Finite Group Databasebeta

← #18674

Subgroups — #18674

Order 256.

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

normal subgroups only

This group has 588 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#3497C211yesmaximal, characteristic, complemented
2#3245C211yesmaximal, characteristic, complemented
3#3169C211yesmaximal, characteristic, complemented
4#3037C211yesmaximal, characteristic, complemented
5#1649C212yesmaximal, complemented
6#1641C212yesmaximal, complemented
7#1627C212yesmaximal, complemented
8#1621C212yesmaximal, complemented
9#1547C211yesmaximal, characteristic, complemented
10#1492C211yesmaximal, characteristic, complemented
11C2×Q8⋊C8C211yesmaximal, characteristic, complemented
12C2×2−1+4—21no—
13C2×#137—21no—
14#550—21no—
15#550C2212yes—
16D4×Q8—21no—
17D4×Q8C2212yes—
18#548—22no—
19#548—22no—
20#537—22no—
21#537—21no—
22#532—22no—
23#532C2211yescharacteristic
24#532C2211yescharacteristic
25#532C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27#500C2211yescharacteristic
28#493C2211yescharacteristic
29#486C2211yescharacteristic
30#486C2211yescharacteristic
31#474—21no—
32#474C2212yes—
33#466—22no—
34#466—22no—
35#464—22no—
36#462C2212yes—
37#460C2212yes—
38#459—21no—
39#459C2212yes—
40C4×D8—22no—
41#423C2211yescharacteristic
42C2×C4⋊C8C2211yescharacteristic
43#417—21no—
44C2×C4×C8C2211yescharacteristic
45#332C2212yes—
46#331C2212yes—
47Q8⋊C8C2212yes—
48Q8⋊C8C2212yes—
49D4⋊C8C2212yes—
50D4⋊C8C2212yes—
512−1+4—44no—
522−1+4—44no—
53C2×#41—24no—
54C2×#41—22no—
55C2×#41—22no—
56C2×#41—21no—
57C2×#41—21no—
58C2×#41—21no—
59C22×Q8—22no—
60C22×Q8—22no—
61C22×Q8D411yescharacteristic
62#137—44no—
63C2×Q16—22no—
64C2×SD16—22no—
65C2×D8—22no—
66C2×D8—22no—
67C2×M16—21no—
68C22×C8—21no—
69C22×C8—21no—
70C4⋊Q8—22no—
71C4⋊Q8D411yescharacteristic
72C4⋊Q8C2311yescharacteristic
73C4⋊Q8C2311yescharacteristic
74#127—22no—
75#127D411yescharacteristic
76#127C2311yescharacteristic
77#127C2311yescharacteristic
78#124—24no—
79#124—21no—
80#124—21no—
81#124—21no—
82C22⋊Q8—22no—
83C22⋊Q8—24no—
84C22⋊Q8—22no—
85C22⋊Q8—24no—
86#121—24no—
87#121—22no—
88#121—24no—
89#121—24no—
90#121—21no—
91#121—21no—
92#121—22no—
93#121—21no—
94#121—21no—
95#121D412yes—
96#121D412yes—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8D412yes—
100C4×Q8C2312yes—
101C4×D4—24no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—21no—
105C4×D4—21no—
106C4×D4—21no—
107C4×D4—21no—
108C4×D4D412yes—
109C4×D4C2312yes—
110C2×C4⋊C4—21no—
111C2×C4⋊C4D411yescharacteristic
112C2×C42—21no—
113C2×C42C2311yescharacteristic
114#107—22no—
115#107—22no—
116C4⋊C8—21no—
117C4⋊C8—21no—
118C4⋊C8C2312yes—
119C4⋊C8C2312yes—
120#103—22no—
121#102—22no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126C4×C8—21no—
127C4×C8C2311yescharacteristic
128C4×C8C2311yescharacteristic
129#41—416no—
130#41—44no—
131#41—48no—
132#41—44no—
133#41—42no—
134#41—42no—
135#41—42no—
136#41—42no—
137#41—42no—
138#41—42no—
139C2×Q8—24no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—22no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—44no—
146C2×Q8—48no—
147C2×Q8—44no—
148C2×Q8D814yescomplemented
149C2×Q8C2×D412yes—
150C2×D4—24no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—24no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—41no—
162C2×D4C2×D412yes—
163C2×D4C2×D412yes—
164C22×C4—21no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—41no—
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174Q16—44no—
175SD16—44no—
176D8—44no—
177D8—82no—
178M16—42no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—42no—
184C2×C8—81no—
185C4⋊C4—24no—
186C4⋊C4—22no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—22no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4C2×D412yes—
194C4⋊C4C2×D412yes—
195C4⋊C4C2×D412yes—
196C22⋊C4—24no—
197C22⋊C4—24no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—22no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C42—22no—
205C42—22no—
206C42C2×D411yescharacteristic
207C42C2×D411yescharacteristic
208C42C2×D411yescharacteristic
209C42C2411yescharacteristic
210C23—22no—
211C23—22no—
212C23—41no—
213C23#12011yescharacteristic
214Q8—22no—
215Q8—24no—
216Q8—48no—
217Q8—41no—
218Q8—41no—
219Q8—42no—
220Q8—41no—
221Q8—41no—
222Q8C2×D814yes—
223D4—22no—
224D4—22no—
225D4—42no—
226D4—48no—
227D4—42no—
228D4—44no—
229D4—44no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—42no—
239D4—81no—
240D4—84no—
241D4—82no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—48no—
251C2×C4—48no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—41no—
256C2×C4—81no—
257C2×C4—82no—
258C2×C4#12011yescharacteristic
259C2×C4#12011yescharacteristic
260C2×C4#12011yescharacteristic
261C2×C4C2×D812yes—
262C2×C4AGL(1,ℤ/8)12yes—
263C2×C4C22×D411yescharacteristic
264C2×C4C22×D411yescharacteristic
265C2×C4C22×D411yescharacteristic
266C8—22no—
267C8—41no—
268C8—42no—
269C8—42no—
270C22—21no—
271C22—21no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22#44712yes—
282C22C2×#12011yescharacteristic
283C4—24no—
284C4—21no—
285C4—21no—
286C4—21no—
287C4—42no—
288C4—42no—
289C4—42no—
290C4—81no—
291C4#44712yes—
292C4C22×D811yescharacteristic
293C4C2×AGL(1,ℤ/8)11yescharacteristic
294C4#57711yescharacteristic
295C4#57811yescharacteristic
296C2—22no—
297C2—82no—
298C2—82no—
299C2—161no—
300C2C2×#44711yescharacteristic
301C2#301311yescharacteristic
302C2#301811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
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
1. #3497C2semidirect
2. #3245C2semidirect
3. #3169C2semidirect
4. #3037C2semidirect
5. #1649C2semidirect
6. #1641C2semidirect
7. #1627C2semidirect
8. #1621C2semidirect
9. #1547C2semidirect
10. #1492C2semidirect
11. C2×Q8⋊C8C2semidirect
148. C2×Q8D8semidirect

Maximal subgroups

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

subgroupindex
#34972
#32452
#31692
#30372
#16492
#16412
#16272
#16212
#15472
#14922
C2×Q8⋊C82