Finite Group Databasebeta

← #11069287

Subgroups — #11069287

Order 768.

2890 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic, complemented
2C2×#5816C212yesmaximal, complemented
3C2×#5551C211yesmaximal, characteristic
4#84065C212yesmaximal, complemented
5#83253C211yesmaximal, characteristic, complemented
6#83240C211yesmaximal, characteristic, complemented
7#83233C214yesmaximal, complemented
8#82828C218yesmaximal, complemented
9#82786C212yesmaximal, complemented
10#82759C212yesmaximal, complemented
11#82475C211yesmaximal, characteristic, complemented
12C4×#947C211yesmaximal, characteristic, complemented
13#82406C214yesmaximal, complemented
14#82319C211yesmaximal, characteristic, complemented
15#39360—31nomaximal, Sylow
16C23×C3×Q8C2212yes—
17C22×#235C2212yes—
18C3×D4×Q8—28no—
19C2×C3×C4⋊Q8C2212yes—
20C2×C3×C4⋊Q8C2211yescharacteristic
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×C22⋊Q8C2214yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2212yes—
25C2×C3×C4×D4C2211yescharacteristic
26#5816—216no—
27C2×#947C2212yes—
28C2×#947C2211yescharacteristic
29C2×#946C2218yes—
30C2×#944C2212yes—
31C2×#944C2212yes—
32C2×#944C2212yes—
33C2×#938C2212yes—
34C2×#929C2218yes—
35C2×#926C2212yes—
36C2×#924C2212yes—
37#5551—28no—
38C2×D3×C4⋊C4C2212yes—
39C2×#883C2214yes—
40C2×C4×D12C2211yescharacteristic
41C2×#869C2211yescharacteristic
42#5230C2218yes—
43#5229C2211yescharacteristic
44#5229C2212yes—
45#5221C2212yes—
46#5210C2218yes—
47C4×#217C2212yes—
48#4992C2214yes—
49#4990C2214yes—
50#4989C2218yes—
51#4988C2211yescharacteristic
52#4987C2214yes—
53#4984C2212yes—
54#4982C2218yes—
55#4981C2214yes—
56#4977C2211yescharacteristic
57#4975C2214yes—
58#4975C2214yes—
59#4973C2212yes—
60#4959C2218yes—
61#4952C2218yes—
62#4951C2214yes—
63C4×(C2×D3)⋊C4C2214yes—
64C4×C4⋊Dic3C2211yescharacteristic
65C42×Dic3C2211yescharacteristic
66#4928C2214yes—
67C2×D4×Q8D311yescharacteristic, complemented
68C2×#547—31no—
69C2×#536—32no—
70#2868—31no—
71#2680—38no—
72#2621—32no—
73#2547—31no—
74#2546—31no—
75#2440—34no—
76#2434—32no—
77#2432—32no—
78#2422—34no—
79C4×#124—31no—
80#2293—31no—
81C22×C3×Q8—28no—
82C22×C3×Q8—28no—
83C22×C3×Q8—44no—
84C22×C3×Q8C2312yes—
85C22×C3×Q8C2314yes—
86C22×C3×Q8C2311yescharacteristic
87C22×C3×D4C2311yescharacteristic
88C23×C3×C4C2314yes—
89C23×C3×C4C2312yes—
90C22×#67C2314yes—
91C2×#235—28no—
92C2×#235—28no—
93C2×#235—44no—
94C22×D12C2311yescharacteristic
95C22×C4×D3C2312yes—
96C22×C4×D3C2314yes—
97C22×Dic6C2312yes—
98C3×C4⋊Q8—28no—
99C3×C4⋊Q8—24no—
100C3×C22⋊Q8—28no—
