Finite Group Databasebeta

← #26110575

Subgroups — #26110575

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#20251957C211yesmaximal, characteristic
2#20243405C211yesmaximal, characteristic
3#20242164C211yesmaximal, characteristic, complemented
4#20241771C211yesmaximal, characteristic
5#20241289C211yesmaximal, characteristic, complemented
6#20241228C211yesmaximal, characteristic, complemented
7C47×#1567C211yesmaximal, characteristic, complemented
8C2×#13094032—21no—
9C4×#10625350C2211yescharacteristic
10#15689176C2211yescharacteristic
11#15689151—21no—
12#15689148—22no—
13#15689145—21no—
14C2×C4⋊(C47⋊C8)C2211yescharacteristic
15C4⋊C8×D47—21no—
16C2×#13093950—21no—
17C4×C4⋊Dic47C2211yescharacteristic
18#15688884—21no—
19#15688875—21no—
20#15688875—21no—
21#15688855C411yescharacteristic, complemented
22#15688855C411yescharacteristic, complemented
23#15688852C411yescharacteristic, complemented
24#15688852C411yescharacteristic, complemented
25C47×#423C2211yescharacteristic
26C47×#423C2211yescharacteristic
27C2×C4×C8×C47C2211yescharacteristic
28C47×#335—22no—
29C2×#10625350D411yescharacteristic
30#13094032—42no—
31#13094032—42no—
32C2×#10625324—22no—
33C22×C47⋊C8—21no—
34C2×C8×D47—22no—
35C4×Dic94C2×C411yescharacteristic
36C4×Dic94Q811yescharacteristic
37C4×D188C2×C411yescharacteristic
38C4×D188Q811yescharacteristic
39C4×C22⋊D47—21no—
40C4×C22⋊D47—21no—
41#13093983D411yescharacteristic
42#13093981—21no—
43C2×C4⋊Dic47—21no—
44C2×C4×Dic47—21no—
45C42×D47—21no—
46C4⋊(C47⋊C8)—21no—
47C4⋊(C47⋊C8)C2×C411yescharacteristic
48C4⋊(C47⋊C8)C2×C411yescharacteristic
49C4⋊(C47⋊C8)D411yescharacteristic
50C4⋊(C47⋊C8)D411yescharacteristic
51Dic47⋊C8—22no—
52Dic47⋊C8—22no—
53#13093950—42no—
54C22⋊(C47⋊C8)—21no—
55C4×C47⋊C8—21no—
56C47⋊#95—21no—
57C2×M16×C47—21no—
58C2×M16×C47—21no—
59C22×C8×C47—21no—
60C2×C42×C47C2311yescharacteristic
61C4⋊C8×C47—21no—
62C4⋊C8×C47—21no—
63C4⋊C8×C47—21no—
64C4⋊C8×C47D412yes—
65C4×C8×C47—21no—
66C4×C8×C47C2×C411yescharacteristic
67C4×C8×C47C2×C411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47#4111yescharacteristic
70C22×C4×C47#4111yescharacteristic
71C2×C22⋊D47—21no—
72C22×Dic47—41no—
73#10625350—21no—
74#10625350—21no—
75#10625350D812yes—
76#10625350Q1612yes—
77C2×D188C4⋊C411yescharacteristic
78C2×C4×D47—21no—
79C2×C4×D47—21no—
80C2×C4×D47—21no—
81C2×Dic94C4⋊C411yescharacteristic
82M16×C47—41no—
83M16×C47—41no—
84M16×C47—41no—
85M16×C47—41no—
86C2×C8×C47—21no—
87C2×C8×C47—21no—
88C2×C8×C47—22no—
89C2×C8×C47—42no—
90C2×C8×C47—41no—
91C42×C47C22×C411yescharacteristic
92C42×C47C2×D411yescharacteristic
93C42×C47C2×D411yescharacteristic
94C42×C47C2×Q811yescharacteristic
95C22⋊Dic47—21no—
96(C2×D47)⋊C4—21no—
97(C2×D47)⋊C4—21no—
98C4⋊Dic47—22no—
99C4⋊Dic47—41no—
100C4⋊Dic47C4⋊C411yescharacteristic
101C4⋊Dic47C4⋊C411yescharacteristic
102Dic47⋊C4—21no—
103Dic47⋊C4—21no—
104C4×Dic47—21no—
105C4×Dic47—21no—
106C4×Dic47—21no—
107C4×Dic47—41no—
108C4×Dic47—41no—
109C2×C47⋊C8—21no—
110C2×C47⋊C8—21no—
111C2×C47⋊C8—21no—
112C2×C47⋊C8—21no—
