Finite Group Databasebeta

← #28359

Subgroups — #28359

Order 256.

524 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic
2C2×#468C211yesmaximal, characteristic
3C2×#451C211yesmaximal, characteristic, complemented
4C2×#419C211yesmaximal, characteristic, complemented
5#2499C211yesmaximal, characteristic, complemented
6#2436C211yesmaximal, characteristic
7#2097C211yesmaximal, characteristic, complemented
8#2095C211yesmaximal, characteristic, complemented
9#2015C211yesmaximal, characteristic, complemented
10#2015C211yesmaximal, characteristic, complemented
11#1971C211yesmaximal, characteristic, complemented
12#1969C211yesmaximal, characteristic, complemented
13#1959C211yesmaximal, characteristic, complemented
14#1958C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×M16C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×#117C2211yescharacteristic
26#482—24no—
27#468—24no—
28#451—24no—
29C2×#107C2211yescharacteristic
30C2×#107C2211yescharacteristic
31C2×#106C2211yescharacteristic
32#419—24no—
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×#97C2211yescharacteristic
45C2×C4×C8C2211yescharacteristic
46#397—22no—
47#393—22no—
48#391C2211yescharacteristic
49#391C2211yescharacteristic
50#391C2211yescharacteristic
51#391C2211yescharacteristic
52#387—22no—
53#386—22no—
54#384C2211yescharacteristic
55#382C2211yescharacteristic
56C4×C22⋊C4C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×Q8—22no—
61C22×Q8—22no—
62C22×Q8—22no—
63C22×Q8—22no—
64C22×Q8—41no—
65C22×Q8—41no—
66C22×Q8C2311yescharacteristic
67C22×Q8C2311yescharacteristic
68C22×Q8C2311yescharacteristic
69C22×D4—21no—
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C23×C4C2311yescharacteristic
73C2×Q16—22no—
74C2×Q16—22no—
75C2×Q16—22no—
76C2×Q16—22no—
77C2×Q16—41no—
78C2×Q16—41no—
79C2×SD16—22no—
80C2×SD16—41no—
81C2×SD16—41no—
82C2×M16—41no—
83C2×M16—41no—
84C2×M16D414yes—
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89#124—22no—
90#124—22no—
91#124—41no—
92#124—41no—
93C22⋊Q8—22no—
94C22⋊Q8—22no—
95#121—22no—
96#121—22no—
97#117—22no—
98#117—22no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—22no—
110C2×C22⋊C4D412yes—
111C2×C42—21no—
112C2×C42C2311yescharacteristic
113C2×C42C2311yescharacteristic
114#107—22no—
115#107—22no—
116#106—22no—
117#103—22no—
118#103—22no—
119#103—22no—
120#103—22no—
121#103—22no—
122#103—22no—
123#103—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127C22⋊C8D414yes—
128#97—41no—
129C4×C8—41no—
130#95—22no—
131#95—22no—
132#95—22no—
133#95—22no—
134#95—22no—
135#95D412yes—
136C24—21no—
137C24C2×D411yescharacteristic
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—28no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—44no—
158C2×Q8—44no—
159C2×Q8—44no—
160C2×Q8—42no—
161C2×Q8#4112yes—
162C2×Q8#4112yes—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—44no—
170C2×D4—42no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4C2×D412yes—
186C22×C4C2×D412yes—
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D412yes—
189C22×C4C2×D412yes—
190C22×C4C2411yescharacteristic
191Q16—44no—
192Q16—44no—
193SD16—44no—
194M16—44no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8C2×D412yes—
204C2×C8C2×D412yes—
205C2×C8C2×D412yes—
206C2×C8C2×D412yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—44no—
216C4⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—44no—
219C22⋊C4—44no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—42no—
228C23—21no—
229C23—21no—
230C23—22no—
231C23—22no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—42no—
236C23—42no—
237C23#12712yes—
238C23C22×D411yescharacteristic
239Q8—22no—
240Q8—22no—
241Q8—22no—
242Q8—22no—
243Q8—48no—
244Q8—41no—
245Q8—41no—
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—41no—
250Q8—41no—
251Q8—41no—
252D4—41no—
253D4—41no—
254D4—41no—
255D4—41no—
256D4—84no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
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—48no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
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—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4#12712yes—
288C2×C4C22×D411yescharacteristic
289C2×C4C22×D411yescharacteristic
290C2×C4C2×#4111yescharacteristic
291C2×C4C2×#4111yescharacteristic
292C2×C4H(5,𝔽2)11yescharacteristic
293C2×C4H(5,𝔽2)11yescharacteristic
294C8—22no—
295C8—22no—
296C8—22no—
297C8—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—42no—
302C22—42no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—82no—
307C22—82no—
308C22C2×#12711yescharacteristic
309C22#54611yescharacteristic
310C22#54611yescharacteristic
311C22#57512yes—
312C22#57812yes—
313C4—22no—
314C4—42no—
315C4—42no—
316C4—42no—
317C4—42no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4#52511yescharacteristic
324C4#53411yescharacteristic
325C4#54611yescharacteristic
326C4#54611yescharacteristic
327C2—42no—
328C2—161no—
329C2#253911yescharacteristic
330C2#315311yescharacteristic
331C2#315611yescharacteristic
332C2#331312yes—
333C2#332712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#451C2semidirect
4. C2×#419C2semidirect
5. #2499C2semidirect
7. #2097C2semidirect
8. #2095C2semidirect
9. #2015C2semidirect
10. #2015C2semidirect
11. #1971C2semidirect
12. #1969C2semidirect
13. #1959C2semidirect
14. #1958C2semidirect
15. #1753C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C2×#4682
C2×#4512
C2×#4192
#24992
#24362
#20972
#20952
#20152
#19712
#19692
#19592
#19582
#17532