Finite Group Databasebeta

← #26134645

Subgroups — #26134645

Order 12032.

510 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
1C22×#13093965C211yesmaximal, characteristic
2C2×#15688948C211yesmaximal, characteristic, complemented
3C2×#15688925C211yesmaximal, characteristic, complemented
4C2×C16×Dic47C211yesmaximal, characteristic, complemented
5#20244255C211yesmaximal, characteristic, complemented
6#20244252C211yesmaximal, characteristic, complemented
7C2×C47×#357C211yesmaximal, characteristic, complemented
8C22×D376C2211yescharacteristic
9C2×#13093965—22no—
10C2×#13093965—22no—
11C2×#13093965C414yescomplemented
12C22×C47⋊C16C2211yescharacteristic
13C2×#13093958C2211yescharacteristic
14C2×#13093945—21no—
15C2×#13093944—21no—
16C2×C8×Dic47C2211yescharacteristic
17#15689026—21no—
18#15689024—21no—
19#15688948—22no—
20#15688925—22no—
21C16×Dic47—22no—
22C22×D8×C47C2211yescharacteristic
23C22×C16×C47C2211yescharacteristic
24C2×#107×C47C2211yescharacteristic
25C47×#357—22no—
26C22×D188—21no—
27C2×D376—21no—
28C2×D376—21no—
29C2×D376—21no—
30C2×D376—21no—
31C2×D376—21no—
32C2×D376C2×C412yes—
33C22×C47⋊C8D411yescharacteristic
34C2×C4⋊Dic47—21no—
35C2×C22⋊Dic47—42no—
36C2×(C2×D47)⋊C4—42no—
37C2×C4×Dic47D411yescharacteristic
38#13093965—28no—
39C2×C47⋊C16C2×C412yes—
40C2×C47⋊C16D412yes—
41C2×C47⋊C16D412yes—
42#13093958—21no—
43#13093958—21no—
44#13093945—42no—
45#13093944—42no—
46C8×Dic47—21no—
47C8×Dic47—21no—
48C22×D4×C47—21no—
49C2×D8×C47—21no—
50C2×D8×C47—21no—
51C2×D8×C47—21no—
52C2×D8×C47—21no—
53C2×D8×C47—21no—
54C2×D8×C47C2×C412yes—
55C22×C8×C47C2311yescharacteristic
56C2×C4⋊C4×C47—21no—
57C2×C16×C47—21no—
58C2×C16×C47—21no—
59C2×C16×C47D412yes—
60#107×C47—21no—
61#107×C47—21no—
62C24×C47—41no—
63C23×D47—41no—
64C2×D4×C47—22no—
65C2×D4×C47—41no—
66C2×D4×C47—41no—
67C2×D4×C47—41no—
68C2×D4×C47—41no—
69C2×D4×C47—41no—
70C22×C4×C47—41no—
71C22×C4×C47C2×D411yescharacteristic
72C22×Dic47—41no—
73C22×Dic47D812yes—
74C2×D188—22no—
75C2×D188—41no—
76C2×D188—41no—
77C2×D188—41no—
78C2×D188—41no—
79C2×D188—41no—
80D8×C47—22no—
81D8×C47—22no—
82D8×C47—41no—
83D8×C47—41no—
84C2×C8×C47C22×C411yescharacteristic
85C2×C8×C47C2×D411yescharacteristic
86C2×C8×C47C2×D411yescharacteristic
87C2×C8×C47C2×D411yescharacteristic
88C2×C8×C47#4111yescharacteristic
89C2×C8×C47#4111yescharacteristic
90C4⋊C4×C47—41no—
91C4⋊C4×C47—41no—
92C22⋊Dic47—84no—
93(C2×D47)⋊C4—84no—
94C4⋊Dic47—41no—
95C4⋊Dic47—41no—
96C4×Dic47—21no—
97C4×Dic47—21no—
98C2×C47⋊C8—21no—
99C2×C47⋊C8—21no—
100C2×C47⋊C8D812yes—
101D376—22no—
102D376—22no—
103D376—41no—
104D376—41no—
105C16×C47—22no—
106C47⋊C16C22⋊C414yescomplemented
107C23×C47—42no—
108C23×C47—81no—
109C23×C47—81no—
110C23×C47—82no—
111C23×C47—82no—
112C23×C47C2×D811yescharacteristic
113C22×D47—42no—
