Finite Group Databasebeta

← #24116

Subgroups — #24116

Order 256.

717 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 717 conjugacy classes of subgroups — too many to draw. Only its 147 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×#547C211yesmaximal, characteristic, complemented
2C2×#474C211yesmaximal, characteristic, complemented
3C2×C4×D8C211yesmaximal, characteristic, complemented
4#2537C211yesmaximal, characteristic, complemented
5#2033C212yesmaximal, complemented
6#1999C212yesmaximal, complemented
7#1990C211yesmaximal, characteristic, complemented
8#1954C211yesmaximal, characteristic, complemented
9#1874C212yesmaximal, complemented
10#1852C212yesmaximal, complemented
11#1802C211yesmaximal, characteristic, complemented
12C22×#41—21no—
13C22×D8C2211yescharacteristic
14#547—24no—
15#547—24no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#123—22no—
18C2×C22⋊Q8—21no—
19C2×#121—21no—
20C2×#121C2212yes—
21C2×#121C2212yes—
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C22×C4⋊C4C2212yes—
26#474—24no—
27C4×D8—24no—
28C2×#107C2211yescharacteristic
29C2×#107C2212yes—
30C2×C4⋊C8C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2212yes—
33C2×#102C2211yescharacteristic
34C2×#102C2212yes—
35C2×C22⋊C8C2212yes—
36C2×C22⋊C8C2212yes—
37C2×C4×C8C2211yescharacteristic
38#394—22no—
39#384C2211yescharacteristic
40#380—21no—
41#380C2212yes—
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C2×#41—28no—
46C2×#41—44no—
47C2×#41—44no—
48C22×Q8—21no—
49C22×D4—22no—
50C22×D4—22no—
51C22×D4D411yescharacteristic
52C22×D4C2311yescharacteristic
53C22×D4C2311yescharacteristic
54C23×C4—22no—
55C23×C4—21no—
56C23×C4—21no—
57C23×C4C2312yes—
58C2×D8—22no—
59C2×D8—22no—
60C2×D8—22no—
61C22×C8C2311yescharacteristic
62C22×C8C2312yes—
63C22×C8C2311yescharacteristic
64C4⋊Q8—22no—
65C4⋊Q8D414yescomplemented
66#123—216no—
67C22⋊Q8—28no—
68#121—24no—
69#121—24no—
70#121—24no—
71#121—42no—
72#121—44no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4D414yescomplemented
79C2×C4⋊C4—22no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2312yes—
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2312yes—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4D412yes—
96C2×C42C2311yescharacteristic
97#107—22no—
98#107—24no—
99C4⋊C8—22no—
100#102—22no—
101#102—22no—
102#102—24no—
103#102—42no—
104C22⋊C8—24no—
105C22⋊C8—42no—
106C4×C8—22no—
107#95—22no—
108C24—21no—
109C24—21no—
110C24C2×D412yes—
111#41—432no—
112C2×Q8—24no—
113C2×Q8—42no—
114C2×Q8—42no—
115C2×D4—28no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—44no—
129C2×D4—44no—
130C2×D4—44no—
131C2×D4—44no—
132C2×D4—82no—
133C2×D4D814yescomplemented
134C2×D4C2×D412yes—
135C2×D4#4112yescomplemented
136C22×C4—28no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4C2×D412yes—
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2411yescharacteristic
159D8—28no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—21no—
166C2×C8—22no—
167C2×C8—21no—
168C2×C8#4112yescomplemented
169C4⋊C4—28no—
170C4⋊C4—24no—
171C4⋊C4—22no—
172C4⋊C4—21no—
173C4⋊C4—22no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—81no—
180C4⋊C4C2×D412yes—
181C4⋊C4C2×D414yes—
182C22⋊C4—28no—
183C22⋊C4—24no—
184C22⋊C4—24no—
185C22⋊C4—44no—
186C22⋊C4—44no—
187C22⋊C4—42no—
188C42C2×D412yes—
189C42C2×D412yes—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23C2×D814yes—
211C23C22×D411yescharacteristic
212Q8—48no—
213D4—22no—
214D4—22no—
215D4—22no—
216D4—22no—
217D4—28no—
218D4—416no—
219D4—41no—
220D4—41no—
221D4—41no—
222D4—41no—
223D4—88no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—416no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—48no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—84no—
248C2×C4#12012yes—
249C2×C4#12012yes—
250C2×C4C2×D812yes—
251C2×C4AGL(1,ℤ/8)12yes—
252C2×C4C22×D411yescharacteristic
253C2×C4C22×D411yescharacteristic
254C2×C4C2×#4111yescharacteristic
255C2×C4C2×#4111yescharacteristic
256C2×C4H(5,𝔽2)11yescharacteristic
257C2×C42−1+411yescharacteristic
258C8—22no—
259C8—41no—
260C8—42no—
261C22—24no—
262C22—24no—
263C22—24no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22C2×#12011yescharacteristic
283C22#54611yescharacteristic
284C22#54711yescharacteristic
285C22C22×D811yescharacteristic
286C22C2×AGL(1,ℤ/8)11yescharacteristic
287C22#57711yescharacteristic
288C22#57811yescharacteristic
289C4—22no—
290C4—42no—
291C4—42no—
292C4—44no—
293C4—81no—
294C4—82no—
295C4#44714yes—
296C4#52511yescharacteristic
297C4#53711yescharacteristic
298C4#54811yescharacteristic
299C4#55011yescharacteristic
300C2—44no—
301C2—82no—
302C2—82no—
303C2#240711yescharacteristic
304C2C2×#44711yescharacteristic
305C2#301311yescharacteristic
306C2#329411yescharacteristic
307C2#333011yescharacteristic
308C2#333411yescharacteristic
309C2#335411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#547C2semidirect
2. C2×#474C2semidirect
3. C2×C4×D8C2semidirect
4. #2537C2semidirect
5. #2033C2semidirect
6. #1999C2semidirect
7. #1990C2semidirect
8. #1954C2semidirect
9. #1874C2semidirect
10. #1852C2semidirect
11. #1802C2semidirect
65. C4⋊Q8D4semidirect
78. C4×D4D4semidirect
133. C2×D4D8semidirect
135. C2×D4#41semidirect
168. C2×C8#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#4742
C2×C4×D82
#25372
#20332
#19992
#19902
#19542
#18742
#18522
#18022