Finite Group Databasebeta

← #10967587

Subgroups — #10967587

Order 768.

2630 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5522C211yesmaximal, characteristic
2C22×#887C211yesmaximal, characteristic, complemented
3C3×#2484C211yesmaximal, characteristic, complemented
4#82254C211yesmaximal, characteristic, complemented
5#82192C211yesmaximal, characteristic
6#78191C212yesmaximal, complemented
7#78131C212yesmaximal, complemented
8#77784C212yesmaximal, complemented
9#77740C212yesmaximal, complemented
10#75371C214yesmaximal, complemented
11#74941C214yesmaximal, complemented
12#74405C212yesmaximal, complemented
13#74380C212yesmaximal, complemented
14#74075C212yesmaximal, complemented
15#74063C211yesmaximal, characteristic
16#74052C212yesmaximal, complemented
17#74051C211yesmaximal, characteristic, complemented
18#34971—31nomaximal, Sylow
19C22×#235C2211yescharacteristic
20C23×Dic6C2211yescharacteristic
21C2×C3×#117C2211yescharacteristic
22C22×C3×C4⋊C4C2211yescharacteristic
23C2×#929C2212yes—
24C22×Dic3⋊C4C2211yescharacteristic
25C22×Dic3⋊C4C2211yescharacteristic
26C22×C4×Dic3C2211yescharacteristic
27C22×C4×Dic3C2211yescharacteristic
28C22×C4×Dic3C2211yescharacteristic
29#5522—28no—
30C2×#896C2214yes—
31C2×#893C2211yescharacteristic
32C2×#893C2211yescharacteristic
33C2×#887—28no—
34C2×#887C2212yes—
35C2×#887C2212yes—
36C2×#887C2211yescharacteristic
37C2×#887C2212yes—
38C2×#887C2211yescharacteristic
39C2×#887C2212yes—
40C2×#886C2212yes—
41C2×#67⋊C4C2212yes—
42C2×#67⋊C4C2212yes—
43C2×#872C2212yes—
44C2×C42×D3C2212yes—
45C3×#400C2212yes—
46C3×#398C2212yes—
47C3×#387C2214yes—
48C3×#385C2212yes—
49C3×#381C2211yescharacteristic
50C3×#381C2212yes—
51#5213C2212yes—
52#5211C2211yescharacteristic
53#4991C2212yes—
54#4990C2214yes—
55#4989C2214yes—
56#4985C2212yes—
57#4983C2212yes—
58#4979C2212yes—
59#4979C2214yes—
60Dic3×C4⋊C4C2214yes—
61#4958C2212yes—
62#4946C2212yes—
63#4943C2212yes—
64#4942C2212yes—
65Dic3×C22⋊C4C2212yes—
66#4933C2214yes—
67#4675C2214yes—
68#4673C2214yes—
69#4672C2212yes—
70#4669C2214yes—
71#4668C2214yes—
72#4667C2214yes—
73#4666C2214yes—
74#4666C2214yes—
75#4657C2214yes—
76#4656C2212yes—
77#4654C2214yes—
78#4652C2214yes—
79#4651C2212yes—
80#4650C2212yes—
81#4649C2212yes—
82#4649C2214yes—
83#4648C2212yes—
84#4647C2214yes—
85#4646C2214yes—
86C2×C4×#41—31no—
87C22×C4×Q8—31no—
88#2599—34no—
89#2581—32no—
90#2536—32no—
91#2486—32no—
92#2484—31no—
93#2484D311yescharacteristic, complemented
