Finite Group Databasebeta

← D4×#14581937 #36335834

Subgroups — D4×#14581937 #36335834

Order 21376.

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×#14581937C212yesmaximal, complemented
2D4×D668C211yesmaximal, characteristic
3#24923643C211yesmaximal, characteristic, complemented
4#24923500C212yesmaximal, complemented
5#24923471C211yesmaximal, characteristic, complemented
6#24923469C211yesmaximal, characteristic, complemented
7#24923419C212yesmaximal, complemented
8#24923408C212yesmaximal, complemented
9C4×#14581937C211yesmaximal, characteristic, complemented
10D4×C167⋊C8C211yesmaximal, characteristic, complemented
11D4×Q8×C167C211yesmaximal, characteristic, complemented
12C22×D668C2212yes—
13C2×D4×D167—21no—
14C2×#14581937—28no—
15C2×#14581937C2212yes—
16C2×#14581937C2211yescharacteristic
17C2×#14581937C2212yescomplemented
18C2×#14581937C2212yescomplemented
19C2×#14581937C2212yescomplemented
20C22×C167⋊C8C2212yescomplemented
21#19128829C2211yescharacteristic
22#19128807—21no—
23#19128806C2212yes—
24#19128802—22no—
25C4×D668C2211yescharacteristic
26#19128763C2211yescharacteristic, complemented
27C4⋊(C167⋊C8)C2211yescharacteristic, complemented
28#19128756C2211yescharacteristic, complemented
29#19128756C2212yescomplemented
30#19128752C2211yescharacteristic
31#19128752C2212yescomplemented
32C22⋊(C167⋊C8)C2212yescomplemented
33C4×C167⋊C8C2211yescharacteristic, complemented
34C22×Q8×C167C2212yescomplemented
35C4⋊Q8×C167—21no—
36C4⋊Q8×C167C2211yescharacteristic, complemented
37C22⋊Q8×C167—22no—
38C22⋊Q8×C167C2212yescomplemented
39C4×Q8×C167C2211yescharacteristic, complemented
40C4×D4×C167—21no—
41C4×D4×C167C2211yescharacteristic
42C23×D167—22no—
43C2×D668—24no—
44C2×D668—21no—
45C2×D668—22no—
46C2×D668—21no—
47C2×D668—42no—
48C2×D668C2312yes—
49C2×D668C2311yescharacteristic
50C2×C22⋊D167—22no—
51D4×D167—44no—
52C2×C4×D167—21no—
53#14581937—24no—
54#14581937—24no—
55#14581937—24no—
56#14581937—42no—
57#14581937D414yesdirect product factor
58C2×C167⋊C8—24no—
59C2×C167⋊C8C2312yes—
60C2×C167⋊C8C2311yescharacteristic
61C2×C167⋊C8C2311yescharacteristic
62C4⋊Dic167C2311yescharacteristic
63(C2×D167)⋊C4—21no—
64(C2×D167)⋊C4—22no—
65C2×Q8×C167—22no—
66C2×Q8×C167—24no—
67C2×Q8×C167—42no—
68C2×Q8×C167C2312yes—
69C2×Q8×C167C2311yescharacteristic
70C2×D4×C167D411yescharacteristic
71C22×C4×C167—22no—
72C22×C4×C167C2312yes—
73C4⋊C4×C167—21no—
74C4⋊C4×C167—21no—
75C4⋊C4×C167—21no—
76C4⋊C4×C167—22no—
77C4⋊C4×C167D411yescharacteristic
78C4⋊C4×C167C2311yescharacteristic
79C22⋊C4×C167—22no—
80C22⋊C4×C167D412yes—
81C42×C167—21no—
82C42×C167C2311yescharacteristic
83C22×D167—21no—
84C22×D167—22no—
85C22×D167—44no—
86C22×D167—44no—
87C22×D167—82no—
88D668—22no—
89D668—21no—
90D668—22no—
91D668—44no—
92D668—42no—
93D668—81no—
94D668C2×D412yes—
