Finite Group Databasebeta

← #29856

Subgroups — #29856

Order 256.

366 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 119 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×#491C211yesmaximal, characteristic, complemented
2C2×#433C211yesmaximal, characteristic
3#2629C211yesmaximal, characteristic
4#2621C211yesmaximal, characteristic
5#2068C212yesmaximal, complemented
6#2066C212yesmaximal, complemented
7#2036C212yesmaximal
8#2035C212yesmaximal
9#1991C211yesmaximal, characteristic, complemented
10#1956C211yesmaximal, characteristic, complemented
11#1847C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×C22⋊Q8C2211yescharacteristic
15C2×C22⋊Q8C2211yescharacteristic
16C2×#117C2211yescharacteristic
17#491—24no—
18#433—24no—
19C2×#107C2212yes—
20C2×#106C2212yes—
21C2×C4⋊C8C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×C22⋊C8C2212yes—
27C2×#97C2211yescharacteristic
28C2×C4×C8C2211yescharacteristic
29#398—22no—
30#398—22no—
31#397—22no—
32#396—22no—
33#391C2211yescharacteristic
34#391C2211yescharacteristic
35#389C2212yes—
36#384C2211yescharacteristic
37#382C2211yescharacteristic
38#380C2212yes—
39#340C2212yes—
40#340C2212yes—
41C22×Q8C2311yescharacteristic
42C22×Q8C2311yescharacteristic
43C23×C4C2311yescharacteristic
44C22×C8C2312yes—
45C22×C8C2312yes—
46C4⋊Q8—22no—
47C4⋊Q8—22no—
48#125—22no—
49#125—22no—
50C22⋊Q8—42no—
51C22⋊Q8—42no—
52#117—22no—
53#117—22no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4D412yes—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2312yes—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4D412yes—
69C2×C42C2311yescharacteristic
70C2×C42C2311yescharacteristic
71#107—42no—
72#106—42no—
73C4⋊C8—24no—
74#103—24no—
75#103—24no—
76#103—42no—
77#103—42no—
78C22⋊C8—24no—
79#97D414yes—
80C4×C8—22no—
81#95—22no—
82#95—22no—
83#95—22no—
84#95—22no—
85#95—22no—
86C24C2×D411yescharacteristic
87C2×Q8—21no—
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4C2×D412yes—
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D412yes—
108C22×C4C2411yescharacteristic
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8C2×D414yes—
115C4⋊C4—24no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—44no—
125C4⋊C4—44no—
126C4⋊C4—82no—
127C4⋊C4—82no—
128C22⋊C4—24no—
129C22⋊C4—42no—
130C22⋊C4—42no—
131C22⋊C4—82no—
132C42—21no—
133C42—21no—
134C42C2×D412yes—
135C42#4112yes—
136C23—21no—
137C23—21no—
138C23—21no—
139C23—22no—
140C23—22no—
141C23C22×D411yescharacteristic
142Q8—42no—
143Q8—42no—
144Q8—42no—
145Q8—42no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4#12114yes—
173C2×C4C22×D411yescharacteristic
174C2×C4C2×#4111yescharacteristic
175C2×C4H(5,𝔽2)12yes—
176C2×C42−1+412yes—
177C8—24no—
178C8—42no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—42no—
183C22—42no—
184C22C2×#12111yescharacteristic
185C22#53511yescharacteristic
186C22#53611yescharacteristic
187C22#57712yes—
188C22#57812yes—
189C4—24no—
190C4—41no—
191C4—42no—
192C4—81no—
193C4—81no—
194C4—81no—
195C4—82no—
196C4—81no—
197C4—82no—
198C4#53712yes—
199C4#54212yes—
200C2—81no—
201C2#262011yescharacteristic
202C2#306312yes—
203C2#322111yescharacteristic
204C2#322311yescharacteristic
205C2#324212yes—
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
1. C2×#491C2semidirect
5. #2068C2semidirect
6. #2066C2semidirect
9. #1991C2semidirect
10. #1956C2semidirect
11. #1847C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4912
C2×#4332
#26292
#26212
#20682
#20662
#20362
#20352
#19912
#19562
#18472