Finite Group Databasebeta

← #10955007

Subgroups — #10955007

Order 768.

3232 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5576C211yesmaximal, characteristic
2C2×C4×D3×Q8C211yesmaximal, characteristic, complemented
3C2×C4×#239C211yesmaximal, characteristic, complemented
4C3×#2351C211yesmaximal, characteristic, complemented
5Q8×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
6#82457C211yesmaximal, characteristic, complemented
7C4×Q8×Dic3C211yesmaximal, characteristic, complemented
8#76206C213yesmaximal, complemented
9#75984C213yesmaximal, complemented
10#75965C213yesmaximal, complemented
11#75964C213yesmaximal, complemented
12#74229C213yesmaximal, complemented
13#74149C213yesmaximal, complemented
14#73655C213yesmaximal, complemented
15C4×#893C213yesmaximal, complemented
16C4×#548—31nomaximal, Sylow
17C22×#239C2211yescharacteristic
18C22×D3×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×Q8×Dic3C2211yescharacteristic
23C2×Q8×Dic3C2211yescharacteristic
24#5576C4116yescomplemented
25C4×#239—28no—
26C4×D3×Q8—28no—
27C2×#897C2213yes—
28C2×#895C2213yes—
29C2×#893C2213yes—
30C2×#893C2213yes—
31C2×#893C2213yes—
32C2×#892C2213yes—
33C2×D3×C4⋊C4C2213yes—
34C2×#887C2213yes—
35C2×#875C2213yes—
36C2×C4×D12C2213yes—
37C2×C4×D12C2213yes—
38C2×C42×D3C2213yes—
39C2×C42×D3C2213yes—
40C2×C4×Dic6C2213yes—
41C3×#389C2213yes—
42C3×#389C2213yes—
43C3×#382C2213yes—
44C3×C4×C4⋊C4C2213yes—
45#4990C2216yes—
46#4989C2216yes—
47#4988C2213yes—
48#4987C2213yes—
49#4987C2213yes—
50Dic3×C4⋊C4C2213yes—
51Dic3×C4⋊C4C2213yes—
52Dic3×C4⋊C4C2213yes—
53Dic3×C4⋊C4C2213yes—
54#4973C2213yes—
55#4939C2213yes—
56C4×(C2×D3)⋊C4C2213yes—
57C4×(C2×D3)⋊C4C2213yes—
58C4×C4⋊Dic3C2213yes—
59C42×Dic3C2213yes—
60#4929C2213yes—
61#4669C2216yes—
62#4669C2216yes—
63#4666C2211yescharacteristic
64#4666C2213yes—
65#4666C2213yes—
66#4666C2213yes—
67#4666C2216yes—
68#4665C2216yes—
69#4654C2216yes—
70C2×#548—31no—
71C2×C4×#41—31no—
72C22×C4×Q8—31no—
73#2367—33no—
74#2362—33no—
75#2361—33no—
76#2352—33no—
77#2351D311yescharacteristic, complemented
78#2341—31no—
79Q8×C22⋊C4—31no—
80C4×#124—33no—
81C4×C22⋊Q8—33no—
82C4×#121—33no—
83C42×Q8—31no—
84C42×D4—33no—
85C22×C3×Q8C2311yescharacteristic
86C2×#239—24no—
87C2×#239—24no—
88C2×#239C2×C418yescomplemented
89C2×D3×Q8—24no—
90C2×D3×Q8—24no—
91C2×D3×Q8C2×C418yescomplemented
92C22×D12C2313yes—
93C22×D12C2313yes—
94C22×C4×D3C2313yes—
95C22×C4×D3C2313yes—
96C22×C4×D3C2313yes—
