Finite Group Databasebeta

← #30962

Subgroups — #30962

Order 256.

388 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#501C211yesmaximal, characteristic, complemented
2C2×#474C211yesmaximal, characteristic
3#2551C211yesmaximal, characteristic, complemented
4#2454C211yesmaximal, characteristic
5#2094C211yesmaximal, characteristic
6#2088C211yesmaximal, characteristic, complemented
7#2083C211yesmaximal, characteristic, complemented
8#2059C211yesmaximal, characteristic, complemented
9#2058C211yesmaximal, characteristic, complemented
10#2023C211yesmaximal, characteristic
11#1936C211yesmaximal, characteristic
12#1926C211yesmaximal, characteristic
13#1921C211yesmaximal, characteristic, complemented
14#1917C211yesmaximal, characteristic, complemented
15#1862C211yesmaximal, characteristic
16C2×C4⋊Q8—21no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#125C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21#501—24no—
22#474—24no—
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#97C2211yescharacteristic
35#399—21no—
36#396—21no—
37#395—21no—
38#395—21no—
39#390C2211yescharacteristic
40#390C2211yescharacteristic
41#389—21no—
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#385—21no—
47#384C2211yescharacteristic
48#380—21no—
49C4×C4⋊C4C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8—21no—
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C4⋊Q8—22no—
70C4⋊Q8—22no—
71C4⋊Q8—41no—
72C4⋊Q8—41no—
73C4⋊Q8—44no—
74#125—22no—
75#125—41no—
76C4×D4—22no—
77C4×D4—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⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4D411yescharacteristic
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C42—21no—
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106#107—22no—
107#107—22no—
108#107D414yes—
109#106—22no—
110#106—22no—
111#106—41no—
112C4⋊C8—22no—
113#102—22no—
114#102—22no—
115#102—41no—
116#102Q814yescomplemented
117#97—22no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95D411yescharacteristic
124#95D411yescharacteristic
125C24—21no—
126C2×Q8—42no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8—42no—
130C2×Q8—81no—
131C2×Q8—81no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4C2×Q812yes—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—81no—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2411yescharacteristic
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×C8—21no—
165C2×C8C2×D412yes—
166C2×C8C2×Q812yes—
167C4⋊C4—22no—
168C4⋊C4—22no—
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—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—81no—
190C4⋊C4C2×D412yes—
191C4⋊C4C2×D412yes—
192C4⋊C4C2×Q812yes—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—81no—
196C22⋊C4—81no—
197C22⋊C4—81no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—42no—
205C23—22no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23C22×D411yescharacteristic
213Q8—84no—
214Q8—84no—
215D4—22no—
216D4—22no—
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—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
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—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—84no—
257C2×C4AGL(1,ℤ/8)12yes—
258C2×C4#13712yes—
259C2×C4C22×D411yescharacteristic
260C2×C4C22×Q811yescharacteristic
261C2×C4H(5,𝔽2)11yescharacteristic
262C2×C42−1+411yescharacteristic
263C2×C42−1+411yescharacteristic
264C2×C42−1+411yescharacteristic
265C8—22no—
266C8—22no—
267C8—41no—
268C8—41no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22#53611yescharacteristic
277C22D4211yescharacteristic
278C22D4×Q811yescharacteristic
279C22C2×AGL(1,ℤ/8)11yescharacteristic
280C22C2×#13711yescharacteristic
281C22#57611yescharacteristic
282C22#57711yescharacteristic
283C4—22no—
284C4—22no—
285C4—42no—
286C4—41no—
287C4—41no—
288C4—42no—
289C4—42no—
290C4—41no—
291C4—41no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4—81no—
296C4#53611yescharacteristic
297C4#54411yescharacteristic
298C4D4×Q811yescharacteristic
299C4#56311yescharacteristic
300C2—82no—
301C2#267711yescharacteristic
302C2#323911yescharacteristic
303C2#324111yescharacteristic
304C2#326411yescharacteristic
305C2#327811yescharacteristic
306C2#338411yescharacteristic
307C2#338511yescharacteristic
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
1. C2×#501C2semidirect
3. #2551C2semidirect
6. #2088C2semidirect
7. #2083C2semidirect
8. #2059C2semidirect
9. #2058C2semidirect
13. #1921C2semidirect
14. #1917C2semidirect
116. #102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5012
C2×#4742
#25512
#24542
#20942
#20882
#20832
#20592
#20582
#20232
#19362
#19262
#19212
#19172
#18622