Finite Group Databasebeta

← #26112405

Subgroups — #26112405

Order 12032.

450 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252032C211yesmaximal, characteristic, complemented
2#20241980C211yesmaximal, characteristic, complemented
3#20241839C211yesmaximal, characteristic, complemented
4#20241827C211yesmaximal, characteristic, complemented
5#20241279C211yesmaximal, characteristic, complemented
6#20241224C211yesmaximal, characteristic, complemented
7C47×#3106C211yesmaximal, characteristic
8#15689184—22no—
9#15689176—22no—
10#15689150—21no—
11#15689147C2211yescharacteristic
12#15689090—21no—
13C4×#10625329C2211yescharacteristic
14C4×#10625329C2211yescharacteristic
15C2×C4×C47⋊C8C2211yescharacteristic
16#15688863—22no—
17Q8⋊(C47⋊C8)—21no—
18D4⋊(C47⋊C8)—22no—
19D4⋊(C47⋊C8)—21no—
20C47×#532C2211yescharacteristic
21C4×#41×C47C2211yescharacteristic
22C47×#482—21no—
23C47×#477C411yescharacteristic
24C47×#476C411yescharacteristic
25C47×#460C411yescharacteristic
26C47×#459C411yescharacteristic
27C47×#449—21no—
28C4×SD16×C47—21no—
29C4×D8×C47—21no—
30C47×#431C2211yescharacteristic
31C2×#10625329—21no—
32C2×#10625329—21no—
33C2×#10625329—21no—
34C22×C47⋊C8—21no—
35C4⋊(C47⋊C8)—24no—
36C4⋊(C47⋊C8)—21no—
37C4⋊(C47⋊C8)—21no—
38C22⋊(C47⋊C8)—24no—
39#13093935—21no—
40#13093935D412yes—
41C4×C47⋊C8—21no—
42C4×C47⋊C8—21no—
43C4×C47⋊C8—21no—
44C4×C47⋊C8D412yes—
45C2×#41×C47—21no—
46C2×#41×C47D411yescharacteristic
47C2×SD16×C47—21no—
48C2×D8×C47—21no—
49C4⋊Q8×C47C2×C411yescharacteristic
50#127×C47C2×C411yescharacteristic
51#124×C47—21no—
52#121×C47—21no—
53#121×C47—21no—
54C4×Q8×C47C2×C411yescharacteristic
55C4×Q8×C47D411yescharacteristic
56C4×D4×C47—21no—
57C4×D4×C47—21no—
58C4×D4×C47—21no—
59C4×D4×C47—21no—
60C4×D4×C47C2×C411yescharacteristic
61C4×D4×C47D411yescharacteristic
62#117×C47—21no—
63#117×C47D411yescharacteristic
64C2×C42×C47—21no—
65C2×C42×C47C2311yescharacteristic
66#107×C47—21no—
67#106×C47—21no—
68C4⋊C8×C47C2×C411yescharacteristic
69C4⋊C8×C47C2×C411yescharacteristic
70#103×C47—21no—
71#102×C47—21no—
72#102×C47—21no—
73#102×C47—21no—
74C22⋊C8×C47—21no—
75C4×C8×C47—21no—
76#41×C47—41no—
77#41×C47—41no—
78#41×C47—81no—
79#41×C47—81no—
80C2×Q8×C47—21no—
81C2×Q8×C47C22⋊C411yescharacteristic
82C2×D4×C47—21no—
83C2×D4×C47—21no—
84C2×D4×C47—21no—
85C2×D4×C47—21no—
86C2×D4×C47—41no—
87C2×D4×C47C22⋊C411yescharacteristic
88C22×C4×C47—21no—
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—41no—
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47#4111yescharacteristic
94C22×C4×C47#4111yescharacteristic
95SD16×C47—42no—
96D8×C47—42no—
97C2×C8×C47—21no—
98C2×C8×C47—21no—
99C4⋊C4×C47—21no—
100C4⋊C4×C47—21no—
101C4⋊C4×C47—21no—
102C4⋊C4×C47—21no—
103C4⋊C4×C47C22⋊C411yescharacteristic
104C4⋊C4×C47C22⋊C411yescharacteristic
105C22⋊C4×C47—21no—
106C22⋊C4×C47—21no—
107C22⋊C4×C47—21no—
108C22⋊C4×C47—41no—
109C22⋊C4×C47—41no—
110C42×C47—21no—
111C42×C47—21no—
112C42×C47—21no—
113C42×C47C22×C411yescharacteristic
114C42×C47C2×D411yescharacteristic
