Finite Group Databasebeta

← #37584016

Subgroups — #37584016

Order 24448.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259347C211yesmaximal, characteristic, complemented
2C22×#15690274C211yesmaximal, characteristic, complemented
3D4×D764C211yesmaximal, characteristic, complemented
4#27133053C211yesmaximal, characteristic, complemented
5#27132911C211yesmaximal, characteristic, complemented
6#27132897C211yesmaximal, characteristic, complemented
7#27132875C211yesmaximal, characteristic, complemented
8#27132863C211yesmaximal, characteristic, complemented
9#27132813C211yesmaximal, characteristic, complemented
10#27132808C211yesmaximal, characteristic, complemented
11#27132802C211yesmaximal, characteristic, complemented
12#27132796C211yesmaximal, characteristic, complemented
13#27132761C211yesmaximal, characteristic, complemented
14#27132735C211yesmaximal, characteristic, complemented
15#548×C191C211yesmaximal, characteristic, complemented
16C22×D764C2211yescharacteristic
17C22×D764C2211yescharacteristic
18C2×D4×D191—21no—
19#20259347—24no—
20C2×#15690274—24no—
21C2×#15690274C2211yescharacteristic
22C2×#15690274C2211yescharacteristic
23C2×#15690274C2211yescharacteristic, complemented
24C2×#15690274C2211yescharacteristic, complemented
25C2×#15690274C2211yescharacteristic, complemented
26C2×#15690274C2211yescharacteristic, complemented
27C2×#15690272C2211yescharacteristic
28C2×#15690272C2211yescharacteristic, complemented
29C2×#15690272C2211yescharacteristic, complemented
30C2×#15690271C2211yescharacteristic, complemented
31C22×C191⋊C8C2211yescharacteristic, complemented
32#20259319C2211yescharacteristic
33#20259297—21no—
34#20259296C2211yescharacteristic
35#20259296C2211yescharacteristic
36#20259292—21no—
37#20259292—21no—
38C4×D764C2211yescharacteristic
39#20259255C2211yescharacteristic, complemented
40C4⋊(C191⋊C8)C2211yescharacteristic, complemented
41#20259246C2211yescharacteristic, complemented
42#20259246C2211yescharacteristic, complemented
43#20259243C2211yescharacteristic, complemented
44#20259242C2211yescharacteristic
45#20259242C2211yescharacteristic, complemented
46#20259242C2211yescharacteristic, complemented
47C22⋊(C191⋊C8)C2211yescharacteristic, complemented
48C22⋊(C191⋊C8)C2211yescharacteristic, complemented
49#20259233C2211yescharacteristic, complemented
50C2×#41×C191C2211yescharacteristic, complemented
51C22×Q8×C191C2211yescharacteristic, complemented
52#124×C191—21no—
53#124×C191C2211yescharacteristic, complemented
54C22⋊Q8×C191—21no—
55C22⋊Q8×C191C2211yescharacteristic, complemented
56#121×C191—21no—
57#121×C191C2211yescharacteristic, complemented
58C4×Q8×C191C2211yescharacteristic, complemented
59C4×D4×C191—21no—
60C4×D4×C191C2211yescharacteristic
61C23×D191—21no—
62C23×D191—21no—
63C2×D764—22no—
64C2×D764—22no—
65C2×D764—21no—
66C2×D764—21no—
67C2×D764—21no—
68C2×D764—21no—
69C2×D764—41no—
70C2×D764—41no—
71C2×D764C2311yescharacteristic
72C2×D764C2311yescharacteristic
73C2×D764C2311yescharacteristic
74C2×C22⋊D191—21no—
75C2×C22⋊D191—21no—
