Finite Group Databasebeta

← #26201630

Subgroups — #26201630

Order 12032.

676 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254511C211yesmaximal, characteristic
2#20252860C211yesmaximal, characteristic
3C2×#15689333C211yesmaximal, characteristic, complemented
4#20242832C214yesmaximal, complemented
5#20242782C214yesmaximal, complemented
6#20242163C211yesmaximal, characteristic, complemented
7#20242059C211yesmaximal, characteristic, complemented
8#20242035C211yesmaximal, characteristic
9C47×#2982C211yesmaximal, characteristic, complemented
10C2×#13094055—21no—
11C22×#10625337—21no—
12#15689673C2211yescharacteristic
13C2×#13094024C2211yescharacteristic
14#15689528C2212yes—
15#15689470C2212yes—
16#15689437—22no—
17#15689404—22no—
18#15689345—22no—
19#15689333—22no—
20#15689333C2212yes—
21#15689333C2214yes—
22#15689323C2212yes—
23#15689147C2211yescharacteristic
24C2×C4⋊(C47⋊C8)C2211yescharacteristic
25#15689122—21no—
26C2×#13093947—21no—
27#15688876C2214yes—
28#15688868C2212yes—
29#15688868C2214yes—
30#15688866C2212yes—
31C2×C4×Q8×C47C2211yescharacteristic
32C8×Q8×C47—22no—
33C8×Q8×C47C2214yes—
34C47×#431—22no—
35C47×#431C2211yescharacteristic
36C47×#431C2211yescharacteristic
37C2×#10625350—21no—
38C22×Dic94—21no—
39#13094055—44no—
40C2×#10625337—24no—
41C2×#10625337—22no—
42C2×#10625337—44no—
43C2×#10625336—22no—
44C2×#10625329—21no—
45C22×C47⋊C8—21no—
46#13094024—21no—
47#13094024—21no—
48#13094024—21no—
49#13094024C2311yescharacteristic
50#13094024C2312yes—
51#13094021C2311yescharacteristic
52#13094012—21no—
53#13093998—21no—
54#13093998—21no—
55C4×D188—21no—
56C4×D188—21no—
57C2×C4⋊Dic47—21no—
58C2×C4⋊Dic47—21no—
59#13093958—24no—
60C4⋊(C47⋊C8)—21no—
61C4⋊(C47⋊C8)—21no—
62C4⋊(C47⋊C8)C2312yes—
63C4⋊(C47⋊C8)C2312yes—
64#13093947—24no—
65#13093947—22no—
66#13093946—24no—
67#13093944—22no—
68C22×Q8×C47D411yescharacteristic
69C4×Q8×C47—21no—
70C4×Q8×C47—21no—
71C4×Q8×C47D412yes—
72C4×Q8×C47C2312yes—
73C2×C4⋊C4×C47—21no—
74C2×C4⋊C4×C47D411yescharacteristic
75C2×C42×C47—21no—
76C2×C42×C47C2311yescharacteristic
77C4⋊C8×C47—24no—
78C4⋊C8×C47D412yes—
79C4⋊C8×C47C2312yes—
80C22⋊C8×C47—21no—
81C22⋊C8×C47D412yes—
82C4×C8×C47—24no—
83C4×C8×C47—21no—
84C4×C8×C47C2312yes—
85C2×Q8×C47—21no—
86C2×Q8×C47—22no—
87C2×Q8×C47D814yes—
88C2×Q8×C47C2×D412yes—
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47#4111yescharacteristic
94C2×C22⋊D47—41no—
95C22×Dic47—41no—
96#10625350—42no—
97#10625350—42no—
98C2×D188—21no—
99C2×D188—21no—
100C2×D188—21no—
101C2×C4×D47—41no—
102C2×Dic94—21no—
103C2×Dic94—22no—
104C2×Dic94—42no—
105C2×Dic94—42no—
106C2×Dic94—41no—
107C2×C8×C47—22no—
108C2×C8×C47C2×D414yes—
109C4⋊C4×C47—22no—
110C4⋊C4×C47—21no—
111C4⋊C4×C47—22no—
112C4⋊C4×C47C2×D412yes—
113C42×C47—22no—
114C42×C47—22no—
115C42×C47C2×D411yescharacteristic
116C42×C47C2×D411yescharacteristic
117C42×C47#4111yescharacteristic
118C42×C47C2411yescharacteristic
119#10625337—48no—
120#10625337—44no—
