Finite Group Databasebeta

← #26112361

Subgroups — #26112361

Order 12032.

478 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253176C211yesmaximal, characteristic
2#20243408C211yesmaximal, characteristic
3#20241888C211yesmaximal, characteristic, complemented
4#20241844C211yesmaximal, characteristic, complemented
5#20241284C211yesmaximal, characteristic, complemented
6#20241228C211yesmaximal, characteristic, complemented
7C8×#41×C47C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9#15689430—21no—
10#15689423—21no—
11#15689418C411yescharacteristic, complemented
12#15689413C411yescharacteristic, complemented
13#15689408C411yescharacteristic, complemented
14#15689403C411yescharacteristic, complemented
15#15689159—21no—
16#15689147C2211yescharacteristic
17C2×C4⋊(C47⋊C8)C2211yescharacteristic
18C2×#13093951—21no—
19#15689119—21no—
20C4×C4⋊Dic47C2211yescharacteristic
21#15688882—21no—
22#15688861—22no—
23#15688855—21no—
24#15688855—21no—
25C4×#41×C47C2211yescharacteristic
26C8×Q8×C47—21no—
27C8×D4×C47—22no—
28C8×D4×C47—21no—
29C47×#431—21no—
30C47×#431C2211yescharacteristic
31C2×C4×C8×C47—21no—
32C2×C4×C8×C47C2211yescharacteristic
33C2×#10625329—21no—
34C22×C47⋊C8—21no—
35#13094024C2×C411yescharacteristic
36#13094024C2×C411yescharacteristic
37#13094016—21no—
38#13094003—21no—
39#13094003—21no—
40C4×Dic94—21no—
41C4×Dic94—21no—
42#13093983—21no—
43C2×C4⋊Dic47—21no—
44C2×C4×Dic47—21no—
45#13093957—21no—
46#13093957—21no—
47#13093955—21no—
48#13093955—21no—
49C4⋊(C47⋊C8)—21no—
50C4⋊(C47⋊C8)—21no—
51C4⋊(C47⋊C8)—21no—
52C4⋊(C47⋊C8)C2×C411yescharacteristic
53C4⋊(C47⋊C8)C2×C411yescharacteristic
54#13093951—42no—
55#13093948—21no—
56#13093948—21no—
57#13093945—21no—
58#13093945—21no—
59C47⋊#95—21no—
60C2×#41×C47D411yescharacteristic
61C22×C8×C47—22no—
62C22×C8×C47—21no—
63C4×Q8×C47C2×C411yescharacteristic
64C4×Q8×C47D411yescharacteristic
65C4×D4×C47—21no—
66C4×D4×C47—21no—
67C4×D4×C47C2×C411yescharacteristic
68C4×D4×C47D411yescharacteristic
69#117×C47—21no—
70#117×C47D411yescharacteristic
71C2×C42×C47—21no—
72C2×C42×C47C2311yescharacteristic
73C4⋊C8×C47—22no—
74C4⋊C8×C47—21no—
75C22⋊C8×C47—22no—
76C22⋊C8×C47D412yes—
77C4×C8×C47—22no—
78C4×C8×C47—21no—
79C4×C8×C47—21no—
80C4×C8×C47D411yescharacteristic
81C4×C8×C47D411yescharacteristic
82#41×C47—21no—
83#41×C47—21no—
84#41×C47SD1612yes—
85#41×C47Q1612yes—
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—
96C2×Dic94—21no—
97C2×Dic94—21no—
98C2×C8×C47—21no—
99C2×C8×C47—22no—
100C2×C8×C47—21no—
101C2×C8×C47—42no—
102C2×C8×C47—44no—
103C4⋊C4×C47—21no—
104C4⋊C4×C47—21no—
105C4⋊C4×C47C22⋊C411yescharacteristic
106C4⋊C4×C47C22⋊C411yescharacteristic
107C22⋊C4×C47—21no—
108C22⋊C4×C47—21no—
109C22⋊C4×C47—21no—
110C42×C47—21no—
111C42×C47—21no—
112C42×C47—21no—
113C42×C47C22×C411yescharacteristic
114C42×C47C2×D411yescharacteristic
115C42×C47C2×D411yescharacteristic
116C42×C47#4111yescharacteristic
117C22⋊Dic47—41no—
118C4⋊Dic47—22no—
119C4⋊Dic47—21no—
120C4⋊Dic47—21no—
121C4⋊Dic47—21no—
122C4⋊Dic47—21no—
123C4⋊Dic47—41no—
124Dic47⋊C4—41no—
125Dic47⋊C4—41no—
126C4×Dic47—41no—
127C4×Dic47—41no—
128C4×Dic47—41no—
129#10625329—41no—
130#10625329—41no—
131C2×C47⋊C8—21no—
132C2×C47⋊C8—21no—
133C2×C47⋊C8—41no—
134C2×C47⋊C8—41no—
