Finite Group Databasebeta

← #23808

Subgroups — #23808

Order 256.

543 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#442C211yesmaximal, characteristic, complemented
2C2×#440C211yesmaximal, characteristic, complemented
3C2×C4×SD16C211yesmaximal, characteristic, complemented
4C2×C4×D8C211yesmaximal, characteristic, complemented
5#2328C211yesmaximal, characteristic, complemented
6#2326C211yesmaximal, characteristic, complemented
7#2097C211yesmaximal, characteristic, complemented
8#2094C211yesmaximal, characteristic, complemented
9#2093C211yesmaximal, characteristic, complemented
10#2092C211yesmaximal, characteristic, complemented
11#2014C211yesmaximal, characteristic, complemented
12#2012C211yesmaximal, characteristic, complemented
13#2012C211yesmaximal, characteristic, complemented
14#2011C211yesmaximal, characteristic
15#1746C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C2×#127C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26#442—24no—
27#440—24no—
28C4×SD16—24no—
29C4×D8—24no—
30C2×#107C2211yescharacteristic
31C2×#107C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×#106C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#97C2211yescharacteristic
43C2×C4×C8C2211yescharacteristic
44#391C2211yescharacteristic
45#390C2211yescharacteristic
46#390C2211yescharacteristic
47#390C2211yescharacteristic
48#388—22no—
49#386—22no—
50#386—22no—
51#385—22no—
52#384C2211yescharacteristic
53#382C2211yescharacteristic
54#376C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4—22no—
61C22×D4C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C2×SD16—22no—
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16—22no—
71C2×SD16—22no—
72C2×SD16—22no—
73C2×D8—22no—
74C2×D8—22no—
75C2×D8—22no—
76C2×D8—22no—
77C2×D8—22no—
78C2×D8—22no—
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83#127—22no—
84#127—22no—
85#124—22no—
86#124—22no—
87C4×Q8—22no—
88C4×Q8—22no—
89C4×D4—22no—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C22⋊C4—22no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C42—21no—
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111C2×C42C2311yescharacteristic
112#107—22no—
113#107—22no—
114#106—22no—
115#106—22no—
116#103—22no—
117#103—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123#102—22no—
124#97D414yescomplemented
125C4×C8D414yescomplemented
126#95—22no—
127#95—22no—
128#95D412yes—
129#95D412yes—
130C24—21no—
131C24—21no—
132C24—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8#4112yes—
139C2×D4—28no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—82no—
156C2×D4—82no—
157C2×D4#4112yes—
158C2×D4#4112yes—
159C2×D4#4112yes—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4—82no—
171C22×C4C2×D412yes—
172C22×C4C2×D412yes—
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2411yescharacteristic
176SD16—28no—
177SD16—28no—
178D8—28no—
179D8—28no—
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×C8C2×D412yes—
189C2×C8C2×D412yes—
190C2×C8C2×D412yes—
191C2×C8C2×D412yes—
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—42no—
201C4⋊C4—82no—
202C22⋊C4—28no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—82no—
207C22⋊C4—82no—
208C22⋊C4—82no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—42no—
214C42C2×D412yes—
215C42C2×D412yes—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23C22×D411yescharacteristic
232Q8—22no—
233Q8—22no—
234Q8—41no—
235Q8—41no—
236D4—22no—
237D4—22no—
238D4—22no—
239D4—22no—
240D4—22no—
241D4—22no—
242D4—41no—
243D4—41no—
244D4—41no—
245D4—41no—
246D4—41no—
247D4—41no—
248D4—88no—
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—22no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—84no—
269C2×C4—84no—
270C2×C4—84no—
271C2×C4#12712yes—
272C2×C4#12712yes—
273C2×C4C22×D411yescharacteristic
274C2×C4C22×D411yescharacteristic
275C2×C4C2×#4111yescharacteristic
276C2×C4C2×#4111yescharacteristic
277C2×C4C2×#4111yescharacteristic
278C2×C4C2×#4111yescharacteristic
279C8—22no—
280C8—22no—
281C8—22no—
282C8—22no—
283C22—44no—
284C22—44no—
285C22—44no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—82no—
290C22—82no—
291C22—82no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22#52511yescharacteristic
299C22#52511yescharacteristic
300C22C2×#12711yescharacteristic
301C22#57612yes—
302C22#57712yes—
303C4—22no—
304C4—22no—
305C4—42no—
306C4—42no—
307C4—42no—
308C4—81no—
309C4—81no—
310C4—81no—
311C4—81no—
312C4#52511yescharacteristic
313C4#52511yescharacteristic
314C4#52511yescharacteristic
315C4#52511yescharacteristic
316C2—82no—
317C2—82no—
318C2—82no—
319C2#239711yescharacteristic
320C2#311812yes—
321C2#311912yes—
322C2#315411yescharacteristic
323C2#315511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2011C2
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×#442C2semidirect
2. C2×#440C2semidirect
3. C2×C4×SD16C2semidirect
4. C2×C4×D8C2semidirect
5. #2328C2semidirect
6. #2326C2semidirect
7. #2097C2semidirect
8. #2094C2semidirect
9. #2093C2semidirect
10. #2092C2semidirect
11. #2014C2semidirect
12. #2012C2semidirect
13. #2012C2semidirect
15. #1746C2semidirect
124. #97D4semidirect
125. C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4422
C2×#4402
C2×C4×SD162
C2×C4×D82
#23282
#23262
#20972
#20942
#20932
#20922
#20142
#20122
#20112
#17462