Finite Group Databasebeta

← #26111508

Subgroups — #26111508

Order 12032.

470 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252337C211yesmaximal, characteristic, complemented
2#20244308C211yesmaximal, characteristic
3#20241887C211yesmaximal, characteristic, complemented
4#20241787C211yesmaximal, characteristic
5#20241228C211yesmaximal, characteristic, complemented
6#20241225C211yesmaximal, characteristic, complemented
7C47×#3167C211yesmaximal, characteristic
8#15689254—21no—
9#15689254—22no—
10Q8×C47⋊C8—21no—
11#15689176C2211yescharacteristic
12#15689150—21no—
13C2×C4⋊(C47⋊C8)C2211yescharacteristic
14C2×#13093951—21no—
15C2×#13093951—21no—
16#15689079—21no—
17C4×#10625329C2211yescharacteristic
18C4×C4⋊Dic47C2211yescharacteristic
19#15688860—22no—
20D4⋊(C47⋊C8)—21no—
21D4⋊(C47⋊C8)—21no—
22C47×#533C2211yescharacteristic
23C47×#532C2211yescharacteristic
24C47×#498C411yescharacteristic, complemented
25C47×#492C411yescharacteristic, complemented
26C47×#487C411yescharacteristic, complemented
27C47×#486C411yescharacteristic, complemented
28C47×#465—22no—
29C4×SD16×C47—22no—
30C2×C4×C8×C47C2211yescharacteristic
31C2×#10625329D411yescharacteristic
32C2×#10625329D411yescharacteristic
33C22×C47⋊C8—21no—
34C2×C4⋊Dic47—21no—
35C2×C4×Dic47—21no—
36C4⋊(C47⋊C8)—22no—
37C4⋊(C47⋊C8)—22no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)D411yescharacteristic
41C4⋊(C47⋊C8)D411yescharacteristic
42#13093951—42no—
43#13093951—42no—
44C22⋊(C47⋊C8)—21no—
45#13093935—22no—
46#13093935—21no—
47C4×C47⋊C8—22no—
48C4×C47⋊C8—21no—
49C4×C47⋊C8—21no—
50C47⋊#95—21no—
51C2×#41×C47—21no—
52C2×SD16×C47—22no—
53C22×C8×C47—21no—
54C4⋊Q8×C47C2×C411yescharacteristic
55C4⋊Q8×C47C2×C411yescharacteristic
56C4⋊Q8×C47C2×C411yescharacteristic
57#127×C47C2×C411yescharacteristic
58#125×C47—21no—
59#124×C47—21no—
60C22⋊Q8×C47—21no—
61C22⋊Q8×C47D412yes—
62#121×C47—21no—
63#121×C47—21no—
64C4×Q8×C47—21no—
65C4×Q8×C47D412yes—
66C4×D4×C47—21no—
67C4×D4×C47—21no—
68#117×C47—21no—
69C2×C42×C47C2311yescharacteristic
70#106×C47—22no—
71#103×C47—22no—
72#102×C47—22no—
73C4×C8×C47—21no—
74C4×C8×C47C2×C411yescharacteristic
75C4×C8×C47C2×C411yescharacteristic
76#41×C47—41no—
77#41×C47—41no—
78#41×C47—41no—
79#41×C47—41no—
80C2×Q8×C47—21no—
81C2×Q8×C47—21no—
82C2×Q8×C47C22⋊C412yes—
83C2×D4×C47—21no—
84C2×D4×C47—21no—
85C2×D4×C47—21no—
86C2×D4×C47—41no—
87C22×C4×C47—41no—
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47#4111yescharacteristic
91C22×Dic47—41no—
92SD16×C47—44no—
93C2×C8×C47—21no—
94C2×C8×C47—21no—
95C2×C8×C47—42no—
96C4⋊C4×C47—21no—
97C4⋊C4×C47—21no—
98C4⋊C4×C47—21no—
99C4⋊C4×C47—22no—
100C4⋊C4×C47—22no—
101C4⋊C4×C47—21no—
102C4⋊C4×C47—21no—
103C4⋊C4×C47C22⋊C412yes—
104C22⋊C4×C47—22no—
105C22⋊C4×C47—41no—
106C22⋊C4×C47—41no—
107C42×C47—22no—
108C42×C47C22×C411yescharacteristic
109C42×C47C2×D411yescharacteristic
110C42×C47C2×D411yescharacteristic
111C42×C47#4111yescharacteristic
112C4⋊Dic47—22no—
113C4⋊Dic47—41no—
114C4×Dic47—41no—
115C4×Dic47—41no—
116#10625329—21no—
117#10625329—21no—
118#10625329—21no—
119#10625329SD1612yescomplemented
120C2×C47⋊C8—21no—
121C2×C47⋊C8—21no—
122C2×C47⋊C8—21no—
123C2×C47⋊C8—21no—
