Finite Group Databasebeta

← #11714582

Subgroups — #11714582

Order 768.

1151 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 1151 conjugacy classes of subgroups — too many to draw. Only its 139 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
1A4×C4⋊Q8C211yesmaximal, characteristic
2#85502C214yesmaximal
3#85303C214yesmaximal
4#85302C214yesmaximal
5C4×A4⋊Q8C212yesmaximal
6#59151—31nomaximal, Sylow
7C2×Q8×A4C2212yes—
8C2×A4⋊Q8C2214yes—
9C2×A4⋊Q8C2212yes—
10#5722—41nomaximal
11A4×C4⋊C4C2214yes—
12C42×A4C2211yescharacteristic
13C4⋊SL(2,ℤ/4)C2212yes—
14SL(2,ℤ/4)⋊C4C2214yes—
15SL(2,ℤ/4)⋊C4C2218yes—
16C4×SL(2,ℤ/4)C2214yes—
17C4×SL(2,ℤ/4)C2214yes—
18C2×Q82—31no—
19C22×C4⋊Q8D311yescharacteristic
20#2734—34no—
21#2567—32no—
22#2551—34no—
23#2451—34no—
24#2440—34no—
25#2434—34no—
26#2369—31no—
27#2360—34no—
28C4×C22⋊Q8—32no—
29Q8×A4—42no—
30C2×C4×A4C2314yes—
31C2×C4×A4C2311yescharacteristic
32C2×C4×A4C2312yes—
33C2×SL(2,ℤ/4)C2314yes—
34C2×SL(2,ℤ/4)C2314yes—
35A4⋊Q8—44no—
36A4⋊Q8Q818yescomplemented
37C3×C4⋊Q8—41no—
38#944—44no—
39#888—44no—
40#887—44no—
41C4×Dic6—42no—
42C23×Q8C2×D312yes—
43Q82—62no—
44Q82—64no—
45Q82—61no—
46Q82—61no—
47C2×C4⋊Q8—31no—
48C2×C4⋊Q8—31no—
49C2×C4⋊Q8—34no—
50C2×C4⋊Q8—32no—
51C2×C4⋊Q8—34no—
52C2×C4⋊Q8—34no—
53C2×C4⋊Q8—34no—
54C2×C22⋊Q8—32no—
55C2×C22⋊Q8—34no—
56C2×C4×Q8—32no—
57C2×C4×Q8—34no—
58C22×C4⋊C4C2×D314yes—
59C22×C42C2×D311yescharacteristic
60#398—38no—
61#395—38no—
62#395—38no—
63#395—38no—
64#391—34no—
65#391—34no—
66#391—34no—
67#391—32no—
68#389—34no—
69#389—34no—
70#389—38no—
71#389—34no—
72#389—32no—
73#387—38no—
74#387—34no—
75#385—38no—
76#385—34no—
77#384—34no—
78#380—32no—
79C4×C4⋊C4—32no—
80C4×C4⋊C4—34no—
81C4×C4⋊C4—34no—
82C4×C22⋊C4—34no—
83C4×C22⋊C4—34no—
84C22×A4C2411yescharacteristic
85C2×C3×Q8—42no—
86C2×Dic6—42no—
87C2×Dic6—44no—
88C4×A4—24no—
89C4×A4—21no—
90C4×A4C2×Q814yes—
91SL(2,ℤ/4)—24no—
92SL(2,ℤ/4)C2×Q818yes—
93C3×C4⋊C4—44no—
94C3×C42—41no—
95C4⋊Dic3—42no—
96Dic3⋊C4—44no—
97Dic3⋊C4—48no—
98C4×Dic3—44no—
99C4×Dic3—44no—
100C22×Q8—32no—
101C22×Q8—32no—
102C22×Q8—34no—
103C22×Q8—32no—
104C22×Q8—34no—
105C22×Q8—42no—
106C22×Q8—122no—
107C23×C4C22×D314yes—
108C23×C4C22×D311yescharacteristic
109C23×C4C22×D312yes—
110C4⋊Q8—32no—
111C4⋊Q8—34no—
112C4⋊Q8—31no—
113C4⋊Q8—62no—
114C4⋊Q8—61no—
115C4⋊Q8—64no—
116C4⋊Q8—68no—
117C4⋊Q8—64no—
118C4⋊Q8—64no—
119C4⋊Q8—64no—
120C4⋊Q8—64no—
121C4⋊Q8—64no—
122C4⋊Q8—64no—
123C4⋊Q8—128no—
124C4⋊Q8S411yescharacteristic
125C22⋊Q8—38no—
126C22⋊Q8—38no—
127C22⋊Q8—124no—
128C22⋊Q8—124no—
129C4×Q8—62no—
130C4×Q8—62no—
131C4×Q8—62no—
132C4×Q8—64no—
133C4×Q8—68no—
134C4×Q8—64no—
135C4×Q8—62no—
136C2×C4⋊C4—32no—
137C2×C4⋊C4—34no—
138C2×C4⋊C4—34no—
139C2×C4⋊C4—34no—
140C2×C4⋊C4—38no—
141C2×C4⋊C4—32no—
142C2×C4⋊C4—34no—
143C2×C4⋊C4—38no—
144C2×C4⋊C4—34no—
145C2×C4⋊C4—32no—
146C2×C4⋊C4—38no—
147C2×C4⋊C4—34no—
148C2×C4⋊C4—38no—
149C2×C22⋊C4—34no—
150C2×C22⋊C4—34no—
151C2×C42—31no—
152C2×C42—32no—
153C2×C42—31no—
154C2×C42—34no—
155C2×C42—34no—
156C2×C42—34no—
