Finite Group Databasebeta

← #8430

Subgroups — #8430

Order 256.

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

normal subgroups only

This group has 380 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#3506C211yesmaximal, characteristic
2#1804C211yesmaximal, characteristic, complemented
3#1540C211yesmaximal, characteristic, complemented
4#1518C211yesmaximal, characteristic, complemented
5#1517C211yesmaximal, characteristic, complemented
6#1481C211yesmaximal, characteristic, complemented
7#1478C211yesmaximal, characteristic, complemented
8#559—22no—
9#556—21no—
10#555C412yescomplemented
11#553—22no—
12#550—21no—
13#547C412yescomplemented
14#538—22no—
15#535—21no—
16#533C2211yescharacteristic
17#532C2211yescharacteristic
18#525—22no—
19#520—21no—
20C2×C4×D4C2211yescharacteristic
21#431C2211yescharacteristic
22#431C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#102—21no—
25#382C2211yescharacteristic
26#340—21no—
27#332—21no—
28#331—21no—
29#328—22no—
30#327—22no—
31Q8⋊C8—22no—
32D4⋊C8—22no—
33C2×#41—22no—
34C2×#41—21no—
35C22×D4D411yescharacteristic
36C23×C4—21no—
37C22×C8—21no—
38C4⋊Q8C2×C411yescharacteristic
39C4⋊Q8C2×C411yescharacteristic
40C4⋊Q8D411yescharacteristic
41#127—22no—
42#127D411yescharacteristic
43#126—24no—
44#125—21no—
45#124—22no—
46#124—24no—
47#124—21no—
48#123—24no—
49#123—24no—
50C22⋊Q8—21no—
51C22⋊Q8—22no—
52C22⋊Q8D412yes—
53#121—24no—
54#121—22no—
55#121—22no—
56#121—22no—
57#121—21no—
58#121D412yes—
59#120—24no—
60C4×Q8—22no—
61C4×Q8—21no—
62C4×Q8D412yes—
63C4×D4—24no—
64C4×D4—22no—
65C4×D4—24no—
66C4×D4—21no—
67C4×D4—21no—
68C4×D4—21no—
69C4×D4C2×C412yes—
70C4×D4C2×C412yes—
71C4×D4D412yes—
72#117—21no—
73#117—22no—
74#117—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4D411yescharacteristic
78C2×C22⋊C4—21no—
79C2×C42C2311yescharacteristic
80C4⋊C8—21no—
81C4⋊C8—21no—
82C4⋊C8—21no—
83C4⋊C8—21no—
84#102—42no—
85C22⋊C8—21no—
86C22⋊C8—21no—
87C4×C8—21no—
88C4×C8—21no—
89#95—21no—
90#95—21no—
91C24—21no—
92#41—41no—
93#41—41no—
94#41—41no—
95#41—41no—
96#41—84no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8C22⋊C412yes—
100C2×D4—22no—
101C2×D4—24no—
102C2×D4—24no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4C22⋊C412yes—
111C2×D4C22⋊C412yes—
112C22×C4—24no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—41no—
118C22×C4—82no—
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C2×C8—41no—
123C2×C8—41no—
124C2×C8—41no—
125C2×C8—41no—
126C2×C8—41no—
127C4⋊C4—24no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4—22no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4C22⋊C412yes—
138C4⋊C4C22⋊C412yes—
139C4⋊C4C22⋊C412yes—
140C22⋊C4—24no—
141C22⋊C4—24no—
142C22⋊C4—24no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4—22no—
147C22⋊C4—22no—
148C42—24no—
149C42—22no—
150C42C22×C411yescharacteristic
151C42C2×D411yescharacteristic
152C42C2×D411yescharacteristic
153C42C2×D411yescharacteristic
154C23—22no—
155C23—22no—
156C23—41no—
157C23—41no—
158C23—81no—
159C23—81no—
160C23#12011yescharacteristic
161Q8—42no—
162Q8—42no—
163Q8—41no—
164Q8—41no—
165D4—22no—
166D4—22no—
167D4—42no—
168D4—42no—
169D4—42no—
170D4—41no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—81no—
175D4—84no—
176D4—84no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—44no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—82no—
187C2×C4—81no—
188C2×C4—84no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4C2×C22⋊C411yescharacteristic
193C2×C4C2×C22⋊C411yescharacteristic
194C2×C4C2×C22⋊C411yescharacteristic
195C2×C4#12011yescharacteristic
196C2×C4#12011yescharacteristic
197C2×C4#12011yescharacteristic
198C2×C4#13511yescharacteristic
199C2×C4#13511yescharacteristic
200C2×C4AGL(1,ℤ/8)11yescharacteristic
201C2×C4#13711yescharacteristic
202C8—81no—
203C8—81no—
204C8—81no—
205C22—21no—
206C22—21no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—82no—
214C22—82no—
215C22#37911yescharacteristic
216C22#44911yescharacteristic
217C22#45211yescharacteristic
218C4—21no—
219C4—21no—
220C4—21no—
221C4—21no—
222C4—42no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—161no—
227C4#41611yescharacteristic
228C4#41711yescharacteristic
229C4#44911yescharacteristic
230C4#45211yescharacteristic
231C2—21no—
232C2—21no—
233C2—82no—
234C2—82no—
235C2#178811yescharacteristic
236C2#179411yescharacteristic
237C2#179511yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3506C2
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. #1804C2semidirect
3. #1540C2semidirect
4. #1518C2semidirect
5. #1517C2semidirect
6. #1481C2semidirect
7. #1478C2semidirect
10. #555C4semidirect
13. #547C4semidirect

Maximal subgroups

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

subgroupindex
#35062
#18042
#15402
#15182
#15172
#14812
#14782