124C2×C47⋊C8—41no—
125C2×C47⋊C8—41no—
126C2×C47⋊C8—41no—
127C23×C47—41no—
128C23×C47#12111yescharacteristic
129Q8×C47—42no—
130Q8×C47—41no—
131Q8×C47—41no—
132Q8×C47—41no—
133Q8×C47—41no—
134D4×C47—42no—
135D4×C47—41no—
136D4×C47—41no—
137D4×C47—41no—
138D4×C47—41no—
139D4×C47—81no—
140D4×C47—81no—
141C2×C4×C47—21no—
142C2×C4×C47—21no—
143C2×C4×C47—22no—
144C2×C4×C47—22no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—81no—
148C2×C4×C47—81no—
149C2×C4×C47C2×C22⋊C411yescharacteristic
150C2×C4×C47C4×D411yescharacteristic
151C2×C4×C47C4×D411yescharacteristic
152C2×C4×C47#12011yescharacteristic
153C2×C4×C47#12111yescharacteristic
154C2×C4×C47#12311yescharacteristic
155C2×C4×C47C2×SD1611yescharacteristic
156C2×C4×C47#13511yescharacteristic
157C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C47#13711yescharacteristic
159C2×Dic47—42no—
160C2×Dic47—81no—
161C2×Dic47—81no—
162C8×C47—22no—
163C8×C47—41no—
164C47⋊C8—22no—
165C47⋊C8—41no—
166C47⋊C8—41no—
167C47⋊C8—41no—
168C47⋊C8—81no—
169#9054—471nomaximal, Sylow
170C22×C47—21no—
171C22×C47—21no—
172C22×C47—81no—
173C22×C47—81no—
174C22×C47—81no—
175C22×C47#38611yescharacteristic
176C22×C47#46011yescharacteristic
177C22×C47#46411yescharacteristic
178C4×C47—21no—
179C4×C47—21no—
180C4×C47—21no—
181C4×C47—42no—
182C4×C47—42no—
183C4×C47—81no—
184C4×C47C4×SD1611yescharacteristic
185C4×C47#44011yescharacteristic
186C4×C47#44311yescharacteristic
187C4×C47#44411yescharacteristic
188C4×C47#44811yescharacteristic
189C4×C47#44911yescharacteristic
190Dic47—82no—
191#3167D4711yescharacteristic
192#2962—471no—
193#1980—471no—
194#1503—471no—
195#1480—471no—
196#1279—471no—
197#1278—471no—
198C2×C47—21no—
199C2×C47—21no—
200C2×C47—161no—
201C2×C47#187611yescharacteristic
202C2×C47#188011yescharacteristic
203C2×C47#188411yescharacteristic
204#533C2×D4711yescharacteristic
205#532C2×D4711yescharacteristic
206#498Dic4711yescharacteristic, complemented
207#492Dic4711yescharacteristic, complemented
208#487Dic4711yescharacteristic, complemented
209#486Dic4711yescharacteristic, complemented
210#465—22no—
211#446—941no—
212#446—942no—
213C8×Q8—941no—
214C4×SD16—22no—
215#431—471no—
216#424—941no—
217C2×C4⋊C8—471no—
218C2×#104—941no—
219C2×#104—941no—
220#405—941no—
221C4×M16—471no—
222C2×C4×C8C2×D4711yescharacteristic
223C4×C4⋊C4—471no—
224#328—942no—
225D4⋊C8—941no—
226D4⋊C8—941no—
227C47#905411yescharacteristic, complemented, Sylow
228C2×#41—21no—
229C2×SD16—22no—
230C2×M16—471no—
231C2×M16—471no—
232C22×C8—21no—
233C22×C8—941no—
234C4⋊Q8C2×Dic4711yescharacteristic
235C4⋊Q8C2×Dic4711yescharacteristic
236C4⋊Q8C2×Dic4711yescharacteristic
237#127C2×Dic4711yescharacteristic
238#125—21no—
239#124—21no—
240C22⋊Q8—21no—
241C22⋊Q8C22⋊D4712yes—
242#121—21no—
243#121—21no—
244C4×Q8—21no—
245C4×Q8C22⋊D4712yes—
246C4×D4—21no—
247C4×D4—21no—
248#117—21no—
249C2×C4⋊C4—941no—
250C2×C42—941no—
251C2×C42C22×D4711yescharacteristic
252#106—22no—
253C4⋊C8—471no—
254C4⋊C8—471no—
255C4⋊C8—942no—
256C4⋊C8—942no—
257C4⋊C8—941no—
258C4⋊C8—941no—
259#104—1882no—
260#104—1882no—
261#103—22no—
262#102—22no—
263C22⋊C8—941no—
264#97—942no—
265#97—941no—
266C4×C8—21no—
