋊C4C2212yes—
63#4973C2212yes—
64#4972C2214yes—
65#4972C2211yescharacteristic
66#4952C2214yes—
67#4952C2214yes—
68#4951C2212yes—
69#4951C2212yes—
70#4949C2214yes—
71#4948C2214yes—
72#4948C2214yes—
73#4945C2214yes—
74#4944C2214yes—
75C4×C4⋊Dic3C2211yescharacteristic
76C4×C4⋊Dic3C2211yescharacteristic
77C4×C4⋊Dic3C2212yes—
78C4×C4⋊Dic3C2212yes—
79C4×Dic3⋊C4C2212yes—
80C42×Dic3C2211yescharacteristic
81#4928C2212yes—
82#4655C2212yes—
83#4654C2212yes—
84#4653C2214yes—
85C2×#541—32no—
86#2868—31no—
87#2721—34no—
88#2721—34no—
89#2621—32no—
90#2552—32no—
91#2552—32no—
92#2542—31no—
93#2542D311yescharacteristic
94#2512—32no—
95#2512—32no—
96#2429—32no—
97#2429—32no—
98#2365—31no—
99#2365—31no—
100C4×#125—31no—
101#2293—31no—
102C22×C3×Q8C2312yes—
103C22×C3×D4C2311yescharacteristic
104C23×C3×C4C2312yes—
105C22×Dic6C2312yes—
106C3×C4⋊Q8—24no—
107C3×C22⋊Q8—28no—
108C3×C22⋊Q8—44no—
109C3×C4×D4—24no—
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C4⋊C4C2314yes—
112C2×C3×C4⋊C4C2312yes—
113C2×C3×C4⋊C4C2312yes—
114C2×C3×C4⋊C4C2311yescharacteristic
115C2×C3×C4⋊C4C2314yes—
116C2×C3×C4⋊C4C2311yescharacteristic
117C2×C3×C22⋊C4C2314yes—
118C2×C3×C22⋊C4C2312yes—
119C2×C3×C22⋊C4C2314yes—
120C2×C3×C42C2312yes—
121C2×C3×C42C2311yescharacteristic
122C2×#217C2312yes—
123C2×#217C2312yes—
124C2×#217C2314yes—
125C2×#217C2312yes—
126C2×#217C2312yes—
127C2×#217C2314yes—
128Q8×Dic3—28no—
129#944—28no—
130#944—44no—
131#926—28no—
132C2×C4⋊Dic3C2312yes—
133C2×C4⋊Dic3C2312yes—
134C2×C4⋊Dic3C2311yescharacteristic
135C2×C4⋊Dic3C2312yes—
136C2×C4⋊Dic3C2311yescharacteristic
137C2×C4⋊Dic3C2311yescharacteristic
138C2×C4⋊Dic3C2314yes—
139C2×C4⋊Dic3C2312yes—
140C2×C4⋊Dic3C2312yes—
141C2×C4⋊Dic3C2312yes—
142C2×C4⋊Dic3C2311yescharacteristic
143#924—28no—
144#924—44no—
145C2×Dic3⋊C4C2314yes—
146C2×Dic3⋊C4C2312yes—
147C2×Dic3⋊C4C2314yes—
148C2×Dic3⋊C4C2314yes—
149C2×Dic3⋊C4C2314yes—
150C2×Dic3⋊C4C2314yes—
151C2×Dic3⋊C4C2314yes—
152C2×Dic3⋊C4C2314yes—
153C2×Dic3⋊C4C2312yes—
154C2×C4×Dic3C2314yes—
155C2×C4×Dic3C2311yescharacteristic
156C2×C4×Dic3C2311yescharacteristic
157C2×C4×Dic3C2311yescharacteristic
158C2×C4×Dic3C2311yescharacteristic
159C2×C4×Dic3C2311yescharacteristic
160C2×C4×Dic3C2312yes—
161C2×C4×Dic3C2312yes—
162C2×C4×Dic3C2314yes—
163C2×C4×Dic3C2311yescharacteristic
164#890—216no—
165#890D418yescomplemented
166#889—216no—
167#887—28no—
168#879—216no—
169#879—216no—
170#869—24no—
171C3×#95C2312yes—
172C3⋊#95C2312yes—
173C3⋊#95C2314yes—
174C3⋊#95C2314yes—
