Finite Group Databasebeta

← #37582698

Subgroups — #37582698

Order 24448.

406 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
1#27133137C211yesmaximal, characteristic, complemented
2#27133001C211yesmaximal, characteristic, complemented
3#27132984C211yesmaximal, characteristic, complemented
4#27132950C211yesmaximal, characteristic, complemented
5#27132940C211yesmaximal, characteristic, complemented
6#27132875C211yesmaximal, characteristic, complemented
7#27132875C211yesmaximal, characteristic, complemented
8#27132813C211yesmaximal, characteristic, complemented
9#27132796C211yesmaximal, characteristic
10#27132773C211yesmaximal, characteristic, complemented
11#27132761C211yesmaximal, characteristic, complemented
12C4×#15690274C211yesmaximal, characteristic, complemented
13C4×#15690272C211yesmaximal, characteristic, complemented
14#27132724C211yesmaximal, characteristic, complemented
15#542×C191C211yesmaximal, characteristic, complemented
16C22×D764C2211yescharacteristic
17C2×#15690274C2211yescharacteristic
18C2×#15690274C2211yescharacteristic
19C2×#15690272C2211yescharacteristic
20C2×#15690272C2211yescharacteristic
21C4×D764C2211yescharacteristic
22C4×D764C2211yescharacteristic
23C4×D764C2211yescharacteristic
24C4×D764C2211yescharacteristic
25#20259284—21no—
26#20259284—21no—
27#20259281C2211yescharacteristic
28C22⋊C4×D191—21no—
29C22⋊C4×D191—21no—
30#20259255C2211yescharacteristic, complemented
31#20259255C2211yescharacteristic, complemented
32#20259253C2211yescharacteristic, complemented
33#20259253C2211yescharacteristic, complemented
34C4⋊(C191⋊C8)C2211yescharacteristic, complemented
35C4⋊(C191⋊C8)C2211yescharacteristic, complemented
36#20259246C2211yescharacteristic, complemented
37#20259246C2211yescharacteristic, complemented
38#20259243C2211yescharacteristic, complemented
39#20259243C2211yescharacteristic, complemented
40#20259242C2211yescharacteristic
41#20259242C2211yescharacteristic
42#20259242C2211yescharacteristic
43#20259242C2211yescharacteristic
44C22⋊(C191⋊C8)C2211yescharacteristic
45C22⋊(C191⋊C8)C2211yescharacteristic
46#20259233C2211yescharacteristic, complemented
47C4×C191⋊C8C2211yescharacteristic, complemented
48C4⋊Q8×C191C2211yescharacteristic, complemented
49#126×C191—21no—
50#124×C191—21no—
51#124×C191C2211yescharacteristic, complemented
52#123×C191—21no—
53C22⋊Q8×C191—21no—
54C22⋊Q8×C191C2211yescharacteristic
55#121×C191C2211yescharacteristic
56C4×Q8×C191C2211yescharacteristic, complemented
57C4×D4×C191C2211yescharacteristic, complemented
58#117×C191C2211yescharacteristic
59C23×D191—21no—
60C2×D764—22no—
61C2×D764—22no—
62C2×D764—21no—
63C2×D764C2311yescharacteristic
64C2×D764C2311yescharacteristic
65C2×C4×D191—21no—
66C2×C4×D191—21no—
67C2×C4×D191—21no—
68C2×C4×D191—21no—
69#15690274—22no—
70#15690274—22no—
71#15690272—22no—
72#15690272—22no—
73C2×C191⋊C8C2311yescharacteristic
74C2×C191⋊C8C2311yescharacteristic
75C2×C191⋊C8C2311yescharacteristic
76C2×C191⋊C8C2311yescharacteristic
77C4⋊Dic191C2311yescharacteristic
78C4⋊Dic191C2311yescharacteristic
79C22⋊Dic191—21no—
80(C2×D191)⋊C4—21no—
81(C2×D191)⋊C4—21no—
82(C2×D191)⋊C4—21no—
83(C2×D191)⋊C4—21no—
84C4×Dic191—21no—
85C2×Q8×C191—21no—
86C2×Q8×C191C2311yescharacteristic
87C2×D4×C191—21no—
88C2×D4×C191C2311yescharacteristic
89C22×C4×C191—21no—
90C22×C4×C191C2311yescharacteristic
91C4⋊C4×C191—21no—
92C4⋊C4×C191—21no—
