Finite Group Databasebeta

← #26376

Subgroups — #26376

Order 256.

450 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#476C211yesmaximal, characteristic
2C2×C4×SD16C211yesmaximal, characteristic
3#2451C211yesmaximal, characteristic, complemented
4D4×C4⋊C4C211yesmaximal, characteristic
5#2096C211yesmaximal, characteristic, complemented
6#2084C211yesmaximal, characteristic, complemented
7#2066C211yesmaximal, characteristic, complemented
8#2058C211yesmaximal, characteristic, complemented
9#1941C211yesmaximal, characteristic, complemented
10#1916C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic
12#1877C211yesmaximal, characteristic
13#1865C211yesmaximal, characteristic, complemented
14#1864C211yesmaximal, characteristic
15#1864C211yesmaximal, characteristic
16C22×SD16C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C22×C4⋊C4—21no—
23#476—24no—
24C4×SD16—24no—
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×C4×C8C2211yescharacteristic
38#398—21no—
39#395—21no—
40#391—21no—
41#391C2211yescharacteristic
42#391C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#387—21no—
48#385—21no—
49#385—21no—
50#384C2211yescharacteristic
51#380—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—
65C23×C4—21no—
66C23×C4—21no—
67C2×SD16—22no—
68C2×SD16—22no—
69C2×SD16Q814yescomplemented
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C4⋊Q8—22no—
75C4⋊Q8—41no—
76C4×Q8—22no—
77C4×Q8—41no—
78C4×D4—22no—
79C4×D4—22no—
80C4×D4—22no—
81C4×D4—22no—
82C4×D4—41no—
83C4×D4—41no—
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—44no—
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×C42—21no—
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108#106—22no—
109#106—22no—
110#106—22no—
111#106—22no—
112#106—41no—
113C4⋊C8—22no—
114#103—22no—
115#103—22no—
116#103—41no—
117#102—22no—
118#102—22no—
119#102D414yes—
120C4×C8—22no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
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—21no—
136C2×Q8—21no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—81no—
140C2×Q8C2×Q812yes—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4C2×D412yes—
145C2×D4C2×Q812yes—
146C2×D4#4112yes—
147C22×C4—24no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—81no—
162C22×C4—81no—
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2411yescharacteristic
170SD16—28no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8C2×D412yes—
182C2×C8C2×Q812yes—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C4⋊C4—81no—
202C4⋊C4SD1614yescomplemented
203C4⋊C4C2×D412yes—
204C4⋊C4#4112yes—
205C22⋊C4—24no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—42no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23C22×D411yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—41no—
226Q8—41no—
227Q8—84no—
228D4—22no—
229D4—22no—
230D4—22no—
231D4C22⋊Q814yes—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—48no—
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—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—84no—
270C2×C4—84no—
271C2×C4C2×SD1612yes—
272C2×C4C2×SD1612yes—
273C2×C4C2×SD1612yes—
274C2×C4C22×D411yescharacteristic
275C2×C4C22×Q811yescharacteristic
276C2×C4C2×#4111yescharacteristic
277C2×C4C2×#4111yescharacteristic
278C2×C4H(5,𝔽2)11yescharacteristic
279C2×C42−1+411yescharacteristic
280C8—22no—
281C8—22no—
282C8—41no—
283C8—41no—
284C22—24no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22#52511yescharacteristic
292C22D4211yescharacteristic
293C22D4×Q811yescharacteristic
294C22C22×SD1611yescharacteristic
295C22#57511yescharacteristic
296C22#57711yescharacteristic
297C22#57711yescharacteristic
298C4—22no—
299C4—22no—
300C4—22no—
301C4—42no—
302C4—41no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—42no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4—81no—
311C4C2×C22⋊Q811yescharacteristic
312C4#53911yescharacteristic
313C4#54711yescharacteristic
314C4#55411yescharacteristic
315C2—44no—
316C2#244911yescharacteristic
317C2#310111yescharacteristic
318C2#312511yescharacteristic
319C2#326711yescharacteristic
320C2D4×SD1611yescharacteristic
321C2Q8×SD1611yescharacteristic
322C2#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
3. #2451C2semidirect
5. #2096C2semidirect
6. #2084C2semidirect
7. #2066C2semidirect
8. #2058C2semidirect
9. #1941C2semidirect
10. #1916C2semidirect
13. #1865C2semidirect
69. C2×SD16Q8semidirect
202. C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4762
C2×C4×SD162
#24512
D4×C4⋊C42
#20962
#20842
#20662
#20582
#19412
#19162
#18772
#18652
#18642