Finite Group Databasebeta

← #26110592

Subgroups — #26110592

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#20243408C211yesmaximal, characteristic
2#20242158C211yesmaximal, characteristic, complemented
3#20242143C211yesmaximal, characteristic, complemented
4#20241770C211yesmaximal, characteristic
5#20241289C211yesmaximal, characteristic, complemented
6#20241227C211yesmaximal, characteristic, complemented
7C47×#2906C211yesmaximal, characteristic
8#15689176C2211yescharacteristic
9#15689147C2211yescharacteristic
10C2×#13093951—21no—
11C2×C4×C47⋊C8C2211yescharacteristic
12C4×C4⋊Dic47C2211yescharacteristic
13#15688882—21no—
14#15688873—22no—
15#15688873—21no—
16#15688873—21no—
17Q8⋊(C47⋊C8)C411yescharacteristic, complemented
18Q8⋊(C47⋊C8)C411yescharacteristic, complemented
19D4⋊(C47⋊C8)C411yescharacteristic, complemented
20D4⋊(C47⋊C8)C411yescharacteristic, complemented
21C2×#131×C47—21no—
22C4×#41×C47C2211yescharacteristic
23C47×#424—21no—
24C47×#424—22no—
25C47×#423—21no—
26C47×#423C2211yescharacteristic
27C2×C4⋊C8×C47—21no—
28C2×C4⋊C8×C47C2211yescharacteristic
29C2×#10625329—21no—
30C22×C47⋊C8—21no—
31C2×C4⋊Dic47—21no—
32C2×C4×Dic47—21no—
33C4⋊(C47⋊C8)—21no—
34C4⋊(C47⋊C8)—21no—
35C4⋊(C47⋊C8)C2×C411yescharacteristic
36C4⋊(C47⋊C8)C2×C411yescharacteristic
37#13093951—42no—
38C22⋊(C47⋊C8)—21no—
39C4×C47⋊C8—21no—
40C4×C47⋊C8—21no—
41C4×C47⋊C8C2×C411yescharacteristic
42C4×C47⋊C8C2×C411yescharacteristic
43C47⋊#95—21no—
44C2×#41×C47D411yescharacteristic
45#131×C47—42no—
46#131×C47—42no—
47C2×M16×C47—22no—
48C2×M16×C47—21no—
49C22×C8×C47—22no—
50C22×C8×C47—21no—
51C4×Q8×C47C2×C411yescharacteristic
52C4×Q8×C47Q811yescharacteristic
53C4×D4×C47—21no—
54C4×D4×C47—21no—
55C4×D4×C47C2×C411yescharacteristic
56C4×D4×C47Q811yescharacteristic
57#117×C47—21no—
58#117×C47D411yescharacteristic
59C2×C42×C47—21no—
60C2×C42×C47C2311yescharacteristic
61C4⋊C8×C47—22no—
62C4⋊C8×C47—22no—
63C4⋊C8×C47—21no—
64C4⋊C8×C47—21no—
65C4⋊C8×C47D411yescharacteristic
66C4⋊C8×C47D411yescharacteristic
67C4⋊C8×C47D412yes—
68#41×C47—21no—
69#41×C47—21no—
70#41×C47SD1612yes—
71#41×C47SD1612yes—
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×C47C2×D411yescharacteristic
79C22×C4×C47#4111yescharacteristic
80C22×C4×C47#4111yescharacteristic
81C22×Dic47—41no—
82M16×C47—44no—
83M16×C47—41no—
84M16×C47—41no—
85C2×C8×C47—21no—
86C2×C8×C47—22no—
87C2×C8×C47—21no—
88C2×C8×C47—44no—
89C2×C8×C47—41no—
90C2×C8×C47—41no—
91C4⋊C4×C47—21no—
92C4⋊C4×C47—21no—
93C4⋊C4×C47C4⋊C411yescharacteristic
94C4⋊C4×C47C4⋊C411yescharacteristic
95C22⋊C4×C47—21no—
96C22⋊C4×C47—21no—
97C22⋊C4×C47—21no—
98C42×C47—21no—
99C42×C47—21no—
100C42×C47—21no—
101C42×C47C22×C411yescharacteristic
102C42×C47C2×D411yescharacteristic
103C42×C47C2×D411yescharacteristic
104C42×C47C2×Q811yescharacteristic
105C4⋊Dic47—22no—
106C4⋊Dic47—41no—
107C4×Dic47—41no—
108C4×Dic47—41no—
109#10625329—41no—
