Finite Group Databasebeta

← #37584125

Subgroups — #37584125

Order 24448.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259346C211yesmaximal, characteristic, complemented
2#27133307C211yesmaximal, characteristic, complemented
3#27133301C211yesmaximal, characteristic, complemented
4#27133044C211yesmaximal, characteristic, complemented
5#27132909C211yesmaximal, characteristic, complemented
6#27132906C211yesmaximal, characteristic, complemented
7#27132880C211yesmaximal, characteristic, complemented
8#27132876C211yesmaximal, characteristic, complemented
9#27132825C211yesmaximal, characteristic, complemented
10#27132822C211yesmaximal, characteristic, complemented
11#27132797C211yesmaximal, characteristic, complemented
12#27132795C211yesmaximal, characteristic, complemented
13D8×Dic191C211yesmaximal, characteristic, complemented
14#27132739C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C191C211yesmaximal, characteristic, complemented
16C2×#15690286C2211yescharacteristic
17C2×D4×D191C2211yescharacteristic, complemented
18#20259346—24no—
19C2×#15690277C2211yescharacteristic, complemented
20C2×#15690276C2211yescharacteristic, complemented
21C2×#15690275C2211yescharacteristic, complemented
22C2×#15690272C2211yescharacteristic
23C2×#15690272C2211yescharacteristic
24C2×#15690270C2211yescharacteristic, complemented
25#20259311C2211yescharacteristic, complemented
26#20259309C2211yescharacteristic, complemented
27#20259305—21no—
28#20259303—21no—
29#20259301C2211yescharacteristic, complemented
30#20259298C2211yescharacteristic, complemented
31#20259296C2211yescharacteristic
32#20259294—21no—
33#20259294—21no—
34#20259293C2211yescharacteristic
35#20259291—21no—
36#20259290—21no—
37D4×Dic191C2211yescharacteristic, complemented
38D4×Dic191C2211yescharacteristic, complemented
39C4×C22⋊D191C2211yescharacteristic
40C2×C22⋊Dic191—21no—
41C2×(C2×D191)⋊C4—21no—
42#20259256C2211yescharacteristic, complemented
43Dic191⋊C8C2211yescharacteristic, complemented
44#20259247C2211yescharacteristic, complemented
45#20259246C2211yescharacteristic, complemented
46#20259244C2211yescharacteristic, complemented
47#20259243C2211yescharacteristic, complemented
48#20259243C2211yescharacteristic, complemented
49#20259243C2211yescharacteristic, complemented
50C22⋊(C191⋊C8)C2211yescharacteristic, complemented
51(C2×D191)⋊C8C2211yescharacteristic, complemented
52C8×Dic191C2211yescharacteristic, complemented
53C2×#41×C191C2211yescharacteristic
54C22×D4×C191C2211yescharacteristic, complemented
55AGL(1,ℤ/8)×C191—24no—
56C2×SD16×C191C2211yescharacteristic, complemented
57C2×SD16×C191C2211yescharacteristic, complemented
58C2×D8×C191C2211yescharacteristic
59C2×D8×C191C2211yescharacteristic
60C2×M16×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62#15690286—22no—
63#15690286—41no—
64C2×Dic382C2311yescharacteristic
65C2×D764C2311yescharacteristic
66C2×C22⋊D191—21no—
67C2×C22⋊D191—21no—
68C2×C22⋊D191—21no—
