Finite Group Databasebeta

← C2×#24923927 #36337603

Subgroups — C2×#24923927 #36337603

Order 21376.

1620 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581949C211yesmaximal, characteristic, complemented
2C22×Q8×D167C211yesmaximal, characteristic, complemented
3#24923927C2116yesmaximal, direct product factor
4C2×#19128818C212yesmaximal, complemented
5C2×#19128809C212yesmaximal, complemented
6C2×#19128808C211yesmaximal, characteristic, complemented
7C2×#19128805C212yesmaximal, complemented
8C2×#19128803C211yesmaximal, characteristic, complemented
9C2×C4×Dic334C211yesmaximal, characteristic, complemented
10C2×C4×D668C211yesmaximal, characteristic, complemented
11C2×(C22⋊D167)⋊C4C212yesmaximal, complemented
12C2×#124×C167C211yesmaximal, characteristic, complemented
13C2×#14581949—28no—
14C2×#14581949C2218yescomplemented
15C22×Dic334C2212yes—
16C22×Dic334C2211yescharacteristic
17C2×Q8×D167—28no—
18C2×Q8×D167C2218yescomplemented
19C22×D668C2211yescharacteristic
20C22×C22⋊D167C2212yes—
21C22×C22⋊D167C2212yes—
22C23×Dic167C2212yes—
23C22×C4×D167C2212yes—
24C22×C4×D167C2211yescharacteristic
25C22×C4×D167C2211yescharacteristic
26#19128818C22116yescomplemented
27#19128809C22116yescomplemented
28#19128808C2218yescomplemented
29#19128805C22116yescomplemented
30#19128803C2218yescomplemented
31C4×Dic334C2218yescomplemented
32C4×D668C2218yescomplemented
33(C22⋊D167)⋊C4C22116yescomplemented
34C2×C4⋊Dic167C2211yescharacteristic
35C2×C4⋊Dic167C2211yescharacteristic
36C2×Dic167⋊C4C2212yes—
37C2×Dic167⋊C4C2212yes—
38C2×C22⋊Dic167C2212yes—
39C2×(C2×D167)⋊C4C2211yescharacteristic
40C2×(C2×D167)⋊C4C2211yescharacteristic
41C2×(C2×D167)⋊C4C2212yes—
42C2×C4×Dic167C2212yes—
43C22×Q8×C167C2211yescharacteristic
44C22×D4×C167C2211yescharacteristic
45#124×C167C2218yescomplemented
46C2×C22⋊C4×C167C2212yes—
47C2×C22⋊C4×C167C2212yes—
48C2×C42×C167C2211yescharacteristic
49C23×D167C2311yescharacteristic
50C23×D167C2311yescharacteristic
51#14581949—232no—
52C2×Dic334—28no—
53C2×Dic334D418yescomplemented
54C2×Dic334C2318yes—
55C2×Dic334C2314yes—
56Q8×D167—232no—
57C2×D668—24no—
58C2×D668C2314yes—
59C2×C22⋊D167—28no—
60C2×C22⋊D167—28no—
61C2×C22⋊D167C2318yes—
62C2×C22⋊D167C2318yes—
63C22×Dic167—28no—
64C22×Dic167C2311yescharacteristic
65C22×Dic167C2311yescharacteristic
66C22×Dic167C2318yes—
67C22×Dic167C2312yes—
68C22×Dic167C2312yes—
69C2×C4×D167—28no—
70C2×C4×D167—24no—
71C2×C4×D167—24no—
72C2×C4×D167C2318yes—
73C2×C4×D167C2314yes—
74C2×C4×D167C2314yes—
75C4⋊Dic167C2314yes—
76C4⋊Dic167C2314yes—
77Dic167⋊C4C2318yes—
78Dic167⋊C4C2318yes—
79C22⋊Dic167C2318yes—
80(C2×D167)⋊C4C2314yes—
81(C2×D167)⋊C4C2314yes—
82(C2×D167)⋊C4C2318yes—
83C4×Dic167C2318yes—
84C24×C167C2312yes—
85C2×Q8×C167—24no—
86C2×Q8×C167C2314yes—
87C2×D4×C167—24no—
88C2×D4×C167C2314yes—
89C22×C4×C167C2312yes—
90C22×C4×C167C2311yescharacteristic
91C22×C4×C167C2311yescharacteristic
92C22×C4×C167C2311yescharacteristic
93C22⋊C4×C167C2318yes—
94C22⋊C4×C167C2318yes—
95C42×C167C2314yes—
96C22×D167—22no—
97C22×D167—22no—
98C22×D167—22no—
