Finite Group Databasebeta

← #26198788

Subgroups — #26198788

Order 12032.

770 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253384C212yesmaximal, complemented
2C2×#15689202C211yesmaximal, characteristic, complemented
3#20242874C212yesmaximal, complemented
4#20242864C212yesmaximal, complemented
5#20242850C214yesmaximal, complemented
6#20242136C211yesmaximal, characteristic, complemented
7#20242037C212yesmaximal, complemented
8C47×#3147C211yesmaximal, characteristic, complemented
9C22×#10625324—21no—
10C2×C4×D188C2211yescharacteristic
11#15689434—22no—
12#15689400C2214yes—
13#15689315C2214yes—
14#15689256—22no—
15C4×#10625334—24no—
16#15689202—21no—
17#15689202—21no—
18#15689202C2212yes—
19#15689202C2212yes—
20#15689176C2212yes—
21C2×C4⋊(C47⋊C8)C2211yescharacteristic
22C2×(C2×D47)⋊C8—21no—
23#15688876C2212yescomplemented
24#15688874C2212yescomplemented
25#15688869C2214yescomplemented
26#15688867C2214yescomplemented
27C2×#135×C47—21no—
28C2×#135×C47—21no—
29C47×#532C2212yes—
30C47×#495—21no—
31C47×#495—21no—
32C47×#494C2211yescharacteristic, complemented
33C47×#493C2211yescharacteristic, complemented
34C47×#492C2212yescomplemented
35C2×C4×C8×C47C2211yescharacteristic
36C22×D188D411yescharacteristic
37C22×C4×D47—21no—
38C2×#10625334—24no—
39C2×#10625324—22no—
40C2×#10625324—22no—
41C2×#10625324—44no—
42C22×C47⋊C8—21no—
43C4×D188—21no—
44C4×D188—21no—
45C4×D188D412yes—
46C4×D188C2312yes—
47C2×C4⋊Dic47D411yescharacteristic
48C2×(C2×D47)⋊C4—21no—
49#13093957—24no—
50C4⋊(C47⋊C8)—21no—
51C4⋊(C47⋊C8)C2314yes—
52C4⋊(C47⋊C8)C2312yes—
53#13093945—24no—
54#13093944—24no—
55C22⋊(C47⋊C8)—22no—
56(C2×D47)⋊C8—22no—
57(C2×D47)⋊C8—22no—
58C4×C47⋊C8—22no—
59C2×#41×C47—22no—
60#135×C47—44no—
61#135×C47—44no—
62C2×Q16×C47—21no—
63C2×Q16×C47—21no—
64C2×SD16×C47—22no—
65C2×SD16×C47—22no—
66C2×D8×C47—21no—
67C2×D8×C47—21no—
68C22×C8×C47D411yescharacteristic
69C22×C8×C47D411yescharacteristic
70C4⋊Q8×C47C2312yes—
71#127×C47C2312yes—
72#124×C47—22no—
73#121×C47—22no—
74#121×C47—22no—
75C4×D4×C47—22no—
76C4×D4×C47—22no—
77C2×C42×C47C2311yescharacteristic
78C4×C8×C47D411yescharacteristic
79C4×C8×C47D411yescharacteristic
80C4×C8×C47C2311yescharacteristic
81C4×C8×C47C2311yescharacteristic
82C23×D47—21no—
83#41×C47—44no—
84#41×C47—44no—
85C2×Q8×C47—22no—
86C2×D4×C47—22no—
87C2×D4×C47—22no—
88C2×D4×C47—22no—
89C2×D4×C47—42no—
90C22×C4×C47—42no—
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47#4111yescharacteristic
94C22×Dic47—21no—
95C2×D188—21no—
96C2×D188—42no—
97C2×D188C2×D412yes—
98C2×C4×D47—22no—
99C2×C4×D47—22no—
100C2×C4×D47—42no—
101C2×C4×D47—42no—
102Q16×C47—42no—
103Q16×C47—42no—
104SD16×C47—44no—
105SD16×C47—44no—
106D8×C47—42no—
107D8×C47—42no—
108C2×C8×C47—22no—
109C2×C8×C47D814yescomplemented
110C2×C8×C47C2×D411yescharacteristic
111C2×C8×C47C2×D411yescharacteristic
112C2×C8×C47C2×D411yescharacteristic
113C2×C8×C47C2×D411yescharacteristic
114C4⋊C4×C47—22no—
115C4⋊C4×C47—22no—
116C22⋊C4×C47—42no—
117C22⋊C4×C47—42no—
118C42×C47C2×D411yescharacteristic
119C42×C47C2×D411yescharacteristic
120C42×C47#4111yescharacteristic
121C42×C47C2411yescharacteristic
122#10625334—48no—
123(C2×D47)⋊C4—22no—
124(C2×D47)⋊C4—22no—
125C4⋊Dic47—21no—
126C4⋊Dic47C2×D412yes—
127C2×C47⋊C8—24no—
