Finite Group Databasebeta

← #17612

Subgroups — #17612

Order 256.

465 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3495C211yesmaximal, characteristic
2#2924C211yesmaximal, characteristic, complemented
3#2347C211yesmaximal, characteristic, complemented
4#1521C212yesmaximal, complemented
5#1520C211yesmaximal, characteristic, complemented
6#1515C212yesmaximal, complemented
7#1513C211yesmaximal, characteristic
8#1509C212yesmaximal, complemented
9#1508C211yesmaximal, characteristic, complemented
10#1503C212yesmaximal, complemented
11#1501C211yesmaximal, characteristic
12#556—21no—
13#554—21no—
14#552—22no—
15#552—21no—
16#552—21no—
17#551—22no—
18#544C411yescharacteristic, complemented
19#542C412yescomplemented
20#541C412yescomplemented
21#536C411yescharacteristic, complemented
22#533C2212yes—
23#529C2212yes—
24#525C2211yescharacteristic
25C2×C22⋊Q8C2211yescharacteristic
26#520C411yescharacteristic, complemented
27#519C411yescharacteristic, complemented
28C2×#117C2211yescharacteristic
29#433—22no—
30#432C2212yes—
31#431C2212yes—
32#424—21no—
33#407C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35#405—21no—
36#389—21no—
37#388—22no—
38#383C2212yes—
39#380C2211yescharacteristic
40C4×C4⋊C4—21no—
41C4×C4⋊C4C2212yes—
42C4×C22⋊C4C2211yescharacteristic
43#328C2211yescharacteristic
44#328C2211yescharacteristic
45#328C2212yes—
46#328C2212yes—
47#328C2212yes—
48#328C2212yes—
49#328C2212yes—
50#327C2211yescharacteristic
51#327C2211yescharacteristic
52#327C2212yes—
53C2×#41C2×C411yescharacteristic
54C22×Q8C2×C411yescharacteristic
55C23×C4C2311yescharacteristic
56C2×M16—21no—
57C22×C8—21no—
58C4⋊Q8—21no—
59C4⋊Q8C2×C412yes—
60C4⋊Q8C2×C412yes—
61#126—22no—
62#126—22no—
63#126—22no—
64#126—22no—
65#125—22no—
66#125—22no—
67#125—21no—
68#125—22no—
69#125C2×C412yes—
70#124—22no—
71#124C2×C412yes—
72#123—21no—
73#123—22no—
74#123—22no—
75#123—22no—
76C22⋊Q8—21no—
77C22⋊Q8—21no—
78C22⋊Q8—22no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81C22⋊Q8—22no—
82C22⋊Q8—22no—
83C22⋊Q8C2×C412yes—
84C22⋊Q8C2×C411yescharacteristic
85C22⋊Q8C2312yes—
86C22⋊Q8C2311yescharacteristic
87#121C2×C411yescharacteristic
88#121C2311yescharacteristic
89#120—21no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8—22no—
93C4×Q8C2×C412yes—
94C4×Q8C2×C412yes—
95C4×Q8C2×C412yes—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4C2×C412yes—
101#117—22no—
102#117—21no—
103#117—22no—
104#117—22no—
105#117C2×C411yescharacteristic
106#117C2×C411yescharacteristic
107#117C2×C411yescharacteristic
108#117C2×C412yes—
109#117D411yescharacteristic
110#117D412yes—
111#117D411yescharacteristic
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4C2×C411yescharacteristic
117C2×C4⋊C4D411yescharacteristic
118C2×C4⋊C4D411yescharacteristic
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4D412yes—
123C2×C42—21no—
124C2×C42—22no—
125C2×C42C2312yes—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C22⋊C8—21no—
129C22⋊C8—21no—
130C22⋊C8C2311yescharacteristic
131C22⋊C8C2311yescharacteristic
132C22⋊C8C2312yes—
133#97—22no—
134C4×C8—22no—
135#95—22no—
136#95—22no—
137#95C2311yescharacteristic
