Finite Group Databasebeta

← #26109595

Subgroups — #26109595

Order 12032.

384 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252364C211yesmaximal, characteristic
2C4×#13093957C211yesmaximal, characteristic, complemented
3#20242149C211yesmaximal, characteristic, complemented
4#20241913C211yesmaximal, characteristic
5#20241244C211yesmaximal, characteristic, complemented
6#20241243C211yesmaximal, characteristic, complemented
7C47×#1509C211yesmaximal, characteristic
8#15689254C411yescharacteristic, complemented
9Q8×C47⋊C8C411yescharacteristic, complemented
10#15689201C411yescharacteristic, complemented
11#15689195—21no—
12D4×C47⋊C8—21no—
13D4×C47⋊C8C411yescharacteristic, complemented
14#15689184—21no—
15#15689180C2211yescharacteristic
16#15689180C2211yescharacteristic
17C2×#13093957—21no—
18#15689079—21no—
19C2×C4×C47⋊C8C2211yescharacteristic
20C4×C4⋊Dic47C2211yescharacteristic
21#15688885—21no—
22#15688875—21no—
23#15688873—21no—
24#15688860—21no—
25#15688857—21no—
26C47×#529C2211yescharacteristic
27C47×#432C2211yescharacteristic
28C4×C4⋊C4×C47C2211yescharacteristic
29C47×#328—21no—
30C47×#327—21no—
31C2×#10625329—21no—
32C22×C47⋊C8—21no—
33C22×C47⋊C8D411yescharacteristic
34C22×C47⋊C8D411yescharacteristic
35C2×C4⋊Dic47—21no—
36C2×C4×Dic47—21no—
37#13093957—24no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)C2×C411yescharacteristic
42C4⋊(C47⋊C8)C2×C411yescharacteristic
43C22⋊(C47⋊C8)—21no—
44C22⋊(C47⋊C8)—21no—
45C22⋊(C47⋊C8)—21no—
46C22⋊(C47⋊C8)—21no—
47#13093935—21no—
48#13093935—21no—
49#13093935—21no—
50C4×C47⋊C8—21no—
51C4×C47⋊C8C2×C411yescharacteristic
52C4×C47⋊C8C2×C411yescharacteristic
53C4×C47⋊C8D411yescharacteristic
54C4×C47⋊C8D411yescharacteristic
55C47⋊#95—21no—
56#126×C47—21no—
57#125×C47Q811yescharacteristic
58#124×C47Q811yescharacteristic
59#123×C47—21no—
60C22⋊Q8×C47D411yescharacteristic
61#121×C47D411yescharacteristic
62C4×Q8×C47C2×C411yescharacteristic
63C4×D4×C47—21no—
64C4×D4×C47C2×C411yescharacteristic
65#117×C47—21no—
66C2×C4⋊C4×C47—21no—
67C2×C42×C47—21no—
68C2×C42×C47C2311yescharacteristic
69C4⋊C8×C47—21no—
70C22⋊C8×C47—21no—
71#97×C47—21no—
72#95×C47—21no—
73C2×Q8×C47C4⋊C411yescharacteristic
74C2×D4×C47—21no—
75C2×D4×C47C4⋊C411yescharacteristic
76C22×C4×C47—21no—
77C22×C4×C47—41no—
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47#4111yescharacteristic
81C22×Dic47—41no—
82C2×C8×C47—41no—
83C4⋊C4×C47—21no—
84C4⋊C4×C47—21no—
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—
94C22⋊C4×C47—21no—
95C22⋊C4×C47—21no—
96C42×C47—21no—
97C42×C47—81no—
98C42×C47C22×C411yescharacteristic
99C42×C47C2×D411yescharacteristic
100C42×C47C2×Q811yescharacteristic
101C42×C47#4111yescharacteristic
102C4⋊Dic47—22no—
103C4⋊Dic47—41no—
104C4×Dic47—81no—
105#10625329—42no—
106C2×C47⋊C8—22no—
107C2×C47⋊C8—21no—
108C2×C47⋊C8—21no—
109C2×C47⋊C8—41no—
110C2×C47⋊C8—42no—
111C2×C47⋊C8—41no—
112C2×C47⋊C8C22⋊C411yescharacteristic
113C2×C47⋊C8C22⋊C411yescharacteristic
114C2×C47⋊C8C22⋊C411yescharacteristic
115C2×C47⋊C8C22⋊C411yescharacteristic
116C23×C47—21no—
117C23×C47#12111yescharacteristic
118Q8×C47—41no—
119D4×C47—41no—
120D4×C47—41no—
121D4×C47—41no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—42no—
133C2×C4×C47—81no—
134C2×C4×C47—81no—
135C2×C4×C47C2×C22⋊C411yescharacteristic
136C2×C4×C47C2×C4⋊C411yescharacteristic
137C2×C4×C47#11711yescharacteristic
138C2×C4×C47#12111yescharacteristic
139C2×C4×C47C22⋊Q811yescharacteristic
140C2×C4×C47C22⋊Q811yescharacteristic
141C2×Dic47—42no—
142C2×Dic47—81no—
143C2×Dic47—81no—
144C8×C47—81no—
145C47⋊C8—21no—
146C47⋊C8—21no—
147C47⋊C8—21no—
148C47⋊C8—41no—
149C47⋊C8—41no—
150C47⋊C8—81no—
151C47⋊C8#10412yes—
152#8697—471nomaximal, Sylow
153C22×C47—21no—
154C22×C47—21no—
155C22×C47—41no—
156C22×C47—41no—
157C22×C47—41no—
158C22×C47#38011yescharacteristic
159C22×C47#47211yescharacteristic
160C22×C47#47311yescharacteristic
161C4×C47—21no—
162C4×C47—21no—
163C4×C47—21no—
164C4×C47—21no—
165C4×C47—21no—
166C4×C47—41no—
167C4×C47—41no—
168C4×C47—41no—
169C4×C47—82no—
170C4×C47C2×#10411yescharacteristic
171C4×C47#42111yescharacteristic
