Finite Group Databasebeta

← #26371373

Subgroups — #26371373

Order 12032.

548 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#20253958C211yesmaximal, characteristic
2#20245590C211yesmaximal, characteristic, complemented
3#20245573C211yesmaximal, characteristic
4#20245552C212yesmaximal, complemented
5#20245450C212yesmaximal, complemented
6#20245322C211yesmaximal, characteristic, complemented
7#20245305C211yesmaximal, characteristic
8#20245257C212yesmaximal, complemented
9#20245250C212yesmaximal, complemented
10#20245191C211yesmaximal, characteristic, complemented
11C2×C4×Q16×C47C211yesmaximal, characteristic, complemented
12#15689657—21no—
13C2×#13093970C2212yes—
14C2×#13093967C2212yes—
15#15689525C2211yescharacteristic
16#15689507C2211yescharacteristic
17#15689478—21no—
18#15689364C2211yescharacteristic
19#15689359C2211yescharacteristic
20#15689352—21no—
21C4×Dic188C2211yescharacteristic
22C4×#10625325—21no—
23C4×D376C2211yescharacteristic
24#15688950C2212yes—
25#15688948C2212yes—
26C4⋊(C47⋊C16)C2212yes—
27#15688927C2212yes—
28#15688927C2212yes—
29#15688926C2212yes—
30#15688923C2212yes—
31C22⋊(C47⋊C16)C2211yescharacteristic
32C22⋊(C47⋊C16)C2211yescharacteristic
33#15688899C2212yes—
34C22×Q16×C47C2211yescharacteristic
35C2×C4×Q8×C47—21no—
36C4×Q16×C47—21no—
37C4×Q16×C47—21no—
38C4×Q16×C47C2212yes—
39C4×Q16×C47C2212yes—
40C2×#107×C47C2211yescharacteristic
41C2×#103×C47—21no—
42C2×C4×C8×C47C2211yescharacteristic
43C2×Dic188C2311yescharacteristic
44C2×#10625325—21no—
45C2×D376C2311yescharacteristic
46#13094018—41no—
47#13094016—21no—
48#13094012—21no—
49#13094007—41no—
50#13094003—21no—
51#13093998—21no—
52C4×Dic94—21no—
53C4×D188—21no—
54C4×C22⋊D47—41no—
55#13093981—41no—
56#13093970—24no—
57#13093967—24no—
58C2×C47⋊C16C2312yes—
59C2×C47⋊C16C2312yes—
60#13093958C2311yescharacteristic
61#13093958C2311yescharacteristic
62#13093956—21no—
63#13093949—21no—
64#13093949—21no—
65#13093946—21no—
66#13093946—21no—
67C22×Q8×C47—21no—
68C2×Q16×C47—24no—
69C2×Q16×C47—21no—
70C2×Q16×C47C2312yes—
71C22×C8×C47D411yescharacteristic
72C22×C8×C47C2311yescharacteristic
73C4×Q8×C47—22no—
74C4×Q8×C47—22no—
75C4×Q8×C47—41no—
76C4×Q8×C47—41no—
77C2×C4⋊C4×C47—21no—
78C2×C4⋊C4×C47—41no—
79C2×C42×C47—41no—
80C2×C42×C47D411yescharacteristic
81#107×C47—21no—
82#107×C47C2312yes—
83#103×C47—22no—
84#103×C47—22no—
85C4×C8×C47D411yescharacteristic
86C4×C8×C47D411yescharacteristic
87C4×C8×C47C2311yescharacteristic
88C4×C8×C47C2311yescharacteristic
89C2×Q8×C47—24no—
90C2×Q8×C47—22no—
91C2×Q8×C47—42no—
92C2×Q8×C47—41no—
93C22×C4×C47—41no—
94C22×C4×C47—41no—
95C22×C4×C47D811yescharacteristic
96C22×C4×C47D811yescharacteristic
97C22×C4×C47C2×D411yescharacteristic
98C2×C22⋊D47—41no—
99C2×D188—21no—
100C2×C4×D47—41no—
101C2×Dic94—21no—
102Q16×C47—24no—
103Q16×C47—22no—
104Q16×C47#4114yes—
105C2×C8×C47—22no—
106C2×C8×C47—41no—
107C2×C8×C47C2×D411yescharacteristic
108C2×C8×C47C2×D411yescharacteristic
109C2×C8×C47C2×D411yescharacteristic
110C2×C8×C47C2411yescharacteristic
111C4⋊C4×C47—22no—
112C4⋊C4×C47—41no—
113C4⋊C4×C47—42no—
114C4⋊C4×C47—42no—