128C2×C47⋊C8—22no—
129C2×C47⋊C8—42no—
130#10625324—44no—
131#10625324—44no—
132C23×C47—42no—
133C23×C47#12111yescharacteristic
134C22×D47—22no—
135C22×D47—42no—
136C22×D47—42no—
137C22×D47—42no—
138Q8×C47—42no—
139Q8×C47—42no—
140D4×C47—42no—
141D4×C47—42no—
142D4×C47—44no—
143D4×C47—82no—
144D4×C47—84no—
145C2×C4×C47—22no—
146C2×C4×C47—41no—
147C2×C4×C47—42no—
148C2×C4×C47—84no—
149C2×C4×C47#12111yescharacteristic
150C2×C4×C47#12111yescharacteristic
151C2×C4×C47#12111yescharacteristic
152C2×C4×C47C2×D812yes—
153C2×C4×C47AGL(1,ℤ/8)12yes—
154C2×C4×C47C22×D411yescharacteristic
155C2×C4×C47C22×D411yescharacteristic
156C2×C4×C47C2×#4111yescharacteristic
157C2×Dic47—22no—
158C2×Dic47—42no—
159D188—24no—
160D188—42no—
161C4×D47—42no—
162C4×D47—42no—
163C4×D47—44no—
164C8×C47—21no—
165C8×C47—21no—
166C8×C47C2×D812yes—
167C8×C47C2×D812yes—
168C47⋊C8—42no—
169C47⋊C8—44no—
170#19133—471nomaximal, Sylow
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—82no—
174C22×C47—82no—
175C22×C47—82no—
176C22×C47#45912yes—
177C22×C47C2×#12111yescharacteristic
178C2×D47—42no—
179C2×D47—42no—
180C2×D47—42no—
181C2×D47—84no—
182C4×C47—21no—
183C4×C47—21no—
184C4×C47—21no—
185C4×C47—82no—
186C4×C47#45912yes—
187C4×C47C22×D811yescharacteristic
188C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
189C4×C47#57512yes—
190Dic47—42no—
191#3147D4711yescharacteristic, complemented
192#3100—472no—
193C2×#436—471no—
194#1689—472no—
195#1687—472no—
196#1685—474no—
197#1564—471no—
198#1540—472no—
199C2×C47—22no—
200C2×C47—162no—
201C2×C47C2×#45911yescharacteristic
202C2×C47#305411yescharacteristic
203C2×C47#306711yescharacteristic
204D47—82no—
205C2×#135—21no—
206C2×#135—21no—
207C22×M16—941no—
208#532C2×D4712yes—
209C2×C4×D4—471no—
210#495—21no—
211#495—21no—
212#494C2×D4711yescharacteristic, complemented
213#493C2×D4711yescharacteristic, complemented
214#492C2×D4712yescomplemented
215#480—942no—
216#474—474no—
217#459—474no—
218#447—942no—
219C4×D8—944no—
220#436—472no—
221#436—472no—
222#436—941no—
223#436—941no—
224#431—472no—
225C2×C4⋊C8—471no—
226C2×C22⋊C8—941no—
227C2×C4×C8C2×D4711yescharacteristic
228#335—472no—
229#334—472no—
230#332—474no—
231#331—474no—
232C47#1913311yescharacteristic, complemented, Sylow
233C2×#41—22no—
234C22×D4—471no—
235C23×C4—941no—
236#135—44no—
237#135—44no—
238C2×Q16—21no—
239C2×Q16—21no—
240C2×SD16—22no—
241C2×SD16—22no—
242C2×D8—21no—
243C2×D8—21no—
244C2×D8—944no—
245C2×M16—942no—
246C2×M16—942no—
247C2×M16—1884no—
248C22×C8—941no—
249C22×C8C22⋊D4711yescharacteristic
250C22×C8C22⋊D4711yescharacteristic
251C4⋊Q8C22×D4712yes—
252#127C22×D4712yes—
253#124—22no—
254#121—22no—
255#121—22no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—472no—
259C4×D4—472no—
260C4×D4—941no—
261C4×D4—941no—
262C2×C4⋊C4—471no—
263C2×C22⋊C4—941no—
264C2×C42C22×D4711yescharacteristic
265#107—944no—
266C4⋊C8—472no—
267C4⋊C8—474no—
268C4⋊C8—941no—
269#102—944no—
270#102—944no—
271C22⋊C8—942no—
272C22⋊C8—942no—
273C22⋊C8—942no—
274C4×C8—942no—
275C4×C8C22⋊D4711yescharacteristic
276C4×C8C22⋊D4711yescharacteristic
277C4×C8C22×D4711yescharacteristic
278C4×C8C22×D4711yescharacteristic
279C24—941no—
280#41—44no—
281#41—44no—
282C2×Q8—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—42no—
