Finite Group Databasebeta

← #26110576

Subgroups — #26110576

Order 12032.

438 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251956C211yesmaximal, characteristic
2#20242164C211yesmaximal, characteristic, complemented
3#20242146C211yesmaximal, characteristic, complemented
4#20241770C211yesmaximal, characteristic
5#20241291C211yesmaximal, characteristic, complemented
6#20241229C211yesmaximal, characteristic, complemented
7C47×#1806C211yesmaximal, characteristic
8C2×#13094031—21no—
9#15689176C2211yescharacteristic
10#15689150—22no—
11#15689150—21no—
12#15689147—21no—
13#15689147C2211yescharacteristic
14C2×C4⋊(C47⋊C8)—21no—
15C2×C4⋊(C47⋊C8)C2211yescharacteristic
16C2×C4×C47⋊C8C2211yescharacteristic
17#15688875—22no—
18#15688875—21no—
19#15688875—21no—
20Q8⋊(C47⋊C8)C411yescharacteristic, complemented
21Q8⋊(C47⋊C8)C411yescharacteristic, complemented
22D4⋊(C47⋊C8)C411yescharacteristic, complemented
23D4⋊(C47⋊C8)C411yescharacteristic, complemented
24C4×#41×C47C2211yescharacteristic
25C47×#423C2211yescharacteristic
26C2×#104×C47—21no—
27C4×C4⋊C4×C47C2211yescharacteristic
28C47×#339—21no—
29#13094031—42no—
30#13094031—42no—
31C2×#10625329—22no—
32C2×#10625329—21no—
33C22×C47⋊C8—22no—
34C22×C47⋊C8—21no—
35C22×C47⋊C8—21no—
36C4⋊(C47⋊C8)—22no—
37C4⋊(C47⋊C8)—22no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)C2×C411yescharacteristic
41C4⋊(C47⋊C8)C2×C411yescharacteristic
42C4⋊(C47⋊C8)D411yescharacteristic
43C4⋊(C47⋊C8)D411yescharacteristic
44C4⋊(C47⋊C8)D412yes—
45C22⋊(C47⋊C8)—21no—
46C4×C47⋊C8—21no—
47C4×C47⋊C8—21no—
48C4×C47⋊C8C2×C411yescharacteristic
49C4×C47⋊C8C2×C411yescharacteristic
50C2×#41×C47D411yescharacteristic
51C2×M16×C47—21no—
52C4×Q8×C47C2×C411yescharacteristic
53C4×Q8×C47Q811yescharacteristic
54C4×D4×C47—21no—
55C4×D4×C47—21no—
56C4×D4×C47C2×C411yescharacteristic
57C4×D4×C47Q811yescharacteristic
58#117×C47—21no—
59#117×C47D411yescharacteristic
60C2×C4⋊C4×C47—21no—
61C2×C42×C47—21no—
62C2×C42×C47—21no—
63C2×C42×C47C2311yescharacteristic
64C4⋊C8×C47—21no—
65C4⋊C8×C47—21no—
66#104×C47—42no—
67#95×C47—21no—
68#41×C47—21no—
69#41×C47—21no—
70#41×C47D812yes—
71#41×C47Q1612yes—
72C2×Q8×C47C4⋊C411yescharacteristic
73C2×D4×C47—21no—
74C2×D4×C47C4⋊C411yescharacteristic
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47—21no—
78C22×C4×C47—41no—
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47#4111yescharacteristic
81C22×C4×C47#4111yescharacteristic
82M16×C47—41no—
83M16×C47—41no—
84C2×C8×C47—41no—
85C4⋊C4×C47—21no—
86C4⋊C4×C47—21no—
87C4⋊C4×C47—22no—
88C4⋊C4×C47—41no—
89C4⋊C4×C47C4⋊C411yescharacteristic
90C4⋊C4×C47C4⋊C411yescharacteristic
91C22⋊C4×C47—21no—
92C22⋊C4×C47—21no—
93C22⋊C4×C47—21no—
94C42×C47—21no—
95C42×C47—21no—
96C42×C47—21no—
97C42×C47—41no—
98C42×C47—41no—
99C42×C47C22×C411yescharacteristic
100C42×C47C2×D411yescharacteristic
101C42×C47C2×D411yescharacteristic
102C42×C47C2×Q811yescharacteristic
103#10625329—44no—
104#10625329—41no—
105#10625329—41no—
