Finite Group Databasebeta

← #11069651

Subgroups — #11069651

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×#5583C212yesmaximal, complemented
2C3×#2365C211yesmaximal, characteristic, complemented
3#83259C211yesmaximal, characteristic
4#83253C211yesmaximal, characteristic, complemented
5#82496C212yesmaximal, complemented
6#76641C212yesmaximal, complemented
7#76625C212yesmaximal, complemented
8#76066C212yesmaximal, complemented
9#76047C212yesmaximal
10#75172C214yesmaximal, complemented
11#75170C214yesmaximal, complemented
12#73800C211yesmaximal, characteristic
13#73716C212yesmaximal, complemented
14#73693C212yesmaximal
15#73651C211yesmaximal, characteristic, complemented
16C4×#890C211yesmaximal, characteristic, complemented
17#73617C211yesmaximal, characteristic
18#39385—31nomaximal, Sylow
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×C4×Q8C2212yes—
21C2×#946C2212yes—
22C2×#946C2212yes—
23C2×#944C2212yes—
24C2×#944C2212yes—
25#5583—216no—
26C2×#898C2214yes—
27C2×#897C2212yes—
28C2×#897C2212yes—
29C2×#890C2211yescharacteristic
30C2×#889C2214yes—
31C2×#876C2214yes—
32C2×C4×D12C2211yescharacteristic
33C2×#872C2212yes—
34C2×#870C2214yes—
35C2×#869C2211yescharacteristic
36C2×C4×Dic6C2212yes—
37C3×#391C2212yes—
38C3×#389C2212yes—
39C3×#387C2214yes—
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×C4×C4⋊C4C2212yes—
43#4994C2214yes—
44#4993C2214yes—
45#4988C2211yescharacteristic
46#4988C2212yes—
47#4988C2211yescharacteristic
48#4984C2211yescharacteristic
49#4981C2214yes—
50#4977C2211yescharacteristic
51#4975C2212yes—
52#4975C2212yes—
53#4975C2212yes—
54#4975C2212yes—
55#4975C2212yes—
56Dic3×C4⋊C4C2212yes—
57#4972C2211yescharacteristic
58#4972C2211yescharacteristic
59#4972C2211yescharacteristic
60#4972C2212yes—
61#4940C2214yes—
62#4939C2212yes—
63C4×(C2×D3)⋊C4C2212yes—
64#4935C2212yes—
65C4×C4⋊Dic3C2212yes—
66C4×C4⋊Dic3C2211yescharacteristic
67C4×Dic3⋊C4C2212yes—
68C42×Dic3C2211yescharacteristic
69#4929C2212yes—
70#4928C2214yes—
71#4677C2214yes—
72#4676C2214yes—
73#4675C2214yes—
74#4675C2214yes—
75#4669C2214yes—
76#4669C2212yes—
77#4669C2212yes—
78#4657C2214yes—
79#4657C2214yes—
80#4657C2214yes—
81#4655C2212yes—
82#4654C2212yes—
83#4654C2214yes—
84#4647C2214yes—
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#2542—31no—
94#2512—32no—
95#2512—32no—
96#2429—32no—
97#2429—32no—
98#2365—31no—
99#2365D311yescharacteristic, complemented
100C4×#125—31no—
101#2293—31no—
102C22×C3×Q8C2312yes—
103C22×D12C2311yescharacteristic
104C22×C4×D3C2312yes—
105C22×Dic6C2312yes—
106C3×C4⋊Q8—24no—
107C3×C4×Q8—28no—
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C4⋊C4C2312yes—
112C2×C3×C4⋊C4C2312yes—
113C2×C3×C4⋊C4C2314yes—
