Finite Group Databasebeta

← #26132898

Subgroups — #26132898

Order 12032.

398 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688952C211yesmaximal, characteristic, complemented
2C2×#15688930C211yesmaximal, characteristic
3#20245095C211yesmaximal, characteristic
4#20244136C211yesmaximal, characteristic, complemented
5#20241485C211yesmaximal, characteristic, complemented
6#20241466C211yesmaximal, characteristic, complemented
7C2×C47×#443C211yesmaximal, characteristic
8C22×C47⋊C16C2211yescharacteristic
9C22×C47⋊C16C2211yescharacteristic
10C2×#13093960C2211yescharacteristic
11#15689161C2211yescharacteristic
12#15689149—21no—
13#15688952—22no—
14#15688930—22no—
15#15688927C411yescharacteristic, complemented
16#15688927C411yescharacteristic, complemented
17#15688924C411yescharacteristic, complemented
18#15688924C411yescharacteristic, complemented
19#15688883—21no—
20C2×#135×C47C2211yescharacteristic
21C2×#131×C47—21no—
22C47×#443—22no—
23C47×#443—22no—
24C2×#108×C47C2211yescharacteristic
25C2×#104×C47—21no—
26C2×C4×C8×C47C2211yescharacteristic
27C2×#10625329—21no—
28#13093983—21no—
29C2×C47⋊C16—21no—
30C2×C47⋊C16—21no—
31C2×C47⋊C16—21no—
32C2×C47⋊C16—21no—
33C2×C47⋊C16—21no—
34C2×C47⋊C16C2×C411yescharacteristic
35C2×C47⋊C16C2×C411yescharacteristic
36#13093960—21no—
37#13093960—21no—
38#13093958C2×C411yescharacteristic
39#13093958C2×C411yescharacteristic
40#13093956—21no—
41Dic47⋊C8—41no—
42Dic47⋊C8—41no—
43C2×#41×C47—21no—
44#135×C47—21no—
45#135×C47—21no—
46#135×C47D412yes—
47#135×C47Q812yes—
48C2×Q16×C47C2×C411yescharacteristic
49C2×SD16×C47—21no—
50C2×D8×C47C2×C411yescharacteristic
51#131×C47—24no—
52#131×C47—42no—
53C2×M16×C47—21no—
54C2×M16×C47—41no—
55C22×C8×C47—41no—
56C22×C8×C47D411yescharacteristic
57C22×C8×C47C2311yescharacteristic
58C2×C42×C47D411yescharacteristic
59#108×C47—21no—
60#108×C47D412yes—
61#104×C47—24no—
62C4×C8×C47—21no—
63C4×C8×C47—21no—
64#41×C47—22no—
65#41×C47—22no—
66#41×C47—41no—
67#41×C47—41no—
68C2×Q8×C47—21no—
69C2×D4×C47—21no—
70C2×D4×C47—41no—
71C22×C4×C47—21no—
72C22×C4×C47—41no—
73C22×C4×C47C2×D411yescharacteristic
74Q16×C47—21no—
75Q16×C47C4⋊C412yes—
76SD16×C47—22no—
77SD16×C47—22no—
78D8×C47—21no—
79D8×C47C4⋊C412yes—
80M16×C47—22no—
81M16×C47—44no—
82M16×C47—41no—
83C2×C8×C47—21no—
84C2×C8×C47—21no—
85C2×C8×C47—44no—
86C2×C8×C47—41no—
87C2×C8×C47D811yescharacteristic
88C2×C8×C47D811yescharacteristic
89C2×C8×C47C22×C411yescharacteristic
90C2×C8×C47C2×D411yescharacteristic
91C2×C8×C47C2×D411yescharacteristic
92C2×C8×C47C2×Q811yescharacteristic
93C2×C8×C47#4111yescharacteristic
94C2×C8×C47#4111yescharacteristic
95C42×C47—21no—
96C42×C47—21no—
97C22⋊Dic47—41no—
98C4⋊Dic47—21no—
99C4⋊Dic47—21no—
100C4×Dic47—41no—
101#10625329—41no—
102#10625329—41no—
103C2×C47⋊C8—41no—
104C47⋊C16—21no—
105C47⋊C16—21no—
106C47⋊C16—41no—
107C23×C47—41no—
108C23×C47#13511yescharacteristic
109Q8×C47—22no—
110Q8×C47—41no—
111D4×C47—22no—
112D4×C47—42no—
113D4×C47—42no—
114D4×C47—41no—
115C2×C4×C47—22no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47C4×D411yescharacteristic
122C2×C4×C47#12011yescharacteristic
123C2×C4×C47C22⋊Q811yescharacteristic
124C2×C4×C47C2×D811yescharacteristic
125C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C47#13711yescharacteristic
127C2×Dic47—41no—
128C2×Dic47—41no—
129C8×C47—21no—
130C8×C47—21no—
131C8×C47—42no—
132C8×C47C2×C4⋊C411yescharacteristic
133C8×C47C4×D411yescharacteristic
134C8×C47C22⋊Q811yescharacteristic
135C8×C47#12311yescharacteristic
136C47⋊C8—81no—
137#12939—471nomaximal, Sylow
138C22×C47—21no—
139C22×C47—21no—
140C22×C47—42no—
141C22×C47—81no—
142C22×C47—81no—
143C22×C47C4×D811yescharacteristic
144C22×C47#45211yescharacteristic
145C22×C47#47811yescharacteristic
146C4×C47—42no—
147C4×C47—42no—
148C4×C47#38511yescharacteristic
149C4×C47#44011yescharacteristic
150C4×C47#44711yescharacteristic
151C4×C47#47411yescharacteristic
152Dic47—81no—
153Dic47—81no—
154C2×#443D4711yescharacteristic
155C2×#368—471no—
156C2×#359—471no—
157#2138—471no—
158#1945—471no—
159#1382—471no—