69C2×C22⋊D191—21no—
70C2×C22⋊D191D411yescharacteristic
71D4×D191—22no—
72D4×D191—41no—
73C22×Dic191—21no—
74C22×Dic191—21no—
75C22×Dic191—21no—
76C2×C4×D191C2311yescharacteristic
77#15690277—41no—
78#15690276—22no—
79#15690275—22no—
80#15690272—22no—
81#15690272—41no—
82#15690270—22no—
83C2×C191⋊C8C2311yescharacteristic
84C2×C191⋊C8C2311yescharacteristic
85C4⋊Dic191C2311yescharacteristic
86C4⋊Dic191C2311yescharacteristic
87Dic191⋊C4—21no—
88Dic191⋊C4—21no—
89Dic191⋊C4D411yescharacteristic
90C22⋊Dic191—21no—
91C22⋊Dic191—21no—
92C22⋊Dic191—21no—
93C22⋊Dic191—21no—
94C22⋊Dic191—21no—
95C22⋊Dic191—21no—
96C22⋊Dic191D411yescharacteristic
97(C2×D191)⋊C4—21no—
98(C2×D191)⋊C4—21no—
99(C2×D191)⋊C4—21no—
100(C2×D191)⋊C4—21no—
101(C2×D191)⋊C4D411yescharacteristic
102C4×Dic191C2311yescharacteristic
103C24×C191—21no—
104#41×C191—22no—
105#41×C191—41no—
106C2×Q8×C191C2311yescharacteristic
107C2×D4×C191—22no—
108C2×D4×C191—21no—
109C2×D4×C191—21no—
110C2×D4×C191—41no—
111C2×D4×C191C2311yescharacteristic
112C2×D4×C191C2311yescharacteristic
113C2×D4×C191C2311yescharacteristic
114C22×C4×C191—21no—
115C22×C4×C191C2311yescharacteristic
116SD16×C191—22no—
117SD16×C191—22no—
118D8×C191—22no—
119D8×C191D414yescomplemented
120M16×C191—22no—
121C2×C8×C191C2311yescharacteristic
122C2×C8×C191C2311yescharacteristic
123C22×D191—21no—
124C22×D191—42no—
125C22×D191—42no—
126C22×D191—81no—
127C22×D191C2×D411yescharacteristic
128Dic382—21no—
129Dic382—21no—
130D764—21no—
131D764—21no—
132C22⋊D191—41no—
133C22⋊D191—42no—
134C22⋊D191—42no—
135C22⋊D191—81no—
136C22⋊D191—81no—
137C2×Dic191—21no—
138C2×Dic191—21no—
139C2×Dic191—21no—
140C2×Dic191—42no—
141C2×Dic191—81no—
142C2×Dic191—81no—
143C2×Dic191C2×D411yescharacteristic
144C2×Dic191C2×D411yescharacteristic
145C2×Dic191C2×D411yescharacteristic
146C4×D191—21no—
147C4×D191—21no—
148C191⋊C8—21no—
149C191⋊C8—21no—
150C23×C191—21no—
151C23×C191—21no—
152C23×C191—21no—
153C23×C191—42no—
154C23×C191—42no—
155C23×C191—81no—
156C23×C191C2×D411yescharacteristic
157Q8×C191—21no—
158Q8×C191—21no—
159D4×C191—21no—
160D4×C191—21no—
161D4×C191—21no—
162D4×C191—21no—
163D4×C191—42no—
164D4×C191—42no—
165D4×C191C2×D412yes—
166D4×C191C2×D412yes—
167C2×C4×C191—21no—
168C2×C4×C191—21no—
169C2×C4×C191—21no—
170C2×C4×C191—42no—
171C2×C4×C191C2×D411yescharacteristic
172C2×C4×C191C2411yescharacteristic
173C8×C191—21no—
174C8×C191C2×D412yes—
175C22×C191—21no—
176C22×C191—22no—
177C22×C191—22no—
178C22×C191—21no—
179C22×C191—21no—
180C22×C191—41no—
181C22×C191—41no—
182C22×C191—41no—
183C22×C191—41no—
184C22×C191—41no—
185C22×C191—41no—
186C22×C191—41no—
187C22×C191—84no—
188C22×C191C22×D411yescharacteristic
189C2×D191—21no—
190C2×D191—21no—
191C2×D191—21no—
192C2×D191—41no—
193C2×D191—41no—
194C2×D191—41no—
195C2×D191—84no—
196C4×C191—21no—
197C4×C191—41no—
198C4×C191C22×D411yescharacteristic
199C4×C191H(5,𝔽2)11yescharacteristic
200Dic191—21no—
201Dic191—21no—
202Dic191—21no—
