Finite Group Databasebeta

← #26126428

Subgroups — #26126428

Order 12032.

448 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15688905C211yesmaximal, characteristic, complemented
2C2×#15688880C211yesmaximal, characteristic, complemented
3#20241376C212yesmaximal, complemented
4#20241373C212yesmaximal, complemented
5C47×#3054C211yesmaximal, characteristic
6C2×#13093983—21no—
7C2×#13093964—21no—
8#15688905—21no—
9#15688905C2212yes—
10#15688880—21no—
11#15688880C2212yes—
12C2×#135×C47C411yescharacteristic, complemented
13C22×M16×C47C2211yescharacteristic
14C47×#525—21no—
15C47×#470C411yescharacteristic, complemented
16C47×#470C2211yescharacteristic
17C47×#469C411yescharacteristic, complemented
18C47×#469C2211yescharacteristic
19C47×#468—21no—
20C47×#468—21no—
21C47×#427C411yescharacteristic, complemented
22C47×#416—21no—
23#13093983—22no—
24#13093983—22no—
25#13093983—41no—
26#13093983—41no—
27C2×C4⋊Dic47—41no—
28C2×C22⋊Dic47—41no—
29C2×C4×Dic47—22no—
30#13093964—22no—
31#13093964—41no—
32C2×C47⋊C16—41no—
33C2×#41×C47—21no—
34C23×C4×C47D411yescharacteristic
35#135×C47—41no—
36#135×C47—41no—
37C2×Q16×C47C2×C411yescharacteristic, complemented
38C2×SD16×C47—21no—
39C2×D8×C47C2×C411yescharacteristic, complemented
40C2×M16×C47—41no—
41C2×M16×C47—41no—
42C2×M16×C47C2×C411yescharacteristic
43C2×M16×C47D411yescharacteristic
44C2×M16×C47D411yescharacteristic
45C2×M16×C47C2311yescharacteristic
46C22×C8×C47C2×C411yescharacteristic
47C22×C8×C47D411yescharacteristic
48#123×C47—41no—
49C22⋊Q8×C47—21no—
50C22⋊Q8×C47—21no—
51#121×C47—21no—
52#121×C47—21no—
53#120×C47—41no—
54C4×D4×C47—41no—
55C4×D4×C47—41no—
56#117×C47—21no—
57#107×C47—21no—
58#106×C47C2×C411yescharacteristic, complemented
59#106×C47C2×C411yescharacteristic, complemented
60#103×C47—21no—
61#103×C47—21no—
62#102×C47—21no—
63#102×C47—21no—
64C24×C47D811yescharacteristic
65#41×C47—41no—
66#41×C47—41no—
67#41×C47—41no—
68#41×C47—41no—
69C2×Q8×C47—21no—
70C2×D4×C47—21no—
71C2×D4×C47—41no—
72C2×D4×C47—41no—
73C2×D4×C47—41no—
74C22×C4×C47—21no—
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47—21no—
78C22×C4×C47—41no—
79C22×C4×C47C22⋊C411yescharacteristic
80C22×C4×C47C22⋊C411yescharacteristic
81C22×C4×C47D811yescharacteristic
82C22×C4×C47SD1611yescharacteristic
83C22×C4×C47SD1611yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85C22×Dic47—42no—
86Q16×C47—41no—
87SD16×C47—42no—
88D8×C47—41no—
89M16×C47—41no—
90M16×C47—41no—
91M16×C47—41no—
92M16×C47—41no—
93C2×C8×C47—21no—
94C2×C8×C47—21no—
95C2×C8×C47—21no—
96C2×C8×C47—21no—
97C2×C8×C47C22⋊C411yescharacteristic
98C2×C8×C47C22⋊C411yescharacteristic
99C2×C8×C47C22×C411yescharacteristic
100C2×C8×C47C2×D411yescharacteristic
101C4⋊C4×C47—21no—
102C4⋊C4×C47—21no—
103C4⋊C4×C47—41no—
104C4⋊C4×C47—41no—
105C22⋊C4×C47—41no—
106C22⋊C4×C47—41no—
107C22⋊C4×C47—41no—
108C22⋊C4×C47—41no—
109C22⋊C4×C47—41no—
110C42×C47—41no—
111C22⋊Dic47—42no—
112C22⋊Dic47—42no—
113C4⋊Dic47—42no—
114C4⋊Dic47—42no—
115C4×Dic47—24no—
116C4×Dic47—42no—
117C47⋊C16—42no—
118C23×C47—21no—
119C23×C47—21no—
120C23×C47—41no—
121C23×C47—81no—
122C23×C47#10211yescharacteristic
