Finite Group Databasebeta

← #18587

Subgroups — #18587

Order 256.

472 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3520C211yesmaximal, characteristic
2#3218C211yesmaximal, characteristic, complemented
3#3165C211yesmaximal, characteristic, complemented
4C2×#474C211yesmaximal, characteristic, complemented
5#1654C212yesmaximal, complemented
6#1652C212yesmaximal, complemented
7#1649C212yesmaximal, complemented
8#1647C212yesmaximal, complemented
9#1552C211yesmaximal, characteristic, complemented
10#1490C211yesmaximal, characteristic, complemented
11#1483C211yesmaximal, characteristic, complemented
12#556C2211yescharacteristic
13#554—21no—
14#552—21no—
15#551—22no—
16D4×Q8C2211yescharacteristic
17#548—21no—
18#547C2211yescharacteristic
19#547C2211yescharacteristic
20#546—21no—
21#542—21no—
22#539—22no—
23#533C2211yescharacteristic
24C2×C4⋊Q8C2211yescharacteristic
25#529C2211yescharacteristic
26#525—21no—
27C2×C22⋊Q8—21no—
28#520—21no—
29C2×C4×D4C2211yescharacteristic
30#501—22no—
31#500C2212yescomplemented
32#498C2212yescomplemented
33#497—21no—
34#491—21no—
35#474—21no—
36#474—21no—
37#474C2212yes—
38#474C2212yes—
39#462C2212yescomplemented
40#461C2212yescomplemented
41#451—21no—
42#448—21no—
43#432C2211yescharacteristic
44#428—22no—
45C2×#107—21no—
46C2×C4⋊C8C2211yescharacteristic
47C2×#102—21no—
48C4×M16C2211yescharacteristic
49#332C2212yescomplemented
50#332C2212yescomplemented
51Q8⋊C8C2212yescomplemented
52D4⋊C8C2212yes—
53D4⋊C8C2212yes—
54D4⋊C8C2212yescomplemented
55C2×#41—21no—
56C2×#41—21no—
57C22×Q8—21no—
58C22×Q8—21no—
59C22×D4D411yescharacteristic
60C23×C4—21no—
61C2×Q16—22no—
62C2×SD16—22no—
63C2×M16—21no—
64C22×C8—21no—
65C4⋊Q8—21no—
66C4⋊Q8—21no—
67C4⋊Q8—21no—
68C4⋊Q8—21no—
69C4⋊Q8C2312yes—
70C4⋊Q8C2311yescharacteristic
71C4⋊Q8C2311yescharacteristic
72#126—22no—
73#126—22no—
74#126—21no—
75#125—22no—
76#125D411yescharacteristic
77#125D412yes—
78#124—22no—
79#124—21no—
80#124D411yescharacteristic
81#123—22no—
82#123—22no—
83#123—21no—
84#123—22no—
85#123—22no—
86C22⋊Q8—22no—
87C22⋊Q8—22no—
88C22⋊Q8—22no—
89C22⋊Q8—21no—
90C22⋊Q8—21no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93C22⋊Q8—21no—
94C22⋊Q8—22no—
95C22⋊Q8—21no—
96C22⋊Q8D411yescharacteristic
97#121—22no—
98#121—21no—
99#121—21no—
100#121D411yescharacteristic
101#120—22no—
102C4×Q8—21no—
103C4×Q8—21no—
104C4×Q8—21no—
105C4×Q8C2311yescharacteristic
106C4×D4—22no—
107C4×D4—22no—
108C4×D4—22no—
109C4×D4—21no—
110C4×D4—21no—
111C4×D4—21no—
112C4×D4—21no—
113C4×D4D411yescharacteristic
114C4×D4D411yescharacteristic
115C4×D4C2311yescharacteristic
116C4×D4C2311yescharacteristic
117C4×D4C2311yescharacteristic
118#117—22no—
119#117—21no—
120#117—21no—
121#117D412yes—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4D411yescharacteristic
128C2×C22⋊C4—22no—
129C2×C22⋊C4—21no—
130C2×C42C2311yescharacteristic
131#107—22no—
132#107—22no—
133#107—24no—
134#106—24no—
135C4⋊C8—21no—
136C4⋊C8C2312yes—
137C4⋊C8C2312yes—
138#103—22no—
139#103—22no—
140#102—22no—
141#102—22no—
142C22⋊C8—21no—
143#97—21no—
144#97—21no—
145C4×C8C2312yes—
146C24—21no—
147#41—42no—
