Finite Group Databasebeta

← #22499

Subgroups — #22499

Order 256.

565 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#552C211yesmaximal, characteristic
2C2×#436C211yesmaximal, characteristic
3#2352C211yesmaximal, characteristic
4#1991C211yesmaximal, characteristic, complemented
5#1989C211yesmaximal, characteristic, complemented
6#1894C211yesmaximal, characteristic, complemented
7#1894C211yesmaximal, characteristic, complemented
8#1873C211yesmaximal, characteristic, complemented
9#1872C211yesmaximal, characteristic, complemented
10#1825C211yesmaximal, characteristic, complemented
11#1824C211yesmaximal, characteristic, complemented
12#1805C211yesmaximal, characteristic, complemented
13C4×#103C211yesmaximal, characteristic, complemented
14#1736C211yesmaximal, characteristic, complemented
15#1736C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18#552—24no—
19#552C418yescomplemented
20C2×#126—21no—
21C2×#125—21no—
22C2×#125C2211yescharacteristic
23C2×#123—21no—
24C2×C22⋊Q8C2211yescharacteristic
25C2×C22⋊Q8C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×#117—21no—
29C2×#117C2211yescharacteristic
30C2×#117C2211yescharacteristic
31C22×C4⋊C4C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33#436—24no—
34C2×C4⋊C8C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C4×C8C2211yescharacteristic
42#388—21no—
43#387—21no—
44#386—21no—
45#382C2211yescharacteristic
46#380C2211yescharacteristic
47#380C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C22⋊C4—21no—
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×D4D411yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C2×M16—22no—
69C2×M16—22no—
70C2×M16—22no—
71C2×M16—22no—
72C2×M16—41no—
73C2×M16—41no—
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78#126—28no—
79#125—28no—
80#125—22no—
81#125C2×C414yescomplemented
82#123—28no—
83C22⋊Q8—22no—
84C22⋊Q8—22no—
85C22⋊Q8C2×C414yes—
86C22⋊Q8C2×C414yescomplemented
87C4×Q8—22no—
88C4×Q8C2×C414yescomplemented
89C4×D4C2×C414yes—
90C4×D4D414yes—
91#117—22no—
92#117—28no—
93#117—22no—
94#117—41no—
95#117C2×C414yes—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4C2×C414yescomplemented
107C2×C4⋊C4D411yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4D411yescharacteristic
122C2×C22⋊C4D411yescharacteristic
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42C2311yescharacteristic
128C4⋊C8—22no—
129#103—22no—
130#103—22no—
131#103—41no—
132#103—41no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C4×C8—22no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141C24C2×D411yescharacteristic
142C24C2×D411yescharacteristic
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8C22×C412yes—
149C2×D4—41no—
150C2×D4—41no—
151C2×D4C22⋊C414yes—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—24no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—81no—
174C22×C4C22×C412yes—
175C22×C4C22×C412yes—
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D412yes—
181C22×C4C2×D412yes—
182C22×C4C2411yescharacteristic
183M16—44no—
184M16—44no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8#4112yes—
197C4⋊C4—24no—
198C4⋊C4—24no—
199C4⋊C4—24no—
200C4⋊C4—24no—
201C4⋊C4—24no—
202C4⋊C4—24no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—42no—
214C4⋊C4C22⋊C414yes—
215C4⋊C4C22×C412yes—
216C4⋊C4C22×C412yes—
217C4⋊C4C22×C412yes—
218C4⋊C4#4112yes—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C22⋊C4—24no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—81no—
225C22⋊C4C22⋊C414yes—
226C22⋊C4C22⋊C414yes—
227C42—24no—
228C42—24no—
229C42—42no—
230C42—81no—
231C42C22×C412yes—
232C42C2×D412yes—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23C2×C22⋊C412yes—
246C23C2×C22⋊C412yes—
247C23C22×D411yescharacteristic
248Q8—22no—
249Q8—22no—
250Q8—41no—
251Q8—41no—
252D4—44no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—48no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—84no—
295C2×C4C2×C22⋊C412yes—
296C2×C4C2×C22⋊C412yes—
297C2×C4C2×C22⋊C412yes—
298C2×C4C2×C22⋊C412yes—
299C2×C4C23×C411yescharacteristic
300C2×C4C22×D411yescharacteristic
301C2×C4C2×#4111yescharacteristic
302C2×C4C2×#4111yescharacteristic
303C2×C4H(5,𝔽2)11yescharacteristic
304C2×C42−1+411yescharacteristic
305C8—22no—
306C8—41no—
307C8—41no—
308C8—41no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22C22×C22⋊C411yescharacteristic
324C22#54611yescharacteristic
325C22#54711yescharacteristic
326C22#57511yescharacteristic
327C22#57511yescharacteristic
328C22#57711yescharacteristic
329C22#57811yescharacteristic
330C4—22no—
331C4—22no—
332C4—22no—
333C4—22no—
334C4—42no—
335C4—42no—
336C4—42no—
337C4—42no—
338C4—42no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4C4×#4111yescharacteristic
343C4#52011yescharacteristic
344C4#54611yescharacteristic
345C4#54711yescharacteristic
346C2—42no—
347C2—42no—
348C2#233911yescharacteristic
349C2#289611yescharacteristic
350C2#289811yescharacteristic
351C2#332311yescharacteristic
352C2#332711yescharacteristic
353C2#335111yescharacteristic
354C2#335211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1991C2semidirect
5. #1989C2semidirect
6. #1894C2semidirect
7. #1894C2semidirect
8. #1873C2semidirect
9. #1872C2semidirect
10. #1825C2semidirect
11. #1824C2semidirect
12. #1805C2semidirect
13. C4×#103C2semidirect
14. #1736C2semidirect
15. #1736C2semidirect
19. #552C4semidirect
81. #125C2×C4semidirect
86. C22⋊Q8C2×C4semidirect
88. C4×Q8C2×C4semidirect
106. C2×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5522
C2×#4362
#23522
#19912
#19892
#18942
#18732
#18722
#18252
#18242
#18052
C4×#1032
#17362