123C23×C47#10211yescharacteristic
124C23×C47C2×D811yescharacteristic
125Q8×C47—41no—
126Q8×C47—41no—
127D4×C47—41no—
128D4×C47—41no—
129D4×C47—82no—
130D4×C47—82no—
131D4×C47—81no—
132D4×C47—81no—
133C2×C4×C47—21no—
134C2×C4×C47—21no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—21no—
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47—21no—
141C2×C4×C47—21no—
142C2×C4×C47—21no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—81no—
147C2×C4×C47—81no—
148C2×C4×C47#9911yescharacteristic
149C2×C4×C47#9911yescharacteristic
150C2×C4×C47#10211yescharacteristic
151C2×C4×C47#10211yescharacteristic
152C2×C4×C47#10411yescharacteristic
153C2×C4×C47#10411yescharacteristic
154C2×C4×C47C2×C22⋊C411yescharacteristic
155C2×C4×C47C2×SD1611yescharacteristic
156C2×Dic47—44no—
157C2×Dic47—84no—
158C8×C47—21no—
159C8×C47—21no—
160C8×C47—21no—
161C8×C47—21no—
162#11685—471nomaximal, Sylow
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—41no—
166C22×C47—41no—
167C22×C47—41no—
168C22×C47—42no—
169C22×C47—81no—
170C22×C47—81no—
171C22×C47—81no—
172C22×C47—81no—
173C22×C47#32711yescharacteristic
174C22×C47#32711yescharacteristic
175C22×C47C2×#10211yescharacteristic
176C4×C47—21no—
177C4×C47—21no—
178C4×C47—81no—
179C4×C47—81no—
180C4×C47—81no—
181C4×C47#32711yescharacteristic
182C4×C47#32711yescharacteristic
183C4×C47C2×#9911yescharacteristic
184C4×C47C2×#10411yescharacteristic
185Dic47—84no—
186#3054D4711yescharacteristic
187C2×#349—471no—
188C2×H(3,ℤ/4)—471no—
189#1340—472no—
190#1339—472no—
191C2×C47—21no—
192C2×C47—21no—
193C2×C47—21no—
194C2×C47—41no—
195C2×C47—41no—
196C2×C47—161no—
197C2×C47C2×#32711yescharacteristic
198C2×#135Dic4711yescharacteristic, complemented
199C22×M16C2×D4711yescharacteristic
200#525—21no—
201C2×#117—941no—
202C2×M32—941no—
203#470Dic4711yescharacteristic, complemented
204#470C2×D4711yescharacteristic
205#469Dic4711yescharacteristic, complemented
206#469C2×D4711yescharacteristic
207#468—21no—
208#468—21no—
209#427Dic4711yescharacteristic, complemented
210#416—21no—
211#349—472no—
212#349—941no—
213H(3,ℤ/4)—472no—
214H(3,ℤ/4)—941no—
215C47#1168511yescharacteristic, complemented, Sylow
216C2×#41—21no—
217C23×C4C22⋊D4711yescharacteristic
218#135—41no—
219#135—41no—
220C2×Q16C2×Dic4711yescharacteristic, complemented
221C2×SD16—21no—
222C2×D8C2×Dic4711yescharacteristic, complemented
223C2×M16—41no—
224C2×M16—41no—
225C2×M16C2×Dic4711yescharacteristic
226C2×M16C22⋊D4711yescharacteristic
227C2×M16C22⋊D4711yescharacteristic
228C2×M16C22×D4711yescharacteristic
229C22×C8C2×Dic4711yescharacteristic
230C22×C8C22⋊D4711yescharacteristic
231#123—41no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234#121—21no—
235#121—21no—
236#120—41no—
237C4×D4—41no—
238C4×D4—41no—
239#117—21no—
240#117—942no—
241#117—942no—
242#117—1881no—
243#117—1881no—
244C2×C4⋊C4—1881no—
245C2×C22⋊C4—1881no—
246C2×C42—942no—
247M32—942no—
248M32—1881no—
249C2×C16—1881no—
250#107—21no—
251#106C2×Dic4711yescharacteristic, complemented
252#106C2×Dic4711yescharacteristic, complemented
253#103—21no—
254#103—21no—
255#102—21no—
256#102—21no—