101C3×C22⋊Q8—216no—
102C3×C4×Q8—24no—
103C3×C4×D4—28no—
104C3×C4×D4D418yes—
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2312yes—
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C4⋊C4C2314yes—
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C22⋊C4C2314yes—
112C2×C3×C22⋊C4C2312yes—
113C2×C3×C42C2312yes—
114C2×C3×C42C2311yescharacteristic
115C2×#217C2314yes—
116C2×#217C2314yes—
117C2×#217C2314yes—
118#947—28no—
119#947D418yescomplemented
120#946—232no—
121#944—28no—
122#944—28no—
123#944—44no—
124#938—28no—
125#929—232no—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2318yes—
128C2×(C2×D3)⋊C4C2312yes—
129C2×(C2×D3)⋊C4C2314yes—
130#926—28no—
131C2×C4⋊Dic3C2312yes—
132C2×C4⋊Dic3C2312yes—
133C2×C4⋊Dic3C2314yes—
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2311yescharacteristic
136#924—28no—
137C2×Dic3⋊C4C2318yes—
138C2×Dic3⋊C4C2318yes—
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2318yes—
141C2×C4×Dic3C2312yes—
142C2×C4×Dic3C2312yes—
143C2×C4×Dic3C2311yescharacteristic
144C2×C4×Dic3C2312yes—
145C2×C4×Dic3C2311yescharacteristic
146D3×C4⋊C4—28no—
147#883—216no—
148C4×D12—24no—
149#869—24no—
150C3⋊#95C2318yes—
151C3⋊#95C2312yes—
152C3⋊#95C2314yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2318yes—
156C22×#41—32no—
157C23×Q8C2×D312yes—
158D4×Q8—28no—
159#547—68no—
160#536—616no—
161C2×C4⋊Q8—32no—
162C2×C4⋊Q8—32no—
163C2×C4⋊Q8—32no—
164C2×C4⋊Q8—31no—
165C2×C4⋊Q8C2×D311yescharacteristic
166C2×C4⋊Q8C2×D312yes—
167C2×#124—31no—
168C2×#124—32no—
169C2×#123—38no—
170C2×#123—34no—
171C2×C22⋊Q8—38no—
172C2×C22⋊Q8—32no—
173C2×C22⋊Q8C2×D312yes—
174C2×C22⋊Q8C2×D314yes—
175C2×#121—32no—
176C2×C4×Q8C2×D311yescharacteristic
177C2×C4×D4—31no—
178C2×C4×D4C2×D312yes—
179C2×C4×D4C2×D311yescharacteristic
180C2×#117—32no—
181C22×C4⋊C4—32no—
182#398—38no—
183#398—34no—
184#397—34no—
185#397—38no—
186#395—38no—
187#394—38no—
188#391—32no—
189#391—31no—
190#391—32no—
191#390—32no—
192#390—31no—
193#389—32no—
194#389—34no—
195#388—34no—
196#388—34no—
197#385—38no—
198#385—38no—
199#384—34no—
200#384—34no—
201#382—31no—
202#380—34no—
203C4×C4⋊C4—31no—
204C4×C22⋊C4—32no—
205C4×C22⋊C4—34no—
206C43—31no—
207C24×C3C2412yes—
208C23×D3C2412yes—
209C2×C3×Q8—22no—
210C2×C3×Q8—24no—
211C2×C3×Q8—28no—
212C2×C3×Q8—28no—
213C2×C3×Q8—24no—
214C2×C3×Q8—22no—
215C2×C3×Q8—24no—
216C2×C3×Q8—44no—
217C2×C3×Q8—432no—
218C2×C3×Q8C2×D414yes—
219C2×C3×D4—24no—
220C2×C3×D4C2×D414yes—
221C22×C3×C4—24no—
222C22×C3×C4—24no—
223C22×C3×C4—28no—
224C22×C3×C4—28no—
225C22×C3×C4—44no—
226C22×C3×C4C2×D418yes—
