Finite Group Databasebeta

← #22659

Subgroups — #22659

Order 256.

521 conjugacy classes of subgroups.

The subgroup lattice has 343 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#435C211yesmaximal, characteristic
2C2×#428C211yesmaximal, characteristic, complemented
3C22×#106C211yesmaximal, characteristic, complemented
4#2352C211yesmaximal, characteristic
5D4×C4⋊C4C211yesmaximal, characteristic
6#1949C211yesmaximal, characteristic, complemented
7#1948C211yesmaximal, characteristic, complemented
8#1854C211yesmaximal, characteristic, complemented
9#1853C211yesmaximal, characteristic, complemented
10#1853C211yesmaximal, characteristic, complemented
11#1852C211yesmaximal, characteristic, complemented
12#1847C211yesmaximal, characteristic, complemented
13#1809C211yesmaximal, characteristic, complemented
14#1808C211yesmaximal, characteristic, complemented
15#1776C211yesmaximal, 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
25#428—24no—
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×#107C2211yescharacteristic
30C2×#106—24no—
31C2×#106C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×#106C2211yescharacteristic
34C2×#106C2211yescharacteristic
35C2×#106C2211yescharacteristic
36C2×#106C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×#97C2211yescharacteristic
41#389—21no—
42#388—21no—
43#387—21no—
44#386—21no—
45#385—21no—
46#385—21no—
47#384C2211yescharacteristic
48#382C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55C4×C22⋊C4—21no—
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62C22×D4D411yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C2×M16—22no—
69C2×M16—22no—
70C2×M16C2×C414yescomplemented
71C22×C8—22no—
72C22×C8—22no—
73C22×C8C2×C414yescomplemented
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C4×D4—44no—
79C4×D4C2×C414yes—
80C4×D4Q814yes—
81#117—22no—
82#117—41no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4D411yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4D411yescharacteristic
109C2×C22⋊C4D411yescharacteristic
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42C2311yescharacteristic
114#107—22no—
115#107—22no—
116#107—22no—
117#107—41no—
118#106—22no—
119#106—22no—
120#106—22no—
121#106—22no—
122#106—41no—
123#106D414yescomplemented
124C4⋊C8C2×C414yescomplemented
125C22⋊C8C2×C414yescomplemented
126C22⋊C8C2×C414yescomplemented
127#97C2×C414yescomplemented
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136C24C2×D411yescharacteristic
137C24C2×D411yescharacteristic
138C2×D4—22no—
139C2×D4—22no—
140C2×D4C4⋊C414yes—
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—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—81no—
164C22×C4—81no—
165C22×C4—81no—
166C22×C4C22×C412yes—
167C22×C4C22×C412yes—
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×Q812yes—
173C22×C4C2×Q812yes—
174C22×C4C2411yescharacteristic
175M16—28no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—28no—
183C2×C8C22×C412yes—
184C2×C8C22×C412yes—
185C2×C8C22×C412yes—
186C2×C8C22×C412yes—
187C2×C8C2×D412yes—
188C2×C8#4112yes—
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—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⋊C4—42no—
207C4⋊C4—81no—
208C4⋊C4C4⋊C414yes—
209C4⋊C4C2×D412yes—
210C4⋊C4C2×D412yes—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—81no—
216C22⋊C4C4⋊C414yes—
217C22⋊C4C4⋊C414yes—
218C42—42no—
219C42—42no—
220C42—81no—
221C42C22×C412yes—
222C42C2×Q812yes—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23C2×C4⋊C412yes—
234C23C2×C4⋊C412yes—
235C23AGL(1,ℤ/8)12yes—
236C23#13712yes—
237C23C22×D411yescharacteristic
238D4—28no—
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—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
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—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—84no—
280C2×C4—84no—
281C2×C4—84no—
282C2×C4C2×C4⋊C412yes—
283C2×C4C2×C4⋊C412yes—
284C2×C4C2×C4⋊C412yes—
285C2×C4C2×C4⋊C412yes—
286C2×C4C23×C411yescharacteristic
287C2×C4C22×D411yescharacteristic
288C2×C4C22×Q811yescharacteristic
289C2×C4C2×#4111yescharacteristic
290C2×C4H(5,𝔽2)11yescharacteristic
291C2×C42−1+411yescharacteristic
292C8—22no—
293C8—22no—
294C8—22no—
295C8C4×D414yescomplemented
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22#42814yescomplemented
310C22C22×C4⋊C411yescharacteristic
311C22D4211yescharacteristic
312C22#54711yescharacteristic
313C22C2×AGL(1,ℤ/8)11yescharacteristic
314C22C2×#13711yescharacteristic
315C22#57711yescharacteristic
316C22#57711yescharacteristic
317C4—22no—
318C4—22no—
319C4—22no—
320C4—22no—
321C4—42no—
322C4—42no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C4—81no—
329C4C2×C4×D411yescharacteristic
330C4#52011yescharacteristic
331C4D4×Q811yescharacteristic
332C4#55411yescharacteristic
333C2—24no—
334C2—42no—
335C2D4×C4⋊C411yescharacteristic
336C2C2×#42811yescharacteristic
337C2#291211yescharacteristic
338C2#326511yescharacteristic
339C2#326611yescharacteristic
340C2#334411yescharacteristic
341C2#334511yescharacteristic
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
2. C2×#428C2semidirect
3. C22×#106C2semidirect
6. #1949C2semidirect
7. #1948C2semidirect
8. #1854C2semidirect
9. #1853C2semidirect
10. #1853C2semidirect
11. #1852C2semidirect
12. #1847C2semidirect
13. #1809C2semidirect
14. #1808C2semidirect
15. #1776C2semidirect
24. #435C4semidirect
70. C2×M16C2×C4semidirect
73. C22×C8C2×C4semidirect
123. #106D4semidirect
124. C4⋊C8C2×C4semidirect
125. C22⋊C8C2×C4semidirect
126. C22⋊C8C2×C4semidirect
127. #97C2×C4semidirect
295. C8C4×D4semidirect
309. C22#428semidirect

Maximal subgroups

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

subgroupindex
C2×#4352
C2×#4282
C22×#1062
#23522
D4×C4⋊C42
#19492
#19482
#18542
#18532
#18522
#18472
#18092
#18082
#17762