97C22×C4×D3C2313yes—
98C22×C4×D3C2313yes—
99C22×C4×D3C2311yescharacteristic
100C22×Dic6C2313yes—
101C3×C4×Q8—24no—
102C3×C4×Q8—24no—
103C3×C4×Q8C2×C418yescomplemented
104C2×C3×C4⋊C4C2313yes—
105C2×C3×C4⋊C4C2313yes—
106C2×C3×C4⋊C4C2313yes—
107C2×C3×C4⋊C4C2313yes—
108C2×C3×C4⋊C4C2313yes—
109C2×C3×C42C2313yes—
110C2×C3×C42C2313yes—
111C2×C3×C42C2313yes—
112Q8×Dic3—24no—
113Q8×Dic3D418yescomplemented
114C2×(C2×D3)⋊C4C2313yes—
115C2×(C2×D3)⋊C4C2316yes—
116C2×(C2×D3)⋊C4C2313yes—
117C2×(C2×D3)⋊C4C2313yes—
118C2×(C2×D3)⋊C4C2316yes—
119C2×(C2×D3)⋊C4C2311yescharacteristic
120C2×(C2×D3)⋊C4C2313yes—
121C2×(C2×D3)⋊C4C2316yes—
122C2×(C2×D3)⋊C4C2311yescharacteristic
123C2×C4⋊Dic3C2313yes—
124C2×C4⋊Dic3C2313yes—
125C2×C4⋊Dic3C2313yes—
126C2×Dic3⋊C4C2316yes—
127C2×C4×Dic3C2313yes—
128C2×C4×Dic3C2311yescharacteristic
129C2×C4×Dic3C2313yes—
130C2×C4×Dic3C2313yes—
131C2×C4×Dic3C2313yes—
132C2×C4×Dic3C2311yescharacteristic
133C2×C4×Dic3C2313yes—
134C2×C4×Dic3C2313yes—
135C2×C4×Dic3C2313yes—
136C2×C4×Dic3C2313yes—
137#897C2×C4124yescomplemented
138#895C2×C4124yescomplemented
139#893—212no—
140#893—212no—
141#893—212no—
142#892—212no—
143D3×C4⋊C4—212no—
144#887—212no—
145#875C2×C4124yescomplemented
146C4×D12—212no—
147C4×D12C2×C4124yescomplemented
148C42×D3—212no—
149C42×D3—212no—
150C4×Dic6—212no—
151C3×#95C2316yes—
152C3×#95C2311yescharacteristic
153C3×#95C2313yes—
154C3⋊#95C2313yes—
155C3⋊#95C2316yes—
156C3⋊#95C2316yes—
157C3⋊#95C2313yes—
158C3⋊#95C2313yes—
159C3⋊#95C2311yescharacteristic
160C22×#41—31no—
161C23×Q8—31no—
162#548—316no—
163C2×#124—33no—
164C2×C22⋊Q8—33no—
165C2×#121—33no—
166C4×#41—68no—
167C2×C4×Q8—31no—
168C2×C4×Q8—31no—
169C2×C4×Q8—33no—
170C2×C4×Q8—33no—
171C2×C4×Q8—68no—
172C2×C4×Q8C2×D311yescharacteristic
173C2×C4×Q8C2×D311yescharacteristic
174C2×C4×Q8C2×D311yescharacteristic
175C2×C4×D4—33no—
176C2×C4×D4—33no—
177C2×C4×D4—33no—
178C2×C4×D4—33no—
179C2×C4×D4—33no—
180C2×#117—33no—
181C22×C4⋊C4—33no—
182C22×C42—33no—
183C22×C42—33no—
184#391—33no—
185#391—33no—
186#390—33no—
187#390—33no—
188#389C2×D313yes—
189#389C2×D313yes—
190#388—36no—
191#388—36no—
192#388—36no—
193#387—36no—
194#386—36no—
195#385—36no—
196#382C2×D313yes—
197#380—33no—
198#380—33no—
199C4×C4⋊C4—33no—
200C4×C4⋊C4—33no—
201C4×C4⋊C4—33no—
202C4×C4⋊C4—33no—
203C4×C4⋊C4—33no—
204C4×C4⋊C4C2×D313yes—
205C4×C22⋊C4—33no—
206C4×C22⋊C4—33no—
207C4×C22⋊C4—33no—
208C4×C22⋊C4—33no—
209C4×C22⋊C4—31no—
210C4×C22⋊C4—33no—
