Finite Group Databasebeta

← #26190465

Subgroups — #26190465

Order 12032.

976 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689754C211yesmaximal, characteristic, complemented
2#20254194C212yesmaximal
3#20242429C214yesmaximal, complemented
4#20242327C214yesmaximal, complemented
5#20241907C212yesmaximal, complemented
6#20241718C211yesmaximal, characteristic, complemented
7C47×#2591C211yesmaximal, characteristic
8#15689754—24no—
9#15689754—22no—
10#15689754C2214yes—
11C2×#13094023—22no—
12C2×#13094015—22no—
13C2×#13094003—21no—
14C2×#13094003C2212yes—
15C2×#13093983—21no—
16#15689458C2214yes—
17#15689434—22no—
18#15689367—24no—
19#15689274C2214yes—
20#15689261—22no—
21#15689087C2212yes—
22#15688859C2218yes—
23#15688849C2214yes—
24C2×#121×C47C2211yescharacteristic
25C22×C4⋊C4×C47C2211yescharacteristic
26C47×#400C2212yes—
27C47×#394C2212yes—
28C47×#388—22no—
29C47×#386—21no—
30C47×#386—21no—
31C4×C22⋊C4×C47C2211yescharacteristic
32C22×Dic94—21no—
33C22×Dic94D412yes—
34C2×#10625335—24no—
35C2×#10625329—22no—
36#13094023—28no—
37#13094023—28no—
38#13094015—28no—
39#13094015—28no—
40#13094003—24no—
41#13094003—22no—
42#13094003—22no—
43#13094003—24no—
44#13094003D414yes—
45#13094003C2314yes—
46#13093983—24no—
47#13093983—24no—
48C2×C4⋊Dic47—21no—
49C2×C4⋊Dic47—21no—
50C2×C4⋊Dic47—21no—
51C2×C4⋊Dic47D412yes—
52C2×Dic47⋊C4—22no—
53C2×Dic47⋊C4—22no—
54C2×C22⋊Dic47—22no—
55C2×C4×Dic47—22no—
56#13093957—24no—
57#13093947—24no—
58#13093945—24no—
59C22⋊(C47⋊C8)—22no—
60C22⋊(C47⋊C8)C2314yes—
61C22×D4×C47—21no—
62C22×D4×C47C2311yescharacteristic
63C23×C4×C47D411yescharacteristic
64C23×C4×C47D411yescharacteristic
65C23×C4×C47C2311yescharacteristic
66#121×C47—22no—
67#121×C47—22no—
68C2×C4⋊C4×C47—22no—
69C2×C4⋊C4×C47—22no—
70C2×C4⋊C4×C47D412yes—
71C2×C4⋊C4×C47C2311yescharacteristic
72C2×C4⋊C4×C47C2312yes—
73C2×C22⋊C4×C47—21no—
74C2×C22⋊C4×C47—21no—
75C2×C22⋊C4×C47—22no—
76C2×C22⋊C4×C47—22no—
77C2×C42×C47—21no—
78#95×C47—22no—
79#95×C47—22no—
80#95×C47C2312yes—
81C24×C47—21no—
82C24×C47C2×D411yescharacteristic
83C2×D4×C47—22no—
84C2×D4×C47—44no—
85C2×D4×C47—41no—
86C2×D4×C47—41no—
87C2×D4×C47—81no—
88C22×C4×C47—21no—
89C22×C4×C47—21no—
90C22×C4×C47—24no—
91C22×C4×C47—21no—
92C22×C4×C47—21no—
93C22×C4×C47—22no—
94C22×C4×C47—21no—
95C22×C4×C47—81no—
96C22×C4×C47—81no—
97C22×C4×C47C2×D412yes—
98C22×C4×C47C2×D412yes—
99C22×C4×C47C2×D411yescharacteristic
100C22×C4×C47C2×D412yes—
101C22×C4×C47C2×D412yes—
102C22×C4×C47C2411yescharacteristic
103C22×Dic47—22no—
104C22×Dic47—22no—
105C2×Dic94—24no—
106C2×Dic94—22no—
107C2×Dic94—82no—
108C2×Dic94—82no—
109C2×Dic94C2×D414yes—
110C4⋊C4×C47—24no—
111C4⋊C4×C47—24no—
112C4⋊C4×C47—22no—
113C22⋊C4×C47—42no—
114C22⋊C4×C47—42no—
115C22⋊C4×C47—44no—
116C22⋊C4×C47—82no—
117C42×C47—81no—
118C22⋊Dic47—24no—
119C22⋊Dic47—24no—
120#10625335—84no—
