Finite Group Databasebeta

← #34055447

Subgroups — #34055447

Order 19072.

468 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 121 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⋊Q8×D149C211yesmaximal, characteristic, complemented
2C2×#18011886C211yesmaximal, characteristic
3#23749955C211yesmaximal, characteristic, complemented
4#23749945C212yesmaximal, complemented
5#23749857C212yesmaximal, complemented
6#23749855C212yesmaximal, complemented
7#23749778C211yesmaximal, characteristic, complemented
8#23749775C211yesmaximal, characteristic, complemented
9#23749765C212yesmaximal, complemented
10C22⋊C8×D149C211yesmaximal, characteristic, complemented
11#483×C149C211yesmaximal, characteristic, complemented
12C2×Q8×D149C2211yescharacteristic, complemented
13C22×D596—21no—
14C22×D596C2211yescharacteristic
15C22×C4×D149C2211yescharacteristic
16C2×#14558725C2212yescomplemented
17C2×#14558723C2212yescomplemented
18C2×C8×D149C2212yescomplemented
19#18011891C2211yescharacteristic, complemented
20#18011890C2211yescharacteristic
21#18011889—21no—
22#18011888—21no—
23#18011887C2211yescharacteristic, complemented
24#18011886—21no—
25#18011886—21no—
26#18011886C2212yes—
27#18011886C2212yes—
28C4⋊C4×D149—21no—
29C4⋊C4×D149C2211yescharacteristic, complemented
30C2×#14558713—21no—
31C22⋊C4×D149—21no—
32#18011842C2212yescomplemented
33#18011834C2212yescomplemented
34#18011833C2212yescomplemented
35#18011831C2212yescomplemented
36C22⋊#11924746C2211yescharacteristic
37#18011823C2212yescomplemented
38C22⋊Q8×C149C2211yescharacteristic
39C2×C4⋊C4×C149C2211yescharacteristic
40#106×C149C2212yescomplemented
41#103×C149C2212yescomplemented
42C22⋊C8×C149C2211yescharacteristic
43C23×D149—21no—
44C23×D149D411yescharacteristic
45C2×Dic298—21no—
46C2×Dic298C2311yescharacteristic
47Q8×D149—42no—
48C2×D596—22no—
49C2×D596—22no—
50C2×D596—21no—
51C2×D596—42no—
52C2×D596—44no—
53C2×D596C2312yes—
54C22×Dic149D411yescharacteristic
55C2×C4×D149—21no—
56C2×C4×D149—21no—
57C2×C4×D149—42no—
58C2×C4×D149D411yescharacteristic
59C2×C4×D149D411yescharacteristic
60C2×C4×D149C2311yescharacteristic
61C2×C4×D149C2311yescharacteristic
62#14558725—42no—
63#14558723—42no—
64C2×#11924746C2312yes—
65C8×D149—42no—
66C4⋊Dic149—21no—
67C4⋊Dic149C2311yescharacteristic
68#14558715—21no—
69#14558715—21no—
70#14558715—21no—
71C22⋊Dic149—21no—
72#14558713—21no—
73#14558713—21no—
74#14558713—22no—
75#14558713—22no—
76C2×Q8×C149C2311yescharacteristic
77C22×C4×C149—21no—
78C22×C4×C149C2311yescharacteristic
79C2×C8×C149C2312yes—
80C4⋊C4×C149—21no—
81C4⋊C4×C149—21no—
82C4⋊C4×C149C2312yes—
83C4⋊C4×C149C2311yescharacteristic
84C22⋊C4×C149—21no—
85C22×D149—22no—
86C22×D149—41no—
87C22×D149—41no—
88C22×D149—42no—
89C22×D149—42no—
90C22×D149—42no—
91C22×D149SD1614yescomplemented
92C22×D149C2×D411yescharacteristic
93C22×D149C2×D411yescharacteristic
94Dic298—22no—
95Dic298—42no—
96D596—22no—
97D596—22no—
98D596—42no—
99D596—44no—
100D596—44no—
101C2×Dic149—21no—
102C2×Dic149—21no—
103C2×Dic149—41no—
104C2×Dic149C2×D411yescharacteristic
105C2×Dic149C2×D411yescharacteristic
106C4×D149—22no—
107C4×D149—22no—
108C4×D149—41no—
109C4×D149—41no—
110C4×D149—42no—
111C4×D149—42no—
112#11924746—22no—
113C23×C149C2×D411yescharacteristic
114Q8×C149—22no—
115C2×C4×C149—22no—
116C2×C4×C149—21no—
117C2×C4×C149—21no—
118C2×C4×C149—22no—
119C2×C4×C149—42no—
120C2×C4×C149C2×D411yescharacteristic
121C2×C4×C149C2411yescharacteristic
122C8×C149—22no—
123C22×C149—21no—
124C22×C149—21no—
125C22×C149C2×SD1612yes—
126C22×C149C22×D411yescharacteristic
127C2×D149—21no—
128C2×D149—21no—
129C2×D149—21no—
130C2×D149—21no—
131C2×D149—44no—
132C2×D149—42no—
133C2×D149—42no—
134C2×D149—42no—
135C2×D149—84no—
136C2×D149C2×SD1612yes—
137C2×D149C2×SD1612yes—
138C4×C149—21no—
139C4×C149—41no—
140C4×C149—42no—
141C4×C149—41no—
142C4×C149H(5,𝔽2)12yes—
143Dic149—21no—
144Dic149—21no—
145Dic149—41no—
146Dic149—41no—
147C2×C149—22no—
148C2×C149#53411yescharacteristic
149C2×C149C22×SD1611yescharacteristic
150C2×C149#57511yescharacteristic
151D149—22no—
