Finite Group Databasebeta

← #26369034

Subgroups — #26369034

Order 12032.

1190 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251871C211yesmaximal, characteristic, complemented
2#20251852C211yesmaximal, characteristic, complemented
3#20245042C216yesmaximal, complemented
4#20245029C216yesmaximal, complemented
5C47×#3486C211yesmaximal, characteristic
6#15689890—21no—
7#15689601—21no—
8#15689107C2216yes—
9C2×#13093943—23no—
10#15689097—23no—
11#15689097C2216yes—
12#15688913C22112yes—
13#15688910C2214yes—
14C2×2−1+4×C47C2211yescharacteristic
15C23×Q8×C47C411yescharacteristic, complemented
16D4×Q8×C47—23no—
17D4×Q8×C47—21no—
18C47×#548—23no—
19C47×#548—21no—
20C47×#536C413yescomplemented
21C47×#536C2213yes—
22C47×#525C413yescomplemented
23C47×#525C2213yes—
24C47×#519C411yescharacteristic, complemented
25#13094031—44no—
26C2×#10625329—26no—
27Q8×Dic47—44no—
28#13093983—26no—
29#13093943—26no—
30#13093941C2314yes—
31#13093939—26no—
32#13093939C2314yes—
332−1+4×C47—44no—
34C2×#41×C47C2×C413yes—
35C2×#41×C47D413yes—
36C2×#41×C47D411yescharacteristic
37C2×#41×C47C2313yes—
38C22×Q8×C47—23no—
39C22×Q8×C47—21no—
40C22×Q8×C47—44no—
41C22×Q8×C47C2×C411yescharacteristic
42C22×Q8×C47C2×C413yes—
43C22×Q8×C47D413yes—
44C22×Q8×C47D411yescharacteristic
45C23×C4×C47C2×C413yes—
46C4⋊Q8×C47—23no—
47C4⋊Q8×C47—23no—
48#124×C47—23no—
49#124×C47—23no—
50#123×C47C2×C416yes—
51#123×C47C2×C413yes—
52#123×C47C2313yes—
53C22⋊Q8×C47—23no—
54C22⋊Q8×C47—23no—
55C22⋊Q8×C47—26no—
56C22⋊Q8×C47—23no—
57C22⋊Q8×C47—23no—
58#121×C47—23no—
59#121×C47—23no—
60#120×C47C2×C413yes—
61#120×C47C2311yescharacteristic
62C4×Q8×C47—23no—
63C4×Q8×C47—21no—
64C4×D4×C47—26no—
65C4×D4×C47—23no—
66C4×D4×C47—23no—
67#117×C47C2×C413yes—
68#117×C47C2×C413yes—
69C24×C47C22×C411yescharacteristic
70#41×C47—24no—
71#41×C47—46no—
72#41×C47—46no—
73#41×C47—46no—
74C2×Q8×C47—24no—
75C2×Q8×C47—24no—
76C2×Q8×C47—23no—
77C2×Q8×C47—23no—
78C2×Q8×C47—46no—
79C2×Q8×C47—46no—
80C2×Q8×C47—412no—
81C2×Q8×C47—44no—
82C2×Q8×C47C22⋊C413yes—
83C2×Q8×C47C22⋊C413yes—
84C2×Q8×C47C22⋊C411yescharacteristic
85C2×Q8×C47C2×D413yes—
86C2×D4×C47—23no—
87C2×D4×C47—21no—
88C2×D4×C47C22⋊C413yes—
89C2×D4×C47C22×C413yes—
90C2×D4×C47C2×D413yes—
91C2×D4×C47C2411yescharacteristic
92C22×C4×C47—23no—
93C22×C4×C47—23no—
94C22×C4×C47—46no—
95C22×C4×C47C22⋊C416yes—
96C22×C4×C47C22×C413yes—
97C22×C4×C47C22×C413yes—
98C22×C4×C47C2×D413yes—
99C22×C4×C47C2×D413yes—
100C4⋊C4×C47—23no—
101C4⋊C4×C47—23no—
102C4⋊C4×C47—26no—
103C4⋊C4×C47—23no—
104C4⋊C4×C47—23no—
105C22⋊C4×C47—26no—
106C22⋊C4×C47—23no—
107C22⋊C4×C47—23no—
108C22⋊C4×C47—23no—
109C22⋊C4×C47—21no—
110C22⋊C4×C47C22×C413yes—
111C22⋊C4×C47C22×C411yescharacteristic
112C42×C47—23no—
113C42×C47—23no—
114C22⋊Dic47—24no—
115C4⋊Dic47—46no—
