Finite Group Databasebeta

← #37581398

Subgroups — #37581398

Order 24448.

484 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259343C211yesmaximal, characteristic
2#27133145C211yesmaximal, characteristic, complemented
3#27133142C211yesmaximal, characteristic, complemented
4#27132773C211yesmaximal, characteristic, complemented
5#27132772C211yesmaximal, characteristic, complemented
6#27132768C211yesmaximal, characteristic, complemented
7#27132767C211yesmaximal, characteristic, complemented
8C4×#15690276C211yesmaximal, characteristic, complemented
9SD16×Dic191C211yesmaximal, characteristic, complemented
10#27132750C211yesmaximal, characteristic, complemented
11#106×D191C211yesmaximal, characteristic, complemented
12#27132663C211yesmaximal, characteristic, complemented
13#27132661C211yesmaximal, characteristic, complemented
14#27132617C211yesmaximal, characteristic, complemented
15C4×SD16×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×#15690286C2211yescharacteristic
18#20259343C418yescomplemented
19C2×#15690277C2211yescharacteristic
20C2×#15690276C2211yescharacteristic
21C2×#15690272C2211yescharacteristic
22C2×#15690270C2211yescharacteristic
23C2×C8×D191C2211yescharacteristic
24Q8×Dic191C2211yescharacteristic
25C4×Dic382C2211yescharacteristic
26#20259287C2211yescharacteristic
27C4×D764—21no—
28C4×D764C2211yescharacteristic
29D4×Dic191C2211yescharacteristic
30#20259284—21no—
31#20259284C2211yescharacteristic
32C4×C22⋊D191—21no—
33(C22⋊D191)⋊C4—21no—
34#20259279—21no—
35#20259279C2211yescharacteristic
36#20259278—21no—
37C2×Dic191⋊C4—21no—
38C4⋊C4×D191—21no—
39C4⋊C4×D191C2211yescharacteristic
40C4⋊C4×D191C2211yescharacteristic
41#20259254C2211yescharacteristic
42#20259253C2211yescharacteristic
43#20259253C2211yescharacteristic
44#20259247C2211yescharacteristic
45#20259246C2211yescharacteristic
46#20259245C2211yescharacteristic
47#20259244C2211yescharacteristic
48#20259243C2211yescharacteristic
49#20259242C2211yescharacteristic
50#20259233C2211yescharacteristic
51#20259232C2211yescharacteristic
52C8×Dic191C2211yescharacteristic
53C2×SD16×C191C2211yescharacteristic
54C4×Q8×C191C2211yescharacteristic
55C4×D4×C191C2211yescharacteristic
56#106×C191C2211yescharacteristic
57#103×C191C2211yescharacteristic
58#102×C191C2211yescharacteristic
59C4×C8×C191C2211yescharacteristic
60#15690288—22no—
61#15690288C2×C414yescomplemented
62#15690286—22no—
63#15690286C2×C414yescomplemented
64C2×Dic382C2311yescharacteristic
65C2×D764—21no—
66C2×D764C2311yescharacteristic
67C2×C22⋊D191—21no—
68C22×Dic191—21no—
69C22×Dic191—21no—
70C2×C4×D191—21no—
71C2×C4×D191—21no—
72C2×C4×D191—21no—
73C2×C4×D191—21no—
74C2×C4×D191—21no—
75C2×C4×D191C2311yescharacteristic
76#15690277C2×C414yescomplemented
77#15690276C2×C414yescomplemented
78#15690272C2×C414yescomplemented
79#15690270—22no—
80C2×C191⋊C8C2311yescharacteristic
81C2×C191⋊C8C2311yescharacteristic
82C8×D191C2×C414yescomplemented
83C4⋊Dic191—21no—
84C4⋊Dic191C2311yescharacteristic
85C4⋊Dic191C2311yescharacteristic
86Dic191⋊C4—21no—
87Dic191⋊C4—21no—
88Dic191⋊C4—21no—
89Dic191⋊C4—21no—
90Dic191⋊C4D411yescharacteristic
91Dic191⋊C4D411yescharacteristic
92C22⋊Dic191—21no—
93(C2×D191)⋊C4—21no—
94(C2×D191)⋊C4—21no—
95(C2×D191)⋊C4D411yescharacteristic
96(C2×D191)⋊C4D411yescharacteristic
