Finite Group Databasebeta

← #26373883

Subgroups — #26373883

Order 12032.

514 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20255811C211yesmaximal, characteristic, complemented
2#20245668C211yesmaximal, characteristic, complemented
3#20245576C212yesmaximal, complemented
4#20245573C212yesmaximal, complemented
5#20245535C211yesmaximal, characteristic, complemented
6#20245522C211yesmaximal, characteristic, complemented
7#20245489C212yesmaximal, complemented
8C4×#13093965C211yesmaximal, characteristic, complemented
9#20245220C211yesmaximal, characteristic, complemented
10#20245174C212yesmaximal, complemented
11C47×#3334C211yesmaximal, characteristic, complemented
12#15689838—21no—
13C2×#13093965C2211yescharacteristic
14C2×#13093964C2212yes—
15#15689528C2211yescharacteristic
16#15689448—22no—
17#15689414—21no—
18#15689359C2212yes—
19C4×D376C2211yescharacteristic
20#15689161C2212yes—
21#15688950C2212yes—
22#15688950C2212yes—
23#15688950C2212yes—
24#15688948C2212yes—
25#15688948C2211yescharacteristic
26C4⋊(C47⋊C16)C2211yescharacteristic
27#15688924C2212yes—
28#15688924C2211yescharacteristic
29#15688923C2212yes—
30#15688923C2211yescharacteristic
31C22⋊(C47⋊C16)C2212yes—
32C4×C47⋊C16C2211yescharacteristic
33C47×#547—21no—
34C47×#500C2211yescharacteristic
35C47×#480—22no—
36C47×#474—21no—
37C47×#466C2212yes—
38C4×D8×C47C2211yescharacteristic
39C8×D4×C47C2211yescharacteristic
40C2×#107×C47C2212yes—
41C2×D376C2311yescharacteristic
42#13094024—21no—
43#13094010—42no—
44#13093998—22no—
45#13093995—41no—
46C4×D188—21no—
47#13093983—22no—
48#13093982—41no—
49#13093965—22no—
50#13093964—24no—
51C2×C47⋊C16C2312yes—
52C2×C47⋊C16C2311yescharacteristic
53C2×C47⋊C16C2311yescharacteristic
54#13093958—21no—
55#13093958C2312yes—
56#13093958C2311yescharacteristic
57#13093956—22no—
58#13093946—22no—
59#13093946—21no—
60C2×#41×C47—41no—
61C2×D8×C47C2311yescharacteristic
62C22×C8×C47C2312yes—
63C4⋊Q8×C47—21no—
64#123×C47—42no—
65C22⋊Q8×C47—41no—
66#121×C47—22no—
67C4×D4×C47—21no—
68C4×D4×C47D411yescharacteristic
69C2×C4⋊C4×C47—22no—
70#107×C47—22no—
71#107×C47—21no—
72#107×C47C2312yes—
73#107×C47C2311yescharacteristic
74C4⋊C8×C47D411yescharacteristic
75#102×C47—22no—
76#102×C47—21no—
77C22⋊C8×C47D412yes—
78C4×C8×C47C2311yescharacteristic
79#41×C47—84no—
80C2×Q8×C47—41no—
81C2×D4×C47—21no—
82C2×D4×C47—42no—
83C2×D4×C47D811yescharacteristic
84C22×C4×C47—42no—
85C22×C4×C47—41no—
86C22×C4×C47C2×D412yes—
87C2×C22⋊D47—42no—
88C2×D188—21no—
89C2×C4×D47—41no—
90C2×Dic94—41no—
91D8×C47—21no—
92D8×C47—21no—
93C2×C8×C47—22no—
94C2×C8×C47—22no—
95C2×C8×C47C2×D412yes—
96C2×C8×C47C2×D411yescharacteristic
97C2×C8×C47C2411yescharacteristic
98C4⋊C4×C47—21no—
99C4⋊C4×C47—22no—
100C4⋊C4×C47—42no—
101C4⋊C4×C47—41no—
102C4⋊C4×C47D811yescharacteristic
103C22⋊C4×C47—41no—
