Finite Group Databasebeta

← D4×#15668437 #37545413

Subgroups — D4×#15668437 #37545413

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×#15668437C212yesmaximal, complemented
2D4×Dic362C211yesmaximal, characteristic, complemented
3#26041651C211yesmaximal, characteristic, complemented
4#26041506C212yesmaximal, complemented
5#26041484C211yesmaximal, characteristic, complemented
6#26041476C211yesmaximal, characteristic, complemented
7#26041430C212yesmaximal, complemented
8#26041417C212yesmaximal, complemented
9C4×#15668437C211yesmaximal, characteristic, complemented
10D4×#13089678C211yesmaximal, characteristic, complemented
11D42×C181C211yesmaximal, characteristic
12C22×Dic362C2212yescomplemented
13C2×#15668437—28no—
14C2×#15668437C2212yes—
15C2×#15668437C2211yescharacteristic
16C2×#15668437C2212yescomplemented
17C2×#15668437C2212yescomplemented
18C2×#15668437C2212yescomplemented
19C22×#13089678C2212yescomplemented
20#19153816C2211yescharacteristic, complemented
21#19153815—21no—
22#19153795C2212yescomplemented
23#19153793—22no—
24C4×Dic362C2211yescharacteristic, complemented
25D4×Dic181—21no—
26#19153747C2211yescharacteristic, complemented
27C4⋊#13089678C2211yescharacteristic, complemented
28#19153739C2211yescharacteristic, complemented
29#19153739C2212yescomplemented
30#19153737C2211yescharacteristic
31#19153737C2212yescomplemented
32C22⋊#13089678C2212yescomplemented
33C4×#13089678C2211yescharacteristic, complemented
34C22×D4×C181—21no—
35C22×D4×C181C2212yes—
36#127×C181C2211yescharacteristic
37#121×C181—21no—
38#121×C181C2212yes—
39#120×C181—22no—
40C4×D4×C181C2211yescharacteristic
41C4×D4×C181C2211yescharacteristic
42C2×Dic362—24no—
43C2×Dic362—22no—
44C2×Dic362—42no—
45C2×Dic362C2312yes—
46C2×Dic362C2311yescharacteristic
47C22×Dic181—22no—
48#15668437—24no—
49#15668437—24no—
50#15668437—24no—
51#15668437—42no—
52#15668437D414yesdirect product factor
53C2×#13089678—24no—
54C2×#13089678C2311yescharacteristic
55C2×#13089678C2312yes—
56C2×#13089678C2311yescharacteristic
57C4⋊Dic181—21no—
58C4⋊Dic181—21no—
59C4⋊Dic181C2311yescharacteristic
60#15668428—21no—
61#15668428—22no—
62C22⋊Dic181—22no—
63C4×Dic181—21no—
64C24×C181—22no—
65C2×D4×C181—24no—
66C2×D4×C181—22no—
67C2×D4×C181—22no—
68C2×D4×C181—21no—
69C2×D4×C181—21no—
70C2×D4×C181—42no—
71C2×D4×C181—44no—
72C2×D4×C181D411yescharacteristic
73C2×D4×C181C2312yes—
74C2×D4×C181C2311yescharacteristic
75C22×C4×C181—21no—
76C22×C4×C181C2312yes—
77C4⋊C4×C181D411yescharacteristic
78C4⋊C4×C181C2311yescharacteristic
79C22⋊C4×C181—21no—
80C22⋊C4×C181—22no—
81C22⋊C4×C181D412yes—
82C42×C181C2311yescharacteristic
83Dic362—21no—
84Dic362—22no—
85Dic362—22no—
86Dic362—44no—
87Dic362C2×D412yescomplemented
88C2×Dic181—21no—
89C2×Dic181—21no—
90C2×Dic181—22no—
91C2×Dic181—44no—
92#13089678—22no—
93#13089678—21no—
94#13089678C2×D412yescomplemented
95C23×C181—21no—
96C23×C181—22no—
97C23×C181—44no—
98C23×C181—44no—
99C23×C181—82no—
100C23×C181C2×D412yes—
101D4×C181—22no—
102D4×C181—21no—
103D4×C181—22no—
104D4×C181—44no—
105D4×C181—44no—
106D4×C181—42no—
107D4×C181—81no—
108D4×C181SD1614yescomplemented
109D4×C181C2×D412yes—
110C2×C4×C181—22no—
111C2×C4×C181—22no—
112C2×C4×C181—21no—
113C2×C4×C181—42no—
114C2×C4×C181C2×D411yescharacteristic
115C2×C4×C181C2×D412yes—
116C2×C4×C181C2×D411yescharacteristic
117C2×C4×C181C2411yescharacteristic
118C22×C181—22no—
119C22×C181—22no—
120C22×C181—22no—
121C22×C181—41no—
122C22×C181—41no—
123C22×C181—42no—
124C22×C181—42no—
125C22×C181—42no—
126C22×C181—88no—
127C22×C181C2×SD1614yescomplemented
128C22×C181C22×D411yescharacteristic
129C4×C181—22no—
130C4×C181—21no—
131C4×C181—41no—
132C4×C181C2×SD1612yescomplemented
133C4×C181C22×D411yescharacteristic
134C4×C181H(5,𝔽2)11yescharacteristic
135Dic181—22no—
136Dic181—42no—
137Dic181—41no—
138C2×C181—24no—
139C2×C181—42no—
140C2×C181—82no—
141C2×C181D4211yescharacteristic
142C2×C181C22×SD1611yescharacteristic
143C2×C181#57711yescharacteristic
144C181D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—1811nomaximal, Sylow