97C4×Dic191—21no—
98C4×Dic191—21no—
99C4×Dic191—21no—
100C4×Dic191C2311yescharacteristic
101C2×Q8×C191C2311yescharacteristic
102C2×D4×C191C2311yescharacteristic
103C22×C4×C191—21no—
104SD16×C191C2×C414yescomplemented
105C2×C8×C191C2311yescharacteristic
106C2×C8×C191C2311yescharacteristic
107C4⋊C4×C191—21no—
108C4⋊C4×C191C2311yescharacteristic
109C4⋊C4×C191C2311yescharacteristic
110C22⋊C4×C191—21no—
111C42×C191—21no—
112C42×C191C2311yescharacteristic
113C22×D191—21no—
114C22×D191C2×D411yescharacteristic
115Dic382—21no—
116Dic382C22×C412yes—
117D764—24no—
118D764—21no—
119D764C22×C412yes—
120C22⋊D191—24no—
121C2×Dic191—21no—
122C2×Dic191—24no—
123C2×Dic191—21no—
124C2×Dic191—21no—
125C2×Dic191—81no—
126C2×Dic191C2×D411yescharacteristic
127C2×Dic191C2×D411yescharacteristic
128C2×Dic191C2×D411yescharacteristic
129C4×D191—24no—
130C4×D191—81no—
131C4×D191—81no—
132C4×D191—81no—
133C4×D191—81no—
134C4×D191C22×C412yes—
135C4×D191#4112yes—
136C191⋊C8—21no—
137C191⋊C8C22×C412yes—
138C23×C191—21no—
139Q8×C191—21no—
140Q8×C191C22×C412yes—
141D4×C191—21no—
142D4×C191C22×C412yes—
143C2×C4×C191—21no—
144C2×C4×C191—21no—
145C2×C4×C191—21no—
146C2×C4×C191—81no—
147C2×C4×C191C2×D411yescharacteristic
148C2×C4×C191C2×D411yescharacteristic
149C2×C4×C191C2411yescharacteristic
150C8×C191—21no—
151C8×C191C22×C412yes—
152C22×C191—22no—
153C22×C191—41no—
154C22×C191—41no—
155C22×C191C22×D411yescharacteristic
156C2×D191—21no—
157C2×D191—21no—
158C2×D191—22no—
159C2×D191—41no—
160C2×D191—41no—
161C2×D191C4×D412yes—
162C4×C191—22no—
163C4×C191—21no—
164C4×C191—21no—
165C4×C191—41no—
166C4×C191—41no—
167C4×C191C23×C411yescharacteristic
168C4×C191C2×#4111yescharacteristic
169Dic191—22no—
170Dic191—21no—
171Dic191—21no—
172Dic191—41no—
173Dic191—41no—
174Dic191C4×D412yes—
175C2×C191—42no—
176C2×C191C2×C4×D411yescharacteristic
177C2×C191#57711yescharacteristic
178C2×C191#57711yescharacteristic
179D191—22no—
180D191—42no—
181C191#295211yescharacteristic, complemented, Sylow
182#2952—1911nomaximal, Sylow
183C2×#135—1911no—
184#520—1911no—
185#520—1911no—
186#442—1911no—
187#442—1911no—
188#441—1911no—
189#441—1911no—
190C4×SD16—1911no—
191C4×SD16—1911no—
192C4×SD16—1911no—
193C4×SD16D19111yescharacteristic, complemented
194C2×#106—1911no—
195#417—1911no—
196#417—1911no—
197#405—1911no—
198C2×#41—1911no—
199C2×#41—1911no—
200#135—1918no—
201C2×Q16—1911no—
202C2×SD16—1911no—
203C2×SD16C2×D19111yescharacteristic
204C2×D8—1911no—
205C2×M16—1911no—
206C22×C8—1911no—
207C4×Q8—1911no—
208C4×Q8—1911no—
209C4×Q8—1911no—
210C4×Q8C2×D19111yescharacteristic
211C4×D4—3821no—
212C4×D4—3821no—
213C4×D4—3821no—
214C4×D4—3821no—
215C4×D4—1911no—
216C4×D4—1911no—
217C4×D4—1911no—
218C4×D4C2×D19111yescharacteristic
219#117—3821no—
220#117—3821no—
221#117—1911no—
222C2×C4⋊C4—3821no—
223C2×C4⋊C4—3821no—
224C2×C4⋊C4—1911no—
225C2×C4⋊C4—1911no—
226#106—1911no—
227#106—1911no—
228#106—1911no—
229#106C2×D19111yescharacteristic
230#103—1911no—
231#103—1911no—
232#103—1911no—
233#103C2×D19111yescharacteristic
234#102—1911no—
235#102—1911no—
236#102—1911no—
237#102C2×D19111yescharacteristic
238#97—1911no—
239#97—1911no—
240C4×C8—1911no—
241C4×C8C2×D19111yescharacteristic
242#41—3822no—
243#41—3822no—
244#41—1914no—
245#41—1914no—
246C2×Q8—1911no—
247C2×Q8C22×D19111yescharacteristic
