Finite Group Databasebeta

← #26110583

Subgroups — #26110583

Order 12032.

392 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252054C211yesmaximal, characteristic
2#20242160C211yesmaximal, characteristic, complemented
3#20242146C211yesmaximal, characteristic, complemented
4#20241770C211yesmaximal, characteristic
5#20241291C211yesmaximal, characteristic, complemented
6#20241229C211yesmaximal, characteristic, complemented
7C47×#1806C211yesmaximal, characteristic
8#15689184—22no—
9#15689180—21no—
10#15689176—21no—
11#15689176C2211yescharacteristic
12#15689147C2211yescharacteristic
13C2×C4⋊(C47⋊C8)C2211yescharacteristic
14#15689090—21no—
15#15689079—21no—
16C2×C4×C47⋊C8C2211yescharacteristic
17#15688875—22no—
18#15688873—21no—
19#15688873—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—
29C2×#10625329—21no—
30C22×C47⋊C8—21no—
31C22×C47⋊C8—21no—
32C4⋊(C47⋊C8)—21no—
33C4⋊(C47⋊C8)—21no—
34C4⋊(C47⋊C8)C2×C411yescharacteristic
35C4⋊(C47⋊C8)C2×C411yescharacteristic
36C4⋊(C47⋊C8)D411yescharacteristic
37C4⋊(C47⋊C8)D411yescharacteristic
38C4⋊(C47⋊C8)D412yes—
39C22⋊(C47⋊C8)—22no—
40C22⋊(C47⋊C8)—22no—
41C22⋊(C47⋊C8)—21no—
42#13093935—22no—
43C4×C47⋊C8—22no—
44C4×C47⋊C8—21no—
45C4×C47⋊C8—21no—
46C4×C47⋊C8C2×C411yescharacteristic
47C4×C47⋊C8C2×C411yescharacteristic
48C2×#41×C47D411yescharacteristic
49C2×M16×C47—21no—
50C4×Q8×C47C2×C411yescharacteristic
51C4×Q8×C47Q811yescharacteristic
52C4×D4×C47—21no—
53C4×D4×C47—21no—
54C4×D4×C47C2×C411yescharacteristic
55C4×D4×C47Q811yescharacteristic
56#117×C47—21no—
57#117×C47D411yescharacteristic
58C2×C4⋊C4×C47—21no—
59C2×C42×C47—21no—
60C2×C42×C47—21no—
61C2×C42×C47C2311yescharacteristic
62C4⋊C8×C47—21no—
63C4⋊C8×C47—21no—
64#104×C47—42no—
65#95×C47—21no—
66#41×C47—21no—
67#41×C47—21no—
68#41×C47—21no—
69#41×C47—21no—
70C2×Q8×C47C4⋊C411yescharacteristic
71C2×D4×C47—21no—
72C2×D4×C47C4⋊C411yescharacteristic
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—21no—
76C22×C4×C47—41no—
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47#4111yescharacteristic
79C22×C4×C47#4111yescharacteristic
80M16×C47—41no—
81M16×C47—41no—
82C2×C8×C47—41no—
83C4⋊C4×C47—21no—
84C4⋊C4×C47—21no—
85C4⋊C4×C47—22no—
86C4⋊C4×C47—41no—
87C4⋊C4×C47C4⋊C411yescharacteristic
88C4⋊C4×C47C4⋊C411yescharacteristic
89C22⋊C4×C47—21no—
90C22⋊C4×C47—21no—
91C22⋊C4×C47—21no—
92C42×C47—21no—
93C42×C47—21no—
94C42×C47—21no—
95C42×C47—41no—
96C42×C47—41no—
97C42×C47C22×C411yescharacteristic
98C42×C47C2×D411yescharacteristic
99C42×C47C2×D411yescharacteristic
100C42×C47C2×Q811yescharacteristic
101#10625329—41no—
102#10625329—41no—
103C2×C47⋊C8—21no—
104C2×C47⋊C8—21no—
105C2×C47⋊C8—21no—
106C2×C47⋊C8—21no—
107C2×C47⋊C8—21no—
108C2×C47⋊C8—22no—
109C2×C47⋊C8—21no—
110C2×C47⋊C8—42no—
111C2×C47⋊C8—41no—
112C2×C47⋊C8—41no—
113C23×C47—21no—
114C23×C47#12311yescharacteristic
115Q8×C47—21no—
116Q8×C47—21no—
117D4×C47—21no—
118D4×C47—21no—
119D4×C47—41no—
120D4×C47—41no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—42no—
132C2×C4×C47—81no—
133C2×C4×C47—81no—
134C2×C4×C47—81no—
135C2×C4×C47C2×C4⋊C411yescharacteristic
136C2×C4×C47C4×D411yescharacteristic
137C2×C4×C47C4×D411yescharacteristic
138C2×C4×C47#12011yescharacteristic
139C2×C4×C47C22⋊Q811yescharacteristic
140C2×C4×C47C22⋊Q811yescharacteristic
141C2×C4×C47#13511yescharacteristic
142C2×C4×C47#13511yescharacteristic
143C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C47#13711yescharacteristic
145C8×C47—81no—
146C47⋊C8—22no—
147C47⋊C8—41no—
148C47⋊C8—41no—
149C47⋊C8—41no—
150C47⋊C8—42no—
151C47⋊C8—41no—
152C47⋊C8—41no—
153#8903—471nomaximal, Sylow
154C22×C47—21no—
155C22×C47—21no—
156C22×C47—41no—
157C22×C47—41no—
158C22×C47—41no—
159C22×C47#38511yescharacteristic
160C22×C47#48211yescharacteristic
161C22×C47#48511yescharacteristic
162C4×C47—21no—
163C4×C47—21no—
164C4×C47—21no—
165C4×C47—41no—
166C4×C47—41no—
167C4×C47—41no—
168C4×C47—82no—
169C4×C47#42711yescharacteristic
170C4×C47#42811yescharacteristic
171C4×C47#44311yescharacteristic