257C24#1062533411yescharacteristic
258#41—41no—
259#41—41no—
260#41—41no—
261#41—41no—
262C2×Q8—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—41no—
272C22×C4—1882no—
273C22×C4#1062533411yescharacteristic
274C22×C4#1062533511yescharacteristic
275C22×C4#1062533511yescharacteristic
276C22×C4C22⋊Dic4711yescharacteristic
277C22×C4C22⋊Dic4711yescharacteristic
278C22×C4C2×C22⋊D4711yescharacteristic
279Q16—41no—
280SD16—42no—
281D8—41no—
282M16—41no—
283M16—41no—
284M16—41no—
285M16—41no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8C22⋊Dic4711yescharacteristic
291C2×C8C22⋊Dic4711yescharacteristic
292C2×C8C22×Dic4711yescharacteristic
293C2×C8C2×C22⋊D4711yescharacteristic
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—1882no—
299C4⋊C4—1882no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—1882no—
306C22⋊C4—1882no—
307C42—41no—
308C42—944no—
309C42—1882no—
310C16—1882no—
311C23—21no—
312C23—21no—
313C23—41no—
314C23—81no—
315C23#1309394511yescharacteristic
316C23#1309394511yescharacteristic
317C23C2×#1062533411yescharacteristic
318Q8—41no—
319Q8—41no—
320D4—41no—
321D4—41no—
322D4—82no—
323D4—82no—
324D4—81no—
325D4—81no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—3764no—
342C2×C4—1884no—
343C2×C4#1309393911yescharacteristic
344C2×C4#1309393911yescharacteristic
345C2×C4#1309394511yescharacteristic
346C2×C4#1309394511yescharacteristic
347C2×C4#1309395111yescharacteristic
348C2×C4#1309395111yescharacteristic
349C2×C4C2×C22⋊Dic4711yescharacteristic
350C2×C4C2×#1062533511yescharacteristic
351C8—21no—
352C8—21no—
353C8—21no—
354C8—21no—
355C22—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22#1568885711yescharacteristic
366C22#1568885711yescharacteristic
367C22C2×#1309394511yescharacteristic
368C4—21no—
369C4—21no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4—3764no—
374C4#1568885711yescharacteristic
375C4#1568885711yescharacteristic
376C4C2×#1309393911yescharacteristic
377C4C2×#1309395111yescharacteristic
378C2—21no—
379C2—21no—
380C2—21no—
381C2—41no—
382C2—41no—
383C2—161no—
384C2C2×#1568885711yescharacteristic
subgroupG/H
centerC4#15688857
commutatorC2×C8×C47C22×C4
FrattiniC2×M16C22×D47
Fitting, prime radicalC47×#3054C2
socleC2×C47C2×#327
cosocleC2×M16×C47C23
hypercenter#3054D47
hypocenterC47#11685
LCSG ▷ C2×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ #3054

Complements

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

NH ≅ G/Ntype
1. C2×#15688905C2semidirect
2. C2×#15688880C2semidirect
3. #20241376C2semidirect
4. #20241373C2semidirect
12. C2×#135×C47C4semidirect
15. C47×#470C4semidirect
17. C47×#469C4semidirect
21. C47×#427C4semidirect
37. C2×Q16×C47C2×C4semidirect
39. C2×D8×C47C2×C4semidirect
58. #106×C47C2×C4semidirect
59. #106×C47C2×C4semidirect
198. C2×#135Dic47semidirect
203. #470Dic47semidirect
205. #469Dic47semidirect
209. #427Dic47semidirect
215. C47#11685semidirect
220. C2×Q16C2×Dic47semidirect
222. C2×D8C2×Dic47semidirect
251. #106C2×Dic47semidirect
252. #106C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156889052
C2×#156888802
#202413762
#202413732
C47×#30542
#1168547