104C22⋊C4×C47—42no—
105C22⋊C4×C47D812yes—
106C42×C47C2×D411yescharacteristic
107C22⋊Dic47—42no—
108(C2×D47)⋊C4—42no—
109(C2×D47)⋊C4—41no—
110C4⋊Dic47—21no—
111C4⋊Dic47—22no—
112C4⋊Dic47—41no—
113C4×Dic47—42no—
114D376—21no—
115D376—21no—
116C47⋊C16—22no—
117C47⋊C16—21no—
118C47⋊C16#4112yescomplemented
119C23×C47—41no—
120C23×C47C2×D812yes—
121C22×D47—41no—
122Q8×C47—82no—
123D4×C47—22no—
124D4×C47—41no—
125D4×C47—41no—
126D4×C47—84no—
127C2×C4×C47—22no—
128C2×C4×C47—22no—
129C2×C4×C47—41no—
130C2×C4×C47—42no—
131C2×C4×C47—84no—
132C2×C4×C47—82no—
133C2×C4×C47C2×D812yes—
134C2×C4×C47C2×D811yescharacteristic
135C2×C4×C47C2×D811yescharacteristic
136C2×C4×C47C22×D411yescharacteristic
137C22⋊D47—162no—
138C2×Dic47—41no—
139C2×Dic47—42no—
140D188—41no—
141D188—41no—
142C4×D47—82no—
143Dic94—161no—
144C8×C47—22no—
145C8×C47—21no—
146C8×C47C2×#4111yescharacteristic
147C8×C472−1+411yescharacteristic
148#33921—471nomaximal, Sylow
149C22×C47—22no—
150C22×C47—22no—
151C22×C47—22no—
152C22×C47—81no—
153C22×C47—81no—
154C22×C47—81no—
155C22×C47C22×D811yescharacteristic
156C2×D47—81no—
157C2×D47—81no—
158C2×D47—81no—
159C4×C47—22no—
160C4×C47—21no—
161C4×C47—81no—
162C4×C47—82no—
163C4×C47#50912yes—
164C4×C47#54711yescharacteristic
165C4×C47#57811yescharacteristic
166Dic47—81no—
167Dic47—82no—
168#3341—471no—
169#3334D4711yescharacteristic, complemented
170#2254—471no—
171#2234—472no—
172#2233—472no—
173#2226—471no—
174#2224—471no—
175#2218—472no—
176C4×D16—471no—
177#2169—471no—
178#2159—472no—
179C2×C47—42no—
180C2×C47—161no—
181C2×C47#333411yescharacteristic
182C2×C47C2×#50911yescharacteristic
183C2×C47#341611yescharacteristic
184D47—161no—
185#555—941no—
186#547—21no—
187C2×D16—471no—
188C2×M32—472no—
189#500—471no—
190#500C2×D4711yescharacteristic
191#482—942no—
192#480—22no—
193#476—941no—
194#474—21no—
195#466—472no—
196#466C2×D4712yes—
197C4×D8—471no—
198C4×D8C2×D4711yescharacteristic
199C8×D4C2×D4711yescharacteristic
200#427—472no—
201C2×#107C2×D4712yes—
202#367—472no—
203#367—472no—
204#367—472no—
205#366—471no—
206#366—472no—
207C4⋊C16—471no—
208#357—471no—
209#357—472no—
210#357—471no—
211#357—472no—
212C22⋊C16—472no—
213C4×C16—471no—
214C47#3392111yescharacteristic, complemented, Sylow
215C2×#41—41no—
216C2×D8—471no—
217C2×D8C22×D4711yescharacteristic
218C22×C8C22×D4712yes—
219C4⋊Q8—21no—
220C4⋊Q8—941no—
221#124—1882no—
222#123—42no—
223C22⋊Q8—41no—
224#121—22no—
225#121—942no—
226#121—1881no—
227C4×D4—21no—
228C4×D4—941no—
229C4×D4C22⋊D4711yescharacteristic
230#117—942no—
231#117—1881no—
232C2×C4⋊C4—22no—
233D16—942no—
234M32—944no—
235C2×C16—471no—
236C2×C16—471no—
237C2×C16—472no—
238#107—22no—
239#107—21no—
240#107—471no—
241#107—472no—
242#107—941no—
243#107C22×D4711yescharacteristic
244#107C22×D4712yes—
245#106—942no—