114C2×C3×C4⋊C4C2311yescharacteristic
115C2×C3×C42C2314yes—
116C2×C3×C42C2312yes—
117C2×C3×C42C2312yes—
118C2×C3×C42C2311yescharacteristic
119#946—28no—
120#946—44no—
121#944—28no—
122#944—44no—
123C2×(C2×D3)⋊C4C2312yes—
124C2×(C2×D3)⋊C4C2312yes—
125C2×(C2×D3)⋊C4C2312yes—
126C2×(C2×D3)⋊C4C2312yes—
127C2×(C2×D3)⋊C4C2312yes—
128C2×(C2×D3)⋊C4C2314yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×(C2×D3)⋊C4C2314yes—
131C2×(C2×D3)⋊C4C2314yes—
132C2×C4⋊Dic3C2312yes—
133C2×C4⋊Dic3C2312yes—
134C2×C4⋊Dic3C2314yes—
135C2×C4⋊Dic3C2311yescharacteristic
136C2×C4⋊Dic3C2311yescharacteristic
137C2×C4⋊Dic3C2312yes—
138C2×C4⋊Dic3C2314yes—
139C2×C4⋊Dic3C2312yes—
140C2×C4⋊Dic3C2311yescharacteristic
141C2×Dic3⋊C4C2312yes—
142C2×Dic3⋊C4C2314yes—
143C2×Dic3⋊C4C2314yes—
144C2×Dic3⋊C4C2312yes—
145C2×Dic3⋊C4C2312yes—
146C2×Dic3⋊C4C2314yes—
147C2×Dic3⋊C4C2314yes—
148C2×Dic3⋊C4C2314yes—
149C2×Dic3⋊C4C2314yes—
150C2×Dic3⋊C4C2314yes—
151C2×Dic3⋊C4C2312yes—
152C2×C4×Dic3C2314yes—
153C2×C4×Dic3C2311yescharacteristic
154C2×C4×Dic3C2311yescharacteristic
155C2×C4×Dic3C2311yescharacteristic
156C2×C4×Dic3C2311yescharacteristic
157C2×C4×Dic3C2311yescharacteristic
158C2×C4×Dic3C2312yes—
159C2×C4×Dic3C2311yescharacteristic
160#898—216no—
161#897—28no—
162#897—44no—
163#890D418yescomplemented
164#889—216no—
165#876—216no—
166C4×D12—24no—
167#872—28no—
168#870—216no—
169#869—24no—
170C4×Dic6—28no—
171C3×#95C2314yes—
172C3×#95C2312yes—
173C3×#95C2312yes—
174C3×#95C2314yes—
175C3⋊#95C2312yes—
176C3⋊#95C2312yes—
177C3⋊#95C2312yes—
178C3⋊#95C2314yes—
179C3⋊#95C2314yes—
180C3⋊#95C2312yes—
181C3⋊#95C2312yes—
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⋊Q8—32no—
197C2×C22⋊Q8—32no—
198C2×C4×Q8—32no—
199C2×C4×Q8C2×D312yes—
200C2×C4×D4—31no—
201C2×#117—32no—
202#399—34no—
203#399—34no—
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#391C2×D312yes—
214#390—32no—
215#390—31no—
216#390—31no—
217#390—32no—
218#389—31no—
219#389—31no—
220#389—32no—
221#389—31no—
222#389—31no—
223#389—34no—
224#389—34no—
225#389C2×D312yes—
226#388—34no—
227#388—34no—
228#388—32no—
229#388—32no—
230#387—32no—
231#387—32no—
232#387—34no—
233#387C2×D314yes—
234#384—32no—
235#384—32no—
236#384—32no—
237#384—32no—
238#384—32no—
239#384—32no—
240#382—31no—
241C4×C4⋊C4—32no—
242C4×C4⋊C4—31no—
243C4×C4⋊C4—32no—
244C4×C4⋊C4—32no—
245C4×C4⋊C4C2×D311yescharacteristic
246C4×C4⋊C4C2×D311yescharacteristic
247C4×C4⋊C4C2×D312yes—
248C4×C22⋊C4—32no—
249C43—31no—
250C23×D3C2412yes—
251C2×C3×Q8—24no—
252C2×C3×Q8—24no—
253C2×C3×Q8—24no—
254C22×C3×C4C2412yes—
255C22×C3×C4C2412yes—
256C22×C3×C4C2412yes—
257C22×C3×C4C2414yes—
258C22×C3×C4C2412yes—
259C22×C3×C4C2411yescharacteristic
260C22×C3×C4C2411yescharacteristic
