Finite Group Databasebeta

← #26133533

Subgroups — #26133533

Order 12032.

508 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689245C211yesmaximal, characteristic, complemented
2C2×#15688936C211yesmaximal, characteristic, complemented
3C2×#15688933C211yesmaximal, characteristic, complemented
4#20244265C211yesmaximal, characteristic, complemented
5#20241487C211yesmaximal, characteristic, complemented
6#20241467C211yesmaximal, characteristic, complemented
7C2×C47×#509C211yesmaximal, characteristic
8C2×#13094031—21no—
9C2×#13093964C2211yescharacteristic
10#15689245—22no—
11#15689245—22no—
12C2×#13093960C2211yescharacteristic
13C2×#13093956C2211yescharacteristic
14C2×#13093951—21no—
15C2×#13093945—21no—
16C2×#13093934C2211yescharacteristic
17#15689026—21no—
18#15688936—22no—
19#15688933—22no—
20C2×#135×C47C2211yescharacteristic
21C22×D8×C47C2211yescharacteristic
22C47×#509—22no—
23C47×#509—22no—
24C2×SD32×C47C412yescomplemented
25C2×D16×C47C412yescomplemented
26C2×M32×C47C2211yescharacteristic
27#13094031—24no—
28#13094031—42no—
29C2×#10625329—21no—
30C2×#10625329—41no—
31C22×C47⋊C8—41no—
32C22×C47⋊C8D411yescharacteristic
33C2×C4⋊Dic47—21no—
34C2×C22⋊Dic47—42no—
35C2×C4×Dic47D411yescharacteristic
36#13093964—21no—
37#13093964—21no—
38C2×C47⋊C16—21no—
39#13093960—21no—
40#13093960D412yes—
41#13093956—21no—
42#13093956—21no—
43#13093951—24no—
44#13093945—42no—
45#13093934—21no—
46#13093934—21no—
47C2×#41×C47—21no—
48C22×D4×C47—21no—
49#135×C47—21no—
50#135×C47—21no—
51#135×C47D412yes—
52#135×C47D412yes—
53C2×Q16×C47C2×C411yescharacteristic
54C2×SD16×C47—21no—
55C2×D8×C47—21no—
56C2×D8×C47—21no—
57C2×D8×C47—21no—
58C2×D8×C47—21no—
59C2×D8×C47—21no—
60C2×D8×C47C2×C412yes—
61C2×D8×C47C2×C411yescharacteristic
62C22×C8×C47C2311yescharacteristic
63SD32×C47—24no—
64D16×C47—24no—
65M32×C47—21no—
66M32×C47—21no—
67C2×C16×C47C2×C412yes—
68C24×C47—41no—
69#41×C47—22no—
70#41×C47—22no—
71#41×C47—41no—
72#41×C47—41no—
73C2×Q8×C47—21no—
74C2×D4×C47—21no—
75C2×D4×C47—22no—
76C2×D4×C47—41no—
77C2×D4×C47—41no—
78C2×D4×C47—41no—
79C2×D4×C47—41no—
80C2×D4×C47—41no—
81C2×D4×C47—41no—
82C22×C4×C47—41no—
83C22×C4×C47C2×D411yescharacteristic
84C22×Dic47—21no—
85C22×Dic47—41no—
86Q16×C47—21no—
87Q16×C47C22⋊C412yescomplemented
88SD16×C47—22no—
89SD16×C47—22no—
90D8×C47—22no—
91D8×C47—21no—
92D8×C47—22no—
93D8×C47—41no—
94D8×C47—41no—
95D8×C47C22⋊C412yescomplemented
96C2×C8×C47C22×C411yescharacteristic
97C2×C8×C47C2×D411yescharacteristic
98C2×C8×C47C2×D411yescharacteristic
99C2×C8×C47C2×D411yescharacteristic
100C2×C8×C47#4111yescharacteristic
101C2×C8×C47#4111yescharacteristic
102C22⋊Dic47—84no—
103C4⋊Dic47—41no—
104C4⋊Dic47—41no—
105C4×Dic47—21no—
106C4×Dic47—21no—
107#10625329—22no—
108#10625329—44no—
109#10625329—41no—
110C2×C47⋊C8—21no—
111C2×C47⋊C8—21no—
112C2×C47⋊C8—44no—
113C2×C47⋊C8—41no—
114C2×C47⋊C8D812yes—
115C16×C47—22no—
