Finite Group Databasebeta

← #29689

Subgroups — #29689

Order 256.

550 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#529C211yesmaximal, characteristic
2C2×#475C211yesmaximal, characteristic, complemented
3C2×#474C211yesmaximal, characteristic, complemented
4#2857C211yesmaximal, characteristic, complemented
5#2059C212yesmaximal, complemented
6#2042C212yesmaximal, complemented
7#2036C212yesmaximal, complemented
8#2034C212yesmaximal
9#1803C211yesmaximal, characteristic, complemented
10#1802C211yesmaximal, characteristic, complemented
11#1797C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13#529—24no—
14#529—24no—
15C2×#126—21no—
16C2×#125C2211yescharacteristic
17C2×#124C2211yescharacteristic
18C2×#123—21no—
19C2×C22⋊Q8—21no—
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4—21no—
24C2×C4×D4C2211yescharacteristic
25C2×#117—21no—
26C22×C42C2211yescharacteristic
27#475—24no—
28#474—24no—
29C2×#107C2212yes—
30C2×#106C2212yes—
31C2×C4⋊C8C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×#102C2212yes—
36C2×#102C2212yes—
37C2×C22⋊C8C2212yes—
38#391—21no—
39#389—21no—
40#389C2212yes—
41#384C2211yescharacteristic
42#380—21no—
43#380C2212yes—
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8C2311yescharacteristic
49C22×D4—21no—
50C22×D4C2311yescharacteristic
51C23×C4—21no—
52C23×C4D411yescharacteristic
53C23×C4D411yescharacteristic
54C23×C4C2311yescharacteristic
55C22×C8C2312yes—
56C22×C8C2312yes—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59#126—28no—
60#125—22no—
61#125—22no—
62#124—22no—
63#124—22no—
64#123—28no—
65C22⋊Q8—22no—
66C22⋊Q8—22no—
67C22⋊Q8—44no—
68#121—22no—
69#121—22no—
70C4×Q8—22no—
71C4×Q8D414yes—
72C4×D4—28no—
73C4×D4—22no—
74C4×D4D414yes—
75#117—28no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4C2312yes—
85C2×C4⋊C4C2312yes—
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—22no—
94C2×C42—21no—
95C2×C42—22no—
96C2×C42—22no—
97C2×C42D411yescharacteristic
98C2×C42D411yescharacteristic
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#107—24no—
102#106—24no—
103C4⋊C8—22no—
104C4⋊C8—22no—
105#103—24no—
106#103—42no—
107#102—24no—
108#102—42no—
109C22⋊C8—42no—
110#95—22no—
111#95—22no—
112C24—21no—
113C24C2×D411yescharacteristic
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—21no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8C2×D412yes—
123C2×D4—24no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—42no—
130C2×D4—42no—
131C2×D4C2×D412yes—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—24no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—41no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2411yescharacteristic
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—22no—
164C4⋊C4—24no—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—44no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4C2×D412yes—
179C4⋊C4C2×D412yes—
180C22⋊C4—24no—
181C22⋊C4—24no—
182C22⋊C4—24no—
183C22⋊C4—24no—
184C22⋊C4—24no—
185C22⋊C4—44no—
186C42—22no—
187C42—22no—
188C42—24no—
189C42—21no—
190C42—21no—
191C42C2×D412yes—
192C42C2×D412yes—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—22no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23#13712yes—
207C23C22×D411yescharacteristic
208Q8—22no—
209Q8—22no—
210Q8—41no—
211Q8—41no—
212Q8—84no—
213D4—22no—
214D4—22no—
215D4—48no—
216D4—41no—
217D4—41no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—44no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—44no—
244C2×C4—48no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4#12012yes—
252C2×C4#12012yes—
253C2×C4#13512yes—
254C2×C4AGL(1,ℤ/8)12yes—
255C2×C4C22×D411yescharacteristic
256C2×C4C22×D411yescharacteristic
257C2×C4H(5,𝔽2)11yescharacteristic
258C2×C42−1+411yescharacteristic
259C2×C42−1+411yescharacteristic
260C2×C42−1+411yescharacteristic
261C8—42no—
262C8—42no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22C2×#12011yescharacteristic
278C22#53611yescharacteristic
279C22#53711yescharacteristic
280C22C2×#13511yescharacteristic
281C22C2×AGL(1,ℤ/8)11yescharacteristic
282C22C2×#13711yescharacteristic
283C22#57511yescharacteristic
284C4—22no—
285C4—41no—
286C4—42no—
287C4—41no—
288C4—42no—
289C4—41no—
290C4—42no—
291C4—82no—
292C4—82no—
293C4#44914yes—
294C4#53611yescharacteristic
295C4#53611yescharacteristic
296C4#53711yescharacteristic
297C4#53711yescharacteristic
298C2—42no—
299C2—82no—
300C2#261611yescharacteristic
301C2C2×#44911yescharacteristic
302C2#300711yescharacteristic
303C2#323311yescharacteristic
304C2#323911yescharacteristic
305C2#324811yescharacteristic
306C2#325111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#475C2semidirect
3. C2×#474C2semidirect
4. #2857C2semidirect
5. #2059C2semidirect
6. #2042C2semidirect
7. #2036C2semidirect
9. #1803C2semidirect
10. #1802C2semidirect
11. #1797C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
C2×#4752
C2×#4742
#28572
#20592
#20422
#20362
#20342
#18032
#18022
#17972