Finite Group Databasebeta

← #26107710

Subgroups — #26107710

Order 12032.

748 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 748 conjugacy classes of subgroups — too many to draw. Only its 123 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×#15689744C211yesmaximal, characteristic
2#20241782C212yesmaximal, complemented
3#20241780C212yesmaximal, complemented
4#20241702C211yesmaximal, characteristic, complemented
5C47×#1458C211yesmaximal, characteristic, complemented
6#15689744—24no—
7#15689744—24no—
8C2×#13094024C411yescharacteristic
9C2×#13094019—21no—
10C2×#13094012C411yescharacteristic
11C2×#13094010—21no—
12C2×#13094006—21no—
13C2×#13094003C411yescharacteristic
14C2×#13094002—21no—
15C2×#13093998C411yescharacteristic
16C2×C4×Dic94C2211yescharacteristic
17C2×C4×D188C2211yescharacteristic
18#15689176C2212yes—
19#15688853—24no—
20#15688852—24no—
21#15688849—21no—
22#15688847—21no—
23C2×#117×C47C2211yescharacteristic
24C47×#431C2212yes—
25C47×#324—21no—
26C47×#323—21no—
27C22×Dic94—21no—
28C22×Dic94C2×C411yescharacteristic
29C22×D188C2×C411yescharacteristic
30C22×C22⋊D47—21no—
31C22×C4×D47—21no—
32#13094024—22no—
33#13094024—22no—
34#13094019—28no—
35#13094012—22no—
36#13094012—22no—
37#13094010—28no—
38#13094006—28no—
39#13094003—22no—
40#13094003—22no—
41#13094002—28no—
42#13093998—22no—
43#13093998—22no—
44C4×Dic94—22no—
45C4×Dic94D414yes—
46C4×D188—22no—
47C4×D188D414yes—
48C2×C4⋊Dic47—21no—
49C2×C4⋊Dic47C2×C411yescharacteristic
50C2×C4⋊Dic47C2×C411yescharacteristic
51C2×Dic47⋊C4—21no—
52C2×Dic47⋊C4—21no—
53C2×C22⋊Dic47—21no—
54C2×(C2×D47)⋊C4—21no—
55C2×(C2×D47)⋊C4—21no—
56C2×(C2×D47)⋊C4—21no—
57C2×(C2×D47)⋊C4—21no—
58C2×C4×Dic47—21no—
59C4⋊(C47⋊C8)—22no—
60C22⋊(C47⋊C8)—22no—
61C4×C47⋊C8—22no—
62C23×C4×C47C2×C411yescharacteristic
63#117×C47—22no—
64#117×C47—22no—
65C2×C4⋊C4×C47D411yescharacteristic
66C2×C4⋊C4×C47D411yescharacteristic
67C2×C22⋊C4×C47D411yescharacteristic
68C2×C22⋊C4×C47D411yescharacteristic
69C2×C42×C47C2×C411yescharacteristic
70C2×C42×C47C2311yescharacteristic
71C4⋊C8×C47—22no—
72C22⋊C8×C47—22no—
73C4×C8×C47—22no—
74C24×C47C22⋊C411yescharacteristic
75C23×D47—21no—
76C22×C4×C47—21no—
77C22×C4×C47—21no—
78C22×C4×C47—22no—
79C22×C4×C47—22no—
80C22×C4×C47C22⋊C411yescharacteristic
81C22×C4×C47C22⋊C411yescharacteristic
82C22×C4×C47C22⋊C411yescharacteristic
83C22×C4×C47C22×C411yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85C22×C4×C47C2×D411yescharacteristic
86C2×C22⋊D47—24no—
87C2×C22⋊D47—82no—
88C22×Dic47—21no—
89C22×Dic47—21no—
90C22×Dic47—21no—
91C2×D188—22no—
92C2×D188—22no—
93C2×D188—21no—
94C2×D188C22⋊C412yes—
95C2×C4×D47—24no—
96C2×C4×D47—82no—
97C2×Dic94—22no—
98C2×Dic94—22no—
99C2×Dic94—21no—
100C2×Dic94—24no—
101C2×Dic94—82no—
102C2×Dic94C22⋊C412yes—
103C2×C8×C47—42no—
104C4⋊C4×C47—22no—
105C4⋊C4×C47—22no—
106C22⋊C4×C47—22no—
107C22⋊C4×C47—22no—
108C42×C47—21no—
109C42×C47—21no—
110C42×C47C2×D412yes—
111C42×C47C2×D412yes—
112C22⋊Dic47—24no—
113(C2×D47)⋊C4—24no—
114(C2×D47)⋊C4—24no—
