Finite Group Databasebeta

← #24713

Subgroups — #24713

Order 256.

397 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#478C211yesmaximal, characteristic, complemented
2C2×#433C211yesmaximal, characteristic, complemented
3#2629C211yesmaximal, characteristic, complemented
4#2324C211yesmaximal, characteristic
5#2077C211yesmaximal, characteristic, complemented
6#2076C211yesmaximal, characteristic, complemented
7#2033C211yesmaximal, characteristic
8#2033C211yesmaximal, characteristic
9#1989C211yesmaximal, characteristic, complemented
10#1947C211yesmaximal, characteristic, complemented
11#1943C211yesmaximal, characteristic, complemented
12#1942C211yesmaximal, characteristic, complemented
13#1936C211yesmaximal, characteristic, complemented
14#1934C211yesmaximal, characteristic, complemented
15#1802C211yesmaximal, characteristic, complemented
16C2×#125C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×#117C2211yescharacteristic
20#478—24no—
21#433—24no—
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×#97C2211yescharacteristic
35C2×C4×C8C2211yescharacteristic
36#398—21no—
37#398—21no—
38#396—21no—
39#396—21no—
40#390—21no—
41#390C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#388—21no—
46#386—21no—
47#385—21no—
48#384C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×D4—21no—
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69#125—22no—
70#125—22no—
71#121—22no—
72#121—22no—
73C4×D4—22no—
74C4×D4—22no—
75#117—22no—
76#117—22no—
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⋊C4D411yescharacteristic
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C22⋊C4—21no—
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×C22⋊C4D411yescharacteristic
101C2×C22⋊C4D411yescharacteristic
102C2×C42—21no—
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105#107—22no—
106#107—22no—
107#107—22no—
108#106—22no—
109C4⋊C8—22no—
110C4⋊C8D414yescomplemented
111#102—22no—
112#102—22no—
113#102—22no—
114#102—22no—
115C22⋊C8—22no—
116C22⋊C8—22no—
117#97—22no—
118C4×C8—22no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126C24—21no—
127C24C2×D411yescharacteristic
128C2×D4—24no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—81no—
135C2×D4—81no—
136C2×D4#4112yes—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
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×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×C8C2×D412yes—
170C2×C8C2×D412yes—
171C4⋊C4—22no—
172C4⋊C4—22no—
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—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—81no—
189C4⋊C4—81no—
190C4⋊C4—81no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4#4112yes—
194C22⋊C4—24no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—42no—
198C22⋊C4—81no—
199C22⋊C4—81no—
200C22⋊C4—81no—
201C22⋊C4—81no—
202C22⋊C4—81no—
203C42—21no—
204C42—21no—
205C42—81no—
206C42C2×D412yes—
207C42#4112yes—
208C23—21no—
209C23—22no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23C22×D411yescharacteristic
223D4—22no—
224D4—22no—
225D4—41no—
226D4—41no—
227D4—84no—
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—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
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—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—84no—
267C2×C4#12112yes—
268C2×C4#12112yes—
269C2×C4C22×D411yescharacteristic
270C2×C4C2×#4111yescharacteristic
271C2×C4C2×#4111yescharacteristic
272C2×C4C2×#4111yescharacteristic
273C2×C42−1+411yescharacteristic
274C2×C42−1+411yescharacteristic
275C8—22no—
276C8—22no—
277C8—41no—
278C8—41no—
279C22—41no—
280C22—42no—
281C22—41no—
282C22—42no—
283C22—41no—
284C22—42no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22C2×#12111yescharacteristic
294C22#52511yescharacteristic
295C22#53611yescharacteristic
296C22#57611yescharacteristic
297C22#57611yescharacteristic
298C22#57711yescharacteristic
299C22#57811yescharacteristic
300C4—22no—
301C4—22no—
302C4—42no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—81no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4—81no—
311C4—81no—
312C4#54711yescharacteristic
313C4#54811yescharacteristic
314C4#55211yescharacteristic
315C4#55611yescharacteristic
316C2—81no—
317C2—82no—
318C2#241511yescharacteristic
319C2#306111yescharacteristic
320C2#306211yescharacteristic
321C2#312011yescharacteristic
322C2#312211yescharacteristic
323C2#324111yescharacteristic
324C2#324311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#478C2semidirect
2. C2×#433C2semidirect
3. #2629C2semidirect
5. #2077C2semidirect
6. #2076C2semidirect
9. #1989C2semidirect
10. #1947C2semidirect
11. #1943C2semidirect
12. #1942C2semidirect
13. #1936C2semidirect
14. #1934C2semidirect
15. #1802C2semidirect
110. C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4782
C2×#4332
#26292
#23242
#20772
#20762
#20332
#19892
#19472
#19432
#19422
#19362
#19342
#18022