Finite Group Databasebeta

← #11069561

Subgroups — #11069561

Order 768.

1954 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5583C212yesmaximal, complemented
2C3×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3#82500C211yesmaximal, characteristic
4#82493C212yesmaximal, complemented
5C4×#944C211yesmaximal, characteristic, complemented
6#76251C212yesmaximal, complemented
7#76214C212yesmaximal, complemented
8#76125C214yesmaximal, complemented
9#76122C214yesmaximal, complemented
10#74369C211yesmaximal, characteristic
11#74355C211yesmaximal, characteristic, complemented
12#74333C212yesmaximal
13#74330C212yesmaximal, complemented
14#73693C212yesmaximal
15#73683C212yesmaximal, complemented
16C4⋊C4×Dic6C211yesmaximal, characteristic, complemented
17#73617C211yesmaximal, characteristic
18#39379—31nomaximal, Sylow
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2212yes—
21C2×#946C2212yes—
22C2×#944C2211yescharacteristic
23#5583—216no—
24C2×#898C2214yes—
25C2×#897C2212yes—
26C2×#890C2212yes—
27C2×#889C2214yes—
28C2×#876C2214yes—
29C2×C4×D12C2211yescharacteristic
30C2×#872C2212yes—
31C2×#870C2214yes—
32C2×C4×Dic6C2211yescharacteristic
33C2×C4×Dic6C2212yes—
34C3×#389C2214yes—
35C3×#389C2212yes—
36C3×#384C2212yes—
37C3×#384C2211yescharacteristic
38C3×C4×C4⋊C4C2211yescharacteristic
39C3×C4×C4⋊C4C2212yes—
40#5229C2211yescharacteristic
41#4994C2214yes—
42#4994C2214yes—
43#4994C2214yes—
44#4992C2214yes—
45#4990C2214yes—
46#4988C2211yescharacteristic
47#4988C2212yes—
48#4988C2211yescharacteristic
49#4987C2212yes—
50#4987C2212yes—
51#4985C2214yes—
52#4984C2212yes—
53#4984C2212yes—
54#4984C2212yes—
55#4983C2214yes—
56#4980C2214yes—
57#4977C2212yes—
58#4976C2212yes—
59#4976C2212yes—
60#4975C2211yescharacteristic
61#4975C2212yes—
62Dic3×C4⋊C4C2212yes—
63#4973C2211yescharacteristic
64#4972C2211yescharacteristic
65#4972C2212yes—
66#4972C2212yes—
67#4972C2211yescharacteristic
68#4940C2214yes—
69#4939C2212yes—
70C4×(C2×D3)⋊C4C2212yes—
71#4935C2211yescharacteristic
72#4933C2214yes—
73C4×Dic3⋊C4C2212yes—
74C4×Dic3⋊C4C2212yes—
75C42×Dic3C2211yescharacteristic
76#4928C2212yes—
77#4928C2212yes—
78#4676C2214yes—
79#4675C2214yes—
80#4669C2212yes—
81#4669C2212yes—
82#4660C2214yes—
83#4657C2214yes—
84#4647C2214yes—
85C2×#541—32no—
86#2859—31no—
87#2725—34no—
88#2725—34no—
89#2629—32no—
90#2538—31no—
91#2538—31no—
92#2512—32no—
93#2512—32no—
94#2506—32no—
95#2506—32no—
96#2421—32no—
97#2421—32no—
98Q8×C4⋊C4—31no—
99Q8×C4⋊C4D311yescharacteristic, complemented
100C4×C4⋊Q8—31no—
101#2293—31no—
102C22×C3×Q8C2311yescharacteristic
103C22×D12C2311yescharacteristic
104C22×C4×D3C2312yes—
105C22×Dic6C2311yescharacteristic
106C3×C4×Q8—28no—
107C3×C4×Q8—24no—
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C4⋊C4C2312yes—
113C2×C3×C4⋊C4C2312yes—
114C2×C3×C4⋊C4C2314yes—
