Finite Group Databasebeta

← #37583282

Subgroups — #37583282

Order 24448.

472 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#27133173C211yesmaximal, characteristic, complemented
2C2×#20259293C211yesmaximal, characteristic, complemented
3#27132999C211yesmaximal, characteristic, complemented
4#27132985C212yesmaximal, complemented
5#27132979C211yesmaximal, characteristic, complemented
6#27132896C212yesmaximal, complemented
7#27132891C212yesmaximal, complemented
8#27132814C211yesmaximal, characteristic, complemented
9#27132805C212yesmaximal, complemented
10C22⋊C8×D191C211yesmaximal, characteristic, complemented
11#447×C191C211yesmaximal, characteristic, complemented
12C2×#15690286C2211yescharacteristic, complemented
13C22×C22⋊D191—21no—
14C22×C4×D191C2211yescharacteristic
15C2×#15690275C2212yescomplemented
16C2×C8×D191C2212yescomplemented
17#20259304—21no—
18#20259302C2211yescharacteristic, complemented
19#20259301C2211yescharacteristic
20#20259293—21no—
21#20259293—21no—
22#20259293C2212yes—
23#20259293C2212yes—
24#20259293C2211yescharacteristic, complemented
25#20259290—21no—
26D4×Dic191—21no—
27(C22⋊D191)⋊C4—21no—
28#20259280C2211yescharacteristic, complemented
29C2×C4⋊Dic191C2211yescharacteristic
30C2×C22⋊Dic191—21no—
31#20259256C2212yescomplemented
32#20259253C2212yescomplemented
33#20259247C2212yescomplemented
34#20259243C2212yescomplemented
35#20259243C2212yescomplemented
36C22⋊(C191⋊C8)C2211yescharacteristic
37(C2×D191)⋊C8C2212yescomplemented
38C22×D4×C191C2211yescharacteristic
39C2×D8×C191C2212yescomplemented
40#121×C191C2211yescharacteristic
41#102×C191C2212yescomplemented
42C22⋊C8×C191C2211yescharacteristic
43C23×D191D411yescharacteristic
44#15690286—42no—
45C2×Dic382C2311yescharacteristic
46C2×C22⋊D191—22no—
47C2×C22⋊D191—22no—
48C2×C22⋊D191—21no—
49C2×C22⋊D191—21no—
50C2×C22⋊D191—44no—
51C22×Dic191—21no—
52C22×Dic191—21no—
53C22×Dic191D411yescharacteristic
54C2×C4×D191—42no—
55C2×C4×D191D411yescharacteristic
56C2×C4×D191D411yescharacteristic
57C2×C4×D191C2311yescharacteristic
58C2×C4×D191C2311yescharacteristic
59#15690275—42no—
60C2×C191⋊C8C2312yes—
61C8×D191—42no—
62C4⋊Dic191—21no—
63C4⋊Dic191—21no—
64C4⋊Dic191C2311yescharacteristic
65C4⋊Dic191C2312yes—
66Dic191⋊C4—21no—
67C22⋊Dic191—22no—
68C22⋊Dic191—22no—
69C22⋊Dic191—21no—
70C22⋊Dic191—21no—
71(C2×D191)⋊C4—21no—
72(C2×D191)⋊C4—21no—
73C4×Dic191—21no—
74C24×C191—21no—
75C2×D4×C191—21no—
76C2×D4×C191—21no—
77C2×D4×C191—42no—
78C2×D4×C191C2312yes—
79C2×D4×C191C2311yescharacteristic
80C22×C4×C191C2311yescharacteristic
81D8×C191—42no—
82C2×C8×C191C2312yes—
83C4⋊C4×C191C2311yescharacteristic
84C22⋊C4×C191—21no—
85C22×D191—22no—
86C22×D191—22no—
87C22×D191—41no—
88C22×D191C2×D411yescharacteristic
89C22×D191C2×D411yescharacteristic
90Dic382—22no—
91C22⋊D191—44no—
92C22⋊D191—42no—
93C22⋊D191—42no—
94C22⋊D191—44no—
95C2×Dic191—21no—
96C2×Dic191—21no—
97C2×Dic191—22no—
98C2×Dic191—41no—
99C2×Dic191—42no—
100C2×Dic191—42no—
101C2×Dic191C2×D411yescharacteristic
102C2×Dic191C2×D411yescharacteristic
103C4×D191—22no—
104C4×D191—22no—
105C4×D191—41no—
106C4×D191—41no—
107C191⋊C8—22no—
108C23×C191—21no—
109C23×C191—22no—
110C23×C191—42no—
111C23×C191—42no—
112C23×C191—42no—
113C23×C191C2×D411yescharacteristic
114D4×C191—22no—
115D4×C191—22no—
116D4×C191—22no—
117D4×C191—42no—
118D4×C191—42no—
119C2×C4×C191—22no—
120C2×C4×C191—21no—
121C2×C4×C191C2×D411yescharacteristic
122C2×C4×C191C2411yescharacteristic
123C8×C191—22no—
124C22×C191—21no—
125C22×C191—21no—
126C22×C191—42no—
127C22×C191—41no—
128C22×C191—41no—
129C22×C191—42no—
130C22×C191—42no—
131C22×C191—41no—
132C22×C191—84no—
133C22×C191#13512yes—
134C22×C191C22×D411yescharacteristic
135C2×D191—21no—
136C2×D191—21no—
137C2×D191—21no—
138C2×D191—21no—
139C2×D191—21no—
140C2×D191—44no—
141C2×D191AGL(1,ℤ/8)12yes—
142C4×C191—21no—
143C4×C191—41no—
144C4×C191H(5,𝔽2)12yes—
145Dic191—21no—
146Dic191—21no—
147Dic191—41no—
148Dic191—42no—
149Dic191—41no—
150C2×C191—22no—
151C2×C191—81no—
152C2×C191—82no—
153C2×C191#53411yescharacteristic
154C2×C191C2×#13511yescharacteristic
155C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