106C2×C47⋊C8—21no—
107C2×C47⋊C8—21no—
108C2×C47⋊C8—21no—
109C2×C47⋊C8—21no—
110C2×C47⋊C8—21no—
111C2×C47⋊C8—22no—
112C2×C47⋊C8—21no—
113C2×C47⋊C8—42no—
114C2×C47⋊C8—44no—
115C2×C47⋊C8—41no—
116C2×C47⋊C8—41no—
117C23×C47—21no—
118C23×C47#12311yescharacteristic
119Q8×C47—21no—
120Q8×C47#10712yes—
121D4×C47—22no—
122D4×C47—22no—
123D4×C47—21no—
124D4×C47#10712yes—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—22no—
129C2×C4×C47—22no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—42no—
136C2×C4×C47—81no—
137C2×C4×C47—81no—
138C2×C4×C47—81no—
139C2×C4×C47C2×C4⋊C411yescharacteristic
140C2×C4×C47C4×D411yescharacteristic
141C2×C4×C47C4×D411yescharacteristic
142C2×C4×C47#12011yescharacteristic
143C2×C4×C47C22⋊Q811yescharacteristic
144C2×C4×C47C22⋊Q811yescharacteristic
145C2×C4×C47C2×D811yescharacteristic
146C2×C4×C47C2×Q1611yescharacteristic
147C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C47#13711yescharacteristic
149C8×C47—81no—
150C47⋊C8—22no—
151C47⋊C8—41no—
152C47⋊C8—41no—
153C47⋊C8—41no—
154C47⋊C8—41no—
155C47⋊C8—42no—
156C47⋊C8—41no—
157#8902—471nomaximal, Sylow
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—22no—
161C22×C47—41no—
162C22×C47—41no—
163C22×C47#38511yescharacteristic
164C22×C47#48011yescharacteristic
165C22×C47#48411yescharacteristic
166C4×C47—22no—
167C4×C47—21no—
168C4×C47—21no—
169C4×C47—21no—
170C4×C47—41no—
171C4×C47—41no—
172C4×C47—82no—
173C4×C47C2×#10711yescharacteristic
174C4×C47#42811yescharacteristic
175C4×C47#44311yescharacteristic
176C4×C47#44411yescharacteristic
177C4×C47#44711yescharacteristic
178C4×C47#45111yescharacteristic
179#2906—471no—
180#1806D4711yescharacteristic
181#1571—471no—
182#1567—471no—
183#1476—471no—
184#1303—471no—
185#1279—471no—
186C2×C47—21no—
187C2×C47—21no—
188C2×C47—42no—
189C2×C47#185211yescharacteristic
190C2×C47#186111yescharacteristic
191C2×C47#186611yescharacteristic
192C2×#131—941no—
193C4×#41C2×D4711yescharacteristic
194#431—471no—
195#424—942no—
196#424—941no—
197#423—471no—
198#423—941no—
199#423C2×D4711yescharacteristic
200C2×C4⋊C8—471no—
201C2×C4⋊C8—941no—
202C2×#104—21no—
203C2×C4×C8—471no—
204C4×C4⋊C4C2×D4711yescharacteristic
205#339—21no—
206#335—942no—
207#335—941no—
208#335—941no—
209Q8⋊C8—471no—
210Q8⋊C8—471no—
211D4⋊C8—471no—
212D4⋊C8—471no—
213C47#890211yescharacteristic, complemented, Sylow
214C2×#41C22⋊D4711yescharacteristic
215#131—1882no—
216#131—1882no—
217C2×M16—21no—
218C2×M16—942no—
219C2×M16—941no—
220C22×C8—942no—
221C22×C8—941no—
222C22×C8—941no—
223C4×Q8Dic9411yescharacteristic
224C4×Q8C4×D4711yescharacteristic
225C4×D4—21no—
226C4×D4—21no—
227C4×D4Dic9411yescharacteristic
228C4×D4C4×D4711yescharacteristic
229#117—21no—
230#117C22⋊D4711yescharacteristic
231C2×C4⋊C4—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42C22×D4711yescharacteristic
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—471no—
238C4⋊C8—471no—
239C4⋊C8—471no—
240C4⋊C8—471no—
241C4⋊C8—472no—