115C2×C3×C4⋊C4C2312yes—
116C2×C3×C4⋊C4C2314yes—
117C2×C3×C42C2312yes—
118C2×C3×C42C2312yes—
119C2×C3×C42C2314yes—
120C2×C3×C42C2311yescharacteristic
121#946—28no—
122#944D418yescomplemented
123C2×(C2×D3)⋊C4C2312yes—
124C2×(C2×D3)⋊C4C2312yes—
125C2×(C2×D3)⋊C4C2312yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2312yes—
129C2×(C2×D3)⋊C4C2312yes—
130C2×(C2×D3)⋊C4C2314yes—
131C2×(C2×D3)⋊C4C2314yes—
132C2×C4⋊Dic3C2314yes—
133C2×C4⋊Dic3C2314yes—
134C2×C4⋊Dic3C2311yescharacteristic
135C2×C4⋊Dic3C2311yescharacteristic
136C2×C4⋊Dic3C2312yes—
137C2×C4⋊Dic3C2312yes—
138C2×C4⋊Dic3C2312yes—
139C2×Dic3⋊C4C2312yes—
140C2×Dic3⋊C4C2314yes—
141C2×Dic3⋊C4C2312yes—
142C2×Dic3⋊C4C2312yes—
143C2×Dic3⋊C4C2314yes—
144C2×Dic3⋊C4C2314yes—
145C2×Dic3⋊C4C2314yes—
146C2×Dic3⋊C4C2312yes—
147C2×Dic3⋊C4C2312yes—
148C2×Dic3⋊C4C2312yes—
149C2×Dic3⋊C4C2312yes—
150C2×C4×Dic3C2314yes—
151C2×C4×Dic3C2311yescharacteristic
152C2×C4×Dic3C2311yescharacteristic
153C2×C4×Dic3C2311yescharacteristic
154C2×C4×Dic3C2312yes—
155C2×C4×Dic3C2311yescharacteristic
156C2×C4×Dic3C2311yescharacteristic
157C2×C4×Dic3C2311yescharacteristic
158#898—216no—
159#897—28no—
160#890—28no—
161#889—216no—
162#876—216no—
163C4×D12—24no—
164#872—28no—
165#870—216no—
166C4×Dic6—28no—
167C4×Dic6—24no—
168C3×#95C2312yes—
169C3×#95C2314yes—
170C3⋊#95C2314yes—
171C3⋊#95C2314yes—
172C3⋊#95C2314yes—
173C3⋊#95C2312yes—
174C3⋊#95C2312yes—
175C3⋊#95C2314yes—
176C3⋊#95C2312yes—
177C3⋊#95C2312yes—
178C3⋊#95C2314yes—
179C3⋊#95C2312yes—
180C3⋊#95C2312yes—
181#541—616no—
182C2×C4⋊Q8—31no—
183C2×#126—34no—
184C2×#126—34no—
185C2×#125—32no—
186C2×#125—34no—
187C2×#125—34no—
188C2×C22⋊Q8—32no—
189C2×C22⋊Q8—32no—
190C2×C4×Q8—31no—
191C2×C4×Q8—32no—
192C2×C4×Q8C2×D311yescharacteristic
193C2×C4×Q8C2×D312yes—
194C2×C4×D4—31no—
195C2×#117—32no—
196#400—34no—
197#400—34no—
198#398—34no—
199#398—34no—
200#397—34no—
201#397—34no—
202#396—34no—
203#396—34no—
204#396—34no—
205#396—34no—
206#391—31no—
207#391—31no—
208#390—32no—
209#390—31no—
210#390—31no—
211#390—32no—
212#389—32no—
213#389—32no—
214#389—32no—
215#389—32no—
216#389—32no—
217#389—32no—
218#389—31no—
219#389—31no—
220#389—32no—
221#389—32no—
222#389—32no—
223#389—34no—
224#389C2×D312yes—
225#389C2×D314yes—
226#388—34no—
227#388—34no—
228#388—32no—
229#388—32no—
230#387—34no—
231#387—34no—
232#384—31no—
233#384—32no—
234#384—31no—
235#384C2×D311yescharacteristic
236#384C2×D312yes—
237#382—32no—
238#380—32no—
239#380—32no—
240C4×C4⋊C4—32no—
241C4×C4⋊C4—32no—
242C4×C4⋊C4—32no—
243C4×C4⋊C4C2×D311yescharacteristic
244C4×C4⋊C4C2×D312yes—
245C4×C22⋊C4—32no—
246C43—31no—
247C23×D3C2412yes—
248C2×C3×Q8—22no—
249C2×C3×Q8—22no—
250C2×C3×Q8C2×D414yes—
251C22×C3×C4C2412yes—
252C22×C3×C4C2412yes—
253C22×C3×C4C2414yes—
254C22×C3×C4C2412yes—
255C22×C3×C4C2412yes—
256C22×C3×C4C2411yescharacteristic
