Finite Group Databasebeta

← #37582199

Subgroups — #37582199

Order 24448.

504 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259346C211yesmaximal, characteristic, complemented
2C2×#20259328C211yesmaximal, characteristic, complemented
3#27133242C211yesmaximal, characteristic, complemented
4#27133229C211yesmaximal, characteristic, complemented
5#27132914C211yesmaximal, characteristic, complemented
6#27132913C211yesmaximal, characteristic, complemented
7#27132910C211yesmaximal, characteristic, complemented
8#27132907C211yesmaximal, characteristic, complemented
9#27132898C211yesmaximal, characteristic, complemented
10#27132892C211yesmaximal, characteristic, complemented
11#27132891C211yesmaximal, characteristic, complemented
12#27132704C211yesmaximal, characteristic, complemented
13#27132656C211yesmaximal, characteristic, complemented
14#102×D191C211yesmaximal, characteristic, complemented
15#466×C191C211yesmaximal, characteristic, complemented
16C2×#15690287C2211yescharacteristic
17C2×#15690286C2211yescharacteristic
18C2×D4×D191C2211yescharacteristic
19#20259346—24no—
20C2×#15690276C2211yescharacteristic
21C2×#15690275C2211yescharacteristic
22C2×#15690272C2211yescharacteristic
23#20259328—24no—
24C2×#15690271C2211yescharacteristic, complemented
25C2×#15690270C2211yescharacteristic
26C2×#15690270C2211yescharacteristic, complemented
27C2×C8×D191C2211yescharacteristic
28C2×C8×D191C2211yescharacteristic, complemented
29#20259318—21no—
30#20259308—21no—
31#20259302C2211yescharacteristic, complemented
32#20259301C2211yescharacteristic, complemented
33#20259299—21no—
34#20259297C2211yescharacteristic
35#20259296C2211yescharacteristic
36#20259295—21no—
37#20259294—21no—
38#20259293—21no—
39#20259293C2211yescharacteristic
40#20259293C2211yescharacteristic, complemented
41#20259292—21no—
42#20259290—21no—
43#20259280C2211yescharacteristic
44C4⋊C4×D191C2211yescharacteristic
45#20259255C2211yescharacteristic, complemented
46#20259254C2211yescharacteristic, complemented
47#20259253C2211yescharacteristic, complemented
48#20259247C2211yescharacteristic, complemented
49#20259245C2211yescharacteristic, complemented
50#20259244C2211yescharacteristic
51#20259243C2211yescharacteristic
52#20259243C2211yescharacteristic, complemented
53#20259242C2211yescharacteristic
54C2×D8×C191C2211yescharacteristic
55C22×C8×C191C2211yescharacteristic, complemented
56#121×C191C2211yescharacteristic
57#121×C191C2211yescharacteristic, complemented
58#107×C191C2211yescharacteristic, complemented
59#102×C191C2211yescharacteristic
60#102×C191C2211yescharacteristic, complemented
61C23×D191—21no—
62#15690287—22no—
63#15690287—22no—
64#15690286—22no—
65#15690286—22no—
66C2×Dic382D411yescharacteristic
67C2×Dic382C2311yescharacteristic
68C2×D764—21no—
69C2×D764D411yescharacteristic
70C2×D764C2311yescharacteristic
71C2×C22⋊D191—21no—
72C2×C22⋊D191—21no—
73C2×C22⋊D191—21no—
74C2×C22⋊D191—21no—
75C2×C22⋊D191—21no—
76C2×C22⋊D191D411yescharacteristic
77C2×C22⋊D191D411yescharacteristic
78D4×D191—22no—
79D4×D191—22no—
80C22×Dic191—21no—
81C2×C4×D191—21no—
82C2×C4×D191C2311yescharacteristic
83C2×C4×D191C2311yescharacteristic
84#15690276—22no—
85#15690275—22no—
86#15690272—22no—
87#15690271—22no—
88#15690270—22no—
89#15690270D414yescomplemented
90C2×C191⋊C8C2311yescharacteristic
91C2×C191⋊C8C2311yescharacteristic
92C8×D191—22no—
93C8×D191—22no—
94C4⋊Dic191—21no—
95C4⋊Dic191C2311yescharacteristic
96C4⋊Dic191C2311yescharacteristic
97Dic191⋊C4—21no—
98Dic191⋊C4—21no—
99C22⋊Dic191—21no—
100C22⋊Dic191—21no—
101C22⋊Dic191—21no—
102(C2×D191)⋊C4—21no—
103(C2×D191)⋊C4—21no—
104(C2×D191)⋊C4—21no—
105(C2×D191)⋊C4—21no—
106C4×Dic191—21no—
107C2×D4×C191—21no—
