Finite Group Databasebeta

← #10955003

Subgroups — #10955003

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×#5817C211yesmaximal, characteristic
2C22×Q8×Dic3C211yesmaximal, characteristic, complemented
3C2×C4×#239C211yesmaximal, characteristic, complemented
4C3×Q8×C22⋊C4C211yesmaximal, characteristic, complemented
5#82461C211yesmaximal, characteristic, complemented
6#82457C211yesmaximal, characteristic, complemented
7C4×Q8×Dic3C211yesmaximal, characteristic, complemented
8#76493C213yesmaximal, complemented
9#76491C213yesmaximal, complemented
10#76480C213yesmaximal, complemented
11#76479C213yesmaximal, complemented
12#73883C213yesmaximal, complemented
13#73865C213yesmaximal, complemented
14#73467C213yesmaximal, complemented
15C4×#67⋊C4C213yesmaximal, complemented
16C4×#548—31nomaximal, Sylow
17C23×C3×Q8C2211yescharacteristic
18C22×#239C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21#5817C4116yescomplemented
22C2×#947C2213yes—
23C2×#946C2213yes—
24C2×Q8×Dic3—28no—
25C2×Q8×Dic3C2211yescharacteristic
26C2×Q8×Dic3C2213yes—
27C2×Q8×Dic3C2213yes—
28C2×Q8×Dic3C2211yescharacteristic
29C2×Q8×Dic3C2211yescharacteristic
30C2×#930C2213yes—
31C2×C4×#67C2213yes—
32C22×C4⋊Dic3C2213yes—
33C22×C4×Dic3C2213yes—
34C4×#239—28no—
35C2×#893C2213yes—
36C2×#892C2213yes—
37C2×#67⋊C4C2213yes—
38C2×#67⋊C4C2213yes—
39C2×C4×D12C2213yes—
40C2×C42×D3C2213yes—
41C3×#391C2213yes—
42C3×#391C2213yes—
43C3×#380C2213yes—
44C3×C4×C22⋊C4C2213yes—
45#4988C2213yes—
46#4987C2213yes—
47#4977C2213yes—
48Dic3×C4⋊C4C2213yes—
49Dic3×C4⋊C4C2213yes—
50#4972C2213yes—
51#4958C2216yes—
52Dic3×C22⋊C4C2213yes—
53Dic3×C22⋊C4C2211yescharacteristic
54#4939C2213yes—
55C4×(C2×D3)⋊C4C2213yes—
56C4×C4⋊Dic3C2213yes—
57C4×Dic3⋊C4C2213yes—
58C42×Dic3C2213yes—
59#4928C2213yes—
60#4670C2216yes—
61#4670C2216yes—
62#4669C2216yes—
63#4668C2216yes—
64#4666C2216yes—
65#4666C2213yes—
66#4666C2213yes—
67#4655C2216yes—
68#4649C2213yes—
69#4649C2213yes—
70C2×#548—31no—
71C2×C4×#41—31no—
72C22×C4×Q8—31no—
73#2367—33no—
74#2362—33no—
75#2361—33no—
76#2352—33no—
77#2351—31no—
78#2341—31no—
79Q8×C22⋊C4D311yescharacteristic, complemented
80C4×#124—33no—
81C4×C22⋊Q8—33no—
82C4×#121—33no—
83C42×Q8—31no—
84C42×D4—33no—
85C22×C3×Q8—24no—
86C22×C3×Q8—24no—
87C22×C3×Q8C2×C418yescomplemented
88C22×C3×Q8C2313yes—
89C22×C3×Q8C2311yescharacteristic
90C23×C3×C4C2313yes—
91C22×#67C2313yes—
92C23×Dic3C2313yes—
93C23×Dic3C2311yescharacteristic
94C2×#239—24no—
95C2×#239—24no—
96C2×#239C2×C418yescomplemented
97C22×D12C2313yes—
98C22×C4×D3C2313yes—
99C22×C4×D3C2313yes—
100C22×C4×D3C2313yes—
101C3×C4×Q8—24no—
102C3×C4×Q8—24no—
103C2×C3×C4⋊C4C2313yes—
104C2×C3×C4⋊C4C2313yes—
105C2×C3×C22⋊C4C2313yes—
106C2×C3×C22⋊C4C2311yescharacteristic
107C2×C3×C42C2313yes—
108C2×C3×C42C2313yes—
109C2×#217C2311yescharacteristic
110#947C2×C4124yescomplemented