211C4×C22⋊C4—36no—
212C43—33no—
213C23×D3C2413yes—
214C23×D3C2411yescharacteristic
215C2×C3×Q8C22×C414yes—
216C2×C3×Q8C2×D414yes—
217C2×C3×Q8#4114yes—
218C22×C3×C4C2413yes—
219C22×C3×C4C2413yes—
220C22×C3×C4C2413yes—
221C22×C3×C4C2411yescharacteristic
222C22×C3×C4C2411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C22×C3×C4C2413yes—
225C22×Dic3C2413yes—
226C22×Dic3C2411yescharacteristic
227C22×Dic3C2413yes—
228C22×Dic3C2413yes—
229C22×Dic3C2411yescharacteristic
230C22×Dic3C2411yescharacteristic
231#239—232no—
232D3×Q8—232no—
233C2×D12—26no—
234C2×D12—26no—
235C2×D12—26no—
236C2×D12—26no—
237C2×D12C22×C4112yes—
238C2×D12C22×C4112yes—
239C2×C4×D3—26no—
240C2×C4×D3—26no—
241C2×C4×D3—26no—
242C2×C4×D3—26no—
243C2×C4×D3—22no—
244C2×C4×D3—22no—
245C2×C4×D3—26no—
246C2×C4×D3—26no—
247C2×C4×D3—26no—
248C2×C4×D3—26no—
249C2×C4×D3—26no—
250C2×C4×D3—26no—
251C2×C4×D3—26no—
252C2×C4×D3—43no—
253C2×C4×D3—43no—
254C2×C4×D3C22×C4112yes—
255C2×C4×D3C22×C4112yes—
256C2×C4×D3#4114yescomplemented
257C2×Dic6—26no—
258C2×Dic6—26no—
259C2×Dic6C22×C4112yes—
260C3×C4⋊C4—26no—
261C3×C4⋊C4—26no—
262C3×C4⋊C4—26no—
263C3×C4⋊C4—26no—
264C3×C4⋊C4C22×C4112yes—
265C3×C42—26no—
266C3×C42—26no—
267C3×C42C22×C4112yes—
268(C2×D3)⋊C4—26no—
269(C2×D3)⋊C4—26no—
270(C2×D3)⋊C4—212no—
271(C2×D3)⋊C4—26no—
272(C2×D3)⋊C4—46no—
273(C2×D3)⋊C4C22×C414yes—
274(C2×D3)⋊C4C22×C4112yes—
275(C2×D3)⋊C4C22×C4124yes—
276(C2×D3)⋊C4#4114yescomplemented
277C4⋊Dic3—26no—
278C4⋊Dic3C22×C4112yes—
279C4⋊Dic3C2×D4112yes—
280Dic3⋊C4—212no—
281C4×Dic3—26no—
282C4×Dic3—26no—
283C4×Dic3—22no—
284C4×Dic3—26no—
285C4×Dic3—26no—
286C4×Dic3—26no—
287C4×Dic3—26no—
288C4×Dic3—22no—
289C4×Dic3—43no—
290C4×Dic3C2×D4112yes—
291C2×#41—38no—
292C2×#41—64no—
293C2×#41—64no—
294C22×Q8—38no—
295C22×Q8—33no—
296C22×Q8—64no—
297C22×Q8—64no—
298C22×Q8C22×D311yescharacteristic
299C22×D4—33no—
300C22×D4—33no—
301C23×C4—33no—
302C23×C4—33no—
303C23×C4—31no—
304C23×C4—33no—
305C23×C4—33no—
306C23×C4—33no—
307#124—324no—
308C22⋊Q8—324no—
309#121—324no—
310C4×Q8—24no—
311C4×Q8—24no—
312C4×Q8—38no—
313C4×Q8—612no—
314C4×Q8—612no—
315C4×Q8—64no—
316C4×Q8C4×D318yescomplemented
317C4×D4—324no—
318C4×D4—612no—
319C4×D4—612no—
320C4×D4—612no—
321C4×D4—612no—
322#117—612no—
323C2×C4⋊C4—33no—
324C2×C4⋊C4—33no—
325C2×C4⋊C4—36no—
326C2×C4⋊C4—33no—
327C2×C4⋊C4—612no—
328C2×C4⋊C4C22×D313yes—
329C2×C4⋊C4C22×D313yes—
330C2×C4⋊C4C22×D313yes—
331C2×C4⋊C4C22×D313yes—
