Finite Group Databasebeta

← #26132897

Subgroups — #26132897

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×#15689241C211yesmaximal, characteristic
2C2×#15688952C211yesmaximal, characteristic, complemented
3C2×#15688929C211yesmaximal, characteristic
4#20245093C211yesmaximal, characteristic
5#20241483C211yesmaximal, characteristic, complemented
6#20241466C211yesmaximal, characteristic, complemented
7C47×#1945C211yesmaximal, characteristic, complemented
8C2×#13094046C2211yescharacteristic
9C2×#13094030—21no—
10C22×C47⋊C16C2211yescharacteristic
11C22×C47⋊C16C2211yescharacteristic
12#15689241—22no—
13#15689241—22no—
14C2×#13093960C2211yescharacteristic
15C2×#13093952—21no—
16#15688952—22no—
17#15688929—22no—
18#15688926C411yescharacteristic, complemented
19#15688926C411yescharacteristic, complemented
20#15688923C411yescharacteristic, complemented
21#15688923C411yescharacteristic, complemented
22C2×#108×C47C2211yescharacteristic
23C47×#427C2211yescharacteristic
24C47×#424—21no—
25C2×C4×C8×C47C2211yescharacteristic
26C47×#339—21no—
27C2×#10625350—21no—
28#13094046—21no—
29#13094046—21no—
30#13094046D412yes—
31#13094046Q812yes—
32C2×Dic188C2×C411yescharacteristic
33C2×#10625325—21no—
34C2×D376C2×C411yescharacteristic
35#13094030—24no—
36#13094030—42no—
37C2×#10625329—21no—
38C2×#10625324—41no—
39C2×C8×D47—41no—
40C2×C47⋊C16—21no—
41C2×C47⋊C16—21no—
42C2×C47⋊C16—21no—
43C2×C47⋊C16—21no—
44C2×C47⋊C16—21no—
45C2×C47⋊C16C2×C411yescharacteristic
46C2×C47⋊C16C2×C411yescharacteristic
47#13093960—21no—
48#13093960D412yes—
49#13093952—24no—
50C2×M16×C47—21no—
51C22×C8×C47D411yescharacteristic
52C22×C8×C47C2311yescharacteristic
53#117×C47—21no—
54C2×C42×C47D411yescharacteristic
55#108×C47—21no—
56#108×C47—21no—
57#107×C47C2×C411yescharacteristic
58#107×C47C2×C411yescharacteristic
59#106×C47—21no—
60C4⋊C8×C47—41no—
61C4⋊C8×C47—41no—
62C4×C8×C47—21no—
63C4×C8×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47C2×D411yescharacteristic
66C2×C22⋊D47—41no—
67#10625350—22no—
68#10625350—22no—
69#10625350—41no—
70#10625350—41no—
71C2×D188—21no—
72C2×C4×D47—41no—
73C2×Dic94—21no—
74M16×C47—41no—
75M16×C47—41no—
76C2×C8×C47—21no—
77C2×C8×C47—21no—
78C2×C8×C47—41no—
79C2×C8×C47D811yescharacteristic
80C2×C8×C47D811yescharacteristic
81C2×C8×C47C22×C411yescharacteristic
82C2×C8×C47C2×D411yescharacteristic
83C2×C8×C47C2×D411yescharacteristic
84C2×C8×C47C2×Q811yescharacteristic
85C2×C8×C47#4111yescharacteristic
86C2×C8×C47#4111yescharacteristic
87C4⋊C4×C47—21no—
88C4⋊C4×C47—21no—
89C22⋊C4×C47—41no—
90C42×C47—21no—
91C42×C47—21no—
92C42×C47—41no—
93#10625329—22no—
94#10625329—41no—
95C2×C47⋊C8—41no—
96Dic188—21no—
97Dic188C4⋊C412yes—
98D376—21no—
99D376C4⋊C412yes—
100#10625325—22no—
101#10625325—22no—
102#10625324—44no—
103C8×D47—44no—
104C47⋊C16—21no—
105C47⋊C16—21no—
106C47⋊C16—41no—
107C23×C47#13511yescharacteristic
108C22×D47—41no—
109C2×C4×C47—22no—
110C2×C4×C47—41no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47C4×D411yescharacteristic
115C2×C4×C47#12011yescharacteristic
116C2×C4×C47C22⋊Q811yescharacteristic
117C2×C4×C47C2×D811yescharacteristic
118C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C47#13711yescharacteristic
120C22⋊D47—42no—
121C22⋊D47—42no—
122C2×Dic47—41no—
123D188—22no—
124D188—41no—
125C4×D47—42no—
126C4×D47—42no—
127Dic94—22no—
128Dic94—41no—
129C8×C47—21no—
130C8×C47—21no—
131C8×C47—81no—
132C8×C47C2×C4⋊C411yescharacteristic
133C8×C47C4×D411yescharacteristic
134C8×C47C22⋊Q811yescharacteristic
135C8×C47#12311yescharacteristic
136C47⋊C8—42no—
137#12939—471nomaximal, Sylow
138C22×C47—21no—
139C22×C47—21no—
140C22×C47C4×D811yescharacteristic
141C22×C47#45211yescharacteristic
142C22×C47#47811yescharacteristic
143C2×D47—42no—
144C2×D47—81no—
145C2×D47—81no—
146C4×C47—42no—
147C4×C47—81no—
148C4×C47—81no—
149C4×C47#38511yescharacteristic
150C4×C47#44011yescharacteristic
151C4×C47#44711yescharacteristic
152C4×C47#47411yescharacteristic
153Dic47—42no—
154C2×#443—471no—
155C2×#368—471no—
156C2×#359—471no—