172C4×C47#44411yescharacteristic
173C4×C47#44911yescharacteristic
174C4×C47#45211yescharacteristic
175#2925—471no—
176#1806D4711yescharacteristic
177#1570—471no—
178#1567—471no—
179#1476—471no—
180#1303—471no—
181#1279—471no—
182C2×C47—21no—
183C2×C47—21no—
184C2×C47—81no—
185C2×C47#185411yescharacteristic
186C2×C47#186111yescharacteristic
187C2×C47#186611yescharacteristic
188C4×#41C2×D4711yescharacteristic
189#433—942no—
190#432—941no—
191#431—471no—
192#431—941no—
193#423—471no—
194#423C2×D4711yescharacteristic
195C2×C4⋊C8—471no—
196C2×#104—21no—
197#408—941no—
198#405—941no—
199C2×C4×C8—471no—
200C4×C4⋊C4C2×D4711yescharacteristic
201#339—21no—
202#335—942no—
203#334—941no—
204#334—941no—
205Q8⋊C8—471no—
206Q8⋊C8—471no—
207D4⋊C8—471no—
208D4⋊C8—471no—
209C47#890311yescharacteristic, complemented, Sylow
210C2×#41C22⋊D4711yescharacteristic
211C2×M16—21no—
212C2×M16—941no—
213C22×C8—941no—
214C22×C8—941no—
215C4×Q8Dic9411yescharacteristic
216C4×Q8C4×D4711yescharacteristic
217C4×D4—21no—
218C4×D4—21no—
219C4×D4Dic9411yescharacteristic
220C4×D4C4×D4711yescharacteristic
221#117—21no—
222#117C22⋊D4711yescharacteristic
223C2×C4⋊C4—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42C22×D4711yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—471no—
230C4⋊C8—471no—
231C4⋊C8—471no—
232C4⋊C8—471no—
233C4⋊C8—472no—
234C4⋊C8—941no—
235C4⋊C8—941no—
236#104—42no—
237C22⋊C8—942no—
238C22⋊C8—942no—
239C22⋊C8—941no—
240#97—942no—
241C4×C8—471no—
242C4×C8—471no—
243C4×C8—942no—
244C4×C8—941no—
245C4×C8—941no—
246#95—21no—
247#41—21no—
248#41—21no—
249#41—21no—
250#41—21no—
251C2×Q8Dic47⋊C411yescharacteristic
252C2×D4—21no—
253C2×D4Dic47⋊C411yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4#1062535211yescharacteristic
259C22×C4#1062535211yescharacteristic
260C22×C4C2×C22⋊D4711yescharacteristic
261M16—41no—
262M16—41no—
263M16—1881no—
264M16—1881no—
265C2×C8—41no—
266C2×C8—941no—
267C2×C8—941no—
268C2×C8—941no—
269C2×C8—941no—
270C2×C8—941no—
271C2×C8—942no—
272C2×C8—941no—
273C2×C8—1882no—
274C2×C8—1881no—
275C2×C8—1881no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4Dic47⋊C411yescharacteristic
281C4⋊C4Dic47⋊C411yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—41no—
289C42—41no—
290C42C2×Dic9411yescharacteristic
291C42C2×C4×D4711yescharacteristic
292C42D4×D4711yescharacteristic
293C42D4×D4711yescharacteristic
294C23—21no—
295C23#1309400511yescharacteristic
296Q8—21no—
297Q8—21no—
298D4—21no—
299D4—21no—
300D4—41no—
301D4—41no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4C2×Dic47⋊C411yescharacteristic
317C2×C4(C22⋊D47)⋊C411yescharacteristic
318C2×C4(C22⋊D47)⋊C411yescharacteristic
319C2×C4#1309399211yescharacteristic
320C2×C4#1309400111yescharacteristic
321C2×C4#1309400111yescharacteristic
322C2×C4#1309404411yescharacteristic
323C2×C4#1309404411yescharacteristic
324C2×C4#1309405011yescharacteristic
325C2×C4#1309405511yescharacteristic
326C8—81no—
327C8—942no—
328C8—1881no—
329C8—1881no—
330C8—1881no—
331C8—1882no—
332C8—1881no—
333C8—1881no—
334C22—21no—
335C22—21no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22#1568899511yescharacteristic
340C22#1568944411yescharacteristic
341C22#1568946111yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—82no—
349C4#1568915911yescharacteristic
350C4#1568916311yescharacteristic
351C4#1568923711yescharacteristic
352C4#1568924211yescharacteristic
353C4#1568926611yescharacteristic
354C4#1568928311yescharacteristic
355C2—21no—
356C2—21no—
357C2—81no—
358C2#2024358511yescharacteristic
359C2#2024362611yescharacteristic
360C2#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#8903
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. #20242160C2semidirect
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
209. C47#8903semidirect

Maximal subgroups

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

subgroupindex
#202520542
#202421602
#202421462
#202417702
#202412912
#202412292
C47×#18062
#890347