Finite Group Databasebeta

← #34058400

Subgroups — #34058400

Order 19072.

1124 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750330C214yesmaximal, complemented
2#23750323C214yesmaximal, complemented
3#23750310C214yesmaximal, complemented
4#23750276C214yesmaximal, complemented
5#23750256C211yesmaximal, characteristic, complemented
6#23750222C214yesmaximal, complemented
7#23750206C211yesmaximal, characteristic, complemented
8#23750136C212yesmaximal, complemented
9#23750136C214yesmaximal, complemented
10#23750106C212yesmaximal, complemented
11C22×C4⋊C4×C149C211yesmaximal, characteristic, complemented
12C2×#14558736C2212yes—
13C2×#14558736C2214yes—
14#18011910C2214yescomplemented
15#18011906C2218yescomplemented
16#18011902C2214yescomplemented
17#18011901C2218yescomplemented
18#18011896C2218yescomplemented
19#18011894C2214yes—
20#18011894C2218yes—
21#18011891C2214yescomplemented
22#18011890C2214yes—
23#18011890C2212yes—
24#18011890C2212yes—
25#18011890C2214yes—
26#18011889C2218yescomplemented
27#18011886C2214yescomplemented
28#18011885C2214yes—
29#18011885C2212yes—
30#18011885C2212yes—
31#18011885C2214yes—
32#18011880C2214yes—
33#18011876C2214yescomplemented
34#18011873C2214yescomplemented
35C4×C22⋊D149C2218yes—
36C4×C22⋊D149C2214yes—
37C4×C22⋊D149C2214yes—
38C4×C22⋊D149C2218yes—
39#18011870C2212yes—
40#18011870C2214yes—
41#18011869C2214yescomplemented
42C4⋊C4×D149C2214yescomplemented
43C23×C4×C149C2211yescharacteristic
44C23×C4×C149C2212yes—
45C2×C4⋊C4×C149C2211yescharacteristic
46C2×C4⋊C4×C149C2212yes—
47C2×C4⋊C4×C149C2214yes—
48C2×C4⋊C4×C149C2214yes—
49C2×C4⋊C4×C149C2211yescharacteristic
50#14558736—28no—
51#14558736—48no—
52C2×Dic298C2312yes—
53C2×Dic298C2314yes—
54C2×D596C2312yes—
55C2×D596C2314yes—
56C2×C22⋊D149C2314yes—
57C2×C22⋊D149C2314yes—
58C2×C22⋊D149C2314yes—
59C2×C4×D149C2314yes—
60C2×C4×D149C2318yes—
61C4⋊Dic149C2314yes—
62C4⋊Dic149C2312yes—
63C4⋊Dic149C2312yes—
64C4⋊Dic149C2314yes—
65#14558715C2314yes—
66#14558715C2318yes—
67#14558715C2318yes—
68#14558715C2314yes—
69C22⋊Dic149C2314yes—
70C22⋊Dic149C2314yes—
71C22⋊Dic149C2314yes—
72#14558713C2314yes—
73#14558713C2318yes—
74#14558713C2318yes—
75#14558713C2314yes—
76C4×Dic149C2318yes—
77C4×Dic149C2314yes—
78C24×C149C2311yescharacteristic
79C22×C4×C149—24no—
80C22×C4×C149—44no—
81C22×C4×C149C2314yes—
82C22×C4×C149C2312yes—
83C22×C4×C149C2312yes—
84C22×C4×C149C2311yescharacteristic
85C22×C4×C149C2312yes—
86C22×C4×C149C2314yes—
87C22×C4×C149C2312yes—
88C22×C4×C149C2311yescharacteristic
89C4⋊C4×C149C2318yes—
90C4⋊C4×C149C2314yes—
91C4⋊C4×C149C2314yes—
92C22×D149C2414yes—
93Dic298—24no—
94Dic298—44no—
95D596—24no—
96D596—44no—
97C22⋊D149—28no—
98C22⋊D149—28no—
99C22⋊D149—28no—
100C2×Dic149C2414yes—
101C2×Dic149C2414yes—
102C2×Dic149C2414yes—
103C4×D149—28no—
104C4×D149—48no—
105C23×C149—22no—
106C23×C149—22no—
107C23×C149—22no—
108C23×C149C2411yescharacteristic
109C23×C149C2411yescharacteristic
110C23×C149C2411yescharacteristic
111C2×C4×C149—22no—
112C2×C4×C149—28no—
113C2×C4×C149—24no—
114C2×C4×C149—24no—
115C2×C4×C149—42no—
116C2×C4×C149—44no—
117C2×C4×C149—42no—
118C2×C4×C149#4114yescomplemented
119C2×C4×C149C2414yes—
120C2×C4×C149C2414yes—
121C2×C4×C149C2412yes—
122C2×C4×C149C2412yes—
123C22×C149—28no—
124C22×C149—21no—
125C22×C149—21no—
126C22×C149—21no—
127C22×C149—21no—
128C22×C149—21no—
129C22×C149—21no—
130C22×C149C2×#4112yes—
131C22×C149H(5,𝔽2)12yes—
132C22×C1492−1+412yes—
133C22×C149C2511yescharacteristic
134C2×D149—24no—
135C2×D149—24no—
136C2×D149—24no—
137C4×C149—22no—
138C4×C149—24no—
139C4×C149—24no—
140C4×C149C2×#4114yes—
141Dic149—24no—
142Dic149—24no—
143Dic149—24no—
144C2×C149—22no—
145C2×C149—22no—
146C2×C149—22no—
147C2×C149C22×#4111yescharacteristic
148C2×C149C2×H(5,𝔽2)11yescharacteristic
149C2×C149C2×2−1+411yescharacteristic
150D149—44no—
151C149#349211yescharacteristic, complemented, Sylow
152#3492—1491nomaximal, Sylow
153#554—1494no—