116C47⋊C16—41no—
117C23×C47—41no—
118C23×C47—42no—
119C23×C47—82no—
120C23×C47—82no—
121C23×C47—81no—
122C23×C47—81no—
123C23×C47AGL(1,ℤ/8)11yescharacteristic
124Q8×C47—22no—
125Q8×C47—41no—
126D4×C47—22no—
127D4×C47—42no—
128D4×C47—42no—
129D4×C47—42no—
130D4×C47—41no—
131D4×C47—42no—
132D4×C47—81no—
133D4×C47—81no—
134C2×C4×C47—41no—
135C2×C4×C47—42no—
136C2×C4×C47—42no—
137C2×C4×C47C4×D411yescharacteristic
138C2×C4×C47#12011yescharacteristic
139C2×C4×C47#12111yescharacteristic
140C2×C4×C47C2×D811yescharacteristic
141C2×C4×C47#13511yescharacteristic
142C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
143C2×Dic47—42no—
144C2×Dic47—41no—
145C2×Dic47—82no—
146C2×Dic47—81no—
147C8×C47C2×C22⋊C411yescharacteristic
148C8×C47C4×D411yescharacteristic
149C8×C47#12111yescharacteristic
150C8×C47#12311yescharacteristic
151C47⋊C8—22no—
152C47⋊C8—42no—
153#13040—471nomaximal, Sylow
154C22×C47—22no—
155C22×C47—42no—
156C22×C47—82no—
157C22×C47—82no—
158C22×C47—84no—
159C22×C47—162no—
160C22×C47—161no—
161C22×C47—161no—
162C22×C47#44211yescharacteristic
163C22×C47#44911yescharacteristic
164C22×C47#45911yescharacteristic
165C4×C47—42no—
166C4×C47#38611yescharacteristic
167C4×C47C4×D811yescharacteristic
168C4×C47#44711yescharacteristic
169C4×C47#46311yescharacteristic
170Dic47—81no—
171Dic47—161no—
172C2×#509D4711yescharacteristic
173C2×#444—471no—
174C2×#361—471no—
175C2×#360—471no—
176#1972—471no—
177#1383—471no—
178#1375—471no—
179C2×C47—21no—
180C2×C47—21no—
181C2×C47—82no—
182C2×C47—162no—
183C2×C47#187911yescharacteristic
184C2×C47AGL(1,ℤ/16)12yes—
185C2×#135C2×D4711yescharacteristic
186C22×D8C2×D4711yescharacteristic
187C2×#131—941no—
188#509—22no—
189#509—22no—
190C2×SD32Dic4712yescomplemented
191C2×D16Dic4712yescomplemented
192C2×M32—471no—
193C2×M32C2×D4711yescharacteristic
194#444—942no—
195#444—942no—
196C2×#108—471no—
197C2×#106—471no—
198C2×#104—941no—
199C2×#102—941no—
200C2×#97—471no—
201#390—941no—
202#361—942no—
203#360—942no—
204C47#1304011yescharacteristic, complemented, Sylow
205C2×#41—21no—
206C22×D4—21no—
207#135—21no—
208#135—21no—
209#135C22⋊D4712yes—
210#135C22⋊D4712yes—
211C2×Q16C2×Dic4711yescharacteristic
212C2×SD16—21no—
213C2×D8—21no—
214C2×D8—21no—
215C2×D8—21no—
216C2×D8—21no—
217C2×D8—21no—
218C2×D8C2×Dic4712yes—
219C2×D8C2×Dic4711yescharacteristic
220#131—944no—
221#131—1882no—
222C2×M16—941no—
223C2×M16—1881no—
224C22×C8—471no—
225C22×C8—1881no—
226C22×C8C22×D4711yescharacteristic
227C2×C4⋊C4—941no—
228C2×C22⋊C4—1882no—
229C2×C42—471no—
230SD32—24no—
231D16—24no—
232M32—21no—
233M32—21no—
234M32—941no—
235M32—941no—
236C2×C16—941no—
237C2×C16C2×Dic4712yes—
238#108—472no—
239#108—941no—
240#106—941no—
241#106—941no—
242#104—944no—
243#102—1882no—
244#97—941no—
245#97—941no—
246C24—41no—
247#41—22no—
248#41—22no—
249#41—41no—
250#41—41no—
251C2×Q8—21no—
252C2×D4—21no—
