Finite Group Databasebeta

← #37582140

Subgroups — #37582140

Order 24448.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#20259350C211yesmaximal, characteristic, complemented
2C2×M16×D191C211yesmaximal, characteristic, complemented
3#27133175C211yesmaximal, characteristic, complemented
4C2×#20259297C211yesmaximal, characteristic, complemented
5#27132917C212yesmaximal, complemented
6#27132910C212yesmaximal, complemented
7#27132900C212yesmaximal, complemented
8#27132893C212yesmaximal, complemented
9#27132667C211yesmaximal, characteristic, complemented
10#27132662C211yesmaximal, characteristic, complemented
11#428×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13C22×D764—21no—
14C22×D764C2211yescharacteristic
15C22×C4×D191C2211yescharacteristic
16#20259350—24no—
17C2×#15690276C2212yescomplemented
18C2×D1528C2212yescomplemented
19C2×#15690271C2211yescharacteristic
20C2×#15690270C2212yescomplemented
21M16×D191—24no—
22C2×C8×D191C2212yescomplemented
23#20259306—21no—
24#20259302C2211yescharacteristic, complemented
25#20259302C2211yescharacteristic, complemented
26#20259297—21no—
27#20259297—21no—
28#20259297C2212yes—
29#20259297C2212yes—
30#20259297C2211yescharacteristic, complemented
31#20259297C2211yescharacteristic, complemented
32#20259292—21no—
33C4×D764—21no—
34C4×D764—21no—
35C2×(C2×D191)⋊C4—21no—
36#20259245C2212yescomplemented
37#20259242C2212yescomplemented
38#20259242C2212yescomplemented
39#20259242C2212yescomplemented
40C2×M16×C191C2211yescharacteristic
41#117×C191C2211yescharacteristic
42C2×C4⋊C4×C191C2211yescharacteristic
43#107×C191C2212yescomplemented
44#106×C191C2212yescomplemented
45C23×D191—21no—
46C23×D191D411yescharacteristic
47#15690287—22no—
48#15690287—22no—
49C2×Dic382C2311yescharacteristic
50C2×D764—22no—
51C2×D764—22no—
52C2×D764—21no—
53C2×D764—21no—
54C2×D764—22no—
55C2×D764—21no—
56C2×D764—22no—
57C2×D764—44no—
58C2×D764C2311yescharacteristic
59C2×D764C2312yes—
60C2×C22⋊D191—21no—
61C22×Dic191D411yescharacteristic
62C2×C4×D191—22no—
63C2×C4×D191—21no—
64C2×C4×D191—22no—
65C2×C4×D191D411yescharacteristic
66C2×C4×D191D411yescharacteristic
67C2×C4×D191C2311yescharacteristic
68C2×C4×D191C2311yescharacteristic
69#15690276—24no—
70D1528—24no—
71#15690271—22no—
72#15690270—24no—
73C2×C191⋊C8C2312yes—
74C8×D191—24no—
75C4⋊Dic191—21no—
76C4⋊Dic191—21no—
77(C2×D191)⋊C4—21no—
78(C2×D191)⋊C4—21no—
79(C2×D191)⋊C4—22no—
80(C2×D191)⋊C4—22no—
81(C2×D191)⋊C4—21no—
82(C2×D191)⋊C4—21no—
83C22×C4×C191—21no—
84C22×C4×C191C2311yescharacteristic
85M16×C191D414yescomplemented
86C2×C8×C191C2312yes—
87C4⋊C4×C191—21no—
88C4⋊C4×C191C2311yescharacteristic
89C4⋊C4×C191C2312yes—
90C4⋊C4×C191C2311yescharacteristic
91C22⋊C4×C191—21no—
92C42×C191—21no—
93C22×D191—22no—
94C22×D191—22no—
95C22×D191—21no—
96C22×D191—22no—
97C22×D191—41no—
98C22×D191—42no—
99C22×D191—42no—
100C22×D191—42no—
101C22×D191C2×D411yescharacteristic
102C22×D191C2×D411yescharacteristic
103Dic382—21no—
104Dic382—21no—
105D764—21no—
