Finite Group Databasebeta

← #11660199

Subgroups — #11660199

Order 768.

1626 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86274C211yesmaximal, characteristic
2C3×#2961C211yesmaximal, characteristic, complemented
3#83328C211yesmaximal, characteristic, complemented
4#79968C214yesmaximal, complemented
5#79967C214yesmaximal, complemented
6#79966C214yesmaximal, complemented
7#79965C214yesmaximal, complemented
8#79078C212yesmaximal, complemented
9#79072C212yesmaximal
10#79044C211yesmaximal, characteristic, complemented
11#79039C211yesmaximal, characteristic
12#78551C212yesmaximal
13#78544C212yesmaximal, complemented
14C2×#4709C211yesmaximal, characteristic
15#78480C211yesmaximal, characteristic, complemented
16#60252—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18#5729C412yes—
19#5729C412yes—
20#5689C412yes—
21#5689C412yes—
22#5634C412yes—
23#5634C412yes—
24#5623C412yes—
25#5623C412yes—
26#5578C2214yes—
27#5578C2214yes—
28D3×#117C2212yes—
29#5517C2212yes—
30C2×#872C2211yescharacteristic
31#5470C2211yescharacteristic
32C3×#446C2214yescomplemented
33C3×#446C2214yescomplemented
34C3×#424C2212yes—
35C3×#423C2211yescharacteristic
36C3×#405C2212yes—
37C2×C3×#97C2211yescharacteristic
38#5126C2212yes—
39#5122C2212yes—
40#5119C2212yes—
41Dic3×M16C2212yes—
42C2×(C2×D3)⋊C8C2211yescharacteristic
43C2×(C2×D3)⋊C8C2211yescharacteristic
44#5102C2212yes—
45#5101C2212yes—
46C2×Dic3⋊C8C2211yescharacteristic
47C2×Dic3⋊C8C2211yescharacteristic
48#5024C2212yescomplemented
49#5024C2212yescomplemented
50Q8×C3⋊C8C2212yescomplemented
51Q8×C3⋊C8C2212yescomplemented
52#4999C2212yes—
53#4997C2212yes—
54#4924C2211yescharacteristic
55C2×C4×C3⋊C8C2211yescharacteristic
56#4840C2214yes—
57#4840C2214yes—
58#4840C2214yes—
59#4840C2214yes—
60#4835C2214yes—
61#4833C2214yes—
62#4831C2212yescomplemented
63#4831C2214yescomplemented
64#4831C2212yescomplemented
65#4830C2214yescomplemented
66#4830C2212yescomplemented
67#4830C2212yescomplemented
68#4828C2214yescomplemented
69#4828C2214yescomplemented
70#4709C2212yes—
71#4709C2214yes—
72#4709C2212yes—
73#4709C2212yes—
74#4709C2212yes—
75#4705C2214yes—
76#4701C2214yescomplemented
77#4701C2214yescomplemented
78#4701C2214yescomplemented
79#4701C2214yescomplemented
80#4690C2214yes—
81#4685C2214yes—
82#4679C2214yescomplemented
83C8×Dic6C2214yescomplemented
84#3458—31no—
85#2987—34no—
86#2986—34no—
87#2985—34no—
88#2985—34no—
89#2962—32no—
90#2962—32no—
91#2961—31no—
92#2961D311yescharacteristic, complemented
93#2960—31no—
94#2960—31no—
