Finite Group Databasebeta

← #10967586

Subgroups — #10967586

Order 768.

2630 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×C4×D3×Q8C211yesmaximal, characteristic, complemented
2C2×C4×#239C211yesmaximal, characteristic
3C3×#2399C211yesmaximal, characteristic, complemented
4#82492C211yesmaximal, characteristic, complemented
5#82467C211yesmaximal, characteristic, complemented
6#77954C212yesmaximal, complemented
7#77810C212yesmaximal, complemented
8#77784C212yesmaximal, complemented
9#77764C212yesmaximal, complemented
10#77249C212yesmaximal
11#77196C212yesmaximal, complemented
12#77191C212yesmaximal, complemented
13#77172C212yesmaximal, complemented
14C4×#896C214yesmaximal, complemented
15C4×#894C214yesmaximal, complemented
16#73419C211yesmaximal, characteristic, complemented
17#73413C211yesmaximal, characteristic, complemented
18#34971—31nomaximal, Sylow
19C22×#239C2211yescharacteristic
20C22×D3×Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C2×Q8×Dic3C2211yescharacteristic
24C2×Q8×Dic3C2211yescharacteristic
25C4×#239—28no—
26C4×D3×Q8—28no—
27C2×#896C2214yes—
28C2×#894C2214yes—
29C2×#893C2212yes—
30C2×#893C2211yescharacteristic
31C2×#892C2211yescharacteristic
32C2×#892C2212yes—
33C2×D3×C4⋊C4C2212yes—
34C2×D3×C4⋊C4C2211yescharacteristic
35C2×#887C2212yes—
36C2×#887C2211yescharacteristic
37C2×C4×D12C2212yes—
38C2×C4×D12C2211yescharacteristic
39C2×C42×D3C2212yes—
40C2×C42×D3C2212yes—
41C2×C42×D3C2211yescharacteristic
42C2×C4×Dic6C2212yes—
43C2×C4×Dic6C2211yescharacteristic
44C3×#398C2212yes—
45C3×#398C2212yes—
46C3×#395C2212yes—
47C3×#395C2212yes—
48C3×C4×C4⋊C4C2214yes—
49C3×#376C2211yescharacteristic
50#5231C2212yes—
51#5230C2212yes—
52#4992C2212yes—
53#4991C2212yes—
54#4990C2214yes—
55#4989C2214yes—
56#4989C2214yes—
57#4986C2212yes—
58#4983C2212yes—
59#4981C2212yes—
60#4980C2214yes—
61#4980C2214yes—
62#4979C2212yes—
63#4976C2214yes—
64C4×(C2×D3)⋊C4C2212yes—
65C4×(C2×D3)⋊C4C2214yes—
66C4×(C2×D3)⋊C4C2214yes—
67#4932C2211yescharacteristic
68C4×Dic3⋊C4C2214yes—
69C4×Dic3⋊C4C2212yes—
70#4674C2212yes—
71#4673C2212yes—
72#4672C2212yes—
73#4669C2214yes—
74#4668C2214yes—
75#4667C2214yes—
76#4667C2212yes—
77#4665C2214yes—
78#4664C2214yes—
79#4664C2212yes—
80#4659C2212yes—
81#4655C2214yes—
82#4652C2214yes—
83#4651C2212yes—
84#4650C2212yes—
85#4646C2214yes—
86C2×C4×#41—31no—
87C22×C4×Q8—31no—
88#2599—32no—
89#2599—32no—
90#2581—32no—
91#2536—32no—
92#2486—32no—
93#2484—32no—