76D4×D191—44no—
77C2×C4×D191—21no—
78#15690274—22no—
79#15690274—22no—
80#15690274—22no—
81#15690274—22no—
82#15690274—41no—
83#15690274D414yescomplemented
84#15690272—22no—
85#15690272—22no—
86#15690272—41no—
87#15690271—22no—
88C2×C191⋊C8—22no—
89C2×C191⋊C8C2311yescharacteristic
90C2×C191⋊C8C2311yescharacteristic
91C2×C191⋊C8C2311yescharacteristic
92C2×C191⋊C8C2311yescharacteristic
93C4⋊Dic191C2311yescharacteristic
94(C2×D191)⋊C4—21no—
95(C2×D191)⋊C4—21no—
96(C2×D191)⋊C4—21no—
97#41×C191—22no—
98#41×C191—41no—
99C2×Q8×C191—21no—
100C2×Q8×C191—22no—
101C2×Q8×C191—41no—
102C2×Q8×C191C2311yescharacteristic
103C2×Q8×C191C2311yescharacteristic
104C2×D4×C191—21no—
105C2×D4×C191—21no—
106C2×D4×C191D411yescharacteristic
107C2×D4×C191C2311yescharacteristic
108C22×C4×C191—21no—
109C22×C4×C191—21no—
110C22×C4×C191C2311yescharacteristic
111C22×C4×C191C2311yescharacteristic
112C4⋊C4×C191—21no—
113C4⋊C4×C191—21no—
114C4⋊C4×C191D411yescharacteristic
115C4⋊C4×C191C2311yescharacteristic
116C22⋊C4×C191—21no—
117C22⋊C4×C191—21no—
118C22⋊C4×C191—21no—
119C22⋊C4×C191—21no—
120C22⋊C4×C191D411yescharacteristic
121C22⋊C4×C191D411yescharacteristic
122C42×C191—21no—
123C42×C191C2311yescharacteristic
124C22×D191—21no—
125C22×D191—21no—
126C22×D191—21no—
127C22×D191—42no—
128C22×D191—42no—
129C22×D191—42no—
130C22×D191—42no—
131C22×D191—81no—
132C22×D191—81no—
133D764—21no—
134D764—21no—
135D764—21no—
136D764—21no—
137D764—21no—
138D764—42no—
139D764—42no—
140D764—42no—
141D764—81no—
142D764C2×D412yes—
143C22⋊D191—42no—
144C22⋊D191—42no—
145C2×Dic191—21no—
146C4×D191—42no—
147C191⋊C8—21no—
148C191⋊C8—21no—
149C191⋊C8—21no—
150C191⋊C8C2×D412yescomplemented
151C23×C191—21no—
152C23×C191C2×D411yescharacteristic
153C23×C191C2×D411yescharacteristic
154Q8×C191—21no—
155Q8×C191—21no—
156Q8×C191—21no—
157Q8×C191—42no—
158Q8×C191C2×D412yescomplemented
159D4×C191—22no—
160D4×C191—21no—
161D4×C191—21no—
162D4×C191—42no—
163D4×C191—42no—
164C2×C4×C191—21no—
165C2×C4×C191—21no—
166C2×C4×C191—21no—
167C2×C4×C191—21no—
168C2×C4×C191—21no—
169C2×C4×C191—21no—
170C2×C4×C191—21no—
171C2×C4×C191—42no—
172C2×C4×C191—42no—
173C2×C4×C191C2×D411yescharacteristic
174C2×C4×C191C2×D411yescharacteristic
175C2×C4×C191C2×D411yescharacteristic
176C2×C4×C191C2×D411yescharacteristic
177C2×C4×C191C2411yescharacteristic
178C22×C191—21no—
179C22×C191—21no—
180C22×C191—21no—
181C22×C191—21no—
182C22×C191—21no—
183C22×C191—41no—
184C22×C191—41no—
185C22×C191—41no—
186C22×C191AGL(1,ℤ/8)12yescomplemented
187C22×C191C22×D411yescharacteristic
188C2×D191—22no—
189C2×D191—41no—
190C2×D191—41no—
191C2×D191—41no—
192C2×D191—41no—
193C2×D191—41no—
194C2×D191—41no—
195C2×D191—41no—
196C2×D191—41no—
197C2×D191—84no—
198C2×D191—84no—
199C4×C191—21no—
200C4×C191—21no—
201C4×C191—22no—
202C4×C191—21no—
203C4×C191—21no—
204C4×C191—41no—
205C4×C191—41no—
206C4×C191C22×D411yescharacteristic
207C4×C191H(5,𝔽2)11yescharacteristic
208Dic191—41no—
209C2×C191—22no—
210C2×C191—41no—
211C2×C191—81no—
212C2×C191D4211yescharacteristic