248C2×D4—3821no—
249C2×D4—3821no—
250C2×D4—1911no—
251C2×D4C22×D19111yescharacteristic
252C22×C4—21no—
253C22×C4—3821no—
254C22×C4—3821no—
255C22×C4—3821no—
256C22×C4—3821no—
257C22×C4—3821no—
258C22×C4—3821no—
259C22×C4—3821no—
260C22×C4—1911no—
261Q16—1914no—
262SD16—1914no—
263SD16C4×D19114yescomplemented
264D8—1914no—
265M16—3822no—
266C2×C8—1911no—
267C2×C8—1911no—
268C2×C8—1914no—
269C2×C8C22×D19111yescharacteristic
270C2×C8C22×D19111yescharacteristic
271C4⋊C4—21no—
272C4⋊C4—3821no—
273C4⋊C4—3821no—
274C4⋊C4—3821no—
275C4⋊C4—3821no—
276C4⋊C4—3821no—
277C4⋊C4—1911no—
278C4⋊C4—1911no—
279C4⋊C4—1911no—
280C4⋊C4—1911no—
281C4⋊C4C22×D19111yescharacteristic
282C4⋊C4C22×D19111yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—3821no—
285C22⋊C4—3821no—
286C22⋊C4—3821no—
287C22⋊C4—1911no—
288C22⋊C4—1911no—
289C42—21no—
290C42—3821no—
291C42—3821no—
292C42—3821no—
293C42—1911no—
294C42C22×D19111yescharacteristic
295C23—21no—
296C23—3821no—
297C23—1911no—
298Q8—21no—
299Q8—3821no—
300Q8—1912no—
301Q8C2×C4×D19112yes—
302D4—21no—
303D4—3824no—
304D4—3824no—
305D4—3821no—
306D4—1912no—
307D4C2×C4×D19112yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—81no—
312C2×C4—3824no—
313C2×C4—3821no—
314C2×C4—3824no—
315C2×C4—3821no—
316C2×C4—3821no—
317C2×C4—1911no—
318C2×C4—1911no—
319C2×C4—1911no—
320C2×C4—1912no—
321C2×C4—1912no—
322C2×C4—15281no—
323C2×C4—15281no—
324C2×C4—15281no—
325C2×C4—15281no—
326C2×C4—15281no—
327C2×C4D4×D19111yescharacteristic
328C2×C4D4×D19111yescharacteristic
329C2×C4C23×D19111yescharacteristic
330C8—21no—
331C8—3821no—
332C8—1912no—
333C8C2×C4×D19112yes—
334C22—22no—
335C22—41no—
336C22—41no—
337C22—3821no—
338C22—3821no—
339C22—3822no—
340C22—1912no—
341C22—7641no—
342C22—7641no—
343C22C2×D4×D19111yescharacteristic
344C4—21no—
345C4—21no—
346C4—22no—
347C4—41no—
348C4—41no—
349C4—3821no—
350C4—3821no—
351C4—3822no—
352C4—1912no—
353C4—7641no—
354C4—7641no—
355C4C22×C4×D19111yescharacteristic
356C4#41×D19111yescharacteristic
357C2—42no—
358C2—3822no—
359C2—7642no—
360C2C4×D4×D19111yescharacteristic
361C2#2713346911yescharacteristic
362C2#2713347011yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutatorC4×C191C23×C4
FrattiniC2×C4C23×D191
FittingC4×SD16×C191C2
socleC22×C191C22×D4
prime radicalC2×#20259343C2
cosocleC2×C4×C191C24
hypercenterC4×SD16D191
hypocenterC191#2952
LCSG ▷ C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #27133145C2semidirect
3. #27133142C2semidirect
4. #27132773C2semidirect
5. #27132772C2semidirect
6. #27132768C2semidirect
7. #27132767C2semidirect
8. C4×#15690276C2semidirect
9. SD16×Dic191C2semidirect
10. #27132750C2semidirect
11. #106×D191C2semidirect
12. #27132663C2semidirect
13. #27132661C2semidirect
14. #27132617C2semidirect
15. C4×SD16×C191C2semidirect
18. #20259343C4semidirect
61. #15690288C2×C4semidirect
63. #15690286C2×C4semidirect
76. #15690277C2×C4semidirect
77. #15690276C2×C4semidirect
78. #15690272C2×C4semidirect
82. C8×D191C2×C4semidirect
104. SD16×C191C2×C4semidirect
181. C191#2952semidirect
193. C4×SD16D191semidirect
263. SD16C4×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593432
#271331452
#271331422
#271327732
#271327722
#271327682
#271327672
C4×#156902762
SD16×Dic1912
#271327502
#106×D1912
#271326632
#271326612
#271326172
C4×SD16×C1912
#2952191