Finite Group Databasebeta

← #26369634

Subgroups — #26369634

Order 12032.

660 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689563C211yesmaximal, characteristic
2#20252264C211yesmaximal, characteristic
3#20252255C211yesmaximal, characteristic, complemented
4#20245256C212yesmaximal, complemented
5#20245249C212yesmaximal, complemented
6#20245244C212yesmaximal, complemented
7#20245243C212yesmaximal, complemented
8#20245177C211yesmaximal, characteristic, complemented
9C2×#15688928C211yesmaximal, characteristic, complemented
10#20244984C211yesmaximal, characteristic, complemented
11C47×#2138C211yesmaximal, characteristic
12C22×Dic188C2211yescharacteristic
13#15689606—21no—
14#15689600—21no—
15#15689563C418yescomplemented
16C2×#13093970C2212yes—
17C2×#13093968C2212yes—
18C2×#13093964C2211yescharacteristic
19#15689229—22no—
20#15689224—22no—
21Q16×Dic47C2212yes—
22#15689217C2212yes—
23#15689217C2212yes—
24D8×Dic47C2212yes—
25C2×#13093958C2211yescharacteristic
26#15689131—21no—
27#15689124—21no—
28#15689078C2211yescharacteristic
29#15688950C2212yes—
30#15688948C2212yes—
31#15688928C2212yes—
32#15688928C2212yes—
33#15688898C2212yes—
34C16×Dic47C2212yes—
35C2×#135×C47C2211yescharacteristic
36C22×C16×C47C2211yescharacteristic
37C47×#427C2211yescharacteristic
38C47×#358C2211yescharacteristic
39C47×#358C2211yescharacteristic
40C47×#357C2211yescharacteristic
41C47×#357C2211yescharacteristic
42C22×Dic94—21no—
43C2×Dic188—21no—
44C2×Dic188—22no—
45C2×Dic188C2×C414yescomplemented
46C2×Dic188C2312yes—
47C2×#10625329D411yescharacteristic
48Q8×Dic47—22no—
49C4×Dic94—41no—
50C4×Dic94—41no—
51#13093989—22no—
52#13093989—22no—
53D4×Dic47—22no—
54D4×Dic47—42no—
55#13093983—41no—
56#13093983D411yescharacteristic
57#13093980—42no—
58C2×C4⋊Dic47—21no—
59C2×C4⋊Dic47—41no—
60#13093970C2×C418yescomplemented
61#13093968C2×C418yescomplemented
62#13093964C2×C414yescomplemented
63C2×C47⋊C16C2312yes—
64#13093958—21no—
65#13093958C2312yes—
66#13093948—22no—
67#13093947—22no—
68#13093947—22no—
69#13093945—22no—
70#13093934D412yes—
71C8×Dic47C2312yes—
72C2×#41×C47—21no—
73#135×C47—22no—
74#135×C47C2×C414yes—
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—
86#41×C47—24no—
87#41×C47—42no—
88C2×Q8×C47—21no—
89C2×D4×C47—21no—
90C2×D4×C47—41no—
91C22×C4×C47—41no—
92C22×C4×C47C2×D411yescharacteristic
93C22×Dic47—41no—
94C22×Dic47—41no—
95C22×Dic47—42no—
96C2×Dic94—24no—
97C2×Dic94—22no—
98C2×Dic94—41no—
99C2×Dic94—42no—
100Q16×C47—21no—
101Q16×C47C22×C412yes—
102SD16×C47—24no—
103D8×C47—21no—
104D8×C47C22×C412yes—
105C2×C8×C47C22×C412yes—
106C2×C8×C47C2×D411yescharacteristic
107C2×C8×C47#4112yes—
108C2×C8×C47C2411yescharacteristic
109C4⋊C4×C47—21no—
110C4⋊C4×C47—21no—
111C22⋊C4×C47—41no—
112C42×C47—41no—
113C22⋊Dic47—42no—
114C22⋊Dic47D812yes—
115C4⋊Dic47—22no—
116C4⋊Dic47—41no—
117C4⋊Dic47—42no—
118C4⋊Dic47D811yescharacteristic
119C4⋊Dic47D811yescharacteristic