121#10625336—44no—
122(C2×D47)⋊C4—41no—
123(C2×D47)⋊C4—41no—
124C4⋊Dic47—22no—
125C4⋊Dic47—21no—
126C4⋊Dic47—22no—
127C4⋊Dic47—21no—
128C4⋊Dic47—41no—
129C4⋊Dic47—41no—
130#10625329—42no—
131C2×C47⋊C8—22no—
132C2×C47⋊C8—22no—
133C2×C47⋊C8—42no—
134C23×C47#12111yescharacteristic
135C22×D47—41no—
136Q8×C47—22no—
137Q8×C47—24no—
138Q8×C47C2×D814yes—
139C2×C4×C47—22no—
140C2×C4×C47—24no—
141C2×C4×C47—22no—
142C2×C4×C47—22no—
143C2×C4×C47—41no—
144C2×C4×C47—81no—
145C2×C4×C47#12111yescharacteristic
146C2×C4×C47#12111yescharacteristic
147C2×C4×C47#12111yescharacteristic
148C2×C4×C47C2×D812yes—
149C2×C4×C47#13512yes—
150C2×C4×C47C22×D411yescharacteristic
151C2×C4×C47C22×D411yescharacteristic
152C2×C4×C47C2×#4111yescharacteristic
153C22⋊D47—82no—
154C2×Dic47—42no—
155C2×Dic47—41no—
156C2×Dic47—82no—
157D188—41no—
158D188—41no—
159D188—42no—
160D188—81no—
161C4×D47—82no—
162Dic94—42no—
163Dic94—41no—
164Dic94—42no—
165Dic94—41no—
166Dic94—82no—
167C8×C47—24no—
168C8×C47—42no—
169C47⋊C8—42no—
170C47⋊C8—42no—
171#19404—471nomaximal, Sylow
172C22×C47—21no—
173C22×C47—21no—
174C22×C47#46612yes—
175C22×C47C2×#12111yescharacteristic
176C2×D47—81no—
177C2×D47—81no—
178C2×D47—81no—
179C4×C47—21no—
180C4×C47—24no—
181C4×C47—21no—
182C4×C47—21no—
183C4×C47—42no—
184C4×C47#46612yes—
185C4×C47C22×D811yescharacteristic
186C4×C47C2×#13511yescharacteristic
187C4×C47#57812yes—
188Dic47—82no—
189Dic47—81no—
190#3226—471no—
191#3037—471no—
192C2×#462—471no—
193#2982D4711yescharacteristic, complemented
194#1682—474no—
195#1674—474no—
196#1571—471no—
197#1546—471no—
198#1540—471no—
199C2×C47—22no—
200C2×C47#304411yescharacteristic
201C2×C47C2×#46611yescharacteristic
202C2×C47#306911yescharacteristic
203D47—161no—
204C2×#137—941no—
205C22×Q16—941no—
206#532—471no—
207C2×C4⋊Q8—471no—
208C2×C4×Q8C2×D4711yescharacteristic
209#500—472no—
210#486—472no—
211#480—942no—
212#474—942no—
213#464—942no—
214#462—472no—
215#462—474no—
216#462—942no—
217#460—472no—
218C8×Q8—22no—
219C8×Q8C2×D4714yes—
220#431—22no—
221#431C2×D4711yescharacteristic
222#431C2×D4711yescharacteristic
223#423—471no—
224C2×C4⋊C8—471no—
225#417—941no—
226C2×#103—941no—
227#335—474no—
228#332—472no—
229#332—474no—
230#331—472no—
231C47#1940411yescharacteristic, complemented, Sylow
232C2×#41—941no—
233C22×Q8—941no—
234C22×Q8D18811yescharacteristic
235#137—1884no—
236C2×Q16—944no—
237C2×Q16—942no—
238C2×Q16—1884no—
239C2×SD16—942no—
240C2×M16—941no—
241C22×C8—941no—
242C4⋊Q8—472no—
243C4⋊Q8—471no—
244C4⋊Q8—941no—
245C4⋊Q8—941no—
246C4⋊Q8—941no—
247#127—471no—
248#124—941no—
249#121—941no—
250#121—941no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8D18812yes—
254C4×Q8C22×D4712yes—
255C4×D4—941no—
256C4×D4—941no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—941no—
259C2×C4⋊C4—941no—
260C2×C4⋊C4D18811yescharacteristic
261C2×C42—21no—
262C2×C42C22×D4711yescharacteristic
263#107—944no—
264C4⋊C8—24no—
265C4⋊C8—472no—
266C4⋊C8—472no—
267C4⋊C8—941no—
