Finite Group Databasebeta

← D4×#14581938 #36335830

Subgroups — D4×#14581938 #36335830

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×#14581938C212yesmaximal, complemented
2D4×Dic334C211yesmaximal, characteristic, complemented
3#24923641C211yesmaximal, characteristic, complemented
4#24923496C212yesmaximal, complemented
5#24923474C211yesmaximal, characteristic, complemented
6#24923466C211yesmaximal, characteristic, complemented
7#24923420C212yesmaximal, complemented
8#24923407C212yesmaximal, complemented
9C4×#14581938C211yesmaximal, characteristic, complemented
10D4×C167⋊C8C211yesmaximal, characteristic, complemented
11D42×C167C211yesmaximal, characteristic
12C22×Dic334C2212yescomplemented
13C2×#14581938—28no—
14C2×#14581938C2212yes—
15C2×#14581938C2211yescharacteristic
16C2×#14581938C2212yescomplemented
17C2×#14581938C2212yescomplemented
18C2×#14581938C2212yescomplemented
19C22×C167⋊C8C2212yescomplemented
20#19128832C2211yescharacteristic, complemented
21#19128831—21no—
22#19128811C2212yescomplemented
23#19128809—22no—
24C4×Dic334C2211yescharacteristic, complemented
25D4×Dic167—21no—
26#19128763C2211yescharacteristic, complemented
27C4⋊(C167⋊C8)C2211yescharacteristic, complemented
28#19128755C2211yescharacteristic, complemented
29#19128755C2212yescomplemented
30#19128753C2211yescharacteristic
31#19128753C2212yescomplemented
32C22⋊(C167⋊C8)C2212yescomplemented
33C4×C167⋊C8C2211yescharacteristic, complemented
34C22×D4×C167—21no—
35C22×D4×C167C2212yes—
36#127×C167C2211yescharacteristic
37#121×C167—21no—
38#121×C167C2212yes—
39#120×C167—22no—
40C4×D4×C167C2211yescharacteristic
41C4×D4×C167C2211yescharacteristic
42C2×Dic334—24no—
43C2×Dic334—22no—
44C2×Dic334—42no—
45C2×Dic334C2312yes—
46C2×Dic334C2311yescharacteristic
47C22×Dic167—22no—
48#14581938—24no—
49#14581938—24no—
50#14581938—24no—
51#14581938—42no—
52#14581938D414yesdirect product factor
53C2×C167⋊C8—24no—
54C2×C167⋊C8C2312yes—
55C2×C167⋊C8C2311yescharacteristic
56C2×C167⋊C8C2311yescharacteristic
57C4⋊Dic167—21no—
58C4⋊Dic167—21no—
59C4⋊Dic167C2311yescharacteristic
60Dic167⋊C4—21no—
61Dic167⋊C4—22no—
62C22⋊Dic167—22no—
63C4×Dic167—21no—
64C24×C167—22no—
65C2×D4×C167—24no—
66C2×D4×C167—22no—
67C2×D4×C167—22no—
68C2×D4×C167—21no—
69C2×D4×C167—21no—
70C2×D4×C167—42no—
71C2×D4×C167—44no—
72C2×D4×C167D411yescharacteristic
73C2×D4×C167C2312yes—
74C2×D4×C167C2311yescharacteristic
75C22×C4×C167—21no—
76C22×C4×C167C2312yes—
77C4⋊C4×C167D411yescharacteristic
78C4⋊C4×C167C2311yescharacteristic
79C22⋊C4×C167—21no—
80C22⋊C4×C167—22no—
81C22⋊C4×C167D412yes—
82C42×C167C2311yescharacteristic
83Dic334—21no—
84Dic334—22no—
85Dic334—22no—
86Dic334—44no—
87Dic334C2×D412yescomplemented
88C2×Dic167—21no—
89C2×Dic167—21no—
90C2×Dic167—22no—
91C2×Dic167—44no—
92C167⋊C8—22no—
93C167⋊C8—21no—
94C167⋊C8C2×D412yescomplemented
95C23×C167—21no—
96C23×C167—22no—
97C23×C167—44no—
98C23×C167—44no—
99C23×C167—82no—
100C23×C167C2×D412yes—
101D4×C167—22no—
102D4×C167—21no—
103D4×C167—22no—
104D4×C167—44no—
105D4×C167—44no—
106D4×C167—42no—
107D4×C167—81no—
108D4×C167SD1614yescomplemented
109D4×C167C2×D412yes—
110C2×C4×C167—22no—
111C2×C4×C167—22no—
112C2×C4×C167—21no—
113C2×C4×C167—42no—
114C2×C4×C167C2×D411yescharacteristic
115C2×C4×C167C2×D412yes—
116C2×C4×C167C2×D411yescharacteristic
117C2×C4×C167C2411yescharacteristic
118C22×C167—22no—
119C22×C167—22no—
120C22×C167—22no—
121C22×C167—41no—
122C22×C167—41no—
123C22×C167—42no—
124C22×C167—42no—
125C22×C167—42no—
126C22×C167—88no—
127C22×C167C2×SD1614yescomplemented
128C22×C167C22×D411yescharacteristic
129C4×C167—22no—
130C4×C167—21no—
131C4×C167—41no—
132C4×C167C2×SD1612yescomplemented
133C4×C167C22×D411yescharacteristic
134C4×C167H(5,𝔽2)11yescharacteristic
135Dic167—22no—
136Dic167—42no—
137Dic167—41no—