108C2×D4×C191—21no—
109C2×D4×C191C2311yescharacteristic
110C2×D4×C191C2311yescharacteristic
111C22×C4×C191C2311yescharacteristic
112D8×C191—22no—
113C2×C8×C191—22no—
114C2×C8×C191C2311yescharacteristic
115C2×C8×C191C2311yescharacteristic
116C4⋊C4×C191C2311yescharacteristic
117C4⋊C4×C191C2311yescharacteristic
118C22⋊C4×C191—21no—
119C22⋊C4×C191—21no—
120C22×D191—21no—
121C22×D191—42no—
122C22×D191—81no—
123C22×D191—81no—
124C22×D191C2×D411yescharacteristic
125C22×D191C2×D411yescharacteristic
126Dic382—22no—
127Dic382—21no—
128Dic382—21no—
129D764—22no—
130D764—21no—
131D764—21no—
132D764—81no—
133C22⋊D191—22no—
134C22⋊D191—22no—
135C22⋊D191—42no—
136C22⋊D191—42no—
137C22⋊D191—81no—
138C22⋊D191—81no—
139C22⋊D191—81no—
140C2×Dic191—21no—
141C2×Dic191—21no—
142C2×Dic191—21no—
143C2×Dic191—42no—
144C2×Dic191C2×D411yescharacteristic
145C2×Dic191C2×D411yescharacteristic
146C4×D191—21no—
147C4×D191—21no—
148C4×D191—81no—
149C4×D191C2×D412yes—
150C4×D191#4112yes—
151C191⋊C8—21no—
152C191⋊C8C2×D412yes—
153C23×C191—21no—
154C23×C191—21no—
155C23×C191C2×D411yescharacteristic
156D4×C191—21no—
157D4×C191—21no—
158D4×C191—21no—
159D4×C191—21no—
160D4×C191—81no—
161D4×C191—81no—
162C2×C4×C191—21no—
163C2×C4×C191—21no—
164C2×C4×C191—21no—
165C2×C4×C191—21no—
166C2×C4×C191C2×D411yescharacteristic
167C2×C4×C191C2411yescharacteristic
168C8×C191—21no—
169C8×C191C2×D412yes—
170C22×C191—21no—
171C22×C191—21no—
172C22×C191—21no—
173C22×C191—41no—
174C22×C191—41no—
175C22×C191—41no—
176C22×C191—41no—
177C22×C191—41no—
178C22×C191—41no—
179C22×C191C22×D411yescharacteristic
180C2×D191—21no—
181C2×D191—21no—
182C2×D191—21no—
183C2×D191—21no—
184C2×D191—21no—
185C2×D191—41no—
186C2×D191—41no—
187C2×D191—41no—
188C2×D191—84no—
189C2×D191#12112yes—
190C4×C191—21no—
191C4×C191—41no—
192C4×C191—41no—
193C4×C191C22×D411yescharacteristic
194C4×C191C2×#4111yescharacteristic
195Dic191—21no—
196Dic191—41no—
197Dic191—41no—
198Dic191—41no—
199Dic191#12112yes—
200C2×C191—41no—
201C2×C191—81no—
202C2×C191—81no—
203C2×C191C2×#12111yescharacteristic
204C2×C191#57611yescharacteristic
205C2×C191#57711yescharacteristic
206D191—22no—
207D191—41no—
208D191—81no—
209C191#306211yescharacteristic, complemented, Sylow
210#3062—1911nomaximal, Sylow
211C2×AGL(1,ℤ/8)—1911no—
212C2×#131—1911no—
213#537—1911no—
214#535—1911no—
215#469—1911no—
216#469—1911no—
217#468—1911no—
218#468—1911no—
219#466—1911no—
220#466D19111yescharacteristic, complemented
221#465—1911no—
222#465—1911no—
223#428—1911no—
224#416—1911no—
225C2×#102—1911no—
226C2×#41—1911no—
227C2×#41—1911no—
228C22×D4—1911no—
229AGL(1,ℤ/8)—3824no—
230C2×SD16—1911no—
231C2×SD16—1911no—
232C2×D8—1911no—
233C2×D8C2×D19111yescharacteristic
234#131—3824no—
235C2×M16—1911no—
236C2×M16—1911no—
237C2×M16—1911no—
238C22×C8—1911no—
239C22×C8—1911no—
240C22×C8C2×D19111yescharacteristic, complemented
241#127—3821no—
242#124—3821no—
243C22⋊Q8—3821no—
244C22⋊Q8—1911no—
245C22⋊Q8—1911no—
246#121—3821no—
247#121—3821no—
248#121—3821no—
249#121—1911no—
250#121—1911no—
251#121—1911no—
252#121—1911no—
253#121C2×D19111yescharacteristic
254#121C2×D19111yescharacteristic, complemented
255#120—3821no—
256#120—3821no—
257#117—1911no—
258C2×C4⋊C4—1911no—
259#107—1911no—
260#107C2×D19111yescharacteristic, complemented
261#106—1911no—
262#106—1911no—
263#103—1911no—
264#103—1911no—
265#102—1911no—
266#102—1911no—
267#102—1911no—
268#102—1911no—
269#102C2×D19111yescharacteristic
270#102C2×D19111yescharacteristic, complemented
271C24—3821no—
272#41—3822no—
273#41—3822no—
274#41—3822no—
275#41—3822no—
276C2×Q8—1911no—
277C2×Q8—1911no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—3821no—
281C2×D4—3821no—
282C2×D4—3821no—
