Finite Group Databasebeta

← #26109341

Subgroups — #26109341

Order 12032.

526 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
1C2×#15689188C211yesmaximal, characteristic
2#20243515C211yesmaximal, characteristic, complemented
3#20243389C211yesmaximal, characteristic
4#20243385C211yesmaximal, characteristic
5C2×#15688873C211yesmaximal, characteristic, complemented
6#20241272C211yesmaximal, characteristic, complemented
7C47×#1296C211yesmaximal, characteristic, complemented
8C22×C8×D47—21no—
9C2×C4×D188C2211yescharacteristic
10#15689188—22no—
11#15689188C414yescomplemented
12C2×C4⋊(C47⋊C8)C2211yescharacteristic
13C2×#13093946—21no—
14C2×#13093944—21no—
15C2×(C2×D47)⋊C8—21no—
16C2×C4×C47⋊C8C2211yescharacteristic
17#15688993C2211yescharacteristic
18#15688886—21no—
19#15688885—21no—
20#15688885—21no—
21#15688885—21no—
22#15688873—22no—
23C2×C4⋊C8×C47C2211yescharacteristic
24C2×C4⋊C8×C47C2211yescharacteristic
25C47×#384C2211yescharacteristic
26C22×D188D411yescharacteristic
27C22×C4×D47—21no—
28C22×C47⋊C8—21no—
29C22×C47⋊C8D411yescharacteristic
30C22×C47⋊C8D411yescharacteristic
31C2×C8×D47—24no—
32C2×C8×D47—42no—
33C2×C8×D47—42no—
34C4×D188—21no—
35C4×D188—21no—
36C4×D188C2×C412yes—
37C4×D188Q812yes—
38C2×C4⋊Dic47—21no—
39C2×C4⋊Dic47—21no—
40C2×C4⋊Dic47—21no—
41C2×C4⋊Dic47D411yescharacteristic
42C2×(C2×D47)⋊C4—21no—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)—21no—
45#13093946—42no—
46#13093944—42no—
47(C2×D47)⋊C8—24no—
48C4×C47⋊C8C2×C412yes—
49C4×C47⋊C8D412yes—
50C22×C8×C47—21no—
51C22×C8×C47—21no—
52C2×C4⋊C4×C47—21no—
53C2×C4⋊C4×C47—21no—
54C2×C4⋊C4×C47—21no—
55C2×C42×C47C2311yescharacteristic
56C4⋊C8×C47—21no—
57C4⋊C8×C47—21no—
58C4⋊C8×C47—21no—
59C4⋊C8×C47C2×C412yes—
60C23×D47—21no—
61C22×C4×C47—41no—
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47#4111yescharacteristic
65C22×Dic47—21no—
66C22×Dic47—41no—
67C2×D188—21no—
68C2×D188—21no—
69C2×D188—21no—
70C2×D188C4⋊C412yes—
71C2×D188SD1612yes—
72C2×D188SD1612yes—
73C2×C4×D47—22no—
74C2×C4×D47—22no—
75C2×C4×D47—41no—
76C2×C4×D47—41no—
77C2×C4×D47—41no—
78C2×C4×D47—41no—
79C2×C8×C47—22no—
80C2×C8×C47—41no—
81C2×C8×C47—41no—
82C2×C8×C47—41no—
83C2×C8×C47—41no—
84C2×C8×C47—41no—
85C4⋊C4×C47—41no—
86C4⋊C4×C47—41no—
87C4⋊C4×C47—41no—
88C4⋊C4×C47—41no—
89C4⋊C4×C47—81no—
90C42×C47C22×C411yescharacteristic
91C42×C47C2×D411yescharacteristic
92C42×C47C2×Q811yescharacteristic
93C42×C47#4111yescharacteristic
94(C2×D47)⋊C4—22no—
95(C2×D47)⋊C4—22no—
96C4⋊Dic47—21no—
97C4⋊Dic47—41no—
98C4⋊Dic47—41no—
99C4⋊Dic47—41no—
100C4⋊Dic47—41no—
101C4⋊Dic47—81no—
102C4⋊Dic47C4⋊C412yes—
103C2×C47⋊C8—21no—
104C2×C47⋊C8—21no—
105C2×C47⋊C8—21no—
106C2×C47⋊C8—21no—
107C2×C47⋊C8—41no—
108C2×C47⋊C8—41no—
109C2×C47⋊C8—41no—
110C2×C47⋊C8C22⋊C412yes—
111C2×C47⋊C8C22⋊C412yes—
112C8×D47—48no—
113C23×C47#12111yescharacteristic
114C22×D47—22no—
115C22×D47—22no—
116C22×D47—22no—
117C22×D47—41no—
118C22×D47—41no—
119C22×D47—41no—
120C22×D47—41no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—81no—
124C2×C4×C47—81no—
125C2×C4×C47—81no—
126C2×C4×C47C2×C22⋊C411yescharacteristic
127C2×C4×C47C2×C4⋊C411yescharacteristic
128C2×C4×C47#11711yescharacteristic
