Finite Group Databasebeta

← #17522

Subgroups — #17522

Order 256.

504 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3481C211yesmaximal, characteristic
2#2991C211yesmaximal, characteristic, complemented
3#2922C211yesmaximal, characteristic, complemented
4#1495C212yesmaximal, complemented
5#1492C211yesmaximal, characteristic, complemented
6#1489C211yesmaximal, characteristic, complemented
7#1486C212yesmaximal, complemented
8#1484C211yesmaximal, characteristic, complemented
9#1482C212yesmaximal, complemented
10#1478C212yesmaximal, complemented
11#1476C211yesmaximal, characteristic, complemented
12#559C411yescharacteristic
13#554C411yescharacteristic
14#554C411yescharacteristic
15#553—21no—
16#553C412yes—
17#551—22no—
18#546—21no—
19#546C412yes—
20D42C411yescharacteristic
21#532C2211yescharacteristic
22#529—21no—
23#529C2212yes—
24#525—22no—
25C4×#41C2211yescharacteristic
26C4×#41C2211yescharacteristic
27C2×C4×D4—21no—
28C2×C4×D4C2212yes—
29C8×Q8—21no—
30#436—21no—
31#436—22no—
32#433—22no—
33#431C2211yescharacteristic, complemented
34#431C2212yescomplemented
35#431C2212yescomplemented
36#431C2212yescomplemented
37#424—21no—
38#423C2212yescomplemented
39#423C2211yescharacteristic, complemented
40#408—21no—
41C4×M16C2211yescharacteristic, complemented
42C2×C4×C8C2211yescharacteristic, complemented
43Q8⋊C8C2211yescharacteristic, complemented
44Q8⋊C8C2211yescharacteristic, complemented
45Q8⋊C8C2212yescomplemented
46D4⋊C8C2211yescharacteristic
47D4⋊C8C2211yescharacteristic
48D4⋊C8C2212yes—
49D4⋊C8C2212yescomplemented
50D4⋊C8C2212yescomplemented
51D4⋊C8C2212yescomplemented
52D4⋊C8C2212yescomplemented
53C2×#41—21no—
54C2×#41C2×C411yescharacteristic
55C2×#41D411yescharacteristic
56C22×D4—21no—
57C22×D4C2×C412yes—
58C23×C4—22no—
59C2×M16—21no—
60C2×M16—22no—
61C2×M16—21no—
62C22×C8—21no—
63C4⋊Q8C2×C411yescharacteristic
64#127C2×C411yescharacteristic
65#126—22no—
66#126—22no—
67#125—21no—
68#125C2×C412yes—
69#124—21no—
70#124—21no—
71#124C2×C412yes—
72#123—22no—
73#123—22no—
74#123—22no—
75#123—22no—
76C22⋊Q8—21no—
77C22⋊Q8—22no—
78C22⋊Q8—22no—
79C22⋊Q8C2×C412yes—
80#121—22no—
81#121—21no—
82#121—21no—
83#121—22no—
84#121C2×C412yes—
85#121C2×C412yes—
86#120—22no—
87#120—22no—
88C4×Q8C2×C411yescharacteristic
89C4×Q8C2×C411yescharacteristic
90C4×Q8D411yescharacteristic
91C4×Q8C2311yescharacteristic
92C4×D4—21no—
93C4×D4—22no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4—22no—
98C4×D4—21no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4C2×C412yes—
102C4×D4C2×C412yes—
103C4×D4C2×C411yescharacteristic
104C4×D4C2×C411yescharacteristic
105C4×D4C2×C412yes—
106C4×D4D411yescharacteristic
107C4×D4D412yes—
108C4×D4C2311yescharacteristic
109C4×D4C2312yes—
110#117—21no—
111#117—22no—
112#117C2×C411yescharacteristic
113#117D412yes—
114#117D411yescharacteristic
115C2×C4⋊C4—21no—
116C2×C4⋊C4C2×C412yes—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C42—21no—
120C2×C42C2312yes—
121C2×C42C2311yescharacteristic
122C4⋊C8—21no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8C2311yescharacteristic
127C4⋊C8C2312yes—
128C4⋊C8C2311yescharacteristic
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—21no—
133#97—21no—
134C4×C8—21no—
135C4×C8—21no—
136C4×C8—22no—
137C4×C8C2311yescharacteristic
138C4×C8C2311yescharacteristic
139C4×C8C2312yes—
140C24—22no—
141#41—22no—
142#41—41no—
143#41—42no—
144#41—81no—
145#41—81no—
146C2×Q8—21no—
147C2×Q8C22⋊C411yescharacteristic
148C2×Q8C22×C411yescharacteristic
149C2×D4—24no—