95#2924—32no—
96#2924—32no—
97#2921—31no—
98C2×#433—31no—
99C22×C4×D3C2311yescharacteristic
100C2×C3×M16C2312yes—
101C22×C3×C8C2311yescharacteristic
102C22×C3×C8C2311yescharacteristic
103C3×C4⋊Q8C2×C411yescharacteristic
104C3×C4⋊Q8C2×C411yescharacteristic
105C3×#125C2×C412yes—
106C3×C22⋊Q8C2×C412yes—
107C3×C22⋊Q8C2×C412yes—
108C3×C4×Q8C2312yes—
109C3×C4×Q8C2312yes—
110C3×#117C2312yes—
111C2×C3×C42C2311yescharacteristic
112#946C2×C414yes—
113#946C2×C414yes—
114#944C2×C414yes—
115#944C2×C414yes—
116C2×(C2×D3)⋊C4C2×C411yescharacteristic
117C2×(C2×D3)⋊C4C2×C411yescharacteristic
118#926C2312yes—
119#924C2×C412yes—
120#924C2×C412yes—
121C2×Dic3⋊C4C2×C411yescharacteristic
122C2×Dic3⋊C4C2×C411yescharacteristic
123C2×C4×Dic3C2311yescharacteristic
124C2×#208C2312yes—
125C22×C3⋊C8C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127#897C2×C414yes—
128#897C2×C414yes—
129#896C2×C414yes—
130#896C2×C414yes—
131#896C2×C414yes—
132#896C2×C414yes—
133#892C2×C412yes—
134#892C2×C412yes—
135D3×C4⋊C4C2×C412yes—
136D3×C4⋊C4C2×C412yes—
137#890C2×C412yes—
138#890C2×C412yes—
139#889C2×C414yes—
140#889C2×C414yes—
141#889C2×C414yes—
142#889C2×C414yes—
143#888C2×C412yes—
144#888C2×C412yes—
145#888C2×C414yes—
146#888C2×C414yes—
147D3×C22⋊C4C2×C414yes—
148#878C2×C414yes—
149#878C2×C414yes—
150#877C2×C414yes—
151#872C2×C412yes—
152#872C2×C412yes—
153#872C2314yes—
154#872C2312yes—
155#872C2312yes—
156C42×D3C2314yes—
157#870C2×C412yes—
158#869C2×C411yescharacteristic
159#869C2×C411yescharacteristic
160C4×Dic6C2314yes—
161C4×Dic6C2314yes—
162C4×Dic6C2312yes—
163C4×Dic6C2312yes—
164C3×C4⋊C8C2312yes—
165C3×C4⋊C8C2312yes—
166C3×C4⋊C8C2314yes—
167C3×C4⋊C8C2314yes—
168C3×#97C2314yes—
169C3×#97C2312yes—
170C3×#97C2312yes—
171C3×C4×C8C2314yes—
172(C2×D3)⋊C8C2314yes—
173(C2×D3)⋊C8C2314yes—
174(C2×D3)⋊C8C2312yes—
175(C2×D3)⋊C8C2312yes—
176(C2×D3)⋊C8C2312yes—
177(C2×D3)⋊C8C2312yes—
178#815C2314yes—
179#815C2314yes—
180#815C2314yes—
181Dic3⋊C8C2312yes—
182Dic3⋊C8C2314yes—
183Dic3⋊C8C2312yes—
184Dic3⋊C8C2314yes—
185Dic3⋊C8C2314yes—
186Dic3⋊C8C2314yes—
187Dic3⋊C8C2312yes—
188Dic3⋊C8C2312yes—
189C8×Dic3C2314yes—
190C8×Dic3C2314yes—
191C8×Dic3C2314yes—
192C4⋊(C3⋊C8)C2312yes—
193C4⋊(C3⋊C8)C2312yes—
194C4⋊(C3⋊C8)C2314yes—
195C4⋊(C3⋊C8)C2314yes—
196#803C2314yes—
197C4×C3⋊C8C2311yescharacteristic
198C4×C3⋊C8C2311yescharacteristic
199C4×C3⋊C8C2311yescharacteristic
200C4×C3⋊C8C2311yescharacteristic
201C4×C3⋊C8C2314yes—
202#544—32no—
203#544—32no—
204#544—32no—
205#544—32no—
206#544—32no—
207#544—32no—
208#543—32no—
