Finite Group Databasebeta

← #29158

Subgroups — #29158

Order 256.

392 conjugacy classes of subgroups.

The subgroup lattice has 311 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2851C211yesmaximal, characteristic
2C4×C22⋊Q8C211yesmaximal, characteristic
3#2101C211yesmaximal, characteristic, complemented
4#2097C211yesmaximal, characteristic, complemented
5#2059C211yesmaximal, characteristic, complemented
6#2026C211yesmaximal, characteristic, complemented
7#1920C211yesmaximal, characteristic, complemented
8#1918C211yesmaximal, characteristic, complemented
9#1892C211yesmaximal, characteristic
10#1892C211yesmaximal, characteristic
11#1865C211yesmaximal, characteristic, complemented
12#1863C211yesmaximal, characteristic, complemented
13#1808C211yesmaximal, characteristic
14#1807C211yesmaximal, characteristic
15C2×#336C211yesmaximal, characteristic
16C23×C8C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C22×C42C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30#391C2211yescharacteristic
31#389C2211yescharacteristic
32#389C2211yescharacteristic
33#389C2211yescharacteristic
34#387—21no—
35#387—21no—
36#387—21no—
37#385—21no—
38#385—21no—
39#385—21no—
40#382C2211yescharacteristic
41#380C2211yescharacteristic
42#380C2211yescharacteristic
43#380C2211yescharacteristic
44C4×C4⋊C4—21no—
45C4×C4⋊C4C2211yescharacteristic
46C4×C22⋊C4—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
73#125—41no—
74#125—41no—
75C22⋊Q8—22no—
76C22⋊Q8Q814yes—
77C4×Q8—41no—
78C4×Q8—41no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—41no—
100C2×C42—41no—
101C2×C42D411yescharacteristic
102C2×C42D411yescharacteristic
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105#107—22no—
106#107—41no—
107#106—22no—
108#106—41no—
109#103—22no—
110#103—22no—
111#103—41no—
112#103—41no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124C24C2×D411yescharacteristic
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8C2×Q812yes—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×Q812yes—
155C22×C4#4112yes—
156C22×C4C2411yescharacteristic
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—44no—
168C2×C8—44no—
169C2×C8#4112yes—
170C2×C8#4112yes—
171C4⋊C4—24no—
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—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4C2×Q812yes—
192C22⋊C4—24no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—41no—
201C42—41no—
202C42—42no—
203C42—42no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23C22⋊Q812yes—
209C23#13512yes—
210C23#13712yes—
211C23C22×D411yescharacteristic
212Q8—22no—
213Q8—22no—
214Q8—41no—
215Q8—41no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—44no—
236C2×C4—44no—
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—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4C22⋊Q812yes—
260C2×C4#13512yes—
261C2×C4C22×Q811yescharacteristic
262C2×C4C2×#4111yescharacteristic
263C2×C4C2×#4111yescharacteristic
264C2×C4C2×#4111yescharacteristic
265C2×C4H(5,𝔽2)11yescharacteristic
266C2×C4H(5,𝔽2)11yescharacteristic
267C8—22no—
268C8—22no—
269C8—41no—
270C8—41no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22#47914yes—
278C22C2×C22⋊Q811yescharacteristic
279C22#53211yescharacteristic
280C22#53411yescharacteristic
281C22C2×#13511yescharacteristic
282C22C2×#13511yescharacteristic
283C22C2×#13711yescharacteristic
284C22#57511yescharacteristic
285C4—22no—
286C4—22no—
287C4—42no—
288C4—42no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4—81no—
298C4#55111yescharacteristic
299C4#55111yescharacteristic
300C4#55411yescharacteristic
301C4#55411yescharacteristic
302C2—24no—
303C2#256411yescharacteristic
304C2C2×#47911yescharacteristic
305C2#307611yescharacteristic
306C2#316811yescharacteristic
307C2#318111yescharacteristic
308C2#319011yescharacteristic
309C2#319111yescharacteristic
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. #2059C2semidirect
6. #2026C2semidirect
7. #1920C2semidirect
8. #1918C2semidirect
11. #1865C2semidirect
12. #1863C2semidirect

Maximal subgroups

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

subgroupindex
#28512
C4×C22⋊Q82
#21012
#20972
#20592
#20262
#19202
#19182
#18922
#18652
#18632
#18082
#18072
C2×#3362