Finite Group Databasebeta

← #30294

Subgroups — #30294

Order 256.

385 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 139 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
1#2857C211yesmaximal, characteristic
2#2851C211yesmaximal, characteristic
3#2085C212yesmaximal, complemented
4#2084C211yesmaximal, characteristic, complemented
5#2083C211yesmaximal, characteristic, complemented
6#2059C212yesmaximal, complemented
7#2058C211yesmaximal, characteristic, complemented
8#2057C211yesmaximal, characteristic, complemented
9#1808C212yesmaximal
10#1807C212yesmaximal
11C2×#336C211yesmaximal, characteristic
12C23×C8C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×#125C2211yescharacteristic
15C2×C22⋊Q8—21no—
16C22×C42C2211yescharacteristic
17C2×#107C2212yes—
18C2×#107C2212yes—
19C2×#106C2212yes—
20C2×#106C2212yes—
21#391—21no—
22#389—21no—
23#389C2211yescharacteristic
24#389C2211yescharacteristic
25#389C2212yes—
26#387—21no—
27#387—21no—
28#385—21no—
29#385—21no—
30#384C2211yescharacteristic
31#382C2211yescharacteristic
32#380—21no—
33#380C2211yescharacteristic
34#380C2212yes—
35#380C2211yescharacteristic
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#336C2211yescharacteristic
41#336C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C23×C4D411yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4C2311yescharacteristic
47C22×C8—42no—
48C22×C8—42no—
49C22×C8Q818yescomplemented
50C22×C8C2312yes—
51C22×C8C2312yes—
52C4⋊Q8—42no—
53#125—42no—
54C22⋊Q8—44no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C42—42no—
73C2×C42D411yescharacteristic
74C2×C42D411yescharacteristic
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77#107—42no—
78#107—42no—
79#106—42no—
80#106—42no—
81#95—22no—
82#95—21no—
83#95—21no—
84#95—22no—
85#95—21no—
86#95—21no—
87#95—21no—
88#95—21no—
89C24C2×D411yescharacteristic
90C2×Q8—42no—
91C2×Q8—42no—
92C2×Q8—42no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×Q814yes—
114C22×C4C2411yescharacteristic
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—48no—
120C2×C8C2×Q814yes—
121C2×C8C2×Q814yes—
122C4⋊C4—21no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—21no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—42no—
133C4⋊C4—44no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C22⋊C4—42no—
141C22⋊C4—44no—
142C22⋊C4—42no—
143C42—22no—
144C42—22no—
145C42—41no—
146C42—41no—
147C23—21no—
148C23—21no—
149C23—21no—
150C23—21no—
151C23C4⋊Q812yes—
152C23AGL(1,ℤ/8)12yes—
153C23#13712yes—
154C23C22×D411yescharacteristic
155Q8—84no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—44no—
171C2×C4—44no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4C4⋊Q812yes—
191C2×C4#13512yes—
192C2×C4#13512yes—
193C2×C4C22×Q812yes—
194C2×C4H(5,𝔽2)11yescharacteristic
195C2×C4H(5,𝔽2)12yes—
196C2×C42−1+411yescharacteristic
197C8—24no—
198C8—24no—
199C22—22no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22—22no—
205C22#50114yes—
206C22C2×C4⋊Q811yescharacteristic
207C22#53411yescharacteristic
208C22#53711yescharacteristic
209C22C2×#13511yescharacteristic
210C22C2×#13511yescharacteristic
211C22C2×AGL(1,ℤ/8)11yescharacteristic
212C22C2×#13711yescharacteristic
213C4—22no—
214C4—22no—
215C4—22no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—82no—
220C4—81no—
221C4—82no—
222C4—81no—
223C4—81no—
224C4#54312yes—
225C4#54412yes—
226C2—24no—
227C2#265911yescharacteristic
228C2#316011yescharacteristic
229C2C2×#50111yescharacteristic
230C2#318811yescharacteristic
231C2#319011yescharacteristic
232C2#324711yescharacteristic
233C2#324811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #2085C2semidirect
4. #2084C2semidirect
5. #2083C2semidirect
6. #2059C2semidirect
7. #2058C2semidirect
8. #2057C2semidirect
49. C22×C8Q8semidirect

Maximal subgroups

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

subgroupindex
#28572
#28512
#20852
#20842
#20832
#20592
#20582
#20572
#18082
#18072
C2×#3362