115(C2×D47)⋊C4—24no—
116(C2×D47)⋊C4—24no—
117C4⋊Dic47—24no—
118C4⋊Dic47—21no—
119C4⋊Dic47—21no—
120C4⋊Dic47C22⋊C412yes—
121C4⋊Dic47C22⋊C412yes—
122Dic47⋊C4—24no—
123Dic47⋊C4—24no—
124C4×Dic47—24no—
125C2×C47⋊C8—42no—
126C23×C47—21no—
127C23×C47—21no—
128C23×C47—21no—
129C23×C47—41no—
130C23×C47—41no—
131C23×C47C2×C22⋊C411yescharacteristic
132C22×D47—22no—
133C22×D47—42no—
134C22×D47—42no—
135C22×D47—41no—
136C22×D47—41no—
137C2×C4×C47—21no—
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47—21no—
141C2×C4×C47—21no—
142C2×C4×C47—41no—
143C2×C4×C47—41no—
144C2×C4×C47—42no—
145C2×C4×C47—42no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47C2×C22⋊C411yescharacteristic
149C2×C4×C47C2×C22⋊C411yescharacteristic
150C2×C4×C47#12012yes—
151C2×C4×C47#12012yes—
152C2×C4×C47#13112yes—
153C2×C4×C47#13112yes—
154C22⋊D47—84no—
155C2×Dic47—22no—
156C2×Dic47—22no—
157C2×Dic47—22no—
158C2×Dic47—42no—
159C2×Dic47—42no—
160C2×Dic47—41no—
161C2×Dic47—41no—
162D188—24no—
163D188—42no—
164C4×D47—84no—
165Dic94—24no—
166Dic94—42no—
167Dic94—84no—
168C8×C47—82no—
169C47⋊C8—82no—
170#8325—471nomaximal, Sylow
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—41no—
174C22×C47—41no—
175C22×C47—41no—
176C22×C47—81no—
177C22×C47—81no—
178C22×C47#37911yescharacteristic
179C22×C47#40811yescharacteristic
180C22×C47#40811yescharacteristic
181C2×D47—42no—
182C2×D47—42no—
183C2×D47—42no—
184C2×D47—82no—
185C2×D47—82no—
186C4×C47—22no—
187C4×C47—22no—
188C4×C47—41no—
189C4×C47—41no—
190C4×C47—41no—
191C4×C47#40812yes—
192C4×C47#40812yes—
193Dic47—42no—
194Dic47—42no—
195Dic47—42no—
196C2×#542—471no—
197#1479—472no—
198#1478—472no—
199#1458—471no—
200#1458D4711yescharacteristic, complemented
201C2×C47—21no—
202C2×C47—21no—
203C2×C47—81no—
204C2×C47#178011yescharacteristic
205C2×C47#179511yescharacteristic
206C2×C47#179611yescharacteristic
207D47—82no—
208#542—944no—
209#542—944no—
210C2×C4⋊Q8—471no—
211C2×#126—941no—
212C2×#124—471no—
213C2×#124—941no—
214C2×#123—941no—
215C2×C22⋊Q8—471no—
216C2×C22⋊Q8—941no—
217C2×#121—471no—
218C2×C4×Q8—471no—
219C2×C4×D4—471no—
220C2×#117C2×D4711yescharacteristic
221#431—472no—
222#431C2×D4712yes—
223Q8⋊C8—944no—
224D4⋊C8—944no—
225#324—21no—
226#324—941no—
227#323—21no—
228#323—941no—
229C47#832511yescharacteristic, complemented, Sylow
230C22×Q8—471no—
231C22×Q8—941no—
232C22×D4—471no—
233C22×D4—941no—
234C23×C4—941no—
235C23×C4C4×D4711yescharacteristic
236C4⋊Q8—942no—
237C4⋊Q8—942no—
238#126—948no—
239#124—942no—
240#124—942no—
241#124—948no—
242#123—948no—
243C22⋊Q8—948no—
244C22⋊Q8—942no—
245C22⋊Q8—942no—
246#121—942no—
247#121—942no—
248C4×Q8—474no—
249C4×Q8—942no—
250C4×D4—474no—
251C4×D4—942no—
252#117—22no—
253#117—22no—
254C2×C4⋊C4—471no—
255C2×C4⋊C4—471no—
256C2×C4⋊C4—941no—
257C2×C4⋊C4—941no—
258C2×C4⋊C4—941no—
259C2×C4⋊C4D18811yescharacteristic
260C2×C4⋊C4C22⋊D4711yescharacteristic
261C2×C22⋊C4—941no—
262C2×C22⋊C4—941no—
263C2×C22⋊C4—941no—
264C2×C22⋊C4—941no—
265C2×C22⋊C4—941no—
266C2×C22⋊C4D18811yescharacteristic