94#2409—32no—
95#2406—31no—
96#2406—31no—
97#2399—32no—
98#2398—32no—
99C4×C22⋊Q8—34no—
100C4×#123—32no—
101C4×#123—32no—
102#2283—31no—
103C23×C3×C4C2311yescharacteristic
104C23×C3×C4C2311yescharacteristic
105C23×C3×C4C2311yescharacteristic
106C22×#67C2312yes—
107C23×Dic3C2311yescharacteristic
108C23×Dic3C2311yescharacteristic
109C23×Dic3C2311yescharacteristic
110C23×Dic3C2311yescharacteristic
111C2×#235—24no—
112C2×#235—28no—
113C22×D12C2311yescharacteristic
114C22×C4×D3C2312yes—
115C22×C4×D3C2312yes—
116C22×C4×D3C2312yes—
117C22×Dic6—24no—
118C22×Dic6—28no—
119C22×Dic6C2311yescharacteristic
120C22×Dic6C2312yes—
121C22×Dic6C2312yes—
122C3×#117—24no—
123C2×C3×C4⋊C4—24no—
124C2×C3×C4⋊C4C2314yes—
125C2×C3×C4⋊C4C2312yes—
126C2×C3×C4⋊C4C2311yescharacteristic
127C2×C3×C4⋊C4C2312yes—
128C2×C3×C4⋊C4C2311yescharacteristic
129C2×C3×C22⋊C4C2312yes—
130C2×C3×C22⋊C4C2312yes—
131C2×C3×C42C2312yes—
132C2×#217C2312yes—
133#929—28no—
134C2×(C2×D3)⋊C4C2314yes—
135C2×(C2×D3)⋊C4C2314yes—
136C2×(C2×D3)⋊C4C2314yes—
137C2×(C2×D3)⋊C4C2314yes—
138C2×(C2×D3)⋊C4C2312yes—
139C2×(C2×D3)⋊C4C2312yes—
140C2×C4⋊Dic3C2314yes—
141C2×Dic3⋊C4—24no—
142C2×Dic3⋊C4—24no—
143C2×Dic3⋊C4C2314yes—
144C2×Dic3⋊C4C2312yes—
145C2×Dic3⋊C4C2314yes—
146C2×Dic3⋊C4C2314yes—
147C2×Dic3⋊C4C2312yes—
148C2×Dic3⋊C4C2312yes—
149C2×Dic3⋊C4C2312yes—
150C2×Dic3⋊C4C2312yes—
151C2×Dic3⋊C4C2312yes—
152C2×C4×Dic3—24no—
153C2×C4×Dic3—24no—
154C2×C4×Dic3D418yes—
155C2×C4×Dic3C2312yes—
156C2×C4×Dic3C2314yes—
157C2×C4×Dic3C2312yes—
158C2×C4×Dic3C2312yes—
159C2×C4×Dic3C2312yes—
160C2×C4×Dic3C2312yes—
161C2×C4×Dic3C2312yes—
162C2×C4×Dic3C2311yescharacteristic
163C2×C4×Dic3C2311yescharacteristic
164C2×C4×Dic3C2311yescharacteristic
165C2×C4×Dic3C2311yescharacteristic
166#896—216no—
167#893—24no—
168#893—24no—
169#887—28no—
170#887—28no—
171#887—28no—
172#887—28no—
173#887—24no—
174#887—24no—
175#886—28no—
176#67⋊C4—28no—
177#67⋊C4—28no—
178#872—28no—
179C42×D3—28no—
180C3×#95C2314yes—
181C3×#95C2314yes—
182C3×#95C2312yes—
183C3×#95C2312yes—
184C3×#95C2314yes—
185C3⋊#95C2314yes—
186C3⋊#95C2314yes—
187C3⋊#95C2314yes—
188C3⋊#95C2314yes—
189C3⋊#95C2312yes—
190C3⋊#95C2314yes—
191C3⋊#95C2312yes—
192C3⋊#95C2314yes—
193C3⋊#95C2314yes—
194C22×#41—31no—
195C23×Q8—31no—
196C2×#123—32no—
197C2×#123—32no—
198C2×C22⋊Q8—34no—
199C4×#41—68no—
200C2×C4×Q8—31no—
201C2×C4×Q8—32no—
202C2×C4×Q8—32no—
203C2×C4×Q8—32no—
204C2×C4×Q8—31no—
205C2×C4×Q8—32no—
