Finite Group Databasebeta

← #27971

Subgroups — #27971

Order 256.

339 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2628C211yesmaximal, characteristic
2#2317C211yesmaximal, characteristic
3#2101C211yesmaximal, characteristic, complemented
4#2101C211yesmaximal, characteristic, complemented
5#2100C211yesmaximal, characteristic, complemented
6#2099C211yesmaximal, characteristic, complemented
7#1993C211yesmaximal, characteristic, complemented
8#1990C211yesmaximal, characteristic, complemented
9#1836C211yesmaximal, characteristic, complemented
10#1835C211yesmaximal, characteristic, complemented
11#1825C211yesmaximal, characteristic
12#1825C211yesmaximal, characteristic
13#1824C211yesmaximal, characteristic
14#1823C211yesmaximal, characteristic
15#1750C211yesmaximal, characteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×#117C2211yescharacteristic
18C2×#117C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C2×C22⋊C8C2211yescharacteristic
21C2×C22⋊C8C2211yescharacteristic
22C2×#97C2211yescharacteristic
23C2×C4×C8C2211yescharacteristic
24#400—21no—
25#399—21no—
26#395—21no—
27#393—21no—
28#389—21no—
29#389—21no—
30#389C2211yescharacteristic
31#389C2211yescharacteristic
32#385—21no—
33#385—21no—
34#384C2211yescharacteristic
35#384C2211yescharacteristic
36#382C2211yescharacteristic
37#382C2211yescharacteristic
38#382C2211yescharacteristic
39#380C2211yescharacteristic
40#380C2211yescharacteristic
41#340C2211yescharacteristic
42#340C2211yescharacteristic
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#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C23×C4—21no—
61C23×C4C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C4⋊Q8—22no—
67C4⋊Q8—22no—
68#117—22no—
69#117—22no—
70#117—22no—
71#117—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
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—21no—
81C2×C4⋊C4—21no—
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×C4⋊C4C2311yescharacteristic
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4D411yescharacteristic
96C2×C22⋊C4D411yescharacteristic
97C2×C42—21no—
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100C22⋊C8—22no—
101C22⋊C8—22no—
102#97—22no—
103C4×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
115C2×Q8—42no—
116C2×Q8—81no—
117C2×Q8—81no—
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—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—42no—
131C22×C4—81no—
132C22×C4—81no—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4#4112yes—
139C22×C4#4112yes—
140C22×C4C2411yescharacteristic
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—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C4⋊C4—22no—
154C4⋊C4—22no—
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—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—81no—
174C4⋊C4—81no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C4⋊C4—81no—
178C4⋊C4—81no—
179C22⋊C4—22no—
180C22⋊C4—22no—
181C22⋊C4—42no—
182C22⋊C4—81no—
183C22⋊C4—81no—
184C42—21no—
185C42—21no—
186C42—42no—
187C42#4112yes—
188C42#4112yes—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23#12412yes—
196C23C22×D411yescharacteristic
197Q8—84no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—41no—
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—41no—
238C2×C4—84no—
239C2×C4#12312yes—
240C2×C4#12312yes—
241C2×C4#12412yes—
242C2×C4C2×#4111yescharacteristic
243C2×C4C2×#4111yescharacteristic
244C2×C4C2×#4111yescharacteristic
245C2×C4C2×#4111yescharacteristic
246C2×C42−1+411yescharacteristic
247C2×C42−1+411yescharacteristic
248C8—41no—
249C8—41no—
250C8—41no—
251C8—41no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22C2×#12311yescharacteristic
260C22C2×#12411yescharacteristic
261C22#53611yescharacteristic
262C22#57611yescharacteristic
263C22#57711yescharacteristic
264C22#57711yescharacteristic
265C22#57811yescharacteristic
266C4—22no—
267C4—22no—
268C4—22no—
269C4—41no—
270C4—41no—
271C4—81no—
272C4—81no—
273C4—81no—
274C4—81no—
275C4—81no—
276C4—81no—
277C4—81no—
278C4—81no—
279C4#55211yescharacteristic
280C4#55211yescharacteristic
281C4#55211yescharacteristic
282C4#55211yescharacteristic
283C2—42no—
284C2#250311yescharacteristic
285C2#309611yescharacteristic
286C2#309711yescharacteristic
287C2#313611yescharacteristic
288C2#313711yescharacteristic
289C2#324311yescharacteristic
290C2#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. #2100C2semidirect
6. #2099C2semidirect
7. #1993C2semidirect
8. #1990C2semidirect
9. #1836C2semidirect
10. #1835C2semidirect

Maximal subgroups

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

subgroupindex
#26282
#23172
#21012
#21002
#20992
#19932
#19902
#18362
#18352
#18252
#18242
#18232
#17502