Finite Group Databasebeta

← #17767

Subgroups — #17767

Order 256.

431 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3509C211yesmaximal, characteristic
2#2989C211yesmaximal, characteristic
3#2925C211yesmaximal, characteristic
4#1561C211yesmaximal, characteristic, complemented
5#1559C211yesmaximal, characteristic, complemented
6#1559C211yesmaximal, characteristic, complemented
7#1556C211yesmaximal, characteristic, complemented
8#1554C211yesmaximal, characteristic, complemented
9#1554C211yesmaximal, characteristic, complemented
10#1552C211yesmaximal, characteristic, complemented
11#1552C211yesmaximal, characteristic, complemented
12#1550C211yesmaximal, characteristic, complemented
13#1549C211yesmaximal, characteristic, complemented
14#1544C211yesmaximal, characteristic, complemented
15#1542C211yesmaximal, characteristic, complemented
16#557—21no—
17#557C412yes—
18#556—21no—
19#556C412yes—
20#552—21no—
21#552—21no—
22#552C412yes—
23#552C412yes—
24#541—21no—
25#541—21no—
26#536—21no—
27#533C2211yescharacteristic
28#533C2211yescharacteristic
29#533C2211yescharacteristic
30#533C2211yescharacteristic
31#533C2211yescharacteristic
32C2×C4⋊Q8C2211yescharacteristic
33#519—21no—
34C4×#41C2211yescharacteristic
35#446—21no—
36#436—21no—
37#435—21no—
38#435—21no—
39#433—21no—
40#433—21no—
41#432C2211yescharacteristic
42#432C2211yescharacteristic
43#432C2211yescharacteristic
44#432C2211yescharacteristic
45#431C2211yescharacteristic
46#431C2211yescharacteristic
47#431C2211yescharacteristic
48#423C2211yescharacteristic
49#423C2211yescharacteristic
50#423C2211yescharacteristic
51C2×C4⋊C8C2211yescharacteristic
52C2×C4⋊C8C2211yescharacteristic
53#408—21no—
54#405—21no—
55#333C2211yescharacteristic
56#333C2211yescharacteristic
57#333C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62#332C2211yescharacteristic
63#332C2211yescharacteristic
64#332C2211yescharacteristic
65#332C2211yescharacteristic
66#332C2211yescharacteristic
67#332C2211yescharacteristic
68#332C2211yescharacteristic
69#332C2211yescharacteristic
70#332C2211yescharacteristic
71C2×#41D411yescharacteristic
72C22×Q8—21no—
73C2×M16—21no—
74C2×M16—21no—
75C2×M16—21no—
76C22×C8—21no—
77C22×C8—21no—
78C4⋊Q8—22no—
79C4⋊Q8—21no—
80C4⋊Q8—21no—
81C4⋊Q8C2×C412yes—
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86#126—22no—
87#126—22no—
88#126—22no—
89#126—22no—
90#125—22no—
91#125—22no—
92#125—22no—
93#125—22no—
94#125—21no—
95#125—21no—
96#125—21no—
97#125C2×C412yes—
98#125C2×C412yes—
99#124—22no—
100#123—22no—
101#123—22no—
102C22⋊Q8—21no—
103C22⋊Q8—21no—
104C22⋊Q8—21no—
105C22⋊Q8—21no—
106C22⋊Q8—21no—
107C22⋊Q8—21no—
108C22⋊Q8—21no—
109C22⋊Q8C2×C412yes—
110C22⋊Q8C2×C412yes—
111C22⋊Q8C2×C412yes—
112C4×Q8—22no—
113C4×Q8—22no—
114C4×Q8—21no—
115C4×Q8—21no—
116C4×Q8—21no—
117C4×Q8—21no—
118C4×Q8—21no—
119C4×Q8—21no—
120C4×Q8—21no—
121C4×Q8C2×C412yes—
122C4×Q8C2×C412yes—
123C4×Q8C2×C412yes—
124C4×Q8C2×C411yescharacteristic
125C4×Q8D411yescharacteristic
126C4×D4C2×C411yescharacteristic
127C4×D4C2×C411yescharacteristic
128C4×D4C2×C411yescharacteristic
129C4×D4D411yescharacteristic
130C4×D4D411yescharacteristic
131C4×D4D411yescharacteristic
132#117—21no—
133#117—21no—
134#117—21no—
135#117—22no—
136#117—22no—
137#117C2×C412yes—
138#117C2×C412yes—
139#117D411yescharacteristic
140#117D411yescharacteristic
141#117D411yescharacteristic
142C2×C4⋊C4—21no—
143C2×C4⋊C4C2×C412yes—
144C2×C42C2311yescharacteristic
145C2×C42C2311yescharacteristic
146C2×C42C2311yescharacteristic
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—21no—
150C4⋊C8—21no—
151C4⋊C8—21no—
152C4⋊C8C2311yescharacteristic
153C4⋊C8C2311yescharacteristic
154C4⋊C8C2311yescharacteristic
