Finite Group Databasebeta

← #37582018

Subgroups — #37582018

Order 24448.

526 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259346C211yesmaximal, characteristic, complemented
2C2×D8×D191C211yesmaximal, characteristic, complemented
3#27133226C211yesmaximal, characteristic, complemented
4#27133135C211yesmaximal, characteristic, complemented
5#27132879C211yesmaximal, characteristic, complemented
6#27132878C211yesmaximal, characteristic, complemented
7#27132877C211yesmaximal, characteristic, complemented
8#27132876C211yesmaximal, characteristic, complemented
9#27132857C211yesmaximal, characteristic, complemented
10#27132856C211yesmaximal, characteristic, complemented
11#27132855C211yesmaximal, characteristic, complemented
12#27132688C211yesmaximal, characteristic, complemented
13#27132666C211yesmaximal, characteristic, complemented
14#27132656C211yesmaximal, characteristic, complemented
15#459×C191C211yesmaximal, characteristic, complemented
16C2×#15690286C2211yescharacteristic
17C2×D4×D191C2211yescharacteristic
18C2×D4×D191C2211yescharacteristic
19#20259346—24no—
20C2×#15690276C2211yescharacteristic, complemented
21C2×#15690275C2211yescharacteristic, complemented
22C2×D1528C2211yescharacteristic
23C2×#15690272C2211yescharacteristic
24C2×#15690272C2211yescharacteristic
25C2×#15690272C2211yescharacteristic, complemented
26D8×D191—24no—
27C2×#15690270C2211yescharacteristic
28C2×C8×D191C2211yescharacteristic
29#20259318C2211yescharacteristic, complemented
30#20259310C2211yescharacteristic, complemented
31#20259309C2211yescharacteristic, complemented
32#20259294—21no—
33#20259294—21no—
34#20259290—21no—
35#20259290—21no—
36C4×D764C2211yescharacteristic, complemented
37D4×Dic191C2211yescharacteristic, complemented
38#20259284C2211yescharacteristic, complemented
39C4×C22⋊D191—21no—
40(C22⋊D191)⋊C4—21no—
41#20259280C2211yescharacteristic
42#20259279C2211yescharacteristic
43C2×(C2×D191)⋊C4—21no—
44C22⋊C4×D191—21no—
45C4⋊(C191⋊C8)C2211yescharacteristic, complemented
46Dic191⋊C8C2211yescharacteristic, complemented
47Dic191⋊C8C2211yescharacteristic, complemented
48#20259247C2211yescharacteristic, complemented
49#20259245C2211yescharacteristic, complemented
50#20259244C2211yescharacteristic, complemented
51#20259243C2211yescharacteristic, complemented
52#20259243C2211yescharacteristic, complemented
53#20259242C2211yescharacteristic, complemented
54C2×D8×C191C2211yescharacteristic
55C2×D8×C191C2211yescharacteristic, complemented
56#127×C191C2211yescharacteristic, complemented
57C4×D4×C191C2211yescharacteristic, complemented
58C4⋊C8×C191C2211yescharacteristic, complemented
59#102×C191C2211yescharacteristic, complemented
60#102×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62C23×D191—21no—
63#15690286—22no—
64#15690286—22no—
65C2×Dic382C2311yescharacteristic
66C2×D764C2311yescharacteristic
67C2×D764C2311yescharacteristic
