Finite Group Databasebeta

← #26370918

Subgroups — #26370918

Order 12032.

546 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
1C4×#13094046C211yesmaximal, characteristic
2#20245549C212yesmaximal, complemented
3#20245542C212yesmaximal, complemented
4#20245537C211yesmaximal, characteristic, complemented
5#20245532C211yesmaximal, characteristic, complemented
6#20245527C211yesmaximal, characteristic, complemented
7#20245522C211yesmaximal, characteristic, complemented
8#20245170C212yesmaximal, complemented
9C2×C4⋊(C47⋊C16)C211yesmaximal, characteristic, complemented
10#20245093C212yesmaximal
11C47×#3160C211yesmaximal, characteristic, complemented
12C2×#13094046C2211yescharacteristic
13C4×#10625350—21no—
14C22×C47⋊C16C2212yes—
15C4×Dic188C2211yescharacteristic
16C4×Dic188C2211yescharacteristic
17C4×#10625325—21no—
18C4×#10625325—21no—
19C4×D376C2211yescharacteristic
20C4×D376C2211yescharacteristic
21#15689161C2211yescharacteristic
22#15689120—21no—
23#15688950C2212yes—
24#15688950C2212yes—
25#15688948C2212yes—
26#15688948C2212yes—
27C4⋊(C47⋊C16)C2211yescharacteristic
28C4⋊(C47⋊C16)C2211yescharacteristic
29C4⋊(C47⋊C16)C2212yes—
30#15688926C2212yes—
31#15688926C2212yes—
32#15688923C2212yes—
33#15688923C2212yes—
34C47×#533—21no—
35C47×#500C2211yescharacteristic
36C47×#500C2211yescharacteristic
37C47×#499—21no—
38C47×#499C2212yes—
39C47×#498—21no—
40C47×#427—21no—
41C47×#427C2212yes—
42C2×C4×C8×C47C2211yescharacteristic
43C2×#10625350—21no—
44#13094046—22no—
45#13094046Q814yescomplemented
46C2×Dic188C2311yescharacteristic
47C2×#10625325—21no—
48C2×D376C2311yescharacteristic
49C4×Dic94—21no—
50C4×Dic94—21no—
51C4×D188—21no—
52C4×D188—21no—
53C4×C22⋊D47—41no—
54C4×C22⋊D47—41no—
55#13093983—21no—
56#13093981—41no—
57C42×D47—41no—
58C2×C47⋊C16—24no—
59C2×C47⋊C16C2312yes—
60C2×C47⋊C16C2312yes—
61#13093958C2311yescharacteristic
62#13093958C2311yescharacteristic
63#13093956—21no—
64#13093949—21no—
65#13093949—21no—
66#13093946—21no—
67#13093946—21no—
68C22×C8×C47D411yescharacteristic
69C22×C8×C47C2311yescharacteristic
70C4⋊Q8×C47—21no—
71C4⋊Q8×C47—21no—
72#125×C47—22no—
73C22⋊Q8×C47—41no—
74C22⋊Q8×C47—41no—
75C4×Q8×C47—41no—
76C4×Q8×C47—41no—
77#117×C47—22no—
78C2×C42×C47D411yescharacteristic
79#107×C47—21no—
80#107×C47—21no—
81#107×C47C2312yes—
82#107×C47C2312yes—
83#106×C47—22no—
84#106×C47—22no—
85C4×C8×C47D411yescharacteristic
86C4×C8×C47D411yescharacteristic
87C4×C8×C47C2311yescharacteristic
88C4×C8×C47C2311yescharacteristic
89C2×Q8×C47—41no—
90C2×Q8×C47—41no—
91C22×C4×C47D811yescharacteristic
92C22×C4×C47D811yescharacteristic
93C22×C4×C47C2×D411yescharacteristic
94C2×C22⋊D47—41no—
95#10625350—24no—
96#10625350—42no—
97C2×D188—21no—
98C2×C4×D47—41no—
99C2×C4×D47—41no—
100C2×C4×D47—41no—
101C2×Dic94—21no—
102C2×C8×C47—22no—
103C2×C8×C47C2×D411yescharacteristic
104C2×C8×C47C2×D411yescharacteristic
105C2×C8×C47C2×D411yescharacteristic
106C2×C8×C47C2×Q812yes—
107C2×C8×C47#4112yes—
108C2×C8×C47C2411yescharacteristic
109C4⋊C4×C47—22no—
110C4⋊C4×C47—22no—
111C4⋊C4×C47—42no—
112C4⋊C4×C47—42no—
113C4⋊C4×C47—41no—
114C4⋊C4×C47—41no—
115C22⋊C4×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—
