Finite Group Databasebeta

← #26184334

Subgroups — #26184334

Order 12032.

950 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D47C211yesmaximal, characteristic
2#20252019C212yesmaximal
3#20242024C214yesmaximal, complemented
4#20242013C214yesmaximal, complemented
5#20241983C211yesmaximal, characteristic, complemented
6#20241982C212yesmaximal, complemented
7C47×#1526C211yesmaximal, characteristic, complemented
8#15689680C2211yescharacteristic
9#15689679C2212yes—
10#15689678C412yes—
11#15689677—22no—
12#15689676C412yes—
13#15689675—22no—
14C4⋊Q8×D47C412yes—
15#125×D47C2212yes—
16C22⋊Q8×D47—22no—
17C4×Q8×D47—22no—
18#117×D47C412yes—
19C2×C42×D47C2211yescharacteristic
20#15689178C2214yes—
21#15689178C2214yes—
22#15689174—22no—
23#15689174—22no—
24#15689146—22no—
25#15689086—22no—
26C4×#10625329C2212yes—
27#15688865C2214yescomplemented
28#15688864C2218yescomplemented
29C47×#533C2211yescharacteristic
30C4×M16×C47C2212yes—
31C47×#330C2214yescomplemented
32C2×Q8×D47—22no—
33C22×C4×D47C2×C411yescharacteristic
34C22×C4×D47D412yes—
35C2×#10625329—22no—
36#13094024C2×C412yes—
37#13094023—22no—
38#13094023C2×C414yes—
39#13094022—22no—
40#13094016C2312yes—
41#13094015C2×C414yes—
42#13094014—24no—
43#13094004—22no—
44#13094003—22no—
45#13094002—24no—
46#13094001—24no—
47#13094000—22no—
48Q8×Dic47—22no—
49C4×Dic94—22no—
50C4×Dic94—24no—
51#13093989—22no—
52#13093989—24no—
53#13093983C2×C412yes—
54#13093982C2×C414yes—
55#13093981—22no—
56#13093980—22no—
57C4⋊C4×D47—22no—
58C4⋊C4×D47—24no—
59C4⋊C4×D47C2×C414yes—
60C22⋊C4×D47—22no—
61C2×C4×Dic47C2×C411yescharacteristic
62C2×C4×Dic47D412yes—
63C42×D47—22no—
64C42×D47C2×C411yescharacteristic
65C42×D47C2×C411yescharacteristic
66C42×D47D412yes—
67C42×D47D412yes—
68C42×D47C2311yescharacteristic
69C42×D47C2311yescharacteristic
70Dic47⋊C8—24no—
71Dic47⋊C8—24no—
72(C2×D47)⋊C8—24no—
73(C2×D47)⋊C8—24no—
74#13093935C2314yes—
75C4×C47⋊C8—22no—
76C2×M16×C47—22no—
77C4⋊Q8×C47C2×C412yes—
78#125×C47C2312yes—
79C22⋊Q8×C47—22no—
80C4×Q8×C47—22no—
81#117×C47C2×C412yes—
82C2×C42×C47C2311yescharacteristic
83#97×C47C2314yes—
84C4×C8×C47—22no—
85C23×D47C22⋊C411yescharacteristic
86C2×Q8×C47—22no—
87C22×C4×C47C22×C411yescharacteristic
88C22×C4×C47C2×D412yes—
89C22×Dic47C22⋊C411yescharacteristic
90C22×Dic47C22⋊C412yes—
91Q8×D47—48no—
92C2×C4×D47—24no—
93C2×C4×D47—24no—
94C2×C4×D47—44no—
95C2×C4×D47C22⋊C412yes—
96C2×C4×D47C22⋊C412yes—
97C2×C4×D47C22⋊C411yescharacteristic
98C2×C4×D47C22⋊C411yescharacteristic
99C2×C4×D47C22×C411yescharacteristic
100C2×C4×D47C22×C411yescharacteristic
101C2×C4×D47C2×D412yes—
102C2×C4×D47C2×D412yes—
103C2×Dic94—22no—
104C2×Dic94—24no—
105M16×C47—44no—
106C2×C8×C47—24no—
107C4⋊C4×C47—22no—
108C4⋊C4×C47—24no—
109C4⋊C4×C47C22×C414yes—
110C22⋊C4×C47—22no—
111C42×C47—22no—
112C42×C47C22×C411yescharacteristic
113C42×C47C2×D412yes—
114C42×C47C2411yescharacteristic
115C22⋊Dic47—22no—
116(C2×D47)⋊C4—24no—
