Finite Group Databasebeta

← #17548

Subgroups — #17548

Order 256.

551 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#3520C211yesmaximal, characteristic
2#2974C211yesmaximal, characteristic, complemented
3#2275C211yesmaximal, characteristic, complemented
4#1509C211yesmaximal, characteristic, complemented
5#1508C212yesmaximal, complemented
6#1506C211yesmaximal, characteristic, complemented
7#1504C212yesmaximal, complemented
8#1503C211yesmaximal, characteristic, complemented
9#1501C212yesmaximal, complemented
10#1500C211yesmaximal, characteristic, complemented
11C2×#328C212yesmaximal, complemented
12#556C411yescharacteristic, complemented
13#554—21no—
14#552C412yescomplemented
15#551—22no—
16D4×Q8C411yescharacteristic, complemented
17#548C412yescomplemented
18#547C411yescharacteristic, complemented
19#547C411yescharacteristic, complemented
20#546—21no—
21#542—21no—
22#539—22no—
23#533C2211yescharacteristic
24#529C2211yescharacteristic
25#525C2212yes—
26C2×C22⋊Q8C2212yes—
27#520—21no—
28C2×C4×D4—21no—
29C2×C4×D4C2211yescharacteristic
30#436—21no—
31#435—22no—
32C8×D4—21no—
33#432C2211yescharacteristic
34#431C2211yescharacteristic
35#407C2212yes—
36C2×C22⋊C8C2212yes—
37C4×C4⋊C4—21no—
38C4×C4⋊C4C2211yescharacteristic
39C4×C22⋊C4—22no—
40C4×C22⋊C4C2212yes—
41#376C2211yescharacteristic
42C2×#95C2212yes—
43#328C2211yescharacteristic
44#328C2212yes—
45#328C2212yes—
46#328C2212yes—
47#328C2211yescharacteristic
48#328C2212yes—
49#328C2212yes—
50#327C2211yescharacteristic
51#327C2211yescharacteristic
52#327C2212yes—
53C2×#41—21no—
54C2×#41C2×C412yescomplemented
55C22×Q8C2×C412yescomplemented
56C22×D4D411yescharacteristic
57C23×C4—22no—
58C23×C4C2312yes—
59C2×M16—22no—
60C22×C8—22no—
61C4⋊Q8—21no—
62C4⋊Q8C2×C411yescharacteristic, complemented
63C4⋊Q8C2×C411yescharacteristic, complemented
64#126—22no—
65#126—22no—
66#126—21no—
67#125—22no—
68#125—21no—
69#125C2×C411yescharacteristic, complemented
70#124—21no—
71#124—22no—
72#124C2×C411yescharacteristic, complemented
73#123—22no—
74#123—22no—
75#123—21no—
76#123—22no—
77#123—22no—
78C22⋊Q8—22no—
79C22⋊Q8—22no—
80C22⋊Q8—21no—
81C22⋊Q8—22no—
82C22⋊Q8—21no—
83C22⋊Q8—22no—
84C22⋊Q8C2×C412yescomplemented
85C22⋊Q8C2×C412yescomplemented
86C22⋊Q8C2×C412yescomplemented
87C22⋊Q8C2311yescharacteristic
88C22⋊Q8C2312yes—
89#121—22no—
90#121—21no—
91#121C2×C412yescomplemented
92#121C2311yescharacteristic
93#120—22no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8C2×C411yescharacteristic, complemented
97C4×Q8C2×C412yescomplemented
98C4×D4—28no—
99C4×D4—21no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—21no—
103C4×D4—22no—
104C4×D4C2×C411yescharacteristic
105C4×D4C2×C411yescharacteristic
106C4×D4C2×C412yes—
107C4×D4C2×C411yescharacteristic, complemented
108C4×D4D411yescharacteristic
109C4×D4D412yes—
110C4×D4D411yescharacteristic
111#117—21no—
112#117—22no—
113#117—21no—
114#117C2×C412yescomplemented
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4C2×C412yescomplemented
119C2×C4⋊C4D411yescharacteristic
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4D412yes—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—22no—
126C2×C42C2311yescharacteristic
127C4⋊C8—21no—
128C4⋊C8—21no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8C2311yescharacteristic
132C22⋊C8C2312yes—
133C22⋊C8C2311yescharacteristic
134#97—21no—
135C4×C8—21no—
136#95—22no—
137#95—21no—
138#95C2311yescharacteristic
139#95C2312yes—
140#95C2311yescharacteristic
141C24C2×D412yes—
142#41—44no—
143#41—82no—
144C2×Q8—21no—
145C2×Q8—22no—
146C2×Q8—82no—
147C2×Q8C22×C411yescharacteristic
148C2×Q8C22×C412yes—
149C2×D4—24no—
150C2×D4—22no—
151C2×D4—21no—
152C2×D4—22no—
153C2×D4C22⋊C412yes—