106D764—22no—
107D764—22no—
108D764—21no—
109D764—42no—
110D764—44no—
111D764—42no—
112D764—44no—
113D764—42no—
114C22⋊D191—42no—
115C2×Dic191—22no—
116C2×Dic191—21no—
117C2×Dic191C2×D411yescharacteristic
118C2×Dic191C2×D411yescharacteristic
119C4×D191—22no—
120C4×D191—22no—
121C4×D191—21no—
122C4×D191—21no—
123C4×D191—21no—
124C4×D191—21no—
125C4×D191—42no—
126C191⋊C8—22no—
127C23×C191C2×D411yescharacteristic
128C2×C4×C191—21no—
129C2×C4×C191—21no—
130C2×C4×C191—22no—
131C2×C4×C191—42no—
132C2×C4×C191C2×D411yescharacteristic
133C2×C4×C191C2×D412yes—
134C2×C4×C191#4112yes—
135C2×C4×C191C2411yescharacteristic
136C8×C191C2×D414yescomplemented
137C22×C191—21no—
138C22×C191—21no—
139C22×C191#12112yes—
140C22×C191C22×D411yescharacteristic
141C2×D191—21no—
142C2×D191—21no—
143C2×D191—21no—
144C2×D191—21no—
145C2×D191—21no—
146C2×D191—44no—
147C2×D191—41no—
148C2×D191—42no—
149C2×D191—42no—
150C2×D191—41no—
151C2×D191—41no—
152C2×D191—42no—
153C2×D191—84no—
154C2×D191AGL(1,ℤ/8)12yescomplemented
155C4×C191—42no—
156C4×C191—41no—
157C4×C191—41no—
158C4×C191#12112yes—
159C4×C191C22×D411yescharacteristic
160C4×C191C2×#4111yescharacteristic
161Dic191—21no—
162Dic191—21no—
163Dic191—41no—
164C2×C191—22no—
165C2×C191C2×#12111yescharacteristic
166C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C191#57511yescharacteristic
168D191—22no—
169D191—41no—
170D191—81no—
171D191—82no—
172C191#305511yescharacteristic, complemented, Sylow
173#3055—1911nomaximal, Sylow
174C2×AGL(1,ℤ/8)—1911no—
175C22×M16—1911no—
176#525—1911no—
177C2×#121—1911no—
178#469—1912no—
179#468—1912no—
180#466—1912no—
181#465—1912no—
182#428D19111yescharacteristic, complemented
183#418—1911no—
184#417—1911no—
185C2×#41—1911no—
186C22×D4—3821no—
187C22×D4—1911no—
188C23×C4—1911no—
189AGL(1,ℤ/8)—3824no—
190C2×SD16—1912no—
191C2×D8—1912no—
192C2×M16—3824no—
193C2×M16—1911no—
194C2×M16—1912no—
195C2×M16C2×D19111yescharacteristic
196C22×C8—1912no—
197#123—3821no—
198C22⋊Q8—1911no—
199C22⋊Q8—1911no—
200#121—3821no—
201#121—3821no—
202#121—1912no—
203#121—1912no—
204#121—1911no—
205#121—1911no—
206#120—3821no—
207C4×D4—3821no—
208C4×D4—3821no—
209#117C2×D19111yescharacteristic
210C2×C4⋊C4C2×D19111yescharacteristic
211C2×C22⋊C4—3821no—
212#107C2×D19112yescomplemented
213#106C2×D19112yescomplemented
214#103—1912no—
215#102—1912no—
216#102—1912no—
217#102—1912no—
218C24—3821no—
219C24—1911no—
220#41—3822no—
221#41—3822no—
222C2×Q8—1911no—
223C2×D4—3821no—
224C2×D4—3821no—
225C2×D4—3822no—
226C2×D4—3822no—
227C2×D4—3821no—
228C2×D4—3821no—
229C2×D4—3822no—
230C2×D4—3822no—
231C2×D4—1911no—
232C2×D4—1912no—
233C2×D4—7644no—
234C22×C4—21no—
235C22×C4—3822no—
236C22×C4—3822no—
237C22×C4—3821no—
238C22×C4—1911no—
239C22×C4—1911no—
240C22×C4—1911no—
241C22×C4—1911no—
242C22×C4—1911no—
243C22×C4C22×D19111yescharacteristic
244SD16—3824no—
245D8—3824no—
246M16—3822no—
247M16—3824no—
248M16D76414yescomplemented
249C2×C8—3824no—
250C2×C8—1912no—
251C2×C8C22×D19112yes—
