Finite Group Databasebeta

← #26120130

Subgroups — #26120130

Order 12032.

474 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689410C211yesmaximal, characteristic
2#20244182C211yesmaximal, characteristic
3#20243385C211yesmaximal, characteristic
4C2×#15688876C211yesmaximal, characteristic, complemented
5C2×#15688872C211yesmaximal, characteristic, complemented
6#20241271C211yesmaximal, characteristic, complemented
7C47×#1990C211yesmaximal, characteristic, complemented
8C2×C4×D188C2211yescharacteristic
9#15689410—22no—
10#15689410—22no—
11C2×#13093955—21no—
12C2×C4⋊(C47⋊C8)C2211yescharacteristic
13C2×C4⋊(C47⋊C8)C2211yescharacteristic
14C2×C4⋊(C47⋊C8)C2211yescharacteristic
15C2×#13093944—21no—
16C2×#13093944—21no—
17C2×(C2×D47)⋊C8—21no—
18C2×(C2×D47)⋊C8—21no—
19#15688885—21no—
20#15688876—22no—
21#15688872—22no—
22C2×C4⋊Q8×C47C2211yescharacteristic
23C2×C4×C8×C47C2211yescharacteristic
24C47×#384C2211yescharacteristic
25C47×#340—21no—
26C47×#340—21no—
27C22×D188D411yescharacteristic
28C22×C4×D47—21no—
29C22×C47⋊C8—21no—
30C22×C47⋊C8—21no—
31C22×C47⋊C8—21no—
32C4×D188—21no—
33C4×D188—21no—
34C4×D188—21no—
35C4×D188—21no—
36C2×C4⋊Dic47D411yescharacteristic
37C2×(C2×D47)⋊C4—21no—
38#13093955—24no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)D412yes—
45#13093944—24no—
46#13093944—42no—
47(C2×D47)⋊C8—42no—
48(C2×D47)⋊C8—42no—
49C22×Q8×C47—21no—
50C22×C8×C47D411yescharacteristic
51C22×C8×C47D411yescharacteristic
52C4⋊Q8×C47—21no—
53C4⋊Q8×C47—21no—
54C4⋊Q8×C47—21no—
55C4⋊Q8×C47D412yes—
56C2×C4⋊C4×C47—21no—
57C2×C4⋊C4×C47—21no—
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47—21no—
60C2×C4⋊C4×C47—21no—
61C2×C42×C47C2311yescharacteristic
62C4×C8×C47—21no—
63C4×C8×C47—21no—
64C23×D47—21no—
65C2×Q8×C47—22no—
66C2×Q8×C47—41no—
67C2×Q8×C47—81no—
68C2×Q8×C47—81no—
69C22×C4×C47—41no—
70C22×C4×C47—41no—
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47#4111yescharacteristic
74C22×Dic47—21no—
75C2×D188—21no—
76C2×D188—21no—
77C2×D188—21no—
78C2×D188—21no—
79C2×D188—21no—
80C2×D188Q1612yes—
81C2×C4×D47—41no—
82C2×C4×D47—41no—
83C2×C4×D47—41no—
84C2×C4×D47—41no—
85C2×C4×D47—41no—
86C2×C4×D47—41no—
87C2×C8×C47—21no—
88C2×C8×C47—21no—
89C2×C8×C47—21no—
90C2×C8×C47—21no—
91C2×C8×C47—21no—
92C2×C8×C47SD1612yes—
93C4⋊C4×C47—22no—
94C4⋊C4×C47—22no—
95C4⋊C4×C47—41no—
96C4⋊C4×C47—41no—
97C4⋊C4×C47—41no—
98C4⋊C4×C47—41no—
99C4⋊C4×C47—41no—
100C4⋊C4×C47—41no—
101C4⋊C4×C47—81no—
102C42×C47C2×D411yescharacteristic
103C42×C47C2×D411yescharacteristic
104C42×C47#4111yescharacteristic
105C42×C47#4111yescharacteristic
106(C2×D47)⋊C4—41no—
107(C2×D47)⋊C4—41no—
108C4⋊Dic47—21no—
109C4⋊Dic47—21no—
110C2×C47⋊C8—22no—
111C2×C47⋊C8—41no—
112C2×C47⋊C8—41no—
113C2×C47⋊C8—41no—
114C2×C47⋊C8—41no—
115C2×C47⋊C8—41no—
116C2×C47⋊C8—41no—
117C2×C47⋊C8—41no—
118C2×C47⋊C8—41no—
119C23×C47#12111yescharacteristic
120C22×D47—22no—
121C22×D47—41no—
122C22×D47—41no—
123C22×D47—41no—
124C22×D47—41no—
125C22×D47—41no—
126C22×D47—41no—
127Q8×C47—82no—
128Q8×C47—82no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—42no—
132C2×C4×C47—81no—
133C2×C4×C47—81no—
134C2×C4×C47—81no—
135C2×C4×C47—81no—
136C2×C4×C47—81no—