257C22×C3×C4C2411yescharacteristic
258C22×C3×C4C2411yescharacteristic
259C22×Dic3C2411yescharacteristic
260C22×Dic3C2411yescharacteristic
261C22×Dic3C2412yes—
262C22×Dic3C2412yes—
263C22×Dic3C2414yes—
264C22×Dic3C2412yes—
265C22×Dic3C2411yescharacteristic
266C22×Dic3C2411yescharacteristic
267C2×D12—22no—
268C2×D12—22no—
269C2×D12—22no—
270C2×C4×D3—24no—
271C2×C4×D3—24no—
272C2×C4×D3—24no—
273C2×Dic6—22no—
274C2×Dic6—22no—
275C2×Dic6C2×D414yes—
276C3×C4⋊C4—28no—
277C3×C4⋊C4—24no—
278C3×C4⋊C4—24no—
279C3×C4⋊C4—22no—
280C3×C4⋊C4—22no—
281C3×C4⋊C4—22no—
282C3×C4⋊C4—24no—
283C3×C4⋊C4—42no—
284C3×C4⋊C4—44no—
285C3×C42—24no—
286C3×C42—28no—
287C3×C42—24no—
288C3×C42#4114yes—
289(C2×D3)⋊C4—28no—
290(C2×D3)⋊C4—24no—
291(C2×D3)⋊C4—24no—
292(C2×D3)⋊C4—24no—
293(C2×D3)⋊C4—28no—
294(C2×D3)⋊C4—42no—
295(C2×D3)⋊C4—42no—
296(C2×D3)⋊C4—44no—
297(C2×D3)⋊C4—44no—
298C4⋊Dic3—24no—
299C4⋊Dic3—22no—
300C4⋊Dic3—22no—
301C4⋊Dic3—24no—
302C4⋊Dic3—28no—
303C4⋊Dic3—28no—
304C4⋊Dic3—42no—
305Dic3⋊C4—28no—
306Dic3⋊C4—24no—
307Dic3⋊C4—28no—
308Dic3⋊C4—24no—
309Dic3⋊C4—24no—
310Dic3⋊C4—28no—
311Dic3⋊C4—42no—
312Dic3⋊C4—42no—
313Dic3⋊C4—44no—
314Dic3⋊C4C2×D418yes—
315Dic3⋊C4C2×D418yes—
316C4×Dic3—24no—
317C4×Dic3—22no—
318C4×Dic3—22no—
319C4×Dic3—22no—
320C4×Dic3—28no—
321C4×Dic3—41no—
322C4×Dic3—41no—
323C4×Dic3C2×D414yes—
324C22×Q8—31no—
325C22×Q8C22×D311yescharacteristic
326C22×D4—31no—
327C23×C4—32no—
328C4⋊Q8—38no—
329#126—616no—
330#126—616no—
331#125—616no—
332#125—68no—
333#125—616no—
334C22⋊Q8—68no—
335C22⋊Q8—68no—
336C4×Q8—28no—
337C4×Q8—24no—
338C4×Q8—68no—
339C4×Q8—64no—
340C4×D4—64no—
341#117—68no—
342C2×C4⋊C4—32no—
343C2×C4⋊C4—32no—
344C2×C4⋊C4—32no—
345C2×C4⋊C4—32no—
346C2×C4⋊C4—31no—
347C2×C4⋊C4—32no—
348C2×C4⋊C4—34no—
349C2×C4⋊C4—31no—
350C2×C4⋊C4—32no—
351C2×C4⋊C4—34no—
352C2×C4⋊C4—34no—
353C2×C4⋊C4—34no—
354C2×C4⋊C4—34no—
355C2×C4⋊C4—32no—
356C2×C4⋊C4—32no—
357C2×C4⋊C4—32no—
358C2×C4⋊C4—34no—
359C2×C4⋊C4—32no—
360C2×C4⋊C4C22×D312yes—
361C2×C4⋊C4C22×D314yes—
362C2×C4⋊C4C22×D311yescharacteristic
363C2×C4⋊C4C22×D312yes—
364C2×C4⋊C4C22×D311yescharacteristic
365C2×C4⋊C4C22×D311yescharacteristic
366C2×C4⋊C4C22×D312yes—
367C2×C4⋊C4C22×D314yes—
368C2×C4⋊C4C22×D312yes—
369C2×C22⋊C4—32no—
370C2×C22⋊C4—32no—
371C2×C22⋊C4—32no—
372C2×C22⋊C4—34no—
373C2×C22⋊C4—34no—
374C2×C22⋊C4—32no—
375C2×C22⋊C4—32no—
376C2×C22⋊C4—34no—
377C2×C22⋊C4—34no—
378C2×C42—31no—
379C2×C42—34no—
380C2×C42—32no—
381C2×C42—31no—
382C2×C42—31no—
383C2×C42—31no—
384C2×C42—31no—
385C2×C42—31no—
386C2×C42C22×D312yes—
387C2×C42C22×D314yes—
388C2×C42C22×D312yes—
389C2×C42C22×D311yescharacteristic
390#95—32no—
391#95—32no—
392#95—32no—
393#95—32no—
394#95—32no—
395#95—32no—
396#95—34no—
397#95—34no—
398#95—34no—
399#95—34no—
400#95—34no—
401#95C22×D312yes—
402#95C22×D314yes—