95C22⋊D167—44no—
96C2×Dic167—21no—
97C4×D167—42no—
98C167⋊C8—22no—
99C167⋊C8—21no—
100C167⋊C8C2×D412yescomplemented
101C23×C167C2×D412yes—
102Q8×C167—21no—
103Q8×C167—22no—
104Q8×C167—22no—
105Q8×C167—44no—
106Q8×C167C2×D412yescomplemented
107D4×C167SD1614yescomplemented
108C2×C4×C167—21no—
109C2×C4×C167—22no—
110C2×C4×C167—22no—
111C2×C4×C167—21no—
112C2×C4×C167—22no—
113C2×C4×C167—44no—
114C2×C4×C167C2×D411yescharacteristic
115C2×C4×C167C2×D412yes—
116C2×C4×C167C2×D411yescharacteristic
117C2×C4×C167C2411yescharacteristic
118C22×C167—22no—
119C22×C167—22no—
120C22×C167C2×SD1614yescomplemented
121C22×C167C22×D411yescharacteristic
122C2×D167—22no—
123C2×D167—41no—
124C2×D167—41no—
125C2×D167—42no—
126C2×D167—42no—
127C2×D167—42no—
128C2×D167—88no—
129C4×C167—22no—
130C4×C167—22no—
131C4×C167—21no—
132C4×C167—42no—
133C4×C167—41no—
134C4×C167C2×SD1612yescomplemented
135C4×C167C22×D411yescharacteristic
136C4×C167H(5,𝔽2)11yescharacteristic
137Dic167—41no—
138C2×C167—24no—
139C2×C167D4211yescharacteristic
140C2×C167C22×SD1611yescharacteristic
141C2×C167#57711yescharacteristic
142D167—42no—
143D167—82no—
144C167D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—1671nomaximal, Sylow
146C22×SD16—1672no—
147D4×Q8D16711yescharacteristic, complemented
148D42—1671no—
149#492—1671no—
150#465—1672no—
151#461—1671no—
152#460—1671no—
153#450—1672no—
154#448—1672no—
155C4×SD16—1671no—
156C8×D4—1671no—
157C22×Q8C2×D16712yescomplemented
158C22×D4—3341no—
159C22×D4—1672no—
160C2×SD16—3348no—
161C2×SD16—1672no—
162C2×SD16—1672no—
163C2×SD16—1672no—
164C2×SD16—1672no—
165C2×SD16—1671no—
166C22×C8—1672no—
167C4⋊Q8—21no—
168C4⋊Q8C2×D16711yescharacteristic, complemented
169#127—1671no—
170C22⋊Q8—22no—
171C22⋊Q8C2×D16712yescomplemented
172#121—3341no—
173#121—1672no—
174#120—3342no—
175C4×Q8C2×D16711yescharacteristic, complemented
176C4×D4—21no—
177C4×D4—1671no—
178C4×D4C2×D16711yescharacteristic
179#106—1671no—
180C4⋊C8—1671no—
181#103—1671no—
182#103—1672no—
183#102—1671no—
184#102—1672no—
185C22⋊C8—1672no—
186C4×C8—1671no—
187C24—3342no—
188C2×Q8—22no—
189C2×Q8—24no—
190C2×Q8—42no—
191C2×Q8C22×D16712yes—
192C2×Q8C22×D16711yescharacteristic
193C2×D4—3344no—
194C2×D4—3342no—
195C2×D4—3341no—
196C2×D4—3342no—
197C2×D4—3341no—
198C2×D4—6682no—
199C2×D4—6684no—
200C2×D4—1672no—
201C2×D4—1671no—
202C2×D4C22⋊D16711yescharacteristic
203C22×C4—22no—
204C22×C4—3341no—
205C22×C4C22×D16712yes—
206SD16—3344no—
207SD16—3344no—
208SD16—3344no—
209SD16—6682no—
210SD16—1674no—
211C2×C8—3344no—
212C2×C8—1671no—
213C2×C8—1672no—
214C2×C8—1671no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—22no—
219C4⋊C4—1671no—
220C4⋊C4C22⋊D16711yescharacteristic
221C4⋊C4C22×D16711yescharacteristic
222C22⋊C4—22no—
223C22⋊C4—3341no—
224C22⋊C4—3342no—