206C2×C4×Q8—68no—
207C2×C4×D4—31no—
208C2×C4×D4—31no—
209C2×C4×D4—32no—
210C2×C4×D4—32no—
211C2×#117—32no—
212C2×#117C2×D311yescharacteristic
213C22×C4⋊C4—31no—
214C22×C4⋊C4—31no—
215C22×C4⋊C4C2×D311yescharacteristic
216C22×C42—31no—
217C22×C42—31no—
218C22×C42—31no—
219C22×C42—32no—
220#400—32no—
221#400C2×D312yes—
222#398—32no—
223#398—34no—
224#398C2×D312yes—
225#397—34no—
226#395—32no—
227#395—34no—
228#395—32no—
229#394—32no—
230#394—32no—
231#393—34no—
232#391—34no—
233#389—34no—
234#388—34no—
235#388—34no—
236#387—34no—
237#387—34no—
238#387—34no—
239#387C2×D314yes—
240#386—32no—
241#386—32no—
242#385—34no—
243#385—32no—
244#385—34no—
245#385C2×D312yes—
246#383—32no—
247#382—32no—
248#381—31no—
249#381—32no—
250#381C2×D311yescharacteristic
251#381C2×D312yes—
252#380—32no—
253#380—34no—
254C4×C4⋊C4—34no—
255C4×C4⋊C4—34no—
256C4×C4⋊C4—32no—
257C4×C22⋊C4—32no—
258C4×C22⋊C4—34no—
259C4×C22⋊C4—34no—
260#376—32no—
261C24×C3C2411yescharacteristic
262C23×D3C2412yes—
263C22×C3×C4—22no—
264C22×C3×C4—24no—
265C22×C3×C4—24no—
266C22×C3×C4—22no—
267C22×C3×C4—24no—
268C22×C3×C4C2×D414yes—
269C22×C3×C4C2412yes—
270C22×C3×C4C2414yes—
271C22×C3×C4C2412yes—
272C22×C3×C4C2412yes—
273C22×C3×C4C2411yescharacteristic
274C22×C3×C4C2411yescharacteristic
275C22×C3×C4C2412yes—
276C2×#67—24no—
277C2×#67—24no—
278C2×#67#4118yes—
279C22×Dic3—22no—
280C22×Dic3—24no—
281C22×Dic3—24no—
282C22×Dic3—24no—
283C22×Dic3—22no—
284C22×Dic3—24no—
285C22×Dic3C2×D414yes—
286C22×Dic3C2×D414yes—
287C22×Dic3C2414yes—
288C22×Dic3C2412yes—
289C22×Dic3C2411yescharacteristic
290C22×Dic3C2411yescharacteristic
291C22×Dic3C2412yes—
292C22×Dic3C2412yes—
293C22×Dic3C2411yescharacteristic
294C22×Dic3C2411yescharacteristic
295#235—232no—
296C2×D12—24no—
297C2×D12—22no—
298C2×C4×D3—24no—
299C2×C4×D3—24no—
300C2×C4×D3—24no—
301C2×C4×D3—24no—
302C2×C4×D3—24no—
303C2×C4×D3—28no—
304C2×C4×D3—28no—
305C2×Dic6—232no—
306C2×Dic6—22no—
307C2×Dic6—24no—
308C2×Dic6—24no—
309C2×Dic6—24no—
310C2×Dic6—24no—
311C2×Dic6—24no—
312C2×Dic6—24no—
313C2×Dic6#4118yescomplemented
314C3×C4⋊C4—24no—
315C3×C4⋊C4—24no—
316C3×C4⋊C4—22no—
317C3×C4⋊C4—22no—
318C3×C4⋊C4—28no—
319C3×C22⋊C4—24no—
320C3×C22⋊C4—24no—
321C3×C42—24no—
322#217—24no—
323(C2×D3)⋊C4—28no—
324(C2×D3)⋊C4—28no—
325(C2×D3)⋊C4—28no—
326(C2×D3)⋊C4—24no—
327(C2×D3)⋊C4—24no—
328(C2×D3)⋊C4—28no—
329C4⋊Dic3—28no—
330Dic3⋊C4—24no—
331Dic3⋊C4—24no—
332Dic3⋊C4—24no—
333Dic3⋊C4—24no—
334Dic3⋊C4—24no—