99C22×D167—22no—
100C22×D167—22no—
101C22×D167#4114yescomplemented
102C22×D167C2412yes—
103C22×D167C2412yes—
104Dic334—216no—
105Dic334C2×D4116yescomplemented
106D668—28no—
107C22⋊D167—216no—
108C22⋊D167—216no—
109C2×Dic167—22no—
110C2×Dic167—22no—
111C2×Dic167—24no—
112C2×Dic167—216no—
113C2×Dic167C2×D418yes—
114C2×Dic167C2414yes—
115C2×Dic167C2412yes—
116C2×Dic167C2412yes—
117C2×Dic167C2414yes—
118C4×D167—28no—
119C4×D167—28no—
120C4×D167—216no—
121C23×C167—24no—
122C23×C167—24no—
123C23×C167—24no—
124C23×C167C2414yes—
125C23×C167C2411yescharacteristic
126Q8×C167—28no—
127D4×C167—28no—
128C2×C4×C167—24no—
129C2×C4×C167—22no—
130C2×C4×C167—22no—
131C2×C4×C167C2×D414yes—
132C2×C4×C167C2412yes—
133C2×C4×C167C2412yes—
134C2×C4×C167C2414yes—
135C2×C4×C167C2412yes—
136C22×C167—24no—
137C22×C167—24no—
138C22×C167—24no—
139C22×C167—48no—
140C22×C167C22×D412yes—
141C22×C167C2×#4112yes—
142C22×C1672−1+412yes—
143C22×C167C2511yescharacteristic
144C2×D167—28no—
145C2×D167—22no—
146C2×D167—22no—
147C2×D167—22no—
148C2×D167—22no—
149C2×D167—22no—
150C2×D167—44no—
151C2×D167C2×#4114yescomplemented
152C4×C167—24no—
153C4×C167—22no—
154C4×C167—22no—
155C4×C167C22×D414yes—
156Dic167—22no—
157Dic167—22no—
158Dic167—24no—
159Dic167C22×D418yes—
160C2×C167—44no—
161C2×C167#54814yescomplemented
162C2×C167C23×D411yescharacteristic
163C2×C167C22×#4111yescharacteristic
164C2×C167C2×2−1+411yescharacteristic
165D167—24no—
166D167—42no—
167C167C2×#54811yescharacteristic, complemented, Sylow
168C2×#548—1671nomaximal, Sylow
169C22×#41—1671no—
170C23×Q8—1671no—
171#548—16716no—
172C2×#124—1672no—
173C2×#124D16711yescharacteristic, complemented
174C2×C22⋊Q8—1671no—
175C2×C22⋊Q8—1672no—
176C2×#121—1671no—
177C2×#121—1672no—
178C2×C4×Q8—1671no—
179C2×C4×D4—1672no—
180C2×C4×D4—1671no—
181C2×#41—3348no—
182C2×#41—1678no—
183C22×Q8—3348no—
184C22×Q8—1678no—
185C22×Q8—1672no—
186C22×Q8—1671no—
187C22×Q8C2×D16711yescharacteristic
188C22×D4—1672no—
189C22×D4—1672no—
190C22×D4—1671no—
191C22×D4C2×D16711yescharacteristic
192C23×C4—1672no—
193C23×C4—1672no—
194C23×C4—1671no—
195C23×C4—1671no—
196#124—16716no—
197#124C2×D16718yescomplemented
198C22⋊Q8—1678no—
199C22⋊Q8—16716no—
200#121—1678no—
201#121—16716no—
202C4×Q8—1678no—
203C4×D4—16716no—
204C4×D4—1678no—
205C2×C4⋊C4—1672no—
206C2×C4⋊C4—1671no—
207C2×C4⋊C4—1672no—
208C2×C4⋊C4—1671no—
209C2×C22⋊C4—1672no—
210C2×C22⋊C4—1671no—
211C2×C22⋊C4—1671no—
212C2×C22⋊C4—1672no—
213C2×C22⋊C4C2×D16712yes—
214C2×C22⋊C4C2×D16712yes—
215C2×C42—1672no—
216C2×C42C2×D16711yescharacteristic
217C24—1671no—
218C24—1671no—
219C24C22×D16712yes—
220#41—33432no—
221C2×Q8—24no—
222C2×Q8—3348no—
223C2×Q8—33432no—
224C2×Q8—1678no—
225C2×Q8—1678no—
226C2×Q8—1674no—
227C2×Q8C22×D16714yes—
228C2×D4—24no—
229C2×D4—3344no—
230C2×D4—3348no—
231C2×D4—3348no—
232C2×D4—1678no—
233C2×D4—1678no—
234C2×D4—1674no—
235C2×D4C22×D16714yes—
236C22×C4—3348no—
237C22×C4—3348no—
238C22×C4—3344no—
239C22×C4—3344no—
240C22×C4—1678no—
241C22×C4—1672no—
242C22×C4—1671no—
243C22×C4—1671no—
244C22×C4—1678no—