94#2409—32no—
95#2406—32no—
96#2399—31no—
97#2399D311yescharacteristic, complemented
98#2398—31no—
99#2398—31no—
100C4×C22⋊Q8—34no—
101C4×#123—34no—
102#2283—31no—
103C22×C3×Q8C2311yescharacteristic
104C22×C3×Q8C2311yescharacteristic
105C2×#239—28no—
106C2×#239—24no—
107C2×D3×Q8—24no—
108C2×D3×Q8—28no—
109C22×D12C2311yescharacteristic
110C22×D12C2312yes—
111C22×C4×D3C2312yes—
112C22×C4×D3C2312yes—
113C22×C4×D3C2312yes—
114C22×C4×D3C2312yes—
115C22×C4×D3C2312yes—
116C22×C4×D3C2311yescharacteristic
117C22×C4×D3C2311yescharacteristic
118C22×C4×D3C2311yescharacteristic
119C22×Dic6C2311yescharacteristic
120C22×Dic6C2312yes—
121C3×C4×Q8—24no—
122C3×C4×Q8—24no—
123C2×C3×C4⋊C4C2314yes—
124C2×C3×C4⋊C4C2312yes—
125C2×C3×C4⋊C4C2314yes—
126C2×C3×C4⋊C4C2311yescharacteristic
127C2×C3×C4⋊C4C2312yes—
128C2×C3×C4⋊C4C2311yescharacteristic
129C2×C3×C42C2312yes—
130C2×C3×C42C2312yes—
131C2×C3×C42C2312yes—
132C2×C3×C42C2311yescharacteristic
133Q8×Dic3—24no—
134Q8×Dic3—24no—
135C2×(C2×D3)⋊C4C2314yes—
136C2×(C2×D3)⋊C4C2312yes—
137C2×(C2×D3)⋊C4C2314yes—
138C2×(C2×D3)⋊C4C2314yes—
139C2×(C2×D3)⋊C4C2314yes—
140C2×(C2×D3)⋊C4C2314yes—
141C2×(C2×D3)⋊C4C2312yes—
142C2×(C2×D3)⋊C4C2312yes—
143C2×C4⋊Dic3C2312yes—
144C2×C4⋊Dic3C2311yescharacteristic
145C2×C4⋊Dic3C2312yes—
146C2×C4⋊Dic3C2311yescharacteristic
147C2×Dic3⋊C4C2314yes—
148C2×Dic3⋊C4C2312yes—
149C2×Dic3⋊C4C2314yes—
150C2×Dic3⋊C4C2314yes—
151C2×Dic3⋊C4C2312yes—
152C2×Dic3⋊C4C2312yes—
153C2×C4×Dic3C2312yes—
154C2×C4×Dic3C2312yes—
155C2×C4×Dic3C2312yes—
156C2×C4×Dic3C2312yes—
157C2×C4×Dic3C2312yes—
158C2×C4×Dic3C2312yes—
159C2×C4×Dic3C2311yescharacteristic
160C2×C4×Dic3C2311yescharacteristic
161C2×C4×Dic3C2311yescharacteristic
162#896—216no—
163#894—216no—
164#893—28no—
165#893—24no—
166#892—24no—
167#892—28no—
168D3×C4⋊C4—28no—
169D3×C4⋊C4—24no—
170#887—24no—
171#887—28no—
172C4×D12—28no—
173C4×D12—24no—
174C42×D3—28no—
175C42×D3—28no—
176C42×D3D418yes—
177C4×Dic6—24no—
178C4×Dic6—28no—
179C3×#95C2314yes—
180C3×#95C2312yes—
181C3×#95C2312yes—
182C3×#95C2314yes—
183C3⋊#95C2314yes—
184C3⋊#95C2314yes—
185C3⋊#95C2314yes—
186C3⋊#95C2314yes—
187C3⋊#95C2312yes—
188C3⋊#95C2314yes—
189C3⋊#95C2312yes—
190C3⋊#95C2314yes—
191C3⋊#95C2314yes—
192C3⋊#95C2314yes—
193C22×#41—31no—
194C23×Q8—31no—
195C2×#123—34no—
196C2×C22⋊Q8—34no—
197C4×#41—68no—
198C2×C4×Q8—31no—
199C2×C4×Q8—31no—
200C2×C4×Q8—32no—
201C2×C4×Q8—31no—
202C2×C4×Q8—31no—
203C2×C4×Q8—32no—
204C2×C4×Q8—68no—