68C2×C22⋊D191—21no—
69C2×C22⋊D191—21no—
70C2×C22⋊D191—21no—
71C2×C22⋊D191—21no—
72C2×C22⋊D191—21no—
73D4×D191—22no—
74D4×D191—22no—
75D4×D191—22no—
76D4×D191D414yescomplemented
77C22×Dic191—21no—
78C22×Dic191—21no—
79C2×C4×D191—21no—
80C2×C4×D191—21no—
81C2×C4×D191C2311yescharacteristic
82#15690276—22no—
83#15690275—22no—
84D1528—22no—
85#15690272—22no—
86#15690272—22no—
87#15690272—22no—
88#15690270—22no—
89C2×C191⋊C8C2311yescharacteristic
90C2×C191⋊C8C2311yescharacteristic
91C8×D191—22no—
92C4⋊Dic191C2311yescharacteristic
93Dic191⋊C4D411yescharacteristic
94Dic191⋊C4D411yescharacteristic
95C22⋊Dic191—21no—
96C22⋊Dic191—21no—
97C22⋊Dic191—21no—
98(C2×D191)⋊C4—21no—
99(C2×D191)⋊C4—21no—
100(C2×D191)⋊C4—21no—
101(C2×D191)⋊C4—21no—
102(C2×D191)⋊C4—21no—
103(C2×D191)⋊C4D411yescharacteristic
104(C2×D191)⋊C4D411yescharacteristic
105C4×Dic191—21no—
106C4×Dic191C2311yescharacteristic
107C2×D4×C191—21no—
108C2×D4×C191—21no—
109C2×D4×C191C2311yescharacteristic
110C2×D4×C191C2311yescharacteristic
111C2×D4×C191C2311yescharacteristic
112C22×C4×C191—21no—
113D8×C191—22no—
114D8×C191—22no—
115C2×C8×C191C2311yescharacteristic
116C2×C8×C191C2311yescharacteristic
117C4⋊C4×C191C2311yescharacteristic
118C22⋊C4×C191—21no—
119C42×C191C2311yescharacteristic
120C22×D191—21no—
121C22×D191—24no—
122C22×D191—21no—
123C22×D191—42no—
124C22×D191—81no—
125C22×D191—81no—
126C22×D191—81no—
127C22×D191—81no—
128C22×D191C2×D411yescharacteristic
129Dic382—21no—
130Dic382—21no—
131D764—21no—
132D764—21no—
133D764—21no—
134D764C2×D412yes—
135C22⋊D191—24no—
136C22⋊D191—42no—
137C22⋊D191—42no—
138C22⋊D191—81no—
139C22⋊D191—81no—
140C2×Dic191—21no—
141C2×Dic191—21no—
142C2×Dic191—42no—
143C2×Dic191—81no—
144C2×Dic191C2×D411yescharacteristic
145C2×Dic191C2×D411yescharacteristic
146C2×Dic191C2×D411yescharacteristic
147C4×D191—81no—
148C4×D191—81no—
149C4×D191C2×D412yes—
150C4×D191#4112yes—
151C191⋊C8—21no—
152C191⋊C8—21no—
153C23×C191—21no—
154C23×C191—21no—
155C23×C191—21no—
156D4×C191—21no—
157D4×C191—21no—
158D4×C191—21no—
159D4×C191—21no—
160D4×C191—21no—
161D4×C191—81no—
162D4×C191—81no—
163D4×C191C2×D412yes—
164C2×C4×C191—21no—
165C2×C4×C191—81no—
166C2×C4×C191C2×D411yescharacteristic
167C2×C4×C191C2×D411yescharacteristic
168C2×C4×C191C2411yescharacteristic
169C8×C191—21no—
170C8×C191—21no—
171C22×C191—22no—
172C22×C191—41no—
173C22×C191—41no—
174C22×C191—41no—
175C22×C191—41no—
176C22×C191—41no—
177C22×C191—41no—
178C22×C191—41no—
179C22×C191—41no—
180C22×C191C22×D411yescharacteristic
181C2×D191—21no—
182C2×D191—21no—
183C2×D191—22no—
184C2×D191—41no—
185C2×D191—41no—
186C2×D191—41no—
187C2×D191—41no—
188C2×D191—41no—
189C2×D191—84no—
190C2×D191—84no—
191C2×D191#12112yes—
192C4×C191—21no—
193C4×C191—21no—
194C4×C191—41no—
195C4×C191C22×D411yescharacteristic
196C4×C191C2×#4111yescharacteristic
197Dic191—21no—
198Dic191—21no—
199Dic191—41no—
200Dic191—41no—
