Finite Group Databasebeta

← #37582246

Subgroups — #37582246

Order 24448.

466 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133459C211yesmaximal, characteristic, complemented
2C2×#20259346C211yesmaximal, characteristic, complemented
3AGL(1,ℤ/8)×D191C211yesmaximal, characteristic, complemented
4#27133428C211yesmaximal, characteristic, complemented
5#27132932C211yesmaximal, characteristic, complemented
6#27132931C211yesmaximal, characteristic, complemented
7#27132930C211yesmaximal, characteristic, complemented
8#27132927C211yesmaximal, characteristic, complemented
9#27132926C211yesmaximal, characteristic, complemented
10#27132925C211yesmaximal, characteristic, complemented
11#27132924C211yesmaximal, characteristic, complemented
12#27132710C211yesmaximal, characteristic, complemented
13#27132644C211yesmaximal, characteristic, complemented
14#100×D191C211yesmaximal, characteristic, complemented
15#472×C191C211yesmaximal, characteristic, complemented
16C2×#15690286C2211yescharacteristic
17#41×D191C2211yescharacteristic
18C2×D4×D191C2211yescharacteristic
19#20259355C2211yescharacteristic, complemented
20#20259352C2211yescharacteristic, complemented
21#20259351—21no—
22#20259350C2211yescharacteristic
23#20259349—21no—
24#20259348C2211yescharacteristic
25#20259348C2211yescharacteristic, complemented
26#20259347C2211yescharacteristic
27#20259346—21no—
28#20259346—21no—
29#20259346—21no—
30#20259346—21no—
31#20259346—21no—
32#20259346—21no—
33#20259343—21no—
34#20259341—21no—
35C2×#15690276C2211yescharacteristic, complemented
36C2×#15690275C2211yescharacteristic, complemented
37SD16×D191—21no—
38C2×#15690272C2211yescharacteristic, complemented
39D8×D191—21no—
40#20259330—21no—
41#20259330C2211yescharacteristic, complemented
42#20259329C2211yescharacteristic, complemented
43#20259328—21no—
44#20259328C2211yescharacteristic, complemented
45C2×#15690270—21no—
46C2×#15690270C2211yescharacteristic
47M16×D191—21no—
48M16×D191C2211yescharacteristic
49M16×D191C2211yescharacteristic
50M16×D191C2211yescharacteristic
51#20259258C2211yescharacteristic, complemented
52#20259257C2211yescharacteristic, complemented
53#20259257C2211yescharacteristic, complemented
54#20259240C2211yescharacteristic, complemented
55#20259240C2211yescharacteristic, complemented
56#20259239C2211yescharacteristic, complemented
57#20259239C2211yescharacteristic, complemented
58#20259238C2211yescharacteristic, complemented
59#20259238C2211yescharacteristic, complemented
60AGL(1,ℤ/8)×C191C2211yescharacteristic
61AGL(1,ℤ/8)×C191C2211yescharacteristic, complemented
62C2×D8×C191C2211yescharacteristic, complemented
63#131×C191C2211yescharacteristic, complemented
64#108×C191C2211yescharacteristic, complemented
65#100×C191C2211yescharacteristic, complemented
66#100×C191C2211yescharacteristic, complemented
67C23×D191—21no—
68#15690288D411yescharacteristic
69#15690287—21no—
70#15690287C2311yescharacteristic
71#15690286—21no—
72#15690286—21no—
73#15690286—21no—
74#15690286—21no—
75#15690286—21no—
76#15690286D411yescharacteristic
77C2×Dic382C2311yescharacteristic
