Finite Group Databasebeta

← #29849

Subgroups — #29849

Order 256.

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

normal subgroups only

This group has 387 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×#488C211yesmaximal, characteristic, complemented
2C2×#433C211yesmaximal, characteristic, complemented
3#2629C211yesmaximal, characteristic, complemented
4#2620C211yesmaximal, characteristic
5#2068C212yesmaximal, complemented
6#2067C212yesmaximal, complemented
7#2036C212yesmaximal, complemented
8#2033C212yesmaximal
9#1992C211yesmaximal, characteristic, complemented
10#1959C211yesmaximal, characteristic, complemented
11#1845C211yesmaximal, characteristic, complemented
12C2×#125C2211yescharacteristic
13C2×#124C2211yescharacteristic
14C2×C22⋊Q8C2211yescharacteristic
15C2×#121C2211yescharacteristic
16C2×#117C2211yescharacteristic
17#488—24no—
18#433—24no—
19C2×#107C2212yes—
20C2×#107C2212yes—
21C2×C4⋊C8C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×C22⋊C8C2212yes—
27C2×#97C2211yescharacteristic
28C2×C4×C8C2211yescharacteristic
29#398—22no—
30#396—22no—
31#396—22no—
32#394—22no—
33#391C2211yescharacteristic
34#390C2211yescharacteristic
35#389C2212yes—
36#384C2211yescharacteristic
37#384C2211yescharacteristic
38#380C2212yes—
39#340C2212yes—
40#340C2212yes—
41C22×Q8C2311yescharacteristic
42C22×D4—21no—
43C22×D4C2311yescharacteristic
44C23×C4C2311yescharacteristic
45C22×C8C2312yes—
46C22×C8C2312yes—
47#125—22no—
48#125—22no—
49#124—22no—
50#124—22no—
51C22⋊Q8—42no—
52#121—42no—
53#117—22no—
54#117—22no—
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⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4D412yes—
71C2×C42C2311yescharacteristic
72C2×C42C2311yescharacteristic
73#107—42no—
74#107—42no—
75C4⋊C8—24no—
76#103—24no—
77#103—42no—
78#102—24no—
79#102—42no—
80C22⋊C8—24no—
81#97—22no—
82C4×C8D414yescomplemented
83#95—22no—
84#95—22no—
85#95—22no—
86C24—21no—
87C24C2×D411yescharacteristic
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×D4—21no—
93C2×D4—21no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—42no—
97C2×D4—82no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4C2×D412yes—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D412yes—
110C22×C4C2411yescharacteristic
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8C2×D414yes—
117C4⋊C4—24no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—42no—
125C4⋊C4—44no—
126C4⋊C4—82no—
127C4⋊C4—82no—
128C4⋊C4—82no—
129C22⋊C4—24no—
130C22⋊C4—42no—
131C22⋊C4—42no—
132C22⋊C4—44no—
133C22⋊C4—82no—
134C22⋊C4—82no—
135C42—21no—
136C42—21no—
137C42C2×D412yes—
138C42#4112yes—
139C23—21no—
140C23—21no—
141C23—21no—
142C23—22no—
143C23—22no—
144C23—41no—
145C23—41no—
146C23—41no—
147C23—42no—
148C23—42no—
149C23C22×D411yescharacteristic
150Q8—42no—
151Q8—42no—
152D4—42no—
153D4—42no—
154D4—84no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4#12114yes—
179C2×C4C22×D411yescharacteristic
180C2×C4C2×#4111yescharacteristic
181C2×C4H(5,𝔽2)12yes—
182C2×C42−1+412yes—
183C8—24no—
184C8—42no—
185C22—41no—
186C22—41no—
187C22—41no—
188C22—42no—
189C22—42no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22—82no—
194C22—82no—
195C22C2×#12111yescharacteristic
196C22#53511yescharacteristic
197C22#53611yescharacteristic
198C22#57612yes—
199C22#57812yes—
200C4—24no—
201C4—41no—
202C4—42no—
203C4—81no—
204C4—81no—
205C4—82no—
206C4—81no—
207C4—82no—
208C4#53712yes—
209C4#54212yes—
210C2—81no—
211C2—161no—
212C2#262011yescharacteristic
213C2#306112yes—
214C2#322011yescharacteristic
215C2#322311yescharacteristic
216C2#324312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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×#488C2semidirect
2. C2×#433C2semidirect
3. #2629C2semidirect
5. #2068C2semidirect
6. #2067C2semidirect
7. #2036C2semidirect
9. #1992C2semidirect
10. #1959C2semidirect
11. #1845C2semidirect
82. C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4882
C2×#4332
#26292
#26202
#20682
#20672
#20362
#20332
#19922
#19592
#18452