Finite Group Databasebeta

← #26371626

Subgroups — #26371626

Order 12032.

570 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
1#20253882C211yesmaximal, characteristic
2#20245635C212yesmaximal, complemented
3#20245627C212yesmaximal, complemented
4#20245622C212yesmaximal, complemented
5#20245617C212yesmaximal, complemented
6#20245107C211yesmaximal, characteristic, complemented
7#20245098C211yesmaximal, characteristic
8#20245093C211yesmaximal, characteristic
9C2×#15688926C211yesmaximal, characteristic, complemented
10#20244985C211yesmaximal, characteristic, complemented
11C47×#3164C211yesmaximal, characteristic, complemented
12C2×#13094057—21no—
13C2×#13094046C2211yescharacteristic
14C22×Dic188C2211yescharacteristic
15#15689699—21no—
16C2×#13093964C2211yescharacteristic
17C22×C47⋊C16C2211yescharacteristic
18#15689517—22no—
19#15689507C2212yes—
20#15689503C2212yes—
21#15688926C2212yes—
22#15688926C2212yes—
23#15688926C2211yescharacteristic
24#15688926C2211yescharacteristic
25#15688926C2212yes—
26#15688926C2212yes—
27#15688926C2212yes—
28#15688923C2211yescharacteristic
29#15688923C2211yescharacteristic
30#15688923C2212yes—
31#15688899C2212yes—
32C4×C47⋊C16C2212yes—
33C47×#544—21no—
34C47×#501—22no—
35C47×#500C2212yes—
36C47×#499C2212yes—
37C47×#428—21no—
38C47×#427C2211yescharacteristic
39C2×#107×C47C2211yescharacteristic
40C47×#405C2211yescharacteristic
41C2×#10625350—21no—
42C22×Dic94—21no—
43#13094057—44no—
44#13094046—22no—
45#13094046—22no—
46C2×Dic188—22no—
47C2×Dic188—22no—
48C2×Dic188—22no—
49C2×Dic188—21no—
50C2×Dic188C2311yescharacteristic
51C2×Dic188C2312yes—
52C2×#10625325—22no—
53C2×#10625325—21no—
54C2×D376C2311yescharacteristic
55#13094024—22no—
56#13094012—22no—
57#13094008—42no—
58#13094004—41no—
59#13094004—41no—
60#13093964—22no—
61C2×C47⋊C16—22no—
62C2×C47⋊C16C2311yescharacteristic
63C2×C47⋊C16C2311yescharacteristic
64C2×C47⋊C16C2312yes—
65C2×M16×C47D411yescharacteristic
66C22×C8×C47C2311yescharacteristic
67C4⋊Q8×C47—22no—
68C4⋊Q8×C47—41no—
69#125×C47—22no—
70#125×C47—41no—
71C22⋊Q8×C47—42no—
72#117×C47—21no—
73#117×C47D411yescharacteristic
74C2×C4⋊C4×C47—21no—
75#107×C47—22no—
76#107×C47—21no—
77#107×C47C2311yescharacteristic
78#107×C47C2312yes—
79#107×C47C2311yescharacteristic
80#106×C47—22no—
81#106×C47—21no—
82#97×C47D412yes—
83C4×C8×C47C2312yes—
84C2×Q8×C47—42no—
85C22×C4×C47—41no—
86C22×C4×C47C2×D411yescharacteristic
87C2×C22⋊D47—41no—
88C22×Dic47—41no—
89#10625350—42no—
90#10625350—42no—
91C2×D188—21no—
92C2×C4×D47—41no—
93C2×Dic94—22no—
94C2×Dic94—21no—
95C2×Dic94—42no—
96C2×Dic94—42no—
97C2×Dic94—41no—
98M16×C47SD1614yes—
99C2×C8×C47C2×D411yescharacteristic
100C2×C8×C47C2×D412yes—
101C2×C8×C47#4112yes—
102C2×C8×C47#4112yes—
103C2×C8×C47C2411yescharacteristic
104C4⋊C4×C47—21no—
105C4⋊C4×C47—22no—
106C4⋊C4×C47—21no—
107C4⋊C4×C47—41no—
108C4⋊C4×C47—42no—
109C4⋊C4×C47—42no—
110C4⋊C4×C47—42no—
111C4⋊C4×C47D811yescharacteristic
