Finite Group Databasebeta

← #27769

Subgroups — #27769

Order 256.

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

normal subgroups only

This group has 411 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×#487C212yesmaximal
2#2551C211yesmaximal, characteristic
3#2546C211yesmaximal, characteristic
4#2043C212yesmaximal
5#2024C212yesmaximal
6#1981C211yesmaximal, characteristic
7#1956C211yesmaximal, characteristic
8#1933C212yesmaximal
9#1803C211yesmaximal, characteristic
10C8×C4⋊C4C211yesmaximal, characteristic
11C4×#103C211yesmaximal, characteristic
12C2×C4⋊Q8—21no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2212yes—
15C2×C4⋊Q8C2212yes—
16C2×C4×Q8C2211yescharacteristic
17#487—28no—
18C2×C4⋊C8C2211yescharacteristic
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#103C2211yescharacteristic
24C2×#103C2212yes—
25C2×#103C2211yescharacteristic
26C2×C4×C8C2212yes—
27C2×C4×C8C2211yescharacteristic
28#398—22no—
29#395—22no—
30#391C2212yes—
31#389—21no—
32#389—21no—
33#389—21no—
34#389C2212yes—
35#384C2211yescharacteristic
36C4×C4⋊C4C2211yescharacteristic
37C4×C4⋊C4C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2211yescharacteristic
41#336C2212yes—
42C22×Q8—22no—
43C22×Q8C2312yes—
44C22×Q8C2311yescharacteristic
45C22×C8C2311yescharacteristic
46C22×C8C2312yes—
47C22×C8C2311yescharacteristic
48C4⋊Q8—24no—
49C4⋊Q8—24no—
50C4⋊Q8—22no—
51C4⋊Q8—22no—
52C4⋊Q8—42no—
53C4⋊Q8—42no—
54C4⋊Q8—44no—
55C4×Q8—22no—
56C4×Q8—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4D411yescharacteristic
67C2×C4⋊C4D411yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C42—21no—
73C2×C42—21no—
74C2×C42C2312yes—
75C2×C42C2311yescharacteristic
76C4⋊C8—22no—
77#103—24no—
78#103—24no—
79#103—24no—
80#103—22no—
81#103—24no—
82#103—42no—
83#103D414yes—
84C4×C8—24no—
85C4×C8—22no—
86#95—21no—
87#95—22no—
88#95—22no—
89#95—22no—
90#95D412yes—
91C2×Q8—21no—
92C2×Q8—22no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—21no—
101C2×Q8—22no—
102C2×Q8—44no—
103C2×Q8—44no—
104C2×Q8—82no—
105C2×Q8C2×D412yes—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D412yes—
114C22×C4C2×D412yes—
115C22×C4C2×D411yescharacteristic
116C22×C4C2411yescharacteristic
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—22no—
120C2×C8—21no—
121C2×C8—22no—
122C2×C8—21no—
123C2×C8—22no—
124C2×C8—21no—
125C2×C8C2×D412yes—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—22no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—44no—
133C4⋊C4—44no—
134C4⋊C4—44no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—44no—
139C4⋊C4—42no—
140C4⋊C4—81no—
141C4⋊C4SD1614yes—
142C4⋊C4Q1614yes—
143C4⋊C4C2×D412yes—
144C4⋊C4#4112yes—
145C42—22no—
146C42—22no—
147C42—42no—
148C42—42no—
149C42#4112yes—
150C42#4112yes—
151C23C22×D411yescharacteristic
152Q8—22no—
153Q8—22no—
154Q8—41no—
155Q8—41no—
156Q8—42no—
157Q8—42no—
158Q8—42no—
159Q8—42no—
160Q8—88no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4#12412yes—
186C2×C4#12412yes—
187C2×C4C2×SD1614yes—
188C2×C4C2×SD1612yes—
189C2×C4C2×Q1614yes—
190C2×C4C2×Q1612yes—
191C2×C4C22×D411yescharacteristic
192C2×C4C2×#4111yescharacteristic
193C2×C4C2×#4111yescharacteristic
194C2×C4C2×#4111yescharacteristic
195C2×C4H(5,𝔽2)11yescharacteristic
196C2×C42−1+411yescharacteristic
197C8—22no—
198C8—42no—
199C8—41no—
200C22C2×#12411yescharacteristic
201C22D4211yescharacteristic
202C22#54811yescharacteristic
203C22C22×SD1611yescharacteristic
204C22C22×Q1611yescharacteristic
205C22#57711yescharacteristic
206C22#57811yescharacteristic
207C4—22no—
208C4—24no—
209C4—24no—
210C4—42no—
211C4—42no—
212C4—42no—
213C4—81no—
214C4—82no—
215C4—82no—
216C4#48714yescomplemented
217C4#53211yescharacteristic
218C4#54211yescharacteristic
219C4#54711yescharacteristic
220C4#55511yescharacteristic
221C2#249111yescharacteristic
222C2C2×#48711yescharacteristic
223C2#313711yescharacteristic
224C2D4×SD1611yescharacteristic
225C2D4×Q1611yescharacteristic
226C2#335911yescharacteristic
227C2#336311yescharacteristic
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
216. C4#487semidirect

Maximal subgroups

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

subgroupindex
C2×#4872
#25512
#25462
#20432
#20242
#19812
#19562
#19332
#18032
C8×C4⋊C42
C4×#1032