Finite Group Databasebeta

← #26111708

Subgroups — #26111708

Order 12032.

512 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252469C211yesmaximal, characteristic, complemented
2C2×#15689221C211yesmaximal, characteristic
3#20241778C211yesmaximal, characteristic
4#20241708C211yesmaximal, characteristic
5#20241232C211yesmaximal, characteristic, complemented
6#20241218C211yesmaximal, characteristic, complemented
7C47×#1479C211yesmaximal, characteristic, complemented
8C22×#10625336C411yescharacteristic
9C2×C4×D188C2211yescharacteristic
10Q8×C47⋊C8—22no—
11Q8×C47⋊C8—22no—
12#15689221—22no—
13#15689221—22no—
14#15689176—22no—
15#15689176C2211yescharacteristic
16#15689176C2211yescharacteristic
17C2×#13093957C411yescharacteristic
18C2×#13093948C411yescharacteristic
19C2×#13093944C411yescharacteristic
20C2×#13093935C2211yescharacteristic
21#15688850—22no—
22#15688846—22no—
23C2×C4×Q8×C47C2211yescharacteristic
24C47×#431C2211yescharacteristic
25C47×#431C2211yescharacteristic
26C47×Q8⋊C8—22no—
27C22×D188C2×C411yescharacteristic
28C22×C4×D47—21no—
29C2×#10625336—22no—
30C2×#10625336—22no—
31C2×#10625336C814yescomplemented
32C22×C47⋊C8C2×C411yescharacteristic
33C22×C47⋊C8C2×C411yescharacteristic
34C4×D188—21no—
35C4×D188—21no—
36C4×D188—21no—
37C4×D188—21no—
38C2×C4⋊Dic47C2×C411yescharacteristic
39C2×(C2×D47)⋊C4—21no—
40#13093957—22no—
41C4⋊(C47⋊C8)—24no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)—21no—
44#13093948—22no—
45#13093944—22no—
46C22⋊(C47⋊C8)—21no—
47C22⋊(C47⋊C8)D412yes—
48#13093935—21no—
49#13093935—21no—
50C4×C47⋊C8—24no—
51C4×C47⋊C8—21no—
52C4×C47⋊C8D412yes—
53C22×Q8×C47C2×C411yescharacteristic
54C4×Q8×C47—21no—
55C4×Q8×C47—21no—
56C4×Q8×C47D412yes—
57C4×Q8×C47D412yes—
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47C2×C411yescharacteristic
60C2×C42×C47—21no—
61C2×C42×C47C2311yescharacteristic
62C4⋊C8×C47—21no—
63C4⋊C8×C47—21no—
64C22⋊C8×C47—21no—
65C22⋊C8×C47—21no—
66C4×C8×C47—21no—
67C4×C8×C47—21no—
68C23×D47—21no—
69C2×Q8×C47—21no—
70C2×Q8×C47—21no—
71C2×Q8×C47—21no—
72C2×Q8×C47C22⋊C412yes—
73C2×Q8×C47C2×C812yes—
74C2×Q8×C47M1612yes—
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47C22×C411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47#4111yescharacteristic
80C22×Dic47—21no—
81C2×D188—21no—
82C2×D188—21no—
83C2×D188—21no—
84C2×D188—21no—
85C2×D188—21no—
86C2×D188C2×C812yes—
87C2×C4×D47—41no—
88C2×C4×D47—41no—
89C2×C4×D47—82no—
90C2×C8×C47—41no—
91C2×C8×C47—41no—
92C4⋊C4×C47—21no—
93C4⋊C4×C47—22no—
94C4⋊C4×C47—22no—
95C4⋊C4×C47C22⋊C412yes—
96C42×C47—22no—
97C42×C47—22no—
98C42×C47C2×D411yescharacteristic
99C42×C47C2×D411yescharacteristic
100C42×C47C2×D411yescharacteristic
101C42×C47#4111yescharacteristic
102#10625336—28no—
103(C2×D47)⋊C4—41no—
104(C2×D47)⋊C4—41no—
105C4⋊Dic47—21no—
106C4⋊Dic47—21no—
107C2×C47⋊C8—21no—
108C2×C47⋊C8—21no—
109C2×C47⋊C8—21no—
110C2×C47⋊C8—22no—
111C2×C47⋊C8—22no—
112C2×C47⋊C8—21no—
113C2×C47⋊C8—21no—
