Finite Group Databasebeta

← #27619

Subgroups — #27619

Order 256.

335 conjugacy classes of subgroups.

The subgroup lattice has 289 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2317C211yesmaximal, characteristic
2#2314C211yesmaximal, characteristic
3#2101C211yesmaximal, characteristic, complemented
4#2101C211yesmaximal, characteristic, complemented
5#2084C211yesmaximal, characteristic, complemented
6#2083C211yesmaximal, characteristic, complemented
7#1921C211yesmaximal, characteristic, complemented
8#1920C211yesmaximal, characteristic, complemented
9#1919C211yesmaximal, characteristic, complemented
10#1918C211yesmaximal, characteristic, complemented
11#1854C211yesmaximal, characteristic
12#1854C211yesmaximal, characteristic
13#1824C211yesmaximal, characteristic
14#1823C211yesmaximal, characteristic
15#1818C211yesmaximal, characteristic
16C2×#117C2211yescharacteristic
17C2×#117C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×C22⋊C8C2211yescharacteristic
24C2×C22⋊C8C2211yescharacteristic
25#389C2211yescharacteristic
26#389C2211yescharacteristic
27#389C2211yescharacteristic
28#389C2211yescharacteristic
29#388—21no—
30#388—21no—
31#387—21no—
32#387—21no—
33#385—21no—
34#385—21no—
35#384—21no—
36#384C2211yescharacteristic
37#382—21no—
38#382C2211yescharacteristic
39#380C2211yescharacteristic
40#380C2211yescharacteristic
41C4×C4⋊C4C2211yescharacteristic
42C4×C4⋊C4C2211yescharacteristic
43#340C2211yescharacteristic
44#340C2211yescharacteristic
45#340C2211yescharacteristic
46#340C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C23×C4—21no—
60C23×C4C2311yescharacteristic
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65#117—22no—
66#117—22no—
67#117—22no—
68#117—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—21no—
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⋊C4D411yescharacteristic
79C2×C4⋊C4D411yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4D411yescharacteristic
92C2×C22⋊C4D411yescharacteristic
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98#107—22no—
99#107—22no—
100#106—22no—
101#106—22no—
102C22⋊C8—22no—
103C22⋊C8—22no—
104#95—21no—
105#95—21no—
106#95—21no—
107#95—21no—
108#95—21no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114C24C2×D411yescharacteristic
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—42no—
128C22×C4—81no—
129C22×C4—81no—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4#4112yes—
136C22×C4#4112yes—
137C22×C4C2411yescharacteristic
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—42no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C4⋊C4—81no—
171C4⋊C4—81no—
172C4⋊C4—81no—
173C4⋊C4#4112yes—
174C4⋊C4#4112yes—
175C22⋊C4—24no—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C22⋊C4—81no—
179C22⋊C4—81no—
180C42—24no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—81no—
186C42—81no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23#12312yes—
194C23C22×D411yescharacteristic
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
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—84no—
238C2×C4#12312yes—
239C2×C4C2×#4111yescharacteristic
240C2×C4C2×#4111yescharacteristic
241C2×C4C2×#4111yescharacteristic
242C2×C4C2×#4111yescharacteristic
243C2×C42−1+411yescharacteristic
244C2×C42−1+411yescharacteristic
245C8—41no—
246C8—41no—
247C8—41no—
248C8—41no—
249C22—22no—
250C22—22no—
251C22—22no—
252C22—41no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22C2×#12311yescharacteristic
257C22#52511yescharacteristic
258C22#53611yescharacteristic
259C22#57611yescharacteristic
260C22#57711yescharacteristic
261C22#57711yescharacteristic
262C22#57811yescharacteristic
263C4—22no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—41no—
268C4—42no—
269C4—42no—
270C4—81no—
271C4—81no—
272C4—81no—
273C4—81no—
274C4—81no—
275C4—81no—
276C4#55211yescharacteristic
277C4#55211yescharacteristic
278C4#55211yescharacteristic
279C4#55211yescharacteristic
280C2—42no—
281C2#248411yescharacteristic
282C2#309611yescharacteristic
283C2#309711yescharacteristic
284C2#311911yescharacteristic
285C2#312011yescharacteristic
286C2#324311yescharacteristic
287C2#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. #2101C2semidirect
5. #2084C2semidirect
6. #2083C2semidirect
7. #1921C2semidirect
8. #1920C2semidirect
9. #1919C2semidirect
10. #1918C2semidirect

Maximal subgroups

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

subgroupindex
#23172
#23142
#21012
#20842
#20832
#19212
#19202
#19192
#19182
#18542
#18242
#18232
#18182