267C4×C8—942no—
268C4×C8—941no—
269C4×C8—941no—
270C4×C8C2×Dic4711yescharacteristic
271C4×C8C2×Dic4711yescharacteristic
272#95—941no—
273#41—41no—
274#41—41no—
275#41—41no—
276#41—41no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8C22⋊Dic4712yes—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C22×C4—41no—
285C22×C4—1881no—
286C22×C4D4×D4711yescharacteristic
287C22×C4#1062535211yescharacteristic
288C22×C4C2×C22⋊D4711yescharacteristic
289SD16—44no—
290M16—472no—
291M16—941no—
292M16—941no—
293M16—941no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—42no—
297C2×C8—941no—
298C2×C8—941no—
299C2×C8—941no—
300C2×C8—941no—
301C2×C8—1881no—
302C2×C8—1881no—
303C2×C8—1881no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—942no—
312C4⋊C4—1881no—
313C4⋊C4C22⋊Dic4712yes—
314C22⋊C4—22no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C42—22no—
318C42—1881no—
319C42—1881no—
320C42D4×D4711yescharacteristic
321C42#1062535211yescharacteristic
322C42C22×Dic4711yescharacteristic
323C42C2×C22⋊D4711yescharacteristic
324C23—41no—
325C23#1309399611yescharacteristic
326Q8—42no—
327Q8—41no—
328Q8—41no—
329Q8—41no—
330Q8—41no—
331D4—42no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—81no—
337D4—81no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—3761no—
347C2×C4—3761no—
348C2×C4—1882no—
349C2×C4C2×C22⋊Dic4711yescharacteristic
350C2×C4D4×Dic4711yescharacteristic
351C2×C4D4×Dic4711yescharacteristic
352C2×C4#1309399211yescharacteristic
353C2×C4#1309399611yescharacteristic
354C2×C4#1309400511yescharacteristic
355C2×C4SD16×D4711yescharacteristic
356C2×C4#1309404511yescharacteristic
357C2×C4#1309405111yescharacteristic
358C2×C4#1309405311yescharacteristic
359C8—22no—
360C8—41no—
361C8—942no—
362C8—3761no—
363C8—1881no—
364C8—1881no—
365C8—1881no—
366C22—21no—
367C22—21no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22#1568900311yescharacteristic
372C22#1568932111yescharacteristic
373C22#1568934311yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—42no—
379C4—81no—
380C4—3762no—
381C4SD16×Dic4711yescharacteristic
382C4#1568922411yescharacteristic
383C4#1568924011yescharacteristic
384C4#1568924511yescharacteristic
385C4#1568926011yescharacteristic
386C4#1568926711yescharacteristic
387C2—21no—
388C2—21no—
389C2—161no—
390C2#2024371711yescharacteristic
391C2#2024374511yescharacteristic
392C2#2024377211yescharacteristic
subgroupG/H
centerC2×C4#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#3167C2
socleC22×C47#386
prime radicalC47×#532C22
cosocleC2×C42×C47C23
hypercenter#3167D47
hypocenterC47#9054
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #20252337C2semidirect
3. #20241887C2semidirect
5. #20241228C2semidirect
6. #20241225C2semidirect
24. C47×#498C4semidirect
25. C47×#492C4semidirect
26. C47×#487C4semidirect
27. C47×#486C4semidirect
119. #10625329SD16semidirect
206. #498Dic47semidirect
207. #492Dic47semidirect
208. #487Dic47semidirect
209. #486Dic47semidirect
227. C47#9054semidirect

Maximal subgroups

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

subgroupindex
#202523372
#202443082
#202418872
#202417872
#202412282
#202412252
C47×#31672
#905447