Finite Group Databasebeta

← #10972898

Subgroups — #10972898

Order 768.

2654 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5725C211yesmaximal, characteristic
2C2×C4×D3×Q8C211yesmaximal, characteristic, complemented
3C3×#2567C211yesmaximal, characteristic, complemented
4#83243C211yesmaximal, characteristic, complemented
5#82476C211yesmaximal, characteristic, complemented
6#77249C214yesmaximal, complemented
7#77203C214yesmaximal, complemented
8#76500C212yesmaximal, complemented
9#76484C212yesmaximal, complemented
10#76286C214yesmaximal, complemented
11#76075C214yesmaximal, complemented
12#74328C212yesmaximal, complemented
13#74321C212yesmaximal, complemented
14#73752C211yesmaximal, characteristic, complemented
15#73712C211yesmaximal, characteristic, complemented
16#35249—31nomaximal, Sylow
17C22×#239C2212yes—
18C22×D3×Q8C2211yescharacteristic
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×#947C2212yes—
22C2×#947C2212yes—
23C2×Q8×Dic3C2211yescharacteristic
24#5725—28no—
25C4×D3×Q8—28no—
26C2×#895C2212yes—
27C2×#895C2212yes—
28C2×#894C2218yes—
29C2×#893C2212yes—
30C2×#893C2212yes—
31C2×D3×C4⋊C4C2211yescharacteristic
32C2×D3×C4⋊C4C2212yes—
33C2×#887C2211yescharacteristic
34C2×#887C2212yes—
35C2×#875C2212yes—
36C2×#874C2211yescharacteristic
37C2×C42×D3C2212yes—
38C2×C42×D3C2211yescharacteristic
39C2×C4×Dic6C2212yes—
40C2×C4×Dic6C2211yescharacteristic
41C3×#395C2214yes—
42C3×#391C2212yes—
43C3×#389C2214yes—
44C3×#384C2212yes—
45C3×C4×C4⋊C4C2211yescharacteristic
46#5230C2214yes—
47#4993C2218yes—
48#4991C2214yes—
49#4990C2218yes—
50#4989C2218yes—
51#4988C2214yes—
52#4988C2212yes—
53#4987C2212yes—
54#4977C2212yes—
55#4975C2212yes—
56#4973C2212yes—
57#4972C2214yes—
58#4939C2214yes—
59#4939C2212yes—
60C4×(C2×D3)⋊C4C2212yes—
61C4×C4⋊Dic3C2211yescharacteristic
62#4932C2212yes—
63#4673C2218yes—
64#4672C2214yes—
65#4669C2218yes—
66#4668C2218yes—
67#4665C2218yes—
68#4664C2212yes—
69#4651C2212yes—
70C2×#532—31no—
71C22×C4×Q8—31no—
72#2854—34no—
73#2632—34no—
74#2580—34no—
75#2567—31no—
76#2567D311yescharacteristic, complemented
77#2548—31no—
78#2548—31no—
79#2501—32no—
80#2436—32no—
81#2406—34no—
82#2326—32no—
83#2326—32no—
84#2283—31no—
85C22×C3×Q8C2311yescharacteristic
86C22×C3×Q8C2312yes—
87C2×#239—28no—
88C2×#239—48no—
89C2×D3×Q8—24no—
90C2×D3×Q8—24no—
91C2×D3×Q8—24no—
92C22×D12C2314yes—
93C22×D12C2312yes—
94C22×D12C2312yes—
95C22×D12C2312yes—
96C22×C4×D3C2312yes—
97C22×C4×D3C2312yes—
98C22×C4×D3C2311yescharacteristic
99C22×C4×D3C2311yescharacteristic
100C22×C4×D3C2311yescharacteristic
101C22×C4×D3C2314yes—
102C22×Dic6C2312yes—
103C22×Dic6C2311yescharacteristic
104C3×C4⋊Q8—24no—