205C2×C4×Q8C2×D311yescharacteristic
206C2×C4×Q8C2×D311yescharacteristic
207C2×C4×D4—32no—
208C2×C4×D4—31no—
209C2×C4×D4—31no—
210C2×C4×D4—32no—
211C2×#117—32no—
212C2×#117—31no—
213C22×C4⋊C4—32no—
214C22×C4⋊C4—31no—
215C22×C42—31no—
216C22×C42—32no—
217C22×C42—32no—
218#400—32no—
219#400—32no—
220#398—32no—
221#398—32no—
222#398C2×D312yes—
223#398C2×D312yes—
224#397—32no—
225#397—32no—
226#395—32no—
227#395—32no—
228#395C2×D312yes—
229#395C2×D312yes—
230#394—32no—
231#394—32no—
232#393—32no—
233#393—32no—
234#391—34no—
235#389—34no—
236#388—34no—
237#388—34no—
238#387—34no—
239#387—34no—
240#387—34no—
241#387—34no—
242#386—34no—
243#385—34no—
244#385—34no—
245#385—34no—
246#383—32no—
247#382—32no—
248#381—32no—
249#381—34no—
250#380—32no—
251#380—34no—
252C4×C4⋊C4—34no—
253C4×C4⋊C4—32no—
254C4×C4⋊C4C2×D314yes—
255C4×C22⋊C4—34no—
256C4×C22⋊C4—34no—
257C4×C22⋊C4—32no—
258#376—31no—
259#376C2×D311yescharacteristic
260C23×D3C2412yes—
261C23×D3C2411yescharacteristic
262C2×C3×Q8—22no—
263C2×C3×Q8—22no—
264C2×C3×Q8—24no—
265C2×C3×Q8—24no—
266C22×C3×C4C2414yes—
267C22×C3×C4C2412yes—
268C22×C3×C4C2412yes—
269C22×C3×C4C2411yescharacteristic
270C22×C3×C4C2411yescharacteristic
271C22×C3×C4C2412yes—
272C22×C3×C4C2412yes—
273C22×C3×C4C2411yescharacteristic
274C22×Dic3C2412yes—
275C22×Dic3C2414yes—
276C22×Dic3C2412yes—
277C22×Dic3C2411yescharacteristic
278C22×Dic3C2411yescharacteristic
279C22×Dic3C2412yes—
280C22×Dic3C2411yescharacteristic
281#239—232no—
282D3×Q8—232no—
283C2×D12—24no—
284C2×D12—24no—
285C2×D12—24no—
286C2×D12—22no—
287C2×D12#4118yes—
288C2×C4×D3—24no—
289C2×C4×D3—24no—
290C2×C4×D3—24no—
291C2×C4×D3—24no—
292C2×C4×D3—24no—
293C2×C4×D3—24no—
294C2×C4×D3—24no—
295C2×C4×D3—24no—
296C2×C4×D3—24no—
297C2×C4×D3—24no—
298C2×C4×D3—28no—
299C2×C4×D3—24no—
300C2×C4×D3—28no—
301C2×C4×D3—24no—
302C2×C4×D3—28no—
303C2×C4×D3C2×D414yes—
304C2×C4×D3C2×D414yes—
305C2×C4×D3C2×D414yes—
306C2×Dic6—22no—
307C2×Dic6—24no—
308C2×Dic6—24no—
309C2×Dic6—24no—
310C2×Dic6#4118yescomplemented
311C3×C4⋊C4—24no—
312C3×C4⋊C4—22no—
313C3×C4⋊C4—24no—
314C3×C4⋊C4—22no—
315C3×C4⋊C4—28no—
316C3×C4⋊C4—28no—
317C3×C42—24no—
318C3×C42—24no—
319C3×C42—24no—
320C3×C42C2×D414yes—
321(C2×D3)⋊C4—24no—
322(C2×D3)⋊C4—28no—
323(C2×D3)⋊C4—28no—
324(C2×D3)⋊C4—28no—
325(C2×D3)⋊C4—24no—
326(C2×D3)⋊C4—28no—
327(C2×D3)⋊C4—24no—
328(C2×D3)⋊C4—28no—
329C4⋊Dic3—22no—
330C4⋊Dic3—22no—
331C4⋊Dic3—24no—
332C4⋊Dic3—24no—
333Dic3⋊C4—24no—