146C22×SD16—1812no—
147D4×Q8—1811no—
148D42D18111yescharacteristic
149#492—1811no—
150#465—1812no—
151#461—1811no—
152#460—1811no—
153#450—1812no—
154#448—1812no—
155C4×SD16—1811no—
156C8×D4—1811no—
157C22×Q8—1812no—
158C22×D4—21no—
159C22×D4C2×D18112yes—
160C2×SD16—3628no—
161C2×SD16—1812no—
162C2×SD16—1812no—
163C2×SD16—1812no—
164C2×SD16—1811no—
165C2×SD16—1812no—
166C22×C8—1812no—
167C4⋊Q8—3621no—
168C4⋊Q8—1811no—
169#127C2×D18111yescharacteristic
170C22⋊Q8—3622no—
171C22⋊Q8—1812no—
172#121—21no—
173#121C2×D18112yes—
174#120—22no—
175C4×Q8—1811no—
176C4×D4—3621no—
177C4×D4C2×D18111yescharacteristic
178C4×D4C2×D18111yescharacteristic
179#106—1811no—
180C4⋊C8—1811no—
181#103—1811no—
182#103—1812no—
183#102—1811no—
184#102—1812no—
185C22⋊C8—1812no—
186C4×C8—1811no—
187C24—22no—
188C2×Q8—3624no—
189C2×Q8—3622no—
190C2×Q8—1812no—
191C2×Q8—1811no—
192C2×Q8—7242no—
193C2×D4—24no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—42no—
199C2×D4—44no—
200C2×D4C22⋊D18111yescharacteristic
201C2×D4C22×D18112yes—
202C2×D4C22×D18111yescharacteristic
203C22×C4—21no—
204C22×C4—3622no—
205C22×C4C22×D18112yes—
206SD16—3624no—
207SD16—3624no—
208SD16—3624no—
209SD16—1814no—
210SD16—7242no—
211C2×C8—3624no—
212C2×C8—1811no—
213C2×C8—1812no—
214C2×C8—1811no—
215C4⋊C4—3621no—
216C4⋊C4—3621no—
217C4⋊C4—3621no—
218C4⋊C4—3622no—
219C4⋊C4—1811no—
220C4⋊C4C22⋊D18111yescharacteristic
221C4⋊C4C22×D18111yescharacteristic
222C22⋊C4—21no—
223C22⋊C4—22no—
224C22⋊C4—3622no—
225C22⋊C4C22⋊D18112yes—
226C42—3621no—
227C42C22×D18111yescharacteristic
228C23—21no—
229C23—22no—
230C23—44no—
231C23—44no—
232C23—82no—
233C23C2×C22⋊D18112yes—
234Q8—3621no—
235Q8—3622no—
236Q8—3622no—
237Q8—1812no—
238Q8—7244no—
239D4—22no—
240D4—21no—
241D4—22no—
242D4—44no—
243D4—44no—
244D4—42no—
245D4—81no—
246D4#1566843714yesdirect product factor
247D4D4×D18112yes—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—3621no—
253C2×C4—3621no—
254C2×C4—3622no—
255C2×C4—7244no—
256C2×C4C2×C22⋊D18111yescharacteristic
257C2×C4C2×C22⋊D18112yes—
258C2×C4C2×C22⋊D18111yescharacteristic
259C2×C4C23×D18111yescharacteristic
260C8—3622no—
261C8—3621no—
262C8—1812no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—88no—
272C22C2×#1566843714yescomplemented
273C22C22×C22⋊D18111yescharacteristic
274C4—22no—
275C4—21no—
276C4—41no—
277C4—3622no—
278C4—7242no—
279C4—7241no—
280C4C2×#1566843712yescomplemented
281C4C2×D4×D18111yescharacteristic
282C4#1915386111yescharacteristic
283C2—24no—
284C2—42no—
285C2—82no—
286C2D4×C22⋊D18111yescharacteristic
287C2C22×#1566843711yescharacteristic
288C2#2604209011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting, prime radicalD42×C181C2
socleC22×C181C22×D4
hypercenterD42D181
hypocenterC181D4×SD16
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C22×#15668437C2semidirect
2. D4×Dic362C2semidirect
3. #26041651C2semidirect
4. #26041506C2semidirect
5. #26041484C2semidirect
6. #26041476C2semidirect
7. #26041430C2semidirect
8. #26041417C2semidirect
9. C4×#15668437C2semidirect
10. D4×#13089678C2semidirect
12. C22×Dic362C22semidirect
16. C2×#15668437C22semidirect
17. C2×#15668437C22semidirect
18. C2×#15668437C22semidirect
19. C22×#13089678C22semidirect
20. #19153816C22semidirect
22. #19153795C22semidirect
24. C4×Dic362C22semidirect
26. #19153747C22semidirect
27. C4⋊#13089678C22semidirect
28. #19153739C22semidirect
29. #19153739C22semidirect
31. #19153737C22semidirect
32. C22⋊#13089678C22semidirect
33. C4×#13089678C22semidirect
52. #15668437D4direct (N × H)
87. Dic362C2×D4semidirect
94. #13089678C2×D4semidirect
108. D4×C181SD16semidirect
127. C22×C181C2×SD16semidirect
132. C4×C181C2×SD16semidirect
144. C181D4×SD16semidirect
246. D4#15668437direct (N × H)
272. C22C2×#15668437semidirect
280. C4C2×#15668437semidirect

Maximal subgroups

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

subgroupindex
C22×#156684372
D4×Dic3622
#260416512
#260415062
#260414842
#260414762
#260414302
#260414172
C4×#156684372
D4×#130896782
D42×C1812
D4×SD16181