110#10625329—41no—
111C2×C47⋊C8—21no—
112C2×C47⋊C8—21no—
113C2×C47⋊C8—21no—
114C2×C47⋊C8—21no—
115C2×C47⋊C8—42no—
116C2×C47⋊C8—41no—
117C23×C47—21no—
118C23×C47#12311yescharacteristic
119Q8×C47—21no—
120Q8×C47#10612yes—
121D4×C47—22no—
122D4×C47—22no—
123D4×C47—21no—
124D4×C47#10612yes—
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—81no—
136C2×C4×C47C2×C4⋊C411yescharacteristic
137C2×C4×C47C4×D411yescharacteristic
138C2×C4×C47C4×D411yescharacteristic
139C2×C4×C47#12011yescharacteristic
140C2×C4×C47C22⋊Q811yescharacteristic
141C2×C4×C47C22⋊Q811yescharacteristic
142C2×C4×C47C2×SD1611yescharacteristic
143C2×C4×C47C2×SD1611yescharacteristic
144C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C47#13711yescharacteristic
146C2×Dic47—42no—
147C2×Dic47—81no—
148C2×Dic47—81no—
149C8×C47—41no—
150C8×C47—42no—
151C8×C47—41no—
152C47⋊C8—22no—
153C47⋊C8—41no—
154C47⋊C8—41no—
155C47⋊C8—41no—
156C47⋊C8—81no—
157#8904—471nomaximal, Sylow
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—22no—
161C22×C47—41no—
162C22×C47—41no—
163C22×C47#38511yescharacteristic
164C22×C47#48111yescharacteristic
165C22×C47#48311yescharacteristic
166C4×C47—22no—
167C4×C47—21no—
168C4×C47—21no—
169C4×C47—21no—
170C4×C47—41no—
171C4×C47—41no—
172C4×C47C2×#10611yescharacteristic
173C4×C47#42811yescharacteristic
174C4×C47#44311yescharacteristic
175C4×C47#44411yescharacteristic
176C4×C47#44811yescharacteristic
177C4×C47#45011yescharacteristic
178Dic47—82no—
179#2906D4711yescharacteristic
180#1806—471no—
181#1570—471no—
182#1566—471no—
183#1476—471no—
184#1303—471no—
185#1279—471no—
186C2×C47—21no—
187C2×C47—21no—
188C2×C47—42no—
189C2×C47#185311yescharacteristic
190C2×C47#186111yescharacteristic
191C2×C47#186611yescharacteristic
192C2×#131—21no—
193C4×#41C2×D4711yescharacteristic
194#431—471no—
195#424—21no—
196#424—22no—
197#423—21no—
198#423—471no—
199#423C2×D4711yescharacteristic
200C2×C4⋊C8—21no—
201C2×C4⋊C8C2×D4711yescharacteristic
202C2×#104—941no—
203C2×C4×C8—471no—
204C4×C4⋊C4—471no—
205#339—941no—
206#334—942no—
207#334—941no—
208#334—941no—
209Q8⋊C8—471no—
210Q8⋊C8—471no—
211D4⋊C8—471no—
212D4⋊C8—471no—
213C47#890411yescharacteristic, complemented, Sylow
214C2×#41C22⋊D4711yescharacteristic
215#131—42no—
216#131—42no—
217C2×M16—22no—
218C2×M16—21no—
219C2×M16—941no—
220C22×C8—22no—
221C22×C8—21no—
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—941no—
232C2×C42—21no—
233C2×C42—941no—
234C2×C42C22×D4711yescharacteristic
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—471no—
240C4⋊C8—471no—
241C4⋊C8—941no—
242C4⋊C8—941no—
243C4⋊C8C22⋊D4711yescharacteristic
244C4⋊C8C22⋊D4711yescharacteristic
245C4⋊C8C22⋊D4712yes—
246#104—1882no—
247C22⋊C8—941no—
248C4×C8—471no—
249C4×C8—471no—
250C4×C8—941no—
251C4×C8—941no—
252#95—941no—
253#41—21no—
254#41—21no—
255#41#1062533512yes—
256#41#1062533612yes—
257C2×Q8Dic47⋊C411yescharacteristic
258C2×D4—21no—