332C2×C4⋊C4C22×D313yes—
333C2×C22⋊C4—31no—
334C2×C22⋊C4—31no—
335C2×C22⋊C4—33no—
336C2×C22⋊C4—36no—
337C2×C22⋊C4—36no—
338C2×C22⋊C4—33no—
339C2×C22⋊C4—33no—
340C2×C22⋊C4—33no—
341C2×C22⋊C4—36no—
342C2×C42—33no—
343C2×C42—33no—
344C2×C42—33no—
345C2×C42—33no—
346C2×C42—33no—
347C2×C42—33no—
348C2×C42—33no—
349C2×C42—31no—
350C2×C42—31no—
351C2×C42—33no—
352C2×C42—612no—
353C2×C42—612no—
354C2×C42C22×D313yes—
355C2×C42C22×D313yes—
356C2×C42C22×D313yes—
357#95—31no—
358#95—33no—
359#95—33no—
360#95—33no—
361#95—36no—
362#95—36no—
363#95C22×D311yescharacteristic
364#95C22×D313yes—
365#95C22×D316yes—
366C23×C3C2511yescharacteristic
367C22×D3—21no—
368C22×D3—21no—
369C22×D3—21no—
370C22×D3—21no—
371C22×D3—23no—
372C22×D3—23no—
373C22×D3—23no—
374C22×D3—23no—
375C22×D3—23no—
376C22×D3—23no—
377C22×D3C23×C416yes—
378C22×D3C23×C412yes—
379C22×D3C2×#4112yes—
380C22×D3C2×#4112yes—
381C3×Q8C4×D4116yescomplemented
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—23no—
385C2×C3×C4—23no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—23no—
391C2×C3×C4—23no—
392C2×C3×C4—23no—
393C2×C3×C4—23no—
394C2×C3×C4—23no—
395C2×C3×C4—23no—
396C2×C3×C4C23×C416yes—
397C2×C3×C4C23×C412yes—
398C2×C3×C4C23×C416yes—
399C2×C3×C4C22×D416yes—
400C2×C3×C4C2×#4116yes—
401C2×C3×C4C2×#4112yes—
402C2×C3×C4C2×#4112yes—
403C2×Dic3—21no—
404C2×Dic3—23no—
405C2×Dic3—21no—
406C2×Dic3—23no—
407C2×Dic3—23no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—23no—
411C2×Dic3—23no—
412C2×Dic3—23no—
413C2×Dic3—23no—
414C2×Dic3C23×C412yes—
415C2×Dic3C23×C416yes—
416C2×Dic3C22×D416yes—
417C2×Dic3C22×D412yes—
418C2×Dic3C2×#4112yes—
419C2×Dic3C2×#4112yes—
420C2×Dic32−1+412yes—
421D12—224no—
422D12—224no—
423C4×D3—224no—
424C4×D3—224no—
425C4×D3—28no—
426C4×D3—412no—
427C4×D3—412no—
428C4×D3—412no—
429Dic6—224no—
430C24—31no—
431C24—33no—
432#41—632no—
433C2×Q8—312no—
434C2×Q8—632no—
435C2×Q8—66no—
436C2×Q8—66no—
437C2×Q8C2×C4×D314yes—
438C2×Q8D3×D414yes—
439C2×Q8#23914yes—
440C2×D4—312no—
441C2×D4—312no—
442C2×D4—66no—
443C2×D4—66no—
444C2×D4—66no—
445C2×D4—66no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—33no—
450C22×C4—33no—
451C22×C4—312no—
452C22×C4—312no—
453C22×C4—34no—
454C22×C4—33no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—66no—
458C22×C4—66no—
459C22×C4—66no—
460C22×C4—66no—
461C22×C4—66no—
462C22×C4—66no—
463C22×C4—66no—
464C22×C4—66no—
465C22×C4—66no—
466C22×C4—66no—
467C22×C4—66no—