117C4⋊Dic47—22no—
118C4⋊Dic47—24no—
119C4⋊Dic47C22×C414yes—
120Dic47⋊C4—24no—
121Dic47⋊C4—24no—
122Dic47⋊C4—24no—
123Dic47⋊C4—24no—
124C4×Dic47—22no—
125C4×Dic47—24no—
126C4×Dic47C22⋊C412yes—
127C4×Dic47C22⋊C412yes—
128C4×Dic47C22⋊C412yes—
129C4×Dic47C22×C411yescharacteristic
130C4×Dic47C22×C411yescharacteristic
131C4×Dic47C2×D412yes—
132C4×Dic47C2×D412yes—
133#10625329—44no—
134C2×C47⋊C8—24no—
135C23×C47C2×C22⋊C411yescharacteristic
136C22×D47—22no—
137C22×D47—42no—
138C22×D47C2×C22⋊C411yescharacteristic
139C22×D47C2×C22⋊C411yescharacteristic
140Q8×C47—44no—
141C2×C4×C47—22no—
142C2×C4×C47—24no—
143C2×C4×C47—24no—
144C2×C4×C47C2×C22⋊C412yes—
145C2×C4×C47C2×C22⋊C411yescharacteristic
146C2×C4×C47C23×C411yescharacteristic
147C2×C4×C47C22×D412yes—
148C2×Dic47—22no—
149C2×Dic47—24no—
150C2×Dic47—42no—
151C2×Dic47C2×C22⋊C412yes—
152C2×Dic47C2×C22⋊C412yes—
153C2×Dic47C2×C22⋊C411yescharacteristic
154C2×Dic47C2×C22⋊C411yescharacteristic
155C4×D47—22no—
156C4×D47—22no—
157C4×D47—24no—
158C4×D47—24no—
159C4×D47—22no—
160C4×D47—42no—
161C4×D47—42no—
162C4×D47—48no—
163C4×D47#10114yes—
164Dic94—48no—
165Dic94—44no—
166C8×C47—44no—
167C47⋊C8—44no—
168#17664—471nomaximal, Sylow
169C22×C47—21no—
170C22×C47—22no—
171C22×C47C22×C22⋊C411yescharacteristic
172C2×D47—22no—
173C2×D47—21no—
174C2×D47—22no—
175C2×D47—44no—
176C2×D47C2×#10112yes—
177C4×C47—21no—
178C4×C47—22no—
179C4×C47—44no—
180C4×C47C2×#10112yes—
181C4×C47#42114yes—
182Dic47—21no—
183Dic47—22no—
184Dic47—22no—
185Dic47—44no—
186Dic47C2×#10112yes—
187C2×#533—471no—
188#2920—472no—
189#1537—474no—
190#1534—474no—
191#1526—472no—
192#1526—471no—
193#1526D4711yescharacteristic, complemented
194C2×C47—41no—
195C2×C47C22×#10111yescharacteristic
196C2×C47C2×#42112yes—
197D47—22no—
198D47—41no—
199#533—472no—
200#533—472no—
201#533—472no—
202#533—471no—
203#533—942no—
204#533—942no—
205#533C2×D4711yescharacteristic
206C2×C4⋊Q8—472no—
207C2×#125—472no—
208C2×C22⋊Q8—942no—
209C2×C4×Q8—942no—
210C2×#117—472no—
211C22×C42—471no—
212#432—474no—
213#432—474no—
214#431—942no—
215#431—942no—
216#423—942no—
217#407—942no—
218C4×M16—472no—
219C4×M16C2×D4712yes—
220#330—474no—
221#330—478no—
222#330C2×D4714yescomplemented
223C47#1766411yescharacteristic, complemented, Sylow
224C22×Q8—942no—
225C23×C4—471no—
226C23×C4—472no—
227C2×M16—22no—
228C2×M16—942no—
229C4⋊Q8—474no—
230C4⋊Q8—472no—
231C4⋊Q8—942no—
232C4⋊Q8—942no—
233C4⋊Q8C4×D4712yes—
234#125—474no—
235#125—472no—
236#125—944no—
237#125C22×D4712yes—
238C22⋊Q8—22no—
239C22⋊Q8—944no—
240C22⋊Q8—944no—
241C22⋊Q8—942no—
242C22⋊Q8—942no—
243C22⋊Q8—942no—
244C4×Q8—22no—
245C4×Q8—944no—
246C4×Q8—942no—
247C4×Q8—944no—
248C4×Q8—942no—
249C4×Q8—942no—
250#117—472no—
251#117—474no—
252#117—942no—
253#117—942no—
254#117C4×D4712yes—
255C2×C4⋊C4—474no—
256C2×C4⋊C4—944no—
257C2×C4⋊C4—942no—
258C2×C22⋊C4—942no—
259C2×C42—471no—
260C2×C42—472no—
261C2×C42—471no—
