Finite Group Databasebeta

← D4×#15668438 #37545419

Subgroups — D4×#15668438 #37545419

Order 23168.

546 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#15668438C212yesmaximal, complemented
2D4×Dic362C211yesmaximal, characteristic, complemented
3D4×D724C211yesmaximal, characteristic
4#26041652C211yesmaximal, characteristic, complemented
5#26041508C212yesmaximal, complemented
6#26041485C211yesmaximal, characteristic, complemented
7#26041480C211yesmaximal, characteristic, complemented
8#26041432C212yesmaximal, complemented
9#26041420C212yesmaximal, complemented
10C4×#15668438C211yesmaximal, characteristic, complemented
11C8×D4×C181C211yesmaximal, characteristic, complemented
12C22×Dic362C2212yescomplemented
13C22×D724C2212yes—
14C2×D4×D181—21no—
15C2×#15668438—28no—
16C2×#15668438C2212yes—
17C2×#15668438C2211yescharacteristic
18C2×#15668438C2212yescomplemented
19C2×#15668438C2212yescomplemented
20C2×#15668438C2212yescomplemented
21#19153816C2211yescharacteristic, complemented
22#19153815—21no—
23#19153813C2211yescharacteristic
24#19153795C2212yescomplemented
25#19153793—22no—
26#19153791—21no—
27#19153790C2212yes—
28#19153786—22no—
29C4×Dic362C2211yescharacteristic, complemented
30C4×D724C2211yescharacteristic
31D4×Dic181—21no—
32#19153748C2211yescharacteristic, complemented
33#19153741C2211yescharacteristic, complemented
34#19153741C2212yescomplemented
35#19153738C2211yescharacteristic
36#19153738C2212yescomplemented
37C22×C8×C181C2212yescomplemented
38C4×D4×C181C2211yescharacteristic
39C4⋊C8×C181C2211yescharacteristic, complemented
40C22⋊C8×C181C2212yescomplemented
41C4×C8×C181C2211yescharacteristic, complemented
42C23×D181—22no—
43C2×Dic362—24no—
44C2×Dic362—22no—
45C2×Dic362—42no—
46C2×Dic362C2312yes—
47C2×Dic362C2311yescharacteristic
48C2×D724—24no—
49C2×D724—22no—
50C2×D724—21no—
51C2×D724—21no—
52C2×D724—42no—
53C2×D724C2312yes—
54C2×D724C2311yescharacteristic
55C2×C22⋊D181—22no—
56D4×D181—44no—
57C22×Dic181—22no—
58C2×C4×D181—21no—
59#15668438—24no—
60#15668438—24no—
61#15668438—24no—
62#15668438—42no—
63#15668438D414yesdirect product factor
64C4⋊Dic181—21no—
65C4⋊Dic181—21no—
66C4⋊Dic181C2311yescharacteristic
67C4⋊Dic181C2311yescharacteristic
68#15668428—21no—
69#15668428—22no—
70C22⋊Dic181—22no—
71#15668426—21no—
72#15668426—22no—
73C4×Dic181—21no—
74C2×D4×C181D411yescharacteristic
75C22×C4×C181C2312yes—
76C2×C8×C181—24no—
77C2×C8×C181C2311yescharacteristic
78C2×C8×C181C2312yes—
79C2×C8×C181C2311yescharacteristic
80C4⋊C4×C181D411yescharacteristic
81C22⋊C4×C181D412yes—
82C42×C181C2311yescharacteristic
83C22×D181—21no—
84C22×D181—22no—
85C22×D181—44no—
86C22×D181—44no—
87C22×D181—82no—
88Dic362—21no—
89Dic362—22no—
90Dic362—22no—
91Dic362—44no—
92Dic362C2×D412yescomplemented
93D724—22no—
94D724—21no—
95D724—22no—
