Finite Group Databasebeta

← #10981910

Subgroups — #10981910

Order 768.

2190 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5572C211yesmaximal, characteristic
2C2×Q8×Dic6C211yesmaximal, characteristic, complemented
3C3×#2351C211yesmaximal, characteristic, complemented
4Q8×(C2×D3)⋊C4C211yesmaximal, characteristic
5Q8×(C2×D3)⋊C4C211yesmaximal, characteristic
6#82461C211yesmaximal, characteristic, complemented
7#82461C211yesmaximal, characteristic, complemented
8#76250C213yesmaximal
9#76223C213yesmaximal
10#76076C213yesmaximal, complemented
11#76008C213yesmaximal, complemented
12#74185C213yesmaximal, complemented
13#74162C213yesmaximal
14#73778C213yesmaximal
15#73674C213yesmaximal, complemented
16#35711—31nomaximal, Sylow
17C22×D3×Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C2×Q8×Dic3C2211yescharacteristic
23C2×Q8×Dic3C2211yescharacteristic
24#5572—28no—
25Q8×Dic6—28no—
26C2×#892C2213yes—
27C2×#888C2213yes—
28C2×#888C2213yes—
29C2×#887C2213yes—
30C2×#872C2213yes—
31C2×#869C2213yes—
32C2×C4×Dic6C2213yes—
33C2×C4×Dic6C2213yes—
34C3×#389C2213yes—
35C3×#389C2213yes—
36C3×#382C2213yes—
37C3×C4×C4⋊C4C2213yes—
38#4994C2213yes—
39#4993C2213yes—
40#4987C2213yes—
41#4987C2213yes—
42#4983C2213yes—
43#4980C2216yes—
44#4979C2213yes—
45#4979C2216yes—
46#4977C2213yes—
47#4977C2213yes—
48#4973C2213yes—
49#4973C2213yes—
50#4972C2213yes—
51#4972C2213yes—
52C4×(C2×D3)⋊C4C2213yes—
53C4×(C2×D3)⋊C4C2213yes—
54C4×Dic3⋊C4C2213yes—
55C4×Dic3⋊C4C2213yes—
56#4929C2213yes—
57#4929C2213yes—
58#4928C2213yes—
59#4928C2213yes—
60#4676C2213yes—
61#4673C2211yescharacteristic
62#4673C2216yes—
63#4658C2216yes—
64#4658C2213yes—
65#4657C2213yes—
66#4652C2216yes—
67#4646C2216yes—
68#4646C2216yes—
69C2×Q82—31no—
70C2×#519—31no—
71#2628—33no—
72#2581—33no—
73#2568—33no—
74#2451—33no—
75#2451—33no—
76#2439—33no—
77#2439—33no—
78#2396—33no—
79#2351—31no—
80#2351—31no—
81#2351D311yescharacteristic, complemented
82Q8×C22⋊C4—31no—
83Q8×C22⋊C4—31no—
84C22×C3×Q8C2311yescharacteristic
85C2×D3×Q8—24no—
86C2×D3×Q8—24no—
87C2×D3×Q8Q818yescomplemented
88C22×C4×D3C2313yes—
89C22×Dic6C2311yescharacteristic
90C22×Dic6C2313yes—
91C22×Dic6C2316yes—
92C3×C4×Q8—24no—
93C3×C4×Q8—24no—
94C3×C4×Q8—24no—
95C2×C3×C4⋊C4C2313yes—
96C2×C3×C4⋊C4C2313yes—
97C2×C3×C4⋊C4C2313yes—
98C2×C3×C4⋊C4C2313yes—
99C2×C3×C4⋊C4C2313yes—
100C2×C3×C42C2313yes—
101C2×C3×C42C2313yes—
102C2×C3×C42C2313yes—
