Finite Group Databasebeta

← #31288

Subgroups — #31288

Order 256.

417 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2496C211yesmaximal, characteristic
2#2434C211yesmaximal, characteristic
3#2096C211yesmaximal, characteristic, complemented
4#2066C211yesmaximal, characteristic, complemented
5#2043C211yesmaximal, characteristic, complemented
6#2043C211yesmaximal, characteristic, complemented
7#2035C211yesmaximal, characteristic
8#2035C211yesmaximal, characteristic
9#1981C211yesmaximal, characteristic, complemented
10#1956C211yesmaximal, characteristic, complemented
11#1933C211yesmaximal, characteristic, complemented
12#1927C211yesmaximal, characteristic, complemented
13#1923C211yesmaximal, characteristic
14#1787C211yesmaximal, characteristic
15#1787C211yesmaximal, characteristic
16C23×Q8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#126—21no—
20C2×C22⋊Q8C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C4×C8C2211yescharacteristic
39#400—21no—
40#397—21no—
41#395—21no—
42#395—21no—
43#391C2211yescharacteristic
44#391C2211yescharacteristic
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#387—21no—
48#385—21no—
49#385—21no—
50#384C2211yescharacteristic
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
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×Q8C2311yescharacteristic
69C23×C4—21no—
70C23×C4C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C4⋊Q8—22no—
76C4⋊Q8—22no—
77C4⋊Q8—41no—
78C4⋊Q8—41no—
79#126—44no—
80C22⋊Q8—41no—
81C22⋊Q8—41no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4D411yescharacteristic
100C2×C22⋊C4D411yescharacteristic
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104#106—41no—
105#106—41no—
106C4⋊C8—41no—
107#103—22no—
108#103—22no—
109#103—41no—
110#103—41no—
111#103—41no—
112#103—41no—
113#103—41no—
114#103—41no—
115#103—41no—
116#103—41no—
117#103—41no—
118#103—41no—
119C22⋊C8—41no—
120C22⋊C8—41no—
121C4×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#95D411yescharacteristic
132#95D411yescharacteristic
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—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—44no—
154C2×Q8—44no—
155C2×Q8—44no—
156C2×Q8—44no—
157C2×Q8#4112yes—
158C2×Q8#4112yes—
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—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2411yescharacteristic
182C2×C8—21no—
183C2×C8—21no—
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—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C22⋊C4—22no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4SD1614yes—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—42no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23C2×SD1612yes—
230C23C22×D411yescharacteristic
231Q8—22no—
232Q8—22no—
233Q8—22no—
234Q8—22no—
235Q8—48no—
236Q8—41no—
237Q8—41no—
238Q8—41no—
239Q8—41no—
240Q8—41no—
241Q8—41no—
242Q8—41no—
243Q8—41no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—48no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
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×C4C2×SD1612yes—
286C2×C4C2×SD1612yes—
287C2×C4#13712yes—
288C2×C4C2×#4111yescharacteristic
289C2×C4C2×#4111yescharacteristic
290C2×C4H(5,𝔽2)11yescharacteristic
291C2×C4H(5,𝔽2)11yescharacteristic
292C2×C4H(5,𝔽2)11yescharacteristic
293C2×C4H(5,𝔽2)11yescharacteristic
294C8—41no—
295C8—41no—
296C8—41no—
297C8—41no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22#53511yescharacteristic
306C22#54611yescharacteristic
307C22#54611yescharacteristic
308C22C22×SD1611yescharacteristic
309C22C2×#13711yescharacteristic
310C22#57811yescharacteristic
311C22#57811yescharacteristic
312C4—22no—
313C4—22no—
314C4—22no—
315C4—41no—
316C4—41no—
317C4—42no—
318C4—42no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4#53811yescharacteristic
326C4#53811yescharacteristic
327C4#55111yescharacteristic
328C4#56011yescharacteristic
329C2—42no—
330C2#268511yescharacteristic
331C2#320611yescharacteristic
332C2#322311yescharacteristic
333C2#329811yescharacteristic
334C2#329811yescharacteristic
335C2#331211yescharacteristic
336C2#332811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2096C2semidirect
4. #2066C2semidirect
5. #2043C2semidirect
6. #2043C2semidirect
9. #1981C2semidirect
10. #1956C2semidirect
11. #1933C2semidirect
12. #1927C2semidirect

Maximal subgroups

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

subgroupindex
#24962
#24342
#20962
#20662
#20432
#20352
#19812
#19562
#19332
#19272
#19232
#17872