129C2×C4×C47#12111yescharacteristic
130C2×C4×C47C22⋊Q811yescharacteristic
131C2×C4×C47C22⋊Q811yescharacteristic
132C2×C4×C47C2×SD1611yescharacteristic
133C2×C4×C47C2×SD1611yescharacteristic
134C2×C4×C47#13511yescharacteristic
135C2×C4×C47#13511yescharacteristic
136C2×C4×C47#13511yescharacteristic
137C2×C4×C47#13511yescharacteristic
138C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
140C2×Dic47—22no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C2×Dic47—81no—
144C2×Dic47—81no—
145C2×Dic47—81no—
146D188—22no—
147D188—22no—
148D188—22no—
149D188#10614yes—
150C4×D47—42no—
151C4×D47—42no—
152C4×D47—42no—
153C4×D47—42no—
154C8×C47—42no—
155C8×C47—81no—
156C47⋊C8—22no—
157C47⋊C8—22no—
158C47⋊C8—81no—
159#8646—471nomaximal, Sylow
160C22×C47#38011yescharacteristic
161C22×C47#46311yescharacteristic
162C22×C47#46311yescharacteristic
163C22×C47#46511yescharacteristic
164C22×C47#46511yescharacteristic
165C22×C47#47211yescharacteristic
166C22×C47#47311yescharacteristic
167C2×D47—24no—
168C2×D47—42no—
169C2×D47—42no—
170C2×D47—42no—
171C2×D47—42no—
172C2×D47—42no—
173C2×D47—42no—
174C4×C47—21no—
175C4×C47—21no—
176C4×C47—161no—
177C4×C47#41611yescharacteristic
178C4×C47#41811yescharacteristic
179C4×C47C2×#10611yescharacteristic
180C4×C47#42711yescharacteristic
181C4×C47#46511yescharacteristic
182C4×C47#46511yescharacteristic
183C4×C47#47811yescharacteristic
184C4×C47#47811yescharacteristic
185Dic47—42no—
186Dic47—161no—
187C2×C8×D4—471no—
188#1834—471no—
189#1802—471no—
190#1802—471no—
191C2×#334—471no—
192#1296—471no—
193#1296D4711yescharacteristic, complemented
194C2×C47#167711yescharacteristic
195C2×C47#167711yescharacteristic
196C2×C47#167911yescharacteristic
197C2×C47#167911yescharacteristic
198C2×C47#180411yescharacteristic
199C2×C47#180711yescharacteristic
200C2×C47#181611yescharacteristic
201D47—44no—
202C23×C8—941no—
203C2×C4×D4—471no—
204C8×D4—474no—
205C8×D4—942no—
206C2×C4⋊C8—471no—
207C2×C4⋊C8C2×D4711yescharacteristic
208C2×C4⋊C8C2×D4711yescharacteristic
209C2×#102—941no—
210C2×#102—941no—
211C2×C22⋊C8—941no—
212C2×C4×C8—471no—
213#384—471no—
214#384C2×D4711yescharacteristic
215#340—941no—
216#340—941no—
217#340—941no—
218#340—941no—
219#334—942no—
220C47#864611yescharacteristic, complemented, Sylow
221C22×D4—471no—
222C23×C4—941no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—471no—
226C22×C8—471no—
227C22×C8—944no—
228C22×C8—941no—
229C22×C8—1882no—
230C22×C8—1882no—
231C4×D4—472no—
232C4×D4—472no—
233C4×D4—941no—
234C4×D4—941no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—471no—
239C2×C4⋊C4—941no—
240C2×C4⋊C4—941no—
241C2×C4⋊C4—941no—
242C2×C22⋊C4—941no—
243C2×C42C22×D4711yescharacteristic
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—941no—
248C4⋊C8—941no—
249C4⋊C8C4×D4712yes—
250#102—1882no—
251#102—1882no—
252C22⋊C8—944no—
253C4×C8—472no—
254C4×C8—472no—
255C24—941no—
256C2×D4—472no—
257C2×D4—472no—
258C2×D4—472no—
259C2×D4—941no—
260C2×D4—941no—
261C2×D4—941no—
262C22×C4—41no—
263C22×C4—941no—
264C22×C4—942no—
265C22×C4—942no—
266C22×C4—1881no—
267C22×C4—1881no—
268C22×C4—1881no—
269C22×C4—1881no—
270C22×C4—1881no—
271C22×C4#1062535011yescharacteristic