227C22×C3×C4C2412yes—
228C22×C3×C4C2414yes—
229C22×C3×C4C2411yescharacteristic
230C22×C3×C4C2412yes—
231C22×C3×C4C2412yes—
232C22×C3×C4C2411yescharacteristic
233C22×C3×C4C2411yescharacteristic
234C2×#67—28no—
235C2×#67—28no—
236C2×#67—44no—
237C22×Dic3C2412yes—
238C22×Dic3C2414yes—
239C22×Dic3C2414yes—
240C22×Dic3C2412yes—
241C22×Dic3C2412yes—
242#235—432no—
243C2×D12—22no—
244C2×D12—22no—
245C2×D12C2×D414yes—
246C2×C4×D3—28no—
247C2×C4×D3—24no—
248C2×C4×D3—28no—
249C2×C4×D3—28no—
250C2×C4×D3—44no—
251C2×Dic6—24no—
252C2×Dic6—24no—
253C2×Dic6—24no—
254C3×C4⋊C4—24no—
255C3×C4⋊C4—24no—
256C3×C4⋊C4—22no—
257C3×C4⋊C4—28no—
258C3×C4⋊C4—24no—
259C3×C4⋊C4C2×D414yes—
260C3×C22⋊C4—28no—
261C3×C22⋊C4C2×D418yes—
262C3×C42—24no—
263C3×C42C2×D414yes—
264#217—28no—
265#217—28no—
266#217—28no—
267(C2×D3)⋊C4—216no—
268(C2×D3)⋊C4—24no—
269(C2×D3)⋊C4—28no—
270(C2×D3)⋊C4C2×D4116yes—
271C4⋊Dic3—24no—
272C4⋊Dic3—24no—
273C4⋊Dic3—22no—
274C4⋊Dic3—24no—
275C4⋊Dic3—28no—
276Dic3⋊C4—28no—
277Dic3⋊C4—216no—
278Dic3⋊C4—216no—
279Dic3⋊C4—48no—
280C4×Dic3—24no—
281C4×Dic3—22no—
282C4×Dic3—24no—
283C4×Dic3—24no—
284C4×Dic3C2×D414yes—
285C2×#41—68no—
286C2×#41—68no—
287C2×#41—124no—
288C22×Q8—28no—
289C22×Q8—28no—
290C22×Q8—32no—
291C22×Q8—44no—
292C22×Q8C22×D312yes—
293C22×Q8C22×D314yes—
294C22×Q8C22×D311yescharacteristic
295C22×D4—34no—
296C22×D4—31no—
297C22×D4C22×D311yescharacteristic
298C23×C4—32no—
299C23×C4—34no—
300C23×C4C22×D314yes—
301C23×C4C22×D312yes—
302C4⋊Q8—28no—
303C4⋊Q8—24no—
304C4⋊Q8—68no—
305C4⋊Q8—68no—
306C4⋊Q8—64no—
307C4⋊Q8—124no—
308#124—38no—
309#124—68no—
310#123—632no—
311#123—616no—
312C22⋊Q8—28no—
313C22⋊Q8—216no—
314C22⋊Q8—632no—
315C22⋊Q8—68no—
316#121—68no—
317C4×Q8—24no—
318C4×D4—28no—
319C4×D4—64no—
320C4×D4#6718yes—
321#117—68no—
322C2×C4⋊C4—32no—
323C2×C4⋊C4—34no—
324C2×C4⋊C4—34no—
325C2×C4⋊C4—31no—
326C2×C4⋊C4—38no—
327C2×C4⋊C4—38no—
328C2×C4⋊C4—32no—
329C2×C4⋊C4—32no—
330C2×C4⋊C4—38no—
331C2×C4⋊C4—68no—
332C2×C4⋊C4C22×D312yes—
333C2×C4⋊C4C22×D311yescharacteristic
334C2×C4⋊C4C22×D312yes—
335C2×C4⋊C4C22×D312yes—
336C2×C4⋊C4C22×D314yes—
337C2×C4⋊C4C22×D311yescharacteristic
338C2×C22⋊C4—34no—
339C2×C22⋊C4—34no—
340C2×C22⋊C4—34no—
341C2×C22⋊C4—34no—
342C2×C22⋊C4—32no—
343C2×C22⋊C4—38no—
344C2×C22⋊C4—34no—
345C2×C22⋊C4C22×D314yes—
346C2×C22⋊C4C22×D312yes—
347C2×C42—31no—
348C2×C42—32no—
349C2×C42—31no—
350C2×C42—32no—
351C2×C42—32no—
352C2×C42C22×D312yes—
353C2×C42C22×D311yescharacteristic
354#95—34no—
355#95—34no—
356#95—34no—
357#95—32no—
358#95—38no—
359#95—38no—
