Finite Group Databasebeta

← #29699

Subgroups — #29699

Order 256.

474 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#541C211yesmaximal, characteristic
2C2×#477C211yesmaximal, characteristic, complemented
3C2×#475C211yesmaximal, characteristic, complemented
4#2628C211yesmaximal, characteristic
5#2059C212yesmaximal, complemented
6#2043C212yesmaximal, complemented
7#2036C212yesmaximal
8#2035C212yesmaximal
9#1805C211yesmaximal, characteristic, complemented
10#1805C211yesmaximal, characteristic, complemented
11#1798C211yesmaximal, characteristic
12#541—24no—
13#541—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#126—21no—
17C2×#126—21no—
18C2×#125—21no—
19C2×#125—21no—
20C2×#125C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×#117C2211yescharacteristic
26#477—24no—
27#475—24no—
28C2×#107C2212yes—
29C2×#106C2212yes—
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×#103C2212yes—
36C2×C22⋊C8C2212yes—
37#400—21no—
38#399—21no—
39#395—21no—
40#393—21no—
41#389C2212yes—
42#382C2211yescharacteristic
43#380C2212yes—
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8C2311yescharacteristic
49C22×Q8C2311yescharacteristic
50C23×C4C2311yescharacteristic
51C22×C8C2312yes—
52C22×C8C2312yes—
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55C4⋊Q8—22no—
56C4⋊Q8—22no—
57#126—28no—
58#126—28no—
59#125—28no—
60#125—28no—
61#125—22no—
62#125—22no—
63C22⋊Q8—22no—
64C22⋊Q8—22no—
65C22⋊Q8—22no—
66C22⋊Q8—22no—
67C4×Q8—22no—
68C4×Q8—22no—
69C4×Q8D414yes—
70C4×Q8D414yes—
71#117—22no—
72#117—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4C2312yes—
86C2×C4⋊C4C2312yes—
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4D411yescharacteristic
93C2×C22⋊C4D411yescharacteristic
94C2×C42—21no—
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98#107—24no—
99#106—24no—
100C4⋊C8—22no—
101C4⋊C8—22no—
102#103—24no—
103#103—24no—
104#103—42no—
105#103—42no—
106C22⋊C8—42no—
107#95—22no—
108#95—21no—
109#95—21no—
110#95—22no—
111C24C2×D411yescharacteristic
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—42no—
123C2×Q8—82no—
124C2×Q8C2×D412yes—
125C2×Q8C2×D412yes—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2411yescharacteristic
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C4⋊C4—24no—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—42no—
167C4⋊C4—82no—
168C4⋊C4C2×D412yes—
169C4⋊C4C2×D412yes—
170C22⋊C4—24no—
171C22⋊C4—24no—
172C22⋊C4—22no—
173C22⋊C4—22no—
174C22⋊C4—82no—
175C42—24no—
176C42—24no—
177C42—21no—
178C42—21no—
179C42C2×D412yes—
180C42C2×D412yes—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—22no—
185C23—22no—
186C23C22×D411yescharacteristic
187Q8—22no—
188Q8—22no—
189Q8—22no—
190Q8—22no—
191Q8—41no—
192Q8—41no—
193Q8—41no—
194Q8—41no—
195Q8—84no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4#12012yes—
226C2×C4#12012yes—
227C2×C4C22×D411yescharacteristic
228C2×C4C22×D411yescharacteristic
229C2×C4H(5,𝔽2)11yescharacteristic
230C2×C42−1+411yescharacteristic
231C2×C42−1+411yescharacteristic
232C2×C42−1+411yescharacteristic
233C8—42no—
234C8—42no—
235C22—41no—
236C22—42no—
237C22—42no—
238C22—41no—
239C22—41no—
240C22C2×#12011yescharacteristic
241C22#53611yescharacteristic
242C22#53711yescharacteristic
243C22#57712yes—
244C22#57812yes—
245C4—22no—
246C4—22no—
247C4—41no—
248C4—42no—
249C4—42no—
250C4—41no—
251C4—42no—
252C4—41no—
253C4—42no—
254C4—82no—
255C4—82no—
256C4#53611yescharacteristic
257C4#53611yescharacteristic
258C4#53711yescharacteristic
259C4#53711yescharacteristic
260C2—81no—
261C2#261611yescharacteristic
262C2#301011yescharacteristic
263C2#301111yescharacteristic
264C2#324212yes—
265C2#325812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#477C2semidirect
3. C2×#475C2semidirect
5. #2059C2semidirect
6. #2043C2semidirect
9. #1805C2semidirect
10. #1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5412
C2×#4772
C2×#4752
#26282
#20592
#20432
#20362
#20352
#18052
#17982