Finite Group Databasebeta

← #17684

Subgroups — #17684

Order 256.

466 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3517C211yesmaximal, characteristic
2#2970C211yesmaximal, characteristic, complemented
3#2923C211yesmaximal, characteristic, complemented
4#1536C212yesmaximal, complemented
5#1535C212yesmaximal, complemented
6#1532C212yesmaximal, complemented
7#1530C212yesmaximal, complemented
8#1530C211yesmaximal, characteristic, complemented
9#1529C211yesmaximal, characteristic, complemented
10#1525C211yesmaximal, characteristic, complemented
11#1524C211yesmaximal, characteristic, complemented
12#557C411yescharacteristic
13#553—21no—
14#553—22no—
15#550—21no—
16#550C411yescharacteristic
17#550C412yes—
18#550C411yescharacteristic
19#547C411yescharacteristic
20#547C412yes—
21#540—21no—
22#540—22no—
23#532C2212yes—
24#532C2211yescharacteristic
25#529C2211yescharacteristic
26#529C2212yes—
27#520—21no—
28C4×#41C2211yescharacteristic
29#446—21no—
30#435—22no—
31C8×D4—21no—
32#433—22no—
33#432C2212yes—
34#432C2212yes—
35#432C2212yes—
36#432C2211yescharacteristic
37#424—21no—
38#408—21no—
39C4×M16C2211yescharacteristic
40C4×M16C2211yescharacteristic
41C2×#97C2211yescharacteristic
42C2×#97C2212yes—
43#330C2211yescharacteristic
44#330C2211yescharacteristic
45#330C2212yes—
46#329C2211yescharacteristic
47#329C2212yes—
48#329C2212yes—
49#329C2212yes—
50#329C2211yescharacteristic
51#329C2212yes—
52#329C2212yes—
53C2×#41—21no—
54C2×#41C2×C412yes—
55C2×#41C2×C412yes—
56C2×#41C2×C412yes—
57C2×#41D411yescharacteristic
58C2×M16—21no—
59C2×M16—21no—
60C22×C8—21no—
61C22×C8—22no—
62C4⋊Q8C2×C412yes—
63C4⋊Q8C2×C411yescharacteristic
64#127—21no—
65#127—22no—
66#127C2×C412yes—
67#127C2×C411yescharacteristic
68#126—21no—
69#126—22no—
70#125—22no—
71#125—21no—
72#125C2311yescharacteristic
73#124C2311yescharacteristic
74#124C2312yes—
75#123—22no—
76#123—21no—
77#123—22no—
78#123—22no—
79#123—22no—
80C22⋊Q8C2×C411yescharacteristic
81C22⋊Q8C2×C412yes—
82#121—21no—
83#121—22no—
84#121—22no—
85#121—21no—
86#121—21no—
87#121—22no—
88#121—22no—
89#121—21no—
90#121—22no—
91#121—22no—
92#121—22no—
93#121C2×C411yescharacteristic
94#121C2×C412yes—
95C4×Q8—21no—
96C4×Q8C2×C411yescharacteristic
97C4×Q8C2×C412yes—
98C4×Q8C2×C411yescharacteristic
99C4×Q8D411yescharacteristic
100C4×D4—21no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—21no—
104C4×D4—21no—
105C4×D4—22no—
106C4×D4—21no—
107C4×D4—22no—
108C4×D4C2×C411yescharacteristic
109C4×D4C2×C412yes—
110C4×D4C2×C411yescharacteristic
111C4×D4C2×C412yes—
112C4×D4D411yescharacteristic
113C4×D4D412yes—
114#117—21no—
115#117—22no—
116#117D411yescharacteristic
117#117D412yes—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C42C2311yescharacteristic
121C2×C42C2312yes—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—21no—
125C4⋊C8—22no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—21no—
130#97—21no—
131#97—22no—
132#97C2311yescharacteristic
133#97C2311yescharacteristic
134#97C2312yes—
135#97C2311yescharacteristic
136#97C2311yescharacteristic
137#97C2312yes—
138C4×C8—21no—
139C4×C8—21no—
140#41—42no—
141#41—44no—
142#41—82no—
143#41—82no—
