Finite Group Databasebeta

← #26199890

Subgroups — #26199890

Order 12032.

636 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254730C211yesmaximal, characteristic, complemented
2#20253762C211yesmaximal, characteristic
3#20252337C211yesmaximal, characteristic, complemented
4#20242954C212yesmaximal, complemented
5#20242926C212yesmaximal, complemented
6#20242830C212yesmaximal, complemented
7#20242802C212yesmaximal, complemented
8#20242141C211yesmaximal, characteristic, complemented
9#20242043C211yesmaximal, characteristic, complemented
10C2×#15688867C211yesmaximal, characteristic
11C47×#3208C211yesmaximal, characteristic
12#15689680C2211yescharacteristic
13#15689487—22no—
14#15689469C2212yes—
15#15689469C2212yes—
16#15689418C2212yescomplemented
17#15689415—21no—
18#15689362—22no—
19#15689333C2212yescomplemented
20#15689330—21no—
21#15689254—22no—
22#15689254C2212yescomplemented
23Q8×C47⋊C8C2212yescomplemented
24#15689150—21no—
25C2×C4⋊(C47⋊C8)C2211yescharacteristic
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27#15689128—21no—
28#15689128—21no—
29#15689079—21no—
30C4×#10625329C2211yescharacteristic
31#15688875C2212yescomplemented
32#15688873C2212yescomplemented
33#15688871C2211yescharacteristic, complemented
34#15688871C2211yescharacteristic, complemented
35#15688870C2211yescharacteristic, complemented
36#15688870C2211yescharacteristic, complemented
37#15688867C2212yes—
38#15688867C2212yes—
39C47×#533C2211yescharacteristic
40C2×#127×C47C2211yescharacteristic
41C47×#492—21no—
42C47×#486—21no—
43C47×#460C2212yes—
44C47×#460C2212yes—
45C47×#450—22no—
46C47×#431C2211yescharacteristic
47C2×#10625337—22no—
48C2×#10625329D411yescharacteristic
49C2×#10625329D411yescharacteristic
50C22×C47⋊C8—21no—
51C22×C47⋊C8—21no—
52#13094024C2311yescharacteristic
53#13094024C2311yescharacteristic
54#13094016—21no—
55#13094003—21no—
56#13094003—21no—
57C4×Dic94—21no—
58C4×Dic94—21no—
59#13093983—21no—
60#13093957—22no—
61C4⋊(C47⋊C8)—21no—
62C4⋊(C47⋊C8)—21no—
63C4⋊(C47⋊C8)—22no—
64C4⋊(C47⋊C8)—22no—
65C4⋊(C47⋊C8)C2311yescharacteristic
66C4⋊(C47⋊C8)C2311yescharacteristic
67C4⋊(C47⋊C8)C2312yes—
68#13093948—22no—
69#13093947—22no—
70#13093945—22no—
71#13093945—22no—
72#13093945—22no—
73#13093945—22no—
74#13093935—22no—
75#13093935D412yes—
76C4×C47⋊C8—22no—
77C4×C47⋊C8C2312yes—
78C22×D4×C47—21no—
79C22×D4×C47—21no—
80C22×D4×C47—21no—
81C2×SD16×C47—22no—
82C2×SD16×C47—22no—
83C4⋊Q8×C47D411yescharacteristic
84C4⋊Q8×C47D411yescharacteristic
85#127×C47—21no—
86#127×C47—21no—
87#127×C47—21no—
88#127×C47C2312yes—
89#125×C47—21no—
90C22⋊Q8×C47—21no—
91C22⋊Q8×C47D412yes—
92C4×Q8×C47—21no—
93C4×Q8×C47C2312yes—
94#117×C47—21no—
95C2×C42×C47C2311yescharacteristic
96C4⋊C8×C47C2311yescharacteristic
97C4⋊C8×C47C2311yescharacteristic
98#102×C47—22no—
99#102×C47—22no—
100C22⋊C8×C47—21no—
101C4×C8×C47—21no—
102C24×C47—41no—
103C2×Q8×C47—21no—
104C2×Q8×C47C2×D412yes—
105C2×D4×C47—22no—
106C2×D4×C47—22no—
107C2×D4×C47—22no—
108C2×D4×C47—41no—
109C2×D4×C47—41no—
110C2×D4×C47—41no—
111C2×D4×C47—42no—
112C2×D4×C47—42no—
113C2×D4×C47—81no—
114C2×D4×C47—81no—
115C2×D4×C47—82no—
116C22×C4×C47C2×D411yescharacteristic
117C22×C4×C47C2×D411yescharacteristic
118C22×C4×C47#4111yescharacteristic
119C2×Dic94—21no—
120C2×Dic94—21no—
121SD16×C47—82no—
122SD16×C47—82no—
123C2×C8×C47—21no—
124C2×C8×C47—21no—
125C4⋊C4×C47—21no—
126C4⋊C4×C47—21no—
127C4⋊C4×C47—21no—
128C4⋊C4×C47—22no—
129C4⋊C4×C47—22no—
