Finite Group Databasebeta

← #30841

Subgroups — #30841

Order 256.

309 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3163C211yesmaximal, characteristic, complemented
2#3137C211yesmaximal, characteristic
3#3097C211yesmaximal, characteristic
4#3083C211yesmaximal, characteristic
5#2097C211yesmaximal, characteristic, complemented
6#2079C211yesmaximal, characteristic
7#2064C211yesmaximal, characteristic, complemented
8#2061C211yesmaximal, characteristic, complemented
9#2038C211yesmaximal, characteristic
10#2028C211yesmaximal, characteristic, complemented
11#1952C211yesmaximal, characteristic, complemented
12#1930C211yesmaximal, characteristic, complemented
13#1929C211yesmaximal, characteristic
14#1922C211yesmaximal, characteristic, complemented
15#1860C211yesmaximal, characteristic
16#544C2211yescharacteristic
17#544C2211yescharacteristic
18#536C2211yescharacteristic
19#520C2211yescharacteristic
20#501—21no—
21#501—21no—
22#499—21no—
23#498—21no—
24#491—21no—
25#489—21no—
26#488—21no—
27#487—21no—
28#485—21no—
29#485—21no—
30#484—21no—
31#483—21no—
32#479—21no—
33#478—21no—
34#477—21no—
35#474—21no—
36C2×#108C2211yescharacteristic
37#428C2211yescharacteristic
38#428C2211yescharacteristic
39#428C2211yescharacteristic
40#428C2211yescharacteristic
41#427C2211yescharacteristic
42C2×#107C2211yescharacteristic
43C2×#106C2211yescharacteristic
44#424C2211yescharacteristic
45C2×#104C2211yescharacteristic
46#419C2211yescharacteristic
47#419C2211yescharacteristic
48#417C2211yescharacteristic
49#417C2211yescharacteristic
50#416C2211yescharacteristic
51C2×#103C2211yescharacteristic
52C2×#103C2211yescharacteristic
53C2×#101C2211yescharacteristic
54#408C2211yescharacteristic
55#405C2211yescharacteristic
56#405C2211yescharacteristic
57#391C2211yescharacteristic
58#382C2211yescharacteristic
59#343C2211yescharacteristic
60#343C2211yescharacteristic
61#343C2211yescharacteristic
62#341C2211yescharacteristic
63#339C2211yescharacteristic
64#339C2211yescharacteristic
65#339C2211yescharacteristic
66#336C2211yescharacteristic
67C2×#41C2311yescharacteristic
68C22×Q8C2311yescharacteristic
69C2×M16C2311yescharacteristic
70C2×M16C2311yescharacteristic
71C2×M16C2311yescharacteristic
72C2×M16C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C4⋊Q8—21no—
76C4⋊Q8—21no—
77C4⋊Q8—21no—
78C4⋊Q8—21no—
79C4⋊Q8—21no—
80#125—21no—
81#125—21no—
82#125—21no—
83#125—21no—
84#124—21no—
85#123—41no—
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88C22⋊Q8—41no—
89C22⋊Q8—41no—
90C4×Q8—21no—
91C4×D4—21no—
92C4×D4—41no—
93#117—21no—
94#117C2311yescharacteristic
95#117C2311yescharacteristic
96#117C2311yescharacteristic
97C2×C4⋊C4—21no—
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C42C2311yescharacteristic
102#108—22no—
103#107—21no—
104#107—21no—
105#107—21no—
106#107—21no—
107#107—21no—
108#107—21no—
109#107D411yescharacteristic
110#107D411yescharacteristic
111#106—21no—
112#106—21no—
113#106—21no—
114#106—21no—
115#106—21no—
116#106—21no—
117#106D411yescharacteristic
118#106D411yescharacteristic
119C4⋊C8—21no—
120C4⋊C8—21no—
121#104—22no—
122#103—21no—
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103—21no—
129#103—21no—
130#103—21no—
131#103—21no—
132#103D411yescharacteristic
133#103D411yescharacteristic
134#102—21no—
135#102—21no—
136#102D411yescharacteristic
137#102D411yescharacteristic
138#101—22no—
139C22⋊C8—21no—
140C22⋊C8—21no—
141#97—21no—
142#97—21no—
143C4×C8—21no—
144C4×C8—21no—
145#95—41no—
146#95—41no—
147#41—21no—
148#41—21no—
149#41—21no—
150#41—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8C2×D411yescharacteristic
160C2×D4—21no—
161C2×D4C2×D411yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4C2411yescharacteristic
170M16—21no—
171M16—21no—
172M16—21no—
173M16—21no—
174M16—21no—
175M16—21no—
176M16—21no—
177M16#4112yes—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8C2×D411yescharacteristic
187C2×C8C2×D411yescharacteristic
188C2×C8C2×D411yescharacteristic
189C2×C8C2×D411yescharacteristic
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4C2×D411yescharacteristic
206C4⋊C4C2×D411yescharacteristic
207C4⋊C4C2×D411yescharacteristic
208C4⋊C4C2×D411yescharacteristic
209C4⋊C4C2×D411yescharacteristic
210C4⋊C4C2×D411yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—41no—
223C23—21no—
224C23H(5,𝔽2)11yescharacteristic
225Q8—21no—
226Q8—21no—
227Q8—41no—
228Q8—41no—
229Q8—41no—
230Q8—41no—
231Q8—161no—
232Q8—161no—
233D4—21no—
234D4—21no—
235D4—41no—
236D4—41no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—161no—
259C2×C4C22×D411yescharacteristic
260C2×C4C22×D411yescharacteristic
261C2×C4C2×#4111yescharacteristic
262C2×C42−1+411yescharacteristic
263C2×C42−1+411yescharacteristic
264C2×C42−1+411yescharacteristic
265C8—21no—
266C8—21no—
267C8—21no—
268C8—21no—
269C8—22no—
270C8—41no—
271C8—41no—
272C8—41no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22#54211yescharacteristic
279C22D4211yescharacteristic
280C22#56311yescharacteristic
281C4—41no—
282C4—41no—
283C4—41no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—81no—
289C4—81no—
290C4—81no—
291C4—81no—
292C4—81no—
293C4#53611yescharacteristic
294C4#53611yescharacteristic
295C4#54711yescharacteristic
296C4#54811yescharacteristic
297C2—21no—
298C2—21no—
299C2—81no—
300C2#267411yescharacteristic
301C2#328911yescharacteristic
302C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3163C2semidirect
5. #2097C2semidirect
7. #2064C2semidirect
8. #2061C2semidirect
10. #2028C2semidirect
11. #1952C2semidirect
12. #1930C2semidirect
14. #1922C2semidirect

Maximal subgroups

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

subgroupindex
#31632
#31372
#30972
#30832
#20972
#20792
#20642
#20612
#20382
#20282
#19522
#19302
#19292
#19222
#18602