268C4⋊C8—941no—
269C4⋊C8C22⋊D4712yes—
270C4⋊C8C22×D4712yes—
271#103—944no—
272#103—942no—
273#102—944no—
274#102—942no—
275C22⋊C8—21no—
276C22⋊C8C22⋊D4712yes—
277C4×C8—24no—
278C4×C8—21no—
279C4×C8C22×D4712yes—
280#41—1882no—
281#41—1882no—
282C2×Q8—21no—
283C2×Q8—22no—
284C2×Q8—941no—
285C2×Q8—942no—
286C2×Q8—1882no—
287C2×Q8—1882no—
288C2×Q8—1881no—
289C2×Q8D37614yes—
290C2×Q8C2×D18812yes—
291C2×D4—941no—
292C2×D4—941no—
293C2×D4—941no—
294C2×D4—1881no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—1881no—
298C22×C4—1881no—
299C22×C4C2×D18811yescharacteristic
300C22×C4#1062535011yescharacteristic
301C22×C4C2×C22⋊D4711yescharacteristic
302Q16—1888no—
303Q16—1884no—
304SD16—1884no—
305M16—1882no—
306C2×C8—22no—
307C2×C8—942no—
308C2×C8—942no—
309C2×C8—1882no—
310C2×C8C2×C22⋊D4714yes—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—942no—
315C4⋊C4—941no—
316C4⋊C4—942no—
317C4⋊C4—941no—
318C4⋊C4—1881no—
319C4⋊C4—1881no—
320C4⋊C4C2×D18812yes—
321C22⋊C4—1881no—
322C22⋊C4—1881no—
323C42—22no—
324C42—22no—
325C42C2×D18811yescharacteristic
326C42#1062535011yescharacteristic
327C42C2×C22⋊D4711yescharacteristic
328C42C23×D4711yescharacteristic
329C23—1881no—
330C23#1309399811yescharacteristic
331Q8—24no—
332Q8—22no—
333Q8—3762no—
334Q8—1882no—
335Q8—1881no—
336Q8—1882no—
337Q8—1881no—
338Q8C2×D37614yes—
339D4—3761no—
340D4—3762no—
341D4—1881no—
342D4—1881no—
343D4—1882no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—3762no—
351C2×C4—3762no—
352C2×C4—1881no—
353C2×C4—1882no—
354C2×C4#1309399811yescharacteristic
355C2×C4#1309399811yescharacteristic
356C2×C4#1309399811yescharacteristic
357C2×C4C2×D37612yes—
358C2×C4#1309404612yes—
359C2×C4C22×C22⋊D4711yescharacteristic
360C2×C4C22×D18811yescharacteristic
361C2×C4C2×#1062535011yescharacteristic
362C8—24no—
363C8—42no—
364C8—1882no—
365C8—1882no—
366C22—21no—
367C22—21no—
368C22—3761no—
369C22—3761no—
370C22—3761no—
371C22#1568935912yes—
372C22C2×#1309399811yescharacteristic
373C4—24no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—3761no—
379C4—3762no—
380C4#1568935912yes—
381C4C22×D37611yescharacteristic
382C4C2×#1309404611yescharacteristic
383C4#1568993812yes—
384C2—22no—
385C2—7521no—
386C2#2025291811yescharacteristic
387C2C2×#1568935911yescharacteristic
388C2#2025312511yescharacteristic
subgroupG/H
centerC22C2×#13093998
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#2982C2
socleC22×C47C2×#121
prime radical#15689673C22
hypercenter#2982D47
hypocenterC47#19404
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
3. C2×#15689333C2semidirect
4. #20242832C2semidirect
5. #20242782C2semidirect
6. #20242163C2semidirect
7. #20242059C2semidirect
9. C47×#2982C2semidirect
193. #2982D47semidirect
231. C47#19404semidirect

Maximal subgroups

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

subgroupindex
#202545112
#202528602
C2×#156893332
#202428322
#202427822
#202421632
#202420592
#202420352
C47×#29822
#1940447