Finite Group Databasebeta

← #26110601

Subgroups — #26110601

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#20243408C211yesmaximal, characteristic
3#20242163C211yesmaximal, characteristic, complemented
4#20242144C211yesmaximal, characteristic, complemented
5#20241290C211yesmaximal, characteristic, complemented
6#20241228C211yesmaximal, characteristic, complemented
7C47×#1476C211yesmaximal, characteristic
8#15689184—22no—
9#15689180—21no—
10#15689176—21no—
11#15689147C2211yescharacteristic
12#15689147C2211yescharacteristic
13C2×C4⋊(C47⋊C8)C2211yescharacteristic
14C2×#13093951—21no—
15#15689090—21no—
16#15689079—21no—
17C4×C4⋊Dic47C2211yescharacteristic
18#15688882—21no—
19#15688876—21no—
20#15688876—21no—
21#15688874—22no—
22C4×#41×C47C2211yescharacteristic
23C47×#431C2211yescharacteristic
24C2×C4×C8×C47C2211yescharacteristic
25C47×Q8⋊C8C411yescharacteristic, complemented
26C47×Q8⋊C8C411yescharacteristic, complemented
27C47×D4⋊C8C411yescharacteristic, complemented
28C47×D4⋊C8C411yescharacteristic, complemented
29C2×#10625329—21no—
30C2×#10625329—21no—
31C22×C47⋊C8—21no—
32C2×C4⋊Dic47—21no—
33C2×C4×Dic47—21no—
34C4⋊(C47⋊C8)—21no—
35C4⋊(C47⋊C8)—21no—
36C4⋊(C47⋊C8)—21no—
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)D412yes—
39C4⋊(C47⋊C8)D411yescharacteristic
40C4⋊(C47⋊C8)D411yescharacteristic
41#13093951—42no—
42C22⋊(C47⋊C8)—22no—
43C22⋊(C47⋊C8)—22no—
44#13093935—22no—
45C4×C47⋊C8—22no—
46C47⋊#95—21no—
47C2×#41×C47D411yescharacteristic
48C22×C8×C47—21no—
49C4×Q8×C47C2×C411yescharacteristic
50C4×Q8×C47Q811yescharacteristic
51C4×D4×C47—21no—
52C4×D4×C47—21no—
53C4×D4×C47C2×C411yescharacteristic
54C4×D4×C47Q811yescharacteristic
55#117×C47—21no—
56#117×C47D411yescharacteristic
57C2×C42×C47—21no—
58C2×C42×C47C2311yescharacteristic
59C4⋊C8×C47C2×C411yescharacteristic
60C4⋊C8×C47C2×C411yescharacteristic
61C22⋊C8×C47—21no—
62C4×C8×C47—21no—
63C4×C8×C47—21no—
64C4×C8×C47C2×C411yescharacteristic
65C4×C8×C47C2×C411yescharacteristic
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×C47C2×D411yescharacteristic
77C22×C4×C47#4111yescharacteristic
78C22×C4×C47#4111yescharacteristic
79C22×Dic47—41no—
80C2×C8×C47—21no—
81C2×C8×C47—21no—
82C2×C8×C47—21no—
83C2×C8×C47—21no—
84C2×C8×C47—42no—
85C4⋊C4×C47—21no—
86C4⋊C4×C47—21no—
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×C47C22×C411yescharacteristic
96C42×C47C2×D411yescharacteristic
97C42×C47C2×D411yescharacteristic
98C42×C47C2×Q811yescharacteristic
99C4⋊Dic47—22no—
100C4⋊Dic47—41no—
101C4×Dic47—41no—
102C4×Dic47—41no—
103#10625329—41no—
104#10625329—41no—
105#10625329—41no—
106#10625329—41no—
107C2×C47⋊C8—21no—
108C2×C47⋊C8—22no—
109C2×C47⋊C8—21no—
110C2×C47⋊C8—41no—
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—81no—
132C2×C4×C47C2×C4⋊C411yescharacteristic
133C2×C4×C47C4×D411yescharacteristic
134C2×C4×C47C4×D411yescharacteristic
135C2×C4×C47#12011yescharacteristic
136C2×C4×C47C22⋊Q811yescharacteristic
137C2×C4×C47C22⋊Q811yescharacteristic
138C2×C4×C47#13511yescharacteristic
139C2×C4×C47#13511yescharacteristic
140C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C47#13711yescharacteristic
142C2×Dic47—42no—
143C2×Dic47—81no—
144C2×Dic47—81no—
145C8×C47—22no—
146C8×C47—41no—
147C8×C47—41no—
148C8×C47—41no—
149C47⋊C8—41no—
150C47⋊C8—41no—
151C47⋊C8—42no—
152C47⋊C8—81no—
153#8905—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#42711yescharacteristic
169C4×C47#42811yescharacteristic
170C4×C47#44311yescharacteristic
171C4×C47#44411yescharacteristic
172C4×C47#44911yescharacteristic
173C4×C47#45211yescharacteristic
174Dic47—82no—
175#2925—471no—
176#1806—471no—
177#1571—471no—
178#1566—471no—
179#1476D4711yescharacteristic
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—941no—
192#431C2×D4711yescharacteristic
193#423—471no—
194#423—471no—
195C2×C4⋊C8—471no—
196C2×#104—941no—
197#408—941no—
198#405—941no—
199C2×C4×C8C2×D4711yescharacteristic
200C4×C4⋊C4—471no—
201#339—941no—
202#335—941no—
203#335—941no—
