Finite Group Databasebeta

← #37582060

Subgroups — #37582060

Order 24448.

504 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259346C211yesmaximal, characteristic, complemented
2C2×#20259343C211yesmaximal, characteristic, complemented
3#27133231C211yesmaximal, characteristic, complemented
4#27133142C211yesmaximal, characteristic, complemented
5#27132888C211yesmaximal, characteristic, complemented
6#27132882C211yesmaximal, characteristic, complemented
7#27132880C211yesmaximal, characteristic, complemented
8#27132864C211yesmaximal, characteristic, complemented
9#27132862C211yesmaximal, characteristic, complemented
10#27132858C211yesmaximal, characteristic, complemented
11#27132856C211yesmaximal, characteristic, complemented
12#27132688C211yesmaximal, characteristic, complemented
13#27132663C211yesmaximal, characteristic, complemented
14#102×D191C211yesmaximal, characteristic, complemented
15#463×C191C211yesmaximal, characteristic, complemented
16C2×#15690288C2211yescharacteristic
17C2×#15690286C2211yescharacteristic
18C2×D4×D191C2211yescharacteristic
19#20259346—24no—
20#20259343—24no—
21C2×#15690277C2211yescharacteristic, complemented
22C2×#15690276C2211yescharacteristic
23C2×#15690276C2211yescharacteristic, complemented
24C2×#15690275C2211yescharacteristic
25C2×#15690272C2211yescharacteristic
26C2×#15690272C2211yescharacteristic, complemented
27C2×#15690270C2211yescharacteristic
28C2×C8×D191C2211yescharacteristic
29#20259319C2211yescharacteristic
30#20259318C2211yescharacteristic
31#20259311C2211yescharacteristic
32#20259295—21no—
33#20259295—21no—
34#20259292—21no—
35#20259292—21no—
36C4×Dic382C2211yescharacteristic, complemented
37#20259287C2211yescharacteristic, complemented
38D4×Dic191C2211yescharacteristic, complemented
39C4×C22⋊D191—21no—
40(C22⋊D191)⋊C4—21no—
41#20259279C2211yescharacteristic
42#20259278—21no—
43C2×Dic191⋊C4—21no—
44C4⋊C4×D191C2211yescharacteristic
45C4⋊(C191⋊C8)C2211yescharacteristic, complemented
46Dic191⋊C8C2211yescharacteristic, complemented
47Dic191⋊C8C2211yescharacteristic, complemented
48#20259246C2211yescharacteristic, complemented
49#20259244C2211yescharacteristic
50#20259244C2211yescharacteristic, complemented
51#20259243C2211yescharacteristic
52#20259242C2211yescharacteristic
53#20259242C2211yescharacteristic, complemented
54C2×SD16×C191C2211yescharacteristic, complemented
55C2×D8×C191C2211yescharacteristic
56#124×C191C2211yescharacteristic
57C4×D4×C191C2211yescharacteristic, complemented
58C4⋊C8×C191C2211yescharacteristic, complemented
59#103×C191C2211yescharacteristic, complemented
60#102×C191C2211yescharacteristic
61C23×D191—21no—
62#15690288—22no—
63#15690288—22no—
64#15690286—22no—
65#15690286D414yescomplemented
66C2×Dic382C2311yescharacteristic
67C2×D764—21no—
68C2×D764—21no—
69C2×D764—21no—
70C2×D764C2311yescharacteristic
71C2×D764C2311yescharacteristic
72C2×C22⋊D191—21no—
73C2×C22⋊D191—21no—
74C2×C22⋊D191—21no—
75C2×C22⋊D191—21no—
76D4×D191—22no—
77D4×D191—22no—
78C22×Dic191—21no—
79C22×Dic191—21no—
80C2×C4×D191—21no—
81C2×C4×D191—21no—
82C2×C4×D191C2311yescharacteristic
83#15690277—22no—
84#15690276—22no—
85#15690276—22no—
86#15690275—22no—
87#15690272—22no—
88#15690272—22no—
89#15690270—22no—
90C2×C191⋊C8C2311yescharacteristic
91C2×C191⋊C8C2311yescharacteristic
92C8×D191—22no—
93C4⋊Dic191C2311yescharacteristic
94Dic191⋊C4—21no—
95Dic191⋊C4—21no—
96Dic191⋊C4—21no—
97Dic191⋊C4D411yescharacteristic
98Dic191⋊C4D411yescharacteristic
99C22⋊Dic191—21no—
100(C2×D191)⋊C4—21no—
101(C2×D191)⋊C4—21no—