78Q8×D191D411yescharacteristic
79C2×D764C2311yescharacteristic
80C2×C22⋊D191—21no—
81C2×C22⋊D191—21no—
82D4×D191—21no—
83D4×D191—21no—
84D4×D191—21no—
85D4×D191—21no—
86D4×D191—21no—
87D4×D191D411yescharacteristic
88C22×Dic191—21no—
89C2×C4×D191—21no—
90C2×C4×D191C2311yescharacteristic
91Dic764—21no—
92#15690277—21no—
93#15690276—21no—
94#15690276—21no—
95#15690276—21no—
96#15690276—21no—
97#15690275—21no—
98#15690275—21no—
99#15690275—21no—
100#15690275—21no—
101#15690275—21no—
102#15690274—21no—
103D1528—21no—
104#15690272—21no—
105#15690272—21no—
106#15690272—21no—
107#15690272—21no—
108#15690272—21no—
109#15690271—21no—
110#15690271C2311yescharacteristic
111#15690271C2311yescharacteristic
112#15690271C2311yescharacteristic
113#15690270—21no—
114#15690270—21no—
115#15690270—21no—
116#15690270—21no—
117#15690270—21no—
118#15690270D411yescharacteristic
119#15690270D411yescharacteristic
120#15690270D411yescharacteristic
121#15690270D411yescharacteristic
122C2×C191⋊C8—21no—
123C2×C191⋊C8C2311yescharacteristic
124C8×D191—21no—
125C8×D191—21no—
126C8×D191—21no—
127C8×D191—21no—
128C8×D191—21no—
129#41×C191C2311yescharacteristic
130C2×D4×C191C2311yescharacteristic
131C2×D4×C191C2311yescharacteristic
132SD16×C191—21no—
133SD16×C191—21no—
134D8×C191—21no—
135D8×C191—21no—
136D8×C191—21no—
137D8×C191—21no—
138M16×C191—21no—
139M16×C191C2311yescharacteristic
140M16×C191C2311yescharacteristic
141M16×C191C2311yescharacteristic
142C2×C8×C191—21no—
143C2×C8×C191C2311yescharacteristic
144C22×D191—21no—
145C22×D191—21no—
146C22×D191—41no—
147C22×D191—41no—
148C22×D191—82no—
149C22×D191#4111yescharacteristic
150Dic382—21no—
151Dic382—21no—
152Dic382—21no—
153Dic382C2×D411yescharacteristic
154D764—21no—
155D764—21no—
156D764—21no—
157D764C2×D411yescharacteristic
158C22⋊D191—21no—
159C22⋊D191—21no—
160C22⋊D191—21no—
161C22⋊D191—41no—
162C22⋊D191—41no—
163C22⋊D191—41no—
164C22⋊D191—41no—
165C2×Dic191—21no—
166C2×Dic191—21no—
167C2×Dic191—41no—
168C2×Dic191—41no—
169C2×Dic191#4111yescharacteristic
170C4×D191—21no—
171C4×D191—21no—
172C4×D191—21no—
173C4×D191C2×D411yescharacteristic
174C4×D191C2×D411yescharacteristic
175C4×D191C2×D411yescharacteristic
176C4×D191C2×D411yescharacteristic
177C191⋊C8—21no—
178C191⋊C8—21no—
179C191⋊C8—21no—
180C191⋊C8C2×D411yescharacteristic
181C191⋊C8C2×D411yescharacteristic
182C23×C191—21no—
183C23×C191—21no—
184Q8×C191C2×D411yescharacteristic
185D4×C191—21no—
186D4×C191—21no—
187D4×C191—21no—
188D4×C191—21no—
189D4×C191—21no—
190D4×C191C2×D411yescharacteristic
191C2×C4×C191—21no—
192C2×C4×C191C2411yescharacteristic
193C8×C191—21no—
194C8×C191—21no—
195C8×C191—21no—
196C8×C191C2×D411yescharacteristic
197C8×C191C2×D411yescharacteristic
198C22×C191—21no—
199C22×C191—41no—
200C22×C191—41no—
201C22×C191—81no—
202C22×C191—81no—
203C22×C191C2×#4111yescharacteristic
204C2×D191—21no—
205C2×D191—41no—
206C2×D191—41no—
207C2×D191—81no—
208C2×D191—82no—
209C2×D191—82no—
210C2×D191—81no—
211C2×D191#12111yescharacteristic
212C2×D191#12111yescharacteristic
213C4×C191—21no—
214C4×C191C22×D411yescharacteristic
215C4×C191C22×D411yescharacteristic