103Q8×Dic3—24no—
104Q8×Dic3—24no—
105Q8×Dic3D418yes—
106C2×(C2×D3)⋊C4C2313yes—
107C2×(C2×D3)⋊C4C2311yescharacteristic
108C2×(C2×D3)⋊C4C2311yescharacteristic
109C2×(C2×D3)⋊C4C2311yescharacteristic
110C2×(C2×D3)⋊C4C2313yes—
111C2×(C2×D3)⋊C4C2313yes—
112C2×C4⋊Dic3C2313yes—
113C2×C4⋊Dic3C2313yes—
114C2×C4⋊Dic3C2313yes—
115C2×C4⋊Dic3C2313yes—
116C2×Dic3⋊C4C2313yes—
117C2×Dic3⋊C4C2313yes—
118C2×Dic3⋊C4C2313yes—
119C2×Dic3⋊C4C2313yes—
120C2×Dic3⋊C4C2313yes—
121C2×Dic3⋊C4C2313yes—
122C2×Dic3⋊C4C2313yes—
123C2×Dic3⋊C4C2313yes—
124C2×C4×Dic3C2316yes—
125C2×C4×Dic3C2313yes—
126C2×C4×Dic3C2313yes—
127C2×C4×Dic3C2313yes—
128#892—212no—
129#888—212no—
130#888—212no—
131#887—212no—
132#872—212no—
133#869—212no—
134C4×Dic6—212no—
135C4×Dic6—212no—
136C3×#95C2313yes—
137C3×#95C2316yes—
138C3×#95C2311yescharacteristic
139C3⋊#95C2313yes—
140C3⋊#95C2313yes—
141C3⋊#95C2313yes—
142C3⋊#95C2316yes—
143C3⋊#95C2316yes—
144C3⋊#95C2316yes—
145C3⋊#95C2313yes—
146C3⋊#95C2316yes—
147C3⋊#95C2313yes—
148C3⋊#95C2311yescharacteristic
149C3⋊#95C2313yes—
150C3⋊#95C2311yescharacteristic
151C23×Q8—31no—
152Q82—68no—
153C2×C4⋊Q8—33no—
154C2×C4⋊Q8—33no—
155C2×C4⋊Q8—33no—
156#519—68no—
157C2×C4×Q8—31no—
158C2×C4×Q8—31no—
159C2×C4×Q8—31no—
160C2×C4×Q8—33no—
161C2×C4×Q8—33no—
162C2×C4×Q8—33no—
163C2×C4×Q8C2×D311yescharacteristic
164C2×C4×Q8C2×D311yescharacteristic
165C2×C4×Q8C2×D311yescharacteristic
166C2×#117—33no—
167C2×#117—33no—
168#400—33no—
169#400—36no—
170#399—33no—
171#398—33no—
172#398—33no—
173#396—33no—
174#396—33no—
175#395—33no—
176#395—36no—
177#393—36no—
178#393—31no—
179#391—33no—
180#391—33no—
181#391—33no—
182#391—33no—
183#391—36no—
184#391—36no—
185#389—33no—
186#389—33no—
187#389—33no—
188#389—33no—
189#389C2×D313yes—
190#389C2×D313yes—
191#387—36no—
192#387—36no—
193#382—33no—
194#382—33no—
195#382C2×D313yes—
196#380—33no—
197#380—33no—
198C4×C4⋊C4—33no—
199C4×C4⋊C4—33no—
200C4×C4⋊C4C2×D313yes—
201C4×C22⋊C4—33no—
202C4×C22⋊C4—33no—
203C23×D3C2411yescharacteristic
204C2×C3×Q8C2×D414yes—
205C2×C3×Q8C2×Q814yes—
206C2×C3×Q8#4114yes—
207C22×C3×C4C2413yes—
208C22×C3×C4C2411yescharacteristic
209C22×C3×C4C2413yes—
210C22×C3×C4C2413yes—
211C22×C3×C4C2411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×C3×C4C2413yes—
214C22×Dic3C2411yescharacteristic
215C22×Dic3C2413yes—
216C22×Dic3C2413yes—
217C22×Dic3C2411yescharacteristic
218C22×Dic3C2413yes—
219C22×Dic3C2413yes—