403C23×C3C2511yescharacteristic
404C22×D3—22no—
405C22×D3—22no—
406C22×D3—22no—
407C22×D3—22no—
408C22×D3—22no—
409C22×D3—22no—
410C22×D3—22no—
411C3×Q8—28no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—21no—
415C2×C3×C4—21no—
416C2×C3×C4—22no—
417C2×C3×C4—22no—
418C2×C3×C4—24no—
419C2×C3×C4—22no—
420C2×C3×C4—22no—
421C2×C3×C4—22no—
422C2×C3×C4—24no—
423C2×C3×C4—22no—
424C2×C3×C4—22no—
425C2×C3×C4—22no—
426C2×C3×C4—22no—
427C2×C3×C4—24no—
428C2×C3×C4C22×D414yes—
429C2×C3×C4C22×D412yes—
430C2×C3×C4C2×#4112yes—
431C2×C3×C4C2×#4112yes—
432C2×C3×C4C2×#4112yes—
433C2×C3×C42−1+412yes—
434C2×C3×C42−1+412yes—
435C2×C3×C42−1+412yes—
436C2×Dic3—22no—
437C2×Dic3—21no—
438C2×Dic3—21no—
439C2×Dic3—22no—
440C2×Dic3—21no—
441C2×Dic3—21no—
442C2×Dic3—24no—
443C2×Dic3—22no—
444C2×Dic3—22no—
445C2×Dic3—22no—
446C2×Dic3—21no—
447C2×Dic3—21no—
448C2×Dic3—21no—
449C2×Dic3—21no—
450C2×Dic3—24no—
451C2×Dic3—22no—
452C2×Dic3—21no—
453C2×Dic3—21no—
454C2×Dic3—22no—
455C2×Dic3—24no—
456C2×Dic3—22no—
457C2×Dic3C22×D414yes—
458C2×Dic3C22×D412yes—
459C2×Dic3C22×D412yes—
460D12—44no—
461C4×D3—48no—
462Dic6—28no—
463C24—32no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8—34no—
467C2×Q8—62no—
468C2×Q8—62no—
469C2×Q8D3×D414yes—
470C2×D4—62no—
471C2×D4—62no—
472C2×D4—62no—
473C22×C4—32no—
474C22×C4—31no—
475C22×C4—31no—
476C22×C4—31no—
477C22×C4—31no—
478C22×C4—32no—
479C22×C4—32no—
480C22×C4—34no—
481C22×C4—64no—
482C22×C4—64no—
483C22×C4—64no—
484C22×C4C23×D312yes—
485C22×C4C23×D312yes—
486C22×C4C23×D314yes—
487C22×C4C23×D312yes—
488C22×C4C23×D311yescharacteristic
489C22×C4C23×D312yes—
490C22×C4C23×D311yescharacteristic
491C22×C4C23×D311yescharacteristic
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—28no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—38no—
500C4⋊C4—38no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—64no—
504C4⋊C4—64no—
505C4⋊C4—68no—
506C4⋊C4—68no—
507C4⋊C4—68no—
508C4⋊C4—68no—
509C4⋊C4—62no—
510C4⋊C4—64no—
511C4⋊C4—62no—
512C4⋊C4—64no—
513C4⋊C4—68no—
514C4⋊C4—64no—
515C4⋊C4—122no—
516C4⋊C4—122no—
517C4⋊C4—124no—
518C4⋊C4—122no—
519C22⋊C4—64no—
520C22⋊C4—64no—
521C22⋊C4—68no—
522C22⋊C4—64no—
523C22⋊C4—68no—
524C22⋊C4—122no—
525C22⋊C4—124no—
526C22⋊C4—124no—
527C22⋊C4—122no—
528C42—24no—
529C42—24no—
530C42—28no—
531C42—34no—
532C42—68no—
533C42—64no—
534C42—62no—
535C42—62no—
536C42—62no—
537C42—121no—
538C42—121no—
539C42#23514yes—
540C22×C3C23×D411yescharacteristic
541C22×C3C22×#4111yescharacteristic
542C22×C3C2×2−1+411yescharacteristic
543C22×C3C2×2−1+411yescharacteristic
544C22×C3C2×2−1+411yescharacteristic
545C22×C3#58511yescharacteristic
546C22×C3#58511yescharacteristic
547C2×D3—42no—
548C2×D3—42no—
549C2×D3—42no—
550C2×D3—42no—
551C2×D3—42no—
552C2×D3—42no—
553C2×D3—42no—
554C3×C4—24no—
555C3×C4—22no—
556C3×C4—42no—
557C3×C4—44no—
