Finite Group Databasebeta

← #30252

Subgroups — #30252

Order 256.

472 conjugacy classes of subgroups.

The subgroup lattice has 329 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#470C211yesmaximal, characteristic, complemented
2C2×#468C211yesmaximal, characteristic
3C2×#467C211yesmaximal, characteristic, complemented
4C2×#465C211yesmaximal, characteristic
5#2621C211yesmaximal, characteristic, complemented
6#2620C211yesmaximal, characteristic
7#2097C211yesmaximal, characteristic, complemented
8#2096C211yesmaximal, characteristic, complemented
9#2095C211yesmaximal, characteristic, complemented
10#2092C211yesmaximal, characteristic, complemented
11#2016C211yesmaximal, characteristic, complemented
12#2015C211yesmaximal, characteristic, complemented
13#2015C211yesmaximal, characteristic, complemented
14#2013C211yesmaximal, characteristic, complemented
15#1729C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×M16C2211yescharacteristic
21C23×C8C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23C2×#124C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×C22⋊Q8C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic
27C2×#121C2211yescharacteristic
28C2×#117C2211yescharacteristic
29#470—24no—
30#468—24no—
31#467—24no—
32#465—24no—
33C2×#107C2211yescharacteristic
34C2×#107C2211yescharacteristic
35C2×#106C2211yescharacteristic
36C2×#106C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45#398—22no—
46#397—22no—
47#396—22no—
48#394—22no—
49#391C2211yescharacteristic
50#391C2211yescharacteristic
51#391C2211yescharacteristic
52#390C2211yescharacteristic
53#384C2211yescharacteristic
54#382C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4—21no—
63C22×D4C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C2×Q16—22no—
66C2×Q16—22no—
67C2×Q16—41no—
68C2×Q16—41no—
69C2×Q16—41no—
70C2×Q16—41no—
71C2×SD16—22no—
72C2×SD16—22no—
73C2×SD16—41no—
74C2×SD16—41no—
75C2×SD16—41no—
76C2×SD16—41no—
77C2×M16—41no—
78C2×M16—41no—
79C2×M16D414yes—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8D414yes—
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C4⋊Q8—41no—
88C4⋊Q8—41no—
89#124—41no—
90#124—41no—
91C22⋊Q8—22no—
92C22⋊Q8—22no—
93C22⋊Q8—22no—
94C22⋊Q8—22no—
95C22⋊Q8—22no—
96C22⋊Q8—22no—
97#121—22no—
98#121—22no—
99#117—41no—
100#117—41no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4D412yes—
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4D412yes—
118C2×C42C2311yescharacteristic
119C2×C42C2311yescharacteristic
120#107—22no—
121#107—22no—
122#106—22no—
123#106—22no—
124#103—22no—
125#103—22no—
126#103—22no—
127#103—22no—
128#103—22no—
129#103—22no—
130#102—22no—
131#102—22no—
132#95—22no—
133#95—22no—
134#95—22no—
135#95—22no—
136C24—21no—
137C24C2×D411yescharacteristic
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
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—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—42no—
163C2×D4—82no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4C2×D412yes—
176C22×C4C2×D412yes—
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D412yes—
179C22×C4C2×D412yes—
180C22×C4C2411yescharacteristic
181Q16—44no—
182Q16—44no—
183SD16—44no—
184SD16—44no—
185M16—44no—
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—44no—
195C2×C8C2×D412yes—
196C2×C8C2×D412yes—
197C2×C8C2×D412yes—
198C2×C8C2×D412yes—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—42no—
208C4⋊C4—44no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—82no—
213C22⋊C4—42no—
214C22⋊C4—44no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—82no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—22no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—42no—
232C23—42no—
233C23#12712yes—
234C23C22×D411yescharacteristic
235Q8—41no—
236Q8—41no—
237Q8—41no—
238Q8—41no—
239Q8—41no—
240Q8—41no—
241Q8—41no—
242Q8—41no—
243Q8—41no—
244Q8—41no—
245Q8—41no—
246Q8—41no—
247D4—41no—
248D4—41no—
249D4—41no—
250D4—41no—
251D4—84no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4#12712yes—
282C2×C4C22×D411yescharacteristic
283C2×C4C22×D411yescharacteristic
284C2×C4H(5,𝔽2)11yescharacteristic
285C2×C4H(5,𝔽2)11yescharacteristic
286C2×C4H(5,𝔽2)11yescharacteristic
287C2×C42−1+411yescharacteristic
288C8—22no—
289C8—22no—
290C8—22no—
291C8—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—42no—
296C22—42no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22—82no—
301C22—82no—
302C22C2×#12711yescharacteristic
303C22#53411yescharacteristic
304C22#53711yescharacteristic
305C22#57712yes—
306C22#57812yes—
307C4—22no—
308C4—42no—
309C4—42no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4—81no—
317C4#53411yescharacteristic
318C4#53411yescharacteristic
319C4#53711yescharacteristic
320C4#53711yescharacteristic
321C2—42no—
322C2—161no—
323C2#265611yescharacteristic
324C2#315411yescharacteristic
325C2#315611yescharacteristic
326C2#320112yes—
327C2#325812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#470C2semidirect
3. C2×#467C2semidirect
5. #2621C2semidirect
7. #2097C2semidirect
8. #2096C2semidirect
9. #2095C2semidirect
10. #2092C2semidirect
11. #2016C2semidirect
12. #2015C2semidirect
13. #2015C2semidirect
14. #2013C2semidirect
15. #1729C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4702
C2×#4682
C2×#4672
C2×#4652
#26212
#26202
#20972
#20962
#20952
#20922
#20162
#20152
#20132
#17292