156D191—22no—
157D191—41no—
158C191#318811yescharacteristic, complemented, Sylow
159#3188—1911nomaximal, Sylow
160#525—1911no—
161C2×#121—1911no—
162#484—1911no—
163#482—1912no—
164#481—1911no—
165#466—1912no—
166#465—1912no—
167#450—1911no—
168#449—1912no—
169#447D19111yescharacteristic, complemented
170C2×C22⋊C8—1911no—
171C2×#41—1911no—
172C22×D4—3821no—
173C22×D4C2×D19111yescharacteristic
174C23×C4—1911no—
175C2×SD16—1912no—
176C2×D8C2×D19112yescomplemented
177C22×C8—1912no—
178#123—3821no—
179C22⋊Q8—1911no—
180C22⋊Q8—1911no—
181#121—3821no—
182#121—3821no—
183#121—1912no—
184#121—1912no—
185#121—1911no—
186#121C2×D19111yescharacteristic
187#120—3821no—
188C4×D4—3821no—
189C4×D4—3821no—
190#117—1911no—
191C2×C4⋊C4—1911no—
192C2×C22⋊C4—3821no—
193#107—1912no—
194#106—1912no—
195#103—1912no—
196#102—1912no—
197#102—1912no—
198#102C2×D19112yescomplemented
199C22⋊C8—1911no—
200C22⋊C8—1912no—
201C22⋊C8C2×D19111yescharacteristic
202C24—21no—
203C24—1911no—
204#41—7642no—
205C2×Q8—1911no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—42no—
209C2×D4—3821no—
210C2×D4—3821no—
211C2×D4—3822no—
212C2×D4—3822no—
213C2×D4—7644no—
214C2×D4C22×D19111yescharacteristic
215C2×D4C22×D19112yes—
216C22×C4—3821no—
217C22×C4—3821no—
218C22×C4—1911no—
219C22×C4—1911no—
220C22×C4—1911no—
221C22×C4—1911no—
222C22×C4—1911no—
223C22×C4—7642no—
224C22×C4C22×D19111yescharacteristic
225SD16—7642no—
226D8—42no—
227C2×C8—1912no—
228C2×C8—7642no—
229C2×C8C22×D19112yes—
230C4⋊C4—3821no—
231C4⋊C4—3821no—
232C4⋊C4—3821no—
233C4⋊C4—1911no—
234C4⋊C4—1912no—
235C4⋊C4C22×D19111yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—3821no—
238C22⋊C4—3821no—
239C22⋊C4—3821no—
240C22⋊C4—3822no—
241C22⋊C4—3822no—
242C22⋊C4—3821no—
243C42—3821no—
244C23—21no—
245C23—22no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—3822no—
250C23—3822no—
251C23—1911no—
252C23—1911no—
253C23—7641no—
254C23D4×D19111yescharacteristic
255Q8—3822no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—7644no—
262D4—7642no—
263D4—7644no—
264D4—7642no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—3822no—
268C2×C4—3822no—
269C2×C4—3821no—
270C2×C4—3821no—
271C2×C4—3822no—
272C2×C4—1911no—
273C2×C4—1911no—
274C2×C4—7641no—
275C2×C4—7641no—
276C2×C4—7641no—
277C2×C4—7642no—
278C2×C4—7642no—
279C2×C4D4×D19111yescharacteristic
280C2×C4C23×D19111yescharacteristic
281C8—22no—
282C8—3822no—
283C22—21no—
284C22—21no—
285C22—41no—
286C22—42no—
287C22—42no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—84no—
292C22—3821no—
293C22—3821no—
294C22—3821no—
295C22—3821no—
296C22—3821no—
297C22—1912no—
298C22—7644no—
299C22#2025934112yes—
300C22C2×D4×D19111yescharacteristic
301C4—21no—
302C4—41no—
303C4—3821no—
304C4—3821no—
305C4—7642no—
306C4—7641no—
307C4—7641no—
308C4#2025936712yes—
309C2—22no—
310C2—82no—
311C2—81no—
312C2—3822no—
313C2—7641no—
314C2#2713322311yescharacteristic
315C2C2×#2025934111yescharacteristic
316C2AGL(1,ℤ/8)×D19111yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#447×C191C2
socleC22×C191C22×D4
hypercenter#447D191
hypocenterC191#3188
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #27133173C2semidirect
2. C2×#20259293C2semidirect
3. #27132999C2semidirect
4. #27132985C2semidirect
5. #27132979C2semidirect
6. #27132896C2semidirect
7. #27132891C2semidirect
8. #27132814C2semidirect
9. #27132805C2semidirect
10. C22⋊C8×D191C2semidirect
11. #447×C191C2semidirect
12. C2×#15690286C22semidirect
15. C2×#15690275C22semidirect
16. C2×C8×D191C22semidirect
18. #20259302C22semidirect
24. #20259293C22semidirect
28. #20259280C22semidirect
31. #20259256C22semidirect
32. #20259253C22semidirect
33. #20259247C22semidirect
34. #20259243C22semidirect
35. #20259243C22semidirect
37. (C2×D191)⋊C8C22semidirect
39. C2×D8×C191C22semidirect
41. #102×C191C22semidirect
158. C191#3188semidirect
169. #447D191semidirect
176. C2×D8C2×D191semidirect
198. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271331732
C2×#202592932
#271329992
#271329852
#271329792
#271328962
#271328912
#271328142
#271328052
C22⋊C8×D1912
#447×C1912
#3188191