137C2×C4×C47#12011yescharacteristic
138C2×C4×C47#12111yescharacteristic
139C2×C4×C47#12111yescharacteristic
140C2×C4×C47#12311yescharacteristic
141C2×C4×C47#12311yescharacteristic
142C2×C4×C47#12411yescharacteristic
143C2×C4×C47C2×SD1611yescharacteristic
144C2×C4×C47C2×Q1611yescharacteristic
145C2×C4×C47#13511yescharacteristic
146C2×C4×C47#13511yescharacteristic
147C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C47#13711yescharacteristic
150C2×C4×C47#13711yescharacteristic
151C2×Dic47—41no—
152C2×Dic47—41no—
153C2×Dic47—41no—
154D188—22no—
155D188—22no—
156D188—41no—
157D188—41no—
158C4×D47—81no—
159C4×D47—81no—
160C4×D47—81no—
161C4×D47—81no—
162C8×C47—22no—
163C8×C47—41no—
164C47⋊C8—42no—
165C47⋊C8—81no—
166C47⋊C8—81no—
167#10481—471nomaximal, Sylow
168C22×C47#39411yescharacteristic
169C22×C47#46111yescharacteristic
170C22×C47#46311yescharacteristic
171C22×C47#46711yescharacteristic
172C22×C47#46811yescharacteristic
173C22×C47#47111yescharacteristic
174C22×C47#47211yescharacteristic
175C2×D47—42no—
176C2×D47—42no—
177C2×D47—42no—
178C2×D47—81no—
179C2×D47—81no—
180C2×D47—81no—
181C2×D47—81no—
182C4×C47—21no—
183C4×C47—21no—
184C4×C47—82no—
185C4×C47—161no—
186C4×C47#44911yescharacteristic
187C4×C47#45111yescharacteristic
188C4×C47#46011yescharacteristic
189C4×C47#46411yescharacteristic
190C4×C47#46511yescharacteristic
191C4×C47#46811yescharacteristic
192C4×C47#48411yescharacteristic
193C4×C47#48511yescharacteristic
194Dic47—81no—
195C2×#476—471no—
196#1990D4711yescharacteristic, complemented
197#1954—471no—
198#1802—471no—
199C2×#335—471no—
200C2×#333—471no—
201#1296—471no—
202C2×C47#167511yescharacteristic
203C2×C47#168311yescharacteristic
204C2×C47#169111yescharacteristic
205C2×C47#169711yescharacteristic
206C2×C47#203611yescharacteristic
207C2×C47#205111yescharacteristic
208C2×C47#205511yescharacteristic
209D47—82no—
210C2×C4⋊Q8C2×D4711yescharacteristic
211C2×C4×D4—471no—
212#476—942no—
213#476—942no—
214C2×#106—941no—
215C2×C4⋊C8—471no—
216C2×C4⋊C8—471no—
217C2×C4⋊C8—471no—
218C2×#102—941no—
219C2×#102—941no—
220C2×C22⋊C8—941no—
221C2×C22⋊C8—941no—
222C2×C4×C8C2×D4711yescharacteristic
223#384C2×D4711yescharacteristic
224#340—21no—
225#340—21no—
226#340—941no—
227#335—942no—
228#333—942no—
229C47#1048111yescharacteristic, complemented, Sylow
230C22×Q8—21no—
231C22×D4—471no—
232C23×C4—941no—
233C22×C8—941no—
234C22×C8—941no—
235C22×C8—941no—
236C22×C8C22⋊D4711yescharacteristic
237C22×C8C22⋊D4711yescharacteristic
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8D18812yes—
242C4×D4—941no—
243C4×D4—941no—
244C4×D4—941no—
245C4×D4—941no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—471no—
252C2×C22⋊C4—941no—
253C2×C42C22×D4711yescharacteristic
254#106—944no—
255C4⋊C8—472no—
256C4⋊C8—941no—
257C4⋊C8—941no—
258C4⋊C8—941no—
259C4⋊C8—941no—
260C4⋊C8—941no—
261#102—944no—
262#102—1882no—
263C22⋊C8—1882no—
264C22⋊C8—1882no—
265C4×C8—21no—
266C4×C8—21no—
267C24—941no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—81no—
271C2×Q8—81no—
272C2×D4—472no—
273C2×D4—941no—
274C2×D4—941no—
275C2×D4—941no—
276C2×D4—941no—
277C2×D4—941no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—941no—
281C22×C4—1881no—