113C2×C47⋊C8—41no—
114C2×C47⋊C8—41no—
115#10625324—44no—
116C8×D47—44no—
117C23×C47#12311yescharacteristic
118C22×D47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—41no—
122C2×C4×C47C2×C4⋊C411yescharacteristic
123C2×C4×C47C4×D411yescharacteristic
124C2×C4×C47C4×D411yescharacteristic
125C2×C4×C47#12011yescharacteristic
126C2×C4×C47C22⋊Q811yescharacteristic
127C2×C4×C47C22⋊Q811yescharacteristic
128C2×C4×C47C2×D811yescharacteristic
129C2×C4×C47C2×Q1611yescharacteristic
130C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C47#13711yescharacteristic
132C22⋊D47—22no—
133C22⋊D47—22no—
134C2×Dic47—21no—
135C2×Dic47—21no—
136C2×Dic47—21no—
137C2×Dic47—42no—
138C2×Dic47—81no—
139C2×Dic47—81no—
140D188—21no—
141D188#10712yes—
142C4×D47—22no—
143C4×D47—22no—
144C4×D47—41no—
145C4×D47—41no—
146C4×D47—81no—
147Dic94—21no—
148Dic94#10712yes—
149C8×C47—22no—
150C8×C47—41no—
151C8×C47—42no—
152C8×C47—81no—
153C47⋊C8—41no—
154C47⋊C8—41no—
155C47⋊C8—41no—
156C47⋊C8—41no—
157#8902—471nomaximal, Sylow
158C22×C47—21no—
159C22×C47—21no—
160C22×C47#38511yescharacteristic
161C22×C47#48011yescharacteristic
162C22×C47#48411yescharacteristic
163C2×D47—22no—
164C2×D47—41no—
165C2×D47—41no—
166C4×C47—21no—
167C4×C47—21no—
168C4×C47—21no—
169C4×C47C2×#10711yescharacteristic
170C4×C47#42811yescharacteristic
171C4×C47#44311yescharacteristic
172C4×C47#44411yescharacteristic
173C4×C47#44711yescharacteristic
174C4×C47#45111yescharacteristic
175Dic47—22no—
176Dic47—41no—
177Dic47—41no—
178Dic47—82no—
179#2906—471no—
180#1806—471no—
181#1571—471no—
182#1567D4711yescharacteristic, complemented
183#1476—471no—
184#1303—471no—
185#1279—471no—
186C2×C47—21no—
187C2×C47—21no—
188C2×C47#185211yescharacteristic
189C2×C47#186111yescharacteristic
190C2×C47#186611yescharacteristic
191D47—42no—
192C2×#131—941no—
193C4×#41—471no—
194#431—471no—
195#424—941no—
196#424—942no—
197#423—941no—
198#423C2×D4711yescharacteristic
199#423C2×D4711yescharacteristic
200C2×C4⋊C8—471no—
201C2×C4⋊C8—941no—
202C2×#104—941no—
203C2×C4×C8C2×D4711yescharacteristic
204C4×C4⋊C4—471no—
205#339—941no—
206#335—22no—
207#335—941no—
208#335—941no—
209Q8⋊C8—471no—
210Q8⋊C8—471no—
211D4⋊C8—471no—
212D4⋊C8—471no—
213C47#890211yescharacteristic, complemented, Sylow
214C2×#41—471no—
215#131—1882no—
216#131—1882no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—942no—
220C22×C8—21no—
221C22×C8—942no—
222C22×C8—941no—
223C4×Q8—471no—
224C4×Q8—471no—
225C4×D4—471no—
226C4×D4—471no—
227C4×D4—941no—
228C4×D4—941no—
229#117—471no—
230#117—941no—
231C2×C4⋊C4—941no—
232C2×C42—941no—
233C2×C42—941no—
234C2×C42C22×D4711yescharacteristic
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—471no—
239C4⋊C8—471no—
240C4⋊C8—471no—
241C4⋊C8—471no—
242C4⋊C8—942no—
243C4⋊C8—942no—
244C4⋊C8—941no—
245C4⋊C8C22⋊D4712yes—
246#104—1882no—
247C22⋊C8—941no—
248C4×C8—21no—
249C4×C8—941no—
250C4×C8C4×D4711yescharacteristic
251C4×C8C4×D4711yescharacteristic
252#95—941no—
253#41—472no—
254#41—472no—
255#41—941no—
256#41—941no—
257C2×Q8—471no—
258C2×D4—471no—
