Finite Group Databasebeta

← #17691

Subgroups — #17691

Order 256.

450 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3510C211yesmaximal, characteristic
2#2971C211yesmaximal, characteristic, complemented
3#2970C211yesmaximal, characteristic, complemented
4#1537C211yesmaximal, characteristic, complemented
5#1535C212yesmaximal, complemented
6#1532C212yesmaximal, complemented
7#1529C212yesmaximal, complemented
8#1527C211yesmaximal, characteristic, complemented
9#1527C211yesmaximal, characteristic, complemented
10#1526C211yesmaximal, characteristic, complemented
11#1524C212yesmaximal, complemented
12#559—21no—
13#559—21no—
14#556—21no—
15#556—21no—
16#551—22no—
17#551—22no—
18#542C412yes—
19#542C412yes—
20#536C411yescharacteristic
21#535C411yescharacteristic
22#533C2211yescharacteristic
23#529C2212yes—
24#529C2212yes—
25C2×#124C2211yescharacteristic
26#519C411yescharacteristic
27#518C411yescharacteristic
28C4×#41C2211yescharacteristic
29#446—21no—
30C8×Q8—21no—
31#436—21no—
32#436—22no—
33#435—22no—
34C8×D4—21no—
35#432C2211yescharacteristic
36#432C2211yescharacteristic
37#432C2212yescomplemented
38#432C2212yescomplemented
39C4×M16C2211yescharacteristic
40C4×M16C2211yescharacteristic
41C4×M16C2212yescomplemented
42C2×#97C2212yescomplemented
43#330C2212yescomplemented
44#330C2212yescomplemented
45#330C2211yescharacteristic, complemented
46#330C2211yescharacteristic, complemented
47#330C2211yescharacteristic, complemented
48#330C2211yescharacteristic, complemented
49#329C2212yes—
50#329C2212yes—
51#329C2212yes—
52#329C2212yes—
53C2×#41C2×C411yescharacteristic
54C22×Q8C2×C411yescharacteristic
55C22×D4C2×C411yescharacteristic
56C2×M16—21no—
57C2×M16—21no—
58C2×M16—22no—
59C22×C8—22no—
60C4⋊Q8—21no—
61C4⋊Q8C2×C411yescharacteristic
62C4⋊Q8C2×C411yescharacteristic
63#127—21no—
64#126—22no—
65#126—22no—
66#126—22no—
67#126—22no—
68#125C2311yescharacteristic
69#125C2311yescharacteristic
70#124—21no—
71#124—21no—
72#124—22no—
73#124—22no—
74#124—21no—
75#124—21no—
76#124C2×C412yes—
77#124C2312yes—
78#123—22no—
79#123—22no—
80#123—22no—
81#123—22no—
82#123—22no—
83C22⋊Q8—21no—
84C22⋊Q8—21no—
85C22⋊Q8—22no—
86C22⋊Q8—22no—
87C22⋊Q8C2×C412yes—
88C22⋊Q8C2×C412yes—
89#121C2×C412yes—
90#121C2×C412yes—
91#120—22no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8C2×C412yes—
97C4×Q8C2×C412yes—
98C4×Q8D411yescharacteristic
99C4×Q8D411yescharacteristic
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—21no—
103C4×D4—21no—
104C4×D4C2×C412yes—
105C4×D4C2×C412yes—
106C4×D4D412yes—
107C4×D4D412yes—
108C4×D4D411yescharacteristic
109C4×D4D411yescharacteristic
110#117—22no—
111#117—22no—
112#117—22no—
113#117C2×C411yescharacteristic
114#117C2×C411yescharacteristic
115#117C2×C411yescharacteristic
116#117C2×C412yes—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—22no—
120C2×C42C2311yescharacteristic
121C2×C42C2312yes—
122C4⋊C8—21no—
123C4⋊C8—21no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C22⋊C8—21no—
127C22⋊C8—21no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130#97—22no—
131#97C2311yescharacteristic
132#97C2312yes—
133#97C2311yescharacteristic
134#97C2311yescharacteristic