121C4⋊Dic47—24no—
122C4⋊Dic47—24no—
123C4⋊Dic47—24no—
124C4⋊Dic47—22no—
125C4⋊Dic47C2×D414yes—
126Dic47⋊C4—28no—
127Dic47⋊C4—24no—
128Dic47⋊C4—24no—
129C4×Dic47—28no—
130#10625329—82no—
131C2×C47⋊C8—24no—
132C23×C47—22no—
133C23×C47—22no—
134C23×C47—41no—
135C23×C47—42no—
136C23×C47—42no—
137C23×C47—41no—
138C23×C47—42no—
139C23×C47#12011yescharacteristic
140C23×C47#12011yescharacteristic
141C23×C47C22×D411yescharacteristic
142D4×C47—44no—
143D4×C47—82no—
144D4×C47—82no—
145C2×C4×C47—24no—
146C2×C4×C47—24no—
147C2×C4×C47—22no—
148C2×C4×C47—22no—
149C2×C4×C47—22no—
150C2×C4×C47—42no—
151C2×C4×C47—41no—
152C2×C4×C47—42no—
153C2×C4×C47—41no—
154C2×C4×C47—42no—
155C2×C4×C47—42no—
156C2×C4×C47—42no—
157C2×C4×C47—42no—
158C2×C4×C47—42no—
159C2×C4×C47—41no—
160C2×C4×C47—42no—
161C2×C4×C47—81no—
162C2×C4×C47#12012yes—
163C2×C4×C47#13714yes—
164C2×C4×C47C22×D412yes—
165C2×Dic47—24no—
166C2×Dic47—24no—
167C2×Dic47—44no—
168C2×Dic47—44no—
169Dic94—44no—
170Dic94—82no—
171Dic94—84no—
172C47⋊C8—44no—
173#18326—471nomaximal, Sylow
174C22×C47—21no—
175C22×C47—21no—
176C22×C47—21no—
177C22×C47—21no—
178C22×C47—21no—
179C22×C47—21no—
180C22×C47—21no—
181C22×C47—44no—
182C22×C47—84no—
183C22×C47—81no—
184C22×C47—81no—
185C22×C47—81no—
186C22×C47#45412yes—
187C22×C47C2×#12011yescharacteristic
188C22×C47C2×#13712yes—
189C4×C47—24no—
190C4×C47—22no—
191C4×C47—22no—
192C4×C47—42no—
193C4×C47—82no—
194C4×C47—81no—
195Dic47—44no—
196Dic47—44no—
197C2×#544—471no—
198#3193—472no—
199#2591D4711yescharacteristic
200#1617—474no—
201#1599—474no—
202#1508—472no—
203#1463—471no—
204C2×C47—22no—
205C2×C47—22no—
206C2×C47—41no—
207C2×C47—161no—
208C2×C47#300411yescharacteristic
209C2×C47C2×#45411yescharacteristic
210C2×C47#302611yescharacteristic
211#544—474no—
212#544—944no—
213#544—942no—
214C2×C4⋊Q8—942no—
215C2×#125—942no—
216C2×C22⋊Q8—472no—
217C2×C22⋊Q8—941no—
218C2×#121C2×D4711yescharacteristic
219C2×#117—941no—
220C22×C4⋊C4C2×D4711yescharacteristic
221#484—474no—
222#480—942no—
223#468—944no—
224#450—474no—
225#448—942no—
226#407—472no—
227#400C2×D4712yes—
228#394C2×D4712yes—
229#388—22no—
230#386—21no—
231#386—21no—
232C4×C22⋊C4C2×D4711yescharacteristic
233#328—478no—
234#324—474no—
235C47#1832611yescharacteristic, complemented, Sylow
236C22×Q8—472no—
237C22×Q8—941no—
238C22×D4—21no—
239C22×D4C22×D4711yescharacteristic
240C23×C4C22⋊D4711yescharacteristic
241C23×C4C22⋊D4711yescharacteristic
242C23×C4C22×D4711yescharacteristic
243C2×SD16—944no—
244C2×M16—942no—
245C4⋊Q8—948no—
246C4⋊Q8—948no—
247#125—948no—
248#125—948no—
249C22⋊Q8—474no—
250C22⋊Q8—474no—
251C22⋊Q8—944no—
252C22⋊Q8—942no—
253C22⋊Q8—942no—
254C22⋊Q8—944no—
255#121—22no—
256#121—22no—
257#117—944no—
258#117—944no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—472no—
262C2×C4⋊C4—941no—
263C2×C4⋊C4—941no—