259C2×D4Dic47⋊C411yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—1881no—
264C22×C4#1062535011yescharacteristic
265C22×C4#1062535011yescharacteristic
266C22×C4C2×C22⋊D4711yescharacteristic
267M16—44no—
268M16—41no—
269M16—41no—
270M16—1881no—
271M16—1881no—
272C2×C8—21no—
273C2×C8—22no—
274C2×C8—21no—
275C2×C8—44no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—941no—
279C2×C8—941no—
280C2×C8—941no—
281C2×C8—941no—
282C2×C8—1882no—
283C2×C8—1881no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—942no—
287C4⋊C4—1881no—
288C4⋊C4Dic47⋊C411yescharacteristic
289C4⋊C4Dic47⋊C411yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—1881no—
297C42—1881no—
298C42C2×Dic9411yescharacteristic
299C42C2×C4×D4711yescharacteristic
300C42C2×C22⋊D4711yescharacteristic
301C42C2×C22⋊D4711yescharacteristic
302C23—21no—
303C23#1309400711yescharacteristic
304Q8—21no—
305Q8#1309395512yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4#1309395512yes—
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—81no—
321C2×C4—3761no—
322C2×C4—3761no—
323C2×C4—1882no—
324C2×C4C2×Dic47⋊C411yescharacteristic
325C2×C4C4×C22⋊D4711yescharacteristic
326C2×C4C4×C22⋊D4711yescharacteristic
327C2×C4#1309399311yescharacteristic
328C2×C4#1309400311yescharacteristic
329C2×C4#1309400311yescharacteristic
330C2×C4C2×#1062533611yescharacteristic
331C2×C4C2×#1062533511yescharacteristic
332C2×C4#1309405011yescharacteristic
333C2×C4#1309405511yescharacteristic
334C8—41no—
335C8—42no—
336C8—41no—
337C8—942no—
338C8—3761no—
339C8—1881no—
340C8—1881no—
341C8—1881no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22#1568899611yescharacteristic
348C22#1568944111yescharacteristic
349C22#1568945311yescharacteristic
350C4—22no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—3762no—
357C4C2×#1309395511yescharacteristic
358C4#1568916311yescharacteristic
359C4#1568923811yescharacteristic
360C4#1568924311yescharacteristic
361C4#1568926211yescharacteristic
362C4#1568927411yescharacteristic
363C2—21no—
364C2—21no—
365C2—42no—
366C2#2024358111yescharacteristic
367C2#2024362811yescharacteristic
368C2#2024366311yescharacteristic
subgroupG/H
centerC2×C4#13093993
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2906C2
socleC22×C47#385
prime radicalC2×#41×C47D4
cosocleC2×C42×C47C23
hypercenter#2906D47
hypocenterC47#8904
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2906

Complements

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

NH ≅ G/Ntype
2. #20242158C2semidirect
3. #20242143C2semidirect
5. #20241289C2semidirect
6. #20241227C2semidirect
17. Q8⋊(C47⋊C8)C4semidirect
18. Q8⋊(C47⋊C8)C4semidirect
19. D4⋊(C47⋊C8)C4semidirect
20. D4⋊(C47⋊C8)C4semidirect
213. C47#8904semidirect

Maximal subgroups

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

subgroupindex
#202434082
#202421582
#202421432
#202417702
#202412892
#202412272
C47×#29062
#890447