Finite Group Databasebeta

← #32970

Subgroups — #32970

Order 256.

597 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 597 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C4×2−1+4C211yesmaximal, characteristic
2#3079C212yesmaximal, complemented
3#2568C211yesmaximal, characteristic, complemented
4#1860C214yesmaximal, complemented
5#1857C212yesmaximal, complemented
6#1806C212yesmaximal, complemented
7#1801C211yesmaximal, characteristic, complemented
8#1793C212yesmaximal
9C2×2−1+4C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11#520—22no—
12#520—22no—
13#520C2212yes—
14#519—22no—
15#519C2211yescharacteristic
16C4×#41—22no—
17C4×#41—22no—
18C4×#41C2212yes—
19C2×C4×Q8—22no—
20C2×C4×Q8C2211yescharacteristic
21#477—22no—
22#476—22no—
23#475—22no—
24#474—22no—
25#428C2214yes—
26#423C2212yes—
27C2×#104C2214yes—
28#417C2214yes—
29C2×#101C2212yes—
30#400—22no—
31#391—22no—
32#391C2212yes—
33#382C2212yes—
34C4×C4⋊C4C2211yescharacteristic
35#343C2212yes—
36#339C2214yes—
372−1+4—24no—
382−1+4—22no—
392−1+4Q814yescomplemented
40C2×#41—22no—
41C2×#41—22no—
42C2×#41C2312yes—
43C22×Q8—22no—
44C22×Q8—21no—
45C22×Q8C2311yescharacteristic
46C2×M16C2314yes—
47C4⋊Q8—22no—
48C4⋊Q8—22no—
49C4×Q8—22no—
50C4×Q8—24no—
51C4×Q8—22no—
52C4×Q8—22no—
53C4×Q8—22no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×Q8D412yes—
58C4×Q8D412yes—
59C4×D4—22no—
60C4×D4—24no—
61C4×D4—22no—
62C4×D4—22no—
63C4×D4—22no—
64C4×D4—24no—
65C4×D4—22no—
66C4×D4D412yes—
67C4×D4D412yes—
68#117—24no—
69#117—24no—
70#117—22no—
71#117—22no—
72#117—22no—
73#117C2312yes—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4D412yes—
79C2×C4⋊C4C2312yes—
80C2×C4⋊C4C2311yescharacteristic
81C2×C42—24no—
82C2×C42—22no—
83C2×C42D412yes—
84C2×C42C2312yes—
85C2×C42C2311yescharacteristic
86#107—24no—
87#106—24no—
88C4⋊C8—22no—
89C4⋊C8—22no—
90#104—28no—
91#103—24no—
92#102—24no—
93#101—24no—
94#95—21no—
95#95—24no—
96#95—24no—
97#41—28no—
98#41—28no—
99#41—28no—
100#41—28no—
101#41—22no—
102#41—22no—
103#41C2×Q814yes—
104#41#4114yes—
105C2×Q8—22no—
106C2×Q8—28no—
107C2×Q8—28no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—44no—
111C2×Q8—41no—
112C2×Q8—41no—
113C2×Q8C2×D412yes—
114C2×Q8C2×Q812yes—
115C2×Q8#4112yes—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4C2×D412yes—
119C22×C4—24no—
120C22×C4—22no—
121C22×C4—24no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D412yes—
128C22×C4C2×D412yes—
129C22×C4C2×D411yescharacteristic
130C22×C4C2411yescharacteristic
131M16—24no—
132M16—24no—
133C2×C8—24no—
134C4⋊C4—24no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—24no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—24no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4C2×D412yes—
145C4⋊C4C2×D412yes—
146C22⋊C4—24no—
147C22⋊C4—22no—
148C22⋊C4—22no—
149C22⋊C4—22no—
150C42—24no—
151C42—24no—
152C42—22no—
153C42—22no—
154C42—22no—
155C42—22no—
156C42—22no—
157C42—24no—
158C42C2×D412yes—
159C42C2×D412yes—
160C23—22no—
161C23C22×D411yescharacteristic
162Q8—28no—
163Q8—24no—
164Q8—22no—
165Q8—24no—
166Q8—82no—
167Q8—82no—
168Q8C22⋊Q814yes—
169D4—28no—
170D4—24no—
171D4—24no—
172D4—22no—
173D4C22⋊Q814yes—
174C2×C4—216no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—44no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—44no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—82no—
193C2×C4—84no—
194C2×C4#12011yescharacteristic
195C2×C4#12012yes—
196C2×C4#12011yescharacteristic
197C2×C4C22⋊Q814yes—
198C2×C4C22×D412yes—
199C2×C4C22×Q811yescharacteristic
200C2×C4C2×#4112yes—
201C2×C4C2×#4111yescharacteristic
202C8—44no—
203C22—21no—
204C22—21no—
205C22—24no—
206C22—42no—
207C22—42no—
208C22C2×#12011yescharacteristic
209C22C2×C22⋊Q811yescharacteristic
210C22#52511yescharacteristic
211C4—24no—
212C4—24no—
213C4—21no—
214C4—22no—
215C4—42no—
216C4—42no—
217C4—42no—
218C4—82no—
219C4—82no—
220C4#45412yes—
221C4C2×C22⋊Q812yes—
222C4#52512yes—
223C2—21no—
224C2—21no—
225C2—44no—
226C2#284811yescharacteristic
227C2C2×#45411yescharacteristic
228C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×2−1+4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3079C2semidirect
3. #2568C2semidirect
4. #1860C2semidirect
5. #1857C2semidirect
6. #1806C2semidirect
7. #1801C2semidirect
39. 2−1+4Q8semidirect

Maximal subgroups

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

subgroupindex
C4×2−1+42
#30792
#25682
#18602
#18572
#18062
#18012
#17932