150C2×D4—21no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—42no—
157C2×D4—44no—
158C2×D4C22⋊C411yescharacteristic
159C2×D4C22⋊C412yes—
160C2×D4C22×C412yes—
161C2×D4C22×C411yescharacteristic
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—82no—
170C22×C4—82no—
171C22×C4C22×C412yes—
172C22×C4C22×C411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D412yes—
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D412yes—
177M16—42no—
178M16—82no—
179M16—81no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—22no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—22no—
186C2×C8—42no—
187C4⋊C4—21no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—21no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4C22⋊C411yescharacteristic
195C4⋊C4C22⋊C411yescharacteristic
196C4⋊C4C22⋊C412yes—
197C4⋊C4C22⋊C412yes—
198C4⋊C4C22×C411yescharacteristic
199C4⋊C4C22×C412yes—
200C4⋊C4C22×C411yescharacteristic
201C22⋊C4—22no—
202C22⋊C4—21no—
203C22⋊C4—22no—
204C22⋊C4—22no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—21no—
208C22⋊C4—22no—
209C22⋊C4—22no—
210C22⋊C4—21no—
211C22⋊C4—22no—
212C22⋊C4C22⋊C412yes—
213C22⋊C4C22⋊C412yes—
214C22⋊C4C22⋊C412yes—
215C42—21no—
216C42—21no—
217C42—21no—
218C42C22×C412yes—
219C42C22×C411yescharacteristic
220C42C2×D411yescharacteristic
221C42C2×D412yes—
222C42C2×D411yescharacteristic
223C42C2×D412yes—
224C42C2411yescharacteristic
225C23—21no—
226C23—22no—
227C23—44no—
228C23—44no—
229C23—82no—
230C23C2×C22⋊C412yes—
231C23C2×C22⋊C411yescharacteristic
232Q8—22no—
233Q8—21no—
234Q8—21no—
235Q8—81no—
236D4—22no—
237D4—21no—
238D4—22no—
239D4—22no—
240D4—42no—
241D4—44no—
242D4—42no—
243D4—44no—
244D4—42no—
245D4—81no—
246D4—82no—
247D4#13112yes—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4C2×C22⋊C411yescharacteristic
272C2×C4C2×C22⋊C412yes—
273C2×C4C2×C22⋊C412yes—
274C2×C4C2×C22⋊C411yescharacteristic
275C2×C4C2×C22⋊C412yes—
276C2×C4C2×C22⋊C411yescharacteristic
277C2×C4C23×C411yescharacteristic
278C2×C4C22×D411yescharacteristic
279C2×C4C22×D411yescharacteristic
280C8—41no—
281C8—41no—
282C8—42no—
283C8—41no—
284C8—42no—
285C8—41no—
286C22—21no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—41no—
294C22—41no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—88no—
299C22C22×C22⋊C411yescharacteristic
300C4—21no—
301C4—22no—
302C4—22no—
303C4—21no—
304C4—21no—
305C4—22no—
306C4—41no—
307C4—41no—
308C4—42no—
309C4—41no—
310C4#42112yes—
311C4C2×#13111yescharacteristic
312C4#56811yescharacteristic
313C4#57611yescharacteristic
314C4#57711yescharacteristic
315C2—41no—
316C2—42no—
317C2—42no—
318C2—82no—
319C2#288011yescharacteristic
320C2#289711yescharacteristic
321C2C2×#42111yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC42C24
socleC22C22×C22⋊C4
prime radical#3481C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2991C2semidirect
3. #2922C2semidirect
4. #1495C2semidirect
5. #1492C2semidirect
6. #1489C2semidirect
7. #1486C2semidirect
8. #1484C2semidirect
9. #1482C2semidirect
10. #1478C2semidirect
11. #1476C2semidirect
33. #431C22semidirect
34. #431C22semidirect
35. #431C22semidirect
36. #431C22semidirect
38. #423C22semidirect
39. #423C22semidirect
41. C4×M16C22semidirect
42. C2×C4×C8C22semidirect
43. Q8⋊C8C22semidirect
44. Q8⋊C8C22semidirect
45. Q8⋊C8C22semidirect
49. D4⋊C8C22semidirect
50. D4⋊C8C22semidirect
51. D4⋊C8C22semidirect
52. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#34812
#29912
#29222
#14952
#14922
#14892
#14862
#14842
#14822
#14782
#14762