Finite Group Databasebeta

← #30824

Subgroups — #30824

Order 256.

383 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#489C211yesmaximal, characteristic
2C2×#475C211yesmaximal, characteristic, complemented
3#2551C211yesmaximal, characteristic, complemented
4#2448C211yesmaximal, characteristic
5#2094C211yesmaximal, characteristic
6#2072C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic
8#2058C211yesmaximal, characteristic, complemented
9#2042C211yesmaximal, characteristic
10#2023C211yesmaximal, characteristic
11#1943C211yesmaximal, characteristic, complemented
12#1928C211yesmaximal, characteristic, complemented
13#1926C211yesmaximal, characteristic
14#1917C211yesmaximal, characteristic, complemented
15#1865C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8—21no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21#489—24no—
22#475—24no—
23C2×#107C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#97C2211yescharacteristic
37#399—21no—
38#397—21no—
39#395—21no—
40#395—21no—
41#391C2211yescharacteristic
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#389—21no—
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#389C2211yescharacteristic
48#388—21no—
49#388—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×Q8—21no—
62C22×Q8—21no—
63C22×Q8C2311yescharacteristic
64C22×D4C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C4⋊Q8—22no—
70C4⋊Q8—22no—
71C4⋊Q8—41no—
72C4⋊Q8—41no—
73C4⋊Q8—44no—
74#124—22no—
75#124—41no—
76C4×Q8—22no—
77C4×Q8—41no—
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⋊C4—21no—
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105#107—22no—
106#106—22no—
107#106—22no—
108#106—41no—
109C4⋊C8—22no—
110#103—22no—
111#103—22no—
112#103—22no—
113#103D414yes—
114#102—22no—
115#102—22no—
116#102—22no—
117#102—41no—
118#97—22no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95D411yescharacteristic
126#95D411yescharacteristic
127C24—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—81no—
138C2×Q8—81no—
139C2×Q8C2×D412yes—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2411yescharacteristic
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8C2×D412yes—
172C4⋊C4—22no—
173C4⋊C4—22no—
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—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—81no—
191C4⋊C4—81no—
192C4⋊C4C2×D412yes—
193C4⋊C4#4112yes—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—81no—
199C22⋊C4—81no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—42no—
207C42—42no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23C22×D411yescharacteristic
216Q8—22no—
217Q8—22no—
218Q8—41no—
219Q8—41no—
220Q8—84no—
221Q8—84no—
222D4—41no—
223D4—41no—
224D4—41no—
225D4—41no—
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—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
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—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4AGL(1,ℤ/8)12yes—
264C2×C4#13712yes—
265C2×C4C22×D411yescharacteristic
266C2×C4C2×#4111yescharacteristic
267C2×C4H(5,𝔽2)11yescharacteristic
268C2×C42−1+411yescharacteristic
269C2×C42−1+411yescharacteristic
270C2×C42−1+411yescharacteristic
271C8—22no—
272C8—41no—
273C8—41no—
274C8—41no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22#53611yescharacteristic
283C22D4211yescharacteristic
284C22#54811yescharacteristic
285C22C2×AGL(1,ℤ/8)11yescharacteristic
286C22C2×#13711yescharacteristic
287C22#57711yescharacteristic
288C22#57711yescharacteristic
289C4—22no—
290C4—22no—
291C4—42no—
292C4—42no—
293C4—41no—
294C4—41no—
295C4—42no—
296C4—41no—
297C4—41no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4#53611yescharacteristic
303C4#54211yescharacteristic
304C4#54711yescharacteristic
305C4#56311yescharacteristic
306C2—161no—
307C2#267411yescharacteristic
308C2#323911yescharacteristic
309C2#324411yescharacteristic
310C2#326511yescharacteristic
311C2#327811yescharacteristic
312C2#336911yescharacteristic
313C2#337311yescharacteristic
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×#475C2semidirect
3. #2551C2semidirect
6. #2072C2semidirect
8. #2058C2semidirect
11. #1943C2semidirect
12. #1928C2semidirect
14. #1917C2semidirect
15. #1865C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4892
C2×#4752
#25512
#24482
#20942
#20722
#20672
#20582
#20422
#20232
#19432
#19282
#19262
#19172
#18652