114C22×D47—81no—
115C22×D47—81no—
116C22×D47—82no—
117C22×D47—82no—
118D4×C47—42no—
119D4×C47—42no—
120D4×C47—81no—
121D4×C47—81no—
122C2×C4×C47—81no—
123C2×C4×C47—82no—
124C2×C4×C47C4×D411yescharacteristic
125C2×C4×C47#12111yescharacteristic
126C2×C4×C47#12711yescharacteristic
127C2×C4×C47C2×D811yescharacteristic
128C2×C4×C47#13511yescharacteristic
129C2×C4×C47#13511yescharacteristic
130C2×Dic47—22no—
131C2×Dic47—22no—
132C2×Dic47—81no—
133C2×Dic47—82no—
134C2×Dic47D1614yescomplemented
135D188—42no—
136D188—42no—
137D188—81no—
138D188—81no—
139C8×C47C2×C22⋊C411yescharacteristic
140C8×C47C4×D411yescharacteristic
141C8×C47#12111yescharacteristic
142C8×C47#12411yescharacteristic
143C47⋊C8—22no—
144#13235—471nomaximal, Sylow
145C22×C47—84no—
146C22×C47—82no—
147C22×C47—162no—
148C22×C47—161no—
149C22×C47—161no—
150C22×C47C4×D811yescharacteristic
151C22×C47#46611yescharacteristic
152C22×C47#49311yescharacteristic
153C22×C47C2×D1612yes—
154C22×C47#50812yes—
155C2×D47—84no—
156C2×D47—82no—
157C2×D47—162no—
158C2×D47—161no—
159C2×D47—161no—
160C4×C47—161no—
161C4×C47#39011yescharacteristic
162C4×C47C4×D811yescharacteristic
163C4×C47#46611yescharacteristic
164C4×C47#49511yescharacteristic
165Dic47—24no—
166Dic47—161no—
167C22×D16—471no—
168C2×#366—471no—
169C2×#357—471no—
170C2×#357D4711yescharacteristic, complemented
171C2×C4×C16—471no—
172#1970—471no—
173#1970—471no—
174C2×C47—162no—
175C2×C47#197011yescharacteristic
176C2×C47C4×D1612yes—
177C2×C47#221512yes—
178C2×C47#224611yescharacteristic
179C2×C47#224911yescharacteristic
180D47—162no—
181C22×D8—471no—
182C22×D8C2×D4711yescharacteristic
183C2×D16—474no—
184C2×D16—942no—
185C2×D16—942no—
186C22×C16—471no—
187C22×C16C2×D4711yescharacteristic
188C2×#107—471no—
189C2×#107C2×D4711yescharacteristic
190C2×#102—941no—
191C2×#102—941no—
192C2×C4×C8—471no—
193#390—941no—
194#390—941no—
195#366—942no—
196#357—22no—
197#357—942no—
198C4×C16—942no—
199C47#1323511yescharacteristic, complemented, Sylow
200C22×D4—21no—
201C22×D4—941no—
202C2×D8—21no—
203C2×D8—21no—
204C2×D8—21no—
205C2×D8—21no—
206C2×D8—21no—
207C2×D8—472no—
208C2×D8—941no—
209C2×D8—941no—
210C2×D8—941no—
211C2×D8—941no—
212C2×D8—941no—
213C2×D8C4×D4712yes—
214C22×C8—471no—
215C22×C8C22×D4711yescharacteristic
216C2×C4⋊C4—21no—
217C2×C4⋊C4—941no—
218C2×C22⋊C4—1882no—
219C2×C22⋊C4—1882no—
220C2×C42—471no—
221D16—948no—
222C2×C16—21no—
223C2×C16—21no—
224C2×C16—472no—
225C2×C16—472no—
226C2×C16—472no—
227C2×C16C22⋊D4712yes—
228#107—21no—
229#107—21no—
230#107—941no—
231#107—941no—
232#102—1882no—
233#102—1882no—
234C4×C8—941no—
235C4×C8—941no—
236C24—41no—
237C24—1881no—
238C2×D4—22no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—942no—
245C2×D4—1881no—
246C2×D4—1881no—
247C2×D4—1881no—
248C2×D4—1881no—
249C2×D4—1881no—
