Finite Group Databasebeta

← #11714441

Subgroups — #11714441

Order 768.

1102 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×Q8×A4C211yesmaximal, characteristic
2Q8×SL(2,ℤ/4)C211yesmaximal, characteristic
3Q8×SL(2,ℤ/4)C211yesmaximal, characteristic
4#85305C213yesmaximal
5#85303C213yesmaximal
6#85284C213yesmaximal
7C4×A4⋊Q8C213yesmaximal
8#36435—31nomaximal, Sylow
9C2×Q8×A4C2211yescharacteristic
10C2×A4⋊Q8C2213yes—
11#5569—41nomaximal
12A4×C4⋊C4C2213yes—
13C42×A4C2213yes—
14C4⋊SL(2,ℤ/4)C2213yes—
15C4⋊SL(2,ℤ/4)C2213yes—
16C4⋊SL(2,ℤ/4)C2213yes—
17SL(2,ℤ/4)⋊C4C2211yescharacteristic
18SL(2,ℤ/4)⋊C4C2213yes—
19SL(2,ℤ/4)⋊C4C2216yes—
20C4×SL(2,ℤ/4)C2213yes—
21C4×SL(2,ℤ/4)C2213yes—
22C2×#557—31no—
23C22×C4×Q8D311yescharacteristic
24#2851—33no—
25#2581—33no—
26#2565—33no—
27#2453—33no—
28#2451—33no—
29#2442—33no—
30#2440—33no—
31#2351—31no—
32#2351—31no—
33Q8×C4⋊C4—31no—
34Q8×C22⋊C4—31no—
35Q8×C22⋊C4—31no—
36C4×C22⋊Q8—33no—
37Q8×A4Q814yescomplemented
38C2×C4×A4C2313yes—
39C2×C4×A4C2311yescharacteristic
40C2×C4×A4C2313yes—
41C2×SL(2,ℤ/4)C2313yes—
42C2×SL(2,ℤ/4)C2311yescharacteristic
43C2×SL(2,ℤ/4)C2313yes—
44C2×SL(2,ℤ/4)C2311yescharacteristic
45A4⋊Q8—26no—
46C3×C4×Q8—41no—
47Q8×Dic3—41no—
48Q8×Dic3—41no—
49#890—43no—
50#888—43no—
51#870—43no—
52C4×Dic6—43no—
53C23×Q8C2×D311yescharacteristic
54#557—61no—
55#557—63no—
56#557—63no—
57#557—61no—
58C2×C4⋊Q8—33no—
59C2×#125—33no—
60C2×#125—33no—
61C2×C22⋊Q8—33no—
62C2×C4×Q8—31no—
63C2×C4×Q8—33no—
64C2×C4×Q8—31no—
65C2×C4×Q8—33no—
66C2×C4×Q8—33no—
67C2×C4×Q8—31no—
68C2×C4×Q8—31no—
69C2×C4×Q8—31no—
70C2×C4×Q8—31no—
71C22×C4⋊C4C2×D313yes—
72C22×C42C2×D313yes—
73#400—36no—
74#398—36no—
75#398—36no—
76#395—36no—
77#391—33no—
78#391—33no—
79#391—33no—
80#391—33no—
81#391—33no—
82#389—33no—
83#389—33no—
84#389—33no—
85#389—33no—
86#389—33no—
87#389—33no—
88#389—33no—
89#387—33no—
90#387—36no—
91#387—33no—
92#387—36no—
93#385—33no—
94#385—36no—
95#385—31no—
96#384—33no—
97#382—33no—
98#382—33no—
99#380—33no—
100#380—33no—
101#380—33no—
102C4×C4⋊C4—33no—
103C4×C4⋊C4—33no—
104C4×C4⋊C4—33no—
105C4×C22⋊C4—33no—
106C4×C22⋊C4—33no—
107C22×A4C2411yescharacteristic
108C2×C3×Q8—41no—
109C2×Dic6—43no—
110C4×A4—23no—
111C4×A4—21no—
112C4×A4C2×Q816yes—
113SL(2,ℤ/4)—23no—
114SL(2,ℤ/4)—21no—
115SL(2,ℤ/4)—23no—
116SL(2,ℤ/4)#4112yes—
117C3×C4⋊C4—43no—
118C3×C42—43no—
119C4⋊Dic3—43no—
120C4⋊Dic3—43no—
121C4⋊Dic3—43no—
122Dic3⋊C4—41no—
123Dic3⋊C4—43no—
124Dic3⋊C4—46no—
125C4×Dic3—43no—
126C4×Dic3—43no—
