Finite Group Databasebeta

← #18699

Subgroups — #18699

Order 256.

439 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 115 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#3552C211yesmaximal, characteristic
2#3239C211yesmaximal, characteristic, complemented
3#3218C211yesmaximal, characteristic, complemented
4C2×#500C211yesmaximal, characteristic, complemented
5#1650C212yesmaximal, complemented
6#1649C212yesmaximal, complemented
7#1648C212yesmaximal, complemented
8#1647C212yesmaximal, complemented
9#1556C211yesmaximal, characteristic, complemented
10#1494C211yesmaximal, characteristic, complemented
11#1494C211yesmaximal, characteristic, complemented
12#563—22no—
13#557C2211yescharacteristic
14#556C2211yescharacteristic
15#554—21no—
16#553—21no—
17D4×Q8—21no—
18D4×Q8C2211yescharacteristic
19#548—21no—
20#547C2211yescharacteristic
21#544—21no—
22#542—21no—
23#541—21no—
24#539—22no—
25#536—21no—
26C2×C4⋊Q8C2211yescharacteristic
27C2×C4⋊Q8C2211yescharacteristic
28#529C2211yescharacteristic
29#529C2211yescharacteristic
30#501—21no—
31#500—22no—
32#500C2212yes—
33#500C2212yes—
34#497—21no—
35#491—21no—
36#489—21no—
37#484—21no—
38#481—21no—
39#475C2212yes—
40#474C2212yes—
41#462C2212yes—
42#461C2212yes—
43#451—21no—
44#448—21no—
45#432C2211yescharacteristic
46#432C2211yescharacteristic
47C2×#107—22no—
48C2×C4×C8C2211yescharacteristic
49#332C2212yes—
50#332C2212yes—
51Q8⋊C8C2212yes—
52Q8⋊C8C2212yes—
53D4⋊C8C2212yes—
54D4⋊C8C2212yes—
55C2×#41—21no—
56C22×Q8—21no—
57C22×Q8—21no—
58C22×Q8—21no—
59C2×Q16—22no—
60C2×SD16—22no—
61C22×C8—21no—
62C4⋊Q8—22no—
63C4⋊Q8—21no—
64C4⋊Q8—21no—
65C4⋊Q8—21no—
66C4⋊Q8—21no—
67C4⋊Q8—21no—
68C4⋊Q8—21no—
69C4⋊Q8—21no—
70C4⋊Q8D412yes—
71C4⋊Q8C2312yes—
72C4⋊Q8C2311yescharacteristic
73C4⋊Q8C2311yescharacteristic
74#126—22no—
75#126—22no—
76#126—22no—
77#126—21no—
78#126—21no—
79#125—22no—
80#125—22no—
81#125—22no—
82#125—21no—
83#125D411yescharacteristic
84#125D411yescharacteristic
85#124—22no—
86#124—21no—
87#124D411yescharacteristic
88#124D411yescharacteristic
89#123—22no—
90#123—22no—
91#123—21no—
92#123—21no—
93#123—22no—
94C22⋊Q8—22no—
95C22⋊Q8—22no—
96C22⋊Q8—22no—
97C22⋊Q8—22no—
98C22⋊Q8—22no—
99C22⋊Q8—21no—
100C22⋊Q8—22no—
101C22⋊Q8—21no—
102C22⋊Q8—21no—
103C22⋊Q8D411yescharacteristic
104C22⋊Q8D411yescharacteristic
105#121—22no—
106#121—21no—
107#121D411yescharacteristic
108#121D411yescharacteristic
109C4×Q8—21no—
110C4×Q8—21no—
111C4×Q8—21no—
112C4×Q8C2311yescharacteristic
113C4×Q8C2311yescharacteristic
114C4×D4—22no—
115C4×D4—22no—
116C4×D4—21no—
117C4×D4—21no—
118C4×D4—21no—
119C4×D4C2311yescharacteristic
120C4×D4C2311yescharacteristic
121#117—21no—
122#117—21no—
123#117—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4D412yes—
129C2×C42C2311yescharacteristic
130#107—24no—
131#107—24no—
132#107—22no—
133#106—22no—
134C4⋊C8C2312yes—
135C4⋊C8C2312yes—
136#103—22no—
137#103—22no—
138#103—22no—
139#102—22no—
140C22⋊C8—21no—
141C22⋊C8—21no—
142#97—21no—
143#97—21no—
144C4×C8—21no—
145C4×C8C2312yes—
146#41—82no—
147C2×Q8—22no—
148C2×Q8—22no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—22no—
152C2×Q8—41no—
153C2×Q8—82no—
154C2×Q8—82no—
155C2×Q8—82no—
156C2×Q8C2×D411yescharacteristic
157C2×Q8C2×D411yescharacteristic
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—22no—
161C2×D4C2×D411yescharacteristic
162C2×D4C2×D411yescharacteristic
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172Q16—82no—
173SD16—82no—
174C2×C8—24no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D411yescharacteristic
203C4⋊C4C2×D411yescharacteristic
204C4⋊C4C2×D412yes—
205C4⋊C4C2×D412yes—
206C22⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—22no—
209C22⋊C4—22no—
210C22⋊C4—21no—
211C22⋊C4—21no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—21no—
218C22⋊C4—21no—
219C22⋊C4—21no—
220C42—22no—
221C42—22no—
222C42—21no—
223C42—21no—
224C42C2×D411yescharacteristic
225C42C2×D411yescharacteristic
226C42C2×D411yescharacteristic
227C42C2411yescharacteristic
228C23—21no—
229C23—21no—
230C23#12011yescharacteristic
231Q8—22no—
232Q8—22no—
233Q8—42no—
234Q8—42no—
235Q8—42no—
236Q8—42no—
237Q8—82no—
238Q8—82no—
239Q8—82no—
240D4—22no—
241D4—22no—
242D4—42no—
243D4—42no—
244D4—82no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4#12011yescharacteristic
264C2×C4#12011yescharacteristic
265C2×C4#12011yescharacteristic
266C2×C4C22×D411yescharacteristic
267C2×C4C22×D411yescharacteristic
268C2×C4C22×D411yescharacteristic
269C8—24no—
270C8—42no—
271C8—42no—
272C22—21no—
273C22—21no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22#45612yes—
281C22C2×#12011yescharacteristic
282C4—21no—
283C4—21no—
284C4—21no—
285C4—21no—
286C4—41no—
287C4—42no—
288C4—41no—
289C4—42no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—82no—
295C4#57612yes—
296C4#57812yes—
297C2—22no—
298C2—81no—
299C2—81no—
300C2#300811yescharacteristic
301C2#301111yescharacteristic
302C2C2×#45611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3552C2
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. #3239C2semidirect
3. #3218C2semidirect
4. C2×#500C2semidirect
5. #1650C2semidirect
6. #1649C2semidirect
7. #1648C2semidirect
8. #1647C2semidirect
9. #1556C2semidirect
10. #1494C2semidirect
11. #1494C2semidirect

Maximal subgroups

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

subgroupindex
#35522
#32392
#32182
C2×#5002
#16502
#16492
#16482
#16472
#15562
#14942