334Dic3⋊C4—28no—
335Dic3⋊C4—24no—
336Dic3⋊C4—24no—
337Dic3⋊C4—28no—
338Dic3⋊C4—28no—
339C4×Dic3—24no—
340C4×Dic3—24no—
341C4×Dic3—24no—
342C4×Dic3—24no—
343C4×Dic3—24no—
344C4×Dic3—24no—
345C4×Dic3C2×D414yes—
346C4×Dic3C2×D414yes—
347C4×Dic3C2×D414yes—
348C2×#41—64no—
349C2×#41—68no—
350C22×Q8—32no—
351C22×Q8—31no—
352C22×Q8—64no—
353C22×Q8—68no—
354C22×Q8C22×D311yescharacteristic
355C22×Q8C22×D311yescharacteristic
356C22×D4—32no—
357C22×D4—31no—
358C23×C4—31no—
359C23×C4—31no—
360C23×C4—31no—
361C23×C4—32no—
362C23×C4—32no—
363C23×C4—32no—
364C23×C4—32no—
365C23×C4—32no—
366#123—616no—
367C22⋊Q8—616no—
368C4×Q8—24no—
369C4×Q8—24no—
370C4×Q8—68no—
371C4×Q8—64no—
372C4×Q8—68no—
373C4×Q8—64no—
374C4×Q8—64no—
375C4×Q8—64no—
376C4×D4—68no—
377C4×D4—64no—
378C4×D4—68no—
379C4×D4—64no—
380#117—68no—
381#117—64no—
382C2×C4⋊C4—32no—
383C2×C4⋊C4—32no—
384C2×C4⋊C4—32no—
385C2×C4⋊C4—32no—
386C2×C4⋊C4—31no—
387C2×C4⋊C4—32no—
388C2×C4⋊C4—31no—
389C2×C4⋊C4—34no—
390C2×C4⋊C4—34no—
391C2×C4⋊C4—34no—
392C2×C4⋊C4—64no—
393C2×C4⋊C4—68no—
394C2×C4⋊C4C22×D312yes—
395C2×C4⋊C4C22×D311yescharacteristic
396C2×C4⋊C4C22×D312yes—
397C2×C4⋊C4C22×D311yescharacteristic
398C2×C4⋊C4C22×D314yes—
399C2×C4⋊C4C22×D314yes—
400C2×C22⋊C4—32no—
401C2×C22⋊C4—32no—
402C2×C22⋊C4—32no—
403C2×C22⋊C4—34no—
404C2×C22⋊C4—34no—
405C2×C22⋊C4—34no—
406C2×C22⋊C4—34no—
407C2×C22⋊C4—34no—
408C2×C42—31no—
409C2×C42—31no—
410C2×C42—31no—
411C2×C42—38no—
412C2×C42—32no—
413C2×C42—32no—
414C2×C42—32no—
415C2×C42—32no—
416C2×C42—32no—
417C2×C42—32no—
418C2×C42—68no—
419C2×C42—68no—
420C2×C42C22×D312yes—
421C2×C42C22×D312yes—
422C2×C42C22×D312yes—
423C2×C42C22×D311yescharacteristic
424#95—32no—
425#95—32no—
426#95—34no—
427#95—34no—
428#95—34no—
429#95—34no—
430#95—34no—
431#95—34no—
432#95—34no—
433#95—34no—
434#95C22×D312yes—
435#95C22×D312yes—
436#95C22×D314yes—
437#95C22×D314yes—
438C23×C3C2511yescharacteristic
439C22×D3—22no—
440C22×D3—22no—
441C22×D3—22no—
442C22×D3—24no—
443C22×D3—22no—
444C22×D3—24no—
445C22×D3C22×D412yes—
446C22×D3C2×#4114yes—
447C22×D3C2×#4114yes—
448C3×Q8—28no—
449C3×Q8—28no—
450C2×C3×C4—22no—
451C2×C3×C4—22no—
452C2×C3×C4—22no—
453C2×C3×C4—22no—
454C2×C3×C4—22no—
455C2×C3×C4—22no—
456C2×C3×C4—22no—
457C2×C3×C4—22no—
458C2×C3×C4—24no—
459C2×C3×C4—24no—
460C2×C3×C4—24no—
461C2×C3×C4—24no—
462C2×C3×C4—24no—
463C2×C3×C4C22×D412yes—
464C2×C3×C4C22×D412yes—
465C2×C3×C4C22×D412yes—
466C2×C3×C4C2×#4114yes—
467C2×C3×C4C2×#4114yes—
