Finite Group Databasebeta

← #32266

Subgroups — #32266

Order 256.

353 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2563C211yesmaximal, characteristic
2#2485C211yesmaximal, characteristic
3#2077C211yesmaximal, characteristic, complemented
4#2077C211yesmaximal, characteristic, complemented
5#2076C211yesmaximal, characteristic, complemented
6#2075C211yesmaximal, characteristic, complemented
7#2058C211yesmaximal, characteristic, complemented
8#2057C211yesmaximal, characteristic, complemented
9#1920C211yesmaximal, characteristic, complemented
10#1918C211yesmaximal, characteristic, complemented
11#1869C211yesmaximal, characteristic
12#1854C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14#1808C211yesmaximal, characteristic, complemented
15#1807C211yesmaximal, characteristic, complemented
16C23×C8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×C22⋊C8C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29#397—21no—
30#394—21no—
31#393—21no—
32#393—21no—
33#390—21no—
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#388—21no—
37#387—21no—
38#382—21no—
39#380C2211yescharacteristic
40#380C2211yescharacteristic
41#380C2211yescharacteristic
42#380C2211yescharacteristic
43#380C2211yescharacteristic
44#380C2211yescharacteristic
45#380C2211yescharacteristic
46C4×C4⋊C4C2211yescharacteristic
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—
60C22×D4D411yescharacteristic
61C23×C4C2311yescharacteristic
62C23×C4C2311yescharacteristic
63C22×C8—22no—
64C22×C8—22no—
65C22×C8—41no—
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C4⋊Q8—41no—
71C4⋊Q8—41no—
72C4×D4—41no—
73C4×D4—41no—
74#117—41no—
75#117—41no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
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⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4D411yescharacteristic
97C2×C22⋊C4D411yescharacteristic
98C2×C42—21no—
99C2×C42—21no—
100C2×C42C2311yescharacteristic
101#107—22no—
102#107—41no—
103#107—41no—
104#106—22no—
105#106—41no—
106#106—41no—
107C22⋊C8—22no—
108C22⋊C8—22no—
109C22⋊C8—41no—
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
121C24C2×D411yescharacteristic
122C2×Q8—42no—
123C2×Q8—81no—
124C2×Q8—81no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—41no—
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—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
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#4112yes—
165C2×C8#4112yes—
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—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—41no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—81no—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—81no—
191C22⋊C4—81no—
192C22⋊C4—81no—
193C22⋊C4—81no—
194C22⋊C4—81no—
195C22⋊C4—81no—
196C42—21no—
197C42—21no—
198C42—42no—
199C42—42no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23#13712yes—
214C23C22×D411yescharacteristic
215Q8—84no—
216D4—44no—
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—21no—
227C2×C4—21no—
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—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4#13512yes—
261C2×C4C2×#4111yescharacteristic
262C2×C4C2×#4111yescharacteristic
263C2×C4H(5,𝔽2)11yescharacteristic
264C2×C4H(5,𝔽2)11yescharacteristic
265C2×C42−1+411yescharacteristic
266C2×C42−1+411yescharacteristic
267C8—22no—
268C8—22no—
269C8—41no—
270C8—41no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22#53411yescharacteristic
286C22#54711yescharacteristic
287C22#54711yescharacteristic
288C22C2×#13511yescharacteristic
289C22C2×#13711yescharacteristic
290C22#57611yescharacteristic
291C22#57711yescharacteristic
292C4—22no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4—81no—
303C4—81no—
304C4#54211yescharacteristic
305C4#54211yescharacteristic
306C4#55111yescharacteristic
307C4#56311yescharacteristic
308C2—42no—
309C2—81no—
310C2#271711yescharacteristic
311C2#319011yescharacteristic
312C2#319811yescharacteristic
313C2#334011yescharacteristic
314C2#334111yescharacteristic
315C2#334411yescharacteristic
316C2#334611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2077C2semidirect
4. #2077C2semidirect
5. #2076C2semidirect
6. #2075C2semidirect
7. #2058C2semidirect
8. #2057C2semidirect
9. #1920C2semidirect
10. #1918C2semidirect
12. #1854C2semidirect
13. #1854C2semidirect
14. #1808C2semidirect
15. #1807C2semidirect

Maximal subgroups

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

subgroupindex
#25632
#24852
#20772
#20762
#20752
#20582
#20572
#19202
#19182
#18692
#18542
#18082
#18072