Finite Group Databasebeta

← #26200110

Subgroups — #26200110

Order 12032.

656 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253947C211yesmaximal, characteristic
2#20253193C211yesmaximal, characteristic, complemented
3#20252860C211yesmaximal, characteristic
4#20242818C212yesmaximal, complemented
5#20242804C212yesmaximal, complemented
6#20242770C212yesmaximal, complemented
7#20242736C212yesmaximal, complemented
8#20242144C211yesmaximal, characteristic, complemented
9#20242076C211yesmaximal, characteristic, complemented
10#20242063C211yesmaximal, characteristic
11C2×C8×Q8×C47C211yesmaximal, characteristic, complemented
12C2×#13094055—21no—
13#15689680C2211yescharacteristic
14#15689673C2211yescharacteristic
15#15689521C2211yescharacteristic
16#15689496C2211yescharacteristic
17#15689474C2211yescharacteristic
18#15689470C2211yescharacteristic
19#15689430—22no—
20#15689418C2212yes—
21#15689408C2212yes—
22#15689352—21no—
23#15689352—21no—
24#15689345—22no—
25#15689333C2212yes—
26#15689323C2212yes—
27C4×#10625325—21no—
28C4×#10625325—21no—
29#15689163—21no—
30#15689147C2211yescharacteristic
31#15689147C2211yescharacteristic
32#15689132—21no—
33#15689122—21no—
34#15688874C2212yes—
35#15688874C2212yes—
36#15688871C2212yes—
37#15688870C2212yes—
38#15688868C2212yes—
39#15688866C2212yes—
40C2×C4×Q8×C47C2211yescharacteristic
41C8×Q8×C47—21no—
42C8×Q8×C47—21no—
43C8×Q8×C47C2212yes—
44C8×Q8×C47C2212yes—
45C2×C4⋊C8×C47—21no—
46C2×C4⋊C8×C47C2211yescharacteristic
47C2×C4×C8×C47—21no—
48C2×C4×C8×C47C2211yescharacteristic
49C2×#10625350—21no—
50#13094055—44no—
51C2×#10625337—22no—
52C2×#10625325—21no—
53C2×#10625325—21no—
54C2×#10625336—22no—
55C2×#10625329—21no—
56C2×#10625329—21no—
57#13094024C2311yescharacteristic
58#13094024C2311yescharacteristic
59#13094024C2311yescharacteristic
60#13094021C2311yescharacteristic
61#13094016—21no—
62#13094012—21no—
63#13094003—21no—
64#13094003—21no—
65#13093998—21no—
66#13093998—21no—
67C4×Dic94—21no—
68C4×Dic94—21no—
69C4×D188—21no—
70C4×D188—21no—
71#13093983—21no—
72#13093957—22no—
73#13093956—21no—
74#13093956—21no—
75#13093955—22no—
76C4⋊(C47⋊C8)—21no—
77C4⋊(C47⋊C8)—21no—
78C4⋊(C47⋊C8)C2312yes—
79C4⋊(C47⋊C8)C2312yes—
80#13093949—21no—
81#13093949—21no—
82#13093948—22no—
83#13093948—22no—
84#13093947—22no—
85#13093946—21no—
86#13093946—21no—
87#13093944—22no—
88C22×Q8×C47D411yescharacteristic
89C22×C8×C47—21no—
90C22×C8×C47D411yescharacteristic
91C22×C8×C47D411yescharacteristic
92C4×Q8×C47—21no—
93C4×Q8×C47—21no—
94C4×Q8×C47D412yes—
95C4×Q8×C47C2312yes—
96C2×C4⋊C4×C47—21no—
97C2×C4⋊C4×C47D411yescharacteristic
98C2×C42×C47—21no—
99C2×C42×C47C2311yescharacteristic
100C4⋊C8×C47—22no—
101C4⋊C8×C47—22no—
102C4⋊C8×C47—21no—
103C4⋊C8×C47C2312yes—
104C4×C8×C47—22no—
105C4×C8×C47—22no—
106C4×C8×C47D411yescharacteristic
107C4×C8×C47D411yescharacteristic
108C4×C8×C47C2311yescharacteristic