138#95C2311yescharacteristic
139#95C2312yes—
140C24C2×D411yescharacteristic
141#41—42no—
142C2×Q8—21no—
143C2×Q8—22no—
144C2×Q8—42no—
145C2×Q8C22×C412yes—
146C2×Q8C22×C411yescharacteristic
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4C22×C411yescharacteristic
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—42no—
159C22×C4C22×C411yescharacteristic
160C22×C4C22×C411yescharacteristic
161C22×C4C2×D412yes—
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D412yes—
165C22×C4C2×D411yescharacteristic
166C22×C4C2411yescharacteristic
167M16—42no—
168C2×C8—21no—
169C2×C8—22no—
170C2×C8—21no—
171C2×C8—42no—
172C4⋊C4—21no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—21no—
185C4⋊C4—42no—
186C4⋊C4—44no—
187C4⋊C4C22⋊C411yescharacteristic
188C4⋊C4C22⋊C411yescharacteristic
189C4⋊C4C22⋊C412yes—
190C4⋊C4C22⋊C411yescharacteristic
191C4⋊C4C22⋊C411yescharacteristic
192C4⋊C4C22⋊C412yes—
193C4⋊C4C22×C412yes—
194C4⋊C4C22×C411yescharacteristic
195C4⋊C4C22×C411yescharacteristic
196C22⋊C4—22no—
197C22⋊C4—21no—
198C22⋊C4—21no—
199C22⋊C4—22no—
200C22⋊C4—21no—
201C22⋊C4—22no—
202C22⋊C4—22no—
203C22⋊C4—21no—
204C22⋊C4—22no—
205C22⋊C4—21no—
206C22⋊C4—42no—
207C22⋊C4—81no—
208C22⋊C4C22⋊C412yes—
209C22⋊C4C22⋊C412yes—
210C22⋊C4C22⋊C412yes—
211C22⋊C4C22⋊C412yes—
212C42—22no—
213C42—22no—
214C42—22no—
215C42—21no—
216C42—42no—
217C42—82no—
218C42C22×C412yes—
219C42C22×C412yes—
220C42C2×D412yes—
221C42C2×D412yes—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—41no—
226C23—41no—
227C23C2×C22⋊C411yescharacteristic
228C23C2×C22⋊C411yescharacteristic
229C23C22×D411yescharacteristic
230Q8—42no—
231Q8—41no—
232Q8—42no—
233Q8—82no—
234D4—41no—
235D4—42no—
236D4—41no—
237D4—41no—
238D4—81no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—82no—
265C2×C4—81no—
266C2×C4C2×C22⋊C412yes—
267C2×C4C2×C22⋊C412yes—
268C2×C4C2×C22⋊C412yes—
269C2×C4C2×C22⋊C412yes—
270C2×C4C2×C22⋊C411yescharacteristic
271C2×C4C2×C22⋊C411yescharacteristic
272C2×C4C23×C411yescharacteristic
273C2×C4C22×D411yescharacteristic
274C8—41no—
275C8—41no—
276C8—42no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—81no—
289C22—81no—
290C22C22×C22⋊C411yescharacteristic
291C22#57711yescharacteristic
292C22#57811yescharacteristic
293C4—21no—
294C4—21no—
295C4—21no—
296C4—22no—
297C4—22no—
298C4—22no—
299C4—22no—
300C4—42no—
301C4—41no—
302C4—42no—
303C4—41no—
304C4—42no—
305C4—41no—
306C4—82no—
307C4—81no—
308C4#41012yes—
309C2—21no—
310C2—21no—
311C2—41no—
312C2—41no—
313C2—81no—
314C2C2×#41011yescharacteristic
315C2#289811yescharacteristic
316C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#525C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2924C2semidirect
3. #2347C2semidirect
4. #1521C2semidirect
5. #1520C2semidirect
6. #1515C2semidirect
8. #1509C2semidirect
9. #1508C2semidirect
10. #1503C2semidirect
18. #544C4semidirect
19. #542C4semidirect
20. #541C4semidirect
21. #536C4semidirect
26. #520C4semidirect
27. #519C4semidirect

Maximal subgroups

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

subgroupindex
#34952
#29242
#23472
#15212
#15202
#15152
#15132
#15092
#15082
#15032
#15012