Finite Group Databasebeta

← #31267

Subgroups — #31267

Order 256.

428 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2562C211yesmaximal, characteristic
2#2409C211yesmaximal, characteristic
3#2077C211yesmaximal, characteristic, complemented
4#2075C211yesmaximal, characteristic, complemented
5#2036C211yesmaximal, characteristic, complemented
6#2035C211yesmaximal, characteristic, complemented
7#2026C211yesmaximal, characteristic, complemented
8#2026C211yesmaximal, characteristic, complemented
9#1928C211yesmaximal, characteristic, complemented
10#1927C211yesmaximal, characteristic, complemented
11#1892C211yesmaximal, characteristic, complemented
12#1892C211yesmaximal, characteristic, complemented
13#1869C211yesmaximal, characteristic
14#1787C211yesmaximal, characteristic, complemented
15#1787C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35#400—21no—
36#395—21no—
37#394—21no—
38#393—21no—
39#391C2211yescharacteristic
40#391C2211yescharacteristic
41#388—21no—
42#388—21no—
43#386—21no—
44#385—21no—
45#380C2211yescharacteristic
46#380C2211yescharacteristic
47#380C2211yescharacteristic
48#380C2211yescharacteristic
49#380C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×Q8—22no—
61C22×Q8—22no—
62C22×Q8—41no—
63C22×Q8—41no—
64C22×Q8—41no—
65C22×Q8—41no—
66C22×Q8C2311yescharacteristic
67C22×Q8C2311yescharacteristic
68C22×D4D411yescharacteristic
69C23×C4—21no—
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C22×C8—41no—
73C22×C8—41no—
74C22×C8—41no—
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79#125—41no—
80#125—41no—
81C22⋊Q8—22no—
82C22⋊Q8—22no—
83C22⋊Q8—41no—
84C22⋊Q8—41no—
85C4×D4—41no—
86C4×D4—41no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4D411yescharacteristic
106C2×C22⋊C4D411yescharacteristic
107C2×C42—21no—
108C2×C42C2311yescharacteristic
109#107—41no—
110#106—41no—
111#103—22no—
112#103—22no—
113#103—41no—
114#103—41no—
115#103—41no—
116#103—41no—
117#103—41no—
118#103—41no—
119C22⋊C8—41no—
120C22⋊C8—41no—
121C22⋊C8—41no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132C24C2×D411yescharacteristic
133C24C2×D411yescharacteristic
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—42no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—44no—
148C2×Q8—44no—
149C2×Q8—44no—
150C2×Q8—44no—
151C2×Q8#4112yes—
152C2×Q8#4112yes—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—41no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2411yescharacteristic
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—44no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—81no—
222C22⋊C4—81no—
223C42—21no—
224C42—21no—
225C42—81no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23#13512yes—
239C23#13712yes—
240C23C22×D411yescharacteristic
241Q8—22no—
242Q8—22no—
243Q8—22no—
244Q8—22no—
245Q8—48no—
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8—41no—
250D4—44no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—48no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4C2×#4111yescharacteristic
297C2×C4C2×#4111yescharacteristic
298C2×C4H(5,𝔽2)11yescharacteristic
299C2×C4H(5,𝔽2)11yescharacteristic
300C2×C4H(5,𝔽2)11yescharacteristic
301C2×C4H(5,𝔽2)11yescharacteristic
302C8—41no—
303C8—41no—
304C8—41no—
305C8—41no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22#53411yescharacteristic
321C22#54611yescharacteristic
322C22#54611yescharacteristic
323C22C2×#13511yescharacteristic
324C22C2×#13711yescharacteristic
325C22#57711yescharacteristic
326C22#57811yescharacteristic
327C4—41no—
328C4—41no—
329C4—41no—
330C4—42no—
331C4—42no—
332C4—41no—
333C4—81no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4#53811yescharacteristic
340C4#53811yescharacteristic
341C4#55111yescharacteristic
342C4H(3,𝔽4)11yescharacteristic
343C2—42no—
344C2—42no—
345C2#268411yescharacteristic
346C2#319011yescharacteristic
347C2#320111yescharacteristic
348C2#330011yescharacteristic
349C2#330111yescharacteristic
350C2#331211yescharacteristic
351C2#332011yescharacteristic
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. #2075C2semidirect
5. #2036C2semidirect
6. #2035C2semidirect
7. #2026C2semidirect
8. #2026C2semidirect
9. #1928C2semidirect
10. #1927C2semidirect
11. #1892C2semidirect
12. #1892C2semidirect
14. #1787C2semidirect
15. #1787C2semidirect

Maximal subgroups

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

subgroupindex
#25622
#24092
#20772
#20752
#20362
#20352
#20262
#19282
#19272
#18922
#18692
#17872