102(C2×D191)⋊C4D411yescharacteristic
103(C2×D191)⋊C4D411yescharacteristic
104C4×Dic191—21no—
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—21no—
122C22×D191—42no—
123C22×D191—81no—
124C22×D191—81no—
125C22×D191C2×D411yescharacteristic
126Dic382—21no—
127Dic382C2×D412yes—
128D764—21no—
129D764—21no—
130D764—21no—
131D764—21no—
132D764—42no—
133D764—81no—
134D764—81no—
135C22⋊D191—24no—
136C22⋊D191—42no—
137C22⋊D191—81no—
138C22⋊D191—81no—
139C2×Dic191—21no—
140C2×Dic191—24no—
141C2×Dic191—21no—
142C2×Dic191—81no—
143C2×Dic191C2×D411yescharacteristic
144C2×Dic191C2×D411yescharacteristic
145C2×Dic191C2×D411yescharacteristic
146C4×D191—42no—
147C4×D191—81no—
148C4×D191C2×D412yes—
149C4×D191#4112yes—
150C191⋊C8—21no—
151C191⋊C8—21no—
152C23×C191—21no—
153C23×C191—21no—
154Q8×C191—21no—
155Q8×C191—21no—
156D4×C191—21no—
157D4×C191—21no—
158D4×C191—21no—
159D4×C191C2×D412yes—
160C2×C4×C191—21no—
161C2×C4×C191—21no—
162C2×C4×C191—81no—
163C2×C4×C191C2×D411yescharacteristic
164C2×C4×C191C2×D411yescharacteristic
165C2×C4×C191C2411yescharacteristic
166C8×C191—21no—
167C8×C191—21no—
168C22×C191—22no—
169C22×C191—41no—
170C22×C191—41no—
171C22×C191—41no—
172C22×C191—41no—
173C22×C191—41no—
174C22×C191C22×D411yescharacteristic
175C2×D191—21no—
176C2×D191—21no—
177C2×D191—41no—
178C2×D191—41no—
179C2×D191—41no—
180C2×D191—41no—
181C2×D191—41no—
182C2×D191—41no—
183C2×D191—84no—
184C2×D191#12112yes—
185C4×C191—21no—
186C4×C191—21no—
187C4×C191—41no—
188C4×C191—41no—
189C4×C191C22×D411yescharacteristic
190C4×C191C2×#4111yescharacteristic
191Dic191—22no—
192Dic191—21no—
193Dic191—21no—
194Dic191—41no—
195Dic191#12112yes—
196C2×C191—42no—
197C2×C191—81no—
198C2×C191C2×#12111yescharacteristic
199C2×C191#57611yescharacteristic
200C2×C191#57711yescharacteristic
201D191—22no—
202D191—81no—
203D191—81no—
204C191#304511yescharacteristic, complemented, Sylow
205#3045—1911nomaximal, Sylow
206C2×AGL(1,ℤ/8)—1911no—
207C2×#135—1911no—
208#535—1911no—
209#520—1911no—
210#464—1911no—
211#464—1911no—
212#463—1911no—
213#463D19111yescharacteristic, complemented
214#460—1911no—
215#460—1911no—
216#459—1911no—
217#459—1911no—
218#424—1911no—
219#417—1911no—
220C2×#102—1911no—
221C2×#41—1911no—
222C2×#41—1911no—
223C22×D4—1911no—
224AGL(1,ℤ/8)—3824no—
225#135—3824no—
226C2×Q16—1911no—
227C2×SD16—1911no—
228C2×SD16—1911no—
229C2×SD16—1911no—
230C2×SD16C2×D19111yescharacteristic, complemented
231C2×D8—1911no—
232C2×D8—1911no—
233C2×D8C2×D19111yescharacteristic
234C2×M16—1911no—
235C22×C8—1911no—
236#127—1911no—
237#127—1911no—
238#124—1911no—
239#124C2×D19111yescharacteristic
240#121—3821no—
241#121—3821no—
242#120—3821no—
243#120—3821no—
244C4×Q8—1911no—
245C4×Q8—1911no—
246C4×D4—3821no—
247C4×D4—3821no—
248C4×D4—1911no—
249C4×D4C2×D19111yescharacteristic, complemented
250#117—3821no—
251#117—1911no—
252C2×C4⋊C4—3821no—
253C2×C4⋊C4—1911no—
254C4⋊C8—1911no—
255C4⋊C8—1911no—
256C4⋊C8—1911no—
257C4⋊C8C2×D19111yescharacteristic, complemented
258#103—1911no—
259#103C2×D19111yescharacteristic, complemented
260#102—1911no—
261#102—1911no—
262#102—1911no—
263#102—1911no—
264#102—1911no—
265#102C2×D19111yescharacteristic
266C24—3821no—
267#41—3822no—
268#41—3822no—
269#41—3822no—
270#41—1914no—
271C2×Q8—1911no—
272C2×Q8C22×D19111yescharacteristic
273C2×D4—3821no—
274C2×D4—3821no—
275C2×D4—3822no—
276C2×D4—3822no—
277C2×D4—3821no—
278C2×D4—3821no—
279C2×D4—3821no—
280C2×D4—3821no—
281C2×D4—3821no—
282C2×D4—1911no—
283C2×D4—1911no—
284C2×D4C22×D19111yescharacteristic