152D149—22no—
153D149—82no—
154C149#318711yescharacteristic, complemented, Sylow
155#3187—1491nomaximal, Sylow
156C2×C22⋊Q8—1491no—
157C2×#121—1491no—
158#483—1491no—
159#483D14911yescharacteristic, complemented
160#481—1492no—
161#465—1492no—
162#465—1492no—
163#450—1491no—
164#450—1491no—
165#448—1492no—
166C2×C22⋊C8—1491no—
167C22×Q8—1491no—
168C22×D4—2981no—
169C22×D4—1491no—
170C23×C4—1491no—
171C2×SD16—1492no—
172C2×SD16—1492no—
173C22×C8—1492no—
174C22⋊Q8—2981no—
175C22⋊Q8—2981no—
176C22⋊Q8—1491no—
177C22⋊Q8—1491no—
178C22⋊Q8—1491no—
179C22⋊Q8C2×D14911yescharacteristic
180#121—2981no—
181#121—2981no—
182#121—1492no—
183#121—1492no—
184C2×C4⋊C4—2981no—
185C2×C4⋊C4—1491no—
186C2×C4⋊C4C2×D14911yescharacteristic
187C2×C22⋊C4—2981no—
188C2×C22⋊C4—2981no—
189#106—1492no—
190#106C2×D14912yescomplemented
191#103—1492no—
192#103C2×D14912yescomplemented
193#102—1492no—
194#102—1492no—
195C22⋊C8—1491no—
196C22⋊C8—1492no—
197C22⋊C8C2×D14911yescharacteristic
198C24—2981no—
199C24—1491no—
200C2×Q8—2981no—
201C2×Q8—5962no—
202C2×Q8—1491no—
203C2×Q8C22×D14911yescharacteristic
204C2×D4—2982no—
205C2×D4—2982no—
206C2×D4—2981no—
207C2×D4—5962no—
208C2×D4—5964no—
209C2×D4—1492no—
210C22×C4—21no—
211C22×C4—2981no—
212C22×C4—2981no—
213C22×C4—5962no—
214C22×C4—1491no—
215C22×C4—1491no—
216C22×C4—1491no—
217C22×C4—1491no—
218C22×C4—1491no—
219C22×C4C22×D14911yescharacteristic
220SD16—5962no—
221SD16—5962no—
222C2×C8—5962no—
223C2×C8—1492no—
224C2×C8C22×D14912yes—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—2981no—
228C4⋊C4—2981no—
229C4⋊C4—2981no—
230C4⋊C4—2981no—
231C4⋊C4—1491no—
232C4⋊C4C22×D14911yescharacteristic
233C4⋊C4C22×D14912yes—
234C22⋊C4—21no—
235C22⋊C4—2981no—
236C22⋊C4—2981no—
237C22⋊C4—2982no—
238C22⋊C4—2982no—
239C22⋊C4—2981no—
240C23—2982no—
241C23—5961no—
242C23—5961no—
243C23—5962no—
244C23—5962no—
245C23—5962no—
246C23—1491no—
247C23—1491no—
248C23—1494no—
249C23D4×D14911yescharacteristic
250Q8—22no—
251Q8—2982no—
252Q8—5962no—
253D4—2982no—
254D4—2982no—
255D4—5962no—
256D4—5964no—
257D4—5964no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—2982no—
264C2×C4—2982no—
265C2×C4—2981no—
266C2×C4—2981no—
267C2×C4—5961no—
268C2×C4—5961no—
269C2×C4—5962no—
270C2×C4—5962no—
271C2×C4—5961no—
272C2×C4—1491no—
273C2×C4—1491no—
274C2×C4D4×D14911yescharacteristic
275C2×C4C23×D14911yescharacteristic
276C8—22no—
277C8—2982no—
278C22—21no—
279C22—21no—
280C22—2981no—
281C22—2981no—
282C22—2981no—
283C22—2981no—
284C22—5964no—
285C22—5962no—
286C22—5962no—
287C22—5962no—
288C22—1492no—
289C22—1492no—
290C22—11924no—
291C22SD16×D14912yes—
292C22C2×D4×D14911yescharacteristic
293C4—21no—
294C4—41no—
295C4—41no—
296C4—42no—
297C4—2981no—
298C4—2981no—
299C4—5961no—
300C4—5961no—
301C4#1801195712yes—
302C2—22no—
303C2—2982no—
304C2—2982no—
305C2—11922no—
306C2#2375018611yescharacteristic
307C2C2×SD16×D14911yescharacteristic
308C2#2375042411yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#483×C149C2
socleC22×C149C22×D4
prime radicalC2×#18011886C2
hypercenter#483D149
hypocenterC149#3187
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D149C2semidirect
3. #23749955C2semidirect
4. #23749945C2semidirect
5. #23749857C2semidirect
6. #23749855C2semidirect
7. #23749778C2semidirect
8. #23749775C2semidirect
9. #23749765C2semidirect
10. C22⋊C8×D149C2semidirect
11. #483×C149C2semidirect
12. C2×Q8×D149C22semidirect
16. C2×#14558725C22semidirect
17. C2×#14558723C22semidirect
18. C2×C8×D149C22semidirect
19. #18011891C22semidirect
23. #18011887C22semidirect
29. C4⋊C4×D149C22semidirect
32. #18011842C22semidirect
33. #18011834C22semidirect
34. #18011833C22semidirect
35. #18011831C22semidirect
37. #18011823C22semidirect
40. #106×C149C22semidirect
41. #103×C149C22semidirect
91. C22×D149SD16semidirect
154. C149#3187semidirect
159. #483D149semidirect
190. #106C2×D149semidirect
192. #103C2×D149semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1492
C2×#180118862
#237499552
#237499452
#237498572
#237498552
#237497782
#237497752
#237497652
C22⋊C8×D1492
#483×C1492
#3187149