Finite Group Databasebeta

← #33358

Subgroups — #33358

Order 256.

626 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 181 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#3449C211yesmaximal, characteristic
2C2×#410C212yesmaximal, complemented
3#2129C212yesmaximal, complemented
4#2128C212yesmaximal, complemented
5#2127C214yesmaximal, complemented
6#2126C214yesmaximal, complemented
7C22×#41C411yescharacteristic, complemented
8C22×#41C2211yescharacteristic
9#537—22no—
10#536C412yescomplemented
11#536C411yescharacteristic, complemented
12#536C2211yescharacteristic
13#535C411yescharacteristic, complemented
14#535C2212yes—
15#535C2211yescharacteristic
16#534—22no—
17#532C412yescomplemented
18#532C2212yes—
19#525—22no—
20#525—22no—
21C2×#117—22no—
22C2×#117C411yescharacteristic, complemented
23#410—22no—
24#410—22no—
25#410C2214yes—
26#410C2214yescomplemented
27C2×#99—22no—
28C2×#99C2214yescomplemented
29#354C2218yescomplemented
30#353C2218yescomplemented
31C2×#41—21no—
32C2×#41—21no—
33C2×#41—42no—
34C2×#41—42no—
35C2×#41—42no—
36C2×#41—42no—
37C2×#41C2×C412yes—
38C2×#41C2×C411yescharacteristic
39C2×#41C2×C411yescharacteristic
40C2×#41C2×C412yescomplemented
41C2×#41D412yes—
42C2×#41D412yes—
43C2×#41C2311yescharacteristic
44C2×#41C2311yescharacteristic
45C22×Q8C2×C411yescharacteristic
46C22×Q8C2×C411yescharacteristic, complemented
47C22×D4—21no—
48C22×D4C2×C411yescharacteristic, complemented
49C22×D4D412yes—
50C22×D4C2311yescharacteristic
51C23×C4—21no—
52C23×C4C2×C411yescharacteristic
53C23×C4D412yes—
54C4⋊Q8C2×C412yescomplemented
55C4⋊Q8C2×C412yescomplemented
56#127C2×C412yes—
57#127C2312yes—
58#124C2×C412yes—
59#124C2×C412yescomplemented
60#124C2×C412yescomplemented
61#124C2312yes—
62#123—22no—
63#123—22no—
64#123—24no—
65C22⋊Q8—24no—
66C22⋊Q8—22no—
67C22⋊Q8—22no—
68#121—24no—
69#121—24no—
70#121—24no—
71#121—22no—
72#121—22no—
73#120—24no—
74#120—22no—
75#120—22no—
76C4×D4—24no—
77C4×D4—24no—
78#117—21no—
79#117—21no—
80#117—24no—
81#117—24no—
82#117—42no—
83#117—42no—
84#117C2×C412yes—
85#117C2×C411yescharacteristic, complemented
86#117C2×C411yescharacteristic, complemented
87C2×C4⋊C4—21no—
88C2×C4⋊C4—42no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—24no—
91C2×C42—42no—
92C2×C42C2×C412yescomplemented
93#99—24no—
94#99—24no—
95#99—24no—
96#99C2318yes—
97C24—21no—
98C24C22⋊C411yescharacteristic
99C24C2×D412yes—
100#41—42no—
101#41—41no—
102#41—41no—
103#41—42no—
104#41—42no—
105#41—42no—
106#41—42no—
107#41—42no—
108#41—41no—
109#41—41no—
110#41—44no—
111#41—44no—
112#41—42no—
113#41—42no—
114#41—42no—
115#41—42no—
116C2×Q8—21no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8C22⋊C412yes—
120C2×Q8C22×C412yes—
121C2×Q8C22×C411yescharacteristic
122C2×Q8C22×C411yescharacteristic
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—24no—
126C2×D4—24no—
127C2×D4—21no—
128C2×D4—42no—
129C2×D4—42no—
130C2×D4—42no—
131C2×D4—42no—
132C2×D4—44no—
133C2×D4C22⋊C412yes—
134C2×D4C22⋊C412yes—
135C2×D4C22×C412yes—
136C2×D4C22×C411yescharacteristic
137C2×D4C2×D412yes—
138C2×D4C2×D412yes—