112C4⋊C4×C47D811yescharacteristic
113C22⋊C4×C47—41no—
114C22⋊C4×C47D812yes—
115C42×C47—41no—
116C42×C47C2×D412yes—
117(C2×D47)⋊C4—42no—
118(C2×D47)⋊C4—42no—
119C4⋊Dic47—42no—
120C4⋊Dic47—42no—
121Dic188—22no—
122Dic188—22no—
123Dic188—21no—
124Dic188—21no—
125Dic188—42no—
126Dic188—44no—
127D376—21no—
128D376—21no—
129#10625325—42no—
130#10625325—44no—
131C47⋊C16—21no—
132C47⋊C16—21no—
133C47⋊C16—22no—
134C23×C47C2×D811yescharacteristic
135C22×D47—41no—
136Q8×C47—162no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—42no—
140C2×C4×C47—161no—
141C2×C4×C47#12412yes—
142C2×C4×C47C2×D812yes—
143C2×C4×C47C2×D812yes—
144C2×C4×C47C2×D811yescharacteristic
145C2×C4×C47C2×SD1612yes—
146C2×C4×C47C22×D411yescharacteristic
147C22⋊D47—82no—
148C2×Dic47—41no—
149C2×Dic47—42no—
150C2×Dic47—82no—
151D188—41no—
152D188—41no—
153C4×D47—82no—
154Dic94—41no—
155Dic94—42no—
156Dic94—42no—
157Dic94—41no—
158Dic94—82no—
159C8×C47#12412yes—
160C8×C47C2×SD1614yes—
161C8×C47C2×#4111yescharacteristic
162C8×C47C2×#4111yescharacteristic
163#33703—471nomaximal, Sylow
164C22×C47—21no—
165C22×C47—21no—
166C22×C47#48612yes—
167C22×C47C22×D811yescharacteristic
168C2×D47—81no—
169C2×D47—81no—
170C2×D47—81no—
171C4×C47—22no—
172C4×C47—22no—
173C4×C47—82no—
174C4×C47—81no—
175C4×C47—81no—
176C4×C47#48612yes—
177C4×C47C2×#12411yescharacteristic
178C4×C47C22×SD1611yescharacteristic
179Dic47—82no—
180Dic47—81no—
181#3164D4711yescharacteristic, complemented
182#3156—471no—
183#2245—472no—
184#2243—472no—
185#2242—472no—
186#2241—472no—
187#2141—471no—
188#2139—471no—
189#2138—471no—
190C2×#358—471no—
191#2108—471no—
192C2×C47—22no—
193C2×C47C2×#48611yescharacteristic
194C2×C47#341611yescharacteristic
195C2×C47#341711yescharacteristic
196D47—161no—
197C2×#137—941no—
198C2×#135—471no—
199C22×Q16—471no—
200#544—21no—
201#536—941no—
202C2×M32—471no—
203C22×C16—471no—
204#501—22no—
205#500C2×D4712yes—
206#499C2×D4712yes—
207#497—942no—
208#495—472no—
209#494—472no—
210#428—21no—
211#427C2×D4711yescharacteristic
212C2×#107C2×D4711yescharacteristic
213#405C2×D4711yescharacteristic
214#358—472no—
215#358—472no—
216#358—471no—
217#358—471no—
218#358—472no—
219#358—472no—
220#358—472no—
221#357—471no—
222#357—471no—
223#357—472no—
224#346—472no—
225C4×C16—472no—
226C47#3370311yescharacteristic, complemented, Sylow
227C2×#41—941no—
228C22×Q8—941no—
229#137—1884no—
230#135—942no—
231#135—942no—
232C2×Q16—472no—
233C2×Q16—471no—
234C2×Q16—942no—
235C2×Q16—942no—
236C2×Q16—941no—
237C2×Q16—942no—
238C2×SD16—941no—
239C2×SD16—942no—
240C2×D8—471no—
241C2×M16C22⋊D4711yescharacteristic
242C22×C8C22×D4711yescharacteristic
243C4⋊Q8—22no—
244C4⋊Q8—41no—
245C4⋊Q8—942no—
246#125—22no—
247#125—41no—
248#124—942no—
249#123—1882no—
250C22⋊Q8—42no—
251C22⋊Q8—1881no—
252C22⋊Q8—1881no—
253#117—21no—
254#117C22⋊D4711yescharacteristic
255C2×C4⋊C4—21no—
256M32—942no—
257C2×C16—471no—
258C2×C16—472no—
259C2×C16—471no—