116C4×Dic47—46no—
117#10625329—24no—
118#10625329—46no—
119C2×C47⋊C8—46no—
120C23×C47—21no—
121C23×C47—41no—
122C23×C47—42no—
123C23×C47C2×C22⋊C413yes—
124C23×C47C23×C411yescharacteristic
125C23×C47C22×D411yescharacteristic
126Q8×C47—26no—
127Q8×C47—26no—
128Q8×C47—43no—
129Q8×C47—46no—
130Q8×C47—46no—
131Q8×C47#9914yes—
132D4×C47—26no—
133D4×C47—43no—
134D4×C47—43no—
135D4×C47—41no—
136D4×C47—46no—
137D4×C47—42no—
138C2×C4×C47—26no—
139C2×C4×C47—26no—
140C2×C4×C47—26no—
141C2×C4×C47—21no—
142C2×C4×C47—23no—
143C2×C4×C47—23no—
144C2×C4×C47—46no—
145C2×C4×C47—46no—
146C2×C4×C47—46no—
147C2×C4×C47—43no—
148C2×C4×C47C2×C22⋊C413yes—
149C2×C4×C47C2×C22⋊C413yes—
150C2×C4×C47C2×C22⋊C413yes—
151C2×C4×C47C22×D411yescharacteristic
152C2×Dic47—44no—
153C47⋊C8—44no—
154#33349—471nomaximal, Sylow
155C22×C47—22no—
156C22×C47—21no—
157C22×C47—21no—
158C22×C47—23no—
159C22×C47—43no—
160C22×C47—41no—
161C22×C47—41no—
162C22×C47—41no—
163C22×C47—42no—
164C22×C47—42no—
165C22×C47—41no—
166C22×C47C22×C22⋊C411yescharacteristic
167C4×C47—21no—
168C4×C47—23no—
169C4×C47—23no—
170C4×C47—41no—
171C4×C47—43no—
172C4×C47—43no—
173C4×C47C2×#9916yes—
174Dic47—84no—
175#3486D4711yescharacteristic
176#2889—471no—
177#2884—471no—
178#2123—476no—
179#2120—476no—
180C2×C47—21no—
181C2×C47—41no—
182C2×C47—43no—
183C2×C47—42no—
184C2×C47—41no—
185C2×C47C22×#9911yescharacteristic
186C2×2−1+4C2×D4711yescharacteristic
187C23×Q8Dic4711yescharacteristic, complemented
188#568—941no—
189D4×Q8—23no—
190D4×Q8—21no—
191#548—23no—
192#548—21no—
193#536Dic4713yescomplemented
194#536C2×D4713yes—
195#525Dic4713yescomplemented
196#525C2×D4713yes—
197#519—941no—
198#519Dic4711yescharacteristic, complemented
199#413—476no—
200C2×#101—943no—
201#410—476no—
202#410—943no—
203#352—4712no—
204AΣL(1,𝔽16)—474no—
205C47#3334911yescharacteristic, complemented, Sylow
2062−1+4—44no—
207C2×#41C2×Dic4713yes—
208C2×#41C22⋊D4713yes—
209C2×#41C22⋊D4711yescharacteristic
210C2×#41C22×D4713yes—
211C22×Q8—23no—
212C22×Q8—21no—
213C22×Q8—44no—
214C22×Q8C2×Dic4711yescharacteristic
215C22×Q8C2×Dic4713yes—
216C22×Q8C22⋊D4713yes—
217C22×Q8C22⋊D4711yescharacteristic
218C23×C4C2×Dic4713yes—
219#131—1884no—
220C2×M16—946no—
221C4⋊Q8—23no—
222C4⋊Q8—23no—
223#124—23no—
224#124—23no—
225#123C2×Dic4716yes—
226#123C2×Dic4713yes—
227#123C22×D4713yes—
228C22⋊Q8—23no—
229C22⋊Q8—23no—
230C22⋊Q8—26no—
231C22⋊Q8—23no—
232C22⋊Q8—23no—
233#121—23no—
234#121—23no—
235#120C2×Dic4713yes—
236#120C22×D4711yescharacteristic
237C4×Q8—23no—
238C4×Q8—21no—
239C4×Q8—1884no—
240C4×D4—26no—
241C4×D4—23no—
242C4×D4—23no—
243#117—946no—
244#117C2×Dic4713yes—
245#117C2×Dic4713yes—
246#101—946no—
247#100—474no—
248#99—474no—
249#99—946no—
250C24C22×Dic4711yescharacteristic
251#41—24no—
252#41—46no—