130C4⋊C4×C47C2×D412yes—
131C22⋊C4×C47—22no—
132C42×C47—22no—
133C42×C47C2×D411yescharacteristic
134C42×C47C2×D411yescharacteristic
135C42×C47#4111yescharacteristic
136C42×C47C2411yescharacteristic
137C22⋊Dic47—41no—
138#10625337—82no—
139C4⋊Dic47—21no—
140C4⋊Dic47—21no—
141C4⋊Dic47—21no—
142C4⋊Dic47—21no—
143Dic47⋊C4—41no—
144Dic47⋊C4—41no—
145C4×Dic47—41no—
146#10625329—22no—
147#10625329—22no—
148C2×C47⋊C8—21no—
149C2×C47⋊C8—21no—
150C2×C47⋊C8—22no—
151C2×C47⋊C8—42no—
152C2×C47⋊C8—81no—
153C2×C47⋊C8C2×D412yes—
154C2×C47⋊C8C2×D412yes—
155C23×C47—42no—
156C23×C47—82no—
157C23×C47—82no—
158C23×C47—82no—
159C23×C47#12111yescharacteristic
160Q8×C47—22no—
161Q8×C47—22no—
162Q8×C47—81no—
163D4×C47—42no—
164D4×C47—42no—
165D4×C47—42no—
166D4×C47—42no—
167D4×C47—82no—
168D4×C47—82no—
169D4×C47—82no—
170D4×C47—82no—
171C2×C4×C47—21no—
172C2×C4×C47—21no—
173C2×C4×C47—22no—
174C2×C4×C47—22no—
175C2×C4×C47—41no—
176C2×C4×C47#12111yescharacteristic
177C2×C4×C47#12111yescharacteristic
178C2×C4×C47#12111yescharacteristic
179C2×C4×C47#13512yes—
180C2×C4×C47#13712yes—
181C2×C4×C47C22×D411yescharacteristic
182C2×C4×C47C22×D411yescharacteristic
183C2×C4×C47C2×#4111yescharacteristic
184C2×Dic47—41no—
185C2×Dic47—41no—
186Dic94—41no—
187Dic94—41no—
188Dic94—81no—
189C8×C47—41no—
190C8×C47—41no—
191C47⋊C8—22no—
192C47⋊C8—22no—
193C47⋊C8—41no—
194C47⋊C8—41no—
195C47⋊C8—42no—
196#19242—471nomaximal, Sylow
197C22×C47—21no—
198C22×C47—21no—
199C22×C47—82no—
200C22×C47—82no—
201C22×C47—82no—
202C22×C47—164no—
203C22×C47#46412yes—
204C22×C47C2×#12111yescharacteristic
205C4×C47—21no—
206C4×C47—21no—
207C4×C47—21no—
208C4×C47—42no—
209C4×C47—42no—
210C4×C47#46412yes—
211C4×C47C2×#13511yescharacteristic
212C4×C47C2×#13711yescharacteristic
213C4×C47#57611yescharacteristic
214C4×C47#57711yescharacteristic
215Dic47—81no—
216Dic47—81no—
217#3249—471no—
218#3208D4711yescharacteristic
219#3138—471no—
220#2962—471no—
221#1702—472no—
222#1698—472no—
223#1682—472no—
224#1678—472no—
225#1565—471no—
226#1542—471no—
227C2×#331—471no—
228C2×C47—22no—
229C2×C47—162no—
230C2×C47C2×#46411yescharacteristic
231C2×C47#306211yescharacteristic
232C2×C47#306811yescharacteristic
233#533—471no—
234#533C2×D4711yescharacteristic
235C2×#127C2×D4711yescharacteristic
236#492—21no—
237#490—942no—
238#486—21no—
239#486—472no—
240#486—472no—
241#477—472no—
242#477—941no—
243#467—942no—
244#462—472no—
245#462—941no—
246#460C2×D4712yes—
247#460C2×D4712yes—
248#450—22no—
249#446—472no—
250#446—942no—
251C8×Q8—472no—
252#431C2×D4711yescharacteristic
253#424—941no—
254C2×C4⋊C8—471no—
255C2×C4⋊C8—471no—
256#418—941no—
257#418—941no—
258#405—941no—
259C4×M16—471no—
260#335—472no—
261#334—472no—
262#333—471no—
263#333—471no—
264#332—471no—
265#332—471no—
266#331—472no—
267#331—472no—
268C47#1924211yescharacteristic, complemented, Sylow
269C22×D4—21no—
270C22×D4—21no—
271C22×D4—21no—
272C2×Q16—942no—
273C2×SD16—22no—
274C2×SD16—22no—
275C2×M16—471no—
276C2×M16—471no—
277C22×C8—941no—
278C22×C8—941no—
279C4⋊Q8—471no—
280C4⋊Q8—471no—
281C4⋊Q8C22⋊D4711yescharacteristic
282C4⋊Q8C22⋊D4711yescharacteristic
283#127—21no—
284#127—21no—
285#127—21no—
286#127C22×D4712yes—
287#125—21no—
288#125—941no—
289C22⋊Q8—21no—
290C22⋊Q8—941no—
291C22⋊Q8—941no—
292C22⋊Q8C22⋊D4712yes—
293C4×Q8—21no—
294C4×Q8—941no—
295C4×Q8—941no—
296C4×Q8C22×D4712yes—
297#117—21no—
298#117—941no—
299C2×C42C22×D4711yescharacteristic
300#107—942no—