115C42×C47—42no—
116C42×C47—42no—
117C42×C47D811yescharacteristic
118C42×C47D811yescharacteristic
119C42×C47C2×D411yescharacteristic
120C42×C47C2×D411yescharacteristic
121C22⋊Dic47—41no—
122(C2×D47)⋊C4—41no—
123(C2×D47)⋊C4—41no—
124(C2×D47)⋊C4—41no—
125(C2×D47)⋊C4—41no—
126C4⋊Dic47—21no—
127C4⋊Dic47—21no—
128Dic47⋊C4—41no—
129Dic47⋊C4—41no—
130Dic47⋊C4—41no—
131Dic47⋊C4—41no—
132C4×Dic47—41no—
133Dic188—22no—
134D376—22no—
135#10625325—42no—
136C47⋊C16—22no—
137C47⋊C16—22no—
138C23×C47C2×D811yescharacteristic
139C22×D47—41no—
140Q8×C47—24no—
141Q8×C47—44no—
142Q8×C47—42no—
143C2×C4×C47—22no—
144C2×C4×C47—21no—
145C2×C4×C47—21no—
146C2×C4×C47—41no—
147C2×C4×C47—44no—
148C2×C4×C47—42no—
149C2×C4×C47—42no—
150C2×C4×C47—82no—
151C2×C4×C47C2×D811yescharacteristic
152C2×C4×C47C2×D811yescharacteristic
153C2×C4×C47C2×D811yescharacteristic
154C2×C4×C47C2×D811yescharacteristic
155C2×C4×C47C2×D811yescharacteristic
156C2×C4×C47C22×D411yescharacteristic
157C22⋊D47—82no—
158C2×Dic47—41no—
159C2×Dic47—41no—
160C2×Dic47—41no—
161D188—41no—
162D188—41no—
163C4×D47—82no—
164Dic94—41no—
165Dic94—41no—
166C8×C47—21no—
167C8×C47—21no—
168C8×C47—21no—
169C8×C47C2×#4112yes—
170#33677—471nomaximal, Sylow
171C22×C47—21no—
172C22×C47—21no—
173C22×C47#51012yes—
174C22×C47C22×D811yescharacteristic
175C2×D47—81no—
176C2×D47—81no—
177C2×D47—81no—
178C4×C47—21no—
179C4×C47—21no—
180C4×C47—21no—
181C4×C47—21no—
182C4×C47—21no—
183C4×C47—44no—
184C4×C47—82no—
185C4×C47#52511yescharacteristic
186C4×C47#57511yescharacteristic
187Dic47—81no—
188Dic47—81no—
189Dic47—81no—
190#3168—471no—
191C2×C4×Q16D4711yescharacteristic, complemented
192#2236—471no—
193#2233—471no—
194#2229—472no—
195#2211—472no—
196#2188—471no—
197#2185—471no—
198#2176—472no—
199#2175—472no—
200#2163—471no—
201C2×C47—22no—
202C2×C47#310011yescharacteristic
203C2×C47C2×#51011yescharacteristic
204C2×C47#341411yescharacteristic
205D47—161no—
206C22×Q16C2×D4711yescharacteristic
207#529—941no—
208C2×C4×Q8—21no—
209C2×Q32—472no—
210C2×SD32—472no—
211#499—471no—
212#495—471no—
213#488—941no—
214#467—471no—
215#466—471no—
216#465—941no—
217C4×Q16—21no—
218C4×Q16—21no—
219C4×Q16—471no—
220C4×Q16C2×D4712yes—
221C4×Q16C2×D4712yes—
222C4×SD16—941no—
223C4×D8—471no—
224C2×#107C2×D4711yescharacteristic
225C2×#103—21no—
226C2×C4×C8C2×D4711yescharacteristic
227#367—472no—
228#366—472no—
229C4⋊C16—472no—
230#358—472no—
231#358—472no—
232#358—472no—
233#357—472no—
234C22⋊C16—471no—
235C22⋊C16—471no—
236#346—472no—
237C47#3367711yescharacteristic, complemented, Sylow
238C22×Q8—21no—
239C2×Q16—24no—
240C2×Q16—21no—
241C2×Q16—471no—
242C2×Q16C22×D4712yes—
243C2×SD16—941no—
244C2×D8—471no—
245C22×C8C22⋊D4711yescharacteristic
246C22×C8C22×D4711yescharacteristic
247#126—1881no—
248#125—941no—
249#124—941no—
250#123—1881no—
251C22⋊Q8—941no—
252#121—941no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—41no—
256C4×Q8—41no—
257C4×Q8—941no—
258C4×D4—941no—
259C4×D4—1881no—
260#117—1881no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—41no—
263C2×C42—41no—
264C2×C42C22⋊D4711yescharacteristic