203Dic191—41no—
204Dic191—41no—
205Dic191—41no—
206C2×C191—42no—
207C2×C191—41no—
208C2×C191—42no—
209C2×C191—81no—
210C2×C191D4211yescharacteristic
211C2×C191#57511yescharacteristic
212C2×C191#57611yescharacteristic
213D191—41no—
214D191—81no—
215C191#327311yescharacteristic, complemented, Sylow
216#3273—1911nomaximal, Sylow
217C2×AGL(1,ℤ/8)—1911no—
218C2×AGL(1,ℤ/8)D19111yescharacteristic, complemented
219#546—1911no—
220#546—1911no—
221#495—1911no—
222#468—1911no—
223#468—1911no—
224#463—1911no—
225#463—1911no—
226#452—1911no—
227#452—1911no—
228#447—1911no—
229#447—1911no—
230C4×D8—1911no—
231#436—1911no—
232C2×#41—1911no—
233C2×#41C2×D19111yescharacteristic
234C22×D4—1911no—
235C22×D4C2×D19111yescharacteristic, complemented
236AGL(1,ℤ/8)—24no—
237AGL(1,ℤ/8)—3824no—
238C2×Q16—1911no—
239C2×SD16—1911no—
240C2×SD16—1911no—
241C2×SD16C2×D19111yescharacteristic, complemented
242C2×SD16C2×D19111yescharacteristic, complemented
243C2×D8—1911no—
244C2×D8—1911no—
245C2×D8C2×D19111yescharacteristic
246C2×D8C2×D19111yescharacteristic
247C2×M16—1911no—
248C2×M16C2×D19111yescharacteristic, complemented
249#124—1911no—
250#124—1911no—
251#123—3821no—
252#123—3821no—
253C22⋊Q8—1911no—
254C22⋊Q8—1911no—
255#121—3821no—
256#121—3821no—
257#121—1911no—
258#121—1911no—
259#120—3821no—
260#120—3821no—
261C4×D4—1911no—
262C4×D4—1911no—
263C4×D4—1911no—
264C2×C22⋊C4—3821no—
265C2×C22⋊C4—3821no—
266#107—1911no—
267C4⋊C8—1911no—
268#103—1911no—
269#103—1911no—
270#102—1911no—
271#102—1911no—
272#102—1911no—
273#102—1911no—
274C22⋊C8—1911no—
275C22⋊C8—1911no—
276C4×C8—1911no—
277C24—21no—
278C24—3821no—
279#41—22no—
280#41—41no—
281#41—3822no—
282#41—7641no—
283C2×Q8—1911no—
284C2×Q8C22×D19111yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—3822no—
290C2×D4—3821no—
291C2×D4—3821no—
292C2×D4—3821no—
293C2×D4—3821no—
294C2×D4—1911no—
295C2×D4—1911no—
296C2×D4—7641no—
297C2×D4C22×D19111yescharacteristic
298C2×D4C22×D19111yescharacteristic
299C2×D4C22×D19111yescharacteristic
300C22×C4—21no—
301C22×C4—3821no—
302C22×C4—3821no—
303C22×C4—3821no—
304C22×C4—1911no—
305C22×C4C22×D19111yescharacteristic
306Q16—7641no—
307SD16—22no—
308SD16—22no—
309SD16—3822no—
310SD16—3822no—
311D8—22no—
312D8—3822no—
313D8—7641no—
314D8C22⋊D19114yescomplemented
315M16—22no—
316M16—3822no—
317C2×C8—1911no—
318C2×C8—1911no—
319C2×C8C22×D19111yescharacteristic
320C2×C8C22×D19111yescharacteristic
321C4⋊C4—3821no—
322C4⋊C4—3821no—
323C4⋊C4—1911no—
324C4⋊C4—1911no—
325C4⋊C4—1911no—
326C22⋊C4—3821no—
327C22⋊C4—3821no—
328C22⋊C4—3821no—
329C22⋊C4—3821no—
330C22⋊C4—3821no—
331C22⋊C4—3821no—
332C22⋊C4—3821no—
333C22⋊C4—3821no—
334C22⋊C4—3821no—
335C22⋊C4—3821no—
336C22⋊C4—1911no—
337C22⋊C4—1911no—
338C42—1911no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—3821no—
346C23—1911no—
347C23—15281no—
348C23—7642no—
349C23—7642no—
350C23D4×D19111yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—3821no—