96D724—44no—
97D724—42no—
98D724—81no—
99D724C2×D412yes—
100C22⋊D181—44no—
101C2×Dic181—21no—
102C2×Dic181—21no—
103C2×Dic181—21no—
104C2×Dic181—22no—
105C2×Dic181—44no—
106C4×D181—42no—
107C23×C181C2×D412yes—
108D4×C181SD1614yescomplemented
109C2×C4×C181—22no—
110C2×C4×C181—22no—
111C2×C4×C181C2×D411yescharacteristic
112C2×C4×C181C2×D412yes—
113C2×C4×C181C2×D411yescharacteristic
114C2×C4×C181C2411yescharacteristic
115C8×C181—22no—
116C8×C181—21no—
117C8×C181C2×D412yescomplemented
118C22×C181—22no—
119C22×C181—22no—
120C22×C181C2×SD1614yescomplemented
121C22×C181C22×D411yescharacteristic
122C2×D181—22no—
123C2×D181—41no—
124C2×D181—41no—
125C2×D181—42no—
126C2×D181—42no—
127C2×D181—42no—
128C2×D181—88no—
129C4×C181—22no—
130C4×C181—21no—
131C4×C181C2×SD1612yescomplemented
132C4×C181C22×D411yescharacteristic
133C4×C181H(5,𝔽2)11yescharacteristic
134Dic181—22no—
135Dic181—42no—
136Dic181—41no—
137Dic181—41no—
138C2×C181—24no—
139C2×C181D4211yescharacteristic
140C2×C181C22×SD1611yescharacteristic
141C2×C181#57711yescharacteristic
142D181—42no—
143D181—82no—
144C181D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—1811nomaximal, Sylow
146C22×SD16—1812no—
147D4×Q8—1811no—
148D42—1811no—
149#492—1811no—
150#465—1812no—
151#461—1811no—
152#460—1811no—
153#450—1812no—
154#448—1812no—
155C4×SD16—1811no—
156C8×D4D18111yescharacteristic, complemented
157C22×Q8—1812no—
158C22×D4—3621no—
159C22×D4—1812no—
160C2×SD16—3628no—
161C2×SD16—1812no—
162C2×SD16—1812no—
163C2×SD16—1812no—
164C2×SD16—1811no—
165C2×SD16—1812no—
166C22×C8C2×D18112yescomplemented
167C4⋊Q8—3621no—
168C4⋊Q8—1811no—
169#127—1811no—
170C22⋊Q8—3622no—
171C22⋊Q8—1812no—
172#121—3621no—
173#121—1812no—
174#120—3622no—
175C4×Q8—1811no—
176C4×D4—3621no—
177C4×D4—1811no—
178C4×D4C2×D18111yescharacteristic
179#106—1811no—
180C4⋊C8C2×D18111yescharacteristic, complemented
181#103—1811no—
182#103—1812no—
183#102—1811no—
184#102—1812no—
185C22⋊C8C2×D18112yescomplemented
186C4×C8C2×D18111yescharacteristic, complemented
187C24—3622no—
188C2×Q8—3624no—
189C2×Q8—3622no—
190C2×Q8—1812no—
191C2×Q8—1811no—
192C2×Q8—7242no—
193C2×D4—3624no—
194C2×D4—3622no—
195C2×D4—3621no—
196C2×D4—3622no—
197C2×D4—3621no—
198C2×D4—1812no—
199C2×D4—1811no—
200C2×D4—7242no—
201C2×D4—7244no—
202C2×D4D72411yescharacteristic
203C22×C4—3622no—
204C22×C4—3621no—
205C22×C4C22×D18112yes—
206SD16—3624no—
207SD16—3624no—
208SD16—3624no—
209SD16—1814no—
210SD16—7242no—
211C2×C8—24no—
212C2×C8C22×D18111yescharacteristic
213C2×C8C22×D18112yes—
214C2×C8C22×D18111yescharacteristic
215C4⋊C4—3621no—
216C4⋊C4—3621no—
217C4⋊C4—3621no—
218C4⋊C4—3622no—
219C4⋊C4—1811no—
220C4⋊C4—1811no—
221C4⋊C4D72411yescharacteristic
222C22⋊C4—3622no—
223C22⋊C4—3621no—
224C22⋊C4—3622no—
225C22⋊C4D72412yes—
226C42—3621no—
