Finite Group Databasebeta

← D4×#15668436 #37545417

Subgroups — D4×#15668436 #37545417

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×#15668436C212yesmaximal, complemented
2D4×D724C211yesmaximal, characteristic
3#26041653C211yesmaximal, characteristic, complemented
4#26041510C212yesmaximal, complemented
5#26041481C211yesmaximal, characteristic, complemented
6#26041479C211yesmaximal, characteristic, complemented
7#26041429C212yesmaximal, complemented
8#26041418C212yesmaximal, complemented
9C4×#15668436C211yesmaximal, characteristic, complemented
10D4×#13089678C211yesmaximal, characteristic, complemented
11D4×Q8×C181C211yesmaximal, characteristic, complemented
12C22×D724C2212yes—
13C2×D4×D181—21no—
14C2×#15668436—28no—
15C2×#15668436C2212yes—
16C2×#15668436C2211yescharacteristic
17C2×#15668436C2212yescomplemented
18C2×#15668436C2212yescomplemented
19C2×#15668436C2212yescomplemented
20C22×#13089678C2212yescomplemented
21#19153813C2211yescharacteristic
22#19153791—21no—
23#19153790C2212yes—
24#19153786—22no—
25C4×D724C2211yescharacteristic
26#19153747C2211yescharacteristic, complemented
27C4⋊#13089678C2211yescharacteristic, complemented
28#19153740C2211yescharacteristic, complemented
29#19153740C2212yescomplemented
30#19153736C2211yescharacteristic
31#19153736C2212yescomplemented
32C22⋊#13089678C2212yescomplemented
33C4×#13089678C2211yescharacteristic, complemented
34C22×Q8×C181C2212yescomplemented
35C4⋊Q8×C181—21no—
36C4⋊Q8×C181C2211yescharacteristic, complemented
37C22⋊Q8×C181—22no—
38C22⋊Q8×C181C2212yescomplemented
39C4×Q8×C181C2211yescharacteristic, complemented
40C4×D4×C181—21no—
41C4×D4×C181C2211yescharacteristic
42C23×D181—22no—
43C2×D724—24no—
44C2×D724—22no—
45C2×D724—21no—
46C2×D724—21no—
47C2×D724—42no—
48C2×D724C2312yes—
49C2×D724C2311yescharacteristic
50C2×C22⋊D181—22no—
51D4×D181—44no—
52C2×C4×D181—21no—
53#15668436—24no—
54#15668436—24no—
55#15668436—24no—
56#15668436—42no—
57#15668436D414yesdirect product factor
58C2×#13089678—24no—
59C2×#13089678C2311yescharacteristic
60C2×#13089678C2312yes—
61C2×#13089678C2311yescharacteristic
62C4⋊Dic181C2311yescharacteristic
63#15668426—21no—
64#15668426—22no—
65C2×Q8×C181—22no—
66C2×Q8×C181—24no—
67C2×Q8×C181—42no—
68C2×Q8×C181C2312yes—
69C2×Q8×C181C2311yescharacteristic
70C2×D4×C181D411yescharacteristic
71C22×C4×C181—22no—
72C22×C4×C181C2312yes—
73C4⋊C4×C181—21no—
74C4⋊C4×C181—21no—
75C4⋊C4×C181—21no—
76C4⋊C4×C181—22no—
77C4⋊C4×C181D411yescharacteristic
78C4⋊C4×C181C2311yescharacteristic
79C22⋊C4×C181—22no—
80C22⋊C4×C181D412yes—
81C42×C181—21no—
82C42×C181C2311yescharacteristic
83C22×D181—21no—
84C22×D181—22no—
85C22×D181—44no—
86C22×D181—44no—
87C22×D181—82no—
88D724—22no—
89D724—21no—
90D724—22no—
91D724—44no—
92D724—42no—
93D724—81no—
94D724C2×D412yes—
95C22⋊D181—44no—
96C2×Dic181—21no—
97C4×D181—42no—
98#13089678—22no—
99#13089678—21no—
100#13089678C2×D412yescomplemented
101C23×C181C2×D412yes—
102Q8×C181—21no—
103Q8×C181—22no—
104Q8×C181—22no—
105Q8×C181—44no—
106Q8×C181C2×D412yescomplemented
107D4×C181SD1614yescomplemented
108C2×C4×C181—21no—
109C2×C4×C181—22no—
110C2×C4×C181—22no—
111C2×C4×C181—21no—
112C2×C4×C181—22no—
113C2×C4×C181—44no—
114C2×C4×C181C2×D411yescharacteristic
115C2×C4×C181C2×D412yes—
116C2×C4×C181C2×D411yescharacteristic
117C2×C4×C181C2411yescharacteristic
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—22no—
131C4×C181—21no—
132C4×C181—42no—
133C4×C181—41no—
134C4×C181C2×SD1612yescomplemented
135C4×C181C22×D411yescharacteristic
136C4×C181H(5,𝔽2)11yescharacteristic
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×Q8D18111yescharacteristic, complemented