93C4⋊C4×C191—21no—
94C4⋊C4×C191D411yescharacteristic
95C4⋊C4×C191D411yescharacteristic
96C4⋊C4×C191C2311yescharacteristic
97C4⋊C4×C191C2311yescharacteristic
98C22⋊C4×C191—21no—
99C22⋊C4×C191—21no—
100C22⋊C4×C191—21no—
101C22⋊C4×C191—21no—
102C22⋊C4×C191—21no—
103C22⋊C4×C191D411yescharacteristic
104C22⋊C4×C191D411yescharacteristic
105C42×C191—21no—
106C42×C191C2311yescharacteristic
107C42×C191C2311yescharacteristic
108C22×D191—21no—
109C22×D191—21no—
110C22×D191—24no—
111C22×D191—81no—
112C22×D191—81no—
113D764—24no—
114D764—21no—
115D764—21no—
116D764#4112yes—
117D764#4112yes—
118C2×Dic191—21no—
119C2×Dic191—21no—
120C4×D191—81no—
121C4×D191—81no—
122C4×D191—81no—
123C4×D191—81no—
124C191⋊C8—21no—
125C191⋊C8—21no—
126C191⋊C8—21no—
127C191⋊C8—21no—
128C23×C191—21no—
129C23×C191C2×D411yescharacteristic
130Q8×C191—21no—
131Q8×C191—21no—
132Q8×C191—81no—
133D4×C191—21no—
134D4×C191—21no—
135D4×C191—81no—
136C2×C4×C191—21no—
137C2×C4×C191—21no—
138C2×C4×C191—21no—
139C2×C4×C191—21no—
140C2×C4×C191—21no—
141C2×C4×C191—81no—
142C2×C4×C191C2×D411yescharacteristic
143C2×C4×C191C2×D411yescharacteristic
144C2×C4×C191C2×D411yescharacteristic
145C2×C4×C191C2×D411yescharacteristic
146C2×C4×C191C2×D411yescharacteristic
147C2×C4×C191C2411yescharacteristic
148C22×C191—21no—
149C22×C191—21no—
150C22×C191—21no—
151C22×C191—41no—
152C22×C191—41no—
153C22×C191—41no—
154C22×C191C22×D411yescharacteristic
155C2×D191—22no—
156C2×D191—22no—
157C2×D191—41no—
158C2×D191—41no—
159C2×D191—41no—
160C2×D191—41no—
161C2×D191—84no—
162C4×C191—21no—
163C4×C191—21no—
164C4×C191—21no—
165C4×C191—21no—
166C4×C191—21no—
167C4×C191—41no—
168C4×C191—41no—
169C4×C191—41no—
170C4×C191C2×#4111yescharacteristic
171C4×C191C2×#4111yescharacteristic
172Dic191—41no—
173Dic191—41no—
174C2×C191—41no—
175C2×C191—81no—
176C2×C191#52511yescharacteristic
177C2×C191#57611yescharacteristic
178C2×C191#57711yescharacteristic
179D191—42no—
180D191—42no—
181C191#311811yescharacteristic, complemented, Sylow
182#3118—1911nomaximal, Sylow
183#542D19111yescharacteristic, complemented
184#518—1911no—
185#485—1911no—
186#482—1911no—
187#476—1911no—
188#474—1911no—
189#463—1911no—
190#463—1911no—
191#450—1911no—
192#447—1911no—
193#442—1911no—
194#440—1911no—
195C4×SD16—1911no—
196C4×D8—1911no—
197#433—1911no—
198C22×D4—1911no—
199C2×SD16—1911no—
200C2×SD16—1911no—
201C2×D8—1911no—
202C2×D8—1911no—
203C4⋊Q8C2×D19111yescharacteristic, complemented
204#126—21no—
205#124—21no—
206#124C2×D19111yescharacteristic, complemented
207#123—21no—
208C22⋊Q8—21no—
209C22⋊Q8C2×D19111yescharacteristic
210#121C2×D19111yescharacteristic
211C4×Q8C2×D19111yescharacteristic, complemented
212C4×D4—3821no—
213C4×D4—3821no—
214C4×D4—1911no—
215C4×D4—1911no—
216C4×D4—1911no—
217C4×D4—1911no—
218C4×D4C2×D19111yescharacteristic, complemented
219#117—1911no—
220#117C2×D19111yescharacteristic
221C2×C22⋊C4—3821no—
222C2×C22⋊C4—3821no—
223#107—1911no—
224#107—1911no—
225#106—1911no—
226#106—1911no—
227C4⋊C8—1911no—
228C4⋊C8—1911no—
229#103—1911no—
230#103—1911no—
231#102—1911no—
232#102—1911no—
233#102—1911no—
234#102—1911no—
235#102—1911no—
236#102—1911no—
237C22⋊C8—1911no—
238C22⋊C8—1911no—
239#97—1911no—
240C4×C8—1911no—
241C24—3821no—
242C2×Q8—21no—
243C2×Q8C22×D19111yescharacteristic
244C2×D4—21no—
245C2×D4—3822no—
246C2×D4—3822no—
247C2×D4—3821no—
248C2×D4—1911no—
249C2×D4—1911no—