220C22×Dic3C2411yescharacteristic
221D3×Q8—232no—
222C2×C4×D3—26no—
223C2×C4×D3—26no—
224C2×C4×D3C2×Q8112yes—
225C2×Dic6—26no—
226C2×Dic6—26no—
227C2×Dic6—22no—
228C2×Dic6—22no—
229C2×Dic6—22no—
230C2×Dic6—212no—
231C2×Dic6—46no—
232C2×Dic6—46no—
233C2×Dic6C2×Q8112yes—
234C3×C4⋊C4—26no—
235C3×C4⋊C4—26no—
236C3×C4⋊C4—26no—
237C3×C4⋊C4—26no—
238C3×C4⋊C4—43no—
239C3×C42—26no—
240C3×C42—26no—
241C3×C42—26no—
242(C2×D3)⋊C4—22no—
243(C2×D3)⋊C4—22no—
244(C2×D3)⋊C4—26no—
245(C2×D3)⋊C4—22no—
246(C2×D3)⋊C4—43no—
247(C2×D3)⋊C4—43no—
248C4⋊Dic3—26no—
249C4⋊Dic3—26no—
250C4⋊Dic3—26no—
251C4⋊Dic3C2×D4112yes—
252Dic3⋊C4—26no—
253Dic3⋊C4—26no—
254Dic3⋊C4—26no—
255Dic3⋊C4—26no—
256Dic3⋊C4—26no—
257Dic3⋊C4—43no—
258Dic3⋊C4—43no—
259Dic3⋊C4—43no—
260C4×Dic3—26no—
261C4×Dic3—26no—
262C4×Dic3—212no—
263C4×Dic3C2×D4112yes—
264C22×Q8—38no—
265C22×Q8—33no—
266C22×Q8—31no—
267C22×Q8—36no—
268C22×Q8—64no—
269C22×Q8—64no—
270C22×Q8C22×D311yescharacteristic
271C23×C4—33no—
272C4⋊Q8—612no—
273C4⋊Q8—612no—
274C4⋊Q8—612no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—38no—
279C4×Q8—612no—
280C4×Q8—612no—
281C4×Q8—612no—
282C4×Q8—64no—
283C4×Q8—64no—
284#117—612no—
285#117—612no—
286C2×C4⋊C4—33no—
287C2×C4⋊C4—33no—
288C2×C4⋊C4—33no—
289C2×C4⋊C4—33no—
290C2×C4⋊C4—33no—
291C2×C4⋊C4—33no—
292C2×C4⋊C4—33no—
293C2×C4⋊C4—33no—
294C2×C4⋊C4—33no—
295C2×C4⋊C4—33no—
296C2×C4⋊C4—33no—
297C2×C4⋊C4—33no—
298C2×C4⋊C4C22×D313yes—
299C2×C4⋊C4C22×D313yes—
300C2×C4⋊C4C22×D313yes—
301C2×C4⋊C4C22×D313yes—
302C2×C4⋊C4C22×D313yes—
303C2×C22⋊C4—33no—
304C2×C22⋊C4—31no—
305C2×C22⋊C4—31no—
306C2×C22⋊C4—31no—
307C2×C22⋊C4—33no—
308C2×C22⋊C4—33no—
309C2×C42—33no—
310C2×C42—36no—
311C2×C42—33no—
312C2×C42—33no—
313C2×C42C22×D313yes—
314C2×C42C22×D313yes—
315C2×C42C22×D313yes—
316#95—33no—
317#95—33no—
318#95—33no—
319#95—33no—
320#95—31no—
321#95—31no—
322#95—33no—
323#95—33no—
324#95—36no—
325#95—36no—
326#95—36no—
327#95—36no—
328#95C22×D311yescharacteristic
329#95C22×D313yes—
330#95C22×D316yes—
331C23×C3C2511yescharacteristic
332C22×D3—21no—
333C22×D3—21no—
334C22×D3—21no—
335C22×D3—21no—
336C22×D3—21no—
337C22×D3—21no—
338C22×D3C22×Q812yes—
339C3×Q8C22⋊Q8116yes—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—23no—
343C2×C3×C4—21no—
344C2×C3×C4—23no—
345C2×C3×C4—21no—
346C2×C3×C4—23no—
347C2×C3×C4—23no—
348C2×C3×C4—21no—
349C2×C3×C4—23no—