105C3×C4×Q8—24no—
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2312yes—
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C4⋊C4C2318yes—
112C2×C3×C42C2312yes—
113C2×C3×C42C2312yes—
114C2×C3×C42C2311yescharacteristic
115#947—28no—
116#947—28no—
117Q8×Dic3—24no—
118C2×(C2×D3)⋊C4C2314yes—
119C2×(C2×D3)⋊C4C2318yes—
120C2×(C2×D3)⋊C4C2318yes—
121C2×(C2×D3)⋊C4C2314yes—
122C2×(C2×D3)⋊C4C2314yes—
123C2×(C2×D3)⋊C4C2318yes—
124C2×(C2×D3)⋊C4C2318yes—
125C2×C4⋊Dic3C2312yes—
126C2×C4⋊Dic3C2312yes—
127C2×C4⋊Dic3C2311yescharacteristic
128C2×Dic3⋊C4C2312yes—
129C2×Dic3⋊C4C2312yes—
130C2×Dic3⋊C4C2312yes—
131C2×Dic3⋊C4C2318yes—
132C2×C4×Dic3C2312yes—
133C2×C4×Dic3C2312yes—
134C2×C4×Dic3C2312yes—
135C2×C4×Dic3C2312yes—
136C2×C4×Dic3C2311yescharacteristic
137C2×C4×Dic3C2311yescharacteristic
138C2×C4×Dic3C2311yescharacteristic
139#895—28no—
140#895—28no—
141#894—416no—
142#893—28no—
143#893—28no—
144D3×C4⋊C4—28no—
145D3×C4⋊C4—24no—
146#887—24no—
147#887—28no—
148#875—28no—
149#874—24no—
150C42×D3—28no—
151C42×D3D418yes—
152C4×Dic6—24no—
153C4×Dic6—28no—
154C3×#95C2312yes—
155C3×#95C2318yes—
156C3⋊#95C2312yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2318yes—
161C22×#41—32no—
162C23×Q8—31no—
163#532—68no—
164C2×C4⋊Q8C2×D311yescharacteristic
165C2×#127—31no—
166C2×#124—32no—
167C2×#124—32no—
168C2×#124—32no—
169C2×#123—38no—
170C2×#121—32no—
171C2×#121—32no—
172C2×C4×Q8—31no—
173C2×C4×Q8—31no—
174C2×C4×Q8—32no—
175C2×C4×Q8—31no—
176C2×C4×Q8—32no—
177C2×C4×Q8—68no—
178C2×C4×Q8C2×D311yescharacteristic
179C2×C4×D4—32no—
180C2×C4×D4—32no—
181C22×C4⋊C4—32no—
182C22×C4⋊C4—31no—
183C22×C42—31no—
184C22×C42—32no—
185#399—38no—
186#395—34no—
187#395C2×D314yes—
188#394—34no—
189#394—34no—
190#393—38no—
191#391—32no—
192#391C2×D312yes—
193#390—34no—
194#390—32no—
195#390—34no—
196#390—32no—
197#389—34no—
198#389C2×D314yes—
199#388—38no—
200#388—38no—
201#386—38no—
202#385—38no—
203#385—38no—
204#384—32no—
205#384C2×D312yes—
206#383—32no—
207#382—32no—
208#381—32no—
209#380—32no—
210C4×C4⋊C4—31no—
211C4×C4⋊C4C2×D311yescharacteristic
212C4×C22⋊C4—32no—
213#376—32no—
214C23×D3C2414yes—
215C23×D3C2411yescharacteristic
216C2×C3×Q8—22no—
217C2×C3×Q8—24no—
218C2×C3×Q8—22no—
219C2×C3×Q8—44no—
220C2×C3×Q8#4114yescomplemented
221C22×C3×C4C2414yes—
222C22×C3×C4C2414yes—
223C22×C3×C4C2412yes—
224C22×C3×C4C2411yescharacteristic
225C22×C3×C4C2412yes—
226C22×C3×C4C2411yescharacteristic
227C22×C3×C4C2411yescharacteristic
228C22×Dic3C2414yes—
229C22×Dic3C2412yes—
230C22×Dic3C2411yescharacteristic
231C22×Dic3C2411yescharacteristic
232C22×Dic3C2411yescharacteristic
233C22×Dic3C2412yes—
234#239—432no—
235D3×Q8—232no—