157#2138—471no—
158#1945D4711yescharacteristic, complemented
159#1382—471no—
160#1374—471no—
161C2×C47—21no—
162C2×C47—21no—
163C2×C47#186211yescharacteristic
164C2×C47#217811yescharacteristic
165C2×C47#217911yescharacteristic
166D47—82no—
167C2×#135—471no—
168C2×#131—941no—
169C22×C16—471no—
170C22×C16—471no—
171#443—942no—
172#443—942no—
173C2×#108—471no—
174C2×#108C2×D4711yescharacteristic
175#427C2×D4711yescharacteristic
176#424—21no—
177C2×#104—941no—
178C2×C4×C8C2×D4711yescharacteristic
179#368—942no—
180#359—942no—
181#358—471no—
182#358—471no—
183#357—471no—
184#357—471no—
185#339—21no—
186C47#1293911yescharacteristic, complemented, Sylow
187C2×#41—941no—
188#135—472no—
189#135—472no—
190#135—941no—
191#135—941no—
192C2×Q16—471no—
193C2×SD16—941no—
194C2×D8—471no—
195#131—944no—
196#131—1882no—
197C2×M16—21no—
198C2×M16—941no—
199C2×M16—1881no—
200C22×C8—1881no—
201C22×C8C22⋊D4711yescharacteristic
202C22×C8C22×D4711yescharacteristic
203#117—21no—
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—21no—
214#108—472no—
215#108—941no—
216#107C4×D4711yescharacteristic
217#107C4×D4711yescharacteristic
218#106—21no—
219C4⋊C8—41no—
220C4⋊C8—41no—
221#104—944no—
222C4×C8—21no—
223C4×C8—21no—
224#41—942no—
225#41—942no—
226#41—1881no—
227#41—1881no—
228C2×Q8—941no—
229C2×D4—941no—
230C2×D4—1881no—
231C22×C4—21no—
232C22×C4—1881no—
233C22×C4C2×C22⋊D4711yescharacteristic
234Q16—472no—
235Q16—941no—
236SD16—942no—
237SD16—942no—
238D8—472no—
239D8—941no—
240M16—41no—
241M16—41no—
242M16—942no—
243M16—1884no—
244M16—1881no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—41no—
248C2×C8—1884no—
249C2×C8—1881no—
250C2×C8#1062533411yescharacteristic
251C2×C8#1062533411yescharacteristic
252C2×C8C2×C4×D4711yescharacteristic
253C2×C8#1062535011yescharacteristic
254C2×C8D4×D4711yescharacteristic
255C2×C8D4×D4711yescharacteristic
256C2×C8Q8×D4711yescharacteristic
257C2×C8#1062535411yescharacteristic
258C4⋊C4—21no—
259C4⋊C4—21no—
260C22⋊C4—41no—
261C42—21no—
262C42—21no—
263C42—41no—
264C16—941no—
265C16—941no—
266C16—1881no—
267C23—1881no—
268C23#1309404411yescharacteristic
269Q8—942no—
270Q8—1881no—
271D4—942no—
272D4—1882no—
273D4—1882no—
274D4—1881no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—1881no—
281C2×C4—1882no—
282C2×C4—1882no—
283C2×C4C4×C22⋊D4711yescharacteristic
284C2×C4#1309399211yescharacteristic
285C2×C4#1309400011yescharacteristic
286C2×C4C2×#1062533411yescharacteristic
287C2×C4#1309404911yescharacteristic
288C2×C4#1309405511yescharacteristic
289C8—21no—
290C8—21no—
291C8—81no—
292C8—1882no—
293C8C4⋊C4×D4711yescharacteristic
294C8#1309398611yescharacteristic
295C8#1309400211yescharacteristic
296C8#1309400911yescharacteristic
297C22—21no—
298C22—21no—
299C22—3761no—
300C22—3761no—
301C22—1882no—
302C22C4×#1062533411yescharacteristic
303C22#1568928311yescharacteristic
304C22#1568942011yescharacteristic
305C4—42no—
306C4—81no—
307C4—81no—
308C4—1882no—
309C4#1568899411yescharacteristic
310C4#1568922111yescharacteristic
311C4#1568925611yescharacteristic
312C4#1568940011yescharacteristic
313C2—21no—
314C2—21no—
315C2—3762no—
316C2#2024363411yescharacteristic
317C2#2024526711yescharacteristic
318C2#2024527211yescharacteristic
subgroupG/H
centerC2×C4C2×#10625334
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#1945C2
socleC22×C47C4×D8
prime radicalC2×#13094046C22
cosocleC22×C8×C47C23
hypercenter#1945D47
hypocenterC47#12939
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ #1945

Complements

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

NH ≅ G/Ntype
2. C2×#15688952C2semidirect
5. #20241483C2semidirect
6. #20241466C2semidirect
7. C47×#1945C2semidirect
18. #15688926C4semidirect
19. #15688926C4semidirect
20. #15688923C4semidirect
21. #15688923C4semidirect
158. #1945D47semidirect
186. C47#12939semidirect

Maximal subgroups

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

subgroupindex
C2×#156892412
C2×#156889522
C2×#156889292
#202450932
#202414832
#202414662
C47×#19452
#1293947