253#41—46no—
254#41—46no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—23no—
258C2×Q8—23no—
259C2×Q8—46no—
260C2×Q8—46no—
261C2×Q8—412no—
262C2×Q8—44no—
263C2×Q8C22⋊Dic4713yes—
264C2×Q8C22⋊Dic4713yes—
265C2×Q8C22⋊Dic4711yescharacteristic
266C2×Q8C2×C22⋊D4713yes—
267C2×D4—23no—
268C2×D4—21no—
269C2×D4C22⋊Dic4713yes—
270C2×D4C22×Dic4713yes—
271C2×D4C2×C22⋊D4713yes—
272C2×D4C23×D4711yescharacteristic
273C22×C4—23no—
274C22×C4—23no—
275C22×C4—46no—
276C22×C4C22⋊Dic4716yes—
277C22×C4C22×Dic4713yes—
278C22×C4C22×Dic4713yes—
279C22×C4C2×C22⋊D4713yes—
280C22×C4C2×C22⋊D4713yes—
281M16—944no—
282M16—1886no—
283C2×C8—1886no—
284C4⋊C4—23no—
285C4⋊C4—23no—
286C4⋊C4—26no—
287C4⋊C4—23no—
288C4⋊C4—23no—
289C4⋊C4—1886no—
290C22⋊C4—26no—
291C22⋊C4—23no—
292C22⋊C4—23no—
293C22⋊C4—23no—
294C22⋊C4—21no—
295C22⋊C4—944no—
296C22⋊C4C22×Dic4713yes—
297C22⋊C4C22×Dic4711yescharacteristic
298C42—23no—
299C42—23no—
300C42—1886no—
301C23—21no—
302C23—41no—
303C23—42no—
304C23C2×C22⋊Dic4713yes—
305C23C23×Dic4711yescharacteristic
306C23C22×C22⋊D4711yescharacteristic
307Q8—26no—
308Q8—26no—
309Q8—43no—
310Q8—46no—
311Q8—46no—
312Q8#1309393914yes—
313D4—26no—
314D4—43no—
315D4—43no—
316D4—41no—
317D4—46no—
318D4—42no—
319C2×C4—26no—
320C2×C4—26no—
321C2×C4—26no—
322C2×C4—21no—
323C2×C4—23no—
324C2×C4—23no—
325C2×C4—46no—
326C2×C4—46no—
327C2×C4—46no—
328C2×C4—43no—
329C2×C4—1884no—
330C2×C4C2×C22⋊Dic4713yes—
331C2×C4C2×C22⋊Dic4713yes—
332C2×C4C2×C22⋊Dic4713yes—
333C2×C4C22×C22⋊D4711yescharacteristic
334C8—1884no—
335C22—22no—
336C22—21no—
337C22—21no—
338C22—23no—
339C22—43no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—42no—
344C22—42no—
345C22—41no—
346C22C22×C22⋊Dic4711yescharacteristic
347C4—21no—
348C4—23no—
349C4—23no—
350C4—41no—
351C4—43no—
352C4—43no—
353C4—3764no—
354C4C2×#1309393916yes—
355C2—21no—
356C2—41no—
357C2—42no—
358C2—41no—
359C2—43no—
360C2C22×#1309393911yescharacteristic
subgroupG/H
centerC2C22×#13093939
commutatorC23×C47C23×C4
FrattiniC2×D4C23×D47
Fitting, prime radicalC47×#3486C2
socleC2×C47C22×#99
cosocleC2×D4×C47C24
hypercenter#3486D47
hypocenterC47#33349
LCSG ▷ C23×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×Q8 ◁ C2×2−1+4 ◁ #3486

Complements

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

NH ≅ G/Ntype
1. #20251871C2semidirect
2. #20251852C2semidirect
3. #20245042C2semidirect
4. #20245029C2semidirect
15. C23×Q8×C47C4semidirect
20. C47×#536C4semidirect
22. C47×#525C4semidirect
24. C47×#519C4semidirect
187. C23×Q8Dic47semidirect
193. #536Dic47semidirect
195. #525Dic47semidirect
198. #519Dic47semidirect
205. C47#33349semidirect

Maximal subgroups

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

subgroupindex
#202518712
#202518522
#202450422
#202450292
C47×#34862
#3334947