282C22×C4—1881no—
283C22×C4—1881no—
284C22×C4—1881no—
285C22×C4—1881no—
286C22×C4—1881no—
287C22×C4D4×D4711yescharacteristic
288C22×C4#1062535211yescharacteristic
289C22×C4C2×C22⋊D4711yescharacteristic
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—942no—
296C2×C8—1881no—
297C2×C8—1881no—
298C2×C8—1881no—
299C2×C8—1881no—
300C2×C8—1881no—
301C2×C8—1881no—
302C2×C8—1881no—
303C2×C8—1881no—
304C2×C8#1062533612yes—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—81no—
314C4⋊C4—941no—
315C4⋊C4—941no—
316C22⋊C4—1881no—
317C22⋊C4—1881no—
318C42C2×D18811yescharacteristic
319C42#1062535011yescharacteristic
320C42D4×D4711yescharacteristic
321C42#1062535211yescharacteristic
322C23—942no—
323C23—1881no—
324C23—1881no—
325C23—1881no—
326C23—1881no—
327C23—1881no—
328C23—1881no—
329C23#1309399511yescharacteristic
330Q8—82no—
331Q8—82no—
332D4—942no—
333D4—942no—
334D4—1881no—
335D4—1881no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—3761no—
345C2×C4—3761no—
346C2×C4—3761no—
347C2×C4—3761no—
348C2×C4—1881no—
349C2×C4—1881no—
350C2×C4—1881no—
351C2×C4#1309399411yescharacteristic
352C2×C4#1309399611yescharacteristic
353C2×C4#1309399811yescharacteristic
354C2×C4#1309400611yescharacteristic
355C2×C4#1309400811yescharacteristic
356C2×C4#1309401011yescharacteristic
357C2×C4C2×#1062533611yescharacteristic
358C2×C4Q16×D4711yescharacteristic
359C2×C4#1309404411yescharacteristic
360C2×C4#1309404711yescharacteristic
361C2×C4#1309404911yescharacteristic
362C2×C4#1309405111yescharacteristic
363C2×C4#1309405311yescharacteristic
364C2×C4#1309405511yescharacteristic
365C8—22no—
366C8—41no—
367C8—3761no—
368C8—3761no—
369C8—1882no—
370C22—3761no—
371C22—3761no—
372C22—3761no—
373C22—3761no—
374C22—1882no—
375C22—1882no—
376C22—1882no—
377C22#1568903911yescharacteristic
378C22#1568932511yescharacteristic
379C22#1568933511yescharacteristic
380C22#1568936111yescharacteristic
381C22#1568936611yescharacteristic
382C22#1568938311yescharacteristic
383C22#1568939011yescharacteristic
384C4—21no—
385C4—21no—
386C4—82no—
387C4—161no—
388C4—3761no—
389C4#1568927111yescharacteristic
390C4#1568928011yescharacteristic
391C4#1568932311yescharacteristic
392C4#1568934511yescharacteristic
393C4#1568935411yescharacteristic
394C4#1568937111yescharacteristic
395C4#1568945711yescharacteristic
396C4#1568946311yescharacteristic
397C2—3762no—
398C2#2024278411yescharacteristic
399C2#2024283611yescharacteristic
400C2#2024288411yescharacteristic
401C2#2024291811yescharacteristic
402C2#2024459011yescharacteristic
403C2#2024468111yescharacteristic
404C2#2024470311yescharacteristic
subgroupG/H
centerC23#13093995
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1990C2
socleC23×C47#121
prime radicalC22×D188D4
hypercenter#1990D47
hypocenterC47#10481
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
4. C2×#15688876C2semidirect
5. C2×#15688872C2semidirect
6. #20241271C2semidirect
7. C47×#1990C2semidirect
196. #1990D47semidirect
229. C47#10481semidirect

Maximal subgroups

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

subgroupindex
C2×#156894102
#202441822
#202433852
C2×#156888762
C2×#156888722
#202412712
C47×#19902
#1048147