Finite Group Databasebeta

← #26369623

Subgroups — #26369623

Order 12032.

696 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689558C211yesmaximal, characteristic
2#20252264C211yesmaximal, characteristic, complemented
3#20252239C211yesmaximal, characteristic, complemented
4#20245261C212yesmaximal, complemented
5#20245255C212yesmaximal, complemented
6#20245237C212yesmaximal, complemented
7#20245232C212yesmaximal, complemented
8#20245177C211yesmaximal, characteristic, complemented
9C2×#15688925C211yesmaximal, characteristic, complemented
10#20244984C211yesmaximal, characteristic, complemented
11C47×#2138C211yesmaximal, characteristic, complemented
12C22×D376C2211yescharacteristic
13#15689606—21no—
14#15689597—21no—
15#15689558C418yescomplemented
16C2×#13093967C2212yes—
17C2×#13093965C2212yes—
18C2×#13093964C2211yescharacteristic
19#15689234—22no—
20#15689224—22no—
21Q16×Dic47C2212yescomplemented
22D8×Dic47C2212yescomplemented
23#15689205C2212yes—
24#15689205C2212yes—
25C2×#13093958C2211yescharacteristic
26#15689127—21no—
27#15689124—21no—
28#15689078C2211yescharacteristic
29#15688950C2212yescomplemented
30#15688948C2212yescomplemented
31#15688925C2212yes—
32#15688925C2212yes—
33#15688898C2212yescomplemented
34C16×Dic47C2212yescomplemented
35C2×#135×C47C2211yescharacteristic
36C22×C16×C47C2211yescharacteristic
37C47×#427C2211yescharacteristic
38C47×#358C2211yescharacteristic, complemented
39C47×#358C2211yescharacteristic, complemented
40C47×#357C2211yescharacteristic, complemented
41C47×#357C2211yescharacteristic, complemented
42C22×D188—21no—
43C2×D376—21no—
44C2×D376—22no—
45C2×D376C2×C414yescomplemented
46C2×D376C2312yes—
47C2×#10625329D411yescharacteristic
48Q8×Dic47—22no—
49C4×D188—41no—
50C4×D188—41no—
51D4×Dic47—22no—
52D4×Dic47—42no—
53#13093986—22no—
54#13093986—22no—
55#13093983—41no—
56#13093983D411yescharacteristic
57C2×C4⋊Dic47—21no—
58C2×C4⋊Dic47—41no—
59C22⋊C4×D47—42no—
60#13093967C2×C418yescomplemented
61#13093965C2×C418yescomplemented
62#13093964C2×C414yescomplemented
63C2×C47⋊C16C2312yes—
64#13093958—21no—
65#13093958C2312yes—
66#13093948—22no—
67#13093945—22no—
68#13093944—22no—
69#13093944—22no—
70#13093934D412yes—
71C8×Dic47C2312yes—
72C2×#41×C47—21no—
73#135×C47—22no—
74#135×C47C2×C414yescomplemented
75C2×Q16×C47C2311yescharacteristic
76C2×SD16×C47—21no—
77C2×D8×C47C2311yescharacteristic
78C22×C8×C47C2311yescharacteristic
79#117×C47—21no—
80C2×C16×C47—22no—
81C2×C16×C47C2311yescharacteristic
82C2×C16×C47C2311yescharacteristic
83#107×C47C2311yescharacteristic
84#107×C47C2311yescharacteristic
85#106×C47—21no—
86C23×D47—41no—
87#41×C47—24no—
88#41×C47—42no—
89C2×Q8×C47—21no—
90C2×D4×C47—21no—
91C2×D4×C47—41no—
92C22×C4×C47—41no—
93C22×C4×C47C2×D411yescharacteristic
94C22×Dic47—41no—
95C22×Dic47—42no—
96C2×D188—24no—
97C2×D188—22no—
98C2×D188—41no—
99C2×D188—42no—
100C2×C4×D47—42no—
101C2×C4×D47—42no—
102Q16×C47—21no—
103Q16×C47C22×C412yes—
104SD16×C47—24no—
105D8×C47—21no—
106D8×C47C22×C412yes—
107C2×C8×C47C22×C412yes—
108C2×C8×C47C2×D411yescharacteristic
109C2×C8×C47#4112yes—
110C2×C8×C47C2411yescharacteristic
111C4⋊C4×C47—21no—
112C4⋊C4×C47—21no—
113C22⋊C4×C47—41no—
114C42×C47—41no—
115C22⋊Dic47—42no—
116C22⋊Dic47D812yes—
117(C2×D47)⋊C4—42no—
118(C2×D47)⋊C4—42no—
119C4⋊Dic47—22no—
120C4⋊Dic47—41no—
121C4⋊Dic47—42no—