260C2×C16—942no—
261#107—21no—
262#107—22no—
263#107C22×D4712yes—
264#107C22×D4711yescharacteristic
265#107C22×D4711yescharacteristic
266#106—21no—
267#106—22no—
268#97C22⋊D4712yes—
269C4×C8C22×D4712yes—
270#41—1882no—
271#41—1882no—
272C2×Q8—42no—
273C2×Q8—942no—
274C2×Q8—941no—
275C2×Q8—1882no—
276C2×Q8—1882no—
277C2×Q8—1881no—
278C2×D4—941no—
279C2×D4—1881no—
280C22×C4—41no—
281C22×C4—1881no—
282C22×C4—1881no—
283C22×C4C2×C22⋊D4711yescharacteristic
284Q16—942no—
285Q16—942no—
286Q16—941no—
287Q16—941no—
288Q16—1882no—
289Q16—1884no—
290SD16—1882no—
291SD16—1884no—
292D8—941no—
293D8—941no—
294M16#1062533614yes—
295C2×C8#1062535412yes—
296C2×C8#1062535412yes—
297C2×C8C2×C22⋊D4711yescharacteristic
298C2×C8C2×C22⋊D4712yes—
299C2×C8C23×D4711yescharacteristic
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—1882no—
308C4⋊C4—1882no—
309C4⋊C4#1062533411yescharacteristic
310C4⋊C4#1062533411yescharacteristic
311C22⋊C4—41no—
312C22⋊C4—1882no—
313C22⋊C4—1882no—
314C22⋊C4#1062533412yes—
315C42—41no—
316C42C2×C22⋊D4712yes—
317C16—941no—
318C16—941no—
319C16—942no—
320C23—1881no—
321C23C2×#1062533411yescharacteristic
322Q8—162no—
323Q8—3762no—
324Q8—1882no—
325Q8—1881no—
326Q8—1881no—
327Q8—1882no—
328D4—3762no—
329D4—1881no—
330D4—1881no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—161no—
335C2×C4—3762no—
336C2×C4—3762no—
337C2×C4—1882no—
338C2×C4—1881no—
339C2×C4#1309401312yes—
340C2×C4C2×#1062533412yes—
341C2×C4C2×#1062533412yes—
342C2×C4C2×#1062533411yescharacteristic
343C2×C4C2×#1062533612yes—
344C2×C4C22×C22⋊D4711yescharacteristic
345C8#1309401312yes—
346C8C2×#1062533614yes—
347C8C2×#1062535411yescharacteristic
348C8C2×#1062535411yescharacteristic
349C22—21no—
350C22—21no—
351C22—3761no—
352C22—3761no—
353C22—3761no—
354C22#1568947112yes—
355C22C22×#1062533411yescharacteristic
356C4—22no—
357C4—22no—
358C4—82no—
359C4—81no—
360C4—81no—
361C4—3761no—
362C4—3762no—
363C4#1568947112yes—
364C4C2×#1309401311yescharacteristic
365C4C22×#1062533611yescharacteristic
366C2—22no—
367C2—7521no—
368C2C2×#1568947111yescharacteristic
369C2#2025653511yescharacteristic
370C2#2025653911yescharacteristic
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#3164C2
socleC22×C47C22×D8
prime radicalC2×#13094046C22
hypercenter#3164D47
hypocenterC47#33703
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3164

Complements

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

NH ≅ G/Ntype
2. #20245635C2semidirect
3. #20245627C2semidirect
4. #20245622C2semidirect
5. #20245617C2semidirect
6. #20245107C2semidirect
9. C2×#15688926C2semidirect
10. #20244985C2semidirect
11. C47×#3164C2semidirect
181. #3164D47semidirect
226. C47#33703semidirect

Maximal subgroups

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

subgroupindex
#202538822
#202456352
#202456272
#202456222
#202456172
#202451072
#202450982
#202450932
C2×#156889262
#202449852
C47×#31642
#3370347