259C2×D4—941no—
260C22×C4—941no—
261C22×C4—941no—
262C22×C4—941no—
263C22×C4—1881no—
264C22×C4#1062535011yescharacteristic
265C22×C4D4×D4711yescharacteristic
266C22×C4#1062535411yescharacteristic
267M16—41no—
268M16—41no—
269M16—41no—
270M16—41no—
271M16—1884no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—22no—
275C2×C8—42no—
276C2×C8—41no—
277C2×C8—941no—
278C2×C8—941no—
279C2×C8—941no—
280C2×C8—941no—
281C2×C8—1884no—
282C2×C8—1881no—
283C2×C8—1881no—
284C4⋊C4—471no—
285C4⋊C4—471no—
286C4⋊C4—942no—
287C4⋊C4—941no—
288C4⋊C4—941no—
289C4⋊C4—1881no—
290C22⋊C4—941no—
291C22⋊C4—941no—
292C22⋊C4—941no—
293C42—941no—
294C42—941no—
295C42—941no—
296C42—1881no—
297C42—1881no—
298C42C2×C4×D4711yescharacteristic
299C42D4×D4711yescharacteristic
300C42Q8×D4711yescharacteristic
301C42C2×C22⋊D4711yescharacteristic
302C23—941no—
303C23#1309400911yescharacteristic
304Q8—472no—
305Q8—941no—
306D4—472no—
307D4—942no—
308D4—942no—
309D4—941no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—941no—
314C2×C4—942no—
315C2×C4—942no—
316C2×C4—941no—
317C2×C4—941no—
318C2×C4—3761no—
319C2×C4—3761no—
320C2×C4—3761no—
321C2×C4—1882no—
322C2×C4—1881no—
323C2×C4—1881no—
324C2×C4C4⋊C4×D4711yescharacteristic
325C2×C4C4×C22⋊D4711yescharacteristic
326C2×C4#1309398611yescharacteristic
327C2×C4#1309399211yescharacteristic
328C2×C4#1309400011yescharacteristic
329C2×C4#1309400211yescharacteristic
330C2×C4D8×D4711yescharacteristic
331C2×C4Q16×D4711yescharacteristic
332C2×C4#1309404811yescharacteristic
333C2×C4#1309405611yescharacteristic
334C8—22no—
335C8—41no—
336C8—42no—
337C8—81no—
338C8—1881no—
339C8—1881no—
340C8—1881no—
341C8—1881no—
342C22—21no—
343C22—21no—
344C22—942no—
345C22—1881no—
346C22—1881no—
347C22#1568899411yescharacteristic
348C22#1568943811yescharacteristic
349C22#1568946011yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—942no—
354C4—3762no—
355C4—1881no—
356C4—1881no—
357C4#107×D4711yescharacteristic
358C4#1568916411yescharacteristic
359C4#1568923911yescharacteristic
360C4#1568924411yescharacteristic
361C4#1568925511yescharacteristic
362C4#1568927711yescharacteristic
363C2—21no—
364C2—21no—
365C2—1882no—
366C2#2024357411yescharacteristic
367C2#2024362711yescharacteristic
368C2#2024366011yescharacteristic
subgroupG/H
centerC2×C4#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1567C2
socleC22×C47#385
prime radicalC2×#10625350D4
cosocleC2×C42×C47C23
hypercenter#1567D47
hypocenterC47#8902
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1567

Complements

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

NH ≅ G/Ntype
3. #20242164C2semidirect
5. #20241289C2semidirect
6. #20241228C2semidirect
7. C47×#1567C2semidirect
21. #15688855C4semidirect
22. #15688855C4semidirect
23. #15688852C4semidirect
24. #15688852C4semidirect
182. #1567D47semidirect
213. C47#8902semidirect

Maximal subgroups

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

subgroupindex
#202519572
#202434052
#202421642
#202417712
#202412892
#202412282
C47×#15672
#890247