148#41—42no—
149#41—82no—
150C2×Q8—22no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—22no—
154C2×Q8—41no—
155C2×Q8—82no—
156C2×Q8—82no—
157C2×Q8C2×D411yescharacteristic
158C2×D4—22no—
159C2×D4—22no—
160C2×D4—21no—
161C2×D4—22no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4C2×D411yescharacteristic
165C2×D4C2×D411yescharacteristic
166C2×D4C2×D411yescharacteristic
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—41no—
177C22×C4—82no—
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181Q16—82no—
182SD16—82no—
183M16—24no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—42no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—21no—
193C4⋊C4—22no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—22no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4C2×D411yescharacteristic
206C4⋊C4C2×D411yescharacteristic
207C4⋊C4C2×D412yes—
208C4⋊C4C2×D411yescharacteristic
209C4⋊C4C2×D411yescharacteristic
210C4⋊C4C2×D412yes—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—22no—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—21no—
217C22⋊C4—22no—
218C22⋊C4—21no—
219C22⋊C4—21no—
220C22⋊C4—21no—
221C22⋊C4—21no—
222C22⋊C4—21no—
223C22⋊C4—21no—
224C22⋊C4—21no—
225C22⋊C4—21no—
226C42—22no—
227C42—22no—
228C42—21no—
229C42C2×D411yescharacteristic
230C42C2×D411yescharacteristic
231C42C2×D411yescharacteristic
232C42C2411yescharacteristic
233C23—21no—
234C23—21no—
235C23—21no—
236C23—42no—
237C23—42no—
238C23—81no—
239C23#12011yescharacteristic
240Q8—22no—
241Q8—41no—
242Q8—41no—
243Q8—42no—
244Q8—42no—
245Q8—42no—
246Q8—82no—
247Q8—82no—
248D4—21no—
249D4—21no—
250D4—21no—
251D4—22no—
252D4—42no—
253D4—44no—
254D4—42no—
255D4—82no—
256D4#13712yes—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—44no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4#12011yescharacteristic
281C2×C4#12011yescharacteristic
282C2×C4#12011yescharacteristic
283C2×C4C22×D411yescharacteristic
284C2×C4C22×D411yescharacteristic
285C2×C4C22×D411yescharacteristic
286C8—24no—
287C8—42no—
288C8—42no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—84no—
301C22#45612yes—
302C22C2×#12011yescharacteristic
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—41no—
308C4—42no—
309C4—41no—
310C4—41no—
311C4—42no—
312C4—41no—
313C4—41no—
314C4—82no—
315C4C2×#13711yescharacteristic
316C4#57511yescharacteristic
317C4#57612yes—
318C2—22no—
319C2—42no—
320C2—81no—
321C2—81no—
322C2#300711yescharacteristic
323C2#300811yescharacteristic
324C2C2×#45611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3520C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3218C2semidirect
3. #3165C2semidirect
4. C2×#474C2semidirect
5. #1654C2semidirect
6. #1652C2semidirect
7. #1649C2semidirect
8. #1647C2semidirect
9. #1552C2semidirect
10. #1490C2semidirect
11. #1483C2semidirect
31. #500C22semidirect
32. #498C22semidirect
39. #462C22semidirect
40. #461C22semidirect
49. #332C22semidirect
50. #332C22semidirect
51. Q8⋊C8C22semidirect
54. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35202
#32182
#31652
C2×#4742
#16542
#16522
#16492
#16472
#15522
#14902
#14832