264C2×C4⋊C4—941no—
265C2×C4⋊C4—942no—
266C2×C4⋊C4—942no—
267C2×C4⋊C4C22⋊D4712yes—
268C2×C4⋊C4C22×D4711yescharacteristic
269C2×C4⋊C4C22×D4712yes—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—942no—
275C2×C42—21no—
276C2×C42—942no—
277#107—944no—
278#103—944no—
279#102—944no—
280C22⋊C8—474no—
281C22⋊C8—942no—
282#95—22no—
283#95—22no—
284#95C22×D4712yes—
285C24—21no—
286C24C2×C22⋊D4711yescharacteristic
287C2×Q8—474no—
288C2×Q8—944no—
289C2×Q8—942no—
290C2×Q8—3762no—
291C2×Q8—3762no—
292C2×D4—22no—
293C2×D4—44no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—81no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—81no—
305C22×C4—81no—
306C22×C4—942no—
307C22×C4—942no—
308C22×C4D4×D4712yes—
309C22×C4D4×D4712yes—
310C22×C4C2×C22⋊D4712yes—
311C22×C4C2×C22⋊D4712yes—
312C22×C4C2×C22⋊D4711yescharacteristic
313C22×C4C23×D4711yescharacteristic
314SD16—3764no—
315M16—3762no—
316C2×C8—944no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4—474no—
321C4⋊C4—948no—
322C4⋊C4—944no—
323C4⋊C4—944no—
324C4⋊C4—944no—
325C4⋊C4—944no—
326C4⋊C4—942no—
327C4⋊C4—944no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—44no—
331C22⋊C4—82no—
332C22⋊C4—944no—
333C22⋊C4—944no—
334C42—81no—
335C42—948no—
336C23—22no—
337C23—22no—
338C23—41no—
339C23—42no—
340C23—42no—
341C23—41no—
342C23—42no—
343C23#1309399211yescharacteristic
344C23#1309399211yescharacteristic
345C23C22×C22⋊D4711yescharacteristic
346Q8—3762no—
347Q8—3764no—
348Q8—1884no—
349D4—44no—
350D4—82no—
351D4—82no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—81no—
369C2×C4—944no—
370C2×C4—944no—
371C2×C4—1884no—
372C2×C4—1884no—
373C2×C4#1309399212yes—
374C2×C4#1309405414yes—
375C2×C4C2×D4×D4712yes—
376C8—1884no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—44no—
385C22—84no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22#1568929512yes—
390C22C2×#1309399211yescharacteristic
391C22C2×#1309405412yes—
392C4—22no—
393C4—22no—
394C4—24no—
395C4—42no—
396C4—82no—
397C4—81no—
398C4—1884no—
399C4—1884no—
400C2—22no—
401C2—22no—
402C2—41no—
403C2—161no—
404C2#2025262111yescharacteristic
405C2C2×#1568929511yescharacteristic
406C2#2025278411yescharacteristic
subgroupG/H
centerC22C2×#13093992
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2591C2
socleC22×C47C2×#120
prime radicalC2×#121×C47C22
hypercenter#2591D47
hypocenterC47#18326
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2591

Complements

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

NH ≅ G/Ntype
1. C2×#15689754C2semidirect
3. #20242429C2semidirect
4. #20242327C2semidirect
5. #20241907C2semidirect
6. #20241718C2semidirect
235. C47#18326semidirect

Maximal subgroups

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

subgroupindex
C2×#156897542
#202541942
#202424292
#202423272
#202419072
#202417182
C47×#25912
#1832647