124C4⋊Dic47—21no—
125C4⋊Dic47—21no—
126Dic47⋊C4—41no—
127Dic47⋊C4—41no—
128C4×Dic47—41no—
129C4×Dic47—41no—
130C4×Dic47—41no—
131Dic188—21no—
132Dic188C2×Q812yes—
133D376—21no—
134D376C2×Q812yes—
135#10625325—24no—
136C47⋊C16—22no—
137C47⋊C16—22no—
138C23×C47C2×D811yescharacteristic
139C22×D47—41no—
140Q8×C47—82no—
141Q8×C47—82no—
142C2×C4×C47—22no—
143C2×C4×C47—41no—
144C2×C4×C47—42no—
145C2×C4×C47—42no—
146C2×C4×C47C22⋊Q812yes—
147C2×C4×C47C2×D811yescharacteristic
148C2×C4×C47C2×D811yescharacteristic
149C2×C4×C47C2×D811yescharacteristic
150C2×C4×C47C2×D811yescharacteristic
151C2×C4×C47C2×D811yescharacteristic
152C2×C4×C47#13712yes—
153C2×C4×C47C22×D411yescharacteristic
154C22⋊D47—44no—
155C2×Dic47—41no—
156C2×Dic47—41no—
157C2×Dic47—41no—
158D188—22no—
159D188—41no—
160C4×D47—44no—
161C4×D47—82no—
162C4×D47—82no—
163Dic94—22no—
164Dic94—41no—
165C8×C47—21no—
166C8×C47—21no—
167C8×C47C22⋊Q812yes—
168C8×C47C22×Q811yescharacteristic
169C8×C47C2×#4111yescharacteristic
170#33626—471nomaximal, Sylow
171C22×C47—21no—
172C22×C47—21no—
173C22×C47#47412yes—
174C22×C47C22×D811yescharacteristic
175C2×D47—42no—
176C2×D47—81no—
177C2×D47—81no—
178C4×C47—21no—
179C4×C47—21no—
180C4×C47—21no—
181C4×C47—82no—
182C4×C47—82no—
183C4×C47#47412yes—
184C4×C47#50812yes—
185C4×C47C2×C22⋊Q811yescharacteristic
186C4×C47C2×#13711yescharacteristic
187Dic47—42no—
188Dic47—81no—
189Dic47—81no—
190#3160D4711yescharacteristic, complemented
191C4×#135—471no—
192#2229—472no—
193#2228—472no—
194#2227—471no—
195#2226—471no—
196#2225—471no—
197#2224—471no—
198#2158—472no—
199C2×C4⋊C16—471no—
200#2138—472no—
201C2×C47—22no—
202C2×C47C2×#47411yescharacteristic
203C2×C47C2×#50811yescharacteristic
204C2×C47#341411yescharacteristic
205D47—82no—
206C2×#135—471no—
207#533—21no—
208C4×#41—941no—
209C22×C16—472no—
210#500C2×D4711yescharacteristic
211#500C2×D4711yescharacteristic
212#499—21no—
213#499C2×D4712yes—
214#498—21no—
215C4×Q16—471no—
216C4×Q16—471no—
217C4×SD16—941no—
218C4×SD16—941no—
219C4×D8—471no—
220C4×D8—471no—
221#427—21no—
222#427—471no—
223#427C2×D4712yes—
224#416—941no—
225C2×C4×C8C2×D4711yescharacteristic
226#367—472no—
227#367—472no—
228#366—472no—
229#366—472no—
230C4⋊C16—471no—
231C4⋊C16—471no—
232C4⋊C16—472no—
233#358—472no—
234#358—472no—
235#357—472no—
236#357—472no—
237C47#3362611yescharacteristic, complemented, Sylow
238C2×#41—941no—
239#135—474no—
240#135—942no—
241C2×Q16—471no—
242C2×SD16—941no—
243C2×D8—471no—
244C22×C8C22⋊D4711yescharacteristic
245C22×C8C22×D4711yescharacteristic
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248#125—22no—
249C22⋊Q8—41no—
250C22⋊Q8—41no—
251C4×Q8—41no—
252C4×Q8—41no—
253C4×Q8—941no—
254C4×Q8—941no—
255C4×D4—941no—
256C4×D4—941no—
257C4×D4—1881no—
258C4×D4—1881no—
259#117—22no—
260#117—941no—
261#117—1881no—
262C2×C42—1881no—
263C2×C42C22⋊D4711yescharacteristic
264C2×C16—472no—
265C2×C16—472no—
266C2×C16—944no—
267#107—21no—
268#107—21no—
269#107—471no—
270#107—471no—
271#107C22×D4712yes—
272#107C22×D4712yes—
273#106—22no—
274#106—22no—
275#106—941no—
276#103—941no—
277#103—941no—
278#102—941no—
279#102—941no—