360C23×C3—22no—
361C23×C3—22no—
362C23×C3—24no—
363C23×C3—24no—
364C23×C3C22×D414yes—
365C23×C3C2511yescharacteristic
366C22×D3—22no—
367C22×D3—22no—
368C22×D3—24no—
369C22×D3—24no—
370C22×D3C22×D414yes—
371C3×Q8—28no—
372C3×Q8—416no—
373C3×Q8—48no—
374C3×D4—28no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—216no—
381C2×C3×C4—24no—
382C2×C3×C4—22no—
383C2×C3×C4—24no—
384C2×C3×C4—24no—
385C2×C3×C4—24no—
386C2×C3×C4—416no—
387C2×C3×C4C22×D414yes—
388C2×C3×C4C22×D412yes—
389C2×C3×C4C22×D414yes—
390C2×C3×C4C22×D412yes—
391C2×C3×C4C22×D412yes—
392C2×C3×C4C22×D412yes—
393C2×C3×C4H(5,𝔽2)12yes—
394C2×C3×C42−1+414yes—
395#67—416no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—24no—
401C2×Dic3—24no—
402C2×Dic3—22no—
403C2×Dic3—24no—
404C2×Dic3—24no—
405C2×Dic3—24no—
406C2×Dic3—24no—
407C2×Dic3—24no—
408C2×Dic3C22×D414yes—
409C2×Dic32−1+414yes—
410D12—28no—
411C4×D3—416no—
412C4×D3—48no—
413Dic6—48no—
414C24—32no—
415C24C23×D312yes—
416#41—1232no—
417C2×Q8—28no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—28no—
423C2×Q8—24no—
424C2×Q8—44no—
425C2×Q8—432no—
426C2×Q8—64no—
427C2×Q8—64no—
428C2×Q8—64no—
429C2×Q8D3×D414yes—
430C2×D4—24no—
431C2×D4—34no—
432C2×D4—68no—
433C2×D4—68no—
434C2×D4—62no—
435C2×D4—62no—
436C2×D4—124no—
437C2×D4C2×#6714yes—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—32no—
443C22×C4—32no—
444C22×C4—32no—
445C22×C4—34no—
446C22×C4—34no—
447C22×C4—44no—
448C22×C4—68no—
449C22×C4—68no—
450C22×C4—64no—
451C22×C4—68no—
452C22×C4—124no—
453C22×C4C2×#6718yes—
454C22×C4C23×D312yes—
455C22×C4C23×D314yes—
456C22×C4C23×D312yes—
457C22×C4C23×D311yescharacteristic
458C22×C4C23×D312yes—
459C22×C4C23×D311yescharacteristic
460C22×C4C23×D311yescharacteristic
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—28no—
465C4⋊C4—24no—
466C4⋊C4—68no—
467C4⋊C4—616no—
468C4⋊C4—616no—
469C4⋊C4—64no—
470C4⋊C4—64no—
471C4⋊C4—68no—
472C4⋊C4—62no—
473C4⋊C4—64no—
474C4⋊C4—128no—
475C4⋊C4C2×#6714yes—
476C22⋊C4—28no—
477C22⋊C4—316no—
478C22⋊C4—68no—
479C22⋊C4—616no—
480C22⋊C4—68no—
481C22⋊C4—68no—
482C22⋊C4—68no—
483C22⋊C4—64no—
484C22⋊C4C2×#6718yes—
485C42—24no—
486C42—34no—
487C42—64no—
488C42—64no—
489C42—62no—
490C42—64no—
491C42C2×#6714yes—
492C22×C3—24no—
493C22×C3—24no—
494C22×C3—24no—
495C22×C3—44no—
496C22×C3—44no—
497C22×C3C23×D411yescharacteristic
498C22×C3C23×D411yescharacteristic
499C22×C3C2×H(5,𝔽2)11yescharacteristic
500C22×C3C2×2−1+412yes—
501C22×C3C2×2−1+412yes—
502C2×D3—24no—
503C2×D3—28no—
504C2×D3—42no—
505C2×D3—42no—
506C2×D3—44no—
507C3×C4—24no—
508C3×C4—24no—