301C4⋊C8—471no—
302C4⋊C8—471no—
303C4⋊C8—472no—
304C4⋊C8—942no—
305C4⋊C8—942no—
306C4⋊C8—941no—
307C4⋊C8—941no—
308C4⋊C8C22×D4711yescharacteristic
309C4⋊C8C22×D4711yescharacteristic
310#103—942no—
311#103—942no—
312#102—22no—
313#102—22no—
314#102—942no—
315#102—942no—
316#102—942no—
317#102—942no—
318C22⋊C8—21no—
319#97—472no—
320#97—942no—
321C4×C8—21no—
322C4×C8—472no—
323C4×C8—942no—
324C24—41no—
325C2×Q8—21no—
326C2×Q8—941no—
327C2×Q8—941no—
328C2×Q8C2×C22⋊D4712yes—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—81no—
338C2×D4—81no—
339C2×D4—82no—
340C22×C4D4×D4711yescharacteristic
341C22×C4#1062535211yescharacteristic
342C22×C4C2×C22⋊D4711yescharacteristic
343Q16—3762no—
344SD16—82no—
345SD16—82no—
346M16—942no—
347M16—942no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—472no—
351C2×C8—472no—
352C2×C8—941no—
353C2×C8—941no—
354C2×C8—942no—
355C2×C8—3761no—
356C2×C8—1882no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—941no—
363C4⋊C4—941no—
364C4⋊C4—941no—
365C4⋊C4—941no—
366C4⋊C4—1881no—
367C4⋊C4—1881no—
368C4⋊C4C2×C22⋊D4712yes—
369C22⋊C4—22no—
370C22⋊C4—1881no—
371C42—22no—
372C42—1881no—
373C42D4×D4711yescharacteristic
374C42#1062535211yescharacteristic
375C42C2×C22⋊D4711yescharacteristic
376C42C23×D4711yescharacteristic
377C23—42no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23#1309399611yescharacteristic
382Q8—22no—
383Q8—22no—
384Q8—81no—
385Q8—3761no—
386Q8—1881no—
387Q8—1881no—
388D4—42no—
389D4—42no—
390D4—42no—
391D4—42no—
392D4—82no—
393D4—82no—
394D4—82no—
395D4—82no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—1881no—
402C2×C4—1881no—
403C2×C4#1309399611yescharacteristic
404C2×C4#1309399611yescharacteristic
405C2×C4#1309399611yescharacteristic
406C2×C4#1309404512yes—
407C2×C4#1309405312yes—
408C2×C4C2×D4×D4711yescharacteristic
409C2×C4C22×C22⋊D4711yescharacteristic
410C2×C4C2×#1062535211yescharacteristic
411C8—41no—
412C8—41no—
413C8—942no—
414C8—942no—
415C8—1881no—
416C8—1881no—
417C8—1882no—
418C22—21no—
419C22—21no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—164no—
424C22#1568934312yes—
425C22C2×#1309399611yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—42no—
430C4—42no—
431C4—3761no—
432C4—3761no—
433C4#1568934312yes—
434C4C2×#1309404511yescharacteristic
435C4C2×#1309405311yescharacteristic
436C4#1568992611yescharacteristic
437C4#1568993411yescharacteristic
438C2—22no—
439C2—162no—
440C2C2×#1568934311yescharacteristic
441C2#2025306411yescharacteristic
442C2#2025311311yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC47×#3208C2
socleC22×C47C2×#121
prime radicalC2×#127×C47C22
hypercenter#3208D47
hypocenterC47#19242
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
1. #20254730C2semidirect
3. #20252337C2semidirect
4. #20242954C2semidirect
5. #20242926C2semidirect
6. #20242830C2semidirect
7. #20242802C2semidirect
8. #20242141C2semidirect
9. #20242043C2semidirect
16. #15689418C22semidirect
19. #15689333C22semidirect
22. #15689254C22semidirect
23. Q8×C47⋊C8C22semidirect
31. #15688875C22semidirect
32. #15688873C22semidirect
33. #15688871C22semidirect
34. #15688871C22semidirect
35. #15688870C22semidirect
36. #15688870C22semidirect
268. C47#19242semidirect

Maximal subgroups

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

subgroupindex
#202547302
#202537622
#202523372
#202429542
#202429262
#202428302
#202428022
#202421412
#202420432
C2×#156888672
C47×#32082
#1924247