Finite Group Databasebeta

← #22709

Subgroups — #22709

Order 256.

485 conjugacy classes of subgroups.

The subgroup lattice has 337 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#435C211yesmaximal, characteristic
2#2352C211yesmaximal, characteristic
3D4×C4⋊C4C211yesmaximal, characteristic
4#1949C211yesmaximal, characteristic, complemented
5#1947C211yesmaximal, characteristic, complemented
6#1854C211yesmaximal, characteristic, complemented
7#1853C211yesmaximal, characteristic, complemented
8#1852C211yesmaximal, characteristic, complemented
9#1852C211yesmaximal, characteristic, complemented
10#1836C211yesmaximal, characteristic, complemented
11#1809C211yesmaximal, characteristic, complemented
12#1808C211yesmaximal, characteristic, complemented
13#1776C211yesmaximal, characteristic, complemented
14#1736C211yesmaximal, characteristic, complemented
15C2×#340C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×C4×D4—21no—
19C2×C4×D4C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24#435C418yescomplemented
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×#97C2211yescharacteristic
35#389—21no—
36#388—21no—
37#387—21no—
38#386—21no—
39#385—21no—
40#385—21no—
41#384C2211yescharacteristic
42#382C2211yescharacteristic
43#380C2211yescharacteristic
44#380C2211yescharacteristic
45#380C2211yescharacteristic
46#380C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C22⋊C4—21no—
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60C22×D4D411yescharacteristic
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C2×M16—41no—
67C2×M16—41no—
68C2×M16C2×C414yescomplemented
69C22×C8—41no—
70C22×C8—41no—
71C22×C8C2×C414yescomplemented
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C4×D4—44no—
77C4×D4C2×C414yes—
78C4×D4Q814yes—
79#117—22no—
80#117—41no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
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×C22⋊C4D411yescharacteristic
107C2×C22⋊C4D411yescharacteristic
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42C2311yescharacteristic
112#107—22no—
113#107—41no—
114#107—41no—
115#107—41no—
116#106—22no—
117#106—41no—
118C4⋊C8C2×C414yescomplemented
119C22⋊C8C2×C414yescomplemented
120C22⋊C8C2×C414yescomplemented
121#97C2×C414yescomplemented
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130C24C2×D411yescharacteristic
131C24C2×D411yescharacteristic
132C2×D4—22no—
133C2×D4—22no—
134C2×D4C4⋊C414yes—
135C22×C4—21no—
136C22×C4—21no—
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—21no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—81no—
158C22×C4—81no—
159C22×C4—81no—
160C22×C4C22×C412yes—
161C22×C4C22×C412yes—
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×Q812yes—
167C22×C4C2×Q812yes—
168C22×C4C2411yescharacteristic
169M16—44no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—44no—
179C2×C8C22×C412yes—
180C2×C8C22×C412yes—
181C2×C8C22×C412yes—
182C2×C8C22×C412yes—
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—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—81no—
202C4⋊C4C4⋊C414yes—
203C4⋊C4#4112yes—
204C4⋊C4#4112yes—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—81no—
210C22⋊C4C4⋊C414yes—
211C22⋊C4C4⋊C414yes—
212C42—42no—
213C42—42no—
214C42—81no—
215C42C22×C412yes—
216C42C2×Q812yes—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23C2×C4⋊C412yes—
228C23C2×C4⋊C412yes—
229C23AGL(1,ℤ/8)12yes—
230C23#13712yes—
231C23C22×D411yescharacteristic
232D4—28no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
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—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—84no—
274C2×C4—84no—
275C2×C4—84no—
276C2×C4C2×C4⋊C412yes—
277C2×C4C2×C4⋊C412yes—
278C2×C4C2×C4⋊C412yes—
279C2×C4C2×C4⋊C412yes—
280C2×C4C23×C411yescharacteristic
281C2×C4C22×Q811yescharacteristic
282C2×C4C2×#4111yescharacteristic
283C2×C4C2×#4111yescharacteristic
284C2×C4H(5,𝔽2)11yescharacteristic
285C2×C4H(5,𝔽2)11yescharacteristic
286C8—22no—
287C8—22no—
288C8—22no—
289C8—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22#42814yes—
304C22C22×C4⋊C411yescharacteristic
305C22#54611yescharacteristic
306C22#54611yescharacteristic
307C22C2×AGL(1,ℤ/8)11yescharacteristic
308C22C2×#13711yescharacteristic
309C22#57611yescharacteristic
310C22#57811yescharacteristic
311C4—22no—
312C4—22no—
313C4—22no—
314C4—22no—
315C4—42no—
316C4—42no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4C4×#4111yescharacteristic
324C4#51811yescharacteristic
325C4#55411yescharacteristic
326C4#55411yescharacteristic
327C2—24no—
328C2—42no—
329C2#234911yescharacteristic
330C2C2×#42811yescharacteristic
331C2#291311yescharacteristic
332C2#330811yescharacteristic
333C2#331211yescharacteristic
334C2#331411yescharacteristic
335C2#331811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1949C2semidirect
5. #1947C2semidirect
6. #1854C2semidirect
7. #1853C2semidirect
8. #1852C2semidirect
9. #1852C2semidirect
10. #1836C2semidirect
11. #1809C2semidirect
12. #1808C2semidirect
13. #1776C2semidirect
14. #1736C2semidirect
15. C2×#340C2semidirect
24. #435C4semidirect
68. C2×M16C2×C4semidirect
71. C22×C8C2×C4semidirect
118. C4⋊C8C2×C4semidirect
119. C22⋊C8C2×C4semidirect
120. C22⋊C8C2×C4semidirect
121. #97C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4352
#23522
D4×C4⋊C42
#19492
#19472
#18542
#18532
#18522
#18362
#18092
#18082
#17762
#17362
C2×#3402