201Dic191#12112yes—
202C2×C191—42no—
203C2×C191—81no—
204C2×C191—81no—
205C2×C191C2×#12111yescharacteristic
206C2×C191#57611yescharacteristic
207C2×C191#57611yescharacteristic
208D191—22no—
209D191—42no—
210D191—81no—
211C191#304111yescharacteristic, complemented, Sylow
212#3041—1911nomaximal, Sylow
213C2×AGL(1,ℤ/8)—1911no—
214C22×D8—1911no—
215#535—1911no—
216#518—1911no—
217#463—1911no—
218#463—1911no—
219#463—1911no—
220#463—1911no—
221#459—1911no—
222#459—1911no—
223#459—1911no—
224#459D19111yescharacteristic, complemented
225#424—1911no—
226#418—1911no—
227#416—1911no—
228C2×#41—1911no—
229C22×D4—1911no—
230C22×D4—1911no—
231AGL(1,ℤ/8)—3824no—
232C2×SD16—1911no—
233C2×SD16—1911no—
234C2×D8—3824no—
235C2×D8—1911no—
236C2×D8—1911no—
237C2×D8—1911no—
238C2×D8—1911no—
239C2×D8C2×D19111yescharacteristic
240C2×D8C2×D19111yescharacteristic, complemented
241C2×M16—1911no—
242C22×C8—1911no—
243#127—1911no—
244#127C2×D19111yescharacteristic, complemented
245#124—1911no—
246#124—1911no—
247#121—3821no—
248#121—3821no—
249#120—3821no—
250#120—3821no—
251C4×D4—3821no—
252C4×D4—3821no—
253C4×D4—1911no—
254C4×D4—1911no—
255C4×D4—1911no—
256C4×D4C2×D19111yescharacteristic, complemented
257#117—1911no—
258#117—1911no—
259C2×C22⋊C4—3821no—
260C2×C22⋊C4—3821no—
261C4⋊C8—1911no—
262C4⋊C8—1911no—
263C4⋊C8—1911no—
264C4⋊C8C2×D19111yescharacteristic, complemented
265#103—1911no—
266#103—1911no—
267#102—1911no—
268#102—1911no—
269#102—1911no—
270#102—1911no—
271#102C2×D19111yescharacteristic, complemented
272#102C2×D19111yescharacteristic, complemented
273C24—3821no—
274C24—3821no—
275#41—3822no—
276#41—3822no—
277C2×Q8—1911no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—3821no—
281C2×D4—3822no—
282C2×D4—3822no—
283C2×D4—3822no—
284C2×D4—3821no—
285C2×D4—3821no—
286C2×D4—3821no—
287C2×D4—3821no—
288C2×D4—1914no—
289C2×D4—1911no—
290C2×D4—1911no—
291C2×D4C22×D19111yescharacteristic
292C2×D4C22×D19111yescharacteristic
293C2×D4C22×D19111yescharacteristic
294C22×C4—21no—
295C22×C4—3821no—
296C22×C4—3821no—
297C22×C4—3821no—
298C22×C4—3821no—
299C22×C4—1911no—
300SD16—3822no—
301SD16—3822no—
302D8—22no—
303D8—22no—
304D8—3822no—
305D8—3822no—
306D8—3822no—
307D8—3822no—
308M16—3822no—
309C2×C8—3822no—
310C2×C8—1911no—
311C2×C8—1911no—
312C2×C8C22×D19111yescharacteristic
313C2×C8C22×D19111yescharacteristic
314C4⋊C4—1911no—
315C4⋊C4—1911no—
316C4⋊C4—1911no—
317C4⋊C4C22×D19111yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—3821no—
320C22⋊C4—3821no—
321C22⋊C4—3821no—
322C22⋊C4—3821no—
323C22⋊C4—3821no—
324C22⋊C4—3821no—
325C22⋊C4—3821no—
326C22⋊C4—3821no—
327C22⋊C4—1911no—
328C22⋊C4—1911no—
329C42—3821no—
330C42—1911no—
331C42C22×D19111yescharacteristic
332C23—21no—
333C23—21no—
334C23—21no—
335C23—3824no—
336C23—3821no—
337C23—3821no—
338C23—1911no—
339C23—15281no—
340C23—15281no—
341C23—15281no—
342C23—15281no—
343C23—7642no—