213C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C191#57711yescharacteristic
215D191—42no—
216D191—81no—
217D191—81no—
218C191#326511yescharacteristic, complemented, Sylow
219#3265—1911nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1911no—
221C22×SD16—1911no—
222#548D19111yescharacteristic, complemented
223D42—1911no—
224#496—1911no—
225#468—1911no—
226#466—1911no—
227#463—1911no—
228#460—1911no—
229#450—1911no—
230#449—1911no—
231#448—1911no—
232#447—1911no—
233#440—1911no—
234#435—1911no—
235C2×#41C2×D19111yescharacteristic, complemented
236C22×Q8C2×D19111yescharacteristic, complemented
237C22×D4—3821no—
238C22×D4—1911no—
239C22×D4—1911no—
240AGL(1,ℤ/8)—3824no—
241C2×SD16—3824no—
242C2×SD16—1911no—
243C2×SD16—1911no—
244C2×SD16—1911no—
245C2×SD16—1911no—
246C2×SD16—1911no—
247C2×SD16—1911no—
248C2×D8—1911no—
249C2×D8—1911no—
250C2×D8—1911no—
251C2×M16—1911no—
252C22×C8—1911no—
253#127—1911no—
254#124—21no—
255#124C2×D19111yescharacteristic, complemented
256C22⋊Q8—21no—
257C22⋊Q8C2×D19111yescharacteristic, complemented
258#121—21no—
259#121—3821no—
260#121—1911no—
261#121—1911no—
262#121C2×D19111yescharacteristic, complemented
263#120—3821no—
264#120—3821no—
265C4×Q8C2×D19111yescharacteristic, complemented
266C4×D4—21no—
267C4×D4—1911no—
268C4×D4C2×D19111yescharacteristic
269#107—1911no—
270C4⋊C8—1911no—
271#103—1911no—
272#103—1911no—
273#102—1911no—
274#102—1911no—
275#102—1911no—
276#102—1911no—
277C22⋊C8—1911no—
278C22⋊C8—1911no—
279#97—1911no—
280C24—3821no—
281C24—3821no—
282#41—22no—
283#41—41no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D19111yescharacteristic
288C2×Q8C22×D19111yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—3822no—
292C2×D4—3822no—
293C2×D4—3821no—
294C2×D4—3821no—
295C2×D4—3821no—
296C2×D4—3821no—
297C2×D4—3821no—
298C2×D4—3821no—
299C2×D4—1911no—
300C2×D4—1911no—
301C2×D4—1911no—
302C2×D4—7641no—
303C2×D4—7641no—
304C2×D4—7644no—
305C2×D4C22⋊D19111yescharacteristic
306C2×D4C22×D19111yescharacteristic
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—3821no—
310C22×C4C22×D19111yescharacteristic
311C22×C4C22×D19111yescharacteristic
312SD16—3822no—
313SD16—3822no—
314SD16—3822no—
315SD16—3822no—
316SD16—1914no—
317SD16—7641no—
318D8—3822no—
319D8—3822no—
320D8—7641no—
321M16—3822no—
322C2×C8—3822no—
323C2×C8—1911no—
324C2×C8—1911no—
325C2×C8—1911no—
326C2×C8—1911no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—1911no—
330C4⋊C4C22⋊D19111yescharacteristic
331C4⋊C4C22×D19111yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—3821no—
337C22⋊C4—3821no—
338C22⋊C4—3821no—
339C22⋊C4C22⋊D19111yescharacteristic
340C22⋊C4C22⋊D19111yescharacteristic
341C42—21no—
342C42C22×D19111yescharacteristic
343C23—21no—
344C23—3821no—
345C23—3821no—
346C23—3821no—
347C23—15281no—
348C23—15281no—
349C23—7642no—
350C23—7642no—
351C23—7642no—
352C23—7642no—
353C23C2×C22⋊D19111yescharacteristic
354C23C2×C22⋊D19111yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8D4×D19112yescomplemented
360D4—22no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—3821no—