139C2×D4C2×D412yes—
140C2×D4C2411yescharacteristic
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—44no—
154C22×C4C22⋊C411yescharacteristic
155C22×C4C22⋊C411yescharacteristic
156C22×C4C22⋊C412yes—
157C22×C4C22⋊C412yes—
158C22×C4C22⋊C412yes—
159C22×C4C22⋊C411yescharacteristic
160C22×C4C22×C411yescharacteristic
161C22×C4C22×C411yescharacteristic
162C22×C4C22×C411yescharacteristic
163C22×C4C2×D412yes—
164C22×C4C2×D412yes—
165C4⋊C4—22no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—24no—
169C4⋊C4—24no—
170C4⋊C4—44no—
171C4⋊C4—44no—
172C22⋊C4—22no—
173C22⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C22⋊C4—24no—
178C22⋊C4—24no—
179C22⋊C4—28no—
180C22⋊C4—44no—
181C42—22no—
182C42—44no—
183C42—44no—
184C42C22×C412yes—
185C42C22×C412yes—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23—42no—
191C23—82no—
192C23—81no—
193C23—82no—
194C23C2×C22⋊C411yescharacteristic
195C23C2×C22⋊C412yes—
196C23C2×C22⋊C411yescharacteristic
197C23C2×C22⋊C411yescharacteristic
198C23C22×D412yes—
199Q8—42no—
200Q8—41no—
201Q8—41no—
202Q8—42no—
203Q8—42no—
204D4—42no—
205D4—42no—
206D4—41no—
207D4—42no—
208D4—42no—
209D4—42no—
210D4—41no—
211D4—42no—
212D4—44no—
213D4—44no—
214D4—42no—
215D4—84no—
216D4—84no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—44no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—48no—
242C2×C4—82no—
243C2×C4—88no—
244C2×C4#9912yes—
245C2×C4#9912yes—
246C2×C4C2×C22⋊C412yes—
247C2×C4C2×C22⋊C412yes—
248C2×C4C2×C22⋊C411yescharacteristic
249C2×C4C2×C22⋊C411yescharacteristic
250C2×C4C2×C22⋊C411yescharacteristic
251C2×C4C23×C411yescharacteristic
252C22—22no—
253C22—21no—
254C22—21no—
255C22—22no—
256C22—41no—
257C22—41no—
258C22—42no—
259C22—42no—
260C22—41no—
261C22—44no—
262C22—44no—
263C22—42no—
264C22—82no—
265C22—82no—
266C22—84no—
267C22C2×#9912yes—
268C22C22×C22⋊C411yescharacteristic
269C4—21no—
270C4—22no—
271C4—22no—
272C4—21no—
273C4—42no—
274C4—42no—
275C4—42no—
276C4—88no—
277C4C2×#9911yescharacteristic
278C4C2×#9912yes—
279C4C2×#9911yescharacteristic
280C2—21no—
281C2—22no—
282C2—42no—
283C2—42no—
284C2—41no—
285C2—41no—
286C2—82no—
287C2C22×#9911yescharacteristic
subgroupG/H
centerC4C2×#99
commutatorC2×C4C23×C4
Frattini, cosocleC2×D4C24
socleC2C22×#99
prime radical#3449C2
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G
LECSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#410C2semidirect
3. #2129C2semidirect
4. #2128C2semidirect
5. #2127C2semidirect
6. #2126C2semidirect
7. C22×#41C4semidirect
10. #536C4semidirect
11. #536C4semidirect
13. #535C4semidirect
17. #532C4semidirect
22. C2×#117C4semidirect
26. #410C22semidirect
28. C2×#99C22semidirect
29. #354C22semidirect
30. #353C22semidirect
40. C2×#41C2×C4semidirect
46. C22×Q8C2×C4semidirect
48. C22×D4C2×C4semidirect
54. C4⋊Q8C2×C4semidirect
55. C4⋊Q8C2×C4semidirect
59. #124C2×C4semidirect
60. #124C2×C4semidirect
85. #117C2×C4semidirect
86. #117C2×C4semidirect
92. C2×C42C2×C4semidirect

Maximal subgroups

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

subgroupindex
#34492
C2×#4102
#21292
#21282
#21272
#21262