Finite Group Databasebeta

← #29848

Subgroups — #29848

Order 256.

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

normal subgroups only

This group has 370 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×#487C211yesmaximal, characteristic, complemented
2C2×#433C211yesmaximal, characteristic
3#2628C211yesmaximal, characteristic
4#2621C211yesmaximal, characteristic
5#2068C212yesmaximal, complemented
6#2066C212yesmaximal, complemented
7#2036C212yesmaximal
8#2035C212yesmaximal
9#1993C211yesmaximal, characteristic, complemented
10#1960C211yesmaximal, characteristic, complemented
11#1846C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C22⋊Q8C2212yes—
15C2×#117C2211yescharacteristic
16#487—24no—
17#433—24no—
18C2×#107C2212yes—
19C2×#106C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×C22⋊C8C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic
28C2×#97C2211yescharacteristic
29C2×C4×C8C2211yescharacteristic
30#400—21no—
31#399—21no—
32#398—21no—
33#398—21no—
34#397—21no—
35#397—21no—
36#395—21no—
37#393—21no—
38#391C2212yes—
39#389C2212yes—
40#382C2211yescharacteristic
41#382C2211yescharacteristic
42#380C2212yes—
43#340C2212yes—
44#340C2212yes—
45C22×Q8—21no—
46C22×Q8C2312yes—
47C23×C4C2311yescharacteristic
48C22×C8C2311yescharacteristic
49C22×C8C2311yescharacteristic
50C22×C8C2311yescharacteristic
51C22×C8C2311yescharacteristic
52C4⋊Q8—22no—
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55C4⋊Q8—22no—
56C22⋊Q8—42no—
57C22⋊Q8—42no—
58#117—22no—
59#117—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4D411yescharacteristic
66C2×C4⋊C4D411yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2312yes—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4D411yescharacteristic
73C2×C22⋊C4D411yescharacteristic
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76#107—42no—
77#106—42no—
78C4⋊C8—22no—
79C4⋊C8—22no—
80#103—24no—
81#103—24no—
82#103—42no—
83#103—42no—
84C22⋊C8—22no—
85C22⋊C8—22no—
86#97—22no—
87C4×C8D414yes—
88#95—22no—
89#95—21no—
90#95—21no—
91#95—21no—
92#95—21no—
93#95—22no—
94#95—22no—
95#95—22no—
96C24C2×D411yescharacteristic
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—42no—
104C2×Q8—82no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2411yescharacteristic
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8C2×D412yes—
132C2×C8C2×D412yes—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—44no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—82no—
146C22⋊C4—22no—
147C22⋊C4—22no—
148C22⋊C4—44no—
149C22⋊C4—82no—
150C42—21no—
151C42—21no—
152C42C2×D412yes—
153C42#4112yes—
154C23—21no—
155C23—21no—
156C23—21no—
157C23—22no—
158C23—22no—
159C23C22×D411yescharacteristic
160Q8—42no—
161Q8—42no—
162Q8—42no—
163Q8—42no—
164Q8—84no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4#12112yes—
191C2×C4#12112yes—
192C2×C4C22×D411yescharacteristic
193C2×C4C2×#4111yescharacteristic
194C2×C4H(5,𝔽2)11yescharacteristic
195C2×C4H(5,𝔽2)11yescharacteristic
196C2×C42−1+411yescharacteristic
197C2×C42−1+411yescharacteristic
198C8—22no—
199C8—22no—
200C8—41no—
201C8—41no—
202C22—41no—
203C22—42no—
204C22—42no—
205C22—41no—
206C22—41no—
207C22C2×#12111yescharacteristic
208C22#53511yescharacteristic
209C22#53611yescharacteristic
210C22#57712yes—
211C22#57812yes—
212C4—22no—
213C4—22no—
214C4—41no—
215C4—41no—
216C4—41no—
217C4—82no—
218C4—82no—
219C4—82no—
220C4—82no—
221C4#53711yescharacteristic
222C4#53711yescharacteristic
223C4#54211yescharacteristic
224C4#54211yescharacteristic
225C2—81no—
226C2#262011yescharacteristic
227C2#305711yescharacteristic
228C2#305911yescharacteristic
229C2#322512yes—
230C2#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×#487C2semidirect
5. #2068C2semidirect
6. #2066C2semidirect
9. #1993C2semidirect
10. #1960C2semidirect
11. #1846C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4872
C2×#4332
#26282
#26212
#20682
#20662
#20362
#20352
#19932
#19602
#18462