468C2×C3×C42−1+414yes—
469C2×Dic3—22no—
470C2×Dic3—22no—
471C2×Dic3—22no—
472C2×Dic3—22no—
473C2×Dic3—22no—
474C2×Dic3—22no—
475C2×Dic3—24no—
476C2×Dic3—24no—
477C2×Dic3—24no—
478C2×Dic3—24no—
479C2×Dic3—24no—
480C2×Dic3C22×D412yes—
481C2×Dic3C22×D412yes—
482C2×Dic3C22×D412yes—
483C2×Dic3C2×#4114yes—
484C2×Dic3C2×#4114yes—
485D12—28no—
486D12—216no—
487C4×D3—216no—
488C4×D3—216no—
489C4×D3—28no—
490C4×D3—28no—
491C4×D3—28no—
492C4×D3—48no—
493C4×D3—48no—
494C4×D3—48no—
495Dic6—28no—
496Dic6—216no—
497C24—31no—
498C24—32no—
499#41—632no—
500C2×Q8—22no—
501C2×Q8—22no—
502C2×Q8—24no—
503C2×Q8—24no—
504C2×Q8—38no—
505C2×Q8—64no—
506C2×Q8—64no—
507C2×Q8—64no—
508C2×Q8—632no—
509C2×Q8—62no—
510C2×D4—38no—
511C2×D4—64no—
512C2×D4—64no—
513C2×D4—64no—
514C2×D4—62no—
515C22×C4—31no—
516C22×C4—31no—
517C22×C4—31no—
518C22×C4—32no—
519C22×C4—34no—
520C22×C4—34no—
521C22×C4—34no—
522C22×C4—32no—
523C22×C4—34no—
524C22×C4—32no—
525C22×C4—64no—
526C22×C4—64no—
527C22×C4—64no—
528C22×C4—64no—
529C22×C4—68no—
530C22×C4—64no—
531C22×C4—64no—
532C22×C4—64no—
533C22×C4—64no—
534C22×C4—68no—
535C22×C4—64no—
536C22×C4—64no—
537C22×C4—64no—
538C22×C4—64no—
539C22×C4—68no—
540C22×C4C23×D314yes—
541C22×C4C23×D312yes—
542C22×C4C23×D312yes—
543C22×C4C23×D311yescharacteristic
544C22×C4C23×D311yescharacteristic
545C22×C4C23×D312yes—
546C22×C4C23×D312yes—
547C22×C4C23×D311yescharacteristic
548C4⋊C4—24no—
549C4⋊C4—22no—
550C4⋊C4—24no—
551C4⋊C4—22no—
552C4⋊C4—28no—
553C4⋊C4—28no—
554C4⋊C4—64no—
555C4⋊C4—64no—
556C4⋊C4—64no—
557C4⋊C4—64no—
558C4⋊C4—68no—
559C4⋊C4—68no—
560C4⋊C4—64no—
561C4⋊C4—62no—
562C4⋊C4—62no—
563C4⋊C4—68no—
564C22⋊C4—64no—
565C22⋊C4—68no—
566C22⋊C4—64no—
567C22⋊C4—68no—
568C22⋊C4—68no—
569C22⋊C4—68no—
570C22⋊C4—64no—
571C22⋊C4—68no—
572C42—24no—
573C42—24no—
574C42—24no—
575C42—34no—
576C42—34no—
577C42—34no—
578C42—64no—
579C42—64no—
580C42—64no—
581C42—64no—
582C42—64no—
583C42—64no—
584C42D3×D414yes—
585C22×C3C23×D411yescharacteristic
586C22×C3C22×#4112yes—
587C22×C3C22×#4112yes—
588C22×C3C2×2−1+412yes—
589C2×D3—22no—
590C2×D3—22no—
591C2×D3—28no—
592C2×D3—24no—
593C2×D3—28no—
594C2×D3—42no—
595C2×D3—42no—
596C2×D3—44no—
597C2×D3#52514yes—
598C3×C4—22no—
599C3×C4—24no—
600C3×C4—22no—
601C3×C4—24no—
602C3×C4—42no—
603C3×C4—42no—
604C3×C4—44no—
605C3×C4#53614yes—
606Dic3—24no—
607Dic3—22no—
608Dic3—22no—
609Dic3—42no—
610Dic3—42no—
611Dic3—44no—
612Dic3#52514yes—
613C23—32no—