209#543—32no—
210#533—34no—
211#533—34no—
212#533—32no—
213#533—31no—
214#533C2×D311yescharacteristic
215C2×#117—32no—
216C2×#117—31no—
217#446—32no—
218#446—32no—
219#446—32no—
220#446—32no—
221#446—34no—
222#446—34no—
223#446—34no—
224#446—34no—
225#446—34no—
226#446—34no—
227#446—34no—
228#446—34no—
229#446C2×D314yescomplemented
230#446C2×D314yescomplemented
231C8×Q8—32no—
232C8×Q8—32no—
233C8×Q8—32no—
234C8×Q8—32no—
235C8×Q8—34no—
236C8×Q8—34no—
237#433—34no—
238#433—34no—
239#433—34no—
240#433—34no—
241#433—32no—
242#433—32no—
243#433—32no—
244#433—32no—
245#433—34no—
246#433—34no—
247#432—34no—
248#432—34no—
249#431—34no—
250#431—34no—
251#424—32no—
252#424—32no—
253#424—32no—
254#424C2×D312yes—
255#423—31no—
256#423C2×D311yescharacteristic
257C2×C4⋊C8—31no—
258C2×C4⋊C8—31no—
259#407—32no—
260C2×C22⋊C8—31no—
261C2×C22⋊C8—31no—
262#405—32no—
263#405—32no—
264#405—32no—
265#405C2×D312yes—
266C4×M16—32no—
267C2×#97C2×D311yescharacteristic
268C2×C4×C8—31no—
269C23×D3C22×C411yescharacteristic
270C2×C3×Q8C22×C412yes—
271C2×C3×Q8C22×C412yes—
272C22×C3×C4C22×C411yescharacteristic
273C22×C3×C4C22×C411yescharacteristic
274C22×C3×C4C2411yescharacteristic
275C22×Dic3C22×C411yescharacteristic
276C22×Dic3C22×C411yescharacteristic
277C22×Dic3C22×C411yescharacteristic
278C2×C4×D3—22no—
279C2×C4×D3—22no—
280C2×C4×D3C22×C412yes—
281C2×C4×D3C22×C414yes—
282C2×C4×D3C22×C414yes—
283C2×C4×D3C2412yes—
284C2×Dic6C22×C412yes—
285C2×Dic6C22×C412yes—
286C2×Dic6C22×C414yes—
287C2×Dic6C22×C414yes—
288C3×M16—24no—
289C2×C3×C8—22no—
290C2×C3×C8—22no—
291C2×C3×C8C2414yes—
292C2×C3×C8C2412yes—
293C2×C3×C8C2412yes—
294C3×C4⋊C4C22×C414yes—
295C3×C4⋊C4C22×C414yes—
296C3×C4⋊C4C22×C412yes—
297C3×C4⋊C4C22×C412yes—
298C3×C4⋊C4C22×C412yes—
299C3×C4⋊C4C22×C412yes—
300C3×C22⋊C4C22×C414yes—
301C3×C42C22×C411yescharacteristic
302C3×C42C22×C411yescharacteristic
303C3×C42C2411yescharacteristic
304C3×C42C2411yescharacteristic
305C3×C42C2414yes—
306#217C22×C414yes—
307(C2×D3)⋊C4C22×C412yes—
308(C2×D3)⋊C4C22×C412yes—
309(C2×D3)⋊C4C22×C412yes—
310(C2×D3)⋊C4C22×C412yes—
311(C2×D3)⋊C4C22×C414yes—
312(C2×D3)⋊C4C22×C414yes—
313C4⋊Dic3C22×C412yes—
314C4⋊Dic3C22×C412yes—
315C4⋊Dic3C22×C414yes—
316C4⋊Dic3C22×C414yes—
317C4⋊Dic3C22×C412yes—
318C4⋊Dic3C22×C412yes—
319Dic3⋊C4C22×C414yes—
320Dic3⋊C4C22×C414yes—
321Dic3⋊C4C22×C414yes—
322Dic3⋊C4C22×C414yes—
323Dic3⋊C4C22×C414yes—
324Dic3⋊C4C22×C414yes—
325Dic3⋊C4C22×C412yes—
326Dic3⋊C4C22×C412yes—
327Dic3⋊C4C22×C412yes—
328Dic3⋊C4C22×C412yes—
329Dic3⋊C4C22×C414yes—