468C22×C4—123no—
469C22×C4—123no—
470C22×C4C23×D313yes—
471C22×C4C23×D313yes—
472C22×C4C23×D313yes—
473C22×C4C23×D311yescharacteristic
474C22×C4C23×D311yescharacteristic
475C22×C4C23×D311yescharacteristic
476C22×C4C23×D313yes—
477C4⋊C4—26no—
478C4⋊C4—26no—
479C4⋊C4—26no—
480C4⋊C4—26no—
481C4⋊C4—312no—
482C4⋊C4—312no—
483C4⋊C4—612no—
484C4⋊C4—66no—
485C4⋊C4C2×C4×D3112yes—
486C22⋊C4—34no—
487C22⋊C4—312no—
488C22⋊C4—324no—
489C22⋊C4—34no—
490C22⋊C4—612no—
491C22⋊C4—66no—
492C22⋊C4—66no—
493C22⋊C4—66no—
494C22⋊C4—126no—
495C42—26no—
496C42—26no—
497C42—312no—
498C42—62no—
499C42—66no—
500C42—66no—
501C42—66no—
502C42—66no—
503C42—66no—
504C42—62no—
505C42—66no—
506C42—123no—
507C42C2×C4×D3112yes—
508C22×C3C24×C411yescharacteristic
509C22×C3C23×D411yescharacteristic
510C22×C3C22×#4111yescharacteristic
511C22×C3C22×#4111yescharacteristic
512C22×C3C22×#4111yescharacteristic
513C22×C3C2×2−1+411yescharacteristic
514C22×C3#58511yescharacteristic
515C2×D3—22no—
516C2×D3—22no—
517C2×D3—22no—
518C2×D3—22no—
519C2×D3—22no—
520C2×D3—22no—
521C2×D3—26no—
522C2×D3—26no—
523C2×D3—26no—
524C2×D3—43no—
525C2×D3—43no—
526C2×D3—43no—
527C2×D3—43no—
528C2×D3C4×#4114yescomplemented
529C3×C4—26no—
530C3×C4—22no—
531C3×C4—22no—
532C3×C4—22no—
533C3×C4—43no—
534C3×C4—43no—
535C3×C4C2×C4×D4112yes—
536Dic3—26no—
537Dic3—22no—
538Dic3—26no—
539Dic3—22no—
540Dic3—43no—
541Dic3#54814yescomplemented
542C23—32no—
543C23—32no—
544C23—32no—
545C23—36no—
546C23—61no—
547C23—61no—
548C23—61no—
549C23—61no—
550C23—63no—
551C23—63no—
552C23—63no—
553C23—63no—
554C23—63no—
555C23—63no—
556C23C24×D311yescharacteristic
557Q8—624no—
558Q8#893116yescomplemented
559D4—624no—
560D4—624no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—23no—
564C2×C4—23no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—23no—
570C2×C4—23no—
571C2×C4—23no—
572C2×C4—23no—
573C2×C4—23no—
574C2×C4—23no—
575C2×C4—32no—
576C2×C4—32no—
577C2×C4—32no—
578C2×C4—36no—
579C2×C4—32no—
580C2×C4—36no—
581C2×C4—32no—
582C2×C4—68no—
583C2×C4—61no—
584C2×C4—63no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—63no—
588C2×C4—63no—
589C2×C4—61no—
590C2×C4—63no—
591C2×C4—624no—
592C2×C4—624no—
593C2×C4—63no—
594C2×C4—63no—
595C2×C4—63no—
596C2×C4—1212no—
597C2×C4—1212no—
598C2×C4—1212no—
599C2×C4C22×C4×D312yes—
600C2×C4C22×C4×D316yes—
601C2×C4C22×C4×D316yes—
602C2×C4C2×D3×D416yes—
603C2×C4C2×#23916yes—
604C2×C4D3×#4112yes—
605C2×C4D3×#4112yes—