172Dic47—82no—
173#2966—471no—
174C4×#107—471no—
175#1568—471no—
176#1509—471no—
177#1509D4711yescharacteristic
178#1285—471no—
179#1285—471no—
180C2×C47—21no—
181C2×C47—21no—
182C2×C47—81no—
183C2×C47#180611yescharacteristic
184C2×C47#181211yescharacteristic
185C2×C47#181611yescharacteristic
186#529C2×D4711yescharacteristic
187#446—471no—
188C8×Q8—471no—
189#436—471no—
190#435—941no—
191C8×D4—471no—
192C8×D4—941no—
193#433—941no—
194#432—471no—
195#432—471no—
196#432C2×D4711yescharacteristic
197C2×#107—941no—
198#405—941no—
199C2×C4×C8—471no—
200C4×C4⋊C4—471no—
201C4×C4⋊C4C2×D4711yescharacteristic
202#340—941no—
203#335—941no—
204#334—941no—
205#328—21no—
206#328—941no—
207#327—21no—
208#327—941no—
209C47#869711yescharacteristic, complemented, Sylow
210C2×M16—941no—
211C22×C8—471no—
212C22×C8—471no—
213C22×C8—941no—
214#126—21no—
215#125Dic9411yescharacteristic
216#124Dic9411yescharacteristic
217#123—21no—
218C22⋊Q8C22⋊D4711yescharacteristic
219#121C22⋊D4711yescharacteristic
220C4×Q8C4×D4711yescharacteristic
221C4×D4—21no—
222C4×D4C4×D4711yescharacteristic
223#117—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—941no—
226C2×C42—21no—
227C2×C42—941no—
228C2×C42C22×D4711yescharacteristic
229#107—944no—
230C4⋊C8—21no—
231C4⋊C8—471no—
232C4⋊C8—471no—
233C4⋊C8—941no—
234C4⋊C8—941no—
235C4⋊C8—941no—
236C22⋊C8—21no—
237C22⋊C8—941no—
238C22⋊C8—941no—
239C22⋊C8—941no—
240C22⋊C8—941no—
241#97—21no—
242#97—941no—
243#97—941no—
244#97—941no—
245C4×C8—471no—
246C4×C8—471no—
247C4×C8—471no—
248C4×C8—471no—
249C4×C8—941no—
250#95—21no—
251#95—941no—
252C2×Q8Dic47⋊C411yescharacteristic
253C2×D4—21no—
254C2×D4Dic47⋊C411yescharacteristic
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—1881no—
258C22×C4D4×D4711yescharacteristic
259C22×C4#1062535211yescharacteristic
260C22×C4C2×C22⋊D4711yescharacteristic
261M16—1882no—
262C2×C8—41no—
263C2×C8—471no—
264C2×C8—471no—
265C2×C8—471no—
266C2×C8—471no—
267C2×C8—942no—
268C2×C8—941no—
269C2×C8—941no—
270C2×C8—1881no—
271C2×C8—1882no—
272C2×C8—1881no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—942no—
280C4⋊C4—1881no—
281C4⋊C4Dic47⋊C411yescharacteristic
282C4⋊C4Dic47⋊C411yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C42—21no—
289C42—81no—
290C42—3761no—
291C42C2×Dic9411yescharacteristic
292C42C2×C4×D4711yescharacteristic
293C42D4×D4711yescharacteristic
294C42#1062535211yescharacteristic
295C23—21no—
296C23#1309399611yescharacteristic
297Q8—41no—
298D4—41no—
299D4—41no—
300D4—41no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—3761no—
315C2×C4—3761no—
316C2×C4—1882no—
317C2×C4C22⋊C4×D4711yescharacteristic
318C2×C4C2×Dic47⋊C411yescharacteristic
319C2×C4#1309398011yescharacteristic
320C2×C4#1309399611yescharacteristic
321C2×C4#1309400111yescharacteristic
322C2×C4#1309400111yescharacteristic
323C8—81no—
324C8—472no—
325C8—941no—
326C8—941no—
327C8—941no—
328C8—3761no—
329C8—1881no—
330C8—1881no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22#1568898111yescharacteristic
337C22#1568939111yescharacteristic
338C22#1568939611yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—82no—
348C4—3762no—
349C4#104×D4711yescharacteristic
350C4#1568913811yescharacteristic
351C2—21no—
352C2—21no—
353C2—81no—
354C2#2024340611yescharacteristic
355C2#2024343211yescharacteristic
356C2#2024345011yescharacteristic
subgroupG/H
centerC22#15688981
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1509C2
socleC22×C47#380
prime radical#121×C47D4
cosocleC2×C42×C47C23
hypercenter#1509D47
hypocenterC47#8697
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1509

Complements

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

NH ≅ G/Ntype
2. C4×#13093957C2semidirect
3. #20242149C2semidirect
5. #20241244C2semidirect
6. #20241243C2semidirect
8. #15689254C4semidirect
9. Q8×C47⋊C8C4semidirect
10. #15689201C4semidirect
13. D4×C47⋊C8C4semidirect
209. C47#8697semidirect

Maximal subgroups

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

subgroupindex
#202523642
C4×#130939572
#202421492
#202419132
#202412442
#202412432
C47×#15092
#869747