Finite Group Databasebeta

← C2×#23752868 #34082573

Subgroups — C2×#23752868 #34082573

Order 19328.

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×#14559003C211yesmaximal, characteristic, complemented
2C23×Dic302C211yesmaximal, characteristic, complemented
3#23752868C2116yesmaximal, direct product factor
4C2×#18012726C211yesmaximal, characteristic, complemented
5C2×#18012725C212yesmaximal, complemented
6C2×#18012719C211yesmaximal, characteristic, complemented
7C2×#18012716C212yesmaximal, complemented
8C2×#18012711C212yesmaximal, complemented
9C2×#18012704C211yesmaximal, characteristic, complemented
10C2×D4×Dic151C212yesmaximal, complemented
11C2×C4×C22⋊D151C211yesmaximal, characteristic, complemented
12C2×#121×C151C211yesmaximal, characteristic, complemented
13C2×#14559003—28no—
14C2×#14559003C2218yescomplemented
15C22×Dic302—28no—
16C22×Dic302C2212yes—
17C22×Dic302C2211yescharacteristic
18C22×Dic302C2211yescharacteristic
19C22×Dic302C2218yescomplemented
20C22×C22⋊D151C2212yes—
21C22×C22⋊D151C2211yescharacteristic
22C23×Dic151C2212yes—
23C23×Dic151C2212yes—
24C22×C4×D151C2211yescharacteristic
25#18012726C2218yescomplemented
26#18012725C22116yescomplemented
27#18012719C2218yescomplemented
28#18012716C22116yescomplemented
29#18012711C22116yescomplemented
30#18012704C2218yescomplemented
31D4×Dic151C22116yescomplemented
32C4×C22⋊D151C2218yescomplemented
33C2×C4⋊Dic151C2212yes—
34C2×Dic151⋊C4C2211yescharacteristic
35C2×Dic151⋊C4C2212yes—
36C2×C22⋊Dic151C2212yes—
37C2×C22⋊Dic151C2212yes—
38C2×C22⋊Dic151C2211yescharacteristic
39C2×(C2×D151)⋊C4C2212yes—
40C2×(C2×D151)⋊C4C2211yescharacteristic
41C2×C4×Dic151C2212yes—
42C2×C4×Dic151C2211yescharacteristic
43C22×D4×C151C2211yescharacteristic
44C22×D4×C151C2212yes—
45C23×C4×C151C2211yescharacteristic
46#121×C151C2218yescomplemented
47C2×C4⋊C4×C151C2211yescharacteristic
48C2×C22⋊C4×C151C2212yes—
49C23×D151C2311yescharacteristic
50#14559003—232no—
51C2×Dic302—232no—
52C2×Dic302—28no—
53C2×Dic302—24no—
54C2×Dic302D418yescomplemented
55C2×Dic302C2318yes—
56C2×Dic302C2314yes—
57C2×Dic302C2314yes—
58C2×C22⋊D151—28no—
59C2×C22⋊D151—24no—
60C2×C22⋊D151C2314yes—
61C2×C22⋊D151C2318yes—
62C22×Dic151—28no—
63C22×Dic151—28no—
64C22×Dic151C2311yescharacteristic
65C22×Dic151C2311yescharacteristic
66C22×Dic151C2312yes—
67C22×Dic151C2311yescharacteristic
68C22×Dic151C2318yes—
69C22×Dic151C2318yes—
70C22×Dic151C2312yes—
71C2×C4×D151—24no—
72C2×C4×D151C2314yes—
73C4⋊Dic151C2318yes—
74Dic151⋊C4C2314yes—
75Dic151⋊C4C2318yes—
76C22⋊Dic151C2314yes—
77C22⋊Dic151C2318yes—
78C22⋊Dic151C2318yes—
79(C2×D151)⋊C4C2314yes—
80(C2×D151)⋊C4C2318yes—
81C4×Dic151C2314yes—
82C4×Dic151C2318yes—
83C24×C151C2312yes—
84C24×C151C2311yescharacteristic
85C2×D4×C151—28no—
86C2×D4×C151—24no—
87C2×D4×C151C2318yes—
88C2×D4×C151C2314yes—
89C22×C4×C151—24no—
90C22×C4×C151C2312yes—
91C22×C4×C151C2311yescharacteristic
92C22×C4×C151C2314yes—
93C22×C4×C151C2311yescharacteristic
94C4⋊C4×C151C2314yes—
95C22⋊C4×C151C2318yes—
96C22×D151—22no—
97C22×D151—22no—
98C22×D151—22no—
99C22×D151C2412yes—
100Dic302—28no—