144#41—82no—
145C2×Q8C22⋊C411yescharacteristic
146C2×Q8C22×C411yescharacteristic
147C2×Q8C22×C412yes—
148C2×D4—21no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—22no—
157C2×D4C22⋊C411yescharacteristic
158C2×D4C22⋊C412yes—
159C2×D4C22×C411yescharacteristic
160C2×D4C22×C412yes—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4C22×C411yescharacteristic
167C22×C4C22×C412yes—
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D412yes—
170C22×C4C2×D412yes—
171C22×C4C2×D411yescharacteristic
172M16—42no—
173M16—42no—
174C2×C8—21no—
175C2×C8—22no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—22no—
180C2×C8—44no—
181C2×C8—42no—
182C4⋊C4—21no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—22no—
188C4⋊C4—21no—
189C4⋊C4—22no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4C22⋊C412yes—
193C4⋊C4C22⋊C411yescharacteristic
194C4⋊C4C22⋊C412yes—
195C4⋊C4C22⋊C411yescharacteristic
196C4⋊C4C22×C411yescharacteristic
197C4⋊C4C22×C411yescharacteristic
198C22⋊C4—21no—
199C22⋊C4—22no—
200C22⋊C4—21no—
201C22⋊C4—22no—
202C22⋊C4—21no—
203C22⋊C4—22no—
204C22⋊C4—21no—
205C22⋊C4—22no—
206C22⋊C4—21no—
207C22⋊C4—22no—
208C22⋊C4C22⋊C412yes—
209C22⋊C4C22⋊C412yes—
210C22⋊C4C22⋊C412yes—
211C42—22no—
212C42—21no—
213C42C22×C412yes—
214C42C22×C411yescharacteristic
215C42C2×D411yescharacteristic
216C42C2×D411yescharacteristic
217C42C2×D412yes—
218C42C2×D412yes—
219C42C2411yescharacteristic
220C23—22no—
221C23—21no—
222C23C2×C22⋊C411yescharacteristic
223C23C2×C22⋊C412yes—
224Q8—22no—
225Q8—41no—
226Q8—42no—
227D4—22no—
228D4—42no—
229D4—41no—
230D4—42no—
231D4—44no—
232D4—42no—
233D4—82no—
234D4—81no—
235D4—82no—
236D4—81no—
237D4—82no—
238D4—82no—
239D4—82no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—44no—
251C2×C4—42no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4C2×C22⋊C412yes—
256C2×C4C2×C22⋊C411yescharacteristic
257C2×C4C2×C22⋊C412yes—
258C2×C4C2×C22⋊C411yescharacteristic
259C2×C4C2×C22⋊C412yes—
260C2×C4C2×C22⋊C411yescharacteristic
261C2×C4C23×C411yescharacteristic
262C2×C4C22×D411yescharacteristic
263C2×C4C22×D411yescharacteristic
264C8—22no—
265C8—41no—
266C8—41no—
267C8—42no—
268C8—42no—
269C8—41no—
270C22—21no—
271C22—22no—
272C22—21no—
273C22—22no—
274C22—22no—
275C22—21no—
276C22—42no—
277C22—41no—
278C22—42no—
279C22—41no—
280C22—42no—
281C22—41no—
282C22C22×C22⋊C411yescharacteristic
283C4—21no—
284C4—21no—
285C4—22no—
286C4—21no—
287C4—22no—
288C4—21no—
289C4—22no—
290C4—21no—
291C4—42no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4#42112yes—
296C2—41no—
297C2—42no—
298C2—82no—
299C2—81no—
300C2#288811yescharacteristic
301C2C2×#42111yescharacteristic
302C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC42C24
prime radical#3517C2
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. #2970C2semidirect
3. #2923C2semidirect
4. #1536C2semidirect
5. #1535C2semidirect
6. #1532C2semidirect
7. #1530C2semidirect
8. #1530C2semidirect
9. #1529C2semidirect
10. #1525C2semidirect
11. #1524C2semidirect

Maximal subgroups

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

subgroupindex
#35172
#29702
#29232
#15362
#15352
#15322
#15302
#15292
#15252
#15242