261C22×C3×C4C2411yescharacteristic
262C22×Dic3C2412yes—
263C22×Dic3C2414yes—
264C22×Dic3C2412yes—
265C22×Dic3C2411yescharacteristic
266C22×Dic3C2411yescharacteristic
267C22×Dic3C2412yes—
268C22×Dic3C2411yescharacteristic
269C22×Dic3C2411yescharacteristic
270C2×D12—22no—
271C2×D12—22no—
272C2×D12—22no—
273C2×C4×D3—24no—
274C2×C4×D3—24no—
275C2×C4×D3—24no—
276C2×Dic6—24no—
277C2×Dic6—24no—
278C2×Dic6—24no—
279C3×C4⋊C4—28no—
280C3×C4⋊C4—24no—
281C3×C4⋊C4—24no—
282C3×C4⋊C4—22no—
283C3×C4⋊C4—22no—
284C3×C4⋊C4—22no—
285C3×C4⋊C4C2×D414yes—
286C3×C42—24no—
287C3×C42—28no—
288C3×C42—24no—
289C3×C42#4114yes—
290(C2×D3)⋊C4—28no—
291(C2×D3)⋊C4—24no—
292(C2×D3)⋊C4—24no—
293(C2×D3)⋊C4—24no—
294(C2×D3)⋊C4—28no—
295(C2×D3)⋊C4—24no—
296(C2×D3)⋊C4—24no—
297(C2×D3)⋊C4—44no—
298(C2×D3)⋊C4—44no—
299C4⋊Dic3—28no—
300C4⋊Dic3—24no—
301C4⋊Dic3—24no—
302C4⋊Dic3—22no—
303C4⋊Dic3—22no—
304C4⋊Dic3—24no—
305C4⋊Dic3—44no—
306C4⋊Dic3C2×D414yes—
307C4⋊Dic3C2×D418yes—
308Dic3⋊C4—28no—
309Dic3⋊C4—24no—
310Dic3⋊C4—28no—
311Dic3⋊C4—28no—
312Dic3⋊C4—24no—
313Dic3⋊C4—28no—
314Dic3⋊C4—24no—
315Dic3⋊C4—44no—
316Dic3⋊C4—44no—
317Dic3⋊C4—44no—
318Dic3⋊C4C2×D418yes—
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×D4—31no—
330C23×C4—32no—
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—68no—
341C22⋊Q8—68no—
342C22⋊Q8—124no—
343C22⋊Q8—124no—
344C4×Q8—28no—
345C4×Q8—68no—
346C4×D4—64no—
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—31no—
354C2×C4⋊C4—34no—
355C2×C4⋊C4—31no—
356C2×C4⋊C4—34no—
357C2×C4⋊C4—34no—
358C2×C4⋊C4—34no—
359C2×C4⋊C4—34no—
360C2×C4⋊C4—32no—
361C2×C4⋊C4—32no—
362C2×C4⋊C4—32no—
363C2×C4⋊C4—32no—
364C2×C4⋊C4—34no—
365C2×C4⋊C4—34no—
366C2×C4⋊C4—34no—
367C2×C4⋊C4—34no—
368C2×C4⋊C4C22×D314yes—
369C2×C4⋊C4C22×D311yescharacteristic
370C2×C4⋊C4C22×D312yes—
371C2×C4⋊C4C22×D312yes—
372C2×C4⋊C4C22×D311yescharacteristic
373C2×C4⋊C4C22×D311yescharacteristic
374C2×C4⋊C4C22×D311yescharacteristic
375C2×C22⋊C4—32no—
376C2×C22⋊C4—32no—
377C2×C22⋊C4—32no—
378C2×C22⋊C4—34no—
379C2×C22⋊C4—34no—
380C2×C22⋊C4—32no—
381C2×C22⋊C4—32no—
382C2×C22⋊C4—34no—
383C2×C22⋊C4—34no—
384C2×C42—31no—
385C2×C42—34no—
386C2×C42—31no—
387C2×C42—31no—
388C2×C42—31no—
389C2×C42—31no—
390C2×C42—31no—
391C2×C42—32no—
392C2×C42C22×D312yes—
393C2×C42C22×D314yes—
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—34no—
404#95—34no—
405#95C22×D312yes—
406#95C22×D312yes—
407#95C22×D314yes—
408#95C22×D314yes—
409C23×C3C2511yescharacteristic
410C22×D3—22no—
411C22×D3—22no—
412C22×D3—22no—
413C22×D3—22no—
414C22×D3—22no—
415C22×D3—22no—
416C22×D3—22no—
417C3×Q8—48no—
418C2×C3×C4—21no—
419C2×C3×C4—21no—
420C2×C3×C4—22no—
421C2×C3×C4—22no—
422C2×C3×C4—22no—
423C2×C3×C4—22no—