245C22×C4—1674no—
246C22×C4—1674no—
247C22×C4—1672no—
248C22×C4C22×D16711yescharacteristic
249C22×C4C22×D16711yescharacteristic
250C22×C4C22×D16712yes—
251C22×C4C22×D16711yescharacteristic
252C4⋊C4—1678no—
253C4⋊C4—1674no—
254C4⋊C4—1678no—
255C4⋊C4—1674no—
256C22⋊C4—1678no—
257C22⋊C4—1674no—
258C22⋊C4—1674no—
259C22⋊C4—1678no—
260C22⋊C4C22×D16718yes—
261C22⋊C4C22×D16718yes—
262C42—1678no—
263C42C22×D16714yes—
264C23—24no—
265C23—24no—
266C23—24no—
267C23—3342no—
268C23—3342no—
269C23—3342no—
270C23—3342no—
271C23—3342no—
272C23—1672no—
273C23—1674no—
274C23—1672no—
275C23C23×D16714yes—
276C23C23×D16711yescharacteristic
277Q8—28no—
278Q8—33416no—
279Q8—16716no—
280D4—28no—
281D4—3348no—
282D4—33416no—
283D4—33416no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—33416no—
288C2×C4—3342no—
289C2×C4—3342no—
290C2×C4—3344no—
291C2×C4—33416no—
292C2×C4—3348no—
293C2×C4—3348no—
294C2×C4—1674no—
295C2×C4—1672no—
296C2×C4—1672no—
297C2×C4—1678no—
298C2×C4—1674no—
299C2×C4D4×D16714yes—
300C2×C4C23×D16712yes—
301C2×C4C23×D16712yes—
302C2×C4C23×D16714yes—
303C2×C4C23×D16712yes—
304C22—24no—
305C22—24no—
306C22—24no—
307C22—48no—
308C22—3348no—
309C22—3342no—
310C22—3342no—
311C22—3342no—
312C22—3342no—
313C22—3342no—
314C22—6684no—
315C22—1674no—
316C22C2×D4×D16712yes—
317C22#41×D16712yes—
318C22#1912887912yes—
319C22C24×D16711yescharacteristic
320C4—24no—
321C4—22no—
322C4—22no—
323C4—3342no—
324C4—3342no—
325C4—3344no—
326C4—1678no—
327C4C2×D4×D16714yes—
328C2—44no—
329C2—3344no—
330C2—6682no—
331C2#2492392714yesdirect product factor
332C2C22×D4×D16711yescharacteristic
333C2C2×#41×D16711yescharacteristic
334C2C2×#1912887911yescharacteristic
subgroupG/H
centerC23C23×D167
commutator, cosocleC22×C167C25
FrattiniC22C24×D167
FittingC2×#124×C167C2
socleC23×C167C24
hypercenterC2×#124D167
hypocenterC167C2×#548
LCSG ▷ C22×C167 ▷ C167
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C22×#14581949C2semidirect
2. C22×Q8×D167C2semidirect
3. #24923927C2direct (N × H)
4. C2×#19128818C2semidirect
5. C2×#19128809C2semidirect
6. C2×#19128808C2semidirect
7. C2×#19128805C2semidirect
8. C2×#19128803C2semidirect
9. C2×C4×Dic334C2semidirect
10. C2×C4×D668C2semidirect
11. C2×(C22⋊D167)⋊C4C2semidirect
12. C2×#124×C167C2semidirect
14. C2×#14581949C22semidirect
18. C2×Q8×D167C22semidirect
26. #19128818C22semidirect
27. #19128809C22semidirect
28. #19128808C22semidirect
29. #19128805C22semidirect
30. #19128803C22semidirect
31. C4×Dic334C22semidirect
32. C4×D668C22semidirect
33. (C22⋊D167)⋊C4C22semidirect
45. #124×C167C22semidirect
53. C2×Dic334D4semidirect
101. C22×D167#41semidirect
105. Dic334C2×D4semidirect
151. C2×D167C2×#41semidirect
161. C2×C167#548semidirect
167. C167C2×#548semidirect
173. C2×#124D167semidirect
197. #124C2×D167semidirect
331. C2#24923927direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145819492
C22×Q8×D1672
#249239272
C2×#191288182
C2×#191288092
C2×#191288082
C2×#191288052
C2×#191288032
C2×C4×Dic3342
C2×C4×D6682
C2×(C22⋊D167)⋊C42
C2×#124×C1672
C2×#548167