283C2×D4—3821no—
284C2×D4—3821no—
285C2×D4—3822no—
286C2×D4—3822no—
287C2×D4—3821no—
288C2×D4—1911no—
289C2×D4—1911no—
290C2×D4—1911no—
291C2×D4—1911no—
292C2×D4C22×D19111yescharacteristic
293C2×D4C22×D19111yescharacteristic
294C22×C4—3821no—
295C22×C4—3821no—
296C22×C4—1911no—
297C22×C4—1911no—
298C22×C4C22×D19111yescharacteristic
299SD16—3822no—
300SD16—3822no—
301D8—22no—
302D8—3822no—
303M16—3822no—
304M16—3822no—
305M16—1914no—
306C2×C8—22no—
307C2×C8—3822no—
308C2×C8—3822no—
309C2×C8—1911no—
310C2×C8—1911no—
311C2×C8C22×D19111yescharacteristic
312C2×C8C22×D19111yescharacteristic
313C4⋊C4—3821no—
314C4⋊C4—3821no—
315C4⋊C4—3821no—
316C4⋊C4—1911no—
317C4⋊C4—1911no—
318C4⋊C4C22×D19111yescharacteristic
319C4⋊C4C22×D19111yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—3821no—
323C22⋊C4—3821no—
324C22⋊C4—3821no—
325C22⋊C4—3821no—
326C22⋊C4—3821no—
327C22⋊C4—3821no—
328C22⋊C4—3821no—
329C42—3821no—
330C23—21no—
331C23—21no—
332C23—3821no—
333C23—1911no—
334C23—1911no—
335C23—15281no—
336C23—15281no—
337C23—7642no—
338C23D4×D19111yescharacteristic
339Q8—3822no—
340Q8—3821no—
341Q8—3821no—
342D4—21no—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—81no—
347D4—81no—
348D4—3822no—
349D4—3822no—
350D4—3822no—
351D4—3821no—
352D4—3821no—
353D4—15281no—
354D4—15281no—
355D4—15281no—
356D4—15281no—
357D4—7642no—
358D4—7642no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—3821no—
364C2×C4—3821no—
365C2×C4—3821no—
366C2×C4—3821no—
367C2×C4—3821no—
368C2×C4—1911no—
369C2×C4—1911no—
370C2×C4—1912no—
371C2×C4—1912no—
372C2×C4—15281no—
373C2×C4—7642no—
374C2×C4D4×D19111yescharacteristic
375C2×C4C23×D19111yescharacteristic
376C8—21no—
377C8—3821no—
378C8—1912no—
379C8D4×D19112yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—3821no—
390C22—3821no—
391C22—3821no—
392C22—3821no—
393C22—3821no—
394C22—1912no—
395C22—15284no—
396C22—7641no—
397C22—7641no—
398C22—7641no—
399C22C2×D4×D19111yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—3821no—
404C4—1912no—
405C4—7641no—
406C4—7641no—
407C4—7641no—
408C4C2×D4×D19111yescharacteristic
409C4#41×D19111yescharacteristic
410C2—41no—
411C2—81no—
412C2—81no—
413C2—3822no—
414C2—15281no—
415C2—7641no—
416C2#121×D19111yescharacteristic
417C2#2713346411yescharacteristic
418C2#2713347011yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#466×C191C2
socleC22×C191C22×D4
hypercenter#466D191
hypocenterC191#3062
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#20259346C2semidirect
2. C2×#20259328C2semidirect
3. #27133242C2semidirect
4. #27133229C2semidirect
5. #27132914C2semidirect
6. #27132913C2semidirect
7. #27132910C2semidirect
8. #27132907C2semidirect
9. #27132898C2semidirect
10. #27132892C2semidirect
11. #27132891C2semidirect
12. #27132704C2semidirect
13. #27132656C2semidirect
14. #102×D191C2semidirect
15. #466×C191C2semidirect
24. C2×#15690271C22semidirect
26. C2×#15690270C22semidirect
28. C2×C8×D191C22semidirect
31. #20259302C22semidirect
32. #20259301C22semidirect
40. #20259293C22semidirect
45. #20259255C22semidirect
46. #20259254C22semidirect
47. #20259253C22semidirect
48. #20259247C22semidirect
49. #20259245C22semidirect
52. #20259243C22semidirect
55. C22×C8×C191C22semidirect
57. #121×C191C22semidirect
58. #107×C191C22semidirect
60. #102×C191C22semidirect
89. #15690270D4semidirect
209. C191#3062semidirect
220. #466D191semidirect
240. C22×C8C2×D191semidirect
254. #121C2×D191semidirect
260. #107C2×D191semidirect
270. #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×#202593282
#271332422
#271332292
#271329142
#271329132
#271329102
#271329072
#271328982
#271328922
#271328912
#271327042
#271326562
#102×D1912
#466×C1912
#3062191