236C2×D12—28no—
237C2×D12—216no—
238C2×D12—24no—
239C2×D12—24no—
240C2×D12—24no—
241C2×D12—24no—
242C2×D12—24no—
243C2×D12—28no—
244C2×D12—44no—
245C2×C4×D3—24no—
246C2×C4×D3—24no—
247C2×C4×D3—24no—
248C2×C4×D3—22no—
249C2×C4×D3—22no—
250C2×C4×D3—24no—
251C2×C4×D3—24no—
252C2×C4×D3—24no—
253C2×C4×D3—28no—
254C2×C4×D3—28no—
255C2×C4×D3—48no—
256C2×C4×D3C2×D414yes—
257C2×C4×D3C2×D414yes—
258C2×C4×D3C2×D414yes—
259C2×Dic6—22no—
260C2×Dic6—24no—
261C2×Dic6—22no—
262C2×Dic6—24no—
263C2×Dic6—24no—
264C2×Dic6#4114yescomplemented
265C3×C4⋊C4—24no—
266C3×C4⋊C4—22no—
267C3×C4⋊C4—24no—
268C3×C4⋊C4—24no—
269C3×C4⋊C4—24no—
270C3×C4⋊C4—48no—
271C3×C42—24no—
272C3×C42—24no—
273C3×C42C2×D414yes—
274(C2×D3)⋊C4—28no—
275(C2×D3)⋊C4—216no—
276(C2×D3)⋊C4—28no—
277(C2×D3)⋊C4—28no—
278(C2×D3)⋊C4—216no—
279(C2×D3)⋊C4—48no—
280(C2×D3)⋊C4—48no—
281C4⋊Dic3—22no—
282C4⋊Dic3—24no—
283C4⋊Dic3—24no—
284Dic3⋊C4—24no—
285Dic3⋊C4—24no—
286Dic3⋊C4—24no—
287Dic3⋊C4—48no—
288C4×Dic3—24no—
289C4×Dic3—24no—
290C4×Dic3—24no—
291C4×Dic3—24no—
292C4×Dic3C2×D414yes—
293C4×Dic3C2×D414yes—
294C4×Dic3C2×D414yes—
295C2×#41—68no—
296C2×#41—128no—
297C22×Q8—31no—
298C22×Q8—32no—
299C22×Q8—64no—
300C22×Q8—64no—
301C22×Q8—64no—
302C22×Q8C22×D311yescharacteristic
303C22×Q8C22×D312yes—
304C22×D4—32no—
305C22×D4—32no—
306C22×D4—34no—
307C22×D4—32no—
308C23×C4—31no—
309C23×C4—31no—
310C23×C4—31no—
311C23×C4—32no—
312C23×C4—32no—
313C23×C4—34no—
314C4⋊Q8—24no—
315#127—64no—
316#124—68no—
317#124—68no—
318#124—68no—
319#123—1216no—
320#121—68no—
321#121—68no—
322C4×Q8—24no—
323C4×Q8—68no—
324C4×Q8—64no—
325C4×Q8—68no—
326C4×Q8—64no—
327C4×Q8—64no—
328C4×D4—68no—
329C4×D4—68no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—32no—
332C2×C4⋊C4—32no—
333C2×C4⋊C4—32no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4—32no—
336C2×C4⋊C4—38no—
337C2×C4⋊C4—64no—
338C2×C4⋊C4—68no—
339C2×C4⋊C4C22×D311yescharacteristic
340C2×C4⋊C4C22×D312yes—
341C2×C4⋊C4C22×D312yes—
342C2×C4⋊C4C22×D312yes—
343C2×C4⋊C4C22×D312yes—
344C2×C4⋊C4C22×D318yes—
345C2×C22⋊C4—34no—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—34no—
348C2×C22⋊C4—38no—
349C2×C22⋊C4—38no—
350C2×C22⋊C4—38no—
351C2×C22⋊C4—38no—
352C2×C42—31no—
353C2×C42—31no—
354C2×C42—31no—
355C2×C42—38no—
356C2×C42—32no—
357C2×C42—32no—
358C2×C42—32no—
359C2×C42—32no—
360C2×C42—68no—
361C2×C42C22×D312yes—
362C2×C42C22×D312yes—
363C2×C42C22×D311yescharacteristic
364#95—34no—
365#95—34no—
366#95—34no—
367#95—32no—
368#95—38no—
369#95C22×D312yes—
370#95C22×D318yes—
371C23×C3C2511yescharacteristic
372C22×D3—21no—
373C22×D3—21no—
374C22×D3—22no—
375C22×D3—24no—
376C22×D3—24no—
377C22×D3—24no—