120Dic47⋊C4—42no—
121Dic47⋊C4—42no—
122C4×Dic47—42no—
123C4×Dic47—42no—
124C4×Dic47—42no—
125C4×Dic47C2×D412yes—
126#10625329—22no—
127C2×C47⋊C8C2×D412yes—
128Dic188—24no—
129Dic188—22no—
130Dic188C22×C414yes—
131C16×C47—21no—
132C16×C47—21no—
133C47⋊C16C22×C414yes—
134C23×C47—41no—
135C23×C47C2×D811yescharacteristic
136Q8×C47—22no—
137Q8×C47—41no—
138D4×C47—22no—
139D4×C47—44no—
140D4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—44no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47C4×D412yes—
146C2×C4×C47C2×D811yescharacteristic
147C2×C4×C47#13512yes—
148C2×C4×C47C22×D411yescharacteristic
149C2×Dic47—42no—
150C2×Dic47—44no—
151C2×Dic47—42no—
152C2×Dic47—162no—
153C2×Dic47—161no—
154C2×Dic47C2×D812yes—
155C2×Dic47C2×D812yes—
156Dic94—24no—
157Dic94—42no—
158Dic94—44no—
159C8×C47C4×D412yes—
160C8×C47C23×C411yescharacteristic
161C8×C47C2×#4111yescharacteristic
162C47⋊C8—22no—
163#33455—471nomaximal, Sylow
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—42no—
167C22×C47—81no—
168C22×C47—81no—
169C22×C47C4×D812yes—
170C22×C47C22×D811yescharacteristic
171C4×C47—42no—
172C4×C47—81no—
173C4×C47—81no—
174C4×C47C4×D812yes—
175C4×C47C2×C4×D411yescharacteristic
176C4×C47C2×#13511yescharacteristic
177Dic47—22no—
178Dic47—22no—
179Dic47—44no—
180Dic47—82no—
181C2×#510—471no—
182#2951—471no—
183#2950—471no—
184#2176—472no—
185#2175—472no—
186C4×Q32—472no—
187C4×SD32—472no—
188#2159—471no—
189#2138D4711yescharacteristic
190C2×#358—471no—
191#2108—471no—
192C2×C47—22no—
193C2×C47—82no—
194C2×C47C2×C4×D811yescharacteristic
195C2×C47#341611yescharacteristic
196C2×C47#341711yescharacteristic
197C2×#135C2×D4711yescharacteristic
198C22×Q16—471no—
199#520—941no—
200#519—941no—
201#510—478no—
202C2×Q32—472no—
203C2×SD32—472no—
204C2×M32—471no—
205C22×C16C2×D4711yescharacteristic
206#441—942no—
207#440—942no—
208C4×Q16—472no—
209C4×Q16—472no—
210C4×Q16—472no—
211C4×D8—472no—
212#427C2×D4711yescharacteristic
213C2×#107—471no—
214#419—941no—
215#417—941no—
216#405—471no—
217#367—472no—
218#366—472no—
219#358—472no—
220#358—472no—
221#358C2×D4711yescharacteristic
222#358C2×D4711yescharacteristic
223#357C2×D4711yescharacteristic
224#357C2×D4711yescharacteristic
225#346—472no—
226C4×C16—472no—
227C47#3345511yescharacteristic, complemented, Sylow
228C2×#41—21no—
229C22×Q8—941no—
230#135—22no—
231#135C4×D4714yes—
232C2×Q16—474no—
233C2×Q16—472no—
234C2×Q16—941no—
235C2×Q16—942no—
236C2×Q16C22×D4711yescharacteristic
237C2×SD16—21no—
238C2×D8C22×D4711yescharacteristic
239C2×M16—471no—
240C22×C8C22×D4711yescharacteristic
241C4×Q8—942no—
242C4×Q8—942no—
243C4×Q8—942no—
244C4×Q8—1881no—
245C4×Q8—1881no—
246C4×D4—942no—
247C4×D4—1882no—
248#117—21no—
249#117—471no—
250#117—1882no—
251#117—1881no—
252C2×C4⋊C4—941no—
253C2×C4⋊C4—1881no—
254Q32—478no—
255SD32—478no—
256M32—474no—
257C2×C16—22no—
258C2×C16—472no—
259C2×C16C22×D4711yescharacteristic
260C2×C16C22×D4711yescharacteristic