424C2×C3×C4—24no—
425C2×C3×C4—24no—
426C2×C3×C4—24no—
427C2×C3×C4—22no—
428C2×C3×C4—22no—
429C2×C3×C4—22no—
430C2×C3×C4—22no—
431C2×C3×C4—22no—
432C2×C3×C4—22no—
433C2×C3×C4—22no—
434C2×C3×C4C22×D414yes—
435C2×C3×C4C22×D412yes—
436C2×C3×C4C2×#4112yes—
437C2×C3×C4C2×#4112yes—
438C2×C3×C4C2×#4112yes—
439C2×C3×C4H(5,𝔽2)12yes—
440C2×C3×C42−1+412yes—
441C2×C3×C42−1+412yes—
442C2×Dic3—21no—
443C2×Dic3—21no—
444C2×Dic3—21no—
445C2×Dic3—22no—
446C2×Dic3—21no—
447C2×Dic3—21no—
448C2×Dic3—21no—
449C2×Dic3—21no—
450C2×Dic3—21no—
451C2×Dic3—22no—
452C2×Dic3—24no—
453C2×Dic3—22no—
454C2×Dic3—24no—
455C2×Dic3—22no—
456C2×Dic3—22no—
457C2×Dic3—22no—
458C2×Dic3—24no—
459C2×Dic3—22no—
460C2×Dic3—22no—
461C2×Dic3C22×D414yes—
462C2×Dic3C22×D412yes—
463C2×Dic3C22×D412yes—
464C2×Dic32−1+412yes—
465C2×Dic32−1+412yes—
466D12—44no—
467C4×D3—48no—
468Dic6—48no—
469C24—32no—
470C2×Q8—24no—
471C2×Q8—24no—
472C2×Q8—24no—
473C2×Q8—64no—
474C2×Q8—64no—
475C2×Q8—64no—
476C2×D4—62no—
477C2×D4—62no—
478C2×D4—62no—
479C22×C4—31no—
480C22×C4—31no—
481C22×C4—31no—
482C22×C4—31no—
483C22×C4—32no—
484C22×C4—32no—
485C22×C4—34no—
486C22×C4—32no—
487C22×C4—64no—
488C22×C4—64no—
489C22×C4—64no—
490C22×C4C23×D312yes—
491C22×C4C23×D312yes—
492C22×C4C23×D312yes—
493C22×C4C23×D314yes—
494C22×C4C23×D311yescharacteristic
495C22×C4C23×D312yes—
496C22×C4C23×D311yescharacteristic
497C22×C4C23×D311yescharacteristic
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—28no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—34no—
505C4⋊C4—38no—
506C4⋊C4—38no—
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—62no—
516C4⋊C4—64no—
517C4⋊C4—64no—
518C4⋊C4—68no—
519C4⋊C4—62no—
520C4⋊C4—124no—
521C4⋊C4—124no—
522C4⋊C4—124no—
523C4⋊C4—124no—
524C4⋊C4D3×D414yes—
525C22⋊C4—64no—
526C22⋊C4—64no—
527C22⋊C4—68no—
528C22⋊C4—64no—
529C22⋊C4—64no—
530C22⋊C4—64no—
531C22⋊C4—68no—
532C22⋊C4—124no—
533C22⋊C4—124no—
534C42—24no—
535C42—28no—
536C42—24no—
537C42—34no—
538C42—68no—
539C42—62no—
540C42—62no—
541C42—62no—
542C42—64no—
543C42—62no—
544C42—62no—
545C42#23514yes—
546C22×C3C23×D411yescharacteristic
547C22×C3C22×#4111yescharacteristic
548C22×C3C2×H(5,𝔽2)11yescharacteristic
549C22×C3C2×2−1+411yescharacteristic
550C22×C3C2×2−1+411yescharacteristic
551C22×C3C2×2−1+411yescharacteristic
552C22×C3C2×2−1+411yescharacteristic
553C2×D3—42no—
554C2×D3—42no—
555C2×D3—42no—
556C2×D3—42no—
557C2×D3—42no—
558C2×D3—42no—
559C2×D3—42no—
560C3×C4—24no—
561C3×C4—22no—
562C3×C4—44no—
563C3×C4—42no—
564C3×C4—42no—
565C3×C4—42no—
566C3×C4#54114yescomplemented
567C3×C4#55014yes—
568Dic3—22no—
569Dic3—22no—
570Dic3—24no—
571Dic3—22no—
572Dic3—22no—
573Dic3—44no—
574Dic3—42no—
575Dic3—42no—
576C23—62no—
577C23—62no—
