Finite Group Databasebeta

← #8411

Subgroups — #8411

Order 256.

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

normal subgroups only

This group has 357 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#3455C211yesmaximal, characteristic
2#1798C211yesmaximal, characteristic, complemented
3#1542C211yesmaximal, characteristic, complemented
4#1540C211yesmaximal, characteristic, complemented
5#1481C211yesmaximal, characteristic, complemented
6#1480C211yesmaximal, characteristic, complemented
7#1458C211yesmaximal, characteristic, complemented
8#542—21no—
9#542—21no—
10#541—21no—
11#540—21no—
12#539—21no—
13#539C412yes—
14#538—21no—
15#538C412yes—
16#533C2211yescharacteristic
17#532C2211yescharacteristic
18#529—22no—
19#529—22no—
20#525—21no—
21C4×#41—21no—
22C2×#117C2211yescharacteristic
23#431C2211yescharacteristic
24#431C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C22⋊C8—21no—
28#336—21no—
29#333—21no—
30#332—21no—
31#332—21no—
32#331—21no—
33Q8⋊C8—22no—
34D4⋊C8—22no—
35#324—21no—
36#323—21no—
37C2×#41—21no—
38C23×C4D411yescharacteristic
39C22×C8—21no—
40C22×C8—21no—
41C4⋊Q8D411yescharacteristic
42C4⋊Q8D411yescharacteristic
43C4⋊Q8D411yescharacteristic
44#127D411yescharacteristic
45#126—22no—
46#126—22no—
47#126—22no—
48#126—22no—
49#125—22no—
50#125—22no—
51#125—21no—
52#124—22no—
53#124—22no—
54#124—21no—
55#123—22no—
56#123—22no—
57#123—22no—
58#123—22no—
59#123—22no—
60C22⋊Q8—21no—
61C22⋊Q8—21no—
62C22⋊Q8—22no—
63C22⋊Q8—22no—
64C22⋊Q8—21no—
65C22⋊Q8C2×C412yes—
66#121—21no—
67#121—21no—
68#121—22no—
69#121—22no—
70#121—21no—
71#121C2×C412yes—
72#120—22no—
73C4×Q8—21no—
74C4×Q8—21no—
75C4×Q8—21no—
76C4×Q8D412yes—
77C4×D4—21no—
78C4×D4—21no—
79C4×D4—22no—
80C4×D4—22no—
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—21no—
84C4×D4D412yes—
85#117—22no—
86#117—21no—
87#117—21no—
88#117—21no—
89#117—21no—
90#117—21no—
91C2×C4⋊C4C2×C411yescharacteristic
92C2×C4⋊C4D411yescharacteristic
93C2×C22⋊C4C2×C411yescharacteristic
94C2×C22⋊C4D411yescharacteristic
95C2×C42—22no—
96C2×C42D411yescharacteristic
97C2×C42C2311yescharacteristic
98C4⋊C8—21no—
99C4⋊C8—21no—
100C4⋊C8—21no—
101C4⋊C8—21no—
102C4⋊C8—21no—
103C4⋊C8—21no—
104C22⋊C8—21no—
105C22⋊C8—21no—
106C22⋊C8—42no—
107C4×C8—21no—
108C4×C8—21no—
109C24C22⋊C411yescharacteristic
110#41—41no—
111#41—41no—
112#41—41no—
113#41—41no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8C22⋊C412yes—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4C22⋊C412yes—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4C22⋊C411yescharacteristic
133C22×C4C22⋊C411yescharacteristic
134C22×C4C22⋊C411yescharacteristic
135C22×C4C22×C411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—41no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—22no—
160C4⋊C4C22⋊C412yes—
161C4⋊C4C22⋊C412yes—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C22⋊C4—21no—
170C22⋊C4—21no—
171C22⋊C4—22no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C22⋊C4—22no—
175C42—22no—
176C42—22no—
177C42—21no—
178C42—21no—
179C42—22no—
180C42C2×D411yescharacteristic
181C42C2×D411yescharacteristic
182C42C2×D411yescharacteristic
183C42C2×D411yescharacteristic
184C23—21no—
185C23—22no—
186C23—21no—
187C23—21no—
188C23—41no—
189C23—41no—
190C23C2×C22⋊C411yescharacteristic
191Q8—42no—
192Q8—41no—
193Q8—41no—
194Q8—81no—
195D4—42no—
196D4—42no—
197D4—42no—
198D4—41no—
199D4—41no—
200D4—81no—
201D4—82no—
202D4—82no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4C2×C22⋊C411yescharacteristic
224C2×C4C2×C22⋊C411yescharacteristic
225C2×C4#12011yescharacteristic
226C2×C4#12011yescharacteristic
227C2×C4#12011yescharacteristic
228C2×C4#12011yescharacteristic
229C2×C4#13511yescharacteristic
230C2×C4#13511yescharacteristic
231C2×C4AGL(1,ℤ/8)11yescharacteristic
232C2×C4#13711yescharacteristic
233C8—81no—
234C8—81no—
235C8—81no—
236C8—81no—
237C22—21no—
238C22—21no—
239C22—42no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—42no—
244C22—42no—
245C22—41no—
246C22—81no—
247C22#37911yescharacteristic
248C22#41611yescharacteristic
249C22#41711yescharacteristic
250C4—21no—
251C4—21no—
252C4—21no—
253C4—21no—
254C4—41no—
255C4—42no—
256C4—41no—
257C4—42no—
258C4—41no—
259C4—42no—
260C4#44911yescharacteristic
261C4#44911yescharacteristic
262C4#45211yescharacteristic
263C4#45211yescharacteristic
264C2—21no—
265C2—21no—
266C2—82no—
267C2—81no—
268C2#178811yescharacteristic
269C2#179111yescharacteristic
270C2#179411yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
socleC22#379
prime radical#3455C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1798C2semidirect
3. #1542C2semidirect
4. #1540C2semidirect
5. #1481C2semidirect
6. #1480C2semidirect
7. #1458C2semidirect

Maximal subgroups

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

subgroupindex
#34552
#17982
#15422
#15402
#14812
#14802
#14582