350C2×C3×C4—23no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—23no—
354C2×C3×C4—23no—
355C2×C3×C4—21no—
356C2×C3×C4—23no—
357C2×C3×C4C22×D416yes—
358C2×C3×C4C22×Q816yes—
359C2×C3×C4C2×#4116yes—
360C2×C3×C42−1+412yes—
361C2×Dic3—21no—
362C2×Dic3—23no—
363C2×Dic3—23no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—23no—
368C2×Dic3—23no—
369C2×Dic3—23no—
370C2×Dic3—21no—
371C2×Dic3—23no—
372C2×Dic3—23no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—23no—
376C2×Dic3—23no—
377C2×Dic3—23no—
378C2×Dic3C22×D416yes—
379C2×Dic3C22×D412yes—
380C2×Dic3C22×Q816yes—
381C2×Dic32−1+412yes—
382C4×D3—224no—
383Dic6—28no—
384Dic6—224no—
385Dic6—812no—
386C24—31no—
387C2×Q8—312no—
388C2×Q8—62no—
389C2×Q8—62no—
390C2×Q8—612no—
391C2×Q8—632no—
392C2×Q8—66no—
393C2×Q8—66no—
394C2×Q8—62no—
395C2×Q8—126no—
396C2×Q8—126no—
397C2×Q8#23514yes—
398C2×Q8D3×D414yes—
399C2×Q8D3×Q814yes—
400C22×C4—33no—
401C22×C4—31no—
402C22×C4—31no—
403C22×C4—33no—
404C22×C4—312no—
405C22×C4—31no—
406C22×C4—33no—
407C22×C4—33no—
408C22×C4—66no—
409C22×C4—66no—
410C22×C4C23×D313yes—
411C22×C4C23×D311yescharacteristic
412C22×C4C23×D313yes—
413C22×C4C23×D313yes—
414C22×C4C23×D311yescharacteristic
415C22×C4C23×D311yescharacteristic
416C22×C4C23×D313yes—
417C4⋊C4—26no—
418C4⋊C4—26no—
419C4⋊C4—26no—
420C4⋊C4—26no—
421C4⋊C4—312no—
422C4⋊C4—43no—
423C4⋊C4—66no—
424C4⋊C4—66no—
425C4⋊C4—66no—
426C4⋊C4—66no—
427C4⋊C4—66no—
428C4⋊C4—66no—
429C4⋊C4—66no—
430C4⋊C4—66no—
431C4⋊C4—123no—
432C4⋊C4—123no—
433C4⋊C4—123no—
434C22⋊C4—62no—
435C22⋊C4—66no—
436C22⋊C4—62no—
437C22⋊C4—62no—
438C22⋊C4—123no—
439C22⋊C4—123no—
440C42—26no—
441C42—26no—
442C42—26no—
443C42—312no—
444C42—612no—
445C42—66no—
446C42—66no—
447C22×C3C23×D411yescharacteristic
448C22×C3C23×Q811yescharacteristic
449C22×C3C22×#4111yescharacteristic
450C22×C3C2×2−1+411yescharacteristic
451C22×C3C2×2−1+411yescharacteristic
452C22×C3#58511yescharacteristic
453C22×C3#58511yescharacteristic
454C2×D3—22no—
455C2×D3—22no—
456C2×D3—22no—
457C2×D3—41no—
458C2×D3—41no—
459C2×D3—41no—
460C2×D3—41no—
461C3×C4—22no—
462C3×C4—43no—
463C3×C4—43no—
464C3×C4—41no—
465C3×C4—43no—
466C3×C4—41no—
467C3×C4C2×C22⋊Q8112yes—
468Dic3—26no—
469Dic3—22no—
470Dic3—22no—
471Dic3—26no—
472Dic3—43no—
473Dic3—41no—
474Dic3—43no—
475C23—32no—
476C23—61no—
477C23—61no—
478C23—61no—
479C23—61no—
480C23—61no—
481C23—61no—
482C23C24×D311yescharacteristic
483Q8—68no—
484Q8—624no—
485Q8—2412no—