122C4⋊Dic47D811yescharacteristic
123C4⋊Dic47D811yescharacteristic
124C4×Dic47—42no—
125C4×Dic47C2×D412yes—
126#10625329—22no—
127C2×C47⋊C8C2×D412yes—
128D376—24no—
129D376—22no—
130D376C22×C414yes—
131C16×C47—21no—
132C16×C47—21no—
133C47⋊C16C22×C414yes—
134C23×C47—41no—
135C23×C47C2×D811yescharacteristic
136C22×D47—42no—
137C22×D47—44no—
138C22×D47—161no—
139C22×D47—161no—
140Q8×C47—22no—
141Q8×C47—41no—
142D4×C47—22no—
143D4×C47—44no—
144D4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—44no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47C4×D412yes—
150C2×C4×C47C2×D811yescharacteristic
151C2×C4×C47#13512yes—
152C2×C4×C47C22×D411yescharacteristic
153C2×Dic47—42no—
154C2×Dic47—162no—
155C2×Dic47—161no—
156C2×Dic47C2×D812yes—
157C2×Dic47C2×D812yes—
158D188—24no—
159D188—42no—
160D188—44no—
161C4×D47—162no—
162C4×D47—162no—
163C8×C47C4×D412yes—
164C8×C47C23×C411yescharacteristic
165C8×C47C2×#4111yescharacteristic
166C47⋊C8—22no—
167#33454—471nomaximal, Sylow
168C22×C47—21no—
169C22×C47—21no—
170C22×C47—42no—
171C22×C47—81no—
172C22×C47—81no—
173C22×C47C4×D812yes—
174C22×C47C22×D811yescharacteristic
175C2×D47—44no—
176C2×D47—82no—
177C2×D47—82no—
178C2×D47—164no—
179C4×C47—42no—
180C4×C47—81no—
181C4×C47—81no—
182C4×C47C4×D812yes—
183C4×C47C2×C4×D411yescharacteristic
184C4×C47C2×#13511yescharacteristic
185Dic47—22no—
186Dic47—22no—
187Dic47—82no—
188C2×#509—471no—
189#2951—471no—
190#2948—471no—
191#2177—472no—
192#2175—472no—
193C4×SD32—472no—
194C4×D16—472no—
195#2159—471no—
196#2138D4711yescharacteristic, complemented
197C2×#357—471no—
198#2108—471no—
199C2×C47—22no—
200C2×C47—82no—
201C2×C47C2×C4×D811yescharacteristic
202C2×C47#341511yescharacteristic
203C2×C47#341611yescharacteristic
204D47—84no—
205C2×#135C2×D4711yescharacteristic
206C22×D8—471no—
207#520—941no—
208#518—941no—
209#509—478no—
210C2×SD32—472no—
211C2×D16—472no—
212C2×M32—471no—
213C22×C16C2×D4711yescharacteristic
214#442—942no—
215#440—942no—
216C4×Q16—472no—
217C4×D8—472no—
218C4×D8—472no—
219C4×D8—472no—
220#427C2×D4711yescharacteristic
221C2×#107—471no—
222#418—941no—
223#417—941no—
224#405—471no—
225#367—472no—
226#366—472no—
227#358C2×D4711yescharacteristic, complemented
228#358C2×D4711yescharacteristic, complemented
229#357—472no—
230#357—472no—
231#357C2×D4711yescharacteristic, complemented
232#357C2×D4711yescharacteristic, complemented
233#346—472no—
234C4×C16—472no—
235C47#3345411yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C22×D4—941no—
238#135—22no—
239#135C4×D4714yescomplemented
240C2×Q16C22×D4711yescharacteristic
241C2×SD16—21no—
242C2×D8—474no—
243C2×D8—472no—
244C2×D8—941no—
245C2×D8—942no—
246C2×D8C22×D4711yescharacteristic
247C2×M16—471no—
248C22×C8C22×D4711yescharacteristic
249C4×Q8—942no—
250C4×D4—942no—
251C4×D4—942no—
252C4×D4—942no—
253C4×D4—1881no—
254C4×D4—1881no—
255C4×D4—1882no—
256#117—21no—
257#117—471no—
258#117—1881no—
259C2×C4⋊C4—941no—
260C2×C4⋊C4—1881no—
261C2×C22⋊C4—1882no—
262SD32—478no—
263D16—478no—
264M32—474no—
265C2×C16—22no—
266C2×C16—472no—
267C2×C16C22×D4711yescharacteristic
268C2×C16C22×D4711yescharacteristic
269#107—472no—
270#107—941no—
271#107C22×D4711yescharacteristic
272#107C22×D4711yescharacteristic
273#106—21no—
274#103—942no—
275#102—942no—
276#102—942no—
277#102—942no—
278#97—472no—
279C4×C8—472no—