154#553—1494no—
155#552—1494no—
156#547—1494no—
157#544—1491no—
158#539—1494no—
159#537—1491no—
160#525—1492no—
161#525—1494no—
162#520—1492no—
163C22×C4⋊C4D14911yescharacteristic, complemented
164C2×#41—1492no—
165C2×#41—1494no—
166C23×C4C2×D14911yescharacteristic
167C23×C4C2×D14912yes—
168C4⋊Q8—1494no—
169#126—1498no—
170#125—1498no—
171#125—1494no—
172#123—1494no—
173#123—1498no—
174#123—1498no—
175C22⋊Q8—1494no—
176C22⋊Q8—1492no—
177C22⋊Q8—1492no—
178C22⋊Q8—1494no—
179C22⋊Q8—1494no—
180C22⋊Q8—1498no—
181#121—1494no—
182#121—1492no—
183#121—1494no—
184#121—1492no—
185#121—1494no—
186#120—1494no—
187C4×Q8—1494no—
188C4×D4—1498no—
189C4×D4—1494no—
190C4×D4—1498no—
191C4×D4—1494no—
192C4×D4—1494no—
193#117—1492no—
194#117—1494no—
195#117—1494no—
196C2×C4⋊C4—1494no—
197C2×C4⋊C4C2×D14911yescharacteristic
198C2×C4⋊C4C2×D14911yescharacteristic
199C2×C4⋊C4C2×D14912yes—
200C2×C4⋊C4C2×D14914yes—
201C2×C4⋊C4C2×D14914yes—
202C24C22×D14911yescharacteristic
203#41—2988no—
204#41—5968no—
205C2×Q8—1492no—
206C2×Q8—1494no—
207C2×D4—1494no—
208C2×D4—1494no—
209C2×D4—1494no—
210C2×D4—1492no—
211C2×D4—1494no—
212C22×C4—24no—
213C22×C4—44no—
214C22×C4—1494no—
215C22×C4—1498no—
216C22×C4C22×D14914yes—
217C22×C4C22×D14912yes—
218C22×C4C22×D14912yes—
219C22×C4C22×D14911yescharacteristic
220C22×C4C22×D14912yes—
221C22×C4C22×D14914yes—
222C22×C4C22×D14912yes—
223C22×C4C22×D14911yescharacteristic
224C4⋊C4—1492no—
225C4⋊C4—1498no—
226C4⋊C4—1492no—
227C4⋊C4—1494no—
228C4⋊C4—1494no—
229C4⋊C4—1494no—
230C4⋊C4—1498no—
231C4⋊C4—1494no—
232C4⋊C4C22×D14914yes—
233C4⋊C4C22×D14914yes—
234C4⋊C4C22×D14918yes—
235C22⋊C4—1494no—
236C22⋊C4—1494no—
237C22⋊C4—1494no—
238C22⋊C4—1498no—
239C22⋊C4—1498no—
240C22⋊C4—1494no—
241C22⋊C4—1494no—
242C42—1498no—
243C42—1494no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—1494no—
248C23C23×D14911yescharacteristic
249C23C23×D14911yescharacteristic
250C23C23×D14911yescharacteristic
251Q8—2984no—
252Q8—5964no—
253D4—2988no—
254D4—2988no—
255D4—2988no—
256D4—2984no—
257D4—5964no—
258C2×C4—22no—
259C2×C4—28no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—2988no—
266C2×C4—5968no—
267C2×C4—1494no—
268C2×C4—1494no—
269C2×C4—1494no—
270C2×C4#1455873614yescomplemented
271C2×C4C23×D14914yes—
272C2×C4C23×D14912yes—
273C2×C4C23×D14914yes—
274C2×C4C23×D14912yes—
275C22—28no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—2984no—
283C22—2984no—
284C22—2984no—
285C22C2×#1455873612yes—
286C22#1801195612yes—
287C22#1801195912yes—
288C22C24×D14911yescharacteristic
289C4—22no—
290C4—24no—
291C4—24no—
292C4—2984no—
293C4—2984no—
294C4—2984no—
295C4C2×#1455873614yes—
296C2—22no—
297C2—22no—
298C2—22no—
299C2—5964no—
300C2C22×#1455873611yescharacteristic
301C2C2×#1801195611yescharacteristic
302C2C2×#1801195911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D149
commutator, socle, cosocleC22×C149C25
FittingC22×C4⋊C4×C149C2
hypercenterC22×C4⋊C4D149
hypocenterC149#3492
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #23750330C2semidirect
2. #23750323C2semidirect
3. #23750310C2semidirect
4. #23750276C2semidirect
5. #23750256C2semidirect
6. #23750222C2semidirect
7. #23750206C2semidirect
8. #23750136C2semidirect
9. #23750136C2semidirect
10. #23750106C2semidirect
11. C22×C4⋊C4×C149C2semidirect
14. #18011910C22semidirect
15. #18011906C22semidirect
16. #18011902C22semidirect
17. #18011901C22semidirect
18. #18011896C22semidirect
21. #18011891C22semidirect
26. #18011889C22semidirect
27. #18011886C22semidirect
33. #18011876C22semidirect
34. #18011873C22semidirect
41. #18011869C22semidirect
42. C4⋊C4×D149C22semidirect
118. C2×C4×C149#41semidirect
151. C149#3492semidirect
163. C22×C4⋊C4D149semidirect
270. C2×C4#14558736semidirect

Maximal subgroups

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

subgroupindex
#237503302
#237503232
#237503102
#237502762
#237502562
#237502222
#237502062
#237501362
#237501062
C22×C4⋊C4×C1492
#3492149