261#107—472no—
262#107—941no—
263#107C22×D4711yescharacteristic
264#107C22×D4711yescharacteristic
265#106—21no—
266#103—942no—
267#103—942no—
268#103—942no—
269#102—942no—
270#97—472no—
271C4×C8—472no—
272#41—24no—
273#41—42no—
274C2×Q8—21no—
275C2×Q8—944no—
276C2×Q8—942no—
277C2×Q8—1881no—
278C2×Q8—1882no—
279C2×D4—21no—
280C2×D4—41no—
281C22×C4—41no—
282C22×C4—1882no—
283C22×C4—1881no—
284C22×C4—1881no—
285C22×C4D4×D4711yescharacteristic
286Q16—21no—
287Q16—474no—
288Q16—944no—
289Q16—942no—
290Q16C2×C4×D4712yes—
291SD16—24no—
292D8—21no—
293D8C2×C4×D4712yes—
294M16—942no—
295C2×C8—472no—
296C2×C8C2×C4×D4712yes—
297C2×C8D4×D4711yescharacteristic
298C2×C8#1062535212yes—
299C2×C8C23×D4711yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—471no—
303C4⋊C4—471no—
304C4⋊C4—942no—
305C4⋊C4—1882no—
306C4⋊C4—1882no—
307C4⋊C4—1882no—
308C4⋊C4—1881no—
309C22⋊C4—41no—
310C22⋊C4—472no—
311C22⋊C4—1882no—
312C42—41no—
313C42—472no—
314C42—1882no—
315C42—1882no—
316C42—1882no—
317C16—21no—
318C16—21no—
319C16—474no—
320C23—41no—
321C23D8×D4711yescharacteristic
322Q8—22no—
323Q8—41no—
324Q8—944no—
325Q8—1882no—
326Q8—1884no—
327D4—22no—
328D4—44no—
329D4—41no—
330C2×C4—41no—
331C2×C4—44no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—472no—
335C2×C4—472no—
336C2×C4—1882no—
337C2×C4—1882no—
338C2×C4—1884no—
339C2×C4—7522no—
340C2×C4—7521no—
341C2×C4(C22⋊D47)⋊C412yes—
342C2×C4D8×D4711yescharacteristic
343C2×C4#1309404512yes—
344C2×C4C2×D4×D4711yescharacteristic
345C8—942no—
346C8(C22⋊D47)⋊C412yes—
347C8C22×C4×D4711yescharacteristic
348C8C2×#1062535211yescharacteristic
349C22—21no—
350C22—21no—
351C22—42no—
352C22—81no—
353C22—81no—
354C22#1568920312yes—
355C22C2×D8×D4711yescharacteristic
356C4—42no—
357C4—81no—
358C4—81no—
359C4—942no—
360C4—942no—
361C4—3762no—
362C4—1884no—
363C4#1568920312yes—
364C4C2×(C22⋊D47)⋊C411yescharacteristic
365C4C2×#1309404511yescharacteristic
366C2—22no—
367C2—82no—
368C2C2×#1568920311yescharacteristic
369C2#2025653111yescharacteristic
370C2#2025653811yescharacteristic
subgroupG/H
centerC22C2×D8×D47
commutatorC8×C47C23×C4
FrattiniC2×C8C23×D47
FittingC47×#2138C2
socleC22×C47C22×D8
prime radicalC2×#135×C47C22
cosocleC2×C8×C47C24
hypercenter#2138D47
hypocenterC47#33455
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
3. #20252255C2semidirect
4. #20245256C2semidirect
5. #20245249C2semidirect
6. #20245244C2semidirect
7. #20245243C2semidirect
8. #20245177C2semidirect
9. C2×#15688928C2semidirect
10. #20244984C2semidirect
15. #15689563C4semidirect
45. C2×Dic188C2×C4semidirect
60. #13093970C2×C4semidirect
61. #13093968C2×C4semidirect
62. #13093964C2×C4semidirect
227. C47#33455semidirect

Maximal subgroups

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

subgroupindex
C2×#156895632
#202522642
#202522552
#202452562
#202452492
#202452442
#202452432
#202451772
C2×#156889282
#202449842
C47×#21382
#3345547