Finite Group Databasebeta

← #24817

Subgroups — #24817

Order 256.

403 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 151 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×#477C211yesmaximal, characteristic
2C2×C4×Q16C211yesmaximal, characteristic
3#2567C211yesmaximal, characteristic
4Q8×C4⋊C4C211yesmaximal, characteristic
5#2083C212yesmaximal
6#2068C212yesmaximal
7#1990C211yesmaximal, characteristic
8#1939C211yesmaximal, characteristic
9#1878C212yesmaximal
10#1803C211yesmaximal, characteristic
11#1803C212yesmaximal
12C22×Q16C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×Q8C2212yes—
16C2×C4×Q8C2211yescharacteristic
17#477—24no—
18C4×Q16—24no—
19C2×#107C2212yes—
20C2×#107C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×#103C2212yes—
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2212yes—
27C2×C4×C8C2211yescharacteristic
28#395—22no—
29#391C2212yes—
30#389—21no—
31#389—21no—
32#389—22no—
33#389C2212yes—
34#384—21no—
35#384C2212yes—
36#384C2211yescharacteristic
37C4×C4⋊C4—21no—
38C4×C4⋊C4C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×Q8C2311yescharacteristic
46C2×Q16—22no—
47C2×Q16—22no—
48C2×Q16Q814yescomplemented
49C22×C8C2312yes—
50C22×C8C2311yescharacteristic
51C22×C8C2311yescharacteristic
52C4⋊Q8—22no—
53C4⋊Q8—22no—
54C4×Q8—22no—
55C4×Q8—24no—
56C4×Q8—22no—
57C4×Q8—22no—
58C4×Q8—42no—
59C4×Q8D414yes—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4D411yescharacteristic
70C2×C4⋊C4D411yescharacteristic
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2312yes—
74C2×C4⋊C4C2311yescharacteristic
75C2×C42—21no—
76C2×C42—22no—
77C2×C42—21no—
78C2×C42C2312yes—
79C2×C42C2311yescharacteristic
80#107—22no—
81#107—24no—
82C4⋊C8—22no—
83C4⋊C8—42no—
84#103—22no—
85#103—24no—
86#103—22no—
87#103—42no—
88C4×C8—22no—
89#95—22no—
90#95—22no—
91#95—22no—
92#95D412yes—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—21no—
101C2×Q8—42no—
102C2×Q8—81no—
103C2×Q8—81no—
104C2×Q8C2×D412yes—
105C2×Q8C2×Q812yes—
106C2×Q8C2×Q812yes—
107C2×Q8#4112yes—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4C2×D412yes—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D412yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2411yescharacteristic
119Q16—28no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—21no—
126C2×C8—22no—
127C2×C8—21no—
128C2×C8C2×Q812yes—
129C4⋊C4—24no—
130C4⋊C4—22no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—44no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—81no—
143C4⋊C4—82no—
144C4⋊C4—82no—
145C4⋊C4—82no—
146C4⋊C4Q1614yescomplemented
147C4⋊C4C2×D412yes—
148C42—24no—
149C42—22no—
150C42—22no—
151C42—44no—
152C42—42no—
153C42C2×D412yes—
154C42#4112yes—
155C23C22×D411yescharacteristic
156Q8—22no—
157Q8—22no—
158Q8—22no—
159Q8—22no—
160Q8—22no—
161Q8—41no—
162Q8—41no—
163Q8—84no—
164Q8C22⋊Q814yes—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4#12112yes—
191C2×C4#12112yes—
192C2×C4C2×Q1614yes—
193C2×C4C2×Q1612yes—
194C2×C4#13712yes—
195C2×C4C22×D411yescharacteristic
196C2×C4C22×Q811yescharacteristic
197C2×C4C2×#4111yescharacteristic
198C2×C4C2×#4111yescharacteristic
199C2×C4H(5,𝔽2)11yescharacteristic
200C2×C42−1+411yescharacteristic
201C8—22no—
202C8—41no—
203C8—42no—
204C22C2×#12111yescharacteristic
205C22#54611yescharacteristic
206C22D4×Q811yescharacteristic
207C22C22×Q1611yescharacteristic
208C22C2×#13711yescharacteristic
209C22#57611yescharacteristic
210C22#57711yescharacteristic
211C4—24no—
212C4—24no—
213C4—22no—
214C4—42no—
215C4—42no—
216C4—42no—
217C4—82no—
218C4—81no—
219C4—82no—
220C4#46214yes—
221C4C2×C22⋊Q811yescharacteristic
222C4#53711yescharacteristic
223C4#55211yescharacteristic
224C4#55411yescharacteristic
225C2#241711yescharacteristic
226C2C2×#46211yescharacteristic
227C2#304511yescharacteristic
228C2#329911yescharacteristic
229C2#333311yescharacteristic
230C2Q8×Q1611yescharacteristic
231C2#338711yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
48. C2×Q16Q8semidirect
146. C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#4772
C2×C4×Q162
#25672
Q8×C4⋊C42
#20832
#20682
#19902
#19392
#18782
#18032