101Dic302—216no—
102Dic302C2×D4116yescomplemented
103C22⋊D151—28no—
104C22⋊D151—216no—
105C2×Dic151—22no—
106C2×Dic151—24no—
107C2×Dic151—22no—
108C2×Dic151—22no—
109C2×Dic151—216no—
110C2×Dic151—216no—
111C2×Dic151C2×D418yes—
112C2×Dic151C2412yes—
113C2×Dic151C2412yes—
114C2×Dic151C2414yes—
115C2×Dic151C2412yes—
116C2×Dic151C2414yes—
117C4×D151—28no—
118C23×C151—22no—
119C23×C151—22no—
120C23×C151—24no—
121C23×C151—24no—
122C23×C151—24no—
123C23×C151#4114yescomplemented
124C23×C151C2414yes—
125C23×C151C2412yes—
126C23×C151C2411yescharacteristic
127D4×C151—216no—
128D4×C151—28no—
129C2×C4×C151—22no—
130C2×C4×C151—24no—
131C2×C4×C151—28no—
132C2×C4×C151C2×D414yes—
133C2×C4×C151C2414yes—
134C2×C4×C151C2412yes—
135C2×C4×C151C2412yes—
136C22×C151—28no—
137C22×C151—22no—
138C22×C151—22no—
139C22×C151—24no—
140C22×C151—24no—
141C22×C151—24no—
142C22×C151—48no—
143C22×C151C22×D412yes—
144C22×C151C2×#4112yes—
145C22×C151C2×#4114yescomplemented
146C22×C1512−1+412yes—
147C22×C151C2511yescharacteristic
148C2×D151—22no—
149C2×D151—22no—
150C2×D151—22no—
151C2×D151—44no—
152C4×C151—22no—
153C4×C151—24no—
154C4×C151C22×D414yes—
155Dic151—22no—
156Dic151—24no—
157Dic151—22no—
158Dic151—22no—
159Dic151C22×D418yes—
160C2×C151—24no—
161C2×C151—44no—
162C2×C151#54814yescomplemented
163C2×C151C23×D411yescharacteristic
164C2×C151C22×#4111yescharacteristic
165C2×C151C2×2−1+411yescharacteristic
166D151—42no—
167C151C2×#54811yescharacteristic, complemented, Sylow
168C2×#548—1511nomaximal, Sylow
169C22×#41—1511no—
170C23×Q8—1511no—
171#548—15116no—
172C2×#124—1511no—
173C2×#124—1512no—
174C2×C22⋊Q8—1511no—
175C2×C22⋊Q8—1512no—
176C2×#121—1512no—
177C2×#121D15111yescharacteristic, complemented
178C2×C4×Q8—1511no—
179C2×C4×D4—1512no—
180C2×C4×D4—1511no—
181C2×#41—3028no—
182C2×#41—1518no—
183C22×Q8—3028no—
184C22×Q8—1511no—
185C22×Q8—1512no—
186C22×Q8—1518no—
187C22×Q8—1511no—
188C22×D4—1511no—
189C22×D4—1512no—
190C22×D4C2×D15112yes—
191C22×D4C2×D15111yescharacteristic
192C23×C4—1512no—
193C23×C4—1512no—
194C23×C4—1511no—
195C23×C4C2×D15111yescharacteristic
196#124—1518no—
197#124—15116no—
198C22⋊Q8—1518no—
199C22⋊Q8—15116no—
200#121—15116no—
201#121C2×D15118yescomplemented
202C4×Q8—1518no—
203C4×D4—15116no—
204C4×D4—1518no—
205C2×C4⋊C4—1511no—
206C2×C4⋊C4—1512no—
207C2×C4⋊C4—1512no—
208C2×C4⋊C4C2×D15111yescharacteristic
209C2×C22⋊C4—1511no—
210C2×C22⋊C4—1512no—
211C2×C22⋊C4—1512no—
212C2×C22⋊C4—1511no—
213C2×C22⋊C4—1512no—
214C2×C22⋊C4C2×D15112yes—
215C2×C42—1511no—
216C2×C42—1512no—
217C24—1511no—
218C24C22×D15112yes—
219C24C22×D15111yescharacteristic
220#41—30232no—
221C2×Q8—30232no—
222C2×Q8—3028no—
223C2×Q8—3024no—
224C2×Q8—1518no—
225C2×Q8—1514no—
226C2×Q8—1518no—
227C2×Q8—1514no—
228C2×D4—28no—
229C2×D4—24no—
230C2×D4—3024no—
231C2×D4—3028no—
232C2×D4—1514no—
233C2×D4—1518no—
234C2×D4C22×D15118yes—
235C2×D4C22×D15114yes—
236C22×C4—24no—
237C22×C4—3028no—
238C22×C4—3028no—
239C22×C4—3024no—
240C22×C4—1518no—
241C22×C4—1518no—
242C22×C4—1511no—
243C22×C4—1511no—
244C22×C4—1512no—
245C22×C4—1511no—
246C22×C4—1514no—