225C22⋊C4C22⋊D16712yes—
226C42—21no—
227C42C22×D16711yescharacteristic
228C23—13362no—
229C23—3341no—
230C23—3342no—
231C23—6684no—
232C23—6684no—
233C23C2×C22⋊D16712yes—
234Q8—21no—
235Q8—22no—
236Q8—22no—
237Q8—44no—
238Q8D4×D16712yescomplemented
239D4—13361no—
240D4—3342no—
241D4—3341no—
242D4—3342no—
243D4—6684no—
244D4—6684no—
245D4—6682no—
246D4—1672no—
247D4#1458193714yesdirect product factor
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—44no—
254C2×C4—3341no—
255C2×C4—6682no—
256C2×C4C2×C22⋊D16711yescharacteristic
257C2×C4C2×C22⋊D16712yes—
258C2×C4C2×C22⋊D16711yescharacteristic
259C2×C4C23×D16711yescharacteristic
260C8—3342no—
261C8—3341no—
262C8—1672no—
263C22—22no—
264C22—22no—
265C22—13368no—
266C22—3342no—
267C22—6682no—
268C22—6682no—
269C22—6682no—
270C22—6681no—
271C22—6681no—
272C22C2×#1458193714yescomplemented
273C22C22×C22⋊D16711yescharacteristic
274C4—21no—
275C4—22no—
276C4—22no—
277C4—41no—
278C4—42no—
279C4—6681no—
280C4C2×#1458193712yescomplemented
281C4C2×D4×D16711yescharacteristic
282C4#1912887711yescharacteristic
283C2—24no—
284C2—13362no—
285C2—6682no—
286C2D4×C22⋊D16711yescharacteristic
287C2C22×#1458193711yescharacteristic
288C2#2492407711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingD4×Q8×C167C2
socleC22×C167C22×D4
prime radicalD4×D668C2
hypercenterD4×Q8D167
hypocenterC167D4×SD16
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14581937C2semidirect
3. #24923643C2semidirect
4. #24923500C2semidirect
5. #24923471C2semidirect
6. #24923469C2semidirect
7. #24923419C2semidirect
8. #24923408C2semidirect
9. C4×#14581937C2semidirect
10. D4×C167⋊C8C2semidirect
11. D4×Q8×C167C2semidirect
17. C2×#14581937C22semidirect
18. C2×#14581937C22semidirect
19. C2×#14581937C22semidirect
20. C22×C167⋊C8C22semidirect
26. #19128763C22semidirect
27. C4⋊(C167⋊C8)C22semidirect
28. #19128756C22semidirect
29. #19128756C22semidirect
31. #19128752C22semidirect
32. C22⋊(C167⋊C8)C22semidirect
33. C4×C167⋊C8C22semidirect
34. C22×Q8×C167C22semidirect
36. C4⋊Q8×C167C22semidirect
38. C22⋊Q8×C167C22semidirect
39. C4×Q8×C167C22semidirect
57. #14581937D4direct (N × H)
100. C167⋊C8C2×D4semidirect
106. Q8×C167C2×D4semidirect
107. D4×C167SD16semidirect
120. C22×C167C2×SD16semidirect
134. C4×C167C2×SD16semidirect
144. C167D4×SD16semidirect
147. D4×Q8D167semidirect
157. C22×Q8C2×D167semidirect
168. C4⋊Q8C2×D167semidirect
171. C22⋊Q8C2×D167semidirect
175. C4×Q8C2×D167semidirect
238. Q8D4×D167semidirect
247. D4#14581937direct (N × H)
272. C22C2×#14581937semidirect
280. C4C2×#14581937semidirect

Maximal subgroups

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

subgroupindex
C22×#145819372
D4×D6682
#249236432
#249235002
#249234712
#249234692
#249234192
#249234082
C4×#145819372
D4×C167⋊C82
D4×Q8×C1672
D4×SD16167