109C4×C8×C47C2311yescharacteristic
110C2×Q8×C47—21no—
111C2×Q8×C47—22no—
112C2×Q8×C47SD1614yescomplemented
113C2×Q8×C47C2×D412yes—
114C22×C4×C47—21no—
115C22×C4×C47—21no—
116C22×C4×C47C2×D411yescharacteristic
117C22×C4×C47C2×D411yescharacteristic
118C22×C4×C47#4111yescharacteristic
119C2×C22⋊D47—41no—
120#10625350—42no—
121#10625350—42no—
122C2×D188—21no—
123C2×D188—21no—
124C2×D188—21no—
125C2×C4×D47—41no—
126C2×Dic94—21no—
127C2×Dic94—21no—
128C2×Dic94—21no—
129C2×C8×C47—22no—
130C2×C8×C47—22no—
131C2×C8×C47—41no—
132C2×C8×C47—42no—
133C2×C8×C47C2×D411yescharacteristic
134C2×C8×C47C2×D411yescharacteristic
135C2×C8×C47C2×D411yescharacteristic
136C2×C8×C47C2×D411yescharacteristic
137C4⋊C4×C47—22no—
138C4⋊C4×C47—21no—
139C4⋊C4×C47—22no—
140C4⋊C4×C47C2×D412yes—
141C42×C47—22no—
142C42×C47—22no—
143C42×C47C2×D411yescharacteristic
144C42×C47C2×D411yescharacteristic
145C42×C47#4111yescharacteristic
146C42×C47C2411yescharacteristic
147C22⋊Dic47—41no—
148#10625337—44no—
149#10625336—44no—
150(C2×D47)⋊C4—41no—
151(C2×D47)⋊C4—41no—
152C4⋊Dic47—21no—
153C4⋊Dic47—21no—
154C4⋊Dic47—21no—
155C4⋊Dic47—21no—
156C4⋊Dic47—21no—
157C4⋊Dic47—21no—
158Dic47⋊C4—41no—
159Dic47⋊C4—41no—
160C4×Dic47—41no—
161#10625329—42no—
162#10625329—42no—
163C2×C47⋊C8—22no—
164C2×C47⋊C8—22no—
165#10625325—42no—
166#10625325—42no—
167C23×C47#12111yescharacteristic
168C22×D47—41no—
169Q8×C47—22no—
170Q8×C47—24no—
171Q8×C47C2×SD1614yes—
172C2×C4×C47—22no—
173C2×C4×C47—24no—
174C2×C4×C47—21no—
175C2×C4×C47—21no—
176C2×C4×C47—22no—
177C2×C4×C47—41no—
178C2×C4×C47—42no—
179C2×C4×C47#12111yescharacteristic
180C2×C4×C47#12111yescharacteristic
181C2×C4×C47#12111yescharacteristic
182C2×C4×C47C2×SD1612yes—
183C2×C4×C47#13512yes—
184C2×C4×C47C22×D411yescharacteristic
185C2×C4×C47C22×D411yescharacteristic
186C2×C4×C47C2×#4111yescharacteristic
187C22⋊D47—82no—
188C2×Dic47—41no—
189C2×Dic47—41no—
190C2×Dic47—41no—
191D188—41no—
192D188—41no—
193D188—41no—
194D188—41no—
195D188—81no—
196C4×D47—82no—
197Dic94—41no—
198Dic94—41no—
199Dic94—41no—
200Dic94—41no—
201Dic94—81no—
202C8×C47—21no—
203C8×C47—21no—
204C8×C47—21no—
205C8×C47—42no—
206C8×C47#13712yescomplemented
207C47⋊C8—42no—
208C47⋊C8—42no—
209#19260—471nomaximal, Sylow
210C22×C47—21no—
211C22×C47—21no—
212C22×C47#46512yes—
213C22×C47C2×#12111yescharacteristic
214C2×D47—81no—
215C2×D47—81no—
216C2×D47—81no—
217C4×C47—21no—
218C4×C47—24no—
219C4×C47—21no—
220C4×C47—21no—
221C4×C47—42no—
222C4×C47#46512yes—
223C4×C47C22×SD1611yescharacteristic
224C4×C47C2×#13511yescharacteristic
225C4×C47C2×#13711yescharacteristic
226C4×C47#57511yescharacteristic
227Dic47—81no—
228Dic47—81no—
229Dic47—81no—
230#3167—471no—
231#3078—471no—
232#3037—471no—
233C2×C8×Q8D4711yescharacteristic, complemented
234#1680—472no—
235#1678—472no—
236#1672—472no—
237#1666—472no—
238#1566—471no—
239#1550—471no—
240#1547—471no—