252C4⋊C4—21no—
253C4⋊C4—3821no—
254C4⋊C4—3821no—
255C4⋊C4C22×D19111yescharacteristic
256C4⋊C4C22×D19111yescharacteristic
257C4⋊C4C22×D19112yes—
258C22⋊C4—21no—
259C22⋊C4—3821no—
260C22⋊C4—3821no—
261C22⋊C4—3822no—
262C22⋊C4—3822no—
263C22⋊C4—3821no—
264C22⋊C4—3821no—
265C42—21no—
266C23—3822no—
267C23—3822no—
268C23—3821no—
269C23—3822no—
270C23—1911no—
271C23—1911no—
272C23—7641no—
273C23—7642no—
274C23—7642no—
275C23—7642no—
276C23D4×D19111yescharacteristic
277Q8—3821no—
278Q8—3821no—
279D4—3821no—
280D4—3822no—
281D4—3822no—
282D4—3821no—
283D4—7644no—
284D4—7642no—
285D4—7644no—
286D4—7642no—
287D4—7642no—
288D4—7642no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—3822no—
294C2×C4—3821no—
295C2×C4—3821no—
296C2×C4—3821no—
297C2×C4—3821no—
298C2×C4—3822no—
299C2×C4—3822no—
300C2×C4—3821no—
301C2×C4—1911no—
302C2×C4—1911no—
303C2×C4—7642no—
304C2×C4D4×D19111yescharacteristic
305C2×C4C2×D76412yes—
306C2×C4#1569028812yes—
307C2×C4C23×D19111yescharacteristic
308C8—3822no—
309C8C2×D76414yescomplemented
310C22—21no—
311C22—21no—
312C22—3821no—
313C22—3821no—
314C22—3821no—
315C22—3821no—
316C22—3821no—
317C22—1912no—
318C22—15284no—
319C22—7644no—
320C22—7642no—
321C22—7641no—
322C22—7641no—
323C22—7642no—
324C22—7641no—
325C22—7642no—
326C22#2025929712yes—
327C22C2×D4×D19111yescharacteristic
328C4—42no—
329C4—41no—
330C4—41no—
331C4—3821no—
332C4—3821no—
333C4—7641no—
334C4#2025929712yes—
335C4C22×D76411yescharacteristic
336C4C2×#1569028811yescharacteristic
337C2—22no—
338C2—3822no—
339C2—15282no—
340C2—15281no—
341C2—7641no—
342C2C2×#2025929711yescharacteristic
343C2AGL(1,ℤ/8)×D19111yescharacteristic
344C2#2713346211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#428×C191C2
socleC22×C191C22×D4
hypercenter#428D191
hypocenterC191#3055
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#20259350C2semidirect
2. C2×M16×D191C2semidirect
3. #27133175C2semidirect
4. C2×#20259297C2semidirect
5. #27132917C2semidirect
6. #27132910C2semidirect
7. #27132900C2semidirect
8. #27132893C2semidirect
9. #27132667C2semidirect
10. #27132662C2semidirect
11. #428×C191C2semidirect
17. C2×#15690276C22semidirect
18. C2×D1528C22semidirect
20. C2×#15690270C22semidirect
22. C2×C8×D191C22semidirect
24. #20259302C22semidirect
25. #20259302C22semidirect
30. #20259297C22semidirect
31. #20259297C22semidirect
36. #20259245C22semidirect
37. #20259242C22semidirect
38. #20259242C22semidirect
39. #20259242C22semidirect
43. #107×C191C22semidirect
44. #106×C191C22semidirect
85. M16×C191D4semidirect
136. C8×C191C2×D4semidirect
154. C2×D191AGL(1,ℤ/8)semidirect
172. C191#3055semidirect
182. #428D191semidirect
212. #107C2×D191semidirect
213. #106C2×D191semidirect
248. M16D764semidirect
309. C8C2×D764semidirect

Maximal subgroups

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

subgroupindex
C2×#202593502
C2×M16×D1912
#271331752
C2×#202592972
#271329172
#271329102
#271329002
#271328932
#271326672
#271326622
#428×C1912
#3055191