Finite Group Databasebeta

← #17682

Subgroups — #17682

Order 256.

438 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3515C211yesmaximal, characteristic
2#2971C211yesmaximal, characteristic, complemented
3#2923C211yesmaximal, characteristic, complemented
4#1535C212yesmaximal, complemented
5#1534C212yesmaximal, complemented
6#1531C211yesmaximal, characteristic, complemented
7#1529C211yesmaximal, characteristic, complemented
8#1529C212yesmaximal, complemented
9#1528C212yesmaximal, complemented
10#1526C211yesmaximal, characteristic, complemented
11#1524C211yesmaximal, characteristic, complemented
12#557C411yescharacteristic
13#554C411yescharacteristic
14#554C411yescharacteristic
15#553C412yes—
16#552—22no—
17#552—21no—
18#552C412yes—
19#547—21no—
20#547C411yescharacteristic
21#541—21no—
22#539—22no—
23#533C2211yescharacteristic
24#529C2211yescharacteristic
25#529C2212yes—
26C2×#125C2212yes—
27#520—21no—
28C4×#41C2211yescharacteristic
29C8×Q8—21no—
30#436—21no—
31#436—22no—
32#433—22no—
33#432C2212yes—
34#432C2212yes—
35#432C2211yescharacteristic
36#432C2212yes—
37#424—21no—
38#408—21no—
39C4×M16C2211yescharacteristic
40C4×M16C2211yescharacteristic
41C4×M16C2212yes—
42C2×#97C2211yescharacteristic
43#330C2211yescharacteristic
44#330C2211yescharacteristic
45#330C2212yes—
46#330C2212yes—
47#330C2212yes—
48#330C2212yes—
49#330C2212yes—
50#329C2211yescharacteristic
51#329C2211yescharacteristic
52#329C2212yes—
53C2×#41—21no—
54C2×#41D411yescharacteristic
55C2×M16—21no—
56C2×M16—22no—
57C2×M16—21no—
58C22×C8—21no—
59C4⋊Q8—21no—
60C4⋊Q8C2×C411yescharacteristic
61C4⋊Q8C2×C411yescharacteristic
62#126—22no—
63#126—22no—
64#126—22no—
65#126—21no—
66#125—22no—
67#125—22no—
68#125—21no—
69#125—22no—
70#125—22no—
71#125C2×C412yes—
72#125C2×C412yes—
73#125C2311yescharacteristic
74#125C2312yes—
75#124C2311yescharacteristic
76#123—22no—
77#123—21no—
78#123—22no—
79#123—22no—
80#123—22no—
81C22⋊Q8—21no—
82C22⋊Q8—21no—
83C22⋊Q8—21no—
84C22⋊Q8—22no—
85C22⋊Q8—21no—
86C22⋊Q8—22no—
87C22⋊Q8C2×C411yescharacteristic
88C22⋊Q8C2×C412yes—
89#121—22no—
90#121C2×C412yes—
91#121C2×C411yescharacteristic
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8C2×C412yes—
96C4×Q8C2×C411yescharacteristic
97C4×Q8C2×C411yescharacteristic
98C4×Q8D411yescharacteristic
99C4×D4—21no—
100C4×D4—22no—
101C4×D4—21no—
102C4×D4—22no—
103C4×D4C2×C411yescharacteristic
104C4×D4C2×C412yes—
105C4×D4C2×C411yescharacteristic
106C4×D4C2×C412yes—
107C4×D4D411yescharacteristic
108C4×D4D412yes—
109#117—22no—
110#117—21no—
111#117—22no—
112#117C2×C412yes—
113#117D411yescharacteristic
114#117D412yes—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4C2×C412yes—
119C2×C4⋊C4C2×C412yes—
120C2×C42C2312yes—
121C2×C42C2311yescharacteristic
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#97C2311yescharacteristic
132#97C2311yescharacteristic
133#97C2311yescharacteristic
134#97C2311yescharacteristic
135#97C2312yes—
136#97C2312yes—
137C4×C8—21no—
138C4×C8—21no—
139C4×C8—22no—
140#41—42no—
141#41—44no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8C22⋊C411yescharacteristic
145C2×Q8C22×C411yescharacteristic
146C2×D4—22no—
147C2×D4—21no—
148C2×D4C22⋊C411yescharacteristic
149C2×D4C22⋊C412yes—
150C2×D4C22×C411yescharacteristic
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4C22×C412yes—
156C22×C4C22×C411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D412yes—
159C22×C4C2×D412yes—
160C22×C4C2×D411yescharacteristic
161M16—44no—
162M16—42no—
163M16—42no—
164C2×C8—21no—
165C2×C8—22no—
166C2×C8—21no—
167C2×C8—22no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—42no—
171C4⋊C4—22no—
172C4⋊C4—21no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—21no—
179C4⋊C4—22no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—22no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4C22⋊C411yescharacteristic
188C4⋊C4C22⋊C411yescharacteristic
189C4⋊C4C22⋊C412yes—
190C4⋊C4C22⋊C412yes—
191C4⋊C4C22×C411yescharacteristic
192C4⋊C4C22×C411yescharacteristic
193C4⋊C4C22×C412yes—
194C4⋊C4C22×C412yes—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—21no—
198C22⋊C4—21no—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4C22⋊C412yes—
205C22⋊C4C22⋊C412yes—
206C22⋊C4C22⋊C412yes—
207C42—22no—
208C42—22no—
209C42—21no—
210C42C22×C412yes—
211C42C22×C411yescharacteristic
212C42C2×D411yescharacteristic
213C42C2×D411yescharacteristic
214C42C2×D412yes—
215C42C2×D412yes—
216C42C2411yescharacteristic
217C23—21no—
218C23C2×C22⋊C411yescharacteristic
219C23C2×C22⋊C412yes—
220Q8—22no—
221Q8—41no—
222Q8—42no—
223Q8—81no—
224D4—22no—
225D4—42no—
226D4—41no—
227D4—44no—
228D4—81no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4C2×C22⋊C411yescharacteristic
245C2×C4C2×C22⋊C412yes—
246C2×C4C2×C22⋊C411yescharacteristic
247C2×C4C2×C22⋊C412yes—
248C2×C4C2×C22⋊C412yes—
249C2×C4C2×C22⋊C411yescharacteristic
250C2×C4C23×C411yescharacteristic
251C2×C4C22×D411yescharacteristic
252C2×C4C22×D411yescharacteristic
253C8—22no—
254C8—41no—
255C8—41no—
256C8—42no—
257C8—41no—
258C8—42no—
259C22—21no—
260C22—22no—
261C22—21no—
262C22—22no—
263C22—22no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22C22×C22⋊C411yescharacteristic
269C4—21no—
270C4—21no—
271C4—22no—
272C4—21no—
273C4—22no—
274C4—21no—
275C4—22no—
276C4—21no—
277C4—41no—
278C4—41no—
279C4—42no—
280C4—41no—
281C4—42no—
282C4#42112yes—
283C2—41no—
284C2—42no—
285C2—81no—
286C2#288911yescharacteristic
287C2C2×#42111yescharacteristic
288C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC42C24
prime radical#3515C2
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. #2923C2semidirect
4. #1535C2semidirect
5. #1534C2semidirect
6. #1531C2semidirect
7. #1529C2semidirect
8. #1529C2semidirect
9. #1528C2semidirect
10. #1526C2semidirect
11. #1524C2semidirect

Maximal subgroups

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

subgroupindex
#35152
#29712
#29232
#15352
#15342
#15312
#15292
#15282
#15262
#15242