216Dic191—21no—
217Dic191—41no—
218Dic191#12111yescharacteristic
219Dic191#12111yescharacteristic
220C2×C191—21no—
221C2×C191—41no—
222C2×C191—81no—
223C2×C191—81no—
224C2×C191C2×#12111yescharacteristic
225D191—21no—
226D191—21no—
227D191—41no—
228D191—81no—
229C191#306811yescharacteristic, complemented, Sylow
230#3068—1911nomaximal, Sylow
231#575—1911no—
232C2×AGL(1,ℤ/8)—1911no—
233C2×AGL(1,ℤ/8)—1911no—
234#568—1911no—
235#472—1911no—
236#472—1911no—
237#472—1911no—
238#472D19111yescharacteristic, complemented
239#471—1911no—
240#471—1911no—
241#471—1911no—
242#471—1911no—
243#430—1911no—
244#413—1911no—
245C2×#100—1911no—
246C2×#41—1911no—
247C2×#41—1911no—
248C22×D4—1911no—
249#137—3821no—
250#137—1911no—
251#137—1911no—
252AGL(1,ℤ/8)—3821no—
253AGL(1,ℤ/8)—3821no—
254AGL(1,ℤ/8)—3821no—
255AGL(1,ℤ/8)—3821no—
256AGL(1,ℤ/8)—3821no—
257AGL(1,ℤ/8)—3821no—
258AGL(1,ℤ/8)—3821no—
259AGL(1,ℤ/8)—1911no—
260AGL(1,ℤ/8)—1911no—
261AGL(1,ℤ/8)—1911no—
262AGL(1,ℤ/8)—1911no—
263AGL(1,ℤ/8)C2×D19111yescharacteristic
264AGL(1,ℤ/8)C2×D19111yescharacteristic, complemented
265#135—3821no—
266#135—3821no—
267C2×SD16—3821no—
268C2×SD16—1911no—
269C2×SD16—1911no—
270C2×D8—3821no—
271C2×D8—1911no—
272C2×D8C2×D19111yescharacteristic, complemented
273#131—3821no—
274#131—3821no—
275#131—1911no—
276#131—1911no—
277#131—1911no—
278#131C2×D19111yescharacteristic, complemented
279C2×M16—3821no—
280C2×M16—3821no—
281C2×M16—1911no—
282C2×M16—1911no—
283C2×M16—1911no—
284C2×M16—1911no—
285#108—1911no—
286#108—1911no—
287#108—1911no—
288#108C2×D19111yescharacteristic, complemented
289#101—1911no—
290#101—1911no—
291#100—1911no—
292#100—1911no—
293#100—1911no—
294#100—1911no—
295#100C2×D19111yescharacteristic, complemented
296#100C2×D19111yescharacteristic, complemented
297C24—3821no—
298#41—3821no—
299#41—3821no—
300#41—3821no—
301#41—3821no—
302#41—3821no—
303#41—3821no—
304#41—1911no—
305#41—1911no—
306#41—1911no—
307#41C22×D19111yescharacteristic
308C2×Q8—1911no—
309C2×Q8—1911no—
310C2×D4—3821no—
311C2×D4—3821no—
312C2×D4—3821no—
313C2×D4—3821no—
314C2×D4—3821no—
315C2×D4—3821no—
316C2×D4—3821no—
317C2×D4—1911no—
318C2×D4—1911no—
319C2×D4C22×D19111yescharacteristic
320C2×D4C22×D19111yescharacteristic
321C22×C4—3821no—
322C22×C4—3821no—
323C22×C4—1911no—
324Q16—3821no—
325Q16—3821no—
326SD16—21no—
327SD16—21no—
328SD16—3821no—
329SD16—3821no—
330SD16—3821no—
331SD16—3821no—
332SD16—3821no—
333SD16—3821no—
334SD16—3821no—
335SD16—3821no—
336SD16—3821no—
337SD16—3821no—
338D8—21no—
339D8—21no—
340D8—21no—
341D8—21no—
342D8—3821no—
343D8—3821no—
344D8—3821no—
345D8—3821no—
346D8—3821no—
347D8—3821no—
348M16—21no—
349M16—3821no—
350M16—3821no—
351M16—3821no—
352M16—3821no—
353M16—3821no—
354M16—3821no—
355M16—1911no—
356M16—1911no—
357M16—1911no—
358M16—1911no—
359M16—1911no—
360M16—1911no—
361M16—1911no—
362M16C22×D19111yescharacteristic
363M16C22×D19111yescharacteristic
364M16C22×D19111yescharacteristic
365C2×C8—21no—
366C2×C8—3821no—
367C2×C8—3821no—
368C2×C8—3821no—
369C2×C8—3821no—
370C2×C8—3821no—
371C2×C8—3821no—
372C2×C8—1911no—
373C2×C8C22×D19111yescharacteristic
