Finite Group Databasebeta

← #23694

Subgroups — #23694

Order 256.

422 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#441C211yesmaximal, characteristic, complemented
2C2×C4×SD16C211yesmaximal, characteristic
3#2326C211yesmaximal, characteristic
4#2320C211yesmaximal, characteristic, complemented
5#2084C211yesmaximal, characteristic, complemented
6#2072C211yesmaximal, characteristic, complemented
7#2066C211yesmaximal, characteristic, complemented
8#2065C211yesmaximal, characteristic
9#2056C211yesmaximal, characteristic, complemented
10#2055C211yesmaximal, characteristic
11#2042C211yesmaximal, characteristic
12#2015C211yesmaximal, characteristic
13#1768C211yesmaximal, characteristic, complemented
14#1762C211yesmaximal, characteristic
15C4×#103C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22#441—24no—
23C4×SD16—24no—
24C2×#106C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#97C2211yescharacteristic
37C2×C4×C8C2211yescharacteristic
38#391C2211yescharacteristic
39#391C2211yescharacteristic
40#391C2211yescharacteristic
41#390C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#388—21no—
45#388—21no—
46#387—21no—
47#387—21no—
48#385—21no—
49#385—21no—
50#384C2211yescharacteristic
51#383—21no—
52#382—21no—
53C4×C4⋊C4C2211yescharacteristic
54#376C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C2×Q16—22no—
66C2×Q16—22no—
67C2×Q16—22no—
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16—22no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75#124—22no—
76#124—22no—
77C4×Q8—22no—
78C4×Q8—22no—
79C4×Q8—22no—
80C4×Q8D414yes—
81C4×D4—22no—
82C4×D4—22no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104C2×C42C2311yescharacteristic
105#106—22no—
106#106—22no—
107C4⋊C8—22no—
108C4⋊C8—22no—
109#103—22no—
110#103—22no—
111#103—22no—
112#103—22no—
113#103—22no—
114#102—22no—
115#102—22no—
116#102—22no—
117#97—22no—
118C4×C8—22no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95D411yescharacteristic
129#95D411yescharacteristic
130#95D411yescharacteristic
131#95D411yescharacteristic
132C24—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8C2×D412yes—
144C2×Q8#4112yes—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4#4112yes—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—42no—
159C22×C4—81no—
160C22×C4—81no—
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2411yescharacteristic
168Q16—28no—
169SD16—28no—
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—21no—
179C2×C8—21no—
180C2×C8#4112yes—
181C2×C8#4112yes—
182C4⋊C4—24no—
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—42no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C4⋊C4—81no—
196C4⋊C4—81no—
197C4⋊C4C2×D412yes—
198C22⋊C4—24no—
199C22⋊C4—24no—
200C22⋊C4—42no—
201C22⋊C4—81no—
202C22⋊C4—81no—
203C42—24no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—42no—
209C42—81no—
210C42C2×D412yes—
211C42#4112yes—
212C23—22no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23C22×D411yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
227Q8—41no—
228D4—22no—
229D4—22no—
230D4—41no—
231D4—41no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
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—84no—
270C2×C4#12112yes—
271C2×C4#12112yes—
272C2×C4C22×D411yescharacteristic
273C2×C4C2×#4111yescharacteristic
274C2×C4C2×#4111yescharacteristic
275C2×C4C2×#4111yescharacteristic
276C2×C4C2×#4111yescharacteristic
277C2×C4C2×#4111yescharacteristic
278C8—22no—
279C8—22no—
280C8—41no—
281C8—41no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22C2×#12111yescharacteristic
290C22#52511yescharacteristic
291C22#53211yescharacteristic
292C22#57711yescharacteristic
293C22#57711yescharacteristic
294C22#57711yescharacteristic
295C22#57811yescharacteristic
296C4—22no—
297C4—22no—
298C4—41no—
299C4—41no—
300C4—42no—
301C4—42no—
302C4—41no—
303C4—41no—
304C4—42no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4—81no—
309C4#52511yescharacteristic
310C4#52511yescharacteristic
311C4#52911yescharacteristic
312C4#52911yescharacteristic
313C2—82no—
314C2#239211yescharacteristic
315C2#304311yescharacteristic
316C2#304611yescharacteristic
317C2#312011yescharacteristic
318C2#312511yescharacteristic
319C2#317311yescharacteristic
320C2#317811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×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
1. C2×#441C2semidirect
4. #2320C2semidirect
5. #2084C2semidirect
6. #2072C2semidirect
7. #2066C2semidirect
9. #2056C2semidirect
13. #1768C2semidirect
15. C4×#103C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4412
C2×C4×SD162
#23262
#23202
#20842
#20722
#20662
#20652
#20562
#20552
#20422
#20152
#17682
#17622
C4×#1032