280C24—1881no—
281#41—24no—
282#41—42no—
283C2×Q8—21no—
284C2×D4—21no—
285C2×D4—41no—
286C2×D4—944no—
287C2×D4—942no—
288C2×D4—1881no—
289C2×D4—1882no—
290C22×C4—41no—
291C22×C4—1882no—
292C22×C4—1882no—
293C22×C4—1882no—
294C22×C4—1881no—
295C22×C4D4×D4711yescharacteristic
296Q16—21no—
297Q16C2×C4×D4712yes—
298SD16—24no—
299D8—21no—
300D8—474no—
301D8—944no—
302D8—942no—
303D8C2×C4×D4712yes—
304M16—942no—
305C2×C8—472no—
306C2×C8C2×C4×D4712yes—
307C2×C8D4×D4711yescharacteristic
308C2×C8#1062535212yes—
309C2×C8C23×D4711yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—471no—
313C4⋊C4—471no—
314C4⋊C4—942no—
315C4⋊C4—1882no—
316C4⋊C4—1881no—
317C22⋊C4—41no—
318C22⋊C4—472no—
319C22⋊C4—1882no—
320C22⋊C4—1882no—
321C22⋊C4—1882no—
322C42—41no—
323C42—472no—
324C42—1882no—
325C16—21no—
326C16—21no—
327C16—474no—
328C23—41no—
329C23—1882no—
330C23—1884no—
331C23—7521no—
332C23—7521no—
333C23D8×D4711yescharacteristic
334Q8—22no—
335Q8—41no—
336D4—22no—
337D4—44no—
338D4—41no—
339D4—944no—
340D4—1882no—
341D4—1884no—
342C2×C4—41no—
343C2×C4—44no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—472no—
347C2×C4—472no—
348C2×C4—1882no—
349C2×C4—7522no—
350C2×C4—7522no—
351C2×C4—7522no—
352C2×C4—7521no—
353C2×C4(C22⋊D47)⋊C412yes—
354C2×C4D8×D4711yescharacteristic
355C2×C4#1309404512yes—
356C2×C4C2×D4×D4711yescharacteristic
357C8—942no—
358C8(C22⋊D47)⋊C412yes—
359C8C22×C4×D4711yescharacteristic
360C8C2×#1062535211yescharacteristic
361C22—21no—
362C22—21no—
363C22—42no—
364C22—81no—
365C22—81no—
366C22—3762no—
367C22—3762no—
368C22—1884no—
369C22—7524no—
370C22#1568920312yes—
371C22C2×D8×D4711yescharacteristic
372C4—42no—
373C4—81no—
374C4—81no—
375C4—942no—
376C4—942no—
377C4—3762no—
378C4#1568920312yes—
379C4C2×(C22⋊D47)⋊C411yescharacteristic
380C4C2×#1309404511yescharacteristic
381C2—22no—
382C2—82no—
383C2—3764no—
384C2C2×#1568920311yescharacteristic
385C2#2025652611yescharacteristic
386C2#2025653111yescharacteristic
subgroupG/H
centerC22C2×D8×D47
commutatorC8×C47C23×C4
FrattiniC2×C8C23×D47
FittingC47×#2138C2
socleC22×C47C22×D8
prime radicalC2×#15689558C2
cosocleC2×C8×C47C24
hypercenter#2138D47
hypocenterC47#33454
LCSG ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
2. #20252264C2semidirect
3. #20252239C2semidirect
4. #20245261C2semidirect
5. #20245255C2semidirect
6. #20245237C2semidirect
7. #20245232C2semidirect
8. #20245177C2semidirect
9. C2×#15688925C2semidirect
10. #20244984C2semidirect
11. C47×#2138C2semidirect
15. #15689558C4semidirect
21. Q16×Dic47C22semidirect
22. D8×Dic47C22semidirect
29. #15688950C22semidirect
30. #15688948C22semidirect
33. #15688898C22semidirect
34. C16×Dic47C22semidirect
38. C47×#358C22semidirect
39. C47×#358C22semidirect
40. C47×#357C22semidirect
41. C47×#357C22semidirect
45. C2×D376C2×C4semidirect
60. #13093967C2×C4semidirect
61. #13093965C2×C4semidirect
62. #13093964C2×C4semidirect
74. #135×C47C2×C4semidirect
196. #2138D47semidirect
227. #358C2×D47semidirect
228. #358C2×D47semidirect
231. #357C2×D47semidirect
232. #357C2×D47semidirect
235. C47#33454semidirect
239. #135C4×D47semidirect

Maximal subgroups

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

subgroupindex
C2×#156895582
#202522642
#202522392
#202452612
#202452552
#202452372
#202452322
#202451772
C2×#156889252
#202449842
C47×#21382
#3345447