Finite Group Databasebeta

← #37582021

Subgroups — #37582021

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×#20259349C211yesmaximal, characteristic, complemented
2C2×D8×D191C211yesmaximal, characteristic, complemented
3#27133231C211yesmaximal, characteristic, complemented
4#27133135C211yesmaximal, characteristic, complemented
5#27132881C211yesmaximal, characteristic, complemented
6#27132880C211yesmaximal, characteristic, complemented
7#27132875C211yesmaximal, characteristic, complemented
8#27132859C211yesmaximal, characteristic, complemented
9#27132858C211yesmaximal, characteristic, complemented
10#27132857C211yesmaximal, characteristic, complemented
11#27132856C211yesmaximal, characteristic, complemented
12#27132688C211yesmaximal, characteristic, complemented
13#27132666C211yesmaximal, characteristic, complemented
14#27132658C211yesmaximal, characteristic, complemented
15#463×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×D4×D191C2211yescharacteristic
18C2×D4×D191C2211yescharacteristic
19#20259349—24no—
20C2×#15690274C2211yescharacteristic, complemented
21C2×D1528C2211yescharacteristic
22C2×D1528C2211yescharacteristic, complemented
23C2×#15690272C2211yescharacteristic
24C2×#15690272C2211yescharacteristic
25C2×#15690272C2211yescharacteristic, complemented
26D8×D191—24no—
27C2×#15690270C2211yescharacteristic
28C2×C8×D191C2211yescharacteristic
29#20259319C2211yescharacteristic, complemented
30#20259318C2211yescharacteristic, complemented
31#20259311C2211yescharacteristic, complemented
32#20259295—21no—
33#20259295—21no—
34#20259292—21no—
35#20259292—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#20259246C2211yescharacteristic, complemented
49#20259244C2211yescharacteristic, complemented
50#20259244C2211yescharacteristic, complemented
51#20259243C2211yescharacteristic, complemented
52#20259242C2211yescharacteristic, complemented
53#20259242C2211yescharacteristic, complemented
54C2×SD16×C191C2211yescharacteristic, complemented
55C2×D8×C191C2211yescharacteristic
56#124×C191C2211yescharacteristic, complemented
57C4×D4×C191C2211yescharacteristic, complemented
58C4⋊C8×C191C2211yescharacteristic, complemented
59#103×C191C2211yescharacteristic, complemented
60#102×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62C23×D191—21no—
63#15690288—22no—
64#15690288—22no—
65C2×D764—21no—
66C2×D764—21no—
67C2×D764—21no—
68C2×D764C2311yescharacteristic
69C2×D764C2311yescharacteristic
70C2×D764C2311yescharacteristic
71C2×C22⋊D191—21no—
72C2×C22⋊D191—21no—
73C2×C22⋊D191—21no—
74C2×C22⋊D191—21no—
75D4×D191—22no—
76D4×D191—22no—
77D4×D191—22no—
78D4×D191D414yescomplemented
79C22×Dic191—21no—
80C2×C4×D191—21no—
81C2×C4×D191—21no—
82C2×C4×D191—21no—
83C2×C4×D191C2311yescharacteristic
84#15690274—22no—
85D1528—22no—
86D1528—22no—
87#15690272—22no—
88#15690272—22no—
89#15690272—22no—
90#15690270—22no—
91C2×C191⋊C8C2311yescharacteristic
92C2×C191⋊C8C2311yescharacteristic
93C8×D191—22no—
94C4⋊Dic191C2311yescharacteristic
95Dic191⋊C4D411yescharacteristic
96Dic191⋊C4D411yescharacteristic
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×Q8×C191C2311yescharacteristic
108C2×D4×C191C2311yescharacteristic
109C2×D4×C191C2311yescharacteristic
110C22×C4×C191—21no—
111SD16×C191—22no—
112D8×C191—22no—
113C2×C8×C191C2311yescharacteristic
114C2×C8×C191C2311yescharacteristic
115C4⋊C4×C191C2311yescharacteristic
116C22⋊C4×C191—21no—
117C22⋊C4×C191—21no—
118C22⋊C4×C191—21no—
119C42×C191C2311yescharacteristic
120C22×D191—21no—
121C22×D191—24no—
122C22×D191—21no—
123C22×D191—21no—
124C22×D191—42no—
125C22×D191—81no—
126C22×D191—81no—
127C22×D191—81no—
128C22×D191—81no—
129C22×D191C2×D411yescharacteristic
130D764—21no—
131D764—21no—
132D764—21no—
133D764—21no—
134D764—21no—
135D764—42no—
136D764—81no—
137D764—81no—
138D764C2×D412yes—
139C22⋊D191—24no—
140C22⋊D191—42no—
141C22⋊D191—81no—
142C22⋊D191—81no—
143C2×Dic191—21no—
144C2×Dic191—81no—
145C2×Dic191C2×D411yescharacteristic
146C2×Dic191C2×D411yescharacteristic
147C2×Dic191C2×D411yescharacteristic
148C4×D191—42no—
149C4×D191—81no—
150C4×D191—81no—
151C4×D191C2×D412yes—
152C4×D191#4112yes—
153C191⋊C8—21no—
154C191⋊C8—21no—
155C23×C191—21no—
156C23×C191—21no—
157Q8×C191—21no—
158Q8×C191—21no—
159D4×C191—21no—
160D4×C191—21no—
161D4×C191—21no—
162D4×C191C2×D412yes—
163C2×C4×C191—21no—
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×C191C22×D411yescharacteristic
178C2×D191—21no—
179C2×D191—21no—
180C2×D191—22no—
181C2×D191—41no—
182C2×D191—41no—
