Finite Group Databasebeta

← #27860

Subgroups — #27860

Order 256.

372 conjugacy classes of subgroups.

The subgroup lattice has 303 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2855C211yesmaximal, characteristic
2#2283C211yesmaximal, characteristic
3#2101C211yesmaximal, characteristic, complemented
4#2097C211yesmaximal, characteristic, complemented
5#2085C211yesmaximal, characteristic, complemented
6#2074C211yesmaximal, characteristic, complemented
7#1928C211yesmaximal, characteristic, complemented
8#1927C211yesmaximal, characteristic, complemented
9#1919C211yesmaximal, characteristic, complemented
10#1918C211yesmaximal, characteristic, complemented
11#1892C211yesmaximal, characteristic
12#1892C211yesmaximal, characteristic
13#1808C211yesmaximal, characteristic
14#1807C211yesmaximal, characteristic
15C2×#336C211yesmaximal, characteristic
16C23×C8C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C22×C42C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28#391C2211yescharacteristic
29#391C2211yescharacteristic
30#389C2211yescharacteristic
31#389C2211yescharacteristic
32#387—21no—
33#387—21no—
34#385—21no—
35#383—21no—
36#382C2211yescharacteristic
37#382C2211yescharacteristic
38#381—21no—
39#381—21no—
40#380C2211yescharacteristic
41#380C2211yescharacteristic
42#380C2211yescharacteristic
43C4×C4⋊C4C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45C4×C22⋊C4—21no—
46#376—21no—
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×C4D411yescharacteristic
61C23×C4D411yescharacteristic
62C23×C4C2311yescharacteristic
63C22×C8—22no—
64C22×C8—22no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42D411yescharacteristic
94C2×C42D411yescharacteristic
95C2×C42C2311yescharacteristic
96C2×C42C2311yescharacteristic
97#107—22no—
98#107—41no—
99#106—22no—
100#106—41no—
101#103—22no—
102#103—22no—
103#103—41no—
104#103—41no—
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—
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—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—41no—
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4#4112yes—
151C22×C4#4112yes—
152C22×C4C2411yescharacteristic
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—
162C2×C8—21no—
163C2×C8—44no—
164C2×C8—44no—
165C2×C8#4112yes—
166C2×C8#4112yes—
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—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—42no—
188C42—22no—
189C42—22no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—42no—
195C42—42no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23#12312yes—
201C23#13512yes—
202C23#13712yes—
203C23C22×D411yescharacteristic
204Q8—41no—
205Q8—41no—
206Q8—41no—
207Q8—41no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
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—44no—
226C2×C4—44no—
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—41no—
251C2×C4#12312yes—
252C2×C4#13512yes—
253C2×C4C2×#4111yescharacteristic
254C2×C4C2×#4111yescharacteristic
255C2×C4C2×#4111yescharacteristic
256C2×C4C2×#4111yescharacteristic
257C2×C4H(5,𝔽2)11yescharacteristic
258C2×C42−1+411yescharacteristic
259C8—22no—
260C8—22no—
261C8—41no—
262C8—41no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22#48514yes—
270C22C2×#12311yescharacteristic
271C22#53211yescharacteristic
272C22#53711yescharacteristic
273C22C2×#13511yescharacteristic
274C22C2×#13511yescharacteristic
275C22C2×#13711yescharacteristic
276C22#57511yescharacteristic
277C4—22no—
278C4—22no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—81no—
283C4—81no—
284C4—81no—
285C4—81no—
286C4—81no—
287C4—81no—
288C4—81no—
289C4—81no—
290C4#55111yescharacteristic
291C4#55111yescharacteristic
292C4#55211yescharacteristic
293C4#55211yescharacteristic
294C2—24no—
295C2#249811yescharacteristic
296C2C2×#48511yescharacteristic
297C2#309111yescharacteristic
298C2#316811yescharacteristic
299C2#318111yescharacteristic
300C2#324811yescharacteristic
301C2#325011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2101C2semidirect
4. #2097C2semidirect
5. #2085C2semidirect
6. #2074C2semidirect
7. #1928C2semidirect
8. #1927C2semidirect
9. #1919C2semidirect
10. #1918C2semidirect

Maximal subgroups

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

subgroupindex
#28552
#22832
#21012
#20972
#20852
#20742
#19282
#19272
#19192
#19182
#18922
#18082
#18072
C2×#3362