Finite Group Databasebeta

← #17629

Subgroups — #17629

Order 256.

505 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3506C211yesmaximal, characteristic
2#2973C211yesmaximal, characteristic, complemented
3#2353C211yesmaximal, characteristic, complemented
4#1516C212yesmaximal, complemented
5#1515C211yesmaximal, characteristic, complemented
6#1514C211yesmaximal, characteristic, complemented
7#1513C212yesmaximal, complemented
8#1506C211yesmaximal, characteristic, complemented
9#1505C211yesmaximal, characteristic, complemented
10#1504C212yesmaximal, complemented
11#1501C212yesmaximal, complemented
12#559—22no—
13#556C412yescomplemented
14#555C412yescomplemented
15#553—22no—
16#550C412yescomplemented
17#547C412yescomplemented
18#538—21no—
19#538—21no—
20#535—21no—
21#533C2211yescharacteristic
22#532C2211yescharacteristic
23#525C2212yes—
24#525C2212yes—
25#520—21no—
26C2×C4×D4C2211yescharacteristic
27#435—22no—
28#435—22no—
29#432C2211yescharacteristic
30#432C2211yescharacteristic
31#407C2212yes—
32C2×C22⋊C8C2212yes—
33#388—21no—
34#387—21no—
35#386—21no—
36#383C2211yescharacteristic
37#381C2211yescharacteristic
38#380C2211yescharacteristic
39C4×C22⋊C4—21no—
40C4×C22⋊C4C2211yescharacteristic
41#376C2211yescharacteristic
42C2×#95C2211yescharacteristic
43#328C2212yes—
44#328C2212yes—
45#328C2212yes—
46#328C2212yes—
47#327C2212yes—
48#327C2212yes—
49#327C2212yes—
50#327C2212yes—
51C2×#41—21no—
52C2×#41C2×C412yes—
53C2×#41C2×C412yescomplemented
54C22×D4D411yescharacteristic
55C23×C4—21no—
56C23×C4C2311yescharacteristic
57C23×C4C2311yescharacteristic
58C2×M16—22no—
59C22×C8—22no—
60C4⋊Q8C2×C411yescharacteristic, complemented
61C4⋊Q8C2×C411yescharacteristic, complemented
62C4⋊Q8C2×C411yescharacteristic, complemented
63#127—22no—
64#127C2×C411yescharacteristic, complemented
65#126—22no—
66#126—22no—
67#125—21no—
68#124—22no—
69#124—22no—
70#124—22no—
71#124—21no—
72#123—22no—
73#123—22no—
74#123—22no—
75#123—22no—
76C22⋊Q8—21no—
77C22⋊Q8C2×C412yes—
78C22⋊Q8C2×C412yescomplemented
79C22⋊Q8C2312yes—
80#121—22no—
81#121—22no—
82#121—21no—
83#121—22no—
84#121—22no—
85#121C2×C412yes—
86#121C2×C412yescomplemented
87#121C2312yes—
88#120—22no—
89#120—22no—
90C4×Q8—22no—
91C4×Q8—21no—
92C4×Q8C2×C412yescomplemented
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—21no—
99C4×D4C2×C412yes—
100C4×D4C2×C412yes—
101C4×D4C2×C412yescomplemented
102C4×D4D412yes—
103C4×D4D412yes—
104#117—21no—
105#117—22no—
106#117C2×C412yes—
107#117C2×C412yescomplemented
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4D411yescharacteristic
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4D411yescharacteristic
115C2×C22⋊C4D411yescharacteristic
116C2×C42—21no—
117C2×C42—21no—
118C2×C42C2311yescharacteristic
119C4⋊C8—21no—
120C4⋊C8—21no—
121C22⋊C8—22no—
122C22⋊C8—22no—
123C22⋊C8C2312yes—
124C22⋊C8C2312yes—
125#97—21no—
126#97—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95C2311yescharacteristic
133#95C2311yescharacteristic
134#95C2311yescharacteristic
135#95C2311yescharacteristic
136C24C2×D411yescharacteristic
137C24C2×D411yescharacteristic
138#41—42no—
139#41—42no—
140#41—82no—