267C2×C22⋊C4C22⋊D4711yescharacteristic
268C2×C42—941no—
269C2×C42C4×D4711yescharacteristic
270C2×C42C22×D4711yescharacteristic
271C4⋊C8—22no—
272C4⋊C8—942no—
273C22⋊C8—22no—
274C22⋊C8—942no—
275C4×C8—22no—
276C4×C8—942no—
277C24—941no—
278C24(C2×D47)⋊C411yescharacteristic
279C2×Q8—472no—
280C2×Q8—944no—
281C2×Q8—941no—
282C2×Q8—942no—
283C2×Q8—942no—
284C2×Q8—3762no—
285C2×D4—472no—
286C2×D4—944no—
287C2×D4—941no—
288C2×D4—942no—
289C2×D4—942no—
290C2×D4—3762no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—941no—
296C22×C4—941no—
297C22×C4—941no—
298C22×C4—944no—
299C22×C4—3762no—
300C22×C4(C2×D47)⋊C411yescharacteristic
301C22×C4(C2×D47)⋊C411yescharacteristic
302C22×C4(C2×D47)⋊C411yescharacteristic
303C22×C4C2×C4×D4711yescharacteristic
304C22×C4C2×D18811yescharacteristic
305C22×C4C2×C22⋊D4711yescharacteristic
306C2×C8—42no—
307C2×C8—1882no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—472no—
311C4⋊C4—472no—
312C4⋊C4—944no—
313C4⋊C4—944no—
314C4⋊C4—941no—
315C4⋊C4—941no—
316C4⋊C4—944no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—944no—
320C22⋊C4—944no—
321C22⋊C4—944no—
322C22⋊C4—944no—
323C22⋊C4—944no—
324C42—21no—
325C42—21no—
326C42—944no—
327C42D4×D4712yes—
328C42D4×D4712yes—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—41no—
333C23—41no—
334C23—942no—
335C23—1882no—
336C23—1882no—
337C23—1881no—
338C23—1881no—
339C23C2×(C2×D47)⋊C411yescharacteristic
340Q8—944no—
341Q8—3764no—
342Q8—1882no—
343D4—944no—
344D4—3764no—
345D4—1882no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—942no—
358C2×C4—942no—
359C2×C4—942no—
360C2×C4—3764no—
361C2×C4—1882no—
362C2×C4—1882no—
363C2×C4—1881no—
364C2×C4—1881no—
365C2×C4C22⋊C4×D4711yescharacteristic
366C2×C4C22⋊C4×D4711yescharacteristic
367C2×C4#1309399212yes—
368C2×C4#1309399412yes—
369C2×C4#1309403012yes—
370C2×C4#1309403212yes—
371C8—82no—
372C8—3762no—
373C22—21no—
374C22—21no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—81no—
379C22—81no—
380C22—3762no—
381C22—3762no—
382C22—1882no—
383C22—1882no—
384C22—1882no—
385C22#1568897811yescharacteristic
386C22#1568908911yescharacteristic
387C22#1568909111yescharacteristic
388C4—22no—
389C4—22no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—1882no—
394C4—1882no—
395C4—1882no—
396C4#1568908812yes—
397C4#1568908812yes—
398C2—21no—
399C2—21no—
400C2—81no—
401C2—3762no—
402C2#2024329911yescharacteristic
403C2#2024335911yescharacteristic
404C2#2024336111yescharacteristic
subgroupG/H
centerC22#15688978
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1458C2
socleC22×C47#379
prime radicalC2×#13093998C4
cosocleC2×C42×C47C23
hypercenter#1458D47
hypocenterC47#8325
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #20241782C2semidirect
3. #20241780C2semidirect
4. #20241702C2semidirect
5. C47×#1458C2semidirect
200. #1458D47semidirect
229. C47#8325semidirect

Maximal subgroups

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

subgroupindex
C2×#156897442
#202417822
#202417802
#202417022
C47×#14582
#832547