Finite Group Databasebeta

← #26466

Subgroups — #26466

Order 256.

405 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 147 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×#477C212yesmaximal
2#2551C211yesmaximal, characteristic
3#2365C211yesmaximal, characteristic
4#2068C212yesmaximal
5#2024C212yesmaximal
6#1981C211yesmaximal, characteristic
7#1956C211yesmaximal, characteristic
8#1947C211yesmaximal, characteristic
9#1939C211yesmaximal, characteristic
10#1863C212yesmaximal
11C4×#107C211yesmaximal, characteristic
12C2×C4⋊Q8—21no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2212yes—
15C2×C4×Q8C2212yes—
16#477—28no—
17C2×#107C2212yes—
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×C4⋊C8C2212yes—
21C2×C4⋊C8C2211yescharacteristic
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×C4×C8C2211yescharacteristic
27#395—22no—
28#391C2212yes—
29#389—21no—
30#389—21no—
31#389C2212yes—
32#387—22no—
33#384C2211yescharacteristic
34C4×C4⋊C4—21no—
35C4×C4⋊C4C2211yescharacteristic
36C4×C4⋊C4C2211yescharacteristic
37C4×C4⋊C4C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2211yescharacteristic
41#336C2212yes—
42C22×Q8—22no—
43C22×Q8C2312yes—
44C22×C8C2311yescharacteristic
45C22×C8C2312yes—
46C22×C8C2311yescharacteristic
47C4⋊Q8—24no—
48C4⋊Q8—22no—
49C4⋊Q8—42no—
50C4⋊Q8—44no—
51C4⋊Q8Q814yescomplemented
52C4×Q8—24no—
53C4×Q8—42no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4D411yescharacteristic
62C2×C4⋊C4D411yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C42—22no—
69C2×C42—21no—
70C2×C42—21no—
71C2×C42C2312yes—
72C2×C42C2311yescharacteristic
73#107—24no—
74#107—22no—
75#107D414yes—
76C4⋊C8—24no—
77C4⋊C8—22no—
78#103—24no—
79#103—24no—
80#103—24no—
81#103—42no—
82C4×C8—22no—
83#95—22no—
84#95—21no—
85#95—21no—
86#95—22no—
87#95—22no—
88#95D412yes—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—44no—
95C2×Q8—44no—
96C2×Q8—82no—
97C2×Q8C2×Q814yes—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4C2×D412yes—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D412yes—
107C22×C4C2×D411yescharacteristic
108C22×C4C2411yescharacteristic
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—22no—
112C2×C8—21no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8C2×D412yes—
118C4⋊C4—24no—
119C4⋊C4—24no—
120C4⋊C4—22no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—22no—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—21no—
128C4⋊C4—44no—
129C4⋊C4—44no—
130C4⋊C4—44no—
131C4⋊C4—42no—
132C4⋊C4—81no—
133C4⋊C4Q1614yes—
134C4⋊C4C2×D412yes—
135C4⋊C4C2×D412yes—
136C4⋊C4#4112yes—
137C42—22no—
138C42—22no—
139C42—44no—
140C42—42no—
141C42—42no—
142C42C2×Q812yes—
143C42#4112yes—
144C23C22×D411yescharacteristic
145Q8—24no—
146Q8—24no—
147Q8—42no—
148Q8—42no—
149Q8—88no—
150C2×C4—22no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—24no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4C22⋊Q812yes—
176C2×C4C22⋊Q812yes—
177C2×C4C2×Q1614yes—
178C2×C4C2×Q1612yes—
179C2×C4AGL(1,ℤ/8)12yes—
180C2×C4C22×D411yescharacteristic
181C2×C4C22×Q811yescharacteristic
182C2×C4C2×#4111yescharacteristic
183C2×C4C2×#4111yescharacteristic
184C2×C4H(5,𝔽2)11yescharacteristic
185C2×C42−1+411yescharacteristic
186C8—22no—
187C8—42no—
188C8—41no—
189C22C2×C22⋊Q811yescharacteristic
190C22D4211yescharacteristic
191C22#54811yescharacteristic
192C22C22×Q1611yescharacteristic
193C22C2×AGL(1,ℤ/8)11yescharacteristic
194C22#57711yescharacteristic
195C22#57811yescharacteristic
196C4—22no—
197C4—24no—
198C4—42no—
199C4—44no—
200C4—42no—
201C4—42no—
202C4—42no—
203C4—81no—
204C4—82no—
205C4#47714yescomplemented
206C4#53211yescharacteristic
207C4#54211yescharacteristic
208C4D4×Q811yescharacteristic
209C4#55411yescharacteristic
210C2#245211yescharacteristic
211C2C2×#47711yescharacteristic
212C2#308211yescharacteristic
213C2#327211yescharacteristic
214C2D4×Q1611yescharacteristic
215C2#336311yescharacteristic
216C2#336711yescharacteristic
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
51. C4⋊Q8Q8semidirect
205. C4#477semidirect

Maximal subgroups

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

subgroupindex
C2×#4772
#25512
#23652
#20682
#20242
#19812
#19562
#19472
#19392
#18632
C4×#1072