378C22×D3—28no—
379C22×D3—28no—
380C22×D3C22×D412yes—
381C22×D3H(5,𝔽2)14yes—
382C3×Q8—28no—
383C3×Q8—48no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—24no—
389C2×C3×C4—24no—
390C2×C3×C4—28no—
391C2×C3×C4—24no—
392C2×C3×C4—24no—
393C2×C3×C4—24no—
394C2×C3×C4C22×D412yes—
395C2×C3×C4C22×D412yes—
396C2×C3×C4C22×D412yes—
397C2×C3×C4C2×#4114yes—
398C2×C3×C4C2×#4112yes—
399C2×C3×C4C2×#4112yes—
400C2×C3×C42−1+414yes—
401C2×Dic3—22no—
402C2×Dic3—21no—
403C2×Dic3—22no—
404C2×Dic3—21no—
405C2×Dic3—22no—
406C2×Dic3—22no—
407C2×Dic3—22no—
408C2×Dic3—24no—
409C2×Dic3—24no—
410C2×Dic3—24no—
411C2×Dic3—24no—
412C2×Dic3C22×D412yes—
413C2×Dic3C22×D412yes—
414C2×Dic3C22×D412yes—
415C2×Dic3C2×#4114yes—
416D12—48no—
417D12—416no—
418D12—48no—
419D12—48no—
420C4×D3—216no—
421C4×D3—28no—
422C4×D3—28no—
423C4×D3—28no—
424C4×D3—416no—
425C4×D3—48no—
426Dic6—216no—
427Dic6—28no—
428C24—31no—
429C24—34no—
430#41—1232no—
431C2×Q8—22no—
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8—34no—
435C2×Q8—44no—
436C2×Q8—64no—
437C2×Q8—64no—
438C2×Q8—632no—
439C2×Q8—62no—
440C2×Q8—64no—
441C2×Q8—62no—
442C2×Q8#23914yescomplemented
443C2×D4—68no—
444C2×D4—616no—
445C2×D4—64no—
446C2×D4—64no—
447C2×D4—64no—
448C2×D4—64no—
449C2×D4—68no—
450C2×D4—64no—
451C2×D4—124no—
452C22×C4—31no—
453C22×C4—31no—
454C22×C4—32no—
455C22×C4—31no—
456C22×C4—34no—
457C22×C4—34no—
458C22×C4—34no—
459C22×C4—34no—
460C22×C4—32no—
461C22×C4—64no—
462C22×C4—68no—
463C22×C4—64no—
464C22×C4—64no—
465C22×C4—64no—
466C22×C4—68no—
467C22×C4—64no—
468C22×C4—62no—
469C22×C4—64no—
470C22×C4—62no—
471C22×C4—128no—
472C22×C4C23×D314yes—
473C22×C4C23×D314yes—
474C22×C4C23×D312yes—
475C22×C4C23×D311yescharacteristic
476C22×C4C23×D312yes—
477C22×C4C23×D311yescharacteristic
478C22×C4C23×D311yescharacteristic
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—48no—
485C4⋊C4—64no—
486C4⋊C4—64no—
487C4⋊C4—64no—
488C4⋊C4—64no—
489C4⋊C4—62no—
490C4⋊C4—64no—
491C4⋊C4—128no—
492C22⋊C4—68no—
493C22⋊C4—68no—
494C22⋊C4—616no—
495C22⋊C4—68no—
496C22⋊C4—616no—
497C22⋊C4—128no—
498C22⋊C4—128no—
499C42—24no—
500C42—24no—
501C42—34no—
502C42—34no—
503C42—34no—
504C42—64no—
505C42—64no—
506C42—64no—
507C42—64no—
508C42D3×D414yes—
509C22×C3C23×D411yescharacteristic
510C22×C3C22×#4111yescharacteristic
511C22×C3C22×#4111yescharacteristic
512C22×C3C2×H(5,𝔽2)12yes—
513C22×C3C2×2−1+412yes—
514C2×D3—22no—
515C2×D3—22no—
516C2×D3—24no—
517C2×D3—24no—
518C2×D3—44no—
519C2×D3—44no—
520C2×D3—44no—
521C2×D3—48no—
522C2×D3—48no—
523C2×D3#53514yes—
524C3×C4—24no—
525C3×C4—22no—
526C3×C4—44no—
527C3×C4—42no—
528C3×C4—44no—
529C3×C4#52514yes—
530C3×C4#53614yes—
531Dic3—22no—