285C2×D4C22×D19111yescharacteristic
286C22×C4—21no—
287C22×C4—3821no—
288C22×C4—3821no—
289C22×C4—3821no—
290C22×C4—3821no—
291C22×C4—1911no—
292Q16—3822no—
293SD16—22no—
294SD16—3822no—
295SD16—3822no—
296SD16—3822no—
297D8—22no—
298D8—3822no—
299D8—3822no—
300M16—3822no—
301C2×C8—3822no—
302C2×C8—1911no—
303C2×C8—1911no—
304C2×C8C22×D19111yescharacteristic
305C2×C8C22×D19111yescharacteristic
306C4⋊C4—3821no—
307C4⋊C4—3821no—
308C4⋊C4—3821no—
309C4⋊C4—1911no—
310C4⋊C4—1911no—
311C4⋊C4—1911no—
312C4⋊C4C22×D19111yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—3821no—
317C22⋊C4—3821no—
318C22⋊C4—3821no—
319C22⋊C4—1911no—
320C22⋊C4—1911no—
321C42—3821no—
322C42—3821no—
323C42—1911no—
324C42C22×D19111yescharacteristic
325C23—21no—
326C23—21no—
327C23—3821no—
328C23—3821no—
329C23—1911no—
330C23—15281no—
331C23—15281no—
332C23—7642no—
333Q8—21no—
334Q8—21no—
335Q8—3821no—
336Q8—1912no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—3824no—
341D4—3821no—
342D4—3821no—
343D4—3821no—
344D4—3821no—
345D4—15281no—
346D4—15281no—
347D4—15281no—
348D4—15281no—
349D4—7642no—
350D4—7642no—
351D4D4×D19112yes—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—81no—
355C2×C4—3821no—
356C2×C4—3824no—
357C2×C4—3821no—
358C2×C4—1911no—
359C2×C4—1911no—
360C2×C4—1911no—
361C2×C4—1912no—
362C2×C4—1912no—
363C2×C4—15281no—
364C2×C4—15281no—
365C2×C4—7642no—
366C2×C4D4×D19111yescharacteristic
367C2×C4D4×D19111yescharacteristic
368C2×C4C23×D19111yescharacteristic
369C8—21no—
370C8—21no—
371C8—3821no—
372C8—3821no—
373C22—22no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—3821no—
380C22—3821no—
381C22—1912no—
382C22—15284no—
383C22—7641no—
384C22—7641no—
385C22—7641no—
386C22—7641no—
387C22—7641no—
388C22—7641no—
389C22C2×D4×D19111yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—3821no—
395C4—3821no—
396C4—3822no—
397C4—1912no—
398C4—7641no—
399C4C2×D4×D19111yescharacteristic
400C4#41×D19111yescharacteristic
401C2—42no—
402C2—81no—
403C2—3822no—
404C2—15281no—
405C2—15281no—
406C2#121×D19111yescharacteristic
407C2#2713346711yescharacteristic
408C2#2713347011yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#463×C191C2
socleC22×C191C22×D4
hypercenter#463D191
hypocenterC191#3045
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×#20259346C2semidirect
2. C2×#20259343C2semidirect
3. #27133231C2semidirect
4. #27133142C2semidirect
5. #27132888C2semidirect
6. #27132882C2semidirect
7. #27132880C2semidirect
8. #27132864C2semidirect
9. #27132862C2semidirect
10. #27132858C2semidirect
11. #27132856C2semidirect
12. #27132688C2semidirect
13. #27132663C2semidirect
14. #102×D191C2semidirect
15. #463×C191C2semidirect
21. C2×#15690277C22semidirect
23. C2×#15690276C22semidirect
26. C2×#15690272C22semidirect
36. C4×Dic382C22semidirect
37. #20259287C22semidirect
38. D4×Dic191C22semidirect
45. C4⋊(C191⋊C8)C22semidirect
46. Dic191⋊C8C22semidirect
47. Dic191⋊C8C22semidirect
48. #20259246C22semidirect
50. #20259244C22semidirect
53. #20259242C22semidirect
54. C2×SD16×C191C22semidirect
57. C4×D4×C191C22semidirect
58. C4⋊C8×C191C22semidirect
59. #103×C191C22semidirect
65. #15690286D4semidirect
204. C191#3045semidirect
213. #463D191semidirect
230. C2×SD16C2×D191semidirect
249. C4×D4C2×D191semidirect
257. C4⋊C8C2×D191semidirect
259. #103C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593462
C2×#202593432
#271332312
#271331422
#271328882
#271328822
#271328802
#271328642
#271328622
#271328582
#271328562
#271326882
#271326632
#102×D1912
#463×C1912
#3045191