335Dic3⋊C4—24no—
336Dic3⋊C4—28no—
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×Dic3—28no—
346C4×Dic3C2×D414yes—
347C4×Dic3C2×D414yes—
348C4×Dic3C2×D414yes—
349C4×Dic3C2×D414yes—
350C2×#41—68no—
351C2×#41—64no—
352C22×Q8—32no—
353C22×Q8—32no—
354C22×Q8—31no—
355C22×Q8—64no—
356C22×Q8—68no—
357C22×D4—32no—
358C22×D4—31no—
359C23×C4—31no—
360C23×C4—31no—
361C23×C4—31no—
362C23×C4—31no—
363C23×C4—32no—
364C23×C4—32no—
365C23×C4—32no—
366C23×C4C22×D311yescharacteristic
367C23×C4C22×D311yescharacteristic
368C23×C4C22×D311yescharacteristic
369#123—68no—
370#123—68no—
371C22⋊Q8—616no—
372C4×Q8—68no—
373C4×Q8—68no—
374C4×Q8—68no—
375C4×Q8—68no—
376C4×Q8—64no—
377C4×Q8—64no—
378C4×D4—64no—
379C4×D4—64no—
380C4×D4—68no—
381C4×D4—68no—
382#117—24no—
383#117—68no—
384C2×C4⋊C4—24no—
385C2×C4⋊C4—32no—
386C2×C4⋊C4—32no—
387C2×C4⋊C4—32no—
388C2×C4⋊C4—32no—
389C2×C4⋊C4—32no—
390C2×C4⋊C4—32no—
391C2×C4⋊C4—34no—
392C2×C4⋊C4—34no—
393C2×C4⋊C4—34no—
394C2×C4⋊C4—34no—
395C2×C4⋊C4—64no—
396C2×C4⋊C4—64no—
397C2×C4⋊C4C22×D312yes—
398C2×C4⋊C4C22×D312yes—
399C2×C4⋊C4C22×D311yescharacteristic
400C2×C4⋊C4C22×D311yescharacteristic
401C2×C4⋊C4C22×D314yes—
402C2×C22⋊C4—32no—
403C2×C22⋊C4—32no—
404C2×C22⋊C4—32no—
405C2×C22⋊C4—34no—
406C2×C22⋊C4—34no—
407C2×C22⋊C4—34no—
408C2×C22⋊C4—34no—
409C2×C22⋊C4C22×D312yes—
410C2×C22⋊C4C22×D312yes—
411C2×C42—38no—
412C2×C42—31no—
413C2×C42—31no—
414C2×C42—31no—
415C2×C42—31no—
416C2×C42—32no—
417C2×C42—32no—
418C2×C42—32no—
419C2×C42—32no—
420C2×C42—32no—
421C2×C42—32no—
422C2×C42—34no—
423C2×C42—68no—
424C2×C42—64no—
425C2×C42—64no—
426C2×C42C22×D312yes—
427#95—34no—
428#95—32no—
429#95—34no—
430#95—34no—
431#95—34no—
432#95—34no—
433#95—32no—
434#95—34no—
435#95—34no—
436#95C22×D312yes—
437#95C22×D314yes—
438#95C22×D314yes—
439#95C22×D314yes—
440#95C22×D312yes—
441C23×C3—22no—
442C23×C3—22no—
443C23×C3C22×D412yes—
444C23×C3C2×#4114yes—
445C23×C3C2511yescharacteristic
446C22×D3—22no—
447C22×D3—24no—
448C22×D3—22no—
449C22×D3—24no—
450C22×D3C2×#4114yes—
451C2×C3×C4—28no—
452C2×C3×C4—28no—
453C2×C3×C4—22no—
454C2×C3×C4—22no—
455C2×C3×C4—22no—
456C2×C3×C4—22no—
457C2×C3×C4—22no—
458C2×C3×C4—22no—
459C2×C3×C4—22no—
460C2×C3×C4—24no—
461C2×C3×C4—24no—
462C2×C3×C4—24no—
463C2×C3×C4—24no—
464C2×C3×C4—24no—
465C2×C3×C4—24no—
466C2×C3×C4—44no—
467C2×C3×C4C22×D412yes—
468C2×C3×C4C22×D412yes—
469C2×C3×C4C2×#4114yes—
470#67—216no—
471C2×Dic3—28no—
472C2×Dic3—22no—