253C2×D4—22no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C22×C4—41no—
261C22×C4—941no—
262C22×C4—1881no—
263C22×C4D4×D4711yescharacteristic
264Q16—21no—
265Q16C22⋊Dic4712yescomplemented
266SD16—22no—
267SD16—22no—
268D8—22no—
269D8—21no—
270D8—22no—
271D8—41no—
272D8—41no—
273D8C22⋊Dic4712yescomplemented
274M16—942no—
275M16—1884no—
276M16—1881no—
277C2×C8—472no—
278C2×C8—941no—
279C2×C8—941no—
280C2×C8—1884no—
281C2×C8—1881no—
282C2×C8D4×D4711yescharacteristic
283C2×C8#1062535211yescharacteristic
284C2×C8#1062535211yescharacteristic
285C2×C8C22×Dic4711yescharacteristic
286C2×C8C2×C22⋊D4711yescharacteristic
287C2×C8C2×C22⋊D4711yescharacteristic
288C4⋊C4—1881no—
289C4⋊C4—1881no—
290C22⋊C4—3764no—
291C42—941no—
292C42—941no—
293C16—22no—
294C16—1881no—
295C23—41no—
296C23—42no—
297C23—82no—
298C23—82no—
299C23—81no—
300C23—81no—
301C23#1309404811yescharacteristic
302Q8—22no—
303Q8—41no—
304D4—22no—
305D4—42no—
306D4—42no—
307D4—42no—
308D4—41no—
309D4—42no—
310D4—81no—
311D4—81no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—3762no—
316C2×C4—3761no—
317C2×C4—1882no—
318C2×C4—1881no—
319C2×C4D4×Dic4711yescharacteristic
320C2×C4#1309399211yescharacteristic
321C2×C4#1309399611yescharacteristic
322C2×C4D8×D4711yescharacteristic
323C2×C4#1309404311yescharacteristic
324C2×C4#1309404811yescharacteristic
325C8—942no—
326C8—1882no—
327C8C2×C22⋊Dic4711yescharacteristic
328C8D4×Dic4711yescharacteristic
329C8#1309399611yescharacteristic
330C8#1309400511yescharacteristic
331C22—22no—
332C22—42no—
333C22—82no—
334C22—82no—
335C22—84no—
336C22—162no—
337C22—161no—
338C22—161no—
339C22#1568923511yescharacteristic
340C22#1568926511yescharacteristic
341C22#1568931611yescharacteristic
342C4—42no—
343C4—3761no—
344C4—7521no—
345C4#1568900311yescharacteristic
346C4D8×Dic4711yescharacteristic
347C4#1568925511yescharacteristic
348C4#1568933611yescharacteristic
349C2—21no—
350C2—21no—
351C2—82no—
352C2—162no—
353C2#2024373811yescharacteristic
354C2#2024528712yes—
subgroupG/H
centerC22#15689235
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
Fitting, prime radicalC2×C47×#509C2
socleC22×C47#442
cosocleC22×C8×C47C23
hypercenterC2×#509D47
hypocenterC47#13040
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#509

Complements

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

NH ≅ G/Ntype
1. C2×#15689245C2semidirect
2. C2×#15688936C2semidirect
3. C2×#15688933C2semidirect
4. #20244265C2semidirect
5. #20241487C2semidirect
6. #20241467C2semidirect
24. C2×SD32×C47C4semidirect
25. C2×D16×C47C4semidirect
87. Q16×C47C22⋊C4semidirect
95. D8×C47C22⋊C4semidirect
190. C2×SD32Dic47semidirect
191. C2×D16Dic47semidirect
204. C47#13040semidirect
265. Q16C22⋊Dic47semidirect
273. D8C22⋊Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156892452
C2×#156889362
C2×#156889332
#202442652
#202414872
#202414672
C2×C47×#5092
#1304047