Finite Group Databasebeta

← #17512

Subgroups — #17512

Order 256.

502 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3475C211yesmaximal, characteristic
2#2965C211yesmaximal, characteristic, complemented
3#2906C211yesmaximal, characteristic, complemented
4#1495C212yesmaximal, complemented
5#1493C211yesmaximal, characteristic, complemented
6#1489C212yesmaximal, complemented
7#1486C211yesmaximal, characteristic, complemented
8#1483C212yesmaximal, complemented
9#1481C211yesmaximal, characteristic, complemented
10#1477C212yesmaximal, complemented
11C2×D4⋊C8C211yesmaximal, characteristic, complemented
12C2×#131—21no—
13#556C411yescharacteristic
14#555C411yescharacteristic
15#554C411yescharacteristic
16#553—21no—
17#552—21no—
18#552C412yes—
19#551—22no—
20#547C411yescharacteristic
21#546C412yes—
22#533C2211yescharacteristic
23#529—22no—
24#529C2212yes—
25#525—21no—
26C4×#41C2212yes—
27C4×#41C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C2×#117—21no—
30#446—21no—
31#436—21no—
32#435—22no—
33#431C2211yescharacteristic
34#431C2212yescomplemented
35#424—22no—
36#424—21no—
37#423C2211yescharacteristic
38#423C2212yescomplemented
39C2×C4⋊C8C2211yescharacteristic
40C2×C4⋊C8C2212yescomplemented
41C4×M16C2212yescomplemented
42C2×C4×C8C2211yescharacteristic
43Q8⋊C8C2212yescomplemented
44Q8⋊C8C2212yescomplemented
45Q8⋊C8C2212yescomplemented
46Q8⋊C8C2212yescomplemented
47D4⋊C8C2211yescharacteristic
48D4⋊C8C2211yescharacteristic
49D4⋊C8C2211yescharacteristic
50D4⋊C8C2211yescharacteristic
51D4⋊C8C2212yes—
52D4⋊C8C2212yes—
53C2×#41C2×C412yes—
54C2×#41D411yescharacteristic
55C22×D4C2×C411yescharacteristic
56C23×C4—21no—
57#131—44no—
58C2×M16—21no—
59C2×M16—22no—
60C2×M16—22no—
61C22×C8—21no—
62C22×C8—22no—
63C22×C8—21no—
64C4⋊Q8C2×C411yescharacteristic
65C4⋊Q8C2×C411yescharacteristic
66#126—22no—
67#126—22no—
68#126—22no—
69#125—22no—
70#125—21no—
71#125C2×C412yes—
72#124—22no—
73#124C2×C412yes—
74#123—22no—
75#123—22no—
76#123—22no—
77#123—22no—
78C22⋊Q8—21no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81C22⋊Q8—22no—
82C22⋊Q8C2×C412yes—
83C22⋊Q8C2×C412yes—
84#121—21no—
85#121—21no—
86#121—22no—
87#121C2×C412yes—
88#120—22no—
89C4×Q8—21no—
90C4×Q8C2×C412yes—
91C4×Q8C2×C411yescharacteristic
92C4×Q8D411yescharacteristic
93C4×Q8C2312yes—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4C2×C411yescharacteristic
100C4×D4C2×C411yescharacteristic
101C4×D4C2×C412yes—
102C4×D4C2×C412yes—
103C4×D4C2×C411yescharacteristic
104C4×D4D412yes—
105C4×D4D411yescharacteristic
106C4×D4C2311yescharacteristic
107C4×D4C2311yescharacteristic
108#117—21no—
109#117—21no—
110#117—21no—
111#117—22no—
112#117—22no—
113#117—22no—
114#117C2×C412yes—
115#117D411yescharacteristic
116#117D412yes—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4C2×C411yescharacteristic
120C2×C22⋊C4—21no—
121C2×C22⋊C4—22no—
122C2×C42—22no—
123C2×C42C2311yescharacteristic
124C2×C42C2312yes—
125C4⋊C8—21no—
126C4⋊C8—22no—
127C4⋊C8—21no—
128C4⋊C8—22no—
129C4⋊C8—21no—
130C4⋊C8—22no—
131C4⋊C8C2311yescharacteristic
132C4⋊C8C2311yescharacteristic
133C4⋊C8C2312yes—
134C22⋊C8—21no—
135C22⋊C8—22no—
136#97—22no—
137C4×C8—21no—
138C4×C8C2311yescharacteristic
139C4×C8C2311yescharacteristic
140C4×C8C2312yes—
141C24—21no—
142#41—22no—
143#41—22no—
144#41—24no—
145#41—42no—
146C2×Q8—21no—
147C2×Q8C22⋊C411yescharacteristic