250C2×D4C22×D19111yescharacteristic
251C22×C4—21no—
252C22×C4—3821no—
253C22×C4—3821no—
254C22×C4—3821no—
255C22×C4—3821no—
256C22×C4C22×D19111yescharacteristic
257SD16—3822no—
258SD16—3822no—
259D8—3822no—
260D8—3822no—
261C2×C8—1911no—
262C2×C8—1911no—
263C2×C8—1911no—
264C2×C8—1911no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—1911no—
269C4⋊C4—1911no—
270C4⋊C4C22⋊D19111yescharacteristic
271C4⋊C4C22⋊D19111yescharacteristic
272C4⋊C4C22×D19111yescharacteristic
273C4⋊C4C22×D19111yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3821no—
280C22⋊C4—3821no—
281C22⋊C4—3821no—
282C22⋊C4—3821no—
283C22⋊C4—3821no—
284C22⋊C4C22⋊D19111yescharacteristic
285C22⋊C4C22⋊D19111yescharacteristic
286C42—21no—
287C42—3821no—
288C42C22×D19111yescharacteristic
289C42C22×D19111yescharacteristic
290C23—21no—
291C23—3824no—
292C23—3821no—
293C23—3821no—
294C23—15281no—
295C23—15281no—
296C23C2×C22⋊D19111yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—81no—
300D4—21no—
301D4—21no—
302D4—81no—
303D4—3824no—
304D4—3821no—
305D4—3821no—
306D4—1912no—
307D4—1912no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—81no—
314C2×C4—3821no—
315C2×C4—3821no—
316C2×C4—15281no—
317C2×C4—15281no—
318C2×C4—15281no—
319C2×C4—15281no—
320C2×C4C2×C22⋊D19111yescharacteristic
321C2×C4C2×C22⋊D19111yescharacteristic
322C2×C4C2×C22⋊D19111yescharacteristic
323C2×C4C2×C22⋊D19111yescharacteristic
324C2×C4C2×C22⋊D19111yescharacteristic
325C2×C4C23×D19111yescharacteristic
326C8—3821no—
327C8—3821no—
328C8—3821no—
329C8—3821no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—3822no—
337C22—3822no—
338C22—15284no—
339C22—7641no—
340C22—7641no—
341C22—7641no—
342C22—7641no—
343C22C22×C22⋊D19111yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—7641no—
353C4—7641no—
354C4#41×D19111yescharacteristic
355C4#41×D19111yescharacteristic
356C2—41no—
357C2—81no—
358C2—7642no—
359C2—7642no—
360C2#2713317011yescharacteristic
361C2#2713346611yescharacteristic
362C2#2713347111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#542×C191C2
socleC22×C191C22×D4
prime radical#27132796C2
hypercenter#542D191
hypocenterC191#3118
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #27133137C2semidirect
2. #27133001C2semidirect
3. #27132984C2semidirect
4. #27132950C2semidirect
5. #27132940C2semidirect
6. #27132875C2semidirect
7. #27132875C2semidirect
8. #27132813C2semidirect
10. #27132773C2semidirect
11. #27132761C2semidirect
12. C4×#15690274C2semidirect
13. C4×#15690272C2semidirect
14. #27132724C2semidirect
15. #542×C191C2semidirect
30. #20259255C22semidirect
31. #20259255C22semidirect
32. #20259253C22semidirect
33. #20259253C22semidirect
34. C4⋊(C191⋊C8)C22semidirect
35. C4⋊(C191⋊C8)C22semidirect
36. #20259246C22semidirect
37. #20259246C22semidirect
38. #20259243C22semidirect
39. #20259243C22semidirect
46. #20259233C22semidirect
47. C4×C191⋊C8C22semidirect
48. C4⋊Q8×C191C22semidirect
51. #124×C191C22semidirect
56. C4×Q8×C191C22semidirect
57. C4×D4×C191C22semidirect
181. C191#3118semidirect
183. #542D191semidirect
203. C4⋊Q8C2×D191semidirect
206. #124C2×D191semidirect
211. C4×Q8C2×D191semidirect
218. C4×D4C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271331372
#271330012
#271329842
#271329502
#271329402
#271328752
#271328132
#271327962
#271327732
#271327612
C4×#156902742
C4×#156902722
#271327242
#542×C1912
#3118191