154C2×D4C22⋊C411yescharacteristic
155C2×D4C22⋊C411yescharacteristic
156C2×D4C22×C411yescharacteristic
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—28no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—82no—
166C22×C4—82no—
167C22×C4—82no—
168C22×C4C22×C412yes—
169C22×C4C22×C412yes—
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D412yes—
173C22×C4C2×D412yes—
174C22×C4C2×D412yes—
175C22×C4C2411yescharacteristic
176M16—82no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—22no—
180C2×C8—82no—
181C4⋊C4—24no—
182C4⋊C4—22no—
183C4⋊C4—21no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—22no—
190C4⋊C4—21no—
191C4⋊C4—22no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—22no—
195C4⋊C4C22⋊C411yescharacteristic
196C4⋊C4C22⋊C412yes—
197C4⋊C4C22⋊C411yescharacteristic
198C4⋊C4C22×C411yescharacteristic
199C4⋊C4C22×C412yes—
200C4⋊C4C22×C411yescharacteristic
201C22⋊C4—28no—
202C22⋊C4—22no—
203C22⋊C4—22no—
204C22⋊C4—22no—
205C22⋊C4—22no—
206C22⋊C4—21no—
207C22⋊C4—22no—
208C22⋊C4—21no—
209C22⋊C4—21no—
210C22⋊C4—22no—
211C22⋊C4—21no—
212C22⋊C4—21no—
213C22⋊C4C22⋊C412yes—
214C22⋊C4C22⋊C412yes—
215C22⋊C4C22⋊C412yes—
216C22⋊C4C22⋊C412yes—
217C42—24no—
218C42—22no—
219C42—22no—
220C42—21no—
221C42—81no—
222C42—82no—
223C42C22×C411yescharacteristic
224C42C22×C411yescharacteristic
225C42C2×D411yescharacteristic
226C42C2×D411yescharacteristic
227C23—22no—
228C23—22no—
229C23—24no—
230C23—21no—
231C23—42no—
232C23C2×C22⋊C412yes—
233C23C2×C22⋊C412yes—
234C23C22×D411yescharacteristic
235Q8—41no—
236Q8—42no—
237Q8—42no—
238Q8—82no—
239D4—24no—
240D4—22no—
241D4—41no—
242D4—41no—
243D4—41no—
244D4—44no—
245D4—82no—
246D4#10414yes—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—44no—
271C2×C4—42no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—88no—
275C2×C4C2×C22⋊C411yescharacteristic
276C2×C4C2×C22⋊C412yes—
277C2×C4C2×C22⋊C411yescharacteristic
278C2×C4C2×C22⋊C411yescharacteristic
279C2×C4C2×C22⋊C412yes—
280C2×C4C2×C22⋊C411yescharacteristic
281C2×C4C23×C411yescharacteristic
282C2×C4C22×D411yescharacteristic
283C8—41no—
284C8—42no—
285C8—41no—
286C22—21no—
287C22—22no—
288C22—22no—
289C22—21no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—44no—
294C22—44no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22C2×#10414yes—
299C22C22×C22⋊C411yescharacteristic
300C22#57711yescharacteristic
301C22#57811yescharacteristic
302C4—21no—
303C4—21no—
304C4—22no—
305C4—22no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—42no—
310C4—44no—
311C4—42no—
312C4—42no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—42no—
317C4C2×#10412yes—
318C2—21no—
319C2—21no—
320C2—24no—
321C2—42no—
322C2—81no—
323C2#288311yescharacteristic
324C2#289811yescharacteristic
325C2C22×#10411yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#3520C2
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. #2974C2semidirect
3. #2275C2semidirect
4. #1509C2semidirect
5. #1508C2semidirect
6. #1506C2semidirect
7. #1504C2semidirect
8. #1503C2semidirect
9. #1501C2semidirect
10. #1500C2semidirect
11. C2×#328C2semidirect
12. #556C4semidirect
14. #552C4semidirect
16. D4×Q8C4semidirect
17. #548C4semidirect
18. #547C4semidirect
19. #547C4semidirect
54. C2×#41C2×C4semidirect
55. C22×Q8C2×C4semidirect
62. C4⋊Q8C2×C4semidirect
63. C4⋊Q8C2×C4semidirect
69. #125C2×C4semidirect
72. #124C2×C4semidirect
84. C22⋊Q8C2×C4semidirect
85. C22⋊Q8C2×C4semidirect
86. C22⋊Q8C2×C4semidirect
91. #121C2×C4semidirect
96. C4×Q8C2×C4semidirect
97. C4×Q8C2×C4semidirect
107. C4×D4C2×C4semidirect
114. #117C2×C4semidirect
118. C2×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#35202
#29742
#22752
#15092
#15082
#15062
#15042
#15032
#15012
#15002
C2×#3282