Finite Group Databasebeta

← #29919

Subgroups — #29919

Order 256.

334 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2625C211yesmaximal, characteristic
2#2317C211yesmaximal, characteristic
3#2101C211yesmaximal, characteristic, complemented
4#2100C211yesmaximal, characteristic, complemented
5#2097C211yesmaximal, characteristic, complemented
6#2096C211yesmaximal, characteristic, complemented
7#2085C211yesmaximal, characteristic, complemented
8#2084C211yesmaximal, characteristic, complemented
9#2074C211yesmaximal, characteristic, complemented
10#2072C211yesmaximal, characteristic, complemented
11#1873C211yesmaximal, characteristic
12#1872C211yesmaximal, characteristic
13#1854C211yesmaximal, characteristic
14#1853C211yesmaximal, characteristic
15#1818C211yesmaximal, characteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×#117C2211yescharacteristic
18C2×#117C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30#400—21no—
31#398—21no—
32#397—21no—
33#397—21no—
34#391C2211yescharacteristic
35#391C2211yescharacteristic
36#389—21no—
37#389—21no—
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#385—21no—
41#385—21no—
42#384C2211yescharacteristic
43#384C2211yescharacteristic
44#382C2211yescharacteristic
45#382C2211yescharacteristic
46#380C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C23×C4—21no—
61C23×C4C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22⋊Q8—22no—
67C22⋊Q8—22no—
68#117—22no—
69#117—22no—
70#117—41no—
71#117—41no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4D411yescharacteristic
95C2×C22⋊C4D411yescharacteristic
96C2×C42—21no—
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99#107—41no—
100#106—41no—
101#106—41no—
102#106—41no—
103#103—41no—
104#103—41no—
105#103—41no—
106#103—41no—
107C22⋊C8—22no—
108C22⋊C8—22no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120C24C2×D411yescharacteristic
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—42no—
140C22×C4—81no—
141C22×C4—81no—
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4#4112yes—
148C22×C4#4112yes—
149C22×C4C2411yescharacteristic
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—81no—
181C4⋊C4—81no—
182C4⋊C4—81no—
183C4⋊C4—81no—
184C4⋊C4—81no—
185C4⋊C4—81no—
186C22⋊C4—22no—
187C22⋊C4—22no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—81no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—42no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23#12312yes—
203C23C22×D411yescharacteristic
204Q8—41no—
205Q8—41no—
206Q8—41no—
207Q8—41no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—84no—
251C2×C4#12312yes—
252C2×C4C2×#4111yescharacteristic
253C2×C4C2×#4111yescharacteristic
254C2×C4H(5,𝔽2)11yescharacteristic
255C2×C4H(5,𝔽2)11yescharacteristic
256C2×C42−1+411yescharacteristic
257C2×C42−1+411yescharacteristic
258C8—41no—
259C8—41no—
260C8—41no—
261C8—41no—
262C22—22no—
263C22—22no—
264C22—22no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22C2×#12311yescharacteristic
270C22#53511yescharacteristic
271C22#53611yescharacteristic
272C22#57711yescharacteristic
273C22#57711yescharacteristic
274C22#57711yescharacteristic
275C22#57811yescharacteristic
276C4—22no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—81no—
282C4—81no—
283C4—81no—
284C4—81no—
285C4—81no—
286C4—81no—
287C4—81no—
288C4—81no—
289C4#53911yescharacteristic
290C4#53911yescharacteristic
291C4#53911yescharacteristic
292C4#53911yescharacteristic
293C2—42no—
294C2#262311yescharacteristic
295C2#309711yescharacteristic
296C2#309911yescharacteristic
297C2#322211yescharacteristic
298C2#322511yescharacteristic
299C2#324211yescharacteristic
300C2#324411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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
3. #2101C2semidirect
4. #2100C2semidirect
5. #2097C2semidirect
6. #2096C2semidirect
7. #2085C2semidirect
8. #2084C2semidirect
9. #2074C2semidirect
10. #2072C2semidirect

Maximal subgroups

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

subgroupindex
#26252
#23172
#21012
#21002
#20972
#20962
#20852
#20842
#20742
#20722
#18732
#18722
#18542
#18532
#18182