175C3⋊#95C2312yes—
176C3⋊#95C2312yes—
177C3⋊#95C2312yes—
178C3⋊#95C2312yes—
179C3⋊#95C2312yes—
180C3⋊#95C2314yes—
181C3⋊#95C2314yes—
182C3⋊#95C2312yes—
183C3⋊#95C2312yes—
184#541—616no—
185C2×C4⋊Q8—32no—
186C2×C4⋊Q8—32no—
187C2×C4⋊Q8—31no—
188C2×C4⋊Q8C2×D311yescharacteristic
189C2×#126—34no—
190C2×#126—34no—
191C2×#125—31no—
192C2×#125—34no—
193C2×#125—34no—
194C2×C22⋊Q8—32no—
195C2×C22⋊Q8—32no—
196C2×C22⋊Q8C2×D312yes—
197C2×C22⋊Q8C2×D312yes—
198C2×C4×Q8—32no—
199C2×C4×Q8—32no—
200C2×C4×D4C2×D311yescharacteristic
201C2×#117—32no—
202#399—34no—
203#399C2×D314yes—
204#398—34no—
205#398—34no—
206#398—34no—
207#398—34no—
208#397—34no—
209#397—34no—
210#396—34no—
211#396—34no—
212#391—32no—
213#391—32no—
214#390—32no—
215#390—31no—
216#390—31no—
217#390C2×D312yes—
218#389—31no—
219#389—31no—
220#389—32no—
221#389—34no—
222#389—34no—
223#389—31no—
224#389—31no—
225#389—32no—
226#388—34no—
227#388—34no—
228#388—32no—
229#388—32no—
230#387—34no—
231#387—32no—
232#387—32no—
233#387—34no—
234#384—32no—
235#384—32no—
236#384—32no—
237#384—32no—
238#384—32no—
239#384C2×D312yes—
240#382—31no—
241C4×C4⋊C4—32no—
242C4×C4⋊C4—31no—
243C4×C4⋊C4—32no—
244C4×C4⋊C4—32no—
245C4×C4⋊C4—31no—
246C4×C4⋊C4—32no—
247C4×C4⋊C4C2×D311yescharacteristic
248C4×C22⋊C4—32no—
249C43—31no—
250C24×C3C2412yes—
251C2×C3×Q8—24no—
252C2×C3×Q8—24no—
253C2×C3×Q8—24no—
254C2×C3×D4—22no—
255C2×C3×D4—22no—
256C2×C3×D4—22no—
257C22×C3×C4—24no—
258C22×C3×C4—24no—
259C22×C3×C4—24no—
260C22×C3×C4C2412yes—
261C22×C3×C4C2414yes—
262C22×C3×C4C2412yes—
263C22×C3×C4C2412yes—
264C22×C3×C4C2411yescharacteristic
265C22×C3×C4C2411yescharacteristic
266C22×C3×C4C2411yescharacteristic
267C22×Dic3C2412yes—
268C22×Dic3C2412yes—
269C22×Dic3C2412yes—
270C22×Dic3C2414yes—
271C22×Dic3C2411yescharacteristic
272C22×Dic3C2411yescharacteristic
273C22×Dic3C2412yes—
274C22×Dic3C2411yescharacteristic
275C22×Dic3C2411yescharacteristic
276C2×Dic6—24no—
277C2×Dic6—24no—
278C2×Dic6—24no—
279C3×C4⋊C4—28no—
280C3×C4⋊C4—22no—
281C3×C4⋊C4—24no—
282C3×C4⋊C4—24no—
283C3×C4⋊C4—22no—
284C3×C4⋊C4—44no—
285C3×C4⋊C4C2×D414yes—
286C3×C22⋊C4—28no—
287C3×C22⋊C4—24no—
288C3×C22⋊C4—44no—
289C3×C42—24no—
290C3×C42#4114yes—
291#217—28no—
292#217—24no—
293#217—24no—
294#217—24no—
295#217—24no—
296#217—44no—
297C4⋊Dic3—24no—
298C4⋊Dic3—24no—
299C4⋊Dic3—22no—
300C4⋊Dic3—22no—
301C4⋊Dic3—22no—
302C4⋊Dic3—28no—
303C4⋊Dic3—24no—
304C4⋊Dic3—24no—
305C4⋊Dic3—24no—
306C4⋊Dic3C2×D418yes—
307C4⋊Dic3C2×D414yes—
