Finite Group Databasebeta

← #24824

Subgroups — #24824

Order 256.

412 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#479C211yesmaximal, characteristic, complemented
2C2×#440C211yesmaximal, characteristic
3#2566C211yesmaximal, characteristic
4Q8×C4⋊C4C211yesmaximal, characteristic, complemented
5#2085C211yesmaximal, characteristic, complemented
6#2083C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic
8#2065C211yesmaximal, characteristic
9#1992C211yesmaximal, characteristic, complemented
10#1940C211yesmaximal, characteristic, complemented
11#1880C211yesmaximal, characteristic
12#1880C211yesmaximal, characteristic
13#1805C211yesmaximal, characteristic, complemented
14#1803C211yesmaximal, characteristic, complemented
15#1802C211yesmaximal, characteristic
16C22×SD16C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22#479—24no—
23#440—24no—
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#97C2211yescharacteristic
37#396—21no—
38#396—21no—
39#390C2211yescharacteristic
40#390C2211yescharacteristic
41#389—21no—
42#389—21no—
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#385—21no—
46#385—21no—
47#384—21no—
48#384C2211yescharacteristic
49#384C2211yescharacteristic
50#382C2211yescharacteristic
51C4×C4⋊C4—21no—
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C2×SD16—22no—
65C2×SD16—22no—
66C2×SD16Q814yes—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71#125—22no—
72#125—22no—
73C4×Q8—22no—
74C4×Q8—22no—
75C4×Q8—22no—
76C4×Q8—41no—
77C4×Q8—41no—
78C4×Q8D414yes—
79C4×D4—22no—
80C4×D4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
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⋊C4D411yescharacteristic
93C2×C4⋊C4D411yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111#107—22no—
112#107—22no—
113#106—22no—
114C4⋊C8—22no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117#103—22no—
118#103—22no—
119#103—22no—
120#102—22no—
121#102—41no—
122#102—41no—
123#97—22no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95D411yescharacteristic
129#95D411yescharacteristic
130C24—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8C2×D412yes—
135C2×Q8C2×Q812yes—
136C2×Q8#4112yes—
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
160SD16—28no—
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—
172C2×C8C2×Q812yes—
173C4⋊C4—24no—
174C4⋊C4—22no—
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—41no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—81no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C4⋊C4—81no—
196C4⋊C4—81no—
197C4⋊C4C2×D412yes—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—81no—
202C22⋊C4—81no—
203C42—24no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—42no—
209C42—42no—
210C42C2×D412yes—
211C42#4112yes—
212C23—22no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23C22×D411yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—22no—
223Q8C22⋊Q814yes—
224D4—22no—
225D4—22no—
226D4—41no—
227D4—41no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
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—84no—
265C2×C4#12112yes—
266C2×C4#12112yes—
267C2×C4#13712yes—
268C2×C4C22×D411yescharacteristic
269C2×C4C22×Q811yescharacteristic
270C2×C4C2×#4111yescharacteristic
271C2×C4C2×#4111yescharacteristic
272C2×C4H(5,𝔽2)11yescharacteristic
273C2×C42−1+411yescharacteristic
274C8—22no—
275C8—41no—
276C8—41no—
277C8—41no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22C2×#12111yescharacteristic
286C22#54611yescharacteristic
287C22D4×Q811yescharacteristic
288C22C2×#13711yescharacteristic
289C22#57511yescharacteristic
290C22#57611yescharacteristic
291C22#57711yescharacteristic
292C4—24no—
293C4—22no—
294C4—22no—
295C4—22no—
296C4—42no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—81no—
301C4—81no—
302C4—81no—
303C4—81no—
304C4—81no—
305C4C2×C22⋊Q811yescharacteristic
306C4#53711yescharacteristic
307C4#55211yescharacteristic
308C4#55411yescharacteristic
309C2—82no—
310C2#241711yescharacteristic
311C2#303711yescharacteristic
312C2#304511yescharacteristic
313C2#330911yescharacteristic
314C2#332011yescharacteristic
315C2#338511yescharacteristic
316C2#338811yescharacteristic
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×#479C2semidirect
4. Q8×C4⋊C4C2semidirect
5. #2085C2semidirect
6. #2083C2semidirect
9. #1992C2semidirect
10. #1940C2semidirect
13. #1805C2semidirect
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×#4402
#25662
Q8×C4⋊C42
#20852
#20832
#20672
#20652
#19922
#19402
#18802
#18052
#18032
#18022