Finite Group Databasebeta

← #26397

Subgroups — #26397

Order 256.

429 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#477C211yesmaximal, characteristic, complemented
2C2×#440C211yesmaximal, characteristic
3#2452C211yesmaximal, characteristic
4Q8×C4⋊C4C211yesmaximal, characteristic, complemented
5#2093C211yesmaximal, characteristic
6#2083C211yesmaximal, characteristic, complemented
7#2065C211yesmaximal, characteristic
8#2057C211yesmaximal, characteristic, complemented
9#1944C211yesmaximal, characteristic, complemented
10#1917C211yesmaximal, characteristic, complemented
11#1880C211yesmaximal, characteristic
12#1880C211yesmaximal, characteristic
13#1865C211yesmaximal, characteristic, complemented
14#1863C211yesmaximal, characteristic, complemented
15#1862C211yesmaximal, characteristic
16C22×SD16C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C22⋊Q8—21no—
19C2×C4×Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23#477—24no—
24#440—24no—
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#97C2211yescharacteristic
38#396—21no—
39#393—21no—
40#390C2211yescharacteristic
41#390C2211yescharacteristic
42#389—21no—
43#389—21no—
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#389C2211yescharacteristic
48#385—21no—
49#384—21no—
50#384C2211yescharacteristic
51C4×C4⋊C4—21no—
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—
65C2×SD16—22no—
66C2×SD16—22no—
67C2×SD16Q814yescomplemented
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C4⋊Q8—22no—
73C4⋊Q8—41no—
74C22⋊Q8—44no—
75C4×Q8—22no—
76C4×Q8—22no—
77C4×Q8—22no—
78C4×Q8—22no—
79C4×Q8—41no—
80C4×Q8—41no—
81C4×D4—22no—
82C4×D4—41no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4D411yescharacteristic
94C2×C4⋊C4D411yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111C2×C42C2311yescharacteristic
112#107—22no—
113#107—22no—
114#107—22no—
115#107—41no—
116#106—22no—
117C4⋊C8—22no—
118#103—22no—
119#103—22no—
120#103D414yes—
121#102—22no—
122#102—22no—
123#102—41no—
124#97—22no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95D411yescharacteristic
130#95D411yescharacteristic
131C24—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—81no—
138C2×Q8C2×D412yes—
139C2×Q8C2×Q812yes—
140C2×Q8#4112yes—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4C2×Q812yes—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—81no—
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2411yescharacteristic
164SD16—28no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8C2×D412yes—
176C2×C8C2×Q812yes—
177C4⋊C4—24no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—81no—
197C4⋊C4—81no—
198C4⋊C4—81no—
199C4⋊C4C2×D412yes—
200C4⋊C4#4112yes—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—81no—
206C42—24no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—42no—
214C42—42no—
215C23—22no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23C22×D411yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—22no—
226Q8—84no—
227Q8C22⋊Q814yescomplemented
228D4—22no—
229D4—22no—
230D4—41no—
231D4—41no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
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—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—84no—
270C2×C4AGL(1,ℤ/8)12yes—
271C2×C4#13712yes—
272C2×C4C22×D411yescharacteristic
273C2×C4C22×Q811yescharacteristic
274C2×C4C2×#4111yescharacteristic
275C2×C4C2×#4111yescharacteristic
276C2×C4H(5,𝔽2)11yescharacteristic
277C2×C42−1+411yescharacteristic
278C8—22no—
279C8—22no—
280C8—41no—
281C8—41no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22#52511yescharacteristic
290C22D4211yescharacteristic
291C22D4×Q811yescharacteristic
292C22C2×AGL(1,ℤ/8)11yescharacteristic
293C22C2×#13711yescharacteristic
294C22#57611yescharacteristic
295C22#57811yescharacteristic
296C4—22no—
297C4—24no—
298C4—22no—
299C4—41no—
300C4—42no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—42no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4—81no—
309C4C2×C22⋊Q811yescharacteristic
310C4#53911yescharacteristic
311C4#54711yescharacteristic
312C4#55411yescharacteristic
313C2—82no—
314C2#244911yescharacteristic
315C2#311211yescharacteristic
316C2#312411yescharacteristic
317C2#327011yescharacteristic
318C2#327411yescharacteristic
319C2#338511yescharacteristic
320C2#338611yescharacteristic
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×#477C2semidirect
4. Q8×C4⋊C4C2semidirect
6. #2083C2semidirect
8. #2057C2semidirect
9. #1944C2semidirect
10. #1917C2semidirect
13. #1865C2semidirect
14. #1863C2semidirect
67. C2×SD16Q8semidirect
227. Q8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4772
C2×#4402
#24522
Q8×C4⋊C42
#20932
#20832
#20652
#20572
#19442
#19172
#18802
#18652
#18632
#18622