204#334—942no—
205Q8⋊C8Dic4711yescharacteristic, complemented
206Q8⋊C8Dic4711yescharacteristic, complemented
207D4⋊C8Dic4711yescharacteristic, complemented
208D4⋊C8Dic4711yescharacteristic, complemented
209C47#890511yescharacteristic, complemented, Sylow
210C2×#41D18811yescharacteristic
211C2×M16—941no—
212C2×M16—941no—
213C22×C8—21no—
214C22×C8—941no—
215C4×Q8Dic9411yescharacteristic
216C4×Q8C2×Dic4711yescharacteristic
217C4×D4—21no—
218C4×D4—21no—
219C4×D4Dic9411yescharacteristic
220C4×D4C2×Dic4711yescharacteristic
221#117—21no—
222#117D18811yescharacteristic
223C2×C4⋊C4—941no—
224C2×C42—21no—
225C2×C42—941no—
226C2×C42C22×D4711yescharacteristic
227C4⋊C8—472no—
228C4⋊C8—471no—
229C4⋊C8—471no—
230C4⋊C8—941no—
231C4⋊C8—941no—
232C4⋊C8—941no—
233C4⋊C8—941no—
234C4⋊C8C2×Dic4711yescharacteristic
235C4⋊C8C2×Dic4711yescharacteristic
236#104—1882no—
237C22⋊C8—21no—
238C22⋊C8—942no—
239C22⋊C8—942no—
240#97—942no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—942no—
244C4×C8C2×Dic4711yescharacteristic
245C4×C8C2×Dic4711yescharacteristic
246#95—941no—
247#41—21no—
248#41—21no—
249#41—21no—
250#41—21no—
251C2×Q8C4⋊Dic4711yescharacteristic
252C2×D4—21no—
253C2×D4C4⋊Dic4711yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—1881no—
258C22×C4C2×D18811yescharacteristic
259C22×C4#1062535211yescharacteristic
260C22×C4#1062535211yescharacteristic
261M16—1881no—
262M16—1881no—
263M16—1881no—
264M16—1881no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—42no—
270C2×C8—941no—
271C2×C8—942no—
272C2×C8—941no—
273C2×C8—1881no—
274C2×C8—1881no—
275C2×C8—1881no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—942no—
279C4⋊C4—1881no—
280C4⋊C4C4⋊Dic4711yescharacteristic
281C4⋊C4C4⋊Dic4711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—1881no—
289C42—1881no—
290C42C2×Dic9411yescharacteristic
291C42D4×D4711yescharacteristic
292C42D4×D4711yescharacteristic
293C42C22×Dic4711yescharacteristic
294C23—21no—
295C23#1309400811yescharacteristic
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—81no—
313C2×C4—3761no—
314C2×C4—3761no—
315C2×C4—1882no—
316C2×C4C2×C4⋊Dic4711yescharacteristic
317C2×C4D4×Dic4711yescharacteristic
318C2×C4D4×Dic4711yescharacteristic
319C2×C4#1309399411yescharacteristic
320C2×C4#1309400111yescharacteristic
321C2×C4#1309400111yescharacteristic
322C2×C4#1309404611yescharacteristic
323C2×C4#1309404611yescharacteristic
324C2×C4#1309405211yescharacteristic
325C2×C4#1309405711yescharacteristic
326C8—22no—
327C8—41no—
328C8—41no—
329C8—41no—
330C8—3761no—
331C8—1881no—
332C8—1881no—
333C8—1882no—
334C22—21no—
335C22—21no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22#1568899811yescharacteristic
340C22#1568944811yescharacteristic
341C22#1568946511yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—3762no—
349C4#1568916111yescharacteristic
350C4#1568916511yescharacteristic
351C4#1568924011yescharacteristic
352C4#1568924511yescharacteristic
353C4#1568927011yescharacteristic
354C4#1568928711yescharacteristic
355C2—21no—
356C2—21no—
357C2—81no—
358C2#2024358911yescharacteristic
359C2#2024363011yescharacteristic
360C2#2024366511yescharacteristic
subgroupG/H
centerC2×C4#13093994
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1476C2
socleC22×C47#385
prime radicalC2×#41×C47D4
cosocleC2×C42×C47C23
hypercenter#1476D47
hypocenterC47#8905
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1476

Complements

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

NH ≅ G/Ntype
3. #20242163C2semidirect
4. #20242144C2semidirect
5. #20241290C2semidirect
6. #20241228C2semidirect
25. C47×Q8⋊C8C4semidirect
26. C47×Q8⋊C8C4semidirect
27. C47×D4⋊C8C4semidirect
28. C47×D4⋊C8C4semidirect
205. Q8⋊C8Dic47semidirect
206. Q8⋊C8Dic47semidirect
207. D4⋊C8Dic47semidirect
208. D4⋊C8Dic47semidirect
209. C47#8905semidirect

Maximal subgroups

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

subgroupindex
#202520542
#202434082
#202421632
#202421442
#202412902
#202412282
C47×#14762
#890547