330Dic3⋊C4C22×C414yes—
331C4×Dic3C22×C414yes—
332C4×Dic3C22×C414yes—
333C4×Dic3C22×C412yes—
334C4×Dic3C2414yes—
335C4×Dic3C2412yes—
336#208—24no—
337C2×C3⋊C8—22no—
338C2×C3⋊C8—22no—
339C2×C3⋊C8C2411yescharacteristic
340C2×C3⋊C8C2411yescharacteristic
341C2×C3⋊C8C2411yescharacteristic
342C2×C3⋊C8C2411yescharacteristic
343C2×C3⋊C8C2414yes—
344C23×C4—31no—
345C2×M16—32no—
346C2×M16C22×D312yes—
347C22×C8—31no—
348C22×C8—31no—
349C22×C8C22×D311yescharacteristic
350C22×C8C22×D311yescharacteristic
351C4⋊Q8—34no—
352C4⋊Q8—34no—
353C4⋊Q8—34no—
354C4⋊Q8—34no—
355C4⋊Q8—32no—
356C4⋊Q8—32no—
357C4⋊Q8—31no—
358C4⋊Q8—31no—
359C4⋊Q8C4×D311yescharacteristic
360C4⋊Q8C4×D311yescharacteristic
361#125—34no—
362#125—34no—
363#125—34no—
364#125—34no—
365#125—32no—
366#125—32no—
367#125—32no—
368#125C4×D312yes—
369C22⋊Q8—34no—
370C22⋊Q8—34no—
371C22⋊Q8—32no—
372C22⋊Q8—32no—
373C22⋊Q8—34no—
374C22⋊Q8—34no—
375C22⋊Q8—34no—
376C22⋊Q8—34no—
377C22⋊Q8—34no—
378C22⋊Q8—34no—
379C22⋊Q8—34no—
380C22⋊Q8—34no—
381C22⋊Q8C4×D312yes—
382C22⋊Q8C4×D312yes—
383C4×Q8—32no—
384C4×Q8—32no—
385C4×Q8—34no—
386C4×Q8—34no—
387C4×Q8C22×D312yes—
388C4×Q8C22×D312yes—
389#117—32no—
390#117—32no—
391#117—32no—
392#117—34no—
393#117—32no—
394#117—32no—
395#117—32no—
396#117—32no—
397#117—34no—
398#117C22×D312yes—
399C2×C4⋊C4—31no—
400C2×C4⋊C4—31no—
401C2×C4⋊C4—32no—
402C2×C4⋊C4—32no—
403C2×C22⋊C4—31no—
404C2×C22⋊C4—31no—
405C2×C22⋊C4—34no—
406C2×C42—31no—
407C2×C42—34no—
408C2×C42C22×D311yescharacteristic
409C4⋊C8—32no—
410C4⋊C8—32no—
411C4⋊C8—32no—
412C4⋊C8—32no—
413C4⋊C8—34no—
414C4⋊C8—34no—
415C4⋊C8—34no—
416C4⋊C8—34no—
417C4⋊C8—32no—
418C4⋊C8—32no—
419C4⋊C8—34no—
420C4⋊C8—34no—
421C4⋊C8C22×D312yes—
422C4⋊C8C22×D312yes—
423C4⋊C8C22×D314yes—
424C4⋊C8C22×D314yes—
425C22⋊C8—32no—
426C22⋊C8—32no—
427C22⋊C8—32no—
428C22⋊C8—32no—
429C22⋊C8—34no—
430C22⋊C8—34no—
431#97—34no—
432#97—34no—
433#97—34no—
434#97—34no—
435#97C22×D312yes—
436#97C22×D312yes—
437#97C22×D314yes—
438C4×C8—34no—
439C4×C8—34no—
440C4×C8—34no—
441C4×C8—31no—
442C4×C8—31no—
443C4×C8—31no—
444C4×C8—31no—
445C4×C8—34no—
446C4×C8C22×D314yes—
447C23×C3C23×C411yescharacteristic
448C22×D3—22no—
449C22×D3—41no—
450C22×D3—41no—
451C22×D3C23×C412yes—
452C3×Q8—42no—
453C3×Q8—42no—
454C2×C3×C4—41no—
455C2×C3×C4—41no—
456C2×C3×C4C23×C414yes—
457C2×C3×C4C23×C414yes—
458C2×C3×C4C23×C411yescharacteristic
459C2×C3×C4C23×C411yescharacteristic
460C2×C3×C4C23×C411yescharacteristic
461C2×C3×C4C23×C411yescharacteristic
462C2×C3×C4C23×C411yescharacteristic