157C2×C42—34no—
158#95—34no—
159#95—34no—
160#95—38no—
161#95—34no—
162#95—38no—
163#95—34no—
164#95—34no—
165#95—38no—
166C2×A4C22×Q812yes—
167C2×A4H(5,𝔽2)11yescharacteristic
168C3×Q8—162no—
169C2×C3×C4—42no—
170C2×C3×C4—44no—
171C2×C3×C4—41no—
172C2×Dic3—44no—
173C2×Dic3—44no—
174Dic6—48no—
175Dic6—164no—
176C24C23×D311yescharacteristic
177C2×Q8—38no—
178C2×Q8—34no—
179C2×Q8—32no—
180C2×Q8—62no—
181C2×Q8—62no—
182C2×Q8—64no—
183C2×Q8—68no—
184C2×Q8—64no—
185C2×Q8—62no—
186C2×Q8—64no—
187C2×Q8—122no—
188C2×Q8—122no—
189C2×Q8—124no—
190C2×Q8—122no—
191C2×Q8—122no—
192C2×Q8—124no—
193C2×Q8—124no—
194C2×Q8C2×S412yes—
195C22×C4—24no—
196C22×C4—21no—
197C22×C4—32no—
198C22×C4—32no—
199C22×C4—31no—
200C22×C4—34no—
201C22×C4—34no—
202C22×C4—34no—
203C22×C4—31no—
204C22×C4—34no—
205C22×C4—34no—
206C22×C4—34no—
207C22×C4—34no—
208C22×C4—61no—
209C22×C4—64no—
210C22×C4D3×Q814yes—
211C4⋊C4—38no—
212C4⋊C4—316no—
213C4⋊C4—38no—
214C4⋊C4—34no—
215C4⋊C4—64no—
216C4⋊C4—64no—
217C4⋊C4—62no—
218C4⋊C4—64no—
219C4⋊C4—68no—
220C4⋊C4—68no—
221C4⋊C4—64no—
222C4⋊C4—68no—
223C4⋊C4—62no—
224C4⋊C4—124no—
225C4⋊C4—128no—
226C4⋊C4—128no—
227C4⋊C4C2×S414yes—
228C22⋊C4—38no—
229C22⋊C4—38no—
230C22⋊C4—64no—
231C22⋊C4—64no—
232C42—31no—
233C42—32no—
234C42—61no—
235C42—68no—
236C42—64no—
237C42—64no—
238C42—64no—
239C42—64no—
240C42—124no—
241C42C2×S411yescharacteristic
242C22×C3—41no—
243A4Q8211yescharacteristic, complemented
244C3×C4—44no—
245C3×C4—84no—
246C3×C4—81no—
247Dic3—48no—
248Dic3—84no—
249C23—31no—
250C23—32no—
251C23—31no—
252C23#100411yescharacteristic
253C23C2×D3×Q812yes—
254Q8—42no—
255Q8—68no—
256Q8—68no—
257Q8—124no—
258Q8—122no—
259Q8—124no—
260Q8—244no—
261Q8—244no—
262C2×C4—32no—
263C2×C4—34no—
264C2×C4—34no—
265C2×C4—34no—
266C2×C4—38no—
267C2×C4—38no—
268C2×C4—32no—
269C2×C4—31no—
270C2×C4—34no—
271C2×C4—64no—
272C2×C4—64no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—62no—
276C2×C4—64no—
277C2×C4—64no—
278C2×C4—64no—
279C2×C4—64no—
280C2×C4—64no—
281C2×C4—64no—
282C2×C4—61no—
283C2×C4—64no—
284C2×C4—64no—
285C2×C4—64no—
286C2×C4—64no—
287C2×C4—64no—
288C2×C4—64no—
289C2×C4C22×S411yescharacteristic
290C2×C4C22×S414yes—
291C2×C4C22×S412yes—
292C2×C3—41no—
293C2×C3—42no—
294C22—31no—
295C22—32no—
296C22—31no—
297C22—31no—
298C22—32no—
299C22—32no—
300C22—61no—
301C22#572211yescharacteristic, complemented
302C22C23×S411yescharacteristic
303C4—21no—
304C4—24no—
305C4—34no—
306C4—68no—
307C4—61no—
308C4—64no—
309C4—68no—
310C4—124no—
311C4—124no—
312C4Q8×S414yes—
313C3—41noSylow
314C2—32no—
315C2—31no—
316C2—31no—
317C2#8747711yescharacteristic
318C2C2×Q8×S412yes—
subgroupG/H
center, FrattiniC22C23×S4
commutator, prime radical, cosocleC22×A4C24
FittingC22×C4⋊Q8D3
socleC24C23×D3
hypercenterC4⋊Q8S4
hypocenterA4Q82
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
36. A4⋊Q8Q8semidirect
243. A4Q82semidirect
301. C22#5722semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊Q82
#855022
#853032
#853022
C4×A4⋊Q82
#591513
#57224