Finite Group Databasebeta

← #17417

Subgroups — #17417

Order 256.

472 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#552C211yesmaximal, characteristic
2#2924C212yesmaximal, complemented
3#1473C212yesmaximal, complemented
4#1470C212yesmaximal, complemented
5#1468C212yesmaximal
6#1467C211yesmaximal, characteristic, complemented
7#1464C211yesmaximal, characteristic, complemented
8#1463C211yesmaximal, characteristic, complemented
9#1461C211yesmaximal, characteristic, complemented
10#1458C211yesmaximal, characteristic, complemented
11C2×#324C211yesmaximal, characteristic, complemented
12#552—22no—
13#552—22no—
14#552—22no—
15#552—22no—
16C2×#126C411yescharacteristic
17C2×#126C2211yescharacteristic
18C2×#125C411yescharacteristic
19C2×#125C411yescharacteristic
20C2×#125C2211yescharacteristic
21C2×#123C411yescharacteristic
22C2×#123C2211yescharacteristic
23C2×C22⋊Q8C411yescharacteristic
24C2×C22⋊Q8C411yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26C2×C4×D4C411yescharacteristic
27C2×#117C411yescharacteristic
28C2×#117C2211yescharacteristic
29C2×#117C2211yescharacteristic
30C22×C4⋊C4C2211yescharacteristic
31#433—22no—
32#433—22no—
33#432C2212yes—
34#432C2212yes—
35#431C2212yes—
36#431C2212yes—
37#424—22no—
38#407C2212yes—
39C2×C22⋊C8C2212yes—
40#405—22no—
41#324C2211yescharacteristic
42#324C2211yescharacteristic
43#324C2212yes—
44#324C2211yescharacteristic
45#324C2211yescharacteristic
46#324C2212yes—
47#324C2212yes—
48#324C2212yes—
49#323C2212yes—
50#323C2212yes—
51C22×Q8C2×C411yescharacteristic
52C22×D4C2×C411yescharacteristic
53C23×C4C2×C411yescharacteristic
54C23×C4C2×C411yescharacteristic
55C23×C4C2×C411yescharacteristic
56C23×C4C2311yescharacteristic
57C2×M16—22no—
58C22×C8—22no—
59#126—22no—
60#126—22no—
61#126—22no—
62#126—22no—
63#125—22no—
64#125—22no—
65#125—22no—
66#125—22no—
67#125—22no—
68#125—22no—
69#123—22no—
70#123—22no—
71#123—22no—
72#123—22no—
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75C22⋊Q8—22no—
76C22⋊Q8—22no—
77C4×Q8—22no—
78C4×Q8—22no—
79C4×D4—21no—
80C4×D4—21no—
81C4×D4—21no—
82C4×D4—21no—
83#117—21no—
84#117—21no—
85#117—21no—
86#117—21no—
87#117—22no—
88#117—22no—
89#117D412yes—
90#117D412yes—
91#117D412yes—
92#117D412yes—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4C2×C411yescharacteristic
96C2×C4⋊C4C2×C411yescharacteristic
97C2×C4⋊C4C2×C411yescharacteristic
98C2×C4⋊C4C2×C411yescharacteristic
99C2×C4⋊C4C2×C411yescharacteristic
100C2×C4⋊C4C2×C411yescharacteristic
101C2×C4⋊C4C2×C411yescharacteristic
102C2×C4⋊C4C2×C411yescharacteristic
103C2×C4⋊C4C2×C411yescharacteristic
104C2×C4⋊C4C2×C411yescharacteristic
105C2×C4⋊C4C2×C411yescharacteristic
106C2×C4⋊C4C2×C411yescharacteristic
107C2×C4⋊C4C2×C411yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C22⋊C4C2×C411yescharacteristic
112C2×C22⋊C4C2×C411yescharacteristic
113C2×C22⋊C4C2×C411yescharacteristic
114C2×C22⋊C4C2×C411yescharacteristic
115C2×C22⋊C4C2×C411yescharacteristic
116C2×C22⋊C4C2×C411yescharacteristic
117C2×C22⋊C4C2×C411yescharacteristic
118C2×C22⋊C4C2311yescharacteristic
119C2×C42C2×C411yescharacteristic
120C2×C42C2×C411yescharacteristic
121C2×C42C2×C411yescharacteristic