115C42×C47C2×D411yescharacteristic
116C42×C47C2×D411yescharacteristic
117#10625329—22no—
118#10625329—41no—
119#10625329—41no—
120#10625329—41no—
121#10625329—81no—
122C2×C47⋊C8—22no—
123C2×C47⋊C8—22no—
124C2×C47⋊C8—41no—
125C2×C47⋊C8—41no—
126C2×C47⋊C8—41no—
127C2×C47⋊C8—41no—
128C23×C47—21no—
129C23×C47—41no—
130C23×C47#12311yescharacteristic
131Q8×C47—21no—
132Q8×C47—21no—
133Q8×C47—81no—
134D4×C47—21no—
135D4×C47—21no—
136D4×C47—42no—
137D4×C47—41no—
138D4×C47—41no—
139D4×C47—41no—
140D4×C47—41no—
141D4×C47—81no—
142D4×C47—82no—
143C2×C4×C47—21no—
144C2×C4×C47—21no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—21no—
148C2×C4×C47—21no—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—42no—
153C2×C4×C47—41no—
154C2×C4×C47—41no—
155C2×C4×C47—41no—
156C2×C4×C47—41no—
157C2×C4×C47—41no—
158C2×C4×C47—81no—
159C2×C4×C47—81no—
160C2×C4×C47C2×C22⋊C411yescharacteristic
161C2×C4×C47C4×D411yescharacteristic
162C2×C4×C47C4×D411yescharacteristic
163C2×C4×C47#12011yescharacteristic
164C2×C4×C47#12111yescharacteristic
165C2×C4×C47#12111yescharacteristic
166C8×C47—41no—
167C8×C47—41no—
168C47⋊C8—42no—
169C47⋊C8—42no—
170C47⋊C8—42no—
171C47⋊C8—81no—
172#9213—471nomaximal, Sylow
173C22×C47—21no—
174C22×C47—21no—
175C22×C47—21no—
176C22×C47—41no—
177C22×C47—41no—
178C22×C47—41no—
179C22×C47—81no—
180C22×C47—81no—
181C22×C47—81no—
182C22×C47#38611yescharacteristic
183C4×C47—21no—
184C4×C47—21no—
185C4×C47—21no—
186C4×C47—41no—
187C4×C47—41no—
188C4×C47—41no—
189C4×C47—81no—
190C4×C47#42111yescharacteristic
191C4×C47#42111yescharacteristic
192C4×C47#44311yescharacteristic
193C4×C47#44411yescharacteristic
194C4×C47#47111yescharacteristic
195C4×C47#47211yescharacteristic
196#3106D4711yescharacteristic
197#2922—471no—
198#1525—471no—
199#1492—471no—
200#1489—471no—
201#1299—471no—
202#1278—471no—
203C2×C47—41no—
204C2×C47—81no—
205C2×C47—161no—
206C2×C47#187611yescharacteristic
207C2×C47#189711yescharacteristic
208C2×C47#190711yescharacteristic
209#532C2×D4711yescharacteristic
210C4×#41C2×D4711yescharacteristic
211#482—21no—
212#477Dic4711yescharacteristic
213#476Dic4711yescharacteristic
214#460Dic4711yescharacteristic
215#459Dic4711yescharacteristic
216#449—21no—
217C4×SD16—21no—
218C4×D8—21no—
219#433—942no—
220#431—942no—
221#431C2×D4711yescharacteristic
222#424—941no—
223#423—471no—
224#408—941no—
225C4×M16—471no—
226C4×M16—471no—
227C2×C4×C8—471no—
228#329—942no—
229Q8⋊C8—941no—
230D4⋊C8—942no—
231D4⋊C8—941no—
232C47#921311yescharacteristic, complemented, Sylow
233C2×#41—21no—
234C2×#41C22⋊D4711yescharacteristic
235C2×SD16—21no—
236C2×D8—21no—
237C2×M16—941no—
238C2×M16—941no—
239C2×M16—941no—
240C22×C8—941no—
241C4⋊Q8C2×Dic4711yescharacteristic
242#127C2×Dic4711yescharacteristic
243#124—21no—
244#121—21no—
245#121—21no—
246C4×Q8C2×Dic4711yescharacteristic
247C4×Q8C22⋊D4711yescharacteristic
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4C2×Dic4711yescharacteristic
253C4×D4C22⋊D4711yescharacteristic
254#117—21no—
