Finite Group Databasebeta

← #28293

Subgroups — #28293

Order 256.

393 conjugacy classes of subgroups.

The subgroup lattice has 313 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#479C211yesmaximal, characteristic, complemented
2C2×#478C211yesmaximal, characteristic
3#2330C211yesmaximal, characteristic, complemented
4#2326C211yesmaximal, characteristic
5#2068C211yesmaximal, characteristic, complemented
6#2067C211yesmaximal, characteristic
7#2066C211yesmaximal, characteristic, complemented
8#2065C211yesmaximal, characteristic
9#1983C211yesmaximal, characteristic
10#1983C211yesmaximal, characteristic
11#1948C211yesmaximal, characteristic, complemented
12#1947C211yesmaximal, characteristic, complemented
13#1843C211yesmaximal, characteristic, complemented
14#1803C211yesmaximal, characteristic, complemented
15#1802C211yesmaximal, characteristic
16C2×#125C2211yescharacteristic
17C2×#124C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20#479—24no—
21#478—24no—
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40#391C2211yescharacteristic
41#390C2211yescharacteristic
42#389—21no—
43#389—21no—
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#388—21no—
47#388—21no—
48#387—21no—
49#387—21no—
50#385—21no—
51#385—21no—
52#384C2211yescharacteristic
53#384C2211yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55C4×C4⋊C4C2211yescharacteristic
56#376C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68#125—22no—
69#125D414yes—
70#124—22no—
71#124Q814yes—
72C4×Q8—22no—
73C4×Q8—22no—
74C4×D4—22no—
75C4×D4—22no—
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⋊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×C42—21no—
98C2×C42—21no—
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101C2×C42C2311yescharacteristic
102#107—22no—
103#107—22no—
104#106—22no—
105#106—22no—
106C4⋊C8—22no—
107C4⋊C8—22no—
108C4⋊C8—22no—
109C4⋊C8—22no—
110C4⋊C8—41no—
111C4⋊C8—41no—
112#103—22no—
113#103—22no—
114#103—41no—
115#103—41no—
116#102—22no—
117#102—22no—
118#102—41no—
119#102—41no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95D411yescharacteristic
127#95D411yescharacteristic
128#95D411yescharacteristic
129#95D411yescharacteristic
130C24—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8C2×Q812yes—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4C2×Q812yes—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—42no—
151C22×C4—81no—
152C22×C4—81no—
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×C8—21no—
172C4⋊C4—24no—
173C4⋊C4—24no—
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—81no—
186C4⋊C4—81no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—81no—
190C4⋊C4—81no—
191C4⋊C4C2×D412yes—
192C4⋊C4C2×D412yes—
193C22⋊C4—24no—
194C22⋊C4—24no—
195C22⋊C4—42no—
196C22⋊C4—81no—
197C22⋊C4—81no—
198C42—21no—
199C42—21no—
200C42—42no—
201C42—81no—
202C42—81no—
203C42C2×D412yes—
204C42C2×Q812yes—
205C42#4112yes—
206C42#4112yes—
207C23—22no—
208C23—41no—
209C23—41no—
210C23—41no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23C22×D411yescharacteristic
215Q8—22no—
216Q8—22no—
217Q8—41no—
218Q8—41no—
219D4—22no—
220D4—22no—
221D4—41no—
222D4—41no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
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—84no—
259C2×C4#12112yes—
260C2×C4#12112yes—
261C2×C4C22⋊Q812yes—
262C2×C4C22⋊Q812yes—
263C2×C4C22×D411yescharacteristic
264C2×C4C22×Q811yescharacteristic
265C2×C4C2×#4111yescharacteristic
266C2×C4C2×#4111yescharacteristic
267C2×C4H(5,𝔽2)11yescharacteristic
268C2×C42−1+411yescharacteristic
269C8—41no—
270C8—41no—
271C8—41no—
272C8—41no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22C2×#12111yescharacteristic
281C22C2×C22⋊Q811yescharacteristic
282C22#53711yescharacteristic
283C22#57611yescharacteristic
284C22#57711yescharacteristic
285C22#57711yescharacteristic
286C22#57811yescharacteristic
287C4—22no—
288C4—22no—
289C4—22no—
290C4—22no—
291C4—41no—
292C4—41no—
293C4—42no—
294C4—42no—
295C4—42no—
296C4—81no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4D4×Q811yescharacteristic
301C4#55011yescharacteristic
302C4#55411yescharacteristic
303C4#55611yescharacteristic
304C2—82no—
305C2#253811yescharacteristic
306C2#304511yescharacteristic
307C2#304611yescharacteristic
308C2#308311yescharacteristic
309C2#308411yescharacteristic
310C2#325711yescharacteristic
311C2#325811yescharacteristic
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×#479C2semidirect
3. #2330C2semidirect
5. #2068C2semidirect
7. #2066C2semidirect
11. #1948C2semidirect
12. #1947C2semidirect
13. #1843C2semidirect
14. #1803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4792
C2×#4782
#23302
#23262
#20682
#20672
#20662
#20652
#19832
#19482
#19472
#18432
#18032
#18022