247C22×C4—1512no—
248C22×C4C22×D15114yes—
249C22×C4C22×D15112yes—
250C22×C4C22×D15111yescharacteristic
251C22×C4C22×D15111yescharacteristic
252C4⋊C4—1514no—
253C4⋊C4—1518no—
254C4⋊C4—1518no—
255C4⋊C4C22×D15114yes—
256C22⋊C4—1514no—
257C22⋊C4—1518no—
258C22⋊C4—1518no—
259C22⋊C4—1514no—
260C22⋊C4—1518no—
261C22⋊C4C22×D15118yes—
262C42—1514no—
263C42—1518no—
264C23—22no—
265C23—22no—
266C23—24no—
267C23—24no—
268C23—24no—
269C23—3022no—
270C23—3022no—
271C23—3022no—
272C23—1512no—
273C23#1455900314yescomplemented
274C23C23×D15114yes—
275C23C23×D15112yes—
276C23C23×D15111yescharacteristic
277Q8—30216no—
278Q8—3028no—
279Q8—15116no—
280D4—216no—
281D4—28no—
282D4—3028no—
283D4—30216no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—28no—
287C2×C4—3022no—
288C2×C4—3024no—
289C2×C4—3022no—
290C2×C4—3022no—
291C2×C4—30216no—
292C2×C4—30216no—
293C2×C4—3028no—
294C2×C4—1512no—
295C2×C4—1512no—
296C2×C4—1514no—
297C2×C4—1512no—
298C2×C4—1518no—
299C2×C4—1514no—
300C2×C4D4×D15114yes—
301C2×C4C23×D15114yes—
302C2×C4C23×D15112yes—
303C2×C4C23×D15112yes—
304C22—22no—
305C22—22no—
306C22—28no—
307C22—24no—
308C22—24no—
309C22—24no—
310C22—48no—
311C22—3022no—
312C22—3022no—
313C22—3022no—
314C22—6044no—
315C22C2×D4×D15112yes—
316C22C2×#1455900312yes—
317C22C2×#1455900314yescomplemented
318C22#1801278612yes—
319C22C24×D15111yescharacteristic
320C4—22no—
321C4—24no—
322C4—3022no—
323C4—3024no—
324C4—3022no—
325C4—3022no—
326C4—1518no—
327C4C2×D4×D15114yes—
328C2—24no—
329C2—44no—
330C2—6042no—
331C2#2375286814yesdirect product factor
332C2C22×D4×D15111yescharacteristic
333C2C22×#1455900311yescharacteristic
334C2C2×#1801278611yescharacteristic
subgroupG/H
centerC23C23×D151
commutator, cosocleC22×C151C25
FrattiniC22C24×D151
FittingC2×#121×C151C2
socleC23×C151C24
hypercenterC2×#121D151
hypocenterC151C2×#548
LCSG ▷ C22×C151 ▷ C151
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#14559003C2semidirect
2. C23×Dic302C2semidirect
3. #23752868C2direct (N × H)
4. C2×#18012726C2semidirect
5. C2×#18012725C2semidirect
6. C2×#18012719C2semidirect
7. C2×#18012716C2semidirect
8. C2×#18012711C2semidirect
9. C2×#18012704C2semidirect
10. C2×D4×Dic151C2semidirect
11. C2×C4×C22⋊D151C2semidirect
12. C2×#121×C151C2semidirect
14. C2×#14559003C22semidirect
19. C22×Dic302C22semidirect
25. #18012726C22semidirect
26. #18012725C22semidirect
27. #18012719C22semidirect
28. #18012716C22semidirect
29. #18012711C22semidirect
30. #18012704C22semidirect
31. D4×Dic151C22semidirect
32. C4×C22⋊D151C22semidirect
46. #121×C151C22semidirect
54. C2×Dic302D4semidirect
102. Dic302C2×D4semidirect
123. C23×C151#41semidirect
145. C22×C151C2×#41semidirect
162. C2×C151#548semidirect
167. C151C2×#548semidirect
177. C2×#121D151semidirect
201. #121C2×D151semidirect
273. C23#14559003semidirect
317. C22C2×#14559003semidirect
331. C2#23752868direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145590032
C23×Dic3022
#237528682
C2×#180127262
C2×#180127252
C2×#180127192
C2×#180127162
C2×#180127112
C2×#180127042
C2×D4×Dic1512
C2×C4×C22⋊D1512
C2×#121×C1512
C2×#548151