578C23—62no—
579C23—62no—
580C23—62no—
581C23—62no—
582C23—62no—
583C23C24×D311yescharacteristic
584Q8—48no—
585Q8—128no—
586D4—124no—
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—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—32no—
604C2×C4—32no—
605C2×C4—34no—
606C2×C4—32no—
607C2×C4—32no—
608C2×C4—61no—
609C2×C4—62no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—62no—
618C2×C4—62no—
619C2×C4—64no—
620C2×C4—64no—
621C2×C4—62no—
622C2×C4—62no—
623C2×C4—62no—
624C2×C4—64no—
625C2×C4—62no—
626C2×C4—62no—
627C2×C4—128no—
628C2×C4C2×#23512yes—
629C2×C4C2×#23512yes—
630C2×C4C2×#23512yes—
631C2×C4C2×D3×D414yes—
632C2×C4C2×D3×D412yes—
633C2×C4#100712yes—
634C2×C4#100912yes—
635C2×C4#101012yes—
636C2×C3C2×#54111yescharacteristic
637C2×C3C2×#55011yescharacteristic
638C2×C3#349911yescharacteristic
639C2×C3#349911yescharacteristic
640C2×C3#350911yescharacteristic
641C2×C3#355211yescharacteristic
642C2×C3#355211yescharacteristic
643D3—82no—
644C22—122no—
645C22—122no—
646C22—122no—
647C22—122no—
648C22—122no—
649C22—122no—
650C22—122no—
651C22C22×#23511yescharacteristic
652C22C22×D3×D411yescharacteristic
653C22C2×#100711yescharacteristic
654C22C2×#100911yescharacteristic
655C22C2×#101011yescharacteristic
656C22D3×2−1+411yescharacteristic
657C22D3×2−1+411yescharacteristic
658C4—22no—
659C4—24no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—62no—
665C4—62no—
666C4—64no—
667C4—62no—
668C4—62no—
669C4—122no—
670C4—122no—
671C4—124no—
672C4#555314yes—
673C4#558314yescomplemented
674C3#3938511yescharacteristic, complemented, Sylow
675C2—242no—
676C2C2×#555311yescharacteristic
677C2C2×#558311yescharacteristic
678C2#8587611yescharacteristic
679C2#8667911yescharacteristic
680C2#8674611yescharacteristic
681C2#8701011yescharacteristic
682C2#8703411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2365C2
prime radicalC22×D12C23
hypercenter#2365D3
hypocenterC3#39385
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2365

Complements

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

NH ≅ G/Ntype
1. C2×#5583C2semidirect
2. C3×#2365C2semidirect
4. #83253C2semidirect
5. #82496C2semidirect
6. #76641C2semidirect
7. #76625C2semidirect
8. #76066C2semidirect
10. #75172C2semidirect
11. #75170C2semidirect
13. #73716C2semidirect
15. #73651C2semidirect
16. C4×#890C2semidirect
99. #2365D3semidirect
163. #890D4semidirect
566. C3×C4#541semidirect
673. C4#5583semidirect
674. C3#39385semidirect

Maximal subgroups

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

subgroupindex
C2×#55832
C3×#23652
#832592
#832532
#824962
#766412
#766252
#760662
#760472
#751722
#751702
#738002
#737162
#736932
#736512
C4×#8902
#736172
#393853