135#97C2311yescharacteristic
136#97C2312yes—
137C4×C8—21no—
138C4×C8—21no—
139C4×C8—22no—
140C24—21no—
141#41—42no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—42no—
146C2×Q8C22⋊C411yescharacteristic
147C2×Q8C22×C412yes—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—21no—
151C2×D4—42no—
152C2×D4C22⋊C411yescharacteristic
153C2×D4C22⋊C412yes—
154C2×D4C22×C412yes—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4C22×C411yescharacteristic
159C22×C4C22×C412yes—
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D412yes—
163C22×C4C2×D412yes—
164M16—42no—
165M16—42no—
166M16—44no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—44no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4C22⋊C411yescharacteristic
185C4⋊C4C22⋊C411yescharacteristic
186C4⋊C4C22⋊C412yes—
187C4⋊C4C22⋊C412yes—
188C4⋊C4C22×C411yescharacteristic
189C4⋊C4C22×C411yescharacteristic
190C4⋊C4C22×C411yescharacteristic
191C4⋊C4C22×C411yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—22no—
202C22⋊C4—22no—
203C22⋊C4—22no—
204C22⋊C4C22⋊C412yes—
205C22⋊C4C22⋊C412yes—
206C22⋊C4C22⋊C412yes—
207C42—22no—
208C42—22no—
209C42—21no—
210C42—21no—
211C42C22×C412yes—
212C42C22×C411yescharacteristic
213C42C2×D412yes—
214C42C2×D411yescharacteristic
215C42C2×D411yescharacteristic
216C42C2×D412yes—
217C42C2411yescharacteristic
218C23—22no—
219C23—42no—
220C23—81no—
221C23—81no—
222C23C2×C22⋊C411yescharacteristic
223C23C2×C22⋊C412yes—
224Q8—22no—
225Q8—42no—
226Q8—42no—
227Q8—81no—
228Q8—81no—
229D4—22no—
230D4—42no—
231D4—42no—
232D4—42no—
233D4—82no—
234D4—82no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4C2×C22⋊C412yes—
250C2×C4C2×C22⋊C412yes—
251C2×C4C2×C22⋊C411yescharacteristic
252C2×C4C2×C22⋊C411yescharacteristic
253C2×C4C2×C22⋊C412yes—
254C2×C4C2×C22⋊C411yescharacteristic
255C2×C4C23×C411yescharacteristic
256C2×C4C22×D411yescharacteristic
257C2×C4C22×D411yescharacteristic
258C8—22no—
259C8—22no—
260C8—42no—
261C8—41no—
262C8—41no—
263C8—42no—
264C22—21no—
265C22—22no—
266C22—22no—
267C22—22no—
268C22—21no—
269C22—21no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—162no—
274C22C22×C22⋊C411yescharacteristic
275C4—21no—
276C4—22no—
277C4—21no—
278C4—21no—
279C4—22no—
280C4—22no—
281C4—21no—
282C4—21no—
283C4—42no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4#41312yes—
289C2—42no—
290C2—41no—
291C2—82no—
292C2C2×#41311yescharacteristic
293C2#290111yescharacteristic
294C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC42C24
prime radical#535C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2971C2semidirect
3. #2970C2semidirect
4. #1537C2semidirect
5. #1535C2semidirect
6. #1532C2semidirect
7. #1529C2semidirect
8. #1527C2semidirect
9. #1527C2semidirect
10. #1526C2semidirect
11. #1524C2semidirect
37. #432C22semidirect
38. #432C22semidirect
41. C4×M16C22semidirect
42. C2×#97C22semidirect
43. #330C22semidirect
44. #330C22semidirect
45. #330C22semidirect
46. #330C22semidirect
47. #330C22semidirect
48. #330C22semidirect

Maximal subgroups

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

subgroupindex
#35102
#29712
#29702
#15372
#15352
#15322
#15292
#15272
#15262
#15242