Finite Group Databasebeta

← #30831

Subgroups — #30831

Order 256.

368 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#489C211yesmaximal, characteristic
2C2×#478C211yesmaximal, characteristic
3#2550C211yesmaximal, characteristic, complemented
4#2447C211yesmaximal, characteristic
5#2097C211yesmaximal, characteristic, complemented
6#2073C211yesmaximal, characteristic
7#2068C211yesmaximal, characteristic, complemented
8#2057C211yesmaximal, characteristic, complemented
9#2042C211yesmaximal, characteristic
10#2026C211yesmaximal, characteristic, complemented
11#1943C211yesmaximal, characteristic, complemented
12#1928C211yesmaximal, characteristic, complemented
13#1925C211yesmaximal, characteristic
14#1917C211yesmaximal, characteristic, complemented
15#1862C211yesmaximal, characteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20#489—24no—
21#478—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×#97C2211yescharacteristic
36#398—21no—
37#397—21no—
38#397—21no—
39#395—21no—
40#391—21no—
41#391C2211yescharacteristic
42#391C2211yescharacteristic
43#390C2211yescharacteristic
44#389—21no—
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#389C2211yescharacteristic
48#385—21no—
49#382C2211yescharacteristic
50#381—21no—
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×Q8C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C4⋊Q8—22no—
70C4⋊Q8—41no—
71#125—22no—
72#125—41no—
73#124—22no—
74#124—41no—
75C4×D4—22no—
76C4×D4—41no—
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⋊C4D411yescharacteristic
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C42—21no—
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100#107—22no—
101#107—22no—
102#107—41no—
103#106—22no—
104C4⋊C8—22no—
105#103—22no—
106#103—22no—
107#103—22no—
108#103—41no—
109#102—22no—
110#102—22no—
111#102—22no—
112#102D414yes—
113#97—22no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
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×Q8—21no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—81no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4C2×D412yes—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—81no—
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2411yescharacteristic
159C2×C8—21no—
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×C8C2×D412yes—
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—21no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4C2×D412yes—
190C4⋊C4#4112yes—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C22⋊C4—81no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—81no—
203C23—22no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23C22×D411yescharacteristic
211Q8—41no—
212Q8—41no—
213Q8—41no—
214Q8—41no—
215Q8—84no—
216D4—22no—
217D4—22no—
218D4—41no—
219D4—41no—
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—21no—
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—21no—
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—41no—
258C2×C4—41no—
259C2×C4—84no—
260C2×C4C22×D411yescharacteristic
261C2×C4C2×#4111yescharacteristic
262C2×C4H(5,𝔽2)11yescharacteristic
263C2×C42−1+411yescharacteristic
264C2×C42−1+411yescharacteristic
265C2×C42−1+411yescharacteristic
266C8—22no—
267C8—41no—
268C8—41no—
269C8—41no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22#53611yescharacteristic
278C22D4211yescharacteristic
279C22#54811yescharacteristic
280C22#57511yescharacteristic
281C22#57511yescharacteristic
282C22#57611yescharacteristic
283C22#57811yescharacteristic
284C4—42no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—42no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4#53611yescharacteristic
298C4#54211yescharacteristic
299C4#54711yescharacteristic
300C4#56311yescharacteristic
301C2—82no—
302C2#267411yescharacteristic
303C2#324011yescharacteristic
304C2#324311yescharacteristic
305C2#326911yescharacteristic
306C2#327311yescharacteristic
307C2#337011yescharacteristic
308C2#337711yescharacteristic
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
3. #2550C2semidirect
5. #2097C2semidirect
7. #2068C2semidirect
8. #2057C2semidirect
10. #2026C2semidirect
11. #1943C2semidirect
12. #1928C2semidirect
14. #1917C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4892
C2×#4782
#25502
#24472
#20972
#20732
#20682
#20572
#20422
#20262
#19432
#19282
#19252
#19172
#18622