606C2×C3C22×C4×D411yescharacteristic
607C2×C3C2×C4×#4111yescharacteristic
608C2×C3C4×2−1+411yescharacteristic
609C2×C3C2×#54811yescharacteristic
610C2×C3D4×#4111yescharacteristic
611C2×C3#348011yescharacteristic
612C2×C3#348211yescharacteristic
613D3—24no—
614D3—46no—
615C22—34no—
616C22—62no—
617C22—62no—
618C22—62no—
619C22—62no—
620C22—62no—
621C22—62no—
622C22—66no—
623C22—66no—
624C22—66no—
625C22—123no—
626C22—123no—
627C22—123no—
628C22—123no—
629C22C23×C4×D311yescharacteristic
630C22C22×D3×D411yescharacteristic
631C22C22×#23911yescharacteristic
632C22C2×D3×#4111yescharacteristic
633C22C2×D3×#4111yescharacteristic
634C22D3×2−1+411yescharacteristic
635C22#596811yescharacteristic
636C4—26no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—34no—
641C4—43no—
642C4—43no—
643C4—66no—
644C4—62no—
645C4—66no—
646C4—62no—
647C4—123no—
648C4C2×#893112yes—
649C3C4×#54811yescharacteristic, complemented, Sylow
650C2—64no—
651C2—126no—
652C2C22×#89311yescharacteristic
653C2C4×D3×#4111yescharacteristic
654C2#8587311yescharacteristic
655C2#8671011yescharacteristic
656C2D3×#54811yescharacteristic
657C2#8701611yescharacteristic
658C2#8701711yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutatorC22×C3C24×C4
FittingC3×#2351C2
socle, cosocleC23×C3C25
prime radicalC2×#5576C2
hypercenter#2351D3
hypocenterC3C4×#548
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
2. C2×C4×D3×Q8C2semidirect
3. C2×C4×#239C2semidirect
4. C3×#2351C2semidirect
5. Q8×(C2×D3)⋊C4C2semidirect
6. #82457C2semidirect
7. C4×Q8×Dic3C2semidirect
8. #76206C2semidirect
9. #75984C2semidirect
10. #75965C2semidirect
11. #75964C2semidirect
12. #74229C2semidirect
13. #74149C2semidirect
14. #73655C2semidirect
15. C4×#893C2semidirect
24. #5576C4semidirect
77. #2351D3semidirect
88. C2×#239C2×C4semidirect
91. C2×D3×Q8C2×C4semidirect
103. C3×C4×Q8C2×C4semidirect
113. Q8×Dic3D4semidirect
137. #897C2×C4semidirect
138. #895C2×C4semidirect
145. #875C2×C4semidirect
147. C4×D12C2×C4semidirect
256. C2×C4×D3#41semidirect
276. (C2×D3)⋊C4#41semidirect
316. C4×Q8C4×D3semidirect
381. C3×Q8C4×D4semidirect
528. C2×D3C4×#41semidirect
541. Dic3#548semidirect
558. Q8#893semidirect
649. C3C4×#548semidirect

Maximal subgroups

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

subgroupindex
C2×#55762
C2×C4×D3×Q82
C2×C4×#2392
C3×#23512
Q8×(C2×D3)⋊C42
#824572
C4×Q8×Dic32
#762062
#759842
#759652
#759642
#742292
#741492
#736552
C4×#8932
C4×#5483