Finite Group Databasebeta

← #26112075

Subgroups — #26112075

Order 12032.

452 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
1#20252402C211yesmaximal, characteristic
2#20243198C211yesmaximal, characteristic
3#20241983C211yesmaximal, characteristic, complemented
4#20241900C211yesmaximal, characteristic, complemented
5#20241280C211yesmaximal, characteristic, complemented
6#20241259C211yesmaximal, characteristic, complemented
7C47×#3116C211yesmaximal, characteristic
8#15689254—21no—
9#15689195—22no—
10D4×C47⋊C8—21no—
11#15689180—21no—
12#15689180C2211yescharacteristic
13C2×#13093951—21no—
14C4×#10625329C2211yescharacteristic
15C4×#10625329C2211yescharacteristic
16C2×#13093935—21no—
17#15688969C2211yescharacteristic
18#15688882—21no—
19#15688865—21no—
20#15688865—21no—
21#15688860—22no—
22C47×#533C2211yescharacteristic
23C4×#41×C47C2211yescharacteristic
24C47×#485—21no—
25C47×#477C411yescharacteristic, complemented
26C47×#476C411yescharacteristic, complemented
27C47×#462C411yescharacteristic, complemented
28C47×#461C411yescharacteristic, complemented
29C47×#452—21no—
30C47×#441—21no—
31C47×#440—21no—
32C47×#432C2211yescharacteristic
33C2×#10625329—21no—
34C2×#10625329—21no—
35C22×C47⋊C8—22no—
36C2×C4×Dic47—21no—
37C4⋊(C47⋊C8)—22no—
38C4⋊(C47⋊C8)—21no—
39#13093951—24no—
40C22⋊(C47⋊C8)—22no—
41C22⋊(C47⋊C8)D412yes—
42#13093935—22no—
43#13093935—21no—
44#13093935—21no—
45#13093935D411yescharacteristic
46#13093935D411yescharacteristic
47C4×C47⋊C8—21no—
48C4×C47⋊C8—21no—
49C47⋊#95—21no—
50C47⋊#95—21no—
51C2×#41×C47D411yescharacteristic
52C2×Q16×C47—21no—
53C2×SD16×C47—21no—
54C4⋊Q8×C47C2×C411yescharacteristic
55C4⋊Q8×C47C2×C411yescharacteristic
56#125×C47—21no—
57C22⋊Q8×C47—21no—
58C22⋊Q8×C47—21no—
59C4×Q8×C47—21no—
60C4×Q8×C47—21no—
61C4×Q8×C47C2×C411yescharacteristic
62C4×Q8×C47D411yescharacteristic
63C4×D4×C47—21no—
64C4×D4×C47—21no—
65C4×D4×C47C2×C411yescharacteristic
66C4×D4×C47D411yescharacteristic
67#117×C47—21no—
68#117×C47—21no—
69#117×C47D411yescharacteristic
70C2×C42×C47—21no—
71C2×C42×C47C2311yescharacteristic
72#107×C47—21no—
73#106×C47—21no—
74C4⋊C8×C47C2×C411yescharacteristic
75C4⋊C8×C47C2×C411yescharacteristic
76#103×C47—21no—
77#103×C47—21no—
78#103×C47—21no—
79#102×C47—21no—
80C22⋊C8×C47—21no—
81#97×C47—21no—
82#41×C47—22no—
83#41×C47—41no—
84C2×Q8×C47—21no—
85C2×Q8×C47—21no—
86C2×Q8×C47C22⋊C411yescharacteristic
87C2×D4×C47—21no—
88C2×D4×C47C22⋊C411yescharacteristic
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47C2×D411yescharacteristic
94C22×C4×C47#4111yescharacteristic
95C22×Dic47—41no—
96Q16×C47—42no—
97SD16×C47—42no—
98C2×C8×C47—21no—
99C2×C8×C47—21no—
100C4⋊C4×C47—21no—
101C4⋊C4×C47—21no—
102C4⋊C4×C47—21no—
103C4⋊C4×C47—21no—
104C4⋊C4×C47—21no—
105C4⋊C4×C47—21no—
106C4⋊C4×C47—41no—
107C4⋊C4×C47—41no—
108C4⋊C4×C47C22⋊C411yescharacteristic
109C4⋊C4×C47C22⋊C411yescharacteristic
110C22⋊C4×C47—21no—
111C22⋊C4×C47—21no—
112C22⋊C4×C47—21no—
113C22⋊C4×C47—41no—
114C42×C47—21no—
115C42×C47—21no—