127C22×Q8—31no—
128C22×Q8—33no—
129C22×Q8—34no—
130C22×Q8—33no—
131C22×Q8Dic614yescomplemented
132C23×C4C22×D313yes—
133C23×C4C22×D311yescharacteristic
134C23×C4C22×D313yes—
135C4⋊Q8—66no—
136C4⋊Q8—63no—
137C4⋊Q8—63no—
138#125—63no—
139#125—63no—
140#125—63no—
141#125—66no—
142#125—63no—
143#125—63no—
144#125—63no—
145C22⋊Q8—66no—
146C22⋊Q8—66no—
147C4×Q8—31no—
148C4×Q8—33no—
149C4×Q8—33no—
150C4×Q8—61no—
151C4×Q8—63no—
152C4×Q8—63no—
153C4×Q8—63no—
154C4×Q8—63no—
155C4×Q8—63no—
156C4×Q8—64no—
157C4×Q8—64no—
158C4×Q8—63no—
159C4×Q8—63no—
160C4×Q8—61no—
161C4×Q8—61no—
162C4×Q8—63no—
163C4×Q8S411yescharacteristic
164C2×C4⋊C4—33no—
165C2×C4⋊C4—33no—
166C2×C4⋊C4—36no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—33no—
169C2×C4⋊C4—33no—
170C2×C4⋊C4—33no—
171C2×C4⋊C4—33no—
172C2×C4⋊C4—33no—
173C2×C4⋊C4—33no—
174C2×C4⋊C4—33no—
175C2×C4⋊C4—33no—
176C2×C4⋊C4—33no—
177C2×C4⋊C4—36no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—33no—
180C2×C4⋊C4—33no—
181C2×C4⋊C4—36no—
182C2×C22⋊C4—31no—
183C2×C22⋊C4—33no—
184C2×C22⋊C4—31no—
185C2×C22⋊C4—33no—
186C2×C42—33no—
187C2×C42—33no—
188C2×C42—33no—
189C2×C42—33no—
190C2×C42—33no—
191C2×C42—33no—
192C2×C42—33no—
193C2×C42—33no—
194#95—33no—
195#95—33no—
196#95—33no—
197#95—33no—
198#95—31no—
199#95—31no—
200#95—33no—
201#95—33no—
202#95—33no—
203#95—33no—
204#95—36no—
205#95—36no—
206#95—36no—
207#95—36no—
208C2×A4C22×Q811yescharacteristic
209C2×A4C2×#4111yescharacteristic
210C2×A42−1+411yescharacteristic
211C3×Q8—44no—
212C2×C3×C4—43no—
213C2×C3×C4—41no—
214C2×C3×C4—43no—
215C2×Dic3—41no—
216C2×Dic3—41no—
217C2×Dic3—43no—
218C2×Dic3—43no—
219Dic6—86no—
220C24C23×D311yescharacteristic
221C2×Q8—34no—
222C2×Q8—34no—
223C2×Q8—312no—
224C2×Q8—33no—
225C2×Q8—61no—
226C2×Q8—66no—
227C2×Q8—63no—
228C2×Q8—66no—
229C2×Q8—63no—
230C2×Q8—123no—
231C2×Q8C2×S411yescharacteristic
232C22×C4—21no—
233C22×C4—23no—
234C22×C4—33no—
235C22×C4—33no—
236C22×C4—31no—
237C22×C4—36no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—31no—
241C22×C4—33no—
242C22×C4—33no—
243C22×C4—31no—
244C22×C4—33no—
245C22×C4—33no—
246C22×C4—31no—
247C22×C4—33no—
248C22×C4—33no—
249C22×C4—61no—
250C22×C4—63no—
251C22×C4C2×Dic616yes—
252C4⋊C4—36no—
253C4⋊C4—33no—
254C4⋊C4—63no—
255C4⋊C4—62no—
256C4⋊C4—66no—
257C4⋊C4—66no—
258C4⋊C4—66no—
259C4⋊C4—61no—
260C4⋊C4—63no—
261C4⋊C4—63no—
262C4⋊C4—63no—
263C4⋊C4—63no—
264C4⋊C4—66no—
265C4⋊C4—66no—
266C4⋊C4—66no—
267C4⋊C4—66no—
268C4⋊C4—61no—
269C4⋊C4—63no—
270C4⋊C4—63no—