242C4⋊C8—942no—
243C4⋊C8—942no—
244C4⋊C8—941no—
245C4⋊C8—941no—
246#104—42no—
247C22⋊C8—941no—
248C4×C8—471no—
249C4×C8—471no—
250C4×C8—941no—
251C4×C8—941no—
252#95—21no—
253#41—21no—
254#41—21no—
255#41#1062533412yes—
256#41#1062533712yes—
257C2×Q8Dic47⋊C411yescharacteristic
258C2×D4—21no—
259C2×D4Dic47⋊C411yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4#1062535211yescharacteristic
265C22×C4#1062535211yescharacteristic
266C22×C4C2×C22⋊D4711yescharacteristic
267M16—41no—
268M16—41no—
269M16—1884no—
270M16—1881no—
271M16—1881no—
272C2×C8—41no—
273C2×C8—941no—
274C2×C8—941no—
275C2×C8—941no—
276C2×C8—941no—
277C2×C8—941no—
278C2×C8—941no—
279C2×C8—942no—
280C2×C8—1882no—
281C2×C8—1884no—
282C2×C8—1881no—
283C2×C8—1881no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4Dic47⋊C411yescharacteristic
289C4⋊C4Dic47⋊C411yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—41no—
297C42—41no—
298C42C2×Dic9411yescharacteristic
299C42C2×C4×D4711yescharacteristic
300C42D4×D4711yescharacteristic
301C42D4×D4711yescharacteristic
302C23—21no—
303C23#1309400511yescharacteristic
304Q8—21no—
305Q8#1309395712yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4#1309395712yes—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4C2×Dic47⋊C411yescharacteristic
325C2×C4(C22⋊D47)⋊C411yescharacteristic
326C2×C4(C22⋊D47)⋊C411yescharacteristic
327C2×C4#1309399211yescharacteristic
328C2×C4#1309400111yescharacteristic
329C2×C4#1309400111yescharacteristic
330C2×C4C2×#1062533411yescharacteristic
331C2×C4C2×#1062533711yescharacteristic
332C2×C4#1309404911yescharacteristic
333C2×C4#1309405411yescharacteristic
334C8—81no—
335C8—942no—
336C8—1881no—
337C8—1881no—
338C8—1881no—
339C8—1881no—
340C8—1882no—
341C8—1881no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22#1568899511yescharacteristic
348C22#1568943411yescharacteristic
349C22#1568945611yescharacteristic
350C4—22no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—82no—
357C4C2×#1309395711yescharacteristic
358C4#1568916611yescharacteristic
359C4#1568923711yescharacteristic
360C4#1568924211yescharacteristic
361C4#1568925611yescharacteristic
362C4#1568927811yescharacteristic
363C2—21no—
364C2—21no—
365C2—42no—
366C2#2024357511yescharacteristic
367C2#2024362611yescharacteristic
368C2#2024366111yescharacteristic
subgroupG/H
centerC2×C4#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1806C2
socleC22×C47#385
prime radicalC2×#41×C47D4
cosocleC2×C42×C47C23
hypercenter#1806D47
hypocenterC47#8902
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #20242164C2semidirect
3. #20242146C2semidirect
5. #20241291C2semidirect
6. #20241229C2semidirect
20. Q8⋊(C47⋊C8)C4semidirect
21. Q8⋊(C47⋊C8)C4semidirect
22. D4⋊(C47⋊C8)C4semidirect
23. D4⋊(C47⋊C8)C4semidirect
213. C47#8902semidirect

Maximal subgroups

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

subgroupindex
#202519562
#202421642
#202421462
#202417702
#202412912
#202412292
C47×#18062
#890247