287C2×D4—472no—
288C2×D4—941no—
289C2×D4—1882no—
290C22×C4—42no—
291C22×C4—941no—
292C22×C4—942no—
293C22×C4—942no—
294C22×C4—1882no—
295C22×C4—1882no—
296C22×C4D4×D4711yescharacteristic
297C22×C4#1062535211yescharacteristic
298C22×C4C2×C22⋊D4711yescharacteristic
299Q16—42no—
300Q16—42no—
301SD16—44no—
302SD16—44no—
303D8—42no—
304D8—42no—
305D8—1888no—
306M16—1884no—
307M16—1884no—
308C2×C8—22no—
309C2×C8—944no—
310C2×C8—942no—
311C2×C8—1882no—
312C2×C8#1062533414yescomplemented
313C2×C8C2×C22⋊D4711yescharacteristic
314C2×C8C2×C22⋊D4711yescharacteristic
315C2×C8C2×C22⋊D4711yescharacteristic
316C2×C8C2×C22⋊D4711yescharacteristic
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—472no—
320C4⋊C4—941no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—942no—
324C22⋊C4—942no—
325C42D4×D4711yescharacteristic
326C42#1062535211yescharacteristic
327C42C2×C22⋊D4711yescharacteristic
328C42C23×D4711yescharacteristic
329C23—42no—
330C23—942no—
331C23—1882no—
332C23—1882no—
333C23—1882no—
334C23#1309399511yescharacteristic
335Q8—42no—
336Q8—42no—
337D4—42no—
338D4—42no—
339D4—44no—
340D4—82no—
341D4—84no—
342D4—944no—
343D4—1882no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—84no—
348C2×C4—942no—
349C2×C4—1882no—
350C2×C4—1884no—
351C2×C4—1882no—
352C2×C4—1882no—
353C2×C4#1309399511yescharacteristic
354C2×C4#1309399511yescharacteristic
355C2×C4#1309399511yescharacteristic
356C2×C4C2×#1062533412yes—
357C2×C4#1309405012yes—
358C2×C4C2×D4×D4711yescharacteristic
359C2×C4C22×C22⋊D4711yescharacteristic
360C2×C4C2×#1062535211yescharacteristic
361C8—21no—
362C8—21no—
363C8—1882no—
364C8—1884no—
365C8C2×#1062533412yes—
366C8C2×#1062533412yes—
367C22—21no—
368C22—21no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22—3764no—
373C22—1882no—
374C22—1882no—
375C22—1882no—
376C22#1568931512yes—
377C22C2×#1309399511yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—82no—
382C4—1882no—
383C4#1568931512yes—
384C4C22×#1062533411yescharacteristic
385C4C2×#1309405011yescharacteristic
386C4#1568992012yes—
387C2—22no—
388C2—162no—
389C2—3762no—
390C2C2×#1568931511yescharacteristic
391C2#2025299311yescharacteristic
392C2#2025310411yescharacteristic
subgroupG/H
centerC2×C4#13093995
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3147C2
socleC22×C47C2×#121
hypercenter#3147D47
hypocenterC47#19133
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3147

Complements

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

NH ≅ G/Ntype
1. #20253384C2semidirect
2. C2×#15689202C2semidirect
3. #20242874C2semidirect
4. #20242864C2semidirect
5. #20242850C2semidirect
6. #20242136C2semidirect
7. #20242037C2semidirect
8. C47×#3147C2semidirect
23. #15688876C22semidirect
24. #15688874C22semidirect
25. #15688869C22semidirect
26. #15688867C22semidirect
32. C47×#494C22semidirect
33. C47×#493C22semidirect
34. C47×#492C22semidirect
109. C2×C8×C47D8semidirect
191. #3147D47semidirect
212. #494C2×D47semidirect
213. #493C2×D47semidirect
214. #492C2×D47semidirect
232. C47#19133semidirect
312. C2×C8#10625334semidirect

Maximal subgroups

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

subgroupindex
#202533842
C2×#156892022
#202428742
#202428642
#202428502
#202421362
#202420372
C47×#31472
#1913347