Finite Group Databasebeta

← #26115998

Subgroups — #26115998

Order 12032.

510 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243426C211yesmaximal, characteristic, complemented
2#20243164C211yesmaximal, characteristic, complemented
3#20241908C211yesmaximal, characteristic, complemented
4#20241904C211yesmaximal, characteristic
5#20241752C211yesmaximal, characteristic, complemented
6#20241748C211yesmaximal, characteristic
7C2×C47×#465C211yesmaximal, characteristic
8C2×#13093960—21no—
9#15689087C2211yescharacteristic
10#15689087C2211yescharacteristic
11C2×C22⋊(C47⋊C8)C2211yescharacteristic
12C2×#13093934—21no—
13C4×C22⋊Dic47C2211yescharacteristic
14#15688888—21no—
15#15688877—21no—
16#15688860—22no—
17#15688857—22no—
18#15688849—22no—
19#15688847—22no—
20C22×SD16×C47C411yescharacteristic, complemented
21C23×C8×C47C2211yescharacteristic
22C2×C22⋊Q8×C47C2211yescharacteristic
23C2×#121×C47C2211yescharacteristic
24C47×#465—21no—
25C47×#465—21no—
26C47×#465—21no—
27C47×#465—21no—
28C2×#106×C47C411yescharacteristic, complemented
29C2×#103×C47C411yescharacteristic, complemented
30C2×#102×C47C411yescharacteristic, complemented
31C2×#10625329—21no—
32C2×#10625329—21no—
33C22×C47⋊C8—21no—
34C2×C22⋊Dic47—21no—
35C2×C4×Dic47—21no—
36#13093960—42no—
37C22⋊(C47⋊C8)—21no—
38C22⋊(C47⋊C8)—21no—
39C22⋊(C47⋊C8)—21no—
40C22⋊(C47⋊C8)—21no—
41C22⋊(C47⋊C8)—21no—
42C22⋊(C47⋊C8)D412yes—
43#13093934—42no—
44C47⋊#95D412yes—
45C22×Q8×C47C2×C411yescharacteristic
46C22×D4×C47—21no—
47C22×D4×C47C2×C411yescharacteristic
48C23×C4×C47C2311yescharacteristic
49C2×SD16×C47—21no—
50C2×SD16×C47—21no—
51C2×SD16×C47—42no—
52C22×C8×C47—42no—
53C22×C8×C47C2×C411yescharacteristic
54C22×C8×C47C2×C411yescharacteristic
55C22×C8×C47D411yescharacteristic
56C22×C8×C47D411yescharacteristic
57C22×C8×C47D411yescharacteristic
58C22×C8×C47D411yescharacteristic
59C22⋊Q8×C47—21no—
60C22⋊Q8×C47—21no—
61C22⋊Q8×C47—21no—
62C22⋊Q8×C47—21no—
63#121×C47—21no—
64#121×C47—21no—
65#121×C47—21no—
66#121×C47—21no—
67C2×C4⋊C4×C47—21no—
68C2×C4⋊C4×C47C2×C411yescharacteristic
69C2×C4⋊C4×C47C2×C411yescharacteristic
70C2×C22⋊C4×C47—21no—
71C2×C22⋊C4×C47—21no—
72#106×C47—21no—
73#106×C47—21no—
74#103×C47—21no—
75#103×C47—21no—
76#102×C47—21no—
77#102×C47—21no—
78C24×C47—21no—
79C24×C47#4111yescharacteristic
80C2×Q8×C47—21no—
81C2×Q8×C47—21no—
82C2×Q8×C47—41no—
83C2×Q8×C47—41no—
84C2×D4×C47—21no—
85C2×D4×C47—21no—
86C2×D4×C47—41no—
87C2×D4×C47—41no—
88C2×D4×C47—41no—
89C2×D4×C47—41no—
90C2×D4×C47—82no—
91C22×C4×C47—21no—
92C22×C4×C47—21no—
93C22×C4×C47—21no—
94C22×C4×C47—41no—
95C22×C4×C47—41no—
96C22×C4×C47C22×C411yescharacteristic
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47C2×D411yescharacteristic
99C22×C4×C47C2×D411yescharacteristic
100C22×C4×C47C2×D411yescharacteristic
101C22×C4×C47#4111yescharacteristic
102C22×Dic47—22no—
103SD16×C47—42no—
104SD16×C47—42no—
105C2×C8×C47—22no—
106C2×C8×C47—22no—
107C2×C8×C47—22no—
108C2×C8×C47—41no—
109C2×C8×C47—41no—
110C2×C8×C47—41no—
111C2×C8×C47C22⋊C411yescharacteristic
112C2×C8×C47C22⋊C411yescharacteristic
113C2×C8×C47C22⋊C411yescharacteristic
114C2×C8×C47C22⋊C411yescharacteristic
115C4⋊C4×C47—21no—
116C4⋊C4×C47—21no—
117C4⋊C4×C47—21no—
118C4⋊C4×C47—21no—
119C4⋊C4×C47—41no—
120C4⋊C4×C47—41no—
121C22⋊C4×C47—41no—