308Dic3⋊C4—28no—
309Dic3⋊C4—28no—
310Dic3⋊C4—28no—
311Dic3⋊C4—28no—
312Dic3⋊C4—24no—
313Dic3⋊C4—44no—
314Dic3⋊C4—44no—
315Dic3⋊C4—44no—
316Dic3⋊C4C2×D418yes—
317C4×Dic3—28no—
318C4×Dic3—24no—
319C4×Dic3—22no—
320C4×Dic3—22no—
321C4×Dic3—24no—
322C4×Dic3—22no—
323C4×Dic3—28no—
324C4×Dic3—22no—
325C4×Dic3—22no—
326C4×Dic3C2×D414yes—
327C22×Q8—32no—
328C22×Q8C22×D312yes—
329C22×D4C22×D311yescharacteristic
330C23×C4C22×D312yes—
331C4⋊Q8—24no—
332C4⋊Q8—68no—
333C4⋊Q8—64no—
334C4⋊Q8—124no—
335#126—616no—
336#126—616no—
337#125—38no—
338#125—616no—
339#125—616no—
340C22⋊Q8—28no—
341C22⋊Q8—44no—
342C22⋊Q8—68no—
343C22⋊Q8—124no—
344C4×Q8—68no—
345C4×Q8—68no—
346C4×D4—24no—
347#117—68no—
348C2×C4⋊C4—31no—
349C2×C4⋊C4—32no—
350C2×C4⋊C4—32no—
351C2×C4⋊C4—32no—
352C2×C4⋊C4—32no—
353C2×C4⋊C4—34no—
354C2×C4⋊C4—34no—
355C2×C4⋊C4—31no—
356C2×C4⋊C4—34no—
357C2×C4⋊C4—34no—
358C2×C4⋊C4—34no—
359C2×C4⋊C4—32no—
360C2×C4⋊C4—32no—
361C2×C4⋊C4—31no—
362C2×C4⋊C4—31no—
363C2×C4⋊C4—32no—
364C2×C4⋊C4—32no—
365C2×C4⋊C4—34no—
366C2×C4⋊C4—34no—
367C2×C4⋊C4—34no—
368C2×C4⋊C4C22×D311yescharacteristic
369C2×C4⋊C4C22×D311yescharacteristic
370C2×C4⋊C4C22×D314yes—
371C2×C4⋊C4C22×D312yes—
372C2×C4⋊C4C22×D312yes—
373C2×C4⋊C4C22×D314yes—
374C2×C4⋊C4C22×D311yescharacteristic
375C2×C22⋊C4—32no—
376C2×C22⋊C4—32no—
377C2×C22⋊C4—34no—
378C2×C22⋊C4—32no—
379C2×C22⋊C4—32no—
380C2×C22⋊C4—34no—
381C2×C22⋊C4C22×D312yes—
382C2×C22⋊C4C22×D314yes—
383C2×C22⋊C4C22×D314yes—
384C2×C42—31no—
385C2×C42—32no—
386C2×C42—34no—
387C2×C42—34no—
388C2×C42—31no—
389C2×C42—31no—
390C2×C42—31no—
391C2×C42—31no—
392C2×C42—31no—
393C2×C42—32no—
394C2×C42C22×D312yes—
395C2×C42C22×D311yescharacteristic
396#95—32no—
397#95—32no—
398#95—32no—
399#95—32no—
400#95—32no—
401#95—32no—
402#95—32no—
403#95—32no—
404#95—34no—
405#95—34no—
406#95—34no—
407#95—34no—
408#95C22×D312yes—
409C23×C3—22no—
410C23×C3—22no—
411C23×C3—22no—
412C23×C3—22no—
413C23×C3—22no—
414C23×C3—22no—
415C23×C3—22no—
416C23×C3C2511yescharacteristic
417C3×Q8—48no—
418C3×D4—44no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—22no—
422C2×C3×C4—22no—
423C2×C3×C4—24no—
424C2×C3×C4—22no—
425C2×C3×C4—22no—
426C2×C3×C4—24no—
427C2×C3×C4—24no—
428C2×C3×C4—22no—
429C2×C3×C4—22no—
430C2×C3×C4—22no—
431C2×C3×C4—22no—
432C2×C3×C4—48no—
433C2×C3×C4C22×D414yes—
434C2×C3×C4C22×D412yes—
435C2×C3×C4C2×#4112yes—
436C2×C3×C4C2×#4112yes—
437C2×C3×C4C2×#4112yes—
438C2×C3×C4H(5,𝔽2)12yes—
439C2×C3×C42−1+412yes—
440C2×C3×C42−1+412yes—
