Finite Group Databasebeta

← #18883

Subgroups — #18883

Order 256.

548 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 129 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#3499C211yesmaximal, characteristic
2#3246C211yesmaximal, characteristic, complemented
3#3229C211yesmaximal, characteristic, complemented
4#3076C211yesmaximal, characteristic, complemented
5#1654C212yesmaximal, complemented
6#1651C212yesmaximal, complemented
7#1637C212yesmaximal, complemented
8#1635C212yesmaximal, complemented
9C2×#332C211yesmaximal, characteristic, complemented
10#1480C211yesmaximal, characteristic, complemented
11#1477C211yesmaximal, characteristic, complemented
12C2×2−1+4—21no—
13D4×Q8—21no—
14D4×Q8C2212yes—
15#547—22no—
16#547—21no—
17#547—22no—
18#547C2212yes—
19#536—22no—
20#536—21no—
21#533C2211yescharacteristic
22#532—21no—
23#532C2211yescharacteristic
24C2×C4⋊Q8—21no—
25C2×C4⋊Q8C2211yescharacteristic
26C4×#41C2211yescharacteristic
27#498C2212yes—
28#487C2212yes—
29#483—22no—
30#479—21no—
31#478—21no—
32#477C2211yescharacteristic
33#476C2211yescharacteristic
34#476C2212yes—
35#475—22no—
36#475—21no—
37#475C2211yescharacteristic
38#474C2211yescharacteristic
39#465—22no—
40#461C2212yes—
41#460—21no—
42#431C2211yescharacteristic
43#427—21no—
44C2×C4⋊C8C2211yescharacteristic
45C2×C4⋊C8C2211yescharacteristic
46#416—21no—
47#332C2212yes—
48#332C2212yes—
49Q8⋊C8C2212yes—
50D4⋊C8C2212yes—
51D4⋊C8C2212yes—
52D4⋊C8C2212yes—
532−1+4—44no—
542−1+4—44no—
55C2×#41—22no—
56C2×#41—24no—
57C2×#41—22no—
58C2×#41—21no—
59C2×#41D411yescharacteristic
60C22×Q8—22no—
61C22×Q8—22no—
62C22×Q8—21no—
63C2×SD16—22no—
64C22×C8—21no—
65C22×C8—21no—
66C4⋊Q8—22no—
67C4⋊Q8—22no—
68C4⋊Q8—22no—
69C4⋊Q8—21no—
70C4⋊Q8—21no—
71C4⋊Q8D411yescharacteristic
72C4⋊Q8D412yes—
73C4⋊Q8C2311yescharacteristic
74C4⋊Q8C2311yescharacteristic
75C4⋊Q8C2312yes—
76#127D411yescharacteristic
77#125—21no—
78#124—22no—
79#124—21no—
80#123—24no—
81#123—24no—
82#123—24no—
83C22⋊Q8—24no—
84C22⋊Q8—22no—
85C22⋊Q8—24no—
86C22⋊Q8—22no—
87C22⋊Q8—21no—
88C22⋊Q8—21no—
89#121—22no—
90#121—22no—
91#121—21no—
92#121D412yes—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8D411yescharacteristic
96C4×Q8C2311yescharacteristic
97C4×D4—22no—
98C4×D4—22no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4D411yescharacteristic
103C4×D4C2311yescharacteristic
104C4×D4C2312yes—
105#117—21no—
106#117—21no—
107#117D411yescharacteristic
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4D412yes—
112C2×C42—21no—
113C2×C42C2311yescharacteristic
114#107—21no—
115#107—21no—
116#106—21no—
117#106—21no—
118#106—22no—
119#106—24no—
120C4⋊C8—21no—
121C4⋊C8—21no—
122C4⋊C8—21no—
123C4⋊C8C2311yescharacteristic
124C4⋊C8C2311yescharacteristic
125C4⋊C8C2312yes—
126#103—21no—
127#103—21no—
128#103—22no—
129#103—24no—
130#102—21no—
131#102—21no—
132#102—22no—
133C22⋊C8—21no—
134C4×C8C2312yes—
135#41—22no—
136#41—48no—
137#41—416no—
138#41—44no—
139#41—44no—
140#41—42no—
141#41—42no—
142#41SD1614yescomplemented
143C2×Q8—24no—
144C2×Q8—22no—
145C2×Q8—21no—
146C2×Q8—22no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—48no—
150C2×Q8—44no—
151C2×Q8—44no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8C2×D411yescharacteristic
155C2×D4—21no—
156C2×D4—24no—
157C2×D4—22no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4C2×D412yes—
161C2×D4C2×D411yescharacteristic
162C22×C4—21no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172SD16—82no—
173C2×C8—22no—
174C2×C8—21no—
175C2×C8—22no—
176C2×C8—21no—
177C2×C8—42no—
178C2×C8—81no—
179C4⋊C4—24no—
180C4⋊C4—24no—
181C4⋊C4—21no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—21no—
185C4⋊C4—22no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4C2×D412yes—
194C4⋊C4C2×D414yes—
195C4⋊C4C2×D411yescharacteristic
196C4⋊C4C2×D411yescharacteristic
197C22⋊C4—24no—
198C22⋊C4—24no—
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—22no—
202C22⋊C4—21no—
203C22⋊C4—22no—
204C22⋊C4—41no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—41no—
209C42C2×D411yescharacteristic
210C42C2×D411yescharacteristic
211C42C2×D411yescharacteristic
212C42C2411yescharacteristic
213C23—21no—
214C23—22no—
215C23#12011yescharacteristic
216Q8—21no—
217Q8—48no—
218Q8—42no—
219Q8—42no—
220Q8—42no—
221Q8—42no—
222Q8—44no—
223Q8—41no—
224Q8—41no—
225Q8C2×SD1612yes—
226D4—24no—
227D4—21no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—48no—
232D4—44no—
233D4—42no—
234D4C2×SD1612yes—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—24no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—48no—
246C2×C4—48no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4#12011yescharacteristic
255C2×C4#12011yescharacteristic
256C2×C4#12011yescharacteristic
257C2×C4C2×SD1612yes—
258C2×C4AGL(1,ℤ/8)12yes—
259C2×C4C22×D411yescharacteristic
260C2×C4C22×D411yescharacteristic
261C2×C4C22×D411yescharacteristic
262C8—42no—
263C8—41no—
264C8—41no—
265C8—42no—
266C22—21no—
267C22—21no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22#45012yes—
275C22C2×#12011yescharacteristic
276C4—22no—
277C4—21no—
278C4—21no—
279C4—21no—
280C4—44no—
281C4—42no—
282C4—41no—
283C4—41no—
284C4—81no—
285C4—81no—
286C4#45012yes—
287C4C22×SD1611yescharacteristic
288C4C2×AGL(1,ℤ/8)11yescharacteristic
289C4#57711yescharacteristic
290C4#57811yescharacteristic
291C2—22no—
292C2—42no—
293C2—82no—
294C2C2×#45011yescharacteristic
295C2#301311yescharacteristic
296C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3499C2
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. #3246C2semidirect
3. #3229C2semidirect
4. #3076C2semidirect
5. #1654C2semidirect
6. #1651C2semidirect
7. #1637C2semidirect
8. #1635C2semidirect
9. C2×#332C2semidirect
10. #1480C2semidirect
11. #1477C2semidirect
142. #41SD16semidirect

Maximal subgroups

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

subgroupindex
#34992
#32462
#32292
#30762
#16542
#16512
#16372
#16352
C2×#3322
#14802
#14772