114C2×C47⋊C8C2×C812yes—
115C23×C47C4×D411yescharacteristic
116C22×D47—22no—
117C22×D47—41no—
118C22×D47—41no—
119C22×D47—81no—
120C22×D47—81no—
121Q8×C47—22no—
122Q8×C47—22no—
123Q8×C47—22no—
124Q8×C47C22⋊C814yes—
125C2×C4×C47—22no—
126C2×C4×C47—22no—
127C2×C4×C47—22no—
128C2×C4×C47—22no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—81no—
132C2×C4×C47C2×C22⋊C411yescharacteristic
133C2×C4×C47C4×D411yescharacteristic
134C2×C4×C47#12011yescharacteristic
135C2×C4×C47#12111yescharacteristic
136C2×C4×C47#12111yescharacteristic
137C2×C4×C47#12311yescharacteristic
138C2×C4×C47C22×C811yescharacteristic
139C2×C4×C47C2×M1611yescharacteristic
140C2×C4×C47#13111yescharacteristic
141C2×C4×C47#13111yescharacteristic
142C2×Dic47—41no—
143C2×Dic47—81no—
144D188—22no—
145D188—22no—
146D188—41no—
147D188—41no—
148C4×D47—82no—
149C4×D47—82no—
150C8×C47—81no—
151C8×C47—81no—
152C47⋊C8—22no—
153C47⋊C8—42no—
154C47⋊C8—42no—
155C47⋊C8—41no—
156#9090—471nomaximal, Sylow
157C22×C47—21no—
158C22×C47—21no—
159C22×C47#38611yescharacteristic
160C22×C47C8×D411yescharacteristic
161C22×C47#43511yescharacteristic
162C2×D47—42no—
163C2×D47—82no—
164C2×D47—81no—
165C2×D47—81no—
166C2×D47—82no—
167C4×C47—24no—
168C4×C47—21no—
169C4×C47—21no—
170C4×C47—21no—
171C4×C47—21no—
172C4×C47—42no—
173C4×C47C2×C22⋊C811yescharacteristic
174C4×C47#40811yescharacteristic
175C4×C47C8×D411yescharacteristic
176C4×C47#43511yescharacteristic
177Dic47—81no—
178#2982—471no—
179C2×#440—471no—
180#1479D4711yescharacteristic, complemented
181#1478—471no—
182#1460—471no—
183#1281—471no—
184#1275—471no—
185C2×C47—21no—
186C2×C47—21no—
187C2×C47#186911yescharacteristic
188C2×C47#190311yescharacteristic
189C2×C47#190511yescharacteristic
190D47—82no—
191C22×SD16—471no—
192C2×C4×Q8C2×D4711yescharacteristic
193C2×C4×D4—471no—
194C8×Q8—942no—
195C8×Q8—942no—
196#440—942no—
197#440—942no—
198#431—471no—
199#431—471no—
200#431—942no—
201#431C2×D4711yescharacteristic
202#431C2×D4711yescharacteristic
203C2×#107—471no—
204C2×#103—471no—
205C2×#102—471no—
206C2×#97—471no—
207Q8⋊C8—22no—
208D4⋊C8—942no—
209#323—942no—
210C47#909011yescharacteristic, complemented, Sylow
211C22×Q8C4×D4711yescharacteristic
212C22×D4—471no—
213C23×C4—941no—
214C2×SD16—474no—
215C2×SD16—942no—
216C2×SD16—942no—
217C22×C8—471no—
218C22×C8—471no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8D18812yes—
222C4×Q8C22⋊D4712yes—
223C4×D4—941no—
224C4×D4—941no—
225C4×D4—941no—
226C4×D4—941no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—471no—
229C2×C4⋊C4C4×D4711yescharacteristic
230C2×C22⋊C4—941no—
231C2×C42—21no—
232C2×C42C22×D4711yescharacteristic
233#107—942no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—944no—
237C4⋊C8—941no—
238C4⋊C8—941no—
239#103—942no—
240#102—942no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8—472no—
244C22⋊C8—941no—
245#97—941no—
246#97—941no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—472no—