344Q8—3821no—
345Q8—3821no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—81no—
352D4—81no—
353D4—3824no—
354D4—3821no—
355D4—3821no—
356D4—3821no—
357D4—1912no—
358D4—15281no—
359D4—15281no—
360D4—7642no—
361D4—7642no—
362D4D4×D19112yes—
363C2×C4—21no—
364C2×C4—81no—
365C2×C4—3821no—
366C2×C4—3821no—
367C2×C4—1911no—
368C2×C4—1911no—
369C2×C4—1911no—
370C2×C4—1912no—
371C2×C4—1912no—
372C2×C4—15281no—
373C2×C4—15281no—
374C2×C4—15281no—
375C2×C4—7642no—
376C2×C4D4×D19111yescharacteristic
377C2×C4D4×D19111yescharacteristic
378C2×C4C23×D19111yescharacteristic
379C8—21no—
380C8—21no—
381C8—3821no—
382C8—3821no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—3821no—
393C22—3821no—
394C22—3822no—
395C22—1912no—
396C22—15284no—
397C22—15284no—
398C22—7641no—
399C22—7641no—
400C22—7641no—
401C22—7641no—
402C22—7641no—
403C22C2×D4×D19111yescharacteristic
404C4—21no—
405C4—21no—
406C4—41no—
407C4—3821no—
408C4—3821no—
409C4—1912no—
410C4—7641no—
411C4—7641no—
412C4C2×D4×D19111yescharacteristic
413C4#41×D19111yescharacteristic
414C2—42no—
415C2—81no—
416C2—81no—
417C2—3822no—
418C2—15281no—
419C2—7642no—
420C2#121×D19111yescharacteristic
421C2#2713346411yescharacteristic
422C2#2713346711yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#459×C191C2
socleC22×C191C22×D4
hypercenter#459D191
hypocenterC191#3041
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#20259346C2semidirect
2. C2×D8×D191C2semidirect
3. #27133226C2semidirect
4. #27133135C2semidirect
5. #27132879C2semidirect
6. #27132878C2semidirect
7. #27132877C2semidirect
8. #27132876C2semidirect
9. #27132857C2semidirect
10. #27132856C2semidirect
11. #27132855C2semidirect
12. #27132688C2semidirect
13. #27132666C2semidirect
14. #27132656C2semidirect
15. #459×C191C2semidirect
20. C2×#15690276C22semidirect
21. C2×#15690275C22semidirect
25. C2×#15690272C22semidirect
29. #20259318C22semidirect
30. #20259310C22semidirect
31. #20259309C22semidirect
36. C4×D764C22semidirect
37. D4×Dic191C22semidirect
38. #20259284C22semidirect
45. C4⋊(C191⋊C8)C22semidirect
46. Dic191⋊C8C22semidirect
47. Dic191⋊C8C22semidirect
48. #20259247C22semidirect
49. #20259245C22semidirect
50. #20259244C22semidirect
51. #20259243C22semidirect
52. #20259243C22semidirect
53. #20259242C22semidirect
55. C2×D8×C191C22semidirect
56. #127×C191C22semidirect
57. C4×D4×C191C22semidirect
58. C4⋊C8×C191C22semidirect
59. #102×C191C22semidirect
60. #102×C191C22semidirect
76. D4×D191D4semidirect
211. C191#3041semidirect
224. #459D191semidirect
240. C2×D8C2×D191semidirect
244. #127C2×D191semidirect
256. C4×D4C2×D191semidirect
264. C4⋊C8C2×D191semidirect
271. #102C2×D191semidirect
272. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593462
C2×D8×D1912
#271332262
#271331352
#271328792
#271328782
#271328772
#271328762
#271328572
#271328562
#271328552
#271326882
#271326662
#271326562
#459×C1912
#3041191