255#117C22⋊D4711yescharacteristic
256C2×C42—21no—
257C2×C42C22×D4711yescharacteristic
258#107—21no—
259#106—21no—
260C4⋊C8—944no—
261C4⋊C8—941no—
262C4⋊C8—941no—
263C4⋊C8C2×Dic4711yescharacteristic
264C4⋊C8C2×Dic4711yescharacteristic
265#103—21no—
266#102—21no—
267#102—21no—
268#102—21no—
269C22⋊C8—21no—
270C22⋊C8—944no—
271#97—472no—
272#97—941no—
273C4×C8—21no—
274C4×C8—472no—
275C4×C8—941no—
276C4×C8—941no—
277C4×C8—941no—
278#41—41no—
279#41—41no—
280#41—81no—
281#41—81no—
282C2×Q8—21no—
283C2×Q8C22⋊Dic4711yescharacteristic
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4C22⋊Dic4711yescharacteristic
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4#1062535211yescharacteristic
295C22×C4#1062535211yescharacteristic
296C22×C4C2×C22⋊D4711yescharacteristic
297SD16—42no—
298D8—42no—
299M16—942no—
300M16—3761no—
301M16—1881no—
302M16—1881no—
303M16—1881no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—942no—
307C2×C8—942no—
308C2×C8—1881no—
309C2×C8—1881no—
310C2×C8—1881no—
311C2×C8—1881no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4C22⋊Dic4711yescharacteristic
317C4⋊C4C22⋊Dic4711yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42D4×D4711yescharacteristic
327C42D4×D4711yescharacteristic
328C42C22×Dic4711yescharacteristic
329C42C2×C22⋊D4711yescharacteristic
330C23—21no—
331C23—41no—
332C23#1309400511yescharacteristic
333Q8—21no—
334Q8—21no—
335Q8—81no—
336D4—21no—
337D4—21no—
338D4—42no—
339D4—41no—
340D4—41no—
341D4—41no—
342D4—41no—
343D4—81no—
344D4—82no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4C2×C22⋊Dic4711yescharacteristic
363C2×C4D4×Dic4711yescharacteristic
364C2×C4D4×Dic4711yescharacteristic
365C2×C4#1309399211yescharacteristic
366C2×C4#1309399611yescharacteristic
367C2×C4#1309399611yescharacteristic
368C8—41no—
369C8—41no—
370C8—3761no—
371C8—1882no—
372C8—1882no—
373C8—1882no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22#1568900311yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—81no—
391C4#1568914011yescharacteristic
392C4#1568914011yescharacteristic
393C4#1568924011yescharacteristic
394C4#1568924511yescharacteristic
395C4#1568938411yescharacteristic
396C4#1568939111yescharacteristic
397C2—41no—
398C2—81no—
399C2—161no—
400C2#2024371711yescharacteristic
401C2#2024383711yescharacteristic
402C2#2024389811yescharacteristic
subgroupG/H
centerC2×C4#13093996
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#3106C2
socleC22×C47#386
cosocleC2×C42×C47C23
hypercenter#3106D47
hypocenterC47#9213
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3106

Complements

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

NH ≅ G/Ntype
1. #20252032C2semidirect
2. #20241980C2semidirect
3. #20241839C2semidirect
4. #20241827C2semidirect
5. #20241279C2semidirect
6. #20241224C2semidirect
232. C47#9213semidirect

Maximal subgroups

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

subgroupindex
#202520322
#202419802
#202418392
#202418272
#202412792
#202412242
C47×#31062
#921347