183C2×D191—41no—
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×C191—41no—
196C4×C191C22×D411yescharacteristic
197C4×C191C2×#4111yescharacteristic
198Dic191—21no—
199Dic191—21no—
200Dic191—41no—
201Dic191#12112yes—
202C2×C191—42no—
203C2×C191—81no—
204C2×C191C2×#12111yescharacteristic
205C2×C191#57611yescharacteristic
206C2×C191#57611yescharacteristic
207D191—22no—
208D191—42no—
209D191—81no—
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#463D19111yescharacteristic, complemented
221#459—1911no—
222#459—1911no—
223#459—1911no—
224#459—1911no—
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×SD16C2×D19111yescharacteristic, complemented
234C2×D8—3824no—
235C2×D8—1911no—
236C2×D8—1911no—
237C2×D8—1911no—
238C2×D8—1911no—
239C2×D8—1911no—
240C2×D8C2×D19111yescharacteristic
241C2×M16—1911no—
242C22×C8—1911no—
243#127—1911no—
244#127—1911no—
245#124—1911no—
246#124C2×D19111yescharacteristic, complemented
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#103C2×D19111yescharacteristic, complemented
267#102—1911no—
268#102—1911no—
269#102—1911no—
270#102—1911no—
271#102—1911no—
272#102C2×D19111yescharacteristic, complemented
273C24—3821no—
274C24—3821no—
275#41—3822no—
276#41—3822no—
277C2×Q8C22×D19111yescharacteristic
278C2×D4—3821no—
279C2×D4—3822no—
280C2×D4—3822no—
281C2×D4—3822no—
282C2×D4—3821no—
283C2×D4—3821no—
284C2×D4—3821no—
285C2×D4—3821no—
286C2×D4—3821no—
287C2×D4—3821no—
288C2×D4—1914no—
289C2×D4—1911no—
290C2×D4—1911no—
291C2×D4—1911no—
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—22no—
301SD16—3822no—
302D8—22no—
303D8—3822no—
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—21no—
320C22⋊C4—21no—
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—3824no—
335C23—3821no—
336C23—3821no—
337C23—3821no—
338C23—1911no—
339C23—15281no—
340C23—15281no—
341C23—15281no—
342C23—15281no—
343C23—7642no—
344Q8—21no—
345Q8—21no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—3824no—
350D4—3821no—
351D4—3821no—
352D4—3821no—
353D4—3821no—
354D4—3821no—
355D4—1912no—
356D4—15281no—
357D4—15281no—
358D4—15281no—
359D4—15281no—
360D4—7642no—
361D4—7642no—
362D4D4×D19112yes—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—81no—
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—3821no—
390C22—3821no—
391C22—3822no—
392C22—1912no—
393C22—15284no—
394C22—15284no—
395C22—7641no—
396C22—7641no—
397C22—7641no—
398C22—7641no—
399C22—7641no—
400C22—7641no—
401C22—7641no—
402C22—7641no—
403C22C2×D4×D19111yescharacteristic
404C4—21no—
405C4—21no—
406C4—41no—
407C4—41no—
408C4—3821no—
409C4—3821no—
410C4—1912no—
411C4—7641no—
412C4C2×D4×D19111yescharacteristic
413C4#41×D19111yescharacteristic
414C2—42no—
415C2—81no—
416C2—3822no—
417C2—15281no—
418C2—15281no—
419C2—7642no—
420C2#121×D19111yescharacteristic
421C2#2713346511yescharacteristic
422C2#2713346711yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#463×C191C2
socleC22×C191C22×D4
hypercenter#463D191
hypocenterC191#3041
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#20259349C2semidirect
2. C2×D8×D191C2semidirect
3. #27133231C2semidirect
4. #27133135C2semidirect
5. #27132881C2semidirect
6. #27132880C2semidirect
7. #27132875C2semidirect
8. #27132859C2semidirect
9. #27132858C2semidirect
10. #27132857C2semidirect
11. #27132856C2semidirect
12. #27132688C2semidirect
13. #27132666C2semidirect
14. #27132658C2semidirect
15. #463×C191C2semidirect
20. C2×#15690274C22semidirect
22. C2×D1528C22semidirect
25. C2×#15690272C22semidirect
29. #20259319C22semidirect
30. #20259318C22semidirect
31. #20259311C22semidirect
36. C4×D764C22semidirect
37. D4×Dic191C22semidirect
38. #20259284C22semidirect
45. C4⋊(C191⋊C8)C22semidirect
46. Dic191⋊C8C22semidirect
47. Dic191⋊C8C22semidirect
48. #20259246C22semidirect
49. #20259244C22semidirect
50. #20259244C22semidirect
51. #20259243C22semidirect
52. #20259242C22semidirect
53. #20259242C22semidirect
54. C2×SD16×C191C22semidirect
56. #124×C191C22semidirect
57. C4×D4×C191C22semidirect
58. C4⋊C8×C191C22semidirect
59. #103×C191C22semidirect
60. #102×C191C22semidirect
78. D4×D191D4semidirect
211. C191#3041semidirect
220. #463D191semidirect
233. C2×SD16C2×D191semidirect
246. #124C2×D191semidirect
256. C4×D4C2×D191semidirect
264. C4⋊C8C2×D191semidirect
266. #103C2×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×#202593492
C2×D8×D1912
#271332312
#271331352
#271328812
#271328802
#271328752
#271328592
#271328582
#271328572
#271328562
#271326882
#271326662
#271326582
#463×C1912
#3041191