441C2×Dic3—21no—
442C2×Dic3—21no—
443C2×Dic3—21no—
444C2×Dic3—22no—
445C2×Dic3—21no—
446C2×Dic3—21no—
447C2×Dic3—21no—
448C2×Dic3—21no—
449C2×Dic3—21no—
450C2×Dic3—22no—
451C2×Dic3—22no—
452C2×Dic3—22no—
453C2×Dic3—24no—
454C2×Dic3—24no—
455C2×Dic3—24no—
456C2×Dic3—22no—
457C2×Dic3—22no—
458C2×Dic3—22no—
459C2×Dic3—22no—
460C2×Dic3—22no—
461C2×Dic3—22no—
462C2×Dic3—22no—
463C2×Dic3C22×D414yes—
464C2×Dic3C22×D412yes—
465C2×Dic3C22×D412yes—
466C2×Dic32−1+412yes—
467C2×Dic32−1+412yes—
468Dic6—48no—
469C24C23×D312yes—
470C2×Q8—24no—
471C2×Q8—24no—
472C2×Q8—24no—
473C2×Q8—64no—
474C2×Q8—64no—
475C2×Q8—64no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—31no—
483C22×C4—31no—
484C22×C4—31no—
485C22×C4—31no—
486C22×C4—32no—
487C22×C4—32no—
488C22×C4—32no—
489C22×C4—32no—
490C22×C4—34no—
491C22×C4C23×D312yes—
492C22×C4C23×D314yes—
493C22×C4C23×D312yes—
494C22×C4C23×D312yes—
495C22×C4C23×D311yescharacteristic
496C22×C4C23×D311yescharacteristic
497C22×C4C23×D311yescharacteristic
498C4⋊C4—28no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—34no—
504C4⋊C4—38no—
505C4⋊C4—38no—
506C4⋊C4—44no—
507C4⋊C4—64no—
508C4⋊C4—64no—
509C4⋊C4—68no—
510C4⋊C4—68no—
511C4⋊C4—68no—
512C4⋊C4—68no—
513C4⋊C4—64no—
514C4⋊C4—64no—
515C4⋊C4—68no—
516C4⋊C4—64no—
517C4⋊C4—64no—
518C4⋊C4—62no—
519C4⋊C4—62no—
520C4⋊C4—62no—
521C4⋊C4—124no—
522C4⋊C4—124no—
523C4⋊C4—124no—
524C4⋊C4D3×D414yes—
525C22⋊C4—28no—
526C22⋊C4—24no—
527C22⋊C4—44no—
528C22⋊C4—64no—
529C22⋊C4—64no—
530C22⋊C4—68no—
531C22⋊C4—64no—
532C22⋊C4—64no—
533C22⋊C4—124no—
534C42—24no—
535C42—34no—
536C42—68no—
537C42—62no—
538C42—62no—
539C42—64no—
540C42—68no—
541C42—62no—
542C42—62no—
543C42—64no—
544C42—62no—
545C42#23714yes—
546C22×C3—42no—
547C22×C3—42no—
548C22×C3—42no—
549C22×C3—42no—
550C22×C3—42no—
551C22×C3—42no—
552C22×C3—42no—
553C22×C3C23×D411yescharacteristic
554C22×C3C22×#4111yescharacteristic
555C22×C3C2×H(5,𝔽2)11yescharacteristic
556C22×C3C2×2−1+411yescharacteristic
557C22×C3C2×2−1+411yescharacteristic
558C22×C3C2×2−1+411yescharacteristic
559C22×C3C2×2−1+411yescharacteristic
560C3×C4—24no—
561C3×C4—22no—
562C3×C4—42no—
563C3×C4—44no—
564C3×C4—42no—
565C3×C4#54114yescomplemented
566C3×C4#55014yes—
567Dic3—22no—
568Dic3—22no—
569Dic3—24no—
570Dic3—22no—
571Dic3—22no—
572Dic3—44no—
573Dic3—42no—
574Dic3—42no—
575Dic3—42no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23C24×D311yescharacteristic
584Q8—48no—
585Q8—128no—
586D4—44no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—32no—
