Finite Group Databasebeta

← #21291

Subgroups — #21291

Order 256.

407 conjugacy classes of subgroups.

The subgroup lattice has 305 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#440C211yesmaximal, characteristic
2#2320C211yesmaximal, characteristic, complemented
3#2319C211yesmaximal, characteristic
4#1973C211yesmaximal, characteristic
5#1968C211yesmaximal, characteristic
6#1943C211yesmaximal, characteristic, complemented
7#1942C211yesmaximal, characteristic, complemented
8#1936C211yesmaximal, characteristic
9#1935C211yesmaximal, characteristic, complemented
10#1934C211yesmaximal, characteristic
11#1933C211yesmaximal, characteristic, complemented
12#1776C211yesmaximal, characteristic, complemented
13C4×#103C211yesmaximal, characteristic, complemented
14C4×#102C211yesmaximal, characteristic
15#1726C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#440C418yescomplemented
20C2×#107C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#97C2211yescharacteristic
30C2×#97C2211yescharacteristic
31C2×#97C2211yescharacteristic
32C2×C4×C8C2211yescharacteristic
33C2×C4×C8C2211yescharacteristic
34C2×C4×C8C2211yescharacteristic
35#391C2211yescharacteristic
36#391C2211yescharacteristic
37#390C2211yescharacteristic
38#390C2211yescharacteristic
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#389C2211yescharacteristic
42#389C2211yescharacteristic
43#388—21no—
44#388—21no—
45#387—21no—
46#387—21no—
47#383—21no—
48#382—21no—
49#381—21no—
50#380—21no—
51C4×C4⋊C4C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#376C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C2×SD16—22no—
64C2×SD16—22no—
65C2×SD16C2×C414yes—
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C4×Q8—22no—
71C4×Q8C2×C414yes—
72C4×D4—22no—
73C4×D4C2×C414yes—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42C2311yescharacteristic
94C2×C42C2311yescharacteristic
95C2×C42C2311yescharacteristic
96#107—22no—
97#107C2×C414yes—
98#106—22no—
99#103—22no—
100#103—22no—
101#103C2×C414yes—
102#102—22no—
103#102—22no—
104#102C2×C414yes—
105#97—22no—
106#97—22no—
107#97C2×C414yes—
108C4×C8—22no—
109C4×C8—22no—
110C4×C8—22no—
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#95—21no—
120#95—21no—
121#95D411yescharacteristic
122#95D411yescharacteristic
123#95D411yescharacteristic
124#95D411yescharacteristic
125C24—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8C22×C412yes—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4C22×C412yes—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—24no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—81no—
147C22×C4—81no—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2411yescharacteristic
155SD16—28no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8C22×C412yes—
163C2×C8C22×C412yes—
164C2×C8#4112yes—
165C2×C8#4112yes—
166C2×C8#4112yes—
167C2×C8#4112yes—
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—21no—
179C4⋊C4—81no—
180C4⋊C4—81no—
181C4⋊C4—81no—
182C4⋊C4—81no—
183C4⋊C4C22×C412yes—
184C4⋊C4C22×C412yes—
185C22⋊C4—24no—
186C22⋊C4—81no—
187C22⋊C4—81no—
188C22⋊C4—81no—
189C22⋊C4—81no—
190C42—24no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—81no—
196C42—81no—
197C42C22×C412yes—
198C42#4112yes—
199C23—22no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23C22×D411yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—41no—
210Q8—41no—
211D4—22no—
212D4—22no—
213D4—41no—
214D4—41no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
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—21no—
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—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—84no—
253C2×C4C4×D412yes—
254C2×C4C4×D412yes—
255C2×C4C23×C411yescharacteristic
256C2×C4C2×#4111yescharacteristic
257C2×C4C2×#4111yescharacteristic
258C2×C4C2×#4111yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C2×C4C2×#4111yescharacteristic
261C8—22no—
262C8—22no—
263C8—22no—
264C8#11714yes—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—81no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22C2×C4×D411yescharacteristic
273C22#53211yescharacteristic
274C22#53211yescharacteristic
275C22#57611yescharacteristic
276C22#57711yescharacteristic
277C22#57711yescharacteristic
278C22#57811yescharacteristic
279C4—22no—
280C4—22no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—41no—
285C4—42no—
286C4—42no—
287C4—42no—
288C4—81no—
289C4—81no—
290C4—81no—
291C4—81no—
292C4C2×#11711yescharacteristic
293C4C4×#4111yescharacteristic
294C4#52911yescharacteristic
295C4#52911yescharacteristic
296C2—82no—
297C2#229311yescharacteristic
298C2#294611yescharacteristic
299C2#294711yescharacteristic
300C2#317411yescharacteristic
301C2#317511yescharacteristic
302C2#317811yescharacteristic
303C2#317911yescharacteristic
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. #2320C2semidirect
6. #1943C2semidirect
7. #1942C2semidirect
9. #1935C2semidirect
11. #1933C2semidirect
12. #1776C2semidirect
13. C4×#103C2semidirect
15. #1726C2semidirect
19. #440C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4402
#23202
#23192
#19732
#19682
#19432
#19422
#19362
#19352
#19342
#19332
#17762
C4×#1032
C4×#1022
#17262