265Q32—944no—
266SD32—944no—
267C2×C16—472no—
268C2×C16—472no—
269#107—21no—
270#107—471no—
271#107—471no—
272#107C22×D4712yes—
273#106—941no—
274#103—22no—
275#103—22no—
276#103—941no—
277#103—941no—
278#102—941no—
279#102—941no—
280C4×C8C22⋊D4711yescharacteristic
281C4×C8C22⋊D4711yescharacteristic
282C4×C8C22×D4711yescharacteristic
283C4×C8C22×D4711yescharacteristic
284C2×Q8—24no—
285C2×Q8—22no—
286C2×Q8—42no—
287C2×Q8—41no—
288C2×Q8—941no—
289C2×D4—941no—
290C2×D4—1881no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—1881no—
294C22×C4#1062533411yescharacteristic
295C22×C4#1062533411yescharacteristic
296C22×C4C2×C22⋊D4711yescharacteristic
297Q16—24no—
298Q16—22no—
299Q16—942no—
300Q16#1062535014yes—
301SD16—1882no—
302D8—942no—
303C2×C8—22no—
304C2×C8—41no—
305C2×C8C2×C22⋊D4711yescharacteristic
306C2×C8C2×C22⋊D4711yescharacteristic
307C2×C8C2×C22⋊D4711yescharacteristic
308C2×C8C23×D4711yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—941no—
314C4⋊C4—941no—
315C4⋊C4—1881no—
316C4⋊C4—1881no—
317C4⋊C4—1881no—
318C4⋊C4—1881no—
319C22⋊C4—1881no—
320C22⋊C4—1881no—
321C22⋊C4—1881no—
322C22⋊C4—1881no—
323C22⋊C4—1881no—
324C42—42no—
325C42—42no—
326C42—1881no—
327C42#1062533411yescharacteristic
328C42#1062533411yescharacteristic
329C42C2×C22⋊D4711yescharacteristic
330C42C2×C22⋊D4711yescharacteristic
331C16—942no—
332C16—942no—
333C23—1881no—
334C23C2×#1062533411yescharacteristic
335Q8—24no—
336Q8—44no—
337Q8—42no—
338Q8—1881no—
339Q8—1881no—
340D4—3762no—
341D4—1881no—
342D4—1881no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—3762no—
352C2×C4—1881no—
353C2×C4—1881no—
354C2×C4—1881no—
355C2×C4C2×#1062533411yescharacteristic
356C2×C4C2×#1062533411yescharacteristic
357C2×C4C2×#1062533411yescharacteristic
358C2×C4C2×#1062533411yescharacteristic
359C2×C4C2×#1062533411yescharacteristic
360C2×C4C22×C22⋊D4711yescharacteristic
361C8—21no—
362C8—21no—
363C8—21no—
364C8C2×#1062535012yes—
365C22—21no—
366C22—21no—
367C22—3761no—
368C22—3761no—
369C22—3761no—
370C22#1568956412yes—
371C22C22×#1062533411yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—44no—
378C4—82no—
379C4—3761no—
380C4—3761no—
381C4—3761no—
382C4#1568963411yescharacteristic
383C4#1568992111yescharacteristic
384C2—22no—
385C2—7521no—
386C2#2025338811yescharacteristic
387C2C2×#1568956411yescharacteristic
388C2#2025652211yescharacteristic
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC2×C4×Q16×C47C2
socleC22×C47C22×D8
prime radical#15689359C22
hypercenterC2×C4×Q16D47
hypocenterC47#33677
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
2. #20245590C2semidirect
4. #20245552C2semidirect
5. #20245450C2semidirect
6. #20245322C2semidirect
8. #20245257C2semidirect
9. #20245250C2semidirect
10. #20245191C2semidirect
11. C2×C4×Q16×C47C2semidirect
191. C2×C4×Q16D47semidirect
237. C47#33677semidirect

Maximal subgroups

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

subgroupindex
#202539582
#202455902
#202455732
#202455522
#202454502
#202453222
#202453052
#202452572
#202452502
#202451912
C2×C4×Q16×C472
#3367747