532Dic3—24no—
533Dic3—22no—
534Dic3—22no—
535Dic3—42no—
536Dic3—44no—
537C23—32no—
538C23—34no—
539C23—61no—
540C23—62no—
541C23—61no—
542C23—64no—
543C23—64no—
544C23—68no—
545C23—64no—
546C23—68no—
547C23C24×D311yescharacteristic
548Q8—28no—
549Q8—48no—
550Q8—616no—
551Q8—68no—
552D4—128no—
553D4—1216no—
554D4—128no—
555D4—128no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—32no—
567C2×C4—32no—
568C2×C4—32no—
569C2×C4—34no—
570C2×C4—68no—
571C2×C4—616no—
572C2×C4—62no—
573C2×C4—61no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—61no—
577C2×C4—62no—
578C2×C4—68no—
579C2×C4—68no—
580C2×C4—62no—
581C2×C4—64no—
582C2×C4—64no—
583C2×C4—64no—
584C2×C4—64no—
585C2×C4—1216no—
586C2×C4—128no—
587C2×C4C2×D3×D412yes—
588C2×C4C2×D3×D412yes—
589C2×C4C2×D3×D412yes—
590C2×C4C2×#23914yes—
591C2×C4C2×#23912yes—
592C2×C4#100714yes—
593C2×C4D3×#4112yes—
594C2×C3C2×#52511yescharacteristic
595C2×C3C2×#53511yescharacteristic
596C2×C3C2×#53611yescharacteristic
597C2×C3#352212yes—
598C2×C3#352212yes—
599D3—24no—
600D3—84no—
601C22—34no—
602C22—64no—
603C22—64no—
604C22—62no—
605C22—62no—
606C22—128no—
607C22—124no—
608C22—128no—
609C22—124no—
610C22—124no—
611C22C22×D3×D411yescharacteristic
612C22C22×#23911yescharacteristic
613C22C2×#100712yes—
614C22C2×D3×#4111yescharacteristic
615C22D3×H(5,𝔽2)12yes—
616C4—22no—
617C4—24no—
618C4—44no—
619C4—42no—
620C4—44no—
621C4—62no—
622C4—62no—
623C4—62no—
624C4—64no—
625C4—124no—
626C4—122no—
627C4#562714yes—
628C4#572414yes—
629C3#3524911yescharacteristic, complemented, Sylow
630C2—64no—
631C2—244no—
632C2C2×#562711yescharacteristic
633C2C2×#572411yescharacteristic
634C2D3×#53511yescharacteristic
635C2#8693612yes—
636C2#8704312yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2567C2
prime radicalC2×#5725C2
hypercenter#2567D3
hypocenterC3#35249
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2567

Complements

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

NH ≅ G/Ntype
2. C2×C4×D3×Q8C2semidirect
3. C3×#2567C2semidirect
4. #83243C2semidirect
5. #82476C2semidirect
6. #77249C2semidirect
7. #77203C2semidirect
8. #76500C2semidirect
9. #76484C2semidirect
10. #76286C2semidirect
11. #76075C2semidirect
12. #74328C2semidirect
13. #74321C2semidirect
14. #73752C2semidirect
15. #73712C2semidirect
76. #2567D3semidirect
220. C2×C3×Q8#41semidirect
264. C2×Dic6#41semidirect
442. C2×Q8#239semidirect
629. C3#35249semidirect

Maximal subgroups

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

subgroupindex
C2×#57252
C2×C4×D3×Q82
C3×#25672
#832432
#824762
#772492
#772032
#765002
#764842
#762862
#760752
#743282
#743212
#737522
#737122
#352493