473C2×Dic3—22no—
474C2×Dic3—22no—
475C2×Dic3—22no—
476C2×Dic3—22no—
477C2×Dic3—22no—
478C2×Dic3—28no—
479C2×Dic3—28no—
480C2×Dic3—22no—
481C2×Dic3—24no—
482C2×Dic3—24no—
483C2×Dic3—24no—
484C2×Dic3—24no—
485C2×Dic3—44no—
486C2×Dic3C22×D412yes—
487C2×Dic3C22×D412yes—
488C2×Dic3C22×D412yes—
489C2×Dic3C22×D412yes—
490C2×Dic3C2×#4114yes—
491C2×Dic3C2×#4114yes—
492C2×Dic3C2×#4114yes—
493C2×Dic32−1+414yes—
494D12—28no—
495C4×D3—216no—
496C4×D3—48no—
497C4×D3—48no—
498Dic6—216no—
499Dic6—28no—
500Dic6—216no—
501C24—32no—
502C24C23×D311yescharacteristic
503#41—632no—
504C2×Q8—38no—
505C2×Q8—64no—
506C2×Q8—64no—
507C2×Q8—64no—
508C2×Q8—64no—
509C2×Q8—64no—
510C2×Q8—632no—
511C2×Q8—62no—
512C2×Q8—64no—
513C2×D4—38no—
514C2×D4—64no—
515C2×D4—64no—
516C2×D4—64no—
517C2×D4—62no—
518C22×C4—22no—
519C22×C4—24no—
520C22×C4—24no—
521C22×C4—22no—
522C22×C4—24no—
523C22×C4—31no—
524C22×C4—31no—
525C22×C4—34no—
526C22×C4—34no—
527C22×C4—31no—
528C22×C4—31no—
529C22×C4—32no—
530C22×C4—32no—
531C22×C4—34no—
532C22×C4—32no—
533C22×C4—64no—
534C22×C4—64no—
535C22×C4—64no—
536C22×C4—68no—
537C22×C4—68no—
538C22×C4—64no—
539C22×C4—64no—
540C22×C4—64no—
541C22×C4—64no—
542C22×C4—62no—
543C22×C4—64no—
544C22×C4—62no—
545C22×C4—64no—
546C22×C4D3×D414yes—
547C22×C4C23×D312yes—
548C22×C4C23×D314yes—
549C22×C4C23×D312yes—
550C22×C4C23×D312yes—
551C22×C4C23×D311yescharacteristic
552C22×C4C23×D311yescharacteristic
553C22×C4C23×D312yes—
554C4⋊C4—24no—
555C4⋊C4—24no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—28no—
559C4⋊C4—64no—
560C4⋊C4—64no—
561C4⋊C4—64no—
562C4⋊C4—64no—
563C4⋊C4—68no—
564C4⋊C4—68no—
565C4⋊C4—68no—
566C4⋊C4—68no—
567C4⋊C4—64no—
568C4⋊C4—64no—
569C22⋊C4—24no—
570C22⋊C4—24no—
571C22⋊C4—64no—
572C22⋊C4—64no—
573C22⋊C4—68no—
574C22⋊C4—68no—
575C22⋊C4—68no—
576C22⋊C4—64no—
577C22⋊C4—68no—
578C42—24no—
579C42—34no—
580C42—34no—
581C42—34no—
582C42—34no—
583C42—64no—
584C42—64no—
585C42—64no—
586C42—64no—
587C42—64no—
588C42—64no—
589C42—68no—
590C22×C3—22no—
591C22×C3—22no—
592C22×C3—28no—
593C22×C3#52514yes—
594C22×C3C23×D411yescharacteristic
595C22×C3C22×#4112yes—
596C22×C3C22×#4112yes—
597C22×C3C2×2−1+412yes—
598C2×D3—24no—
599C2×D3—28no—
600C2×D3—42no—
601C2×D3—42no—
602C2×D3—44no—
603C3×C4—24no—
604C3×C4—22no—
605C3×C4—22no—
606C3×C4—42no—
607C3×C4—42no—
608C3×C4—42no—
609C3×C4—44no—
610Dic3—22no—
611Dic3—24no—
612Dic3—24no—
613Dic3—22no—
614Dic3—42no—
615Dic3—44no—
616Dic3#52514yes—
617Dic3#53614yes—