509C3×C4—22no—
510C3×C4—44no—
511C3×C4—42no—
512C3×C4#53614yescomplemented
513C3×C4D4214yes—
514Dic3—24no—
515Dic3—24no—
516Dic3—42no—
517Dic3—44no—
518Dic3—44no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—34no—
524C23—62no—
525C23—64no—
526C23—64no—
527C23—62no—
528C23C22×#6714yes—
529C23C24×D311yescharacteristic
530Q8—28no—
531Q8—416no—
532Q8—48no—
533Q8—128no—
534D4—28no—
535D4—68no—
536D4—1216no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—216no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—34no—
549C2×C4—34no—
550C2×C4—416no—
551C2×C4—62no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—64no—
556C2×C4—64no—
557C2×C4—64no—
558C2×C4—64no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—62no—
562C2×C4—64no—
563C2×C4—1216no—
564C2×C4—128no—
565C2×C4C2×D3×D414yes—
566C2×C4C2×D3×D412yes—
567C2×C4#100412yes—
568C2×C4#100714yes—
569C2×C4C22×#6712yes—
570C2×C4C22×#6714yes—
571C2×C4C22×#6712yes—
572C2×C4C22×#6712yes—
573C2×C3—44no—
574C2×C3C2×#53611yescharacteristic
575C2×C3C2×D4211yescharacteristic
576C2×C3#349912yes—
577C2×C3#349911yescharacteristic
578C2×C3#355212yes—
579D3—44no—
580C22—24no—
581C22—24no—
582C22—24no—
583C22—44no—
584C22—44no—
585C22—64no—
586C22—68no—
587C22—122no—
588C22—124no—
589C22—122no—
590C22C22×D3×D411yescharacteristic
591C22C2×#100411yescharacteristic
592C22C2×#100712yes—
593C22D3×2−1+412yes—
594C22C23×#6711yescharacteristic
595C4—24no—
596C4—24no—
597C4—22no—
598C4—44no—
599C4—42no—
600C4—64no—
601C4—64no—
602C4—124no—
603C4—122no—
604C4—124no—
605C4D4×#6714yes—
606C4#581614yescomplemented
607C3#3936011yescharacteristic, complemented, Sylow
608C2—44no—
609C2—124no—
610C2#8674612yes—
611C2#8676412yes—
612C2C2×D4×#6711yescharacteristic
613C2C2×#581611yescharacteristic
614C2#8729211yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×D4×Q8C2
prime radicalC2×#5551C2
hypercenterC2×D4×Q8D3
hypocenterC3#39360
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×Q8C2semidirect
2. C2×#5816C2semidirect
4. #84065C2semidirect
5. #83253C2semidirect
6. #83240C2semidirect
7. #83233C2semidirect
8. #82828C2semidirect
9. #82786C2semidirect
10. #82759C2semidirect
11. #82475C2semidirect
12. C4×#947C2semidirect
13. #82406C2semidirect
14. #82319C2semidirect
67. C2×D4×Q8D3semidirect
119. #947D4semidirect
512. C3×C4#536semidirect
606. C4#5816semidirect
607. C3#39360semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×Q82
C2×#58162
C2×#55512
#840652
#832532
#832402
#832332
#828282
#827862
#827592
#824752
C4×#9472
#824062
#823192
#393603