Finite Group Databasebeta

← #22180

Subgroups — #22180

Order 256.

489 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#465C211yesmaximal, characteristic
2C4×C22⋊Q8C211yesmaximal, characteristic, complemented
3C4×#121C211yesmaximal, characteristic
4#1927C211yesmaximal, characteristic, complemented
5#1926C211yesmaximal, characteristic, complemented
6#1920C211yesmaximal, characteristic, complemented
7#1919C211yesmaximal, characteristic, complemented
8#1893C211yesmaximal, characteristic
9#1892C211yesmaximal, characteristic, complemented
10#1880C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1865C211yesmaximal, characteristic, complemented
13#1864C211yesmaximal, characteristic, complemented
14#1807C211yesmaximal, characteristic, complemented
15C2×#336C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C23×C8C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4—21no—
23C2×C4×D4C2211yescharacteristic
24C22×C42C2211yescharacteristic
25#465C418yescomplemented
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35#391C2211yescharacteristic
36#390C2211yescharacteristic
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#388—21no—
40#387—21no—
41#386—21no—
42#385—21no—
43#380C2211yescharacteristic
44#380C2211yescharacteristic
45C4×C4⋊C4—21no—
46C4×C4⋊C4C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C22⋊C4—21no—
49C4×C22⋊C4—21no—
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×D4—21no—
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C23×C4D411yescharacteristic
65C23×C4D411yescharacteristic
66C23×C4C2311yescharacteristic
67C2×SD16—41no—
68C2×SD16—41no—
69C2×SD16C2×C414yescomplemented
70C22×C8—22no—
71C22×C8—41no—
72C22×C8—41no—
73C22×C8—41no—
74C22×C8—41no—
75C22×C8C2×C414yescomplemented
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22⋊Q8—22no—
81C22⋊Q8C2×C414yescomplemented
82#121—22no—
83#121C2×C414yes—
84C4×Q8—41no—
85C4×Q8—41no—
86C4×D4—41no—
87C4×D4—41no—
88C4×D4—44no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
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×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×C42—41no—
110C2×C42—41no—
111C2×C42D411yescharacteristic
112C2×C42D411yescharacteristic
113C2×C42C2311yescharacteristic
114C2×C42C2311yescharacteristic
115#106—41no—
116#106—41no—
117#106C2×C414yescomplemented
118#103—41no—
119#103—41no—
120#103C2×C414yescomplemented
121#102—41no—
122#102—41no—
123#102C2×C414yescomplemented
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132C24—21no—
133C24C2×D411yescharacteristic
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8C22×C412yes—
140C2×D4—24no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4C22×C412yes—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4SD1614yescomplemented
169C22×C4C22×C412yes—
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4#4112yes—
176C22×C4C2411yescharacteristic
177SD16—44no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—44no—
189C2×C8—44no—
190C2×C8C22×C412yes—
191C2×C8C22×C412yes—
192C4⋊C4—24no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4C22×C412yes—
207C4⋊C4C22×C412yes—
208C22⋊C4—24no—
209C22⋊C4—24no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—41no—
220C42—41no—
221C42—42no—
222C42—42no—
223C42—42no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—22no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23C4×D412yes—
236C23C2×SD1612yes—
237C23#13512yes—
238C23C22×D411yescharacteristic
239Q8—22no—
240Q8—22no—
241Q8—41no—
242Q8—41no—
243D4—22no—
244D4—22no—
245D4—48no—
246D4—41no—
247D4—41no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—84no—
288C2×C4C4×D412yes—
289C2×C4C2×SD1612yes—
290C2×C4C2×SD1612yes—
291C2×C4C23×C411yescharacteristic
292C2×C4C2×#4111yescharacteristic
293C2×C4H(5,𝔽2)11yescharacteristic
294C2×C4H(5,𝔽2)11yescharacteristic
295C2×C4H(5,𝔽2)11yescharacteristic
296C2×C42−1+411yescharacteristic
297C8—22no—
298C8—22no—
299C8—41no—
300C8—41no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22C4×SD1614yes—
315C22C2×C4×D411yescharacteristic
316C22#53411yescharacteristic
317C22#53711yescharacteristic
318C22C22×SD1611yescharacteristic
319C22C2×#13511yescharacteristic
320C22#57511yescharacteristic
321C22#57511yescharacteristic
322C4—22no—
323C4—22no—
324C4—22no—
325C4—42no—
326C4—41no—
327C4—41no—
328C4—42no—
329C4—42no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4#51811yescharacteristic
335C4#52011yescharacteristic
336C4#53811yescharacteristic
337C4#53911yescharacteristic
338C2—24no—
339C2—82no—
340C2#232511yescharacteristic
341C2C2×C4×SD1611yescharacteristic
342C2#294311yescharacteristic
343C2#318711yescharacteristic
344C2#319111yescharacteristic
345C2#324611yescharacteristic
346C2#325011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊Q8C2semidirect
4. #1927C2semidirect
5. #1926C2semidirect
6. #1920C2semidirect
7. #1919C2semidirect
9. #1892C2semidirect
10. #1880C2semidirect
11. #1877C2semidirect
12. #1865C2semidirect
13. #1864C2semidirect
14. #1807C2semidirect
15. C2×#336C2semidirect
25. #465C4semidirect
69. C2×SD16C2×C4semidirect
75. C22×C8C2×C4semidirect
81. C22⋊Q8C2×C4semidirect
117. #106C2×C4semidirect
120. #103C2×C4semidirect
123. #102C2×C4semidirect
168. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4652
C4×C22⋊Q82
C4×#1212
#19272
#19262
#19202
#19192
#18932
#18922
#18802
#18772
#18652
#18642
#18072
C2×#3362