250C4×C8—944no—
251C4×C8—941no—
252C24—941no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8C8×D4712yes—
257C2×Q8#1062532412yes—
258C2×Q8(C2×D47)⋊C412yes—
259C2×D4—472no—
260C2×D4—941no—
261C2×D4—941no—
262C2×D4—941no—
263C2×D4—941no—
264C2×D4—941no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—941no—
268C22×C4—3762no—
269C22×C4—1881no—
270C22×C4—1881no—
271C22×C4C2×C4×D4711yescharacteristic
272C22×C4D4×D4711yescharacteristic
273C22×C4#1062535211yescharacteristic
274SD16—948no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—472no—
278C2×C8—941no—
279C2×C8—941no—
280C2×C8—941no—
281C2×C8—942no—
282C2×C8—942no—
283C2×C8—941no—
284C2×C8—941no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—941no—
289C4⋊C4—941no—
290C4⋊C4(C2×D47)⋊C412yes—
291C22⋊C4—1881no—
292C22⋊C4—1881no—
293C42—22no—
294C42—22no—
295C42C2×D18811yescharacteristic
296C42D4×D4711yescharacteristic
297C42#1062535211yescharacteristic
298C42C2×C22⋊D4711yescharacteristic
299C23—942no—
300C23—3761no—
301C23—3761no—
302C23—1881no—
303C23—1881no—
304C23(C22⋊D47)⋊C411yescharacteristic
305Q8—22no—
306Q8—22no—
307Q8—22no—
308Q8(C2×D47)⋊C814yes—
309D4—942no—
310D4—942no—
311D4—1881no—
312D4—1881no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—3762no—
321C2×C4—3762no—
322C2×C4—3761no—
323C2×C4—1881no—
324C2×C4C2×(C2×D47)⋊C411yescharacteristic
325C2×C4(C22⋊D47)⋊C411yescharacteristic
326C2×C4#1309399411yescharacteristic
327C2×C4#1309399611yescharacteristic
328C2×C4#1309399611yescharacteristic
329C2×C4#1309400811yescharacteristic
330C2×C4C2×C8×D4711yescharacteristic
331C2×C4C2×#1062532411yescharacteristic
332C2×C4#1309403211yescharacteristic
333C2×C4#1309403211yescharacteristic
334C8—81no—
335C8—81no—
336C8—942no—
337C8—1882no—
338C8—1882no—
339C8—1881no—
340C22—21no—
341C22—21no—
342C22—3762no—
343C22—3762no—
344C22—3761no—
345C22—3761no—
346C22—1882no—
347C22#1568900011yescharacteristic
348C22(C22⋊D47)⋊C811yescharacteristic
349C22#1568919111yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—24no—
355C4—42no—
356C4—3761no—
357C4C2×(C2×D47)⋊C811yescharacteristic
358C4#1568909111yescharacteristic
359C4(C22⋊D47)⋊C811yescharacteristic
360C4#1568919111yescharacteristic
361C2—21no—
362C2—21no—
363C2—3762no—
364C2#2024367811yescharacteristic
365C2#2024386611yescharacteristic
366C2#2024388011yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC2×C4×C47C22×C8
FrattiniC2×C42C22×D47
FittingC47×#1479C2
socleC22×C47#386
prime radicalC22×D188C2×C4
cosocleC2×C42×C47C23
hypercenter#1479D47
hypocenterC47#9090
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
1. #20252469C2semidirect
5. #20241232C2semidirect
6. #20241218C2semidirect
7. C47×#1479C2semidirect
31. C2×#10625336C8semidirect
180. #1479D47semidirect
210. C47#9090semidirect

Maximal subgroups

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

subgroupindex
#202524692
C2×#156892212
#202417782
#202417082
#202412322
#202412182
C47×#14792
#909047