463C2×C3×C42−1+412yes—
464C2×C3×C42−1+412yes—
465C2×C3×C4C2511yescharacteristic
466C2×Dic3—22no—
467C2×Dic3—41no—
468C2×Dic3—41no—
469C2×Dic3C23×C412yes—
470C2×Dic3C23×C412yes—
471C2×Dic3C23×C412yes—
472C2×Dic3C23×C414yes—
473C2×Dic3C23×C414yes—
474C4×D3—24no—
475C4×D3—22no—
476C4×D3—22no—
477C4×D3—44no—
478C4×D3—44no—
479Dic6—42no—
480Dic6—42no—
481Dic6—44no—
482Dic6—44no—
483C3×C8—22no—
484C3×C8—22no—
485C3×C8—24no—
486C3⋊C8—21no—
487C3⋊C8—21no—
488C3⋊C8—21no—
489C3⋊C8—21no—
490C3⋊C8—24no—
491C24—31no—
492C2×Q8—34no—
493C2×Q8—34no—
494C2×Q8—32no—
495C2×Q8—32no—
496C2×Q8C2×C4×D312yes—
497C2×Q8C2×C4×D312yes—
498C22×C4—31no—
499C22×C4—31no—
500C22×C4—31no—
501C22×C4—32no—
502C22×C4—32no—
503C22×C4—34no—
504C22×C4—34no—
505C22×C4—62no—
506C22×C4—62no—
507C22×C4C2×C4×D311yescharacteristic
508C22×C4C2×C4×D311yescharacteristic
509C22×C4C23×D311yescharacteristic
510M16—24no—
511M16—64no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—31no—
515C2×C8—31no—
516C2×C8—34no—
517C2×C8—31no—
518C2×C8—31no—
519C2×C8—62no—
520C2×C8—62no—
521C2×C8C23×D314yes—
522C2×C8C23×D312yes—
523C2×C8C23×D312yes—
524C4⋊C4—34no—
525C4⋊C4—34no—
526C4⋊C4—32no—
527C4⋊C4—32no—
528C4⋊C4—32no—
529C4⋊C4—32no—
530C4⋊C4—32no—
531C4⋊C4—32no—
532C4⋊C4—34no—
533C4⋊C4—34no—
534C4⋊C4—34no—
535C4⋊C4—34no—
536C4⋊C4—34no—
537C4⋊C4—34no—
538C4⋊C4—34no—
539C4⋊C4—34no—
540C4⋊C4—32no—
541C4⋊C4—32no—
542C4⋊C4C2×C4×D312yes—
543C4⋊C4C2×C4×D312yes—
544C4⋊C4C2×C4×D312yes—
545C4⋊C4C2×C4×D312yes—
546C4⋊C4C2×C4×D314yes—
547C4⋊C4C2×C4×D314yes—
548C22⋊C4—34no—
549C22⋊C4—34no—
550C22⋊C4—34no—
551C22⋊C4—32no—
552C22⋊C4—32no—
553C22⋊C4—32no—
554C22⋊C4—32no—
555C22⋊C4C2×C4×D314yes—
556C42—34no—
557C42—34no—
558C42—32no—
559C42—32no—
560C42—34no—
561C42C2×C4×D311yescharacteristic
562C42C2×C4×D311yescharacteristic
563C42C23×D311yescharacteristic
564C42C23×D311yescharacteristic
565C42C23×D314yes—
566C22×C3—21no—
567C22×C3—21no—
568C22×C3#51912yes—
569C22×C3C24×C411yescharacteristic
570C2×D3—22no—
571C2×D3—22no—
572C2×D3—22no—
573C2×D3—82no—
574C3×C4—21no—
575C3×C4—21no—
576C3×C4—24no—
577C3×C4—24no—
578C3×C4—21no—
579C3×C4—21no—
580C3×C4#51912yes—
581C3×C4C2×2−1+411yescharacteristic
582C3×C4C2×2−1+411yescharacteristic
583Dic3—22no—
584Dic3—24no—
585Dic3—24no—
586Dic3—22no—
587Dic3—22no—
588C23—32no—
589C23—62no—
590C23—121no—
591C23—121no—
592C23C22×C4×D311yescharacteristic
593Q8—42no—
594Q8—42no—
595Q8—122no—
596Q8—122no—
597Q8—124no—
598Q8—124no—
599C2×C4—32no—
600C2×C4—34no—
601C2×C4—34no—
602C2×C4—32no—
