Finite Group Databasebeta

← #11713664

Subgroups — #11713664

Order 768.

503 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 503 conjugacy classes of subgroups — too many to draw. Only its 69 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#85283C211yesmaximal, characteristic
3#73051C214yesmaximal
4C4×A4⋊C8C211yesmaximal, characteristic
5#25811—31nomaximal, Sylow
6C2×Q8×A4—21no—
7C2×A4⋊Q8—21no—
8A4×C4⋊C4—21no—
9A4×C4⋊C4C2212yes—
10C42×A4C2211yescharacteristic
11C4⋊SL(2,ℤ/4)—21no—
12C4⋊SL(2,ℤ/4)C2212yes—
13C2×A4⋊C8C2212yes—
14#5080—41nomaximal
15C22×C4⋊Q8D311yescharacteristic
16C2×#498—31no—
17C2×#498—31no—
18#2857—31no—
19#2058—34no—
20#1990—31no—
21#1990—31no—
22#1853—34no—
23C4×C22⋊C8—31no—
24Q8×A4—42no—
25C2×C4×A4—22no—
26C2×C4×A4D411yescharacteristic
27C2×C4×A4D411yescharacteristic
28C2×C4×A4C2311yescharacteristic
29C2×SL(2,ℤ/4)—22no—
30A4⋊Q8—42no—
31C3×C4⋊Q8—41no—
32#869—41no—
33A4⋊C8Q814yescomplemented
34#808—44no—
35C4×C3⋊C8—41no—
36C23×Q8—21no—
37C2×C4⋊Q8—31no—
38C2×C4⋊Q8—31no—
39C2×C4⋊Q8—32no—
40C2×C4⋊Q8—31no—
41C2×C4⋊Q8—61no—
42C2×C4⋊Q8—61no—
43C2×C22⋊Q8—61no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2×D312yes—
46C22×C42C2×D311yescharacteristic
47#498—62no—
48#498—62no—
49#498—62no—
50#498—61no—
51#498—61no—
52C2×#106—34no—
53C2×#106—34no—
54C2×C22⋊C8—32no—
55C2×C4×C8—31no—
56C2×C4×C8—31no—
57#391—61no—
58#389—32no—
59#389—61no—
60#384—31no—
61#380—32no—
62#380—61no—
63#340—34no—
64#340—34no—
65#336—32no—
66C22×A4C2×D411yescharacteristic
67C2×C3×Q8—81no—
68C2×Dic6—81no—
69C4×A4—42no—
70C4×A4SD1612yes—
71C4×A4SD1612yes—
72C4×A4C2×Q812yes—
73SL(2,ℤ/4)—42no—
74C3×C4⋊C4—42no—
75C3×C4⋊C4—81no—
76C3×C42—41no—
77C4⋊Dic3—42no—
78C4⋊Dic3—81no—
79C2×C3⋊C8—42no—
80C22×Q8—42no—
81C22×Q8—61no—
82C22×Q8—61no—
83C22×Q8—62no—
84C22×Q8—61no—
85C22×Q8—122no—
86C23×C4—22no—
87C23×C4#6711yescharacteristic
88C23×C4#6711yescharacteristic
89C23×C4C22×D311yescharacteristic
90C22×C8—32no—
91C22×C8—32no—
92C4⋊Q8—32no—
93C4⋊Q8—32no—
94C4⋊Q8—32no—
95C4⋊Q8—31no—
96C4⋊Q8—61no—
97C4⋊Q8—61no—
98C4⋊Q8—61no—
99C4⋊Q8—61no—
100C4⋊Q8—61no—
101C4⋊Q8—61no—
102C4⋊Q8—64no—
103C4⋊Q8—61no—
104C4⋊Q8S411yescharacteristic
105C22⋊Q8—122no—
106C22⋊Q8—122no—
107C2×C4⋊C4—32no—
108C2×C4⋊C4—32no—
109C2×C4⋊C4—32no—
110C2×C4⋊C4—32no—
111C2×C4⋊C4—61no—
112C2×C4⋊C4—62no—
113C2×C4⋊C4—62no—
114C2×C4⋊C4—61no—
115C2×C4⋊C4—61no—
116C2×C4⋊C4—62no—
117C2×C4⋊C4—61no—
118C2×C4⋊C4—61no—
119C2×C22⋊C4—62no—
120C2×C42—31no—
121C2×C42—31no—
122C2×C42—31no—
123C2×C42—31no—
124#106—68no—
125#106—64no—
126#106—64no—
127C22⋊C8—34no—
128C22⋊C8—34no—
129C4×C8—61no—
130C4×C8—61no—
131C4×C8—61no—
132C4×C8—61no—
133#95—62no—
134#95—62no—
135C2×A4C4⋊Q811yescharacteristic
136C2×A4C2×SD1611yescharacteristic
137C2×A4C2×SD1611yescharacteristic
138C3×Q8—162no—
139C2×C3×C4—41no—
140C2×C3×C4—41no—
141C2×C3×C4—41no—
142C2×C3×C4—82no—
143C2×Dic3—82no—