618C23—22no—
619C23—22no—
620C23—34no—
621C23—62no—
622C23—64no—
623C23—62no—
624C23—64no—
625C23C2×D3×D412yes—
626C23D3×#4114yes—
627C23C24×D311yescharacteristic
628Q8—616no—
629Q8—616no—
630Q8—68no—
631D4—68no—
632D4—616no—
633C2×C4—28no—
634C2×C4—28no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—32no—
649C2×C4—32no—
650C2×C4—34no—
651C2×C4—34no—
652C2×C4—32no—
653C2×C4—32no—
654C2×C4—34no—
655C2×C4—34no—
656C2×C4—44no—
657C2×C4—616no—
658C2×C4—68no—
659C2×C4—68no—
660C2×C4—68no—
661C2×C4—62no—
662C2×C4—62no—
663C2×C4—62no—
664C2×C4—62no—
665C2×C4—62no—
666C2×C4—62no—
667C2×C4—62no—
668C2×C4—64no—
669C2×C4—64no—
670C2×C4—64no—
671C2×C4—64no—
672C2×C4—128no—
673C2×C4—128no—
674C2×C4—124no—
675C2×C4C2×D3×D412yes—
676C2×C4C2×D3×D412yes—
677C2×C4D3×#4114yes—
678C2×C3—24no—
679C2×C3C2×#52511yescharacteristic
680C2×C3C2×#52511yescharacteristic
681C2×C3C2×#53611yescharacteristic
682C2×C3#348214yes—
683D3—44no—
684C22—22no—
685C22—28no—
686C22—22no—
687C22—68no—
688C22—64no—
689C22—122no—
690C22—124no—
691C22—122no—
692C22#565614yes—
693C22C22×D3×D411yescharacteristic
694C22C2×D3×#4112yes—
695C22C2×D3×#4112yes—
696C22D3×2−1+412yes—
697C4—24no—
698C4—22no—
699C4—22no—
700C4—34no—
701C4—34no—
702C4—42no—
703C4—44no—
704C4—42no—
705C4—42no—
706C4—62no—
707C4—64no—
708C4—62no—
709C4—64no—
710C4—124no—
711C4—122no—
712C3#3497111yescharacteristic, complemented, Sylow
713C2—24no—
714C2—124no—
715C2C2×#565611yescharacteristic
716C2D3×#52511yescharacteristic
717C2D3×#53611yescharacteristic
718C2#8682814yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2484C2
prime radicalC22×#235C22
hypercenter#2484D3
hypocenterC3#34971
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2484

Complements

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

NH ≅ G/Ntype
2. C22×#887C2semidirect
3. C3×#2484C2semidirect
4. #82254C2semidirect
6. #78191C2semidirect
7. #78131C2semidirect
8. #77784C2semidirect
9. #77740C2semidirect
10. #75371C2semidirect
11. #74941C2semidirect
12. #74405C2semidirect
13. #74380C2semidirect
14. #74075C2semidirect
16. #74052C2semidirect
17. #74051C2semidirect
93. #2484D3semidirect
313. C2×Dic6#41semidirect
712. C3#34971semidirect

Maximal subgroups

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

subgroupindex
C2×#55222
C22×#8872
C3×#24842
#822542
#821922
#781912
#781312
#777842
#777402
#753712
#749412
#744052
#743802
#740752
#740632
#740522
#740512
#349713