486Q8#896116yes—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—23no—
491C2×C4—21no—
492C2×C4—23no—
493C2×C4—23no—
494C2×C4—23no—
495C2×C4—23no—
496C2×C4—23no—
497C2×C4—21no—
498C2×C4—23no—
499C2×C4—21no—
500C2×C4—23no—
501C2×C4—23no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—32no—
505C2×C4—36no—
506C2×C4—36no—
507C2×C4—32no—
508C2×C4—63no—
509C2×C4—61no—
510C2×C4—63no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—63no—
514C2×C4—61no—
515C2×C4—63no—
516C2×C4—624no—
517C2×C4—61no—
518C2×C4—63no—
519C2×C4—63no—
520C2×C4—61no—
521C2×C4—63no—
522C2×C4—61no—
523C2×C4—63no—
524C2×C4—63no—
525C2×C4—63no—
526C2×C4C2×#23516yes—
527C2×C4C2×D3×D416yes—
528C2×C4C2×D3×Q816yes—
529C2×C4#101012yes—
530C2×C3C22×C22⋊Q811yescharacteristic
531C2×C3#348611yescharacteristic
532C2×C3#350011yescharacteristic
533C2×C3#350211yescharacteristic
534C2×C3#350211yescharacteristic
535C2×C3#351811yescharacteristic
536C2×C3#351811yescharacteristic
537D3—42no—
538C22—62no—
539C22—62no—
540C22—62no—
541C22—121no—
542C22—121no—
543C22—121no—
544C22—121no—
545C22C22×#23511yescharacteristic
546C22C22×D3×D411yescharacteristic
547C22C22×D3×Q811yescharacteristic
548C22C2×#101011yescharacteristic
549C22#596411yescharacteristic
550C22D3×2−1+411yescharacteristic
551C22#596811yescharacteristic
552C4—22no—
553C4—43no—
554C4—41no—
555C4—43no—
556C4—41no—
557C4—43no—
558C4—66no—
559C4—62no—
560C4—62no—
561C4—66no—
562C4—121no—
563C4—123no—
564C4—123no—
565C4C2×#896112yes—
566C3#3571111yescharacteristic, complemented, Sylow
567C2—122no—
568C2C22×#89611yescharacteristic
569C2#8583211yescharacteristic
570C2#8587811yescharacteristic
571C2#8671511yescharacteristic
572C2#8673511yescharacteristic
573C2#8703111yescharacteristic
574C2#8703211yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2351C2
prime radicalC23×D3C24
hypercenter#2351D3
hypocenterC3#35711
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic6C2semidirect
3. C3×#2351C2semidirect
6. #82461C2semidirect
7. #82461C2semidirect
10. #76076C2semidirect
11. #76008C2semidirect
12. #74185C2semidirect
15. #73674C2semidirect
81. #2351D3semidirect
87. C2×D3×Q8Q8semidirect
566. C3#35711semidirect

Maximal subgroups

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

subgroupindex
C2×#55722
C2×Q8×Dic62
C3×#23512
Q8×(C2×D3)⋊C42
#824612
#762502
#762232
#760762
#760082
#741852
#741622
#737782
#736742
#357113