227C42C22×D18111yescharacteristic
228C23—3621no—
229C23—3622no—
230C23—14482no—
231C23—7244no—
232C23—7244no—
233C23C2×D72412yes—
234Q8—3621no—
235Q8—3622no—
236Q8—3622no—
237Q8—1812no—
238Q8—7244no—
239D4—3622no—
240D4—3621no—
241D4—3622no—
242D4—14481no—
243D4—1812no—
244D4—7244no—
245D4—7244no—
246D4—7242no—
247D4#1566843814yesdirect product factor
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—3621no—
251C2×C4—3621no—
252C2×C4—3621no—
253C2×C4—3622no—
254C2×C4—7244no—
255C2×C4—7242no—
256C2×C4C2×D72411yescharacteristic
257C2×C4C2×D72412yes—
258C2×C4C2×D72411yescharacteristic
259C2×C4C23×D18111yescharacteristic
260C8—22no—
261C8—21no—
262C8D4×D18112yescomplemented
263C22—22no—
264C22—22no—
265C22—3622no—
266C22—14488no—
267C22—7242no—
268C22—7242no—
269C22—7242no—
270C22—7241no—
271C22—7241no—
272C22C2×#1566843814yescomplemented
273C22C22×D72411yescharacteristic
274C4—22no—
275C4—21no—
276C4—3622no—
277C4—7242no—
278C4—7241no—
279C4—7241no—
280C4C2×#1566843812yescomplemented
281C4C2×D4×D18111yescharacteristic
282C4#1915386211yescharacteristic
283C2—24no—
284C2—14482no—
285C2—7242no—
286C2D4×D72411yescharacteristic
287C2C22×#1566843811yescharacteristic
288C2#2604209111yescharacteristic
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC8×D4×C181C2
socleC22×C181C22×D4
prime radicalD4×D724C2
hypercenterC8×D4D181
hypocenterC181D4×SD16
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. C22×#15668438C2semidirect
2. D4×Dic362C2semidirect
4. #26041652C2semidirect
5. #26041508C2semidirect
6. #26041485C2semidirect
7. #26041480C2semidirect
8. #26041432C2semidirect
9. #26041420C2semidirect
10. C4×#15668438C2semidirect
11. C8×D4×C181C2semidirect
12. C22×Dic362C22semidirect
18. C2×#15668438C22semidirect
19. C2×#15668438C22semidirect
20. C2×#15668438C22semidirect
21. #19153816C22semidirect
24. #19153795C22semidirect
29. C4×Dic362C22semidirect
32. #19153748C22semidirect
33. #19153741C22semidirect
34. #19153741C22semidirect
36. #19153738C22semidirect
37. C22×C8×C181C22semidirect
39. C4⋊C8×C181C22semidirect
40. C22⋊C8×C181C22semidirect
41. C4×C8×C181C22semidirect
63. #15668438D4direct (N × H)
92. Dic362C2×D4semidirect
108. D4×C181SD16semidirect
117. C8×C181C2×D4semidirect
120. C22×C181C2×SD16semidirect
131. C4×C181C2×SD16semidirect
144. C181D4×SD16semidirect
156. C8×D4D181semidirect
166. C22×C8C2×D181semidirect
180. C4⋊C8C2×D181semidirect
185. C22⋊C8C2×D181semidirect
186. C4×C8C2×D181semidirect
247. D4#15668438direct (N × H)
262. C8D4×D181semidirect
272. C22C2×#15668438semidirect
280. C4C2×#15668438semidirect

Maximal subgroups

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

subgroupindex
C22×#156684382
D4×Dic3622
D4×D7242
#260416522
#260415082
#260414852
#260414802
#260414322
#260414202
C4×#156684382
C8×D4×C1812
D4×SD16181