155C4⋊C8C2311yescharacteristic
156C4⋊C8C2311yescharacteristic
157C4⋊C8C2311yescharacteristic
158C4⋊C8C2311yescharacteristic
159C4⋊C8C2311yescharacteristic
160C22⋊C8—21no—
161C22⋊C8—21no—
162C22⋊C8—21no—
163C22⋊C8—21no—
164C22⋊C8—21no—
165C22⋊C8—21no—
166C22⋊C8—21no—
167#97—21no—
168#97—21no—
169#97—21no—
170#97—21no—
171C4×C8—21no—
172C4×C8—21no—
173C4×C8—21no—
174#41—41no—
175#41—41no—
176C2×Q8—22no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8C22⋊C411yescharacteristic
183C2×Q8C22×C412yes—
184C2×D4C22⋊C411yescharacteristic
185C2×D4C22⋊C411yescharacteristic
186C2×D4C22⋊C411yescharacteristic
187C22×C4—22no—
188C22×C4C22×C411yescharacteristic
189C22×C4C22×C411yescharacteristic
190C22×C4C22×C411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197M16—42no—
198M16—81no—
199M16—81no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—42no—
209C2×C8—81no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4C22⋊C411yescharacteristic
228C4⋊C4C22⋊C411yescharacteristic
229C4⋊C4C22⋊C411yescharacteristic
230C4⋊C4C22⋊C411yescharacteristic
231C4⋊C4C22⋊C411yescharacteristic
232C4⋊C4C22⋊C411yescharacteristic
233C4⋊C4C22×C412yes—
234C4⋊C4C22×C412yes—
235C4⋊C4C22×C412yes—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—22no—
239C22⋊C4—22no—
240C22⋊C4—22no—
241C22⋊C4C22⋊C411yescharacteristic
242C22⋊C4C22⋊C411yescharacteristic
243C22⋊C4C22⋊C411yescharacteristic
244C22⋊C4C22⋊C411yescharacteristic
245C22⋊C4C22⋊C411yescharacteristic
246C22⋊C4C22⋊C411yescharacteristic
247C42—22no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42C22×C411yescharacteristic
253C42C22×C411yescharacteristic
254C42C22×C411yescharacteristic
255C42C2×D411yescharacteristic
256C42C2×D411yescharacteristic
257C42C2×D411yescharacteristic
258C42C2×D411yescharacteristic
259C42C2×D411yescharacteristic
260C42C2×D411yescharacteristic
261C42C2411yescharacteristic
262C23C2×C22⋊C411yescharacteristic
263C23C2×C22⋊C411yescharacteristic
264C23C2×C22⋊C411yescharacteristic
265Q8—41no—
266Q8—42no—
267Q8—44no—
268Q8—41no—
269Q8—41no—
270Q8—41no—
271Q8—41no—
272Q8—41no—
273Q8—41no—
274D4—41no—
275D4—41no—
276D4—41no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—44no—
297C2×C4C2×C22⋊C411yescharacteristic
298C2×C4C2×C22⋊C411yescharacteristic
299C2×C4C2×C22⋊C411yescharacteristic
300C2×C4C2×C22⋊C411yescharacteristic
301C2×C4C2×C22⋊C411yescharacteristic
302C2×C4C2×C22⋊C411yescharacteristic
303C2×C4C2×C22⋊C411yescharacteristic
304C2×C4C2×C22⋊C411yescharacteristic
305C2×C4C2×C22⋊C411yescharacteristic
306C2×C4C23×C411yescharacteristic
307C2×C4C22×D411yescharacteristic
308C2×C4C22×D411yescharacteristic
309C8—41no—
310C8—41no—
311C8—41no—
312C8—41no—
313C8—41no—
314C8—41no—
315C8—41no—
316C8—41no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22C22×C22⋊C411yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—42no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4#57511yescharacteristic
342C4#57511yescharacteristic
343C4#57711yescharacteristic
344C4#57811yescharacteristic
345C2—41no—
346C2—41no—
347C2—41no—
348C2#288911yescharacteristic
349C2#289611yescharacteristic
350C2#289811yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC42C24
prime radicalC2×#41D4
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
4. #1561C2semidirect
5. #1559C2semidirect
6. #1559C2semidirect
7. #1556C2semidirect
8. #1554C2semidirect
9. #1554C2semidirect
10. #1552C2semidirect
11. #1552C2semidirect
12. #1550C2semidirect
13. #1549C2semidirect
14. #1544C2semidirect
15. #1542C2semidirect

Maximal subgroups

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

subgroupindex
#35092
#29892
#29252
#15612
#15592
#15562
#15542
#15522
#15502
#15492
#15442
#15422