111#946C2×C4124yescomplemented
112Q8×Dic3—24no—
113Q8×Dic3—212no—
114Q8×Dic3—212no—
115Q8×Dic3C2×C418yescomplemented
116Q8×Dic3D418yescomplemented
117#930C2×C4124yescomplemented
118C4×#67C2×C4124yescomplemented
119C2×(C2×D3)⋊C4C2313yes—
120C2×(C2×D3)⋊C4C2313yes—
121C2×(C2×D3)⋊C4C2316yes—
122C2×(C2×D3)⋊C4C2316yes—
123C2×(C2×D3)⋊C4C2313yes—
124C2×(C2×D3)⋊C4C2316yes—
125C2×C4⋊Dic3—212no—
126C2×C4⋊Dic3C2316yes—
127C2×C4⋊Dic3C2313yes—
128C2×C4⋊Dic3C2313yes—
129C2×C4⋊Dic3C2313yes—
130C2×Dic3⋊C4C2313yes—
131C2×Dic3⋊C4C2313yes—
132C2×Dic3⋊C4C2313yes—
133C2×C4×Dic3—212no—
134C2×C4×Dic3C2311yescharacteristic
135C2×C4×Dic3C2313yes—
136C2×C4×Dic3C2313yes—
137C2×C4×Dic3C2313yes—
138C2×C4×Dic3C2313yes—
139C2×C4×Dic3C2313yes—
140C2×C4×Dic3C2311yescharacteristic
141C2×C4×Dic3C2313yes—
142C2×C4×Dic3C2313yes—
143C2×C4×Dic3C2313yes—
144C2×C4×Dic3C2313yes—
145#893—212no—
146#892—212no—
147#67⋊C4—212no—
148#67⋊C4—212no—
149C4×D12—212no—
150C42×D3—212no—
151C3×#95C2313yes—
152C3×#95C2313yes—
153C3×#95C2316yes—
154C3⋊#95C2311yescharacteristic
155C3⋊#95C2313yes—
156C3⋊#95C2316yes—
157C3⋊#95C2316yes—
158C3⋊#95C2311yescharacteristic
159C3⋊#95C2313yes—
160C22×#41—31no—
161C23×Q8C2×D311yescharacteristic
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—31no—
170C2×C4×Q8—33no—
171C2×C4×Q8—33no—
172C2×C4×Q8—68no—
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#391C2×D313yes—
185#391C2×D313yes—
186#390—33no—
187#390—33no—
188#389—33no—
189#389—33no—
190#388—36no—
191#388—36no—
192#388—36no—
193#387—36no—
194#386—36no—
195#385—36no—
196#382—33no—
197#380—33no—
198#380C2×D313yes—
199C4×C4⋊C4—33no—
200C4×C4⋊C4—33no—
201C4×C4⋊C4—33no—
202C4×C4⋊C4—33no—
203C4×C4⋊C4—33no—
204C4×C4⋊C4—33no—
205C4×C22⋊C4—33no—
206C4×C22⋊C4—31no—
207C4×C22⋊C4—33no—
208C4×C22⋊C4—33no—
209C4×C22⋊C4—33no—
210C4×C22⋊C4—36no—
211C4×C22⋊C4C2×D313yes—
212C43—33no—
213C24×C3C2411yescharacteristic
214C23×D3C2413yes—
215C2×C3×Q8—232no—
216C2×C3×Q8—26no—
217C2×C3×Q8—26no—
218C2×C3×Q8C22×C4112yes—
219C2×C3×Q8C22×C414yes—
220C2×C3×Q8C2×D414yes—
221C2×C3×Q8#4114yes—
222C22×C3×C4—26no—
223C22×C3×C4—26no—
224C22×C3×C4C22×C4112yes—
225C22×C3×C4C2413yes—
226C22×C3×C4C2413yes—
227C22×C3×C4C2413yes—
228C22×C3×C4C2411yescharacteristic
229C22×C3×C4C2411yescharacteristic
230C22×C3×C4C2413yes—
231C2×#67—26no—
232C2×#67—26no—
233C2×#67C22×C4112yes—
234C22×Dic3—22no—
235C22×Dic3—22no—
236C22×Dic3—26no—
237C22×Dic3—26no—
238C22×Dic3—43no—
239C22×Dic3#4114yescomplemented
240C22×Dic3C2413yes—
241C22×Dic3C2411yescharacteristic
242C22×Dic3C2411yescharacteristic
243C22×Dic3C2413yes—
244C22×Dic3C2413yes—
245C22×Dic3C2411yescharacteristic
246C22×Dic3C2411yescharacteristic
247#239—232no—
248C2×D12—26no—
249C2×D12—26no—
250C2×D12C22×C4112yes—
251C2×C4×D3—26no—
252C2×C4×D3—26no—
253C2×C4×D3—26no—
254C2×C4×D3—26no—
255C2×C4×D3—26no—
256C2×C4×D3—26no—