135C2×C47⋊C8—41no—
136C23×C47—21no—
137C23×C47#12111yescharacteristic
138Q8×C47—21no—
139Q8×C47#10312yes—
140D4×C47—22no—
141D4×C47—22no—
142D4×C47—21no—
143D4×C47#10312yes—
144C2×C4×C47—21no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—22no—
148C2×C4×C47—22no—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—41no—
152C2×C4×C47—42no—
153C2×C4×C47—41no—
154C2×C4×C47—41no—
155C2×C4×C47C2×C22⋊C411yescharacteristic
156C2×C4×C47C4×D411yescharacteristic
157C2×C4×C47C4×D411yescharacteristic
158C2×C4×C47#12011yescharacteristic
159C2×C4×C47#12111yescharacteristic
160C2×C4×C47#12311yescharacteristic
161C2×C4×C47C2×SD1611yescharacteristic
162C2×C4×C47C2×Q1611yescharacteristic
163C2×C4×C47#13511yescharacteristic
164C2×C4×C47#13511yescharacteristic
165C2×Dic47—42no—
166C2×Dic47—41no—
167C2×Dic47—41no—
168C2×Dic47—81no—
169C2×Dic47—81no—
170Dic94—41no—
171Dic94—41no—
172Dic94—81no—
173C8×C47—22no—
174C8×C47—42no—
175C8×C47—41no—
176C47⋊C8—41no—
177C47⋊C8—41no—
178C47⋊C8—81no—
179#9207—471nomaximal, Sylow
180C22×C47—21no—
181C22×C47—21no—
182C22×C47—22no—
183C22×C47—41no—
184C22×C47—41no—
185C22×C47#38611yescharacteristic
186C22×C47#46511yescharacteristic
187C22×C47#46711yescharacteristic
188C4×C47—21no—
189C4×C47—22no—
190C4×C47—21no—
191C4×C47—21no—
192C4×C47—41no—
193C4×C47—41no—
194C4×C47C2×#10311yescharacteristic
195C4×C47#41611yescharacteristic
196C4×C47#44311yescharacteristic
197C4×C47#44411yescharacteristic
198C4×C47#46511yescharacteristic
199C4×C47#46711yescharacteristic
200Dic47—82no—
201Dic47—81no—
202Dic47—81no—
203#3076—471no—
204C8×#41D4711yescharacteristic
205#1806—471no—
206#1503—471no—
207#1493—471no—
208#1301—471no—
209#1279—471no—
210C2×C47—21no—
211C2×C47—21no—
212C2×C47—42no—
213C2×C47#187611yescharacteristic
214C2×C47#189211yescharacteristic
215C2×C47#190611yescharacteristic
216#533—471no—
217C4×#41C2×D4711yescharacteristic
218#479—941no—
219#478—941no—
220#477—471no—
221#476—471no—
222#475—471no—
223#474—471no—
224C8×Q8—21no—
225C8×D4—21no—
226C8×D4—22no—
227#431—21no—
228#431C2×D4711yescharacteristic
229#427—941no—
230#423—471no—
231C2×C4⋊C8—471no—
232C2×#104—941no—
233#416—941no—
234C2×C4×C8—21no—
235C2×C4×C8C2×D4711yescharacteristic
236C4×C4⋊C4—471no—
237#339—941no—
238#328—942no—
239Q8⋊C8—941no—
240Q8⋊C8—941no—
241C47#920711yescharacteristic, complemented, Sylow
242C2×#41D18811yescharacteristic
243C2×M16—941no—
244C22×C8—22no—
245C22×C8—21no—
246C22×C8—941no—
247C4⋊Q8—471no—
248C4⋊Q8—471no—
249#125—941no—
250C22⋊Q8—941no—
251C22⋊Q8—941no—
252C4×Q8—941no—
253C4×Q8—941no—
254C4×Q8C4×D4711yescharacteristic
255C4×Q8C22⋊D4711yescharacteristic
256C4×D4—21no—
257C4×D4—21no—
258C4×D4C4×D4711yescharacteristic
259C4×D4C22⋊D4711yescharacteristic
260#117—21no—
261#117—941no—
262#117D18811yescharacteristic
263C2×C4⋊C4—941no—
264C2×C42—21no—
265C2×C42—941no—
266C2×C42C22×D4711yescharacteristic
267#107—941no—
268#107—941no—
269#106—941no—
270#106—941no—
271C4⋊C8—22no—
272C4⋊C8—21no—
273C4⋊C8—471no—
274C4⋊C8—471no—
275C4⋊C8—941no—
276C4⋊C8—941no—
277C4⋊C8—941no—
278#104—1882no—
279#103—941no—
280#103—941no—
281#102—941no—
282#102—941no—
283C22⋊C8—22no—
284C22⋊C8C22⋊D4712yes—
285C4×C8—22no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8C22⋊D4711yescharacteristic
