Finite Group Databasebeta

← #26058

Subgroups — #26058

Order 256.

436 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#476C211yesmaximal, characteristic
2C2×#474C211yesmaximal, characteristic
3#2551C211yesmaximal, characteristic, complemented
4#2349C211yesmaximal, characteristic
5#2088C211yesmaximal, characteristic, complemented
6#2087C211yesmaximal, characteristic, complemented
7#2058C211yesmaximal, characteristic, complemented
8#2057C211yesmaximal, characteristic, complemented
9#1949C211yesmaximal, characteristic, complemented
10#1940C211yesmaximal, characteristic, complemented
11#1864C211yesmaximal, characteristic
12#1862C211yesmaximal, characteristic
13#1847C211yesmaximal, characteristic, complemented
14#1802C211yesmaximal, characteristic
15#1762C211yesmaximal, characteristic
16C2×C4⋊Q8—21no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22#476—24no—
23#474—24no—
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#97C2211yescharacteristic
38#395—21no—
39#395—21no—
40#389—21no—
41#389C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#385—21no—
46#385—21no—
47#384C2211yescharacteristic
48#382C2211yescharacteristic
49#380—21no—
50C4×C4⋊C4C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C2×#95—21no—
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8—21no—
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C4⋊Q8—22no—
72C4⋊Q8—22no—
73C4⋊Q8—41no—
74C4⋊Q8—41no—
75C4⋊Q8—44no—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—22no—
79C4×D4—41no—
80C4×D4—41no—
81C4×D4Q814yescomplemented
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⋊C4D411yescharacteristic
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C42—21no—
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106#107—22no—
107#107—22no—
108#107—41no—
109#106—22no—
110#106—22no—
111#106—41no—
112C4⋊C8—22no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115#102—22no—
116#102—22no—
117#102D414yes—
118#102Q814yescomplemented
119#97—22no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95D411yescharacteristic
128#95D411yescharacteristic
129C24—21no—
130C2×Q8—42no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8—42no—
134C2×Q8—81no—
135C2×Q8—81no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4C2×D412yes—
140C2×D4C2×Q812yes—
141C2×D4C2×Q812yes—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—24no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—81no—
155C22×C4—81no—
156C22×C4—81no—
157C22×C4—81no—
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2411yescharacteristic
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—22no—
178C4⋊C4—22no—
179C4⋊C4—21no—
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—81no—
196C4⋊C4—81no—
197C4⋊C4—81no—
198C4⋊C4C2×D412yes—
199C4⋊C4C2×Q812yes—
200C4⋊C4C2×Q812yes—
201C22⋊C4—24no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—42no—
209C42C2×Q812yes—
210C42#4112yes—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23C22×D411yescharacteristic
219Q8—84no—
220Q8—84no—
221D4—22no—
222D4—22no—
223D4—22no—
224D4C4⋊Q814yescomplemented
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
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—84no—
261C2×C4—84no—
262C2×C4—84no—
263C2×C4C22⋊Q812yes—
264C2×C4C22⋊Q812yes—
265C2×C4AGL(1,ℤ/8)12yes—
266C2×C4#13712yes—
267C2×C4C22×D411yescharacteristic
268C2×C4C22×Q811yescharacteristic
269C2×C4C22×Q811yescharacteristic
270C2×C4C2×#4111yescharacteristic
271C2×C4H(5,𝔽2)11yescharacteristic
272C2×C42−1+411yescharacteristic
273C8—22no—
274C8—22no—
275C8—41no—
276C8—41no—
277C22—24no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22C2×C22⋊Q811yescharacteristic
285C22D4211yescharacteristic
286C22D4×Q811yescharacteristic
287C22C2×AGL(1,ℤ/8)11yescharacteristic
288C22C2×#13711yescharacteristic
289C22#57611yescharacteristic
290C22#57711yescharacteristic
291C4—22no—
292C4—22no—
293C4—22no—
294C4—22no—
295C4—42no—
296C4—42no—
297C4—41no—
298C4—42no—
299C4—41no—
300C4—81no—
301C4—81no—
302C4—81no—
303C4—81no—
304C4C2×C4⋊Q811yescharacteristic
305C4#54411yescharacteristic
306C4#54711yescharacteristic
307C4#55411yescharacteristic
308C2—44no—
309C2#244011yescharacteristic
310C2#307911yescharacteristic
311C2#308111yescharacteristic
312C2#326411yescharacteristic
313C2#326611yescharacteristic
314C2#338411yescharacteristic
315C2#338511yescharacteristic
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. #2551C2semidirect
5. #2088C2semidirect
6. #2087C2semidirect
7. #2058C2semidirect
8. #2057C2semidirect
9. #1949C2semidirect
10. #1940C2semidirect
13. #1847C2semidirect
81. C4×D4Q8semidirect
118. #102Q8semidirect
224. D4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4762
C2×#4742
#25512
#23492
#20882
#20872
#20582
#20572
#19492
#19402
#18642
#18622
#18472
#18022
#17622