148C2×Q8C22×C412yes—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—21no—
153C2×D4—41no—
154C2×D4C22⋊C412yes—
155C2×D4C22⋊C411yescharacteristic
156C2×D4C22×C411yescharacteristic
157C2×D4C22×C411yescharacteristic
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4C22×C411yescharacteristic
168C22×C4C22×C412yes—
169C22×C4C2×D412yes—
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D412yes—
172C22×C4C2×D411yescharacteristic
173M16—44no—
174M16—44no—
175M16—42no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—22no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—22no—
182C2×C8—42no—
183C2×C8—44no—
184C2×C8—42no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4—22no—
195C4⋊C4C22⋊C411yescharacteristic
196C4⋊C4C22⋊C411yescharacteristic
197C4⋊C4C22⋊C412yes—
198C4⋊C4C22⋊C412yes—
199C4⋊C4C22×C411yescharacteristic
200C4⋊C4C22×C411yescharacteristic
201C4⋊C4C22×C412yes—
202C22⋊C4—22no—
203C22⋊C4—22no—
204C22⋊C4—22no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—21no—
208C22⋊C4—21no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—21no—
212C22⋊C4—21no—
213C22⋊C4C22⋊C412yes—
214C22⋊C4C22⋊C412yes—
215C22⋊C4C22⋊C412yes—
216C42—22no—
217C42—22no—
218C42—22no—
219C42C22×C411yescharacteristic
220C42C22×C412yes—
221C42C2×D411yescharacteristic
222C42C2×D412yes—
223C42C2×D412yes—
224C42C2×D411yescharacteristic
225C42C2411yescharacteristic
226C23—21no—
227C23—21no—
228C23—42no—
229C23—42no—
230C23—81no—
231C23C2×C22⋊C412yes—
232C23C2×C22⋊C411yescharacteristic
233Q8—22no—
234Q8—22no—
235Q8—22no—
236Q8—81no—
237D4—24no—
238D4—21no—
239D4—21no—
240D4—21no—
241D4—22no—
242D4—42no—
243D4—44no—
244D4—82no—
245D4#13112yescomplemented
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4C2×C22⋊C412yes—
268C2×C4C2×C22⋊C412yes—
269C2×C4C2×C22⋊C411yescharacteristic
270C2×C4C2×C22⋊C412yes—
271C2×C4C2×C22⋊C411yescharacteristic
272C2×C4C2×C22⋊C411yescharacteristic
273C2×C4AGL(1,ℤ/8)12yes—
274C2×C4#13712yes—
275C2×C4C23×C411yescharacteristic
276C2×C4C22×D411yescharacteristic
277C2×C4C22×D411yescharacteristic
278C8—22no—
279C8—41no—
280C8—42no—
281C8—41no—
282C8—42no—
283C8—41no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—84no—
296C22#41712yes—
297C22C22×C22⋊C411yescharacteristic
298C4—22no—
299C4—21no—
300C4—22no—
301C4—21no—
302C4—22no—
303C4—21no—
304C4—42no—
305C4—41no—
306C4—41no—
307C4—42no—
308C4#41712yes—
309C4C2×#13111yescharacteristic
310C4#56811yescharacteristic
311C4C2×AGL(1,ℤ/8)11yescharacteristic
312C4C2×#13711yescharacteristic
313C2—22no—
314C2—42no—
315C2—42no—
316C2—81no—
317C2#288011yescharacteristic
318C2C2×#41711yescharacteristic
319C2#290111yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC42C24
socleC22C22×C22⋊C4
prime radical#3475C2
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. #2965C2semidirect
3. #2906C2semidirect
4. #1495C2semidirect
5. #1493C2semidirect
6. #1489C2semidirect
7. #1486C2semidirect
8. #1483C2semidirect
9. #1481C2semidirect
10. #1477C2semidirect
11. C2×D4⋊C8C2semidirect
34. #431C22semidirect
38. #423C22semidirect
40. C2×C4⋊C8C22semidirect
41. C4×M16C22semidirect
43. Q8⋊C8C22semidirect
44. Q8⋊C8C22semidirect
45. Q8⋊C8C22semidirect
46. Q8⋊C8C22semidirect
245. D4#131semidirect

Maximal subgroups

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

subgroupindex
#34752
#29652
#29062
#14952
#14932
#14892
#14862
#14832
#14812
#14772
C2×D4⋊C82