374C23—21no—
375C23—21no—
376C23—3821no—
377C23—3821no—
378C23—1911no—
379C23—15282no—
380C23—7641no—
381C23—7641no—
382Q8—3821no—
383Q8—3821no—
384Q8—3821no—
385Q8—1911no—
386Q8D4×D19111yescharacteristic
387D4—21no—
388D4—21no—
389D4—21no—
390D4—21no—
391D4—21no—
392D4—3821no—
393D4—3821no—
394D4—3821no—
395D4—3821no—
396D4—3821no—
397D4—3821no—
398D4—1911no—
399D4—7641no—
400D4—7641no—
401D4—7641no—
402D4—7641no—
403D4D4×D19111yescharacteristic
404C2×C4—21no—
405C2×C4—3821no—
406C2×C4—3821no—
407C2×C4—3821no—
408C2×C4—3821no—
409C2×C4—3821no—
410C2×C4—1911no—
411C2×C4—1911no—
412C2×C4—1911no—
413C2×C4—1911no—
414C2×C4—1911no—
415C2×C4—7641no—
416C2×C4—7641no—
417C2×C4C23×D19111yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—3821no—
422C8—3821no—
423C8—3821no—
424C8—1911no—
425C8—1911no—
426C8D4×D19111yescharacteristic
427C8D4×D19111yescharacteristic
428C22—21no—
429C22—41no—
430C22—41no—
431C22—81no—
432C22—81no—
433C22—3821no—
434C22—1911no—
435C22—1911no—
436C22—15281no—
437C22—15281no—
438C22—15282no—
439C22—15282no—
440C22—7641no—
441C22—7641no—
442C22#41×D19111yescharacteristic
443C4—21no—
444C4—3821no—
445C4—1911no—
446C4—1911no—
447C4—7641no—
448C4C2×D4×D19111yescharacteristic
449C4C2×D4×D19111yescharacteristic
450C2—21no—
451C2—41no—
452C2—81no—
453C2—81no—
454C2—3821no—
455C2—3821no—
456C2—15281no—
457C2—7641no—
458C2#121×D19111yescharacteristic
subgroupG/H
centerC2#121×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#472×C191C2
socleC2×C191C2×#121
hypercenter#472D191
hypocenterC191#3068
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #27133459C2semidirect
2. C2×#20259346C2semidirect
3. AGL(1,ℤ/8)×D191C2semidirect
4. #27133428C2semidirect
5. #27132932C2semidirect
6. #27132931C2semidirect
7. #27132930C2semidirect
8. #27132927C2semidirect
9. #27132926C2semidirect
10. #27132925C2semidirect
11. #27132924C2semidirect
12. #27132710C2semidirect
13. #27132644C2semidirect
14. #100×D191C2semidirect
15. #472×C191C2semidirect
19. #20259355C22semidirect
20. #20259352C22semidirect
25. #20259348C22semidirect
35. C2×#15690276C22semidirect
36. C2×#15690275C22semidirect
38. C2×#15690272C22semidirect
41. #20259330C22semidirect
42. #20259329C22semidirect
44. #20259328C22semidirect
51. #20259258C22semidirect
52. #20259257C22semidirect
53. #20259257C22semidirect
54. #20259240C22semidirect
55. #20259240C22semidirect
56. #20259239C22semidirect
57. #20259239C22semidirect
58. #20259238C22semidirect
59. #20259238C22semidirect
61. AGL(1,ℤ/8)×C191C22semidirect
62. C2×D8×C191C22semidirect
63. #131×C191C22semidirect
64. #108×C191C22semidirect
65. #100×C191C22semidirect
66. #100×C191C22semidirect
229. C191#3068semidirect
238. #472D191semidirect
264. AGL(1,ℤ/8)C2×D191semidirect
272. C2×D8C2×D191semidirect
278. #131C2×D191semidirect
288. #108C2×D191semidirect
295. #100C2×D191semidirect
296. #100C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271334592
C2×#202593462
AGL(1,ℤ/8)×D1912
#271334282
#271329322
#271329312
#271329302
#271329272
#271329262
#271329252
#271329242
#271327102
#271326442
#100×D1912
#472×C1912
#3068191