246C4⋊C8C22⋊D4711yescharacteristic
247#102—21no—
248#102—22no—
249#102—941no—
250#102—942no—
251C22⋊C8C22⋊D4712yes—
252C4×C8C22×D4711yescharacteristic
253#41—84no—
254C2×Q8—41no—
255C2×Q8—1881no—
256C2×D4—21no—
257C2×D4—42no—
258C2×D4—941no—
259C2×D4—1882no—
260C2×D4#1062533411yescharacteristic
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—1881no—
264C22×C4C2×C22⋊D4712yes—
265D8—21no—
266D8—21no—
267D8—941no—
268D8—941no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8C2×C22⋊D4711yescharacteristic
272C2×C8C2×C22⋊D4712yes—
273C2×C8C23×D4711yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—22no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—941no—
279C4⋊C4—942no—
280C4⋊C4—1881no—
281C4⋊C4#1062533411yescharacteristic
282C22⋊C4—41no—
283C22⋊C4—42no—
284C22⋊C4—1882no—
285C22⋊C4—1882no—
286C22⋊C4—1881no—
287C22⋊C4#1062533412yes—
288C42—1882no—
289C42C2×C22⋊D4711yescharacteristic
290C16—472no—
291C16—942no—
292C16—941no—
293C23—41no—
294C23—1881no—
295C23C2×#1062533412yes—
296Q8—82no—
297Q8—7521no—
298D4—22no—
299D4—41no—
300D4—41no—
301D4—84no—
302D4—1881no—
303D4—1881no—
304D4—7522no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—84no—
310C2×C4—82no—
311C2×C4—3762no—
312C2×C4—1881no—
313C2×C4—1882no—
314C2×C4C2×#1062533412yes—
315C2×C4C2×#1062533411yescharacteristic
316C2×C4C2×#1062533411yescharacteristic
317C2×C4C22×C22⋊D4711yescharacteristic
318C8—22no—
319C8—21no—
320C8#41×D4711yescharacteristic
321C8#1309407111yescharacteristic
322C22—22no—
323C22—22no—
324C22—22no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—3761no—
329C22—3761no—
330C22—3761no—
331C22C22×#1062533411yescharacteristic
332C4—21no—
333C4—22no—
334C4—81no—
335C4—82no—
336C4—3762no—
337C4—3761no—
338C4#1568955912yes—
339C4#1568977211yescharacteristic
340C4#1568993811yescharacteristic
341C2—42no—
342C2—161no—
343C2—7521no—
344C2#2025573111yescharacteristic
345C2C2×#1568955911yescharacteristic
346C2#2025653211yescharacteristic
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#3334C2
socleC22×C47C22×D8
hypercenter#3334D47
hypocenterC47#33921
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3334

Complements

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

NH ≅ G/Ntype
1. #20255811C2semidirect
2. #20245668C2semidirect
3. #20245576C2semidirect
4. #20245573C2semidirect
5. #20245535C2semidirect
6. #20245522C2semidirect
7. #20245489C2semidirect
8. C4×#13093965C2semidirect
9. #20245220C2semidirect
10. #20245174C2semidirect
11. C47×#3334C2semidirect
118. C47⋊C16#41semidirect
169. #3334D47semidirect
214. C47#33921semidirect

Maximal subgroups

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

subgroupindex
#202558112
#202456682
#202455762
#202455732
#202455352
#202455222
#202454892
C4×#130939652
#202452202
#202451742
C47×#33342
#3392147