262C2×C42—471no—
263C2×C42—472no—
264C2×C42—472no—
265C2×C42—471no—
266C2×C42—471no—
267C2×C42—942no—
268C2×C42C22×D4711yescharacteristic
269C4⋊C8—944no—
270C4⋊C8—944no—
271C22⋊C8—944no—
272C22⋊C8—944no—
273#97—474no—
274#97C22×D4714yes—
275C4×C8—22no—
276C4×C8—942no—
277C24—471no—
278C2×Q8—22no—
279C2×Q8—944no—
280C2×Q8—942no—
281C2×Q8—1888no—
282C22×C4—471no—
283C22×C4—471no—
284C22×C4—472no—
285C22×C4—472no—
286C22×C4—472no—
287C22×C4—471no—
288C22×C4—471no—
289C22×C4—471no—
290C22×C4—472no—
291C22×C4—472no—
292C22×C4—944no—
293C22×C4—944no—
294C22×C4—1884no—
295C22×C4C2×C4×D4711yescharacteristic
296C22×C4D4×D4712yes—
297M16—44no—
298M16—1884no—
299C2×C8—24no—
300C2×C8—944no—
301C4⋊C4—22no—
302C4⋊C4—24no—
303C4⋊C4—474no—
304C4⋊C4—944no—
305C4⋊C4—944no—
306C4⋊C4—944no—
307C4⋊C4—944no—
308C4⋊C4—942no—
309C4⋊C4—944no—
310C4⋊C4C2×C4×D4714yes—
311C22⋊C4—22no—
312C22⋊C4—944no—
313C22⋊C4—942no—
314C42—22no—
315C42—472no—
316C42—472no—
317C42—472no—
318C42—471no—
319C42—471no—
320C42—472no—
321C42—472no—
322C42—944no—
323C42—942no—
324C42C2×C4×D4711yescharacteristic
325C42D4×D4712yes—
326C42C23×D4711yescharacteristic
327C23—471no—
328C23—471no—
329C23—942no—
330C23—1882no—
331C23C22⋊C4×D4711yescharacteristic
332Q8—44no—
333Q8—1888no—
334Q8—1884no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—471no—
339C2×C4—474no—
340C2×C4—472no—
341C2×C4—471no—
342C2×C4—472no—
343C2×C4—942no—
344C2×C4—942no—
345C2×C4—942no—
346C2×C4—942no—
347C2×C4—944no—
348C2×C4—944no—
349C2×C4—944no—
350C2×C4—1882no—
351C2×C4—1882no—
352C2×C4—1888no—
353C2×C4—1882no—
354C2×C4C22⋊C4×D4712yes—
355C2×C4C22⋊C4×D4711yescharacteristic
356C2×C4C22×C4×D4711yescharacteristic
357C2×C4C2×D4×D4712yes—
358C8—44no—
359C8—1884no—
360C22—22no—
361C22—21no—
362C22—472no—
363C22—942no—
364C22—942no—
365C22—941no—
366C22—1884no—
367C22C2×C22⋊C4×D4711yescharacteristic
368C4—22no—
369C4—21no—
370C4—44no—
371C4—472no—
372C4—942no—
373C4—942no—
374C4—941no—
375C4—1884no—
376C4#101×D4712yes—
377C4#1568913814yes—
378C2—41no—
379C2—942no—
380C2—1881no—
381C2C2×#101×D4711yescharacteristic
382C2C2×#1568913812yes—
subgroupG/H
centerC2×C4C22⋊C4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1526C2
socleC22×C47C22×C22⋊C4
prime radicalC23×D47C22⋊C4
cosocleC42×C47C24
hypercenter#1526D47
hypocenterC47#17664
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1526

Complements

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

NH ≅ G/Ntype
3. #20242024C2semidirect
4. #20242013C2semidirect
5. #20241983C2semidirect
6. #20241982C2semidirect
7. C47×#1526C2semidirect
27. #15688865C22semidirect
28. #15688864C22semidirect
31. C47×#330C22semidirect
193. #1526D47semidirect
222. #330C2×D47semidirect
223. C47#17664semidirect

Maximal subgroups

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

subgroupindex
#533×D472
#202520192
#202420242
#202420132
#202419832
#202419822
C47×#15262
#1766447