558C3×C4—42no—
559C3×C4—42no—
560C3×C4#54114yescomplemented
561C3×C4#54714yes—
562Dic3—24no—
563Dic3—22no—
564Dic3—22no—
565Dic3—42no—
566Dic3—44no—
567Dic3—42no—
568Dic3—41no—
569Dic3—41no—
570C23—62no—
571C23—62no—
572C23—62no—
573C23—62no—
574C23—62no—
575C23—62no—
576C23—62no—
577C23C24×D311yescharacteristic
578Q8—28no—
579Q8—68no—
580D4—124no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—32no—
598C2×C4—32no—
599C2×C4—34no—
600C2×C4—62no—
601C2×C4—61no—
602C2×C4—62no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—64no—
609C2×C4—62no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—64no—
614C2×C4—62no—
615C2×C4—61no—
616C2×C4—61no—
617C2×C4—62no—
618C2×C4—61no—
619C2×C4—64no—
620C2×C4—62no—
621C2×C4—128no—
622C2×C4C2×#23512yes—
623C2×C4C2×#23512yes—
624C2×C4C2×#23512yes—
625C2×C4C2×D3×D414yes—
626C2×C4C2×D3×D412yes—
627C2×C4#100712yes—
628C2×C4#100712yes—
629C2×C4#101012yes—
630C2×C3C2×#54111yescharacteristic
631C2×C3C2×#54711yescharacteristic
632C2×C3#349711yescharacteristic
633C2×C3#349711yescharacteristic
634C2×C3#351211yescharacteristic
635C2×C3#355511yescharacteristic
636C2×C3#355511yescharacteristic
637D3—82no—
638C22—122no—
639C22—122no—
640C22—122no—
641C22—122no—
642C22—122no—
643C22—122no—
644C22—122no—
645C22C22×#23511yescharacteristic
646C22C22×D3×D411yescharacteristic
647C22C2×#100711yescharacteristic
648C22C2×#100711yescharacteristic
649C22C2×#101011yescharacteristic
650C22#596611yescharacteristic
651C22#596811yescharacteristic
652C4—22no—
653C4—24no—
654C4—42no—
655C4—44no—
656C4—42no—
657C4—42no—
658C4—62no—
659C4—64no—
660C4—62no—
661C4—121no—
662C4—122no—
663C4—122no—
664C4—124no—
665C4—121no—
666C4#550714yes—
667C4#558314yescomplemented
668C3#3937911yescharacteristic, complemented, Sylow
669C2—242no—
670C2C2×#550711yescharacteristic
671C2C2×#558311yescharacteristic
672C2#8589411yescharacteristic
673C2#8676011yescharacteristic
674C2#8678411yescharacteristic
675C2#8703611yescharacteristic
676C2#8704811yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×Q8×C4⋊C4C2
prime radicalC22×D12C23
hypercenterQ8×C4⋊C4D3
hypocenterC3#39379
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#5583C2semidirect
2. C3×Q8×C4⋊C4C2semidirect
4. #82493C2semidirect
5. C4×#944C2semidirect
6. #76251C2semidirect
7. #76214C2semidirect
8. #76125C2semidirect
9. #76122C2semidirect
11. #74355C2semidirect
13. #74330C2semidirect
15. #73683C2semidirect
16. C4⋊C4×Dic6C2semidirect
99. Q8×C4⋊C4D3semidirect
122. #944D4semidirect
560. C3×C4#541semidirect
667. C4#5583semidirect
668. C3#39379semidirect

Maximal subgroups

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

subgroupindex
C2×#55832
C3×Q8×C4⋊C42
#825002
#824932
C4×#9442
#762512
#762142
#761252
#761222
#743692
#743552
#743332
#743302
#736932
#736832
C4⋊C4×Dic62
#736172
#393793