141#41—82no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8C22×C412yes—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—41no—
153C2×D4—41no—
154C2×D4C22⋊C412yes—
155C2×D4C22⋊C412yes—
156C2×D4C22×C412yes—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—81no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4C22×C412yes—
170C22×C4C22×C412yes—
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D412yes—
176C22×C4C2×D412yes—
177C22×C4C2411yescharacteristic
178M16—82no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—82no—
182C4⋊C4—21no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—81no—
191C4⋊C4C22⋊C412yes—
192C4⋊C4C22⋊C412yes—
193C4⋊C4C22×C412yes—
194C4⋊C4C22×C412yes—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—22no—
202C22⋊C4—22no—
203C22⋊C4—22no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—81no—
207C22⋊C4C22⋊C412yes—
208C22⋊C4C22⋊C412yes—
209C22⋊C4C22⋊C412yes—
210C22⋊C4C22⋊C412yes—
211C42—22no—
212C42—22no—
213C42—22no—
214C42—42no—
215C42—81no—
216C42C22×C411yescharacteristic
217C42C22×C411yescharacteristic
218C42C2×D411yescharacteristic
219C42C2×D411yescharacteristic
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23C2×C22⋊C412yes—
230C23C2×C22⋊C412yes—
231C23C22×D411yescharacteristic
232Q8—42no—
233Q8—42no—
234Q8—81no—
235D4—42no—
236D4—42no—
237D4—42no—
238D4—42no—
239D4—42no—
240D4—42no—
241D4—81no—
242D4—82no—
243D4—82no—
244D4—82no—
245D4—82no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
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—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—84no—
275C2×C4C2×C22⋊C411yescharacteristic
276C2×C4C2×C22⋊C412yes—
277C2×C4C2×C22⋊C412yes—
278C2×C4C2×C22⋊C411yescharacteristic
279C2×C4C2×C22⋊C411yescharacteristic
280C2×C4C2×C22⋊C411yescharacteristic
281C2×C4C23×C411yescharacteristic
282C2×C4C22×D411yescharacteristic
283C8—42no—
284C8—42no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—81no—
299C22—81no—
300C22C22×C22⋊C411yescharacteristic
301C22#57511yescharacteristic
302C22#57511yescharacteristic
303C4—21no—
304C4—21no—
305C4—22no—
306C4—22no—
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—42no—
312C4—42no—
313C4—42no—
314C4—42no—
315C4—81no—
316C4—81no—
317C4—81no—
318C2—21no—
319C2—21no—
320C2—42no—
321C2—42no—
322C2—82no—
323C2#288311yescharacteristic
324C2#289611yescharacteristic
325C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#3506C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2973C2semidirect
3. #2353C2semidirect
4. #1516C2semidirect
5. #1515C2semidirect
6. #1514C2semidirect
7. #1513C2semidirect
8. #1506C2semidirect
9. #1505C2semidirect
10. #1504C2semidirect
11. #1501C2semidirect
13. #556C4semidirect
14. #555C4semidirect
16. #550C4semidirect
17. #547C4semidirect
53. C2×#41C2×C4semidirect
60. C4⋊Q8C2×C4semidirect
61. C4⋊Q8C2×C4semidirect
62. C4⋊Q8C2×C4semidirect
64. #127C2×C4semidirect
78. C22⋊Q8C2×C4semidirect
86. #121C2×C4semidirect
92. C4×Q8C2×C4semidirect
101. C4×D4C2×C4semidirect
107. #117C2×C4semidirect

Maximal subgroups

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

subgroupindex
#35062
#29732
#23532
#15162
#15152
#15142
#15132
#15062
#15052
#15042
#15012