122C22⋊C4×C47—41no—
123C22⋊C4×C47—41no—
124C22⋊C4×C47—41no—
125C22⋊Dic47—42no—
126C4×Dic47—41no—
127C4×Dic47—41no—
128#10625329—41no—
129#10625329—41no—
130#10625329—41no—
131#10625329—41no—
132C2×C47⋊C8—22no—
133C2×C47⋊C8—41no—
134C2×C47⋊C8—41no—
135C2×C47⋊C8—41no—
136C2×C47⋊C8—41no—
137C23×C47—22no—
138C23×C47—22no—
139C23×C47—41no—
140C23×C47—41no—
141C23×C47—42no—
142C23×C47—81no—
143C23×C47—81no—
144C23×C47C4×D411yescharacteristic
145C23×C47#12111yescharacteristic
146C23×C47#12411yescharacteristic
147Q8×C47—41no—
148Q8×C47—41no—
149Q8×C47—81no—
150D4×C47—41no—
151D4×C47—41no—
152D4×C47—81no—
153D4×C47—82no—
154D4×C47—82no—
155C2×C4×C47—21no—
156C2×C4×C47—21no—
157C2×C4×C47—21no—
158C2×C4×C47—21no—
159C2×C4×C47—21no—
160C2×C4×C47—21no—
161C2×C4×C47—41no—
162C2×C4×C47—41no—
163C2×C4×C47—41no—
164C2×C4×C47—41no—
165C2×C4×C47—41no—
166C2×C4×C47—41no—
167C2×C4×C47—42no—
168C2×C4×C47—81no—
169C2×C4×C47—81no—
170C2×C4×C47C2×C22⋊C411yescharacteristic
171C2×C4×C47C4×D411yescharacteristic
172C2×C4×C47#12111yescharacteristic
173C2×C4×C47#12711yescharacteristic
174C2×Dic47—42no—
175C2×Dic47—42no—
176C2×Dic47—42no—
177C8×C47—21no—
178C8×C47—21no—
179C8×C47—21no—
180C8×C47#9912yes—
181C47⋊C8—42no—
182C47⋊C8—81no—
183C47⋊C8—81no—
184#9816—471nomaximal, Sylow
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—21no—
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—21no—
191C22×C47—21no—
192C22×C47—44no—
193C22×C47—81no—
194C22×C47—81no—
195C22×C47—81no—
196C22×C47—162no—
197C22×C47#39011yescharacteristic
198C22×C47#44412yes—
199C22×C47#47112yes—
200C4×C47—21no—
201C4×C47—21no—
202C4×C47—21no—
203C4×C47—81no—
204C4×C47—81no—
205C4×C47—81no—
206C4×C47C2×#9911yescharacteristic
207C4×C47#41011yescharacteristic
208Dic47—82no—
209C2×#465D4711yescharacteristic
210#1810—471no—
211#1752—471no—
212#1508—471no—
213#1507—471no—
214#1472—471no—
215#1471—471no—
216C2×C47—22no—
217C2×C47—22no—
218C2×C47—41no—
219C2×C47—161no—
220C2×C47#196411yescharacteristic
221C2×C47#197511yescharacteristic
222C2×C47#197711yescharacteristic
223C22×SD16Dic4711yescharacteristic, complemented
224C23×C8C2×D4711yescharacteristic
225C2×C22⋊Q8C2×D4711yescharacteristic
226C2×#121C2×D4711yescharacteristic
227#465—21no—
228#465—21no—
229#465—21no—
230#465—21no—
231C2×#108—941no—
232C2×#106Dic4711yescharacteristic, complemented
233C2×#103Dic4711yescharacteristic, complemented
234C2×#102Dic4711yescharacteristic, complemented
235#407—471no—
236#407—471no—
237C2×C22⋊C8—471no—
238C2×#97—941no—
239C4×C22⋊C4—471no—
240#341—941no—
241#336—941no—
242#328—942no—
243#327—942no—
244#324—942no—
245#323—942no—
246C47#981611yescharacteristic, complemented, Sylow
247C22×Q8C2×Dic4711yescharacteristic
248C22×D4—21no—
249C22×D4C2×Dic4711yescharacteristic
250C23×C4C22×D4711yescharacteristic
251C2×SD16—21no—
252C2×SD16—21no—
253C2×SD16—42no—
254C2×M16—941no—
255C2×M16—941no—
256C22×C8—42no—
257C22×C8—941no—
258C22×C8C2×Dic4711yescharacteristic
259C22×C8C2×Dic4711yescharacteristic
260C22×C8C22⋊D4711yescharacteristic
261C22×C8C22⋊D4711yescharacteristic
262C22×C8C22⋊D4711yescharacteristic
263C22×C8C22⋊D4711yescharacteristic
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268#121—21no—
269#121—21no—
270#121—21no—