160#1374—471no—
161C2×C47—21no—
162C2×C47—21no—
163C2×C47—82no—
164C2×C47#186211yescharacteristic
165C2×C47#217811yescharacteristic
166C2×C47#217911yescharacteristic
167C2×#135C2×D4711yescharacteristic
168C2×#131—21no—
169C22×C16—471no—
170C22×C16—471no—
171#443—22no—
172#443—22no—
173C2×#108—471no—
174C2×#108C2×D4711yescharacteristic
175#427—471no—
176#424—941no—
177C2×#104—21no—
178C2×C4×C8C2×D4711yescharacteristic
179#368—942no—
180#359—942no—
181#358—471no—
182#358—471no—
183#357—471no—
184#357—471no—
185#339—941no—
186C47#1293911yescharacteristic, complemented, Sylow
187C2×#41—21no—
188#135—21no—
189#135—21no—
190#135Dic9412yes—
191#135C22⋊D4712yes—
192C2×Q16C4×D4711yescharacteristic
193C2×SD16—21no—
194C2×D8C4×D4711yescharacteristic
195#131—24no—
196#131—42no—
197C2×M16—21no—
198C2×M16—41no—
199C2×M16—941no—
200C22×C8—41no—
201C22×C8C22⋊D4711yescharacteristic
202C22×C8C22×D4711yescharacteristic
203#117—941no—
204C2×C42C22⋊D4711yescharacteristic
205C2×C16—471no—
206C2×C16—471no—
207C2×C16—941no—
208C2×C16—941no—
209C2×C16—941no—
210C2×C16—941no—
211C2×C16—941no—
212#108—21no—
213#108—941no—
214#108—941no—
215#108C22⋊D4712yes—
216#107—471no—
217#107—471no—
218#106—941no—
219C4⋊C8—1881no—
220C4⋊C8—1881no—
221#104—24no—
222C4×C8—21no—
223C4×C8—21no—
224#41—22no—
225#41—22no—
226#41—41no—
227#41—41no—
228C2×Q8—21no—
229C2×D4—21no—
230C2×D4—41no—
231C22×C4—21no—
232C22×C4—41no—
233C22×C4C2×C22⋊D4711yescharacteristic
234Q16—21no—
235Q16Dic47⋊C412yes—
236SD16—22no—
237SD16—22no—
238D8—21no—
239D8Dic47⋊C412yes—
240M16—22no—
241M16—44no—
242M16—41no—
243M16—1881no—
244M16—1881no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—44no—
248C2×C8—41no—
249C2×C8—1881no—
250C2×C8#1062533411yescharacteristic
251C2×C8#1062533411yescharacteristic
252C2×C8C2×Dic9411yescharacteristic
253C2×C8C2×C4×D4711yescharacteristic
254C2×C8#1062535011yescharacteristic
255C2×C8#1062535011yescharacteristic
256C2×C8C2×C22⋊D4711yescharacteristic
257C2×C8C2×C22⋊D4711yescharacteristic
258C4⋊C4—941no—
259C4⋊C4—941no—
260C22⋊C4—1881no—
261C42—21no—
262C42—21no—
263C42—1881no—
264C16—941no—
265C16—941no—
266C16—1881no—
267C23—41no—
268C23#1309404411yescharacteristic
269Q8—22no—
270Q8—41no—
271D4—22no—
272D4—42no—
273D4—42no—
274D4—41no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—1881no—
282C2×C4—1881no—
283C2×C4C4×C22⋊D4711yescharacteristic
284C2×C4#1309399311yescharacteristic
285C2×C4#1309400311yescharacteristic
286C2×C4C2×#1062533411yescharacteristic
287C2×C4#1309405011yescharacteristic
288C2×C4#1309405411yescharacteristic
289C8—21no—
290C8—21no—
291C8—42no—
292C8—3761no—
293C8C2×Dic47⋊C411yescharacteristic
294C8C4×C22⋊D4711yescharacteristic
295C8#1309400311yescharacteristic
296C8#1309400711yescharacteristic
297C22—21no—
298C22—21no—
299C22—42no—
300C22—81no—
301C22—81no—
302C22C4×#1062533411yescharacteristic
303C22#1568928511yescharacteristic
304C22#1568942311yescharacteristic
305C4—42no—
306C4—42no—
307C4—3761no—
308C4—3761no—
309C4#1568899611yescharacteristic
310C4#1568922311yescharacteristic
311C4#1568925711yescharacteristic
312C4#1568940311yescharacteristic
313C2—21no—
314C2—21no—
315C2—82no—
316C2#2024363511yescharacteristic
317C2#2024526711yescharacteristic
318C2#2024527211yescharacteristic
subgroupG/H
centerC2×C4C2×#10625334
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC2×C47×#443C2
socleC22×C47C4×D8
prime radicalC2×#135×C47C22
cosocleC22×C8×C47C23
hypercenterC2×#443D47
hypocenterC47#12939
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×#443

Complements

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

NH ≅ G/Ntype
1. C2×#15688952C2semidirect
4. #20244136C2semidirect
5. #20241485C2semidirect
6. #20241466C2semidirect
15. #15688927C4semidirect
16. #15688927C4semidirect
17. #15688924C4semidirect
18. #15688924C4semidirect
186. C47#12939semidirect

Maximal subgroups

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

subgroupindex
C2×#156889522
C2×#156889302
#202450952
#202441362
#202414852
#202414662
C2×C47×#4432
#1293947