289C4×C8C22⋊D4711yescharacteristic
290#95—941no—
291#41—21no—
292#41—21no—
293#41#1062532512yes—
294#41Dic18812yes—
295C2×Q8—941no—
296C2×Q8—941no—
297C2×Q8(C2×D47)⋊C411yescharacteristic
298C2×D4—21no—
299C2×D4(C2×D47)⋊C411yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—1881no—
304C22×C4C2×D18811yescharacteristic
305C22×C4#1062535011yescharacteristic
306C22×C4C2×C22⋊D4711yescharacteristic
307M16—1881no—
308M16—1881no—
309C2×C8—21no—
310C2×C8—22no—
311C2×C8—21no—
312C2×C8—42no—
313C2×C8—44no—
314C2×C8—941no—
315C2×C8—941no—
316C2×C8—1881no—
317C2×C8—1881no—
318C2×C8—1881no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—942no—
322C4⋊C4—941no—
323C4⋊C4—941no—
324C4⋊C4—941no—
325C4⋊C4—941no—
326C4⋊C4—1881no—
327C4⋊C4—1881no—
328C4⋊C4—1881no—
329C4⋊C4(C2×D47)⋊C411yescharacteristic
330C4⋊C4(C2×D47)⋊C411yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—1881no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—1881no—
339C42—1881no—
340C42—1881no—
341C42C2×C4×D4711yescharacteristic
342C42#1062535011yescharacteristic
343C42C2×C22⋊D4711yescharacteristic
344C42C2×C22⋊D4711yescharacteristic
345C23—21no—
346C23#1309399811yescharacteristic
347Q8—21no—
348Q8—3761no—
349Q8—1881no—
350Q8—1881no—
351Q8#1309394912yes—
352D4—22no—
353D4—22no—
354D4—21no—
355D4#1309394912yes—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—3761no—
368C2×C4—3761no—
369C2×C4—1882no—
370C2×C4—1881no—
371C2×C4—1881no—
372C2×C4C2×(C2×D47)⋊C411yescharacteristic
373C2×C4C4×C22⋊D4711yescharacteristic
374C2×C4C4×C22⋊D4711yescharacteristic
375C2×C4#1309399311yescharacteristic
376C2×C4#1309399811yescharacteristic
377C2×C4#1309400711yescharacteristic
378C2×C4C2×#1062532511yescharacteristic
379C2×C4C2×Dic18811yescharacteristic
380C2×C4#1309404611yescharacteristic
381C2×C4#1309404611yescharacteristic
382C8—22no—
383C8—42no—
384C8—41no—
385C8—3761no—
386C8—1881no—
387C8—1881no—
388C22—21no—
389C22—21no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22#1568900211yescharacteristic
394C22#1568935211yescharacteristic
395C22#1568936411yescharacteristic
396C4—21no—
397C4—22no—
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—3762no—
403C4—3761no—
404C4—3761no—
405C4C2×#1309394911yescharacteristic
406C4#1568912011yescharacteristic
407C4#1568923811yescharacteristic
408C4#1568924311yescharacteristic
409C4#1568935211yescharacteristic
410C4#1568936411yescharacteristic
411C2—21no—
412C2—21no—
413C2—42no—
414C2#2024371511yescharacteristic
415C2#2024381111yescharacteristic
416C2#2024388911yescharacteristic
subgroupG/H
centerC2×C4#13093998
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC8×#41×C47C2
socleC22×C47#386
prime radicalC2×#41×C47D4
cosocleC2×C42×C47C23
hypercenterC8×#41D47
hypocenterC47#9207
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
3. #20241888C2semidirect
4. #20241844C2semidirect
5. #20241284C2semidirect
6. #20241228C2semidirect
11. #15689418C4semidirect
12. #15689413C4semidirect
13. #15689408C4semidirect
14. #15689403C4semidirect
241. C47#9207semidirect

Maximal subgroups

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

subgroupindex
#202531762
#202434082
#202418882
#202418442
#202412842
#202412282
C8×#41×C472
#920747