Finite Group Databasebeta

← #30818

Subgroups — #30818

Order 256.

360 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#488C211yesmaximal, characteristic
2C2×#479C211yesmaximal, characteristic, complemented
3#2552C211yesmaximal, characteristic, complemented
4#2448C211yesmaximal, characteristic
5#2093C211yesmaximal, characteristic
6#2072C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic
8#2059C211yesmaximal, characteristic, complemented
9#2042C211yesmaximal, characteristic
10#2025C211yesmaximal, characteristic
11#1942C211yesmaximal, characteristic, complemented
12#1928C211yesmaximal, characteristic, complemented
13#1926C211yesmaximal, characteristic
14#1916C211yesmaximal, characteristic, complemented
15#1863C211yesmaximal, characteristic, complemented
16C2×#125C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20#488—24no—
21#479—24no—
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×C4×C8C2211yescharacteristic
36#399—21no—
37#398—21no—
38#398—21no—
39#397—21no—
40#391C2211yescharacteristic
41#390C2211yescharacteristic
42#390C2211yescharacteristic
43#389—21no—
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#388—21no—
48#388—21no—
49#384—21no—
50#384C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67#125—22no—
68#125—22no—
69#125—41no—
70#125—41no—
71#124—22no—
72#124—41no—
73C4×Q8—22no—
74C4×Q8—41no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
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⋊C4D411yescharacteristic
86C2×C4⋊C4D411yescharacteristic
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×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104#107—22no—
105#107—22no—
106#107—41no—
107#106—22no—
108C4⋊C8—22no—
109#103—22no—
110#103—22no—
111#103—22no—
112#103D414yes—
113#102—22no—
114#102—22no—
115#102—22no—
116#102—41no—
117C4×C8—22no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95D411yescharacteristic
125#95D411yescharacteristic
126C24—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8C2×D412yes—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
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—21no—
164C2×C8C2×D412yes—
165C4⋊C4—22no—
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—41no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—81no—
184C4⋊C4—81no—
185C4⋊C4—81no—
186C4⋊C4—81no—
187C4⋊C4C2×D412yes—
188C4⋊C4#4112yes—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—81no—
194C22⋊C4—81no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—42no—
202C42—42no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23C22×D411yescharacteristic
211Q8—22no—
212Q8—22no—
213Q8—41no—
214Q8—41no—
215D4—41no—
216D4—41no—
217D4—41no—
218D4—41no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
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#13512yes—
258C2×C4C22×D411yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C2×C4H(5,𝔽2)11yescharacteristic
261C2×C42−1+411yescharacteristic
262C2×C42−1+411yescharacteristic
263C2×C42−1+411yescharacteristic
264C8—22no—
265C8—41no—
266C8—41no—
267C8—41no—
268C22—81no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22#53611yescharacteristic
276C22D4211yescharacteristic
277C22#54811yescharacteristic
278C22C2×#13511yescharacteristic
279C22#57511yescharacteristic
280C22#57611yescharacteristic
281C22#57811yescharacteristic
282C4—22no—
283C4—42no—
284C4—42no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—42no—
289C4—41no—
290C4—41no—
291C4—81no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4#53611yescharacteristic
296C4#54211yescharacteristic
297C4#54711yescharacteristic
298C4#56311yescharacteristic
299C2—161no—
300C2#267411yescharacteristic
301C2#323211yescharacteristic
302C2#324311yescharacteristic
303C2#326911yescharacteristic
304C2#328711yescharacteristic
305C2#336411yescharacteristic
306C2#337011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#479C2semidirect
3. #2552C2semidirect
6. #2072C2semidirect
8. #2059C2semidirect
11. #1942C2semidirect
12. #1928C2semidirect
14. #1916C2semidirect
15. #1863C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4882
C2×#4792
#25522
#24482
#20932
#20722
#20672
#20592
#20422
#20252
#19422
#19282
#19262
#19162
#18632