280C4×C8C22⋊D4711yescharacteristic
281C4×C8C22⋊D4711yescharacteristic
282C4×C8C22×D4711yescharacteristic
283C4×C8C22×D4711yescharacteristic
284#41—944no—
285#41—1882no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—941no—
289C2×D4—941no—
290C2×D4—1881no—
291C22×C4—1881no—
292C22×C4—1881no—
293C22×C4—1881no—
294C22×C4#1062533411yescharacteristic
295C22×C4#1062533411yescharacteristic
296C22×C4C2×C22⋊D4711yescharacteristic
297Q16—472no—
298Q16—941no—
299SD16—944no—
300D8—472no—
301D8—941no—
302C2×C8—22no—
303C2×C8Q8×D4712yes—
304C2×C8#1062535412yes—
305C2×C8C2×C22⋊D4711yescharacteristic
306C2×C8C2×C22⋊D4711yescharacteristic
307C2×C8C2×C22⋊D4711yescharacteristic
308C2×C8C23×D4711yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—941no—
316C4⋊C4—941no—
317C4⋊C4—1881no—
318C4⋊C4—1881no—
319C22⋊C4—42no—
320C22⋊C4—1881no—
321C22⋊C4—1881no—
322C22⋊C4—1881no—
323C42—42no—
324C42—1881no—
325C42—1881no—
326C42—1881no—
327C42#1062533411yescharacteristic
328C42#1062533411yescharacteristic
329C42C2×C22⋊D4711yescharacteristic
330C42C2×C22⋊D4711yescharacteristic
331C16—942no—
332C16—942no—
333C23—1881no—
334C23C2×#1062533411yescharacteristic
335Q8—82no—
336Q8—82no—
337Q8—942no—
338Q8—1881no—
339D4—942no—
340D4—1884no—
341D4—1881no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—3762no—
347C2×C4—3762no—
348C2×C4—1881no—
349C2×C4—1881no—
350C2×C4—1881no—
351C2×C4—1884no—
352C2×C4#1309400012yes—
353C2×C4C2×#1062533411yescharacteristic
354C2×C4C2×#1062533411yescharacteristic
355C2×C4C2×#1062533411yescharacteristic
356C2×C4C2×#1062533411yescharacteristic
357C2×C4C2×#1062533411yescharacteristic
358C2×C4#1309405512yes—
359C2×C4C22×C22⋊D4711yescharacteristic
360C8—21no—
361C8—21no—
362C8#1309400012yes—
363C8C2×Q8×D4711yescharacteristic
364C8C2×#1062535411yescharacteristic
365C22—21no—
366C22—21no—
367C22—3761no—
368C22—3761no—
369C22—1882no—
370C22#1568940012yes—
371C22C22×#1062533411yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—82no—
376C4—82no—
377C4—3761no—
378C4—3761no—
379C4—1882no—
380C4#1568940012yes—
381C4#1568955312yes—
382C4C2×#1309400011yescharacteristic
383C4C2×#1309405511yescharacteristic
384C2—22no—
385C2—3762no—
386C2C2×#1568940011yescharacteristic
387C2C2×#1568955311yescharacteristic
388C2#2025652211yescharacteristic
subgroupG/H
centerC2×C4C2×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#3160C2
socleC22×C47C22×D8
prime radicalC2×#13094046C22
hypercenter#3160D47
hypocenterC47#33626
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3160

Complements

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

NH ≅ G/Ntype
2. #20245549C2semidirect
3. #20245542C2semidirect
4. #20245537C2semidirect
5. #20245532C2semidirect
6. #20245527C2semidirect
7. #20245522C2semidirect
8. #20245170C2semidirect
9. C2×C4⋊(C47⋊C16)C2semidirect
11. C47×#3160C2semidirect
45. #13094046Q8semidirect
190. #3160D47semidirect
237. C47#33626semidirect

Maximal subgroups

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

subgroupindex
C4×#130940462
#202455492
#202455422
#202455372
#202455322
#202455272
#202455222
#202451702
C2×C4⋊(C47⋊C16)2
#202450932
C47×#31602
#3362647