257C2×C4×D3—26no—
258C2×C4×D3—43no—
259C2×C4×D3C22×C4112yes—
260C3×C4⋊C4—26no—
261C3×C4⋊C4—26no—
262C3×C22⋊C4—26no—
263C3×C22⋊C4#4114yescomplemented
264C3×C42—26no—
265C3×C42—26no—
266#217C22×C414yes—
267(C2×D3)⋊C4—26no—
268(C2×D3)⋊C4—212no—
269(C2×D3)⋊C4—26no—
270(C2×D3)⋊C4—46no—
271(C2×D3)⋊C4C22×C4112yes—
272(C2×D3)⋊C4C22×C4124yes—
273C4⋊Dic3—26no—
274C4⋊Dic3—212no—
275C4⋊Dic3C22×C4112yes—
276C4⋊Dic3C2×D4112yes—
277Dic3⋊C4—26no—
278Dic3⋊C4—26no—
279Dic3⋊C4C22×C4112yes—
280C4×Dic3—26no—
281C4×Dic3—26no—
282C4×Dic3—22no—
283C4×Dic3—26no—
284C4×Dic3—26no—
285C4×Dic3—26no—
286C4×Dic3—26no—
287C4×Dic3—22no—
288C4×Dic3—43no—
289C4×Dic3C22×C4112yes—
290C4×Dic3C2×D4112yes—
291C2×#41—38no—
292C2×#41—64no—
293C2×#41—64no—
294C22×Q8—24no—
295C22×Q8—24no—
296C22×Q8C4×D318yescomplemented
297C22×Q8C22×D313yes—
298C22×Q8C22×D311yescharacteristic
299C22×D4—33no—
300C22×D4—33no—
301C23×C4—31no—
302C23×C4—33no—
303C23×C4—33no—
304C23×C4—33no—
305C23×C4—33no—
306C23×C4C22×D313yes—
307#124—324no—
308C22⋊Q8—324no—
309#121—324no—
310C4×Q8—24no—
311C4×Q8—24no—
312C4×Q8—38no—
313C4×Q8—38no—
314C4×Q8—612no—
315C4×Q8—612no—
316C4×Q8—64no—
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—33no—
326C2×C4⋊C4—33no—
327C2×C4⋊C4—36no—
328C2×C4⋊C4—33no—
329C2×C4⋊C4—33no—
330C2×C4⋊C4—612no—
331C2×C4⋊C4C22×D313yes—
332C2×C4⋊C4C22×D313yes—
333C2×C22⋊C4—31no—
334C2×C22⋊C4—33no—
335C2×C22⋊C4—36no—
336C2×C22⋊C4—36no—
337C2×C22⋊C4—33no—
338C2×C22⋊C4—33no—
339C2×C22⋊C4—36no—
340C2×C22⋊C4C22×D313yes—
341C2×C22⋊C4C22×D311yescharacteristic
342C2×C42—33no—
343C2×C42—33no—
344C2×C42—33no—
345C2×C42—33no—
346C2×C42—33no—
347C2×C42—33no—
348C2×C42—33no—
349C2×C42—33no—
350C2×C42—31no—
351C2×C42—31no—
352C2×C42—33no—
353C2×C42—612no—
354C2×C42—612no—
355C2×C42C22×D313yes—
356C2×C42C22×D313yes—
357#95—31no—
358#95—31no—
359#95—33no—
360#95—33no—
361#95—36no—
362#95—36no—
363#95C22×D313yes—
364#95C22×D313yes—
365#95C22×D316yes—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3C23×C412yes—
371C23×C3C2×#4112yes—
372C23×C3C2×#4112yes—
373C23×C3C2511yescharacteristic
374C22×D3—23no—
375C22×D3—23no—
376C22×D3—23no—
377C22×D3—23no—
378C22×D3—23no—
379C22×D3—23no—
380C22×D3C23×C416yes—
381C3×Q8—224no—
382C3×Q8C4×D4116yescomplemented
383C2×C3×C4—23no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—23no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—224no—
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×C416yes—
398C2×C3×C4C22×D416yes—
399C2×C3×C4C2×#4116yes—
400C2×C3×C4C2×#4112yes—
401C2×C3×C4C2×#4112yes—
402#67—224no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—23no—
406C2×Dic3—23no—
407C2×Dic3—23no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—23no—
413C2×Dic3—28no—
414C2×Dic3—23no—
415C2×Dic3—23no—
416C2×Dic3—23no—
417C2×Dic3—412no—
418C2×Dic3C23×C412yes—
419C2×Dic3C23×C416yes—
420C2×Dic3C23×C412yes—