601C2×C4—32no—
602C2×C4—34no—
603C2×C4—32no—
604C2×C4—32no—
605C2×C4—48no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—62no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—62no—
616C2×C4—62no—
617C2×C4—62no—
618C2×C4—64no—
619C2×C4—64no—
620C2×C4—62no—
621C2×C4—64no—
622C2×C4—62no—
623C2×C4—62no—
624C2×C4—62no—
625C2×C4—62no—
626C2×C4—62no—
627C2×C4—62no—
628C2×C4C2×D3×D414yes—
629C2×C4C2×D3×D412yes—
630C2×C4C2×#23712yes—
631C2×C4C2×#23712yes—
632C2×C4C2×#23712yes—
633C2×C4#100412yes—
634C2×C4#100712yes—
635C2×C4#101012yes—
636C2×C3—82no—
637C2×C3C2×#54111yescharacteristic
638C2×C3C2×#55011yescharacteristic
639C2×C3#349911yescharacteristic
640C2×C3#349911yescharacteristic
641C2×C3#350911yescharacteristic
642C2×C3#355211yescharacteristic
643C2×C3#355211yescharacteristic
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22C22×D3×D411yescharacteristic
652C22C22×#23711yescharacteristic
653C22C2×#100411yescharacteristic
654C22C2×#100711yescharacteristic
655C22C2×#101011yescharacteristic
656C22D3×2−1+411yescharacteristic
657C22D3×2−1+411yescharacteristic
658C4—22no—
659C4—24no—
660C4—42no—
661C4—44no—
662C4—42no—
663C4—62no—
664C4—62no—
665C4—62no—
666C4—64no—
667C4—62no—
668C4—122no—
669C4—124no—
670C4—122no—
671C4—122no—
672C4#562514yescomplemented
673C4#571814yes—
674C3#3938511yescharacteristic, complemented, Sylow
675C2—82no—
676C2C2×#562511yescharacteristic
677C2C2×#571811yescharacteristic
678C2#8634311yescharacteristic
679C2#8667911yescharacteristic
680C2#8670911yescharacteristic
681C2#8674611yescharacteristic
682C2#8676411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2542C2
prime radicalC22×C3×D4C23
hypercenter#2542D3
hypocenterC3#39385
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2542

Complements

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

NH ≅ G/Ntype
1. C2×#5625C2semidirect
3. #83253C2semidirect
4. #83229C2semidirect
5. #82786C2semidirect
6. #82758C2semidirect
7. #82757C2semidirect
10. #78070C2semidirect
11. #78069C2semidirect
14. #74337C2semidirect
15. #74336C2semidirect
16. #73650C2semidirect
17. C4×#890C2semidirect
165. #890D4semidirect
565. C3×C4#541semidirect
672. C4#5625semidirect
674. C3#39385semidirect

Maximal subgroups

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

subgroupindex
C2×#56252
C3×#25422
#832532
#832292
#827862
#827582
#827572
#823792
#823192
#780702
#780692
#763392
#763382
#743372
#743362
#736502
C4×#8902
#393853