614C23—34no—
615C23—34no—
616C23—62no—
617C23—62no—
618C23—62no—
619C23—64no—
620C23—64no—
621C23—62no—
622C23C24×D311yescharacteristic
623Q8—28no—
624Q8—28no—
625Q8—616no—
626Q8—68no—
627D4—616no—
628D4—68no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—32no—
643C2×C4—34no—
644C2×C4—32no—
645C2×C4—32no—
646C2×C4—34no—
647C2×C4—68no—
648C2×C4—68no—
649C2×C4—616no—
650C2×C4—616no—
651C2×C4—62no—
652C2×C4—62no—
653C2×C4—62no—
654C2×C4—62no—
655C2×C4—68no—
656C2×C4—62no—
657C2×C4—62no—
658C2×C4—64no—
659C2×C4—64no—
660C2×C4—64no—
661C2×C4—64no—
662C2×C4—64no—
663C2×C4—128no—
664C2×C4—128no—
665C2×C4—128no—
666C2×C4C2×D3×D412yes—
667C2×C4C2×D3×D412yes—
668C2×C4C2×D3×D412yes—
669C2×C4#100714yes—
670C2×C4D3×#4114yes—
671C2×C4D3×#4114yes—
672C2×C3C2×#52511yescharacteristic
673C2×C3C2×#52511yescharacteristic
674C2×C3C2×#53611yescharacteristic
675C2×C3#348214yes—
676D3—24no—
677D3—44no—
678C22—34no—
679C22—68no—
680C22—62no—
681C22—62no—
682C22—68no—
683C22—64no—
684C22—124no—
685C22—122no—
686C22—122no—
687C22C22×D3×D411yescharacteristic
688C22C2×#100712yes—
689C22C2×D3×#4112yes—
690C22C2×D3×#4112yes—
691C4—22no—
692C4—22no—
693C4—24no—
694C4—24no—
695C4—34no—
696C4—44no—
697C4—42no—
698C4—42no—
699C4—62no—
700C4—64no—
701C4—62no—
702C4—124no—
703C4—122no—
704C4—122no—
705C4#572414yes—
706C3#3497111yescharacteristic, complemented, Sylow
707C2—64no—
708C2—124no—
709C2D3×#52511yescharacteristic
710C2D3×#52511yescharacteristic
711C2C2×#572411yescharacteristic
712C2#8627814yes—
subgroupG/H
centerC22×C4C23×D3
commutator, socle, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC3×#2399C2
prime radicalC22×#239C22
hypercenter#2399D3
hypocenterC3#34971
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ #2399

Complements

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

NH ≅ G/Ntype
1. C2×C4×D3×Q8C2semidirect
3. C3×#2399C2semidirect
4. #82492C2semidirect
5. #82467C2semidirect
6. #77954C2semidirect
7. #77810C2semidirect
8. #77784C2semidirect
9. #77764C2semidirect
11. #77196C2semidirect
12. #77191C2semidirect
13. #77172C2semidirect
14. C4×#896C2semidirect
15. C4×#894C2semidirect
16. #73419C2semidirect
17. #73413C2semidirect
97. #2399D3semidirect
310. C2×Dic6#41semidirect
706. C3#34971semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×Q82
C2×C4×#2392
C3×#23992
#824922
#824672
#779542
#778102
#777842
#777642
#772492
#771962
#771912
#771722
C4×#8962
C4×#8942
#734192
#734132
#349713