Finite Group Databasebeta

← #21624

Subgroups — #21624

Order 256.

438 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#474C211yesmaximal, characteristic
2#2331C211yesmaximal, characteristic, complemented
3#2275C211yesmaximal, characteristic
4#1987C211yesmaximal, characteristic, complemented
5#1949C211yesmaximal, characteristic, complemented
6#1921C211yesmaximal, characteristic, complemented
7#1918C211yesmaximal, characteristic, complemented
8#1864C211yesmaximal, characteristic
9#1862C211yesmaximal, characteristic
10#1804C211yesmaximal, characteristic
11#1776C211yesmaximal, characteristic, complemented
12C4×#107C211yesmaximal, characteristic, complemented
13#1768C211yesmaximal, characteristic, complemented
14#1762C211yesmaximal, characteristic
15C4×#102C211yesmaximal, characteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20#474C418yescomplemented
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#97C2211yescharacteristic
32C2×C4×C8C2211yescharacteristic
33#391—21no—
34#391—21no—
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#387—21no—
38#387—21no—
39#382C2211yescharacteristic
40C4×C4⋊C4C2211yescharacteristic
41C4×C4⋊C4C2211yescharacteristic
42C4×C4⋊C4C2211yescharacteristic
43C4×C4⋊C4C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45C4×C22⋊C4—21no—
46C4×C22⋊C4—21no—
47C4×C22⋊C4—21no—
48#376C2211yescharacteristic
49C2×#95—21no—
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60C22×Q8—21no—
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C4⋊Q8—22no—
70C4⋊Q8C2×C414yescomplemented
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4C2×C414yes—
76C4×D4Q814yescomplemented
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42C2311yescharacteristic
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98#107—22no—
99#107C2×C414yescomplemented
100#107C2×C414yescomplemented
101#106—22no—
102C4⋊C8—22no—
103C4⋊C8C2×C414yescomplemented
104#102—22no—
105#102—22no—
106#102C2×C414yes—
107#102C2×C414yes—
108#97—22no—
109C4×C8—22no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95D411yescharacteristic
120#95D411yescharacteristic
121#95D411yescharacteristic
122#95D411yescharacteristic
123C24—21no—
124C2×Q8—24no—
125C2×Q8—81no—
126C2×Q8—81no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4C22×C412yes—
131C2×D4C2×Q812yes—
132C2×D4#4112yes—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—81no—
144C22×C4—81no—
145C22×C4—81no—
146C22×C4—81no—
147C22×C4—81no—
148C22×C4—81no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2411yescharacteristic
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8C22×C412yes—
167C2×C8C22×C412yes—
168C4⋊C4—24no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C4⋊C4C22×C412yes—
181C4⋊C4C22×C412yes—
182C4⋊C4C22×C412yes—
183C4⋊C4C2×Q812yes—
184C4⋊C4#4112yes—
185C22⋊C4—24no—
186C22⋊C4—24no—
187C22⋊C4—24no—
188C42—21no—
189C42—21no—
190C42—81no—
191C42—81no—
192C42—81no—
193C42—81no—
194C42—81no—
195C42C22×C412yes—
196C42C2×Q812yes—
197C42#4112yes—
198C42#4112yes—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23C22×D411yescharacteristic
207Q8—84no—
208D4—22no—
209D4—22no—
210D4—22no—
211D4C4×Q814yes—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—84no—
248C2×C4—84no—
249C2×C4—84no—
250C2×C4C4×D412yes—
251C2×C4C4×D412yes—
252C2×C4C22⋊Q812yes—
253C2×C4C22⋊Q812yes—
254C2×C4C23×C411yescharacteristic
255C2×C4C22×Q811yescharacteristic
256C2×C4C2×#4111yescharacteristic
257C2×C4C2×#4111yescharacteristic
258C2×C4C2×#4111yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C8—22no—
261C8—22no—
262C8—41no—
263C8—41no—
264C22—24no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22C2×C4×D411yescharacteristic
272C22C2×C22⋊Q811yescharacteristic
273C22#52511yescharacteristic
274C22#57611yescharacteristic
275C22#57611yescharacteristic
276C22#57711yescharacteristic
277C22#57811yescharacteristic
278C4—22no—
279C4—22no—
280C4—22no—
281C4—22no—
282C4—42no—
283C4—41no—
284C4—42no—
285C4—41no—
286C4—42no—
287C4—42no—
288C4—42no—
289C4—81no—
290C4—81no—
291C4C2×C4×Q811yescharacteristic
292C4C4×#4111yescharacteristic
293C4#52911yescharacteristic
294C4#53311yescharacteristic
295C2—44no—
296C2C4×C22⋊Q811yescharacteristic
297C2#294711yescharacteristic
298C2#294811yescharacteristic
299C2#308111yescharacteristic
300C2#308311yescharacteristic
301C2#311811yescharacteristic
302C2#312411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2331C2semidirect
4. #1987C2semidirect
5. #1949C2semidirect
6. #1921C2semidirect
7. #1918C2semidirect
11. #1776C2semidirect
12. C4×#107C2semidirect
13. #1768C2semidirect
20. #474C4semidirect
70. C4⋊Q8C2×C4semidirect
76. C4×D4Q8semidirect
99. #107C2×C4semidirect
100. #107C2×C4semidirect
103. C4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4742
#23312
#22752
#19872
#19492
#19212
#19182
#18642
#18622
#18042
#17762
C4×#1072
#17682
#17622
C4×#1022