138C2×C167—24no—
139C2×C167—42no—
140C2×C167—82no—
141C2×C167D4211yescharacteristic
142C2×C167C22×SD1611yescharacteristic
143C2×C167#57711yescharacteristic
144C167D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—1671nomaximal, Sylow
146C22×SD16—1672no—
147D4×Q8—1671no—
148D42D16711yescharacteristic
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×Q8—1672no—
158C22×D4—21no—
159C22×D4C2×D16712yes—
160C2×SD16—3348no—
161C2×SD16—1672no—
162C2×SD16—1672no—
163C2×SD16—1672no—
164C2×SD16—1672no—
165C2×SD16—1671no—
166C22×C8—1672no—
167C4⋊Q8—3341no—
168C4⋊Q8—1671no—
169#127C2×D16711yescharacteristic
170C22⋊Q8—3342no—
171C22⋊Q8—1672no—
172#121—21no—
173#121C2×D16712yes—
174#120—22no—
175C4×Q8—1671no—
176C4×D4—3341no—
177C4×D4C2×D16711yescharacteristic
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—22no—
188C2×Q8—3344no—
189C2×Q8—3342no—
190C2×Q8—6682no—
191C2×Q8—1672no—
192C2×Q8—1671no—
193C2×D4—24no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—42no—
199C2×D4—44no—
200C2×D4C22⋊D16711yescharacteristic
201C2×D4C22×D16712yes—
202C2×D4C22×D16711yescharacteristic
203C22×C4—21no—
204C22×C4—3342no—
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—3341no—
216C4⋊C4—3341no—
217C4⋊C4—3341no—
218C4⋊C4—3342no—
219C4⋊C4—1671no—
220C4⋊C4C22⋊D16711yescharacteristic
221C4⋊C4C22×D16711yescharacteristic
222C22⋊C4—21no—
223C22⋊C4—22no—
224C22⋊C4—3342no—
225C22⋊C4C22⋊D16712yes—
226C42—3341no—
227C42C22×D16711yescharacteristic
228C23—21no—
229C23—22no—
230C23—44no—
231C23—44no—
232C23—82no—
233C23C2×C22⋊D16712yes—
234Q8—3341no—
235Q8—3342no—
236Q8—3342no—
237Q8—6684no—
238Q8—1672no—
239D4—22no—
240D4—21no—
241D4—22no—
242D4—44no—
243D4—44no—
244D4—42no—
245D4—81no—
246D4#1458193814yesdirect product factor
247D4D4×D16712yes—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—3341no—
253C2×C4—3341no—
254C2×C4—3342no—
255C2×C4—6684no—
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—22no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—88no—
272C22C2×#1458193814yescomplemented
273C22C22×C22⋊D16711yescharacteristic
274C4—21no—
275C4—22no—
276C4—41no—
277C4—3342no—
278C4—6681no—
279C4—6682no—
280C4C2×#1458193812yescomplemented
281C4C2×D4×D16711yescharacteristic
282C4#1912887711yescharacteristic
283C2—24no—
284C2—42no—
285C2—82no—
286C2D4×C22⋊D16711yescharacteristic
287C2C22×#1458193811yescharacteristic
288C2#2492408011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting, prime radicalD42×C167C2
socleC22×C167C22×D4
hypercenterD42D167
hypocenterC167D4×SD16
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C22×#14581938C2semidirect
2. D4×Dic334C2semidirect
3. #24923641C2semidirect
4. #24923496C2semidirect
5. #24923474C2semidirect
6. #24923466C2semidirect
7. #24923420C2semidirect
8. #24923407C2semidirect
9. C4×#14581938C2semidirect
10. D4×C167⋊C8C2semidirect
12. C22×Dic334C22semidirect
16. C2×#14581938C22semidirect
17. C2×#14581938C22semidirect
18. C2×#14581938C22semidirect
19. C22×C167⋊C8C22semidirect
20. #19128832C22semidirect
22. #19128811C22semidirect
24. C4×Dic334C22semidirect
26. #19128763C22semidirect
27. C4⋊(C167⋊C8)C22semidirect
28. #19128755C22semidirect
29. #19128755C22semidirect
31. #19128753C22semidirect
32. C22⋊(C167⋊C8)C22semidirect
33. C4×C167⋊C8C22semidirect
52. #14581938D4direct (N × H)
87. Dic334C2×D4semidirect
94. C167⋊C8C2×D4semidirect
108. D4×C167SD16semidirect
127. C22×C167C2×SD16semidirect
132. C4×C167C2×SD16semidirect
144. C167D4×SD16semidirect
246. D4#14581938direct (N × H)
272. C22C2×#14581938semidirect
280. C4C2×#14581938semidirect

Maximal subgroups

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

subgroupindex
C22×#145819382
D4×Dic3342
#249236412
#249234962
#249234742
#249234662
#249234202
#249234072
C4×#145819382
D4×C167⋊C82
D42×C1672
D4×SD16167