272C22×C4D4×D4711yescharacteristic
273C22×C4D4×D4711yescharacteristic
274C2×C8—22no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—472no—
281C2×C8—472no—
282C2×C8—941no—
283C2×C8—941no—
284C2×C8—941no—
285C2×C8—941no—
286C2×C8—1888no—
287C2×C8—1881no—
288C2×C8—1881no—
289C2×C8—1881no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—81no—
295C4⋊C4—472no—
296C4⋊C4—941no—
297C4⋊C4—3761no—
298C4⋊C4—1881no—
299C4⋊C4—1881no—
300C4⋊C4—1881no—
301C4⋊C4—1881no—
302C22⋊C4—942no—
303C22⋊C4—942no—
304C42C2×C4×D4711yescharacteristic
305C42#1062535011yescharacteristic
306C42D4×D4711yescharacteristic
307C42Q8×D4711yescharacteristic
308C23—942no—
309C23—942no—
310C23—942no—
311C23—1881no—
312C23—1881no—
313C23—1881no—
314C23—1881no—
315C23#1309399711yescharacteristic
316D4—474no—
317D4—942no—
318D4—942no—
319D4—942no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—942no—
326C2×C4—3761no—
327C2×C4—3761no—
328C2×C4—3761no—
329C2×C4—1881no—
330C2×C4—1881no—
331C2×C4—1882no—
332C2×C4—1882no—
333C2×C4—1882no—
334C2×C4—1882no—
335C2×C4C22⋊C4×D4711yescharacteristic
336C2×C4C4⋊C4×D4711yescharacteristic
337C2×C4#1309398111yescharacteristic
338C2×C4#1309399711yescharacteristic
339C2×C4#1309400211yescharacteristic
340C2×C4#1309400211yescharacteristic
341C2×C4SD16×D4711yescharacteristic
342C2×C4SD16×D4711yescharacteristic
343C2×C4#1309404311yescharacteristic
344C2×C4#1309404511yescharacteristic
345C2×C4#1309404711yescharacteristic
346C2×C4#1309404711yescharacteristic
347C2×C4#1309404811yescharacteristic
348C2×C4#1309405111yescharacteristic
349C8—42no—
350C8—81no—
351C8—942no—
352C8—942no—
353C8—3761no—
354C22—944no—
355C22—1882no—
356C22—1882no—
357C22—1882no—
358C22—1882no—
359C22—1882no—
360C22—1882no—
361C22#1568898211yescharacteristic
362C22#1568933711yescharacteristic
363C22#1568934111yescharacteristic
364C22#1568935111yescharacteristic
365C22#1568935511yescharacteristic
366C22#1568939211yescharacteristic
367C22#1568939711yescharacteristic
368C4—21no—
369C4—21no—
370C4—161no—
371C4—1882no—
372C4—7521no—
373C4#1568911811yescharacteristic
374C4#1568912611yescharacteristic
375C4#106×D4711yescharacteristic
376C4#1568916011yescharacteristic
377C4#1568935111yescharacteristic
378C4#1568935511yescharacteristic
379C4#1568942211yescharacteristic
380C4#1568942611yescharacteristic
381C2—1884no—
382C2#2024279611yescharacteristic
383C2#2024280011yescharacteristic
384C2#2024281011yescharacteristic
385C2#2024281411yescharacteristic
386C2#2024339711yescharacteristic
387C2#2024341211yescharacteristic
388C2#2024345111yescharacteristic
subgroupG/H
centerC23#13093997
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1296C2
socleC23×C47#121
prime radicalC22×D188D4
cosocleC2×C42×C47C23
hypercenter#1296D47
hypocenterC47#8646
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. #20243515C2semidirect
5. C2×#15688873C2semidirect
6. #20241272C2semidirect
7. C47×#1296C2semidirect
11. #15689188C4semidirect
193. #1296D47semidirect
220. C47#8646semidirect

Maximal subgroups

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

subgroupindex
C2×#156891882
#202435152
#202433892
#202433852
C2×#156888732
#202412722
C47×#12962
#864647