421C2×Dic3C22×D416yes—
422C2×Dic3C22×D412yes—
423C2×Dic3C2×#4112yes—
424C2×Dic3C2×#4112yes—
425C2×Dic32−1+412yes—
426D12—224no—
427C4×D3—224no—
428C4×D3—412no—
429C4×D3—412no—
430C24—33no—
431C24C23×D311yescharacteristic
432#41—632no—
433C2×Q8—232no—
434C2×Q8—26no—
435C2×Q8—26no—
436C2×Q8C2×C4×D3112yes—
437C2×Q8C2×C4×D314yes—
438C2×Q8D3×D414yes—
439C2×Q8#23714yes—
440C2×D4—312no—
441C2×D4—312no—
442C2×D4—66no—
443C2×D4—66no—
444C2×D4—66no—
445C2×D4—66no—
446C22×C4—26no—
447C22×C4—26no—
448C22×C4—31no—
449C22×C4—34no—
450C22×C4—31no—
451C22×C4—31no—
452C22×C4—33no—
453C22×C4—31no—
454C22×C4—33no—
455C22×C4—312no—
456C22×C4—33no—
457C22×C4—66no—
458C22×C4—66no—
459C22×C4—66no—
460C22×C4—66no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—66no—
464C22×C4—66no—
465C22×C4—66no—
466C22×C4—66no—
467C22×C4—66no—
468C22×C4—123no—
469C22×C4—123no—
470C22×C4C2×C4×D3112yes—
471C22×C4C23×D313yes—
472C22×C4C23×D313yes—
473C22×C4C23×D313yes—
474C22×C4C23×D311yescharacteristic
475C22×C4C23×D311yescharacteristic
476C22×C4C23×D313yes—
477C4⋊C4—26no—
478C4⋊C4—26no—
479C4⋊C4—312no—
480C4⋊C4—312no—
481C4⋊C4—312no—
482C4⋊C4—612no—
483C4⋊C4—66no—
484C4⋊C4—66no—
485C4⋊C4—66no—
486C22⋊C4—26no—
487C22⋊C4—34no—
488C22⋊C4—312no—
489C22⋊C4—324no—
490C22⋊C4—612no—
491C22⋊C4—66no—
492C22⋊C4—66no—
493C22⋊C4—126no—
494C22⋊C4#23914yescomplemented
495C42—26no—
496C42—26no—
497C42—312no—
498C42—312no—
499C42—62no—
500C42—66no—
501C42—66no—
502C42—66no—
503C42—66no—
504C42—62no—
505C42—66no—
506C42—66no—
507C42—123no—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—22no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3C4×#4114yescomplemented
515C22×C3C24×C411yescharacteristic
516C22×C3C23×D411yescharacteristic
517C22×C3C22×#4111yescharacteristic
518C22×C3C22×#4111yescharacteristic
519C22×C3C22×#4111yescharacteristic
520C22×C3C2×2−1+411yescharacteristic
521C22×C3#58511yescharacteristic
522C2×D3—26no—
523C2×D3—26no—
524C2×D3—26no—
525C2×D3—43no—
526C2×D3—43no—
527C2×D3—43no—
528C2×D3—43no—
529C3×C4—26no—
530C3×C4—22no—
531C3×C4—22no—
532C3×C4—43no—
533C3×C4—43no—
534C3×C4C2×C4×D4112yes—
535Dic3—22no—
536Dic3—26no—
537Dic3—22no—
538Dic3—22no—
539Dic3—26no—
540Dic3—43no—
541Dic3#54814yescomplemented
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—36no—
547C23—63no—
548C23—63no—
549C23—63no—
550C23—63no—
551C23—63no—
552C23—63no—
553C23C22×C4×D312yes—
554C23C2×#23912yes—
555C23D3×#4112yes—
556C23C24×D311yescharacteristic
557Q8—224no—
558Q8#67⋊C4116yescomplemented
559D4—624no—
560D4—624no—
561C2×C4—21no—
562C2×C4—23no—
563C2×C4—21no—
564C2×C4—23no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—224no—
568C2×C4—23no—
569C2×C4—23no—
570C2×C4—23no—
571C2×C4—23no—
572C2×C4—23no—
573C2×C4—23no—
574C2×C4—32no—
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—61no—
585C2×C4—61no—