603C2×C4—32no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—64no—
610C2×C4—124no—
611C2×C4—124no—
612C2×C4—121no—
613C2×C4—121no—
614C2×C4C22×C4×D314yes—
615C2×C4C22×C4×D314yes—
616C2×C4C22×C4×D311yescharacteristic
617C2×C4C22×C4×D311yescharacteristic
618C2×C4C22×C4×D311yescharacteristic
619C2×C4C22×C4×D311yescharacteristic
620C2×C4C22×C4×D311yescharacteristic
621C2×C4#100712yes—
622C2×C4#101012yes—
623C2×C4C24×D311yescharacteristic
624C8—22no—
625C8—22no—
626C8—24no—
627C8—61no—
628C8—61no—
629C8—64no—
630C8—61no—
631C8—61no—
632C2×C3—22no—
633C2×C3C2×#51911yescharacteristic
634C2×C3#357311yescharacteristic
635C2×C3#357311yescharacteristic
636D3—42no—
637C22—21no—
638C22—21no—
639C22—62no—
640C22—62no—
641C22—62no—
642C22—242no—
643C22#557212yes—
644C22C23×C4×D311yescharacteristic
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—24no—
650C4—24no—
651C4—62no—
652C4—64no—
653C4—64no—
654C4—62no—
655C4—62no—
656C4#557212yes—
657C4C2×#100711yescharacteristic
658C4C2×#101011yescharacteristic
659C3#6025211yescharacteristic, complemented, Sylow
660C2—22no—
661C2—122no—
662C2C2×#557211yescharacteristic
663C2#8716611yescharacteristic
664C2#8716611yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D3
commutator, socleC22×C3C24×C4
FittingC3×#2961C2
prime radicalC23×D3C22×C4
cosocleC2×C3×C4C25
hypercenter#2961D3
hypocenterC3#60252
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #2961

Complements

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

NH ≅ G/Ntype
2. C3×#2961C2semidirect
3. #83328C2semidirect
4. #79968C2semidirect
5. #79967C2semidirect
6. #79966C2semidirect
7. #79965C2semidirect
8. #79078C2semidirect
10. #79044C2semidirect
13. #78544C2semidirect
15. #78480C2semidirect
32. C3×#446C22semidirect
33. C3×#446C22semidirect
48. #5024C22semidirect
49. #5024C22semidirect
50. Q8×C3⋊C8C22semidirect
51. Q8×C3⋊C8C22semidirect
62. #4831C22semidirect
63. #4831C22semidirect
64. #4831C22semidirect
65. #4830C22semidirect
66. #4830C22semidirect
67. #4830C22semidirect
68. #4828C22semidirect
69. #4828C22semidirect
76. #4701C22semidirect
77. #4701C22semidirect
78. #4701C22semidirect
79. #4701C22semidirect
82. #4679C22semidirect
83. C8×Dic6C22semidirect
92. #2961D3semidirect
229. #446C2×D3semidirect
230. #446C2×D3semidirect
659. C3#60252semidirect

Maximal subgroups

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

subgroupindex
#862742
C3×#29612
#833282
#799682
#799672
#799662
#799652
#790782
#790722
#790442
#790392
#785512
#785442
C2×#47092
#784802
#602523