Finite Group Databasebeta

← #28668

Subgroups — #28668

Order 256.

366 conjugacy classes of subgroups.

The subgroup lattice has 307 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#501C211yesmaximal, characteristic, complemented
2C2×#488C211yesmaximal, characteristic
3#2640C211yesmaximal, characteristic
4#2330C211yesmaximal, characteristic, complemented
5#2068C211yesmaximal, characteristic, complemented
6#2067C211yesmaximal, characteristic
7#2066C211yesmaximal, characteristic, complemented
8#2065C211yesmaximal, characteristic
9#1990C211yesmaximal, characteristic, complemented
10#1959C211yesmaximal, characteristic
11#1928C211yesmaximal, characteristic, complemented
12#1927C211yesmaximal, characteristic, complemented
13#1926C211yesmaximal, characteristic
14#1925C211yesmaximal, characteristic
15#1768C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×#124C2211yescharacteristic
19#501—24no—
20#488—24no—
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#97C2211yescharacteristic
36C2×C4×C8C2211yescharacteristic
37#397—21no—
38#397—21no—
39#393—21no—
40#393—21no—
41#391C2211yescharacteristic
42#391C2211yescharacteristic
43#390C2211yescharacteristic
44#390C2211yescharacteristic
45#389—21no—
46#389—21no—
47#389C2211yescharacteristic
48#389C2211yescharacteristic
49#387—21no—
50#387—21no—
51#384C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54#376C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C4⋊Q8—22no—
69C4⋊Q8—22no—
70#125—22no—
71#125D414yes—
72#124—22no—
73#124—22no—
74C2×C4⋊C4—21no—
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⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42C2311yescharacteristic
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98#107—22no—
99#107—22no—
100#106—22no—
101#106—22no—
102C4⋊C8—22no—
103C4⋊C8—22no—
104#103—22no—
105#103—22no—
106#103—22no—
107#103—22no—
108#102—22no—
109#102—22no—
110#102—22no—
111#102—22no—
112#97—22no—
113C4×C8—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#95D411yescharacteristic
123#95D411yescharacteristic
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—81no—
135C2×Q8—81no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2411yescharacteristic
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×C8—21no—
166C2×C8#4112yes—
167C2×C8#4112yes—
168C4⋊C4—24no—
169C4⋊C4—24no—
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—81no—
182C4⋊C4—81no—
183C4⋊C4—81no—
184C4⋊C4—81no—
185C4⋊C4C2×D412yes—
186C4⋊C4C2×D412yes—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—81no—
190C22⋊C4—81no—
191C22⋊C4—81no—
192C22⋊C4—81no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—81no—
198C42—81no—
199C42C2×D412yes—
200C42#4112yes—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23C22×D411yescharacteristic
209Q8—41no—
210Q8—41no—
211Q8—41no—
212Q8—41no—
213Q8—84no—
214D4—41no—
215D4—41no—
216D4—41no—
217D4—41no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
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—21no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—41no—
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#12112yes—
256C2×C4#12112yes—
257C2×C4C22×D411yescharacteristic
258C2×C4C2×#4111yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C2×C4C2×#4111yescharacteristic
261C2×C42−1+411yescharacteristic
262C2×C42−1+411yescharacteristic
263C8—22no—
264C8—22no—
265C8—41no—
266C8—41no—
267C22—81no—
268C22—81no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22C2×#12111yescharacteristic
275C22#53211yescharacteristic
276C22#53611yescharacteristic
277C22#57611yescharacteristic
278C22#57711yescharacteristic
279C22#57711yescharacteristic
280C22#57811yescharacteristic
281C4—22no—
282C4—22no—
283C4—41no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—42no—
288C4—42no—
289C4—81no—
290C4—81no—
291C4—81no—
292C4—81no—
293C4—81no—
294C4#54811yescharacteristic
295C4#54811yescharacteristic
296C4#55611yescharacteristic
297C4#55611yescharacteristic
298C2—161no—
299C2#254511yescharacteristic
300C2#304511yescharacteristic
301C2#304611yescharacteristic
302C2#317811yescharacteristic
303C2#317911yescharacteristic
304C2#324311yescharacteristic
305C2#324411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#501C2semidirect
4. #2330C2semidirect
5. #2068C2semidirect
7. #2066C2semidirect
9. #1990C2semidirect
11. #1928C2semidirect
12. #1927C2semidirect
15. #1768C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5012
C2×#4882
#26402
#23302
#20682
#20672
#20662
#20652
#19902
#19592
#19282
#19272
#19262
#19252
#17682