586C2×C4—63no—
587C2×C4—63no—
588C2×C4—63no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—63no—
593C2×C4—624no—
594C2×C4—63no—
595C2×C4—63no—
596C2×C4—63no—
597C2×C4—1212no—
598C2×C4—1212no—
599C2×C4—1212no—
600C2×C4C22×C4×D316yes—
601C2×C4C22×C4×D316yes—
602C2×C4C2×D3×D416yes—
603C2×C4C2×#23716yes—
604C2×C4C2×#23912yes—
605C2×C4C2×#23912yes—
606C2×C3—24no—
607C2×C3C22×C4×D411yescharacteristic
608C2×C3C2×C4×#4111yescharacteristic
609C2×C3C4×2−1+411yescharacteristic
610C2×C3C2×#54811yescharacteristic
611C2×C3D4×#4111yescharacteristic
612C2×C3#348011yescharacteristic
613C2×C3#348211yescharacteristic
614D3—46no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—66no—
622C22—66no—
623C22—66no—
624C22—123no—
625C22—123no—
626C22—123no—
627C22—123no—
628C22C4×#23914yescomplemented
629C22C23×C4×D311yescharacteristic
630C22C22×D3×D411yescharacteristic
631C22C22×#23711yescharacteristic
632C22C22×#23911yescharacteristic
633C22C2×D3×#4111yescharacteristic
634C22D3×2−1+411yescharacteristic
635C22#596811yescharacteristic
636C4—26no—
637C4—22no—
638C4—22no—
639C4—34no—
640C4—43no—
641C4—43no—
642C4—66no—
643C4—62no—
644C4—62no—
645C4—62no—
646C4—66no—
647C4—123no—
648C4C2×#67⋊C4112yes—
649C3C4×#54811yescharacteristic, complemented, Sylow
650C2—24no—
651C2—126no—
652C2C22×#67⋊C411yescharacteristic
653C2C2×C4×#23911yescharacteristic
654C2#8587311yescharacteristic
655C2#8671111yescharacteristic
656C2D3×#54811yescharacteristic
657C2D4×#23911yescharacteristic
658C2#8674411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutatorC22×C3C24×C4
FittingC3×Q8×C22⋊C4C2
socle, cosocleC23×C3C25
prime radicalC2×#5817C2
hypercenterQ8×C22⋊C4D3
hypocenterC3C4×#548
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×Q8×Dic3C2semidirect
3. C2×C4×#239C2semidirect
4. C3×Q8×C22⋊C4C2semidirect
5. #82461C2semidirect
6. #82457C2semidirect
7. C4×Q8×Dic3C2semidirect
8. #76493C2semidirect
9. #76491C2semidirect
10. #76480C2semidirect
11. #76479C2semidirect
12. #73883C2semidirect
13. #73865C2semidirect
14. #73467C2semidirect
15. C4×#67⋊C4C2semidirect
21. #5817C4semidirect
79. Q8×C22⋊C4D3semidirect
87. C22×C3×Q8C2×C4semidirect
96. C2×#239C2×C4semidirect
110. #947C2×C4semidirect
111. #946C2×C4semidirect
115. Q8×Dic3C2×C4semidirect
116. Q8×Dic3D4semidirect
117. #930C2×C4semidirect
118. C4×#67C2×C4semidirect
239. C22×Dic3#41semidirect
263. C3×C22⋊C4#41semidirect
296. C22×Q8C4×D3semidirect
382. C3×Q8C4×D4semidirect
494. C22⋊C4#239semidirect
514. C22×C3C4×#41semidirect
541. Dic3#548semidirect
558. Q8#67⋊C4semidirect
628. C22C4×#239semidirect
649. C3C4×#548semidirect

Maximal subgroups

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

subgroupindex
C2×#58172
C22×Q8×Dic32
C2×C4×#2392
C3×Q8×C22⋊C42
#824612
#824572
C4×Q8×Dic32
#764932
#764912
#764802
#764792
#738832
#738652
#734672
C4×#67⋊C42
C4×#5483