271#121—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4C2×Dic4711yescharacteristic
274C2×C4⋊C4C2×Dic4711yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—941no—
278C2×C42—941no—
279#108—1882no—
280#106—21no—
281#106—21no—
282#103—21no—
283#103—21no—
284#102—21no—
285#102—21no—
286C22⋊C8—472no—
287C22⋊C8—941no—
288C22⋊C8—941no—
289C22⋊C8—941no—
290C22⋊C8—941no—
291C22⋊C8—941no—
292#97—1882no—
293#95—472no—
294C24—21no—
295C24#1062535211yescharacteristic
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—82no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—942no—
313C22×C4D4×D4711yescharacteristic
314C22×C4D4×D4711yescharacteristic
315C22×C4#1062535211yescharacteristic
316C22×C4C22×Dic4711yescharacteristic
317C22×C4C2×C22⋊D4711yescharacteristic
318C22×C4C2×C22⋊D4711yescharacteristic
319SD16—42no—
320SD16—42no—
321M16—1881no—
322M16—1881no—
323M16—1881no—
324M16—1881no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—942no—
332C2×C8—1881no—
333C2×C8—1881no—
334C2×C8—1881no—
335C2×C8—1881no—
336C2×C8C22⋊Dic4711yescharacteristic
337C2×C8C22⋊Dic4711yescharacteristic
338C2×C8C22⋊Dic4711yescharacteristic
339C2×C8C22⋊Dic4711yescharacteristic
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—1882no—
351C42—1881no—
352C42—1881no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—41no—
357C23—42no—
358C23—81no—
359C23—81no—
360C23D4×Dic4711yescharacteristic
361C23#1309399511yescharacteristic
362C23#1309401111yescharacteristic
363Q8—41no—
364Q8—41no—
365Q8—81no—
366D4—41no—
367D4—41no—
368D4—81no—
369D4—82no—
370D4—82no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—1882no—
387C2×C4—1882no—
388C2×C4—1882no—
389C2×C4C2×C22⋊Dic4711yescharacteristic
390C2×C4D4×Dic4711yescharacteristic
391C2×C4#1309399511yescharacteristic
392C2×C4#1309402011yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8—3761no—
397C8—3761no—
398C8—1882no—
399C8#1309393912yes—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—44no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—162no—
412C22#1568902611yescharacteristic
413C22#1568924512yes—
414C22#1568938312yes—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—81no—
419C4—81no—
420C4—81no—
421C4—3762no—
422C4C2×#1309393911yescharacteristic
423C4#1568909711yescharacteristic
424C2—22no—
425C2—22no—
426C2—41no—
427C2—161no—
428C2#2024422511yescharacteristic
429C2#2024428211yescharacteristic
430C2#2024429311yescharacteristic
subgroupG/H
centerC22#15689026
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#465C2
socleC22×C47#390
prime radicalC2×#121×C47C22
cosocleC23×C4×C47C23
hypercenterC2×#465D47
hypocenterC47#9816
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#465

Complements

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

NH ≅ G/Ntype
1. #20243426C2semidirect
2. #20243164C2semidirect
3. #20241908C2semidirect
5. #20241752C2semidirect
20. C22×SD16×C47C4semidirect
28. C2×#106×C47C4semidirect
29. C2×#103×C47C4semidirect
30. C2×#102×C47C4semidirect
223. C22×SD16Dic47semidirect
232. C2×#106Dic47semidirect
233. C2×#103Dic47semidirect
234. C2×#102Dic47semidirect
246. C47#9816semidirect

Maximal subgroups

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

subgroupindex
#202434262
#202431642
#202419082
#202419042
#202417522
#202417482
C2×C47×#4652
#981647