116C42×C47—21no—
117C42×C47—41no—
118C42×C47C22×C411yescharacteristic
119C42×C47C2×D411yescharacteristic
120C42×C47C2×D411yescharacteristic
121C42×C47#4111yescharacteristic
122C4×Dic47—22no—
123C4×Dic47—41no—
124#10625329—22no—
125#10625329—42no—
126#10625329—41no—
127C2×C47⋊C8—21no—
128C2×C47⋊C8—21no—
129C2×C47⋊C8—22no—
130C2×C47⋊C8—44no—
131C2×C47⋊C8—41no—
132C23×C47—21no—
133C23×C47#12111yescharacteristic
134Q8×C47—21no—
135Q8×C47—21no—
136Q8×C47—41no—
137Q8×C47—41no—
138Q8×C47—81no—
139D4×C47—21no—
140D4×C47—42no—
141D4×C47#10412yes—
142C2×C4×C47—21no—
143C2×C4×C47—21no—
144C2×C4×C47—21no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—42no—
150C2×C4×C47—42no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
153C2×C4×C47—81no—
154C2×C4×C47C2×C22⋊C411yescharacteristic
155C2×C4×C47C4×D411yescharacteristic
156C2×C4×C47C4×D411yescharacteristic
157C2×C4×C47#12011yescharacteristic
158C2×C4×C47#12111yescharacteristic
159C2×C4×C47#12311yescharacteristic
160C2×Dic47—42no—
161C2×Dic47—81no—
162C2×Dic47—81no—
163C8×C47—41no—
164C8×C47—41no—
165C47⋊C8—22no—
166C47⋊C8—42no—
167C47⋊C8—41no—
168C47⋊C8—42no—
169#9159—471nomaximal, Sylow
170C22×C47—21no—
171C22×C47—22no—
172C22×C47—21no—
173C22×C47—21no—
174C22×C47—41no—
175C22×C47—41no—
176C22×C47#38611yescharacteristic
177C4×C47—21no—
178C4×C47—21no—
179C4×C47—21no—
180C4×C47—21no—
181C4×C47—21no—
182C4×C47—41no—
183C4×C47—41no—
184C4×C47—41no—
185C4×C47—81no—
186C4×C47—81no—
187C4×C47C2×#10411yescharacteristic
188C4×C47#42111yescharacteristic
189Dic47—82no—
190#3116D4711yescharacteristic
191#2970—471no—
192#1759—471no—
193#1526—471no—
194#1506—471no—
195#1300—471no—
196#1291—471no—
197C2×C47—41no—
198C2×C47—42no—
199C2×C47#187111yescharacteristic
200C2×C47#190211yescharacteristic
201C2×C47#190611yescharacteristic
202#533C2×D4711yescharacteristic
203C4×#41C2×D4711yescharacteristic
204#485—21no—
205#477Dic4711yescharacteristic, complemented
206#476Dic4711yescharacteristic, complemented
207#462Dic4711yescharacteristic, complemented
208#461Dic4711yescharacteristic, complemented
209#452—21no—
210#446—941no—
211#441—21no—
212#440—21no—
213#435—942no—
214C8×D4—941no—
215#432—471no—
216#432—941no—
217#432C2×D4711yescharacteristic
218C2×#104—941no—
219C4×M16—471no—
220C4×M16—471no—
221C2×#97—941no—
222#376—471no—
223#339—941no—
224#330—941no—
225#330—941no—
226#328—942no—
227C47#915911yescharacteristic, complemented, Sylow
228C2×#41C22⋊D4711yescharacteristic
229C2×Q16—21no—
230C2×SD16—21no—
231C2×M16—941no—
232C2×M16—941no—
233C22×C8—942no—
234C4⋊Q8C2×Dic4711yescharacteristic
235C4⋊Q8C2×Dic4711yescharacteristic
236#125—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8C2×Dic4711yescharacteristic
242C4×Q8C22⋊D4711yescharacteristic
243C4×D4—21no—
244C4×D4—21no—
245C4×D4C2×Dic4711yescharacteristic
246C4×D4C22⋊D4711yescharacteristic
247#117—21no—
248#117—21no—
249#117C22⋊D4711yescharacteristic
250C2×C42—21no—
251C2×C42—941no—
252C2×C42C22×D4711yescharacteristic
253#107—21no—
254#106—21no—
