Finite Group Databasebeta

← #8431

Subgroups — #8431

Order 256.

348 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 79 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#3507C211yesmaximal, characteristic
2#1805C211yesmaximal, characteristic, complemented
3#1540C211yesmaximal, characteristic
4#1518C211yesmaximal, characteristic, complemented
5#1517C211yesmaximal, characteristic
6#1481C211yesmaximal, characteristic, complemented
7#1479C211yesmaximal, characteristic, complemented
8#559—21no—
9#557—21no—
10#556—22no—
11#556C412yescomplemented
12#554C412yescomplemented
13#552—22no—
14#539—22no—
15#536—21no—
16#533C2211yescharacteristic
17#532C2211yescharacteristic
18#529—22no—
19#520—21no—
20C2×C4×Q8C2211yescharacteristic
21#431C2211yescharacteristic
22#431C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103—21no—
25#382C2211yescharacteristic
26#340—21no—
27#332—21no—
28#331—21no—
29#328—22no—
30#327—22no—
31Q8⋊C8—22no—
32Q8⋊C8—22no—
33C2×#41—21no—
34C22×Q8D411yescharacteristic
35C22×C8—21no—
36C4⋊Q8—22no—
37C4⋊Q8C2×C411yescharacteristic
38C4⋊Q8C2×C411yescharacteristic
39C4⋊Q8D411yescharacteristic
40#127D411yescharacteristic
41#126—24no—
42#126—24no—
43#125—24no—
44#125—22no—
45#125—21no—
46#124—22no—
47#124—22no—
48#124—21no—
49#123—24no—
50#123—24no—
51C22⋊Q8—22no—
52C22⋊Q8—22no—
53C22⋊Q8—24no—
54C22⋊Q8—21no—
55C22⋊Q8D412yes—
56#121—21no—
57#121D412yes—
58C4×Q8—22no—
59C4×Q8—22no—
60C4×Q8—21no—
61C4×Q8—21no—
62C4×Q8—21no—
63C4×Q8C2×C412yes—
64C4×Q8D412yes—
65C4×Q8D412yes—
66C4×D4—24no—
67C4×D4—22no—
68C4×D4—22no—
69C4×D4—21no—
70C4×D4C2×C412yes—
71#117—24no—
72#117—21no—
73#117—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4D411yescharacteristic
78C2×C42—21no—
79C2×C42C2311yescharacteristic
80C4⋊C8—21no—
81C4⋊C8—21no—
82C4⋊C8—21no—
83C4⋊C8—21no—
84#103—42no—
85C22⋊C8—21no—
86C22⋊C8—21no—
87C4×C8—21no—
88C4×C8—21no—
89#95—21no—
90#95—21no—
91#41—41no—
92#41—41no—
93#41—41no—
94#41—41no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—21no—
101C2×Q8—21no—
102C2×Q8C22⋊C412yes—
103C2×Q8C22⋊C412yes—
104C2×D4—22no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4C22⋊C412yes—
108C22×C4—21no—
109C22×C4—24no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—41no—
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C2×C8—41no—
117C2×C8—41no—
118C2×C8—41no—
119C2×C8—41no—
120C2×C8—41no—
121C4⋊C4—24no—
122C4⋊C4—24no—
123C4⋊C4—24no—
124C4⋊C4—21no—
125C4⋊C4—22no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—22no—
129C4⋊C4—21no—
130C4⋊C4—22no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4C22⋊C412yes—
136C4⋊C4C22⋊C412yes—
137C4⋊C4C22⋊C412yes—
138C22⋊C4—24no—
139C22⋊C4—24no—
140C22⋊C4—24no—
141C22⋊C4—22no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C42—24no—
145C42—22no—
146C42—22no—
147C42—22no—
148C42C22×C411yescharacteristic
149C42C2×D411yescharacteristic
150C42C2×D411yescharacteristic
151C42C2×D411yescharacteristic
152C23—22no—
153C23#12011yescharacteristic
154Q8—22no—
155Q8—22no—
156Q8—42no—
157Q8—42no—
158Q8—41no—
159Q8—41no—
160Q8—41no—
161Q8—41no—
162D4—42no—
163D4—42no—
164D4—42no—
165D4—41no—
166D4—41no—
167D4—81no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—81no—
180C2×C4—84no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4C2×C22⋊C411yescharacteristic
185C2×C4C2×C22⋊C411yescharacteristic
186C2×C4C2×C22⋊C411yescharacteristic
187C2×C4#12011yescharacteristic
188C2×C4#12011yescharacteristic
189C2×C4#12011yescharacteristic
190C2×C4#13511yescharacteristic
191C2×C4#13511yescharacteristic
192C2×C4AGL(1,ℤ/8)11yescharacteristic
193C2×C4#13711yescharacteristic
194C8—81no—
195C8—81no—
196C8—81no—
197C22—21no—
198C22—21no—
199C22—42no—
200C22—42no—
201C22—42no—
202C22#37911yescharacteristic
203C22#44911yescharacteristic
204C22#45211yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—42no—
210C4—42no—
211C4—42no—
212C4—42no—
213C4—42no—
214C4—161no—
215C4#41611yescharacteristic
216C4#41711yescharacteristic
217C4#44911yescharacteristic
218C4#45211yescharacteristic
219C2—21no—
220C2—21no—
221C2—82no—
222C2#178811yescharacteristic
223C2#179411yescharacteristic
224C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1805C2semidirect
4. #1518C2semidirect
6. #1481C2semidirect
7. #1479C2semidirect
11. #556C4semidirect
12. #554C4semidirect

Maximal subgroups

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

subgroupindex
#35072
#18052
#15402
#15182
#15172
#14812
#14792