250C22×C4—41no—
251C22×C4—472no—
252C22×C4—1881no—
253C22×C4D4×D4711yescharacteristic
254D8—22no—
255D8—22no—
256D8—41no—
257D8—41no—
258D8—942no—
259D8—942no—
260D8—1881no—
261D8—1881no—
262C2×C8—472no—
263C2×C8—941no—
264C2×C8—941no—
265C2×C8C2×C4×D4711yescharacteristic
266C2×C8#1062535011yescharacteristic
267C2×C8D4×D4711yescharacteristic
268C2×C8D4×D4711yescharacteristic
269C2×C8#1062535211yescharacteristic
270C2×C8C2×C22⋊D4711yescharacteristic
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—1881no—
274C4⋊C4—1881no—
275C22⋊C4—3764no—
276C22⋊C4—3764no—
277C42—941no—
278C42—941no—
279C16—22no—
280C16—474no—
281C23—42no—
282C23—81no—
283C23—82no—
284C23—82no—
285C23—81no—
286C23—3761no—
287C23—3762no—
288C23—3762no—
289C23—3761no—
290C23—1882no—
291C23D8×D4711yescharacteristic
292D4—42no—
293D4—42no—
294D4—81no—
295D4—81no—
296D4—3761no—
297D4—3761no—
298D4—1882no—
299D4—1882no—
300C2×C4—81no—
301C2×C4—82no—
302C2×C4—474no—
303C2×C4—942no—
304C2×C4—942no—
305C2×C4—3761no—
306C2×C4—3762no—
307C2×C4(C22⋊D47)⋊C411yescharacteristic
308C2×C4#1309399711yescharacteristic
309C2×C4#1309402011yescharacteristic
310C2×C4D8×D4711yescharacteristic
311C2×C4#1309404511yescharacteristic
312C2×C4#1309404511yescharacteristic
313C8—942no—
314C8C22⋊C4×D4711yescharacteristic
315C8C4×C22⋊D4711yescharacteristic
316C8#1309399611yescharacteristic
317C8#1309401011yescharacteristic
318C22—82no—
319C22—84no—
320C22—161no—
321C22—161no—
322C22—162no—
323C22—3762no—
324C22—3764no—
325C22—7521no—
326C22—7521no—
327C22—7522no—
328C22#1568920311yescharacteristic
329C22#1568935811yescharacteristic
330C22#1568949811yescharacteristic
331C22D16×D4712yes—
332C22#1568955412yes—
333C4—161no—
334C4—944no—
335C4—7521no—
336C4#1568902311yescharacteristic
337C4#1568920311yescharacteristic
338C4#1568935811yescharacteristic
339C4#1568950411yescharacteristic
340C2—162no—
341C2—7522no—
342C2#2024425111yescharacteristic
343C2#2024523212yes—
344C2#2024547312yes—
345C2#2024563611yescharacteristic
346C2#2024564811yescharacteristic
subgroupG/H
centerC23D8×D47
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC2×C47×#357C2
socleC23×C47C2×D8
prime radicalC22×#13093965C2
cosocleC22×C8×C47C23
hypercenterC2×#357D47
hypocenterC47#13235
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×#15688948C2semidirect
3. C2×#15688925C2semidirect
4. C2×C16×Dic47C2semidirect
5. #20244255C2semidirect
6. #20244252C2semidirect
7. C2×C47×#357C2semidirect
11. C2×#13093965C4semidirect
106. C47⋊C16C22⋊C4semidirect
134. C2×Dic47D16semidirect
170. C2×#357D47semidirect
199. C47#13235semidirect

Maximal subgroups

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

subgroupindex
C22×#130939652
C2×#156889482
C2×#156889252
C2×C16×Dic472
#202442552
#202442522
C2×C47×#3572
#1323547