122C2×C42C2311yescharacteristic
123C2×C42C2311yescharacteristic
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8C2312yes—
131C22⋊C8C2312yes—
132C22⋊C8C2312yes—
133C22⋊C8C2312yes—
134#97—22no—
135#97—22no—
136C4×C8—22no—
137C4×C8—22no—
138C24C22×C411yescharacteristic
139C24C22×C411yescharacteristic
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—41no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—42no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4C22×C411yescharacteristic
159C22×C4C22×C411yescharacteristic
160C22×C4C22×C411yescharacteristic
161C22×C4C22×C411yescharacteristic
162C22×C4C22×C411yescharacteristic
163C22×C4C22×C411yescharacteristic
164C22×C4C22×C411yescharacteristic
165C22×C4C22×C411yescharacteristic
166C22×C4C22×C411yescharacteristic
167C22×C4C22×C411yescharacteristic
168C22×C4C22×C411yescharacteristic
169C22×C4C22×C411yescharacteristic
170C22×C4C2×D412yes—
171C22×C4C2×D412yes—
172C22×C4C2411yescharacteristic
173M16—44no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—44no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4C22⋊C412yes—
197C4⋊C4C22⋊C412yes—
198C4⋊C4C22⋊C412yes—
199C4⋊C4C22⋊C412yes—
200C22⋊C4—21no—
201C22⋊C4—21no—
202C22⋊C4—22no—
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4C22⋊C411yescharacteristic
209C22⋊C4C22⋊C411yescharacteristic
210C22⋊C4C22⋊C411yescharacteristic
211C22⋊C4C22⋊C411yescharacteristic
212C22⋊C4C22⋊C412yes—
213C22⋊C4C22⋊C412yes—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42C2×D412yes—
221C42C2×D412yes—
222C42C2×D412yes—
223C42C2×D412yes—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23C2×C22⋊C411yescharacteristic
234C23C2×C22⋊C411yescharacteristic
235C23C23×C411yescharacteristic
236Q8—44no—
237D4—42no—
238D4—42no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—24no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4C2×C22⋊C411yescharacteristic
272C2×C4C2×C22⋊C411yescharacteristic
273C2×C4C2×C22⋊C411yescharacteristic
274C2×C4C2×C22⋊C411yescharacteristic
275C2×C4C2×C22⋊C412yes—
276C2×C4C2×C22⋊C412yes—
277C2×C4C2×C22⋊C412yes—
278C2×C4#13112yes—
279C2×C4C22×D412yes—
280C8—42no—
281C8—42no—
282C8—42no—
283C8—42no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—42no—
292C22—42no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—81no—
297C22—81no—
298C22#41012yes—
299C22C22×C22⋊C411yescharacteristic
300C22C2×#13111yescharacteristic
301C22#56811yescharacteristic
302C4—22no—
303C4—21no—
304C4—21no—
305C4—22no—
306C4—22no—
307C4—22no—
308C4—21no—
309C4—21no—
310C4—22no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C2—21no—
318C2—21no—
319C2—22no—
320C2—41no—
321C2—81no—
322C2#288011yescharacteristic
323C2C2×#41011yescharacteristic
324C2#288911yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC23C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2924C2semidirect
3. #1473C2semidirect
4. #1470C2semidirect
6. #1467C2semidirect
7. #1464C2semidirect
8. #1463C2semidirect
9. #1461C2semidirect
10. #1458C2semidirect
11. C2×#324C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5522
#29242
#14732
#14702
#14682
#14672
#14642
#14632
#14612
#14582
C2×#3242