255C4⋊C8—942no—
256C4⋊C8—941no—
257C4⋊C8C2×Dic4711yescharacteristic
258C4⋊C8C2×Dic4711yescharacteristic
259#104—944no—
260#103—21no—
261#103—21no—
262#103—21no—
263#102—21no—
264C22⋊C8—21no—
265C22⋊C8—472no—
266C22⋊C8—942no—
267#97—21no—
268#97—471no—
269#97—471no—
270#97—942no—
271#97—941no—
272#97—941no—
273C4×C8—941no—
274C4×C8—941no—
275#95—941no—
276#95—941no—
277#41—22no—
278#41—41no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8C22⋊Dic4711yescharacteristic
282C2×D4—21no—
283C2×D4C22⋊Dic4711yescharacteristic
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—1881no—
288C22×C4D4×D4711yescharacteristic
289C22×C4#1062535211yescharacteristic
290C22×C4C2×C22⋊D4711yescharacteristic
291Q16—42no—
292SD16—42no—
293M16—942no—
294M16—1882no—
295M16—1881no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—941no—
299C2×C8—941no—
300C2×C8—942no—
301C2×C8—1884no—
302C2×C8—1881no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4C22⋊Dic4711yescharacteristic
312C4⋊C4C22⋊Dic4711yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—41no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—41no—
321C42—942no—
322C42—1881no—
323C42D4×D4711yescharacteristic
324C42#1062535211yescharacteristic
325C42C22×Dic4711yescharacteristic
326C42C2×C22⋊D4711yescharacteristic
327C23—21no—
328C23#1309399611yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—41no—
332Q8—41no—
333Q8—81no—
334D4—21no—
335D4—42no—
336D4#1309395112yes—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—3761no—
350C2×C4—3761no—
351C2×C4—1882no—
352C2×C4C2×C22⋊Dic4711yescharacteristic
353C2×C4D4×Dic4711yescharacteristic
354C2×C4D4×Dic4711yescharacteristic
355C2×C4#1309399211yescharacteristic
356C2×C4#1309399611yescharacteristic
357C2×C4#1309400511yescharacteristic
358C8—41no—
359C8—41no—
360C8—942no—
361C8—1882no—
362C8—1881no—
363C8—1882no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—22no—
368C22—41no—
369C22—41no—
370C22#1568900311yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—3762no—
382C4C2×#1309395111yescharacteristic
383C4#1568914011yescharacteristic
384C2—41no—
385C2—42no—
386C2#2024369111yescharacteristic
387C2#2024386311yescharacteristic
388C2#2024389111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3116C2
socleC22×C47#386
prime radicalC2×#41×C47D4
cosocleC2×C42×C47C23
hypercenter#3116D47
hypocenterC47#9159
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3116

Complements

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

NH ≅ G/Ntype
3. #20241983C2semidirect
4. #20241900C2semidirect
5. #20241280C2semidirect
6. #20241259C2semidirect
25. C47×#477C4semidirect
26. C47×#476C4semidirect
27. C47×#462C4semidirect
28. C47×#461C4semidirect
205. #477Dic47semidirect
206. #476Dic47semidirect
207. #462Dic47semidirect
208. #461Dic47semidirect
227. C47#9159semidirect

Maximal subgroups

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

subgroupindex
#202524022
#202431982
#202419832
#202419002
#202412802
#202412592
C47×#31162
#915947