271C4⋊C4—63no—
272C4⋊C4—63no—
273C4⋊C4—66no—
274C4⋊C4—123no—
275C4⋊C4—126no—
276C4⋊C4C2×S413yes—
277C22⋊C4—32no—
278C22⋊C4—32no—
279C22⋊C4—61no—
280C22⋊C4—63no—
281C22⋊C4—63no—
282C22⋊C4—61no—
283C22⋊C4—63no—
284C22⋊C4—63no—
285C42—33no—
286C42—33no—
287C42—33no—
288C42—63no—
289C42—66no—
290C42—66no—
291C42—63no—
292C42—63no—
293C42—63no—
294C42—63no—
295C42C2×S413yes—
296C22×C3—41no—
297A4#55711yescharacteristic, complemented
298C3×C4—46no—
299C3×C4—83no—
300C3×C4—81no—
301Dic3—42no—
302Dic3—83no—
303Dic3—81no—
304Dic3—83no—
305C23—31no—
306C23—31no—
307C23—31no—
308C23—31no—
309C23C22×Dic611yescharacteristic
310C23#100711yescharacteristic
311C23D3×#4111yescharacteristic
312Q8—312no—
313Q8—64no—
314Q8—126no—
315Q8—126no—
316Q8A4⋊Q814yescomplemented
317C2×C4—33no—
318C2×C4—36no—
319C2×C4—36no—
320C2×C4—36no—
321C2×C4—32no—
322C2×C4—32no—
323C2×C4—32no—
324C2×C4—31no—
325C2×C4—33no—
326C2×C4—63no—
327C2×C4—63no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—63no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—63no—
336C2×C4—63no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—63no—
340C2×C4—63no—
341C2×C4—63no—
342C2×C4—63no—
343C2×C4—61no—
344C2×C4—63no—
345C2×C4—61no—
346C2×C4—63no—
347C2×C4—63no—
348C2×C4—61no—
349C2×C4—63no—
350C2×C4—61no—
351C2×C4—63no—
352C2×C4—63no—
353C2×C4—61no—
354C2×C4—63no—
355C2×C4—63no—
356C2×C4C22×S411yescharacteristic
357C2×C4C22×S413yes—
358C2×C4C22×S413yes—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—41no—
362C22—31no—
363C22—31no—
364C22—31no—
365C22—31no—
366C22—31no—
367C22—31no—
368C22—31no—
369C22—31no—
370C22—31no—
371C22—61no—
372C22#556911yescharacteristic, complemented
373C22C23×S411yescharacteristic
374C4—21no—
375C4—23no—
376C4—36no—
377C4—62no—
378C4—61no—
379C4—63no—
380C4—62no—
381C4—123no—
382C4—121no—
383C4—123no—
384C4—121no—
385C4—123no—
386C4—123no—
387C4C2×A4⋊Q816yes—
388C3—41noSylow
389C2—31no—
390C2—31no—
391C2—31no—
392C2—31no—
393C2C22×A4⋊Q811yescharacteristic
394C2#8748311yescharacteristic
395C2#41×S411yescharacteristic
subgroupG/H
center, FrattiniC22C23×S4
commutator, prime radical, cosocleC22×A4C24
FittingC22×C4×Q8D3
socleC24C23×D3
hypercenterC4×Q8S4
hypocenterA4#557
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
37. Q8×A4Q8semidirect
131. C22×Q8Dic6semidirect
297. A4#557semidirect
316. Q8A4⋊Q8semidirect
372. C22#5569semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×A42
Q8×SL(2,ℤ/4)2
#853052
#853032
#852842
C4×A4⋊Q82
#364353
#55694