Finite Group Databasebeta

← #29696

Subgroups — #29696

Order 256.

506 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 127 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×#542C211yesmaximal, characteristic
2C2×#477C211yesmaximal, characteristic, complemented
3C2×#474C211yesmaximal, characteristic, complemented
4#2628C211yesmaximal, characteristic, complemented
5#2057C212yesmaximal, complemented
6#2042C212yesmaximal, complemented
7#2036C212yesmaximal, complemented
8#2033C212yesmaximal
9#1805C211yesmaximal, characteristic, complemented
10#1804C211yesmaximal, characteristic, complemented
11#1798C211yesmaximal, characteristic, complemented
12#542—24no—
13#542—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#126—21no—
17C2×#124—21no—
18C2×#124C2211yescharacteristic
19C2×#123—21no—
20C2×C22⋊Q8—21no—
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×#117C2211yescharacteristic
26#477—24no—
27#474—24no—
28C2×#107C2212yes—
29C2×#107C2212yes—
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×C22⋊C8C2212yes—
37#400—21no—
38#399—21no—
39#395—21no—
40#393—21no—
41#389C2212yes—
42#382C2211yescharacteristic
43#380C2212yes—
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8—21no—
49C22×Q8C2311yescharacteristic
50C22×D4—21no—
51C22×D4C2311yescharacteristic
52C23×C4—21no—
53C23×C4C2311yescharacteristic
54C22×C8C2312yes—
55C22×C8C2312yes—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60#126—28no—
61#124—28no—
62#124—22no—
63#124—22no—
64#123—28no—
65C22⋊Q8—28no—
66C22⋊Q8—22no—
67C22⋊Q8—22no—
68#121—22no—
69#121—22no—
70C4×Q8—22no—
71C4×Q8D414yes—
72C4×D4—22no—
73C4×D4D414yes—
74#117—22no—
75#117—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4D411yescharacteristic
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4C2312yes—
84C2×C4⋊C4C2312yes—
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4D411yescharacteristic
94C2×C22⋊C4D411yescharacteristic
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98#107—24no—
99#107—24no—
100C4⋊C8—22no—
101C4⋊C8—22no—
102#103—24no—
103#103—42no—
104#102—24no—
105#102—42no—
106C22⋊C8—42no—
107#95—22no—
108#95—21no—
109#95—21no—
110#95—22no—
111C24—21no—
112C24C2×D411yescharacteristic
113C2×Q8—24no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—21no—
119C2×Q8—42no—
120C2×Q8—82no—
121C2×Q8—82no—
122C2×Q8C2×D412yes—
123C2×D4—24no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—82no—
130C2×D4C2×D412yes—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—24no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—82no—
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2411yescharacteristic
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—24no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—42no—
168C4⋊C4—82no—
169C4⋊C4C2×D412yes—
170C4⋊C4C2×D412yes—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—24no—
174C22⋊C4—24no—
175C22⋊C4—24no—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C22⋊C4—82no—
179C42—24no—
180C42—21no—
181C42—21no—
182C42C2×D412yes—
183C42C2×D412yes—
184C23—21no—
185C23—22no—
186C23—21no—
187C23—21no—
188C23—22no—
189C23—22no—
190C23—41no—
191C23—41no—
192C23—42no—
193C23—42no—
194C23C22×D411yescharacteristic
195Q8—22no—
196Q8—22no—
197Q8—41no—
198Q8—41no—
199Q8—84no—
200Q8—84no—
201D4—22no—
202D4—22no—
203D4—41no—
204D4—41no—
205D4—84no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—84no—
233C2×C4#12012yes—
234C2×C4#12012yes—
235C2×C4C22×D411yescharacteristic
236C2×C4C22×D411yescharacteristic
237C2×C4H(5,𝔽2)11yescharacteristic
238C2×C42−1+411yescharacteristic
239C2×C42−1+411yescharacteristic
240C2×C42−1+411yescharacteristic
241C8—42no—
242C8—42no—
243C22—41no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—82no—
252C22—82no—
253C22C2×#12011yescharacteristic
254C22#53611yescharacteristic
255C22#53711yescharacteristic
256C22#57612yes—
257C22#57812yes—
258C4—22no—
259C4—22no—
260C4—41no—
261C4—42no—
262C4—41no—
263C4—42no—
264C4—42no—
265C4—41no—
266C4—82no—
267C4—82no—
268C4#53611yescharacteristic
269C4#53611yescharacteristic
270C4#53711yescharacteristic
271C4#53711yescharacteristic
272C2—81no—
273C2—82no—
274C2#261611yescharacteristic
275C2#300811yescharacteristic
276C2#301111yescharacteristic
277C2#324312yes—
278C2#326212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#477C2semidirect
3. C2×#474C2semidirect
4. #2628C2semidirect
5. #2057C2semidirect
6. #2042C2semidirect
7. #2036C2semidirect
9. #1805C2semidirect
10. #1804C2semidirect
11. #1798C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#4772
C2×#4742
#26282
#20572
#20422
#20362
#20332
#18052
#18042
#17982