354Q8—3821no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—21no—
359D4—42no—
360D4—42no—
361D4—3821no—
362D4—3821no—
363D4—15281no—
364D4—15281no—
365D4—7641no—
366D4—7642no—
367D4—7642no—
368D4C2×C22⋊D19112yes—
369D4C2×C22⋊D19112yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—3821no—
375C2×C4—3821no—
376C2×C4—3821no—
377C2×C4—3821no—
378C2×C4—3821no—
379C2×C4—1911no—
380C2×C4—1911no—
381C2×C4—1911no—
382C2×C4—15281no—
383C2×C4—15281no—
384C2×C4—7642no—
385C2×C4D4×D19111yescharacteristic
386C2×C4C23×D19111yescharacteristic
387C8—21no—
388C8—3821no—
389C8—3821no—
390C8C2×C22⋊D19112yes—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—22no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—3821no—
405C22—3821no—
406C22—3821no—
407C22—15284no—
408C22—7641no—
409C22—7641no—
410C22—7641no—
411C22C2×D4×D19111yescharacteristic
412C4—21no—
413C4—41no—
414C4—3821no—
415C4—3821no—
416C4—3821no—
417C4—7641no—
418C4—7641no—
419C4—7641no—
420C4C22×C22⋊D19111yescharacteristic
421C4#2025936711yescharacteristic
422C2—41no—
423C2—42no—
424C2—42no—
425C2—81no—
426C2—15281no—
427C2—7641no—
428C2D4×C22⋊D19111yescharacteristic
429C2#2713345911yescharacteristic
430C2#2713346711yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×AGL(1,ℤ/8)×C191C2
socleC22×C191C22×D4
hypercenterC2×AGL(1,ℤ/8)D191
hypocenterC191#3273
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#20259346C2semidirect
2. #27133307C2semidirect
3. #27133301C2semidirect
4. #27133044C2semidirect
5. #27132909C2semidirect
6. #27132906C2semidirect
7. #27132880C2semidirect
8. #27132876C2semidirect
9. #27132825C2semidirect
10. #27132822C2semidirect
11. #27132797C2semidirect
12. #27132795C2semidirect
13. D8×Dic191C2semidirect
14. #27132739C2semidirect
15. C2×AGL(1,ℤ/8)×C191C2semidirect
17. C2×D4×D191C22semidirect
19. C2×#15690277C22semidirect
20. C2×#15690276C22semidirect
21. C2×#15690275C22semidirect
24. C2×#15690270C22semidirect
25. #20259311C22semidirect
26. #20259309C22semidirect
29. #20259301C22semidirect
30. #20259298C22semidirect
37. D4×Dic191C22semidirect
38. D4×Dic191C22semidirect
42. #20259256C22semidirect
43. Dic191⋊C8C22semidirect
44. #20259247C22semidirect
45. #20259246C22semidirect
46. #20259244C22semidirect
47. #20259243C22semidirect
48. #20259243C22semidirect
49. #20259243C22semidirect
50. C22⋊(C191⋊C8)C22semidirect
51. (C2×D191)⋊C8C22semidirect
52. C8×Dic191C22semidirect
54. C22×D4×C191C22semidirect
56. C2×SD16×C191C22semidirect
57. C2×SD16×C191C22semidirect
60. C2×M16×C191C22semidirect
119. D8×C191D4semidirect
215. C191#3273semidirect
218. C2×AGL(1,ℤ/8)D191semidirect
235. C22×D4C2×D191semidirect
241. C2×SD16C2×D191semidirect
242. C2×SD16C2×D191semidirect
248. C2×M16C2×D191semidirect
314. D8C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593462
#271333072
#271333012
#271330442
#271329092
#271329062
#271328802
#271328762
#271328252
#271328222
#271327972
#271327952
D8×Dic1912
#271327392
C2×AGL(1,ℤ/8)×C1912
#3273191