148D42—1811no—
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×Q8C2×D18112yescomplemented
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×C8—1812no—
167C4⋊Q8—21no—
168C4⋊Q8C2×D18111yescharacteristic, complemented
169#127—1811no—
170C22⋊Q8—22no—
171C22⋊Q8C2×D18112yescomplemented
172#121—3621no—
173#121—1812no—
174#120—3622no—
175C4×Q8C2×D18111yescharacteristic, complemented
176C4×D4—21no—
177C4×D4—1811no—
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—3622no—
188C2×Q8—22no—
189C2×Q8—24no—
190C2×Q8—42no—
191C2×Q8C22×D18112yes—
192C2×Q8C22×D18111yescharacteristic
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×D4C22⋊D18111yescharacteristic
203C22×C4—22no—
204C22×C4—3621no—
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—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—22no—
219C4⋊C4—1811no—
220C4⋊C4C22⋊D18111yescharacteristic
221C4⋊C4C22×D18111yescharacteristic
222C22⋊C4—22no—
223C22⋊C4—3621no—
224C22⋊C4—3622no—
225C22⋊C4C22⋊D18112yes—
226C42—21no—
227C42C22×D18111yescharacteristic
228C23—3621no—
229C23—3622no—
230C23—14482no—
231C23—7244no—
232C23—7244no—
233C23C2×C22⋊D18112yes—
234Q8—21no—
235Q8—22no—
236Q8—22no—
237Q8—44no—
238Q8D4×D18112yescomplemented
239D4—3622no—
240D4—3621no—
241D4—3622no—
242D4—14481no—
243D4—1812no—
244D4—7244no—
245D4—7244no—
246D4—7242no—
247D4#1566843614yesdirect product factor
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—44no—
254C2×C4—3621no—
255C2×C4—7242no—
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—3622no—
266C22—14488no—
267C22—7242no—
268C22—7242no—
269C22—7242no—
270C22—7241no—
271C22—7241no—
272C22C2×#1566843614yescomplemented
273C22C22×C22⋊D18111yescharacteristic
274C4—22no—
275C4—22no—
276C4—21no—
277C4—42no—
278C4—41no—
279C4—7241no—
280C4C2×#1566843612yescomplemented
281C4C2×D4×D18111yescharacteristic
282C4#1915386111yescharacteristic
283C2—24no—
284C2—14482no—
285C2—7242no—
286C2D4×C22⋊D18111yescharacteristic
287C2C22×#1566843611yescharacteristic
288C2#2604208711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingD4×Q8×C181C2
socleC22×C181C22×D4
prime radicalD4×D724C2
hypercenterD4×Q8D181
hypocenterC181D4×SD16
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#15668436C2semidirect
3. #26041653C2semidirect
4. #26041510C2semidirect
5. #26041481C2semidirect
6. #26041479C2semidirect
7. #26041429C2semidirect
8. #26041418C2semidirect
9. C4×#15668436C2semidirect
10. D4×#13089678C2semidirect
11. D4×Q8×C181C2semidirect
17. C2×#15668436C22semidirect
18. C2×#15668436C22semidirect
19. C2×#15668436C22semidirect
20. C22×#13089678C22semidirect
26. #19153747C22semidirect
27. C4⋊#13089678C22semidirect
28. #19153740C22semidirect
29. #19153740C22semidirect
31. #19153736C22semidirect
32. C22⋊#13089678C22semidirect
33. C4×#13089678C22semidirect
34. C22×Q8×C181C22semidirect
36. C4⋊Q8×C181C22semidirect
38. C22⋊Q8×C181C22semidirect
39. C4×Q8×C181C22semidirect
57. #15668436D4direct (N × H)
100. #13089678C2×D4semidirect
106. Q8×C181C2×D4semidirect
107. D4×C181SD16semidirect
120. C22×C181C2×SD16semidirect
134. C4×C181C2×SD16semidirect
144. C181D4×SD16semidirect
147. D4×Q8D181semidirect
157. C22×Q8C2×D181semidirect
168. C4⋊Q8C2×D181semidirect
171. C22⋊Q8C2×D181semidirect
175. C4×Q8C2×D181semidirect
238. Q8D4×D181semidirect
247. D4#15668436direct (N × H)
272. C22C2×#15668436semidirect
280. C4C2×#15668436semidirect

Maximal subgroups

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

subgroupindex
C22×#156684362
D4×D7242
#260416532
#260415102
#260414812
#260414792
#260414292
#260414182
C4×#156684362
D4×#130896782
D4×Q8×C1812
D4×SD16181