366D4—3821no—
367D4—3821no—
368D4—3821no—
369D4—3821no—
370D4—1912no—
371D4—15281no—
372D4—7642no—
373D4—7642no—
374D4—7642no—
375D4—7642no—
376D4—7642no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—3821no—
387C2×C4—7642no—
388C2×C4C2×C22⋊D19111yescharacteristic
389C2×C4C2×C22⋊D19111yescharacteristic
390C2×C4C2×C22⋊D19111yescharacteristic
391C2×C4C2×C22⋊D19111yescharacteristic
392C2×C4C23×D19111yescharacteristic
393C8—3821no—
394C8—3821no—
395C8—3821no—
396C8—1912no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—3822no—
406C22—15284no—
407C22—15284no—
408C22—7641no—
409C22—7641no—
410C22—7641no—
411C22—7641no—
412C22—7641no—
413C22—7641no—
414C22—7641no—
415C22—7641no—
416C22#2025934712yescomplemented
417C22C22×C22⋊D19111yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—22no—
423C4—41no—
424C4—41no—
425C4—7641no—
426C4C2×D4×D19111yescharacteristic
427C4#2025936711yescharacteristic
428C2—22no—
429C2—41no—
430C2—81no—
431C2—15281no—
432C2—15281no—
433C2—7642no—
434C2D4×C22⋊D19111yescharacteristic
435C2C2×#2025934711yescharacteristic
436C2#2713347111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#548×C191C2
socleC22×C191C22×D4
hypercenter#548D191
hypocenterC191#3265
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#20259347C2semidirect
2. C22×#15690274C2semidirect
3. D4×D764C2semidirect
4. #27133053C2semidirect
5. #27132911C2semidirect
6. #27132897C2semidirect
7. #27132875C2semidirect
8. #27132863C2semidirect
9. #27132813C2semidirect
10. #27132808C2semidirect
11. #27132802C2semidirect
12. #27132796C2semidirect
13. #27132761C2semidirect
14. #27132735C2semidirect
15. #548×C191C2semidirect
23. C2×#15690274C22semidirect
24. C2×#15690274C22semidirect
25. C2×#15690274C22semidirect
26. C2×#15690274C22semidirect
28. C2×#15690272C22semidirect
29. C2×#15690272C22semidirect
30. C2×#15690271C22semidirect
31. C22×C191⋊C8C22semidirect
39. #20259255C22semidirect
40. C4⋊(C191⋊C8)C22semidirect
41. #20259246C22semidirect
42. #20259246C22semidirect
43. #20259243C22semidirect
45. #20259242C22semidirect
46. #20259242C22semidirect
47. C22⋊(C191⋊C8)C22semidirect
48. C22⋊(C191⋊C8)C22semidirect
49. #20259233C22semidirect
50. C2×#41×C191C22semidirect
51. C22×Q8×C191C22semidirect
53. #124×C191C22semidirect
55. C22⋊Q8×C191C22semidirect
57. #121×C191C22semidirect
58. C4×Q8×C191C22semidirect
83. #15690274D4semidirect
150. C191⋊C8C2×D4semidirect
158. Q8×C191C2×D4semidirect
186. C22×C191AGL(1,ℤ/8)semidirect
218. C191#3265semidirect
222. #548D191semidirect
235. C2×#41C2×D191semidirect
236. C22×Q8C2×D191semidirect
255. #124C2×D191semidirect
257. C22⋊Q8C2×D191semidirect
262. #121C2×D191semidirect
265. C4×Q8C2×D191semidirect
359. Q8D4×D191semidirect
416. C22#20259347semidirect

Maximal subgroups

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

subgroupindex
C2×#202593472
C22×#156902742
D4×D7642
#271330532
#271329112
#271328972
#271328752
#271328632
#271328132
#271328082
#271328022
#271327962
#271327612
#271327352
#548×C1912
#3265191