241C2×C47—22no—
242C2×C47#303711yescharacteristic
243C2×C47C2×#46511yescharacteristic
244C2×C47#306711yescharacteristic
245D47—161no—
246C2×#137—941no—
247#533—471no—
248#532—471no—
249C2×C4×Q8C2×D4711yescharacteristic
250#498—471no—
251#492—471no—
252#487—471no—
253#486—471no—
254#479—942no—
255#477—472no—
256#475—472no—
257#465—941no—
258#465—941no—
259#464—942no—
260#462—472no—
261#460—472no—
262C8×Q8—21no—
263C8×Q8—21no—
264C8×Q8C2×D4712yes—
265C8×Q8C2×D4712yes—
266C4×SD16—941no—
267C4×SD16—941no—
268#428—941no—
269#423—471no—
270#423—471no—
271C2×C4⋊C8—21no—
272C2×C4⋊C8C2×D4711yescharacteristic
273#419—941no—
274#417—941no—
275C2×C4×C8—21no—
276C2×C4×C8C2×D4711yescharacteristic
277#334—472no—
278#334—472no—
279#333—472no—
280#332—472no—
281#332—472no—
282#331—472no—
283C47#1926011yescharacteristic, complemented, Sylow
284C2×#41—941no—
285C22×Q8D18811yescharacteristic
286#137—1884no—
287C2×Q16—942no—
288C2×SD16—942no—
289C2×SD16—941no—
290C2×SD16—941no—
291C2×M16—941no—
292C2×M16—941no—
293C22×C8—21no—
294C22×C8C22⋊D4711yescharacteristic
295C22×C8C22⋊D4711yescharacteristic
296C4⋊Q8—471no—
297C4⋊Q8—471no—
298C4⋊Q8—471no—
299#127—471no—
300#125—941no—
301#124—941no—
302C22⋊Q8—941no—
303C22⋊Q8—941no—
304#121—941no—
305#121—941no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—941no—
309C4×Q8—941no—
310C4×Q8D18812yes—
311C4×Q8C22×D4712yes—
312C4×D4—941no—
313C4×D4—941no—
314#117—941no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4D18811yescharacteristic
317C2×C42—21no—
318C2×C42C22×D4711yescharacteristic
319#107—942no—
320#106—941no—
321#106—941no—
322#106—942no—
323C4⋊C8—22no—
324C4⋊C8—22no—
325C4⋊C8—21no—
326C4⋊C8—472no—
327C4⋊C8—472no—
328C4⋊C8—941no—
329C4⋊C8—941no—
330C4⋊C8C22×D4712yes—
331#103—942no—
332#103—942no—
333#103—941no—
334#103—941no—
335#103—942no—
336#102—941no—
337#102—941no—
338#102—942no—
339C4×C8—22no—
340C4×C8—22no—
341C4×C8C22⋊D4711yescharacteristic
342C4×C8C22⋊D4711yescharacteristic
343C4×C8C22×D4711yescharacteristic
344C4×C8C22×D4711yescharacteristic
345#41—1882no—
346#41—1882no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—941no—
350C2×Q8—941no—
351C2×Q8—941no—
352C2×Q8#1062532514yescomplemented
353C2×Q8C2×D18812yes—
354C2×D4—941no—
355C2×D4—941no—
356C2×D4—941no—
357C2×D4—1881no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—1881no—
361C22×C4C2×D18811yescharacteristic
362C22×C4#1062535011yescharacteristic
363C22×C4C2×C22⋊D4711yescharacteristic
364Q16—1884no—
365SD16—1882no—
366SD16—1884no—
367SD16—1882no—
368M16—1882no—
369M16—1882no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—41no—
373C2×C8—42no—
374C2×C8—942no—
375C2×C8—942no—
376C2×C8C2×C22⋊D4711yescharacteristic
377C2×C8C2×C22⋊D4711yescharacteristic
378C2×C8C2×C22⋊D4711yescharacteristic
379C2×C8C2×C22⋊D4711yescharacteristic
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—941no