144Dic6—162no—
145C3⋊C8—44no—
146C24C2×#6711yescharacteristic
147C2×Q8—21no—
148C2×Q8—62no—
149C2×Q8—62no—
150C2×Q8—61no—
151C2×Q8—122no—
152C2×Q8—122no—
153C2×Q8—122no—
154C2×Q8—124no—
155C2×Q8—122no—
156C2×Q8—122no—
157C2×Q8—121no—
158C2×Q8—122no—
159C2×Q8—124no—
160C2×Q8—121no—
161C22×C4—31no—
162C22×C4—31no—
163C22×C4—31no—
164C22×C4—31no—
165C22×C4—31no—
166C22×C4—32no—
167C22×C4—31no—
168C22×C4—32no—
169C22×C4—32no—
170C22×C4—42no—
171C22×C4—62no—
172C22×C4—62no—
173C22×C4—62no—
174C22×C4—62no—
175C22×C4—122no—
176C22×C4#21412yes—
177C22×C4#21512yes—
178C22×C4D3×Q812yes—
179C2×C8—34no—
180C2×C8—34no—
181C2×C8—62no—
182C2×C8—62no—
183C2×C8—62no—
184C2×C8—62no—
185C4⋊C4—21no—
186C4⋊C4—34no—
187C4⋊C4—32no—
188C4⋊C4—62no—
189C4⋊C4—62no—
190C4⋊C4—62no—
191C4⋊C4—62no—
192C4⋊C4—64no—
193C4⋊C4—61no—
194C4⋊C4—62no—
195C4⋊C4—122no—
196C4⋊C4—122no—
197C4⋊C4—124no—
198C4⋊C4—121no—
199C4⋊C4—121no—
200C4⋊C4C2×S412yes—
201C22⋊C4—122no—
202C22⋊C4—122no—
203C42—31no—
204C42—31no—
205C42—31no—
206C42—61no—
207C42C2×S411yescharacteristic
208C22×C3—41no—
209A4#49811yescharacteristic, complemented
210C3×C4—42no—
211C3×C4—42no—
212C3×C4—42no—
213C3×C4—162no—
214Dic3—162no—
215C23—31no—
216C23—31no—
217C23—31no—
218C23—31no—
219C23C2×#21411yescharacteristic
220C23C2×#21511yescharacteristic
221C23#94411yescharacteristic
222Q8—42no—
223Q8—124no—
224Q8—124no—
225Q8—122no—
226Q8—242no—
227Q8—242no—
228C2×C4—22no—
229C2×C4—31no—
230C2×C4—31no—
231C2×C4—31no—
232C2×C4—32no—
233C2×C4—32no—
234C2×C4—32no—
235C2×C4—32no—
236C2×C4—32no—
237C2×C4—32no—
238C2×C4—32no—
239C2×C4—32no—
240C2×C4—32no—
241C2×C4—61no—
242C2×C4—61no—
243C2×C4—61no—
244C2×C4—62no—
245C2×C4—122no—
246C2×C4—122no—
247C2×C4—122no—
248C2×C4—122no—
249C2×C4—122no—
250C2×C4—122no—
251C2×C4—122no—
252C2×C4—122no—
253C2×C4—122no—
254C2×C4—122no—
255C2×C4GL(2,ℤ/4)11yescharacteristic
256C2×C4GL(2,ℤ/4)11yescharacteristic
257C2×C4C22×S411yescharacteristic
258C8—64no—
259C8—64no—
260C2×C3—41no—
261C2×C3—41no—
262C2×C3—41no—
263C22—31no—
264C22—31no—
265C22—31no—
266C22—31no—
267C22—31no—
268C22—31no—
269C22—31no—
270C22—31no—
271C22—31no—
272C22—61no—
273C22#508011yescharacteristic, complemented
274C22C2×GL(2,ℤ/4)11yescharacteristic
275C4—32no—
276C4—32no—
277C4—32no—
278C4—42no—
279C4—122no—
280C4—242no—
281C4—242no—
282C4#541412yes—
283C4#541712yes—
284C4Q8×S412yes—
285C3—41noSylow
286C2—31no—
287C2—31no—
288C2—31no—
289C2—31no—
290C2C2×#541411yescharacteristic
291C2C2×#541711yescharacteristic
292C2#8550211yescharacteristic
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×C4⋊Q8D3
socleC24C2×#67
prime radicalC22×A4C2×D4
hypercenterC4⋊Q8S4
hypocenterA4#498
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
33. A4⋊C8Q8semidirect
209. A4#498semidirect
273. C22#5080semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊Q82
#852832
#730512
C4×A4⋊C82
#258113
#50804