Finite Group Databasebeta

← #31113

Subgroups — #31113

Order 256.

319 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3099C211yesmaximal, characteristic
2#3098C211yesmaximal, characteristic
3#3084C211yesmaximal, characteristic, complemented
4#3083C211yesmaximal, characteristic, complemented
5#2090C211yesmaximal, characteristic, complemented
6#2089C211yesmaximal, characteristic, complemented
7#2079C211yesmaximal, characteristic, complemented
8#2078C211yesmaximal, characteristic, complemented
9#2060C211yesmaximal, characteristic, complemented
10#2039C211yesmaximal, characteristic
11#1938C211yesmaximal, characteristic, complemented
12#1938C211yesmaximal, characteristic, complemented
13#1860C211yesmaximal, characteristic, complemented
14#1860C211yesmaximal, characteristic, complemented
15#1847C211yesmaximal, characteristic, complemented
16#544C2211yescharacteristic
17#537C2211yescharacteristic
18#520C2211yescharacteristic
19#520C2211yescharacteristic
20#485—21no—
21#485—21no—
22#484—21no—
23#483—21no—
24#482—21no—
25#482—21no—
26#481—21no—
27#480—21no—
28#479—21no—
29#479—21no—
30#478—21no—
31#478—21no—
32#477—21no—
33#476—21no—
34#475—21no—
35#474—21no—
36#428C2211yescharacteristic
37#428C2211yescharacteristic
38#428C2211yescharacteristic
39#428C2211yescharacteristic
40C2×#107C2211yescharacteristic
41C2×#107C2211yescharacteristic
42C2×#106C2211yescharacteristic
43C2×#106C2211yescharacteristic
44#424C2211yescharacteristic
45#424C2211yescharacteristic
46C2×#104C2211yescharacteristic
47C2×#104C2211yescharacteristic
48#417C2211yescharacteristic
49#417C2211yescharacteristic
50#417C2211yescharacteristic
51#417C2211yescharacteristic
52#416C2211yescharacteristic
53#416C2211yescharacteristic
54#408C2211yescharacteristic
55#408C2211yescharacteristic
56C2×#97C2211yescharacteristic
57#384C2211yescharacteristic
58#382C2211yescharacteristic
59#343C2211yescharacteristic
60#343C2211yescharacteristic
61#343C2211yescharacteristic
62#343C2211yescharacteristic
63#341C2211yescharacteristic
64#341C2211yescharacteristic
65#339C2211yescharacteristic
66#339C2211yescharacteristic
67C2×#41C2311yescharacteristic
68C2×#41C2311yescharacteristic
69C2×M16C2311yescharacteristic
70C2×M16C2311yescharacteristic
71C2×M16C2311yescharacteristic
72C2×M16C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C4⋊Q8—21no—
76C4⋊Q8—21no—
77#125—21no—
78#125—21no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81C22⋊Q8—41no—
82#121—21no—
83#121—21no—
84#121—41no—
85C4×Q8—21no—
86C4×Q8—21no—
87C4×D4—21no—
88C4×D4—21no—
89C4×D4—41no—
90C4×D4—41no—
91#117—21no—
92#117—21no—
93#117C2311yescharacteristic
94#117C2311yescharacteristic
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C42C2311yescharacteristic
103#107—21no—
104#107—21no—
105#107—21no—
106#107—21no—
107#107—21no—
108#107—21no—
109#107—21no—
110#107—21no—
111#106—21no—
112#106—21no—
113#106—21no—
114#106—21no—
115#106—21no—
116#106—21no—
117#106—21no—
118#106—21no—
119C4⋊C8—21no—
120C4⋊C8—21no—
121C4⋊C8—21no—
122C4⋊C8—21no—
123#104—22no—
124#104—22no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103—21no—
129#103D411yescharacteristic
130#103D411yescharacteristic
131#103D411yescharacteristic
132#103D411yescharacteristic
133#102—21no—
134#102—21no—
135#102—21no—
136#102—21no—
137#102D411yescharacteristic
138#102D411yescharacteristic
139#102D411yescharacteristic
140#102D411yescharacteristic
141C22⋊C8—21no—
142C22⋊C8—21no—
143C22⋊C8—21no—
144C22⋊C8—21no—
145#97—22no—
146#95—41no—
147#41—21no—
148#41—21no—
149#41—21no—
150#41—21no—
151#41—21no—
152#41—21no—
153#41—21no—
154#41—21no—
155C2×Q8—41no—
156C2×Q8C2×D411yescharacteristic
157C2×Q8C2×D411yescharacteristic
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—41no—
161C2×D4C2×D411yescharacteristic
162C2×D4C2×D411yescharacteristic
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4C2411yescharacteristic
173M16—21no—
174M16—21no—
175M16—21no—
176M16—21no—
177M16—21no—
178M16—21no—
179M16—21no—
180M16—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8C2×D411yescharacteristic
190C2×C8C2×D411yescharacteristic
191C2×C8C2×D411yescharacteristic
192C2×C8C2×D411yescharacteristic
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—82no—
208C4⋊C4C2×D411yescharacteristic
209C4⋊C4C2×D411yescharacteristic
210C4⋊C4C2×D411yescharacteristic
211C4⋊C4C2×D411yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—41no—
222C42—41no—
223C42#4112yes—
224C23—21no—
225C23—21no—
226C23H(5,𝔽2)11yescharacteristic
227Q8—21no—
228Q8—21no—
229Q8—21no—
230Q8—21no—
231Q8—161no—
232D4—21no—
233D4—21no—
234D4—21no—
235D4—21no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—161no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—161no—
263C2×C4—161no—
264C2×C4C22×D411yescharacteristic
265C2×C4C22×D411yescharacteristic
266C2×C4C2×#4111yescharacteristic
267C2×C42−1+411yescharacteristic
268C2×C42−1+411yescharacteristic
269C2×C42−1+411yescharacteristic
270C8—21no—
271C8—21no—
272C8—21no—
273C8—21no—
274C8—41no—
275C8—41no—
276C8—41no—
277C8—41no—
278C22—21no—
279C22—21no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22#53911yescharacteristic
287C22D4211yescharacteristic
288C22#56311yescharacteristic
289C4—42no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4#53611yescharacteristic
301C4#53611yescharacteristic
302C4#54711yescharacteristic
303C4#54711yescharacteristic
304C2—21no—
305C2—21no—
306C2—81no—
307C2—81no—
308C2#268011yescharacteristic
309C2#328811yescharacteristic
310C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radical#537C22
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
3. #3084C2semidirect
4. #3083C2semidirect
5. #2090C2semidirect
6. #2089C2semidirect
7. #2079C2semidirect
8. #2078C2semidirect
9. #2060C2semidirect
11. #1938C2semidirect
12. #1938C2semidirect
13. #1860C2semidirect
14. #1860C2semidirect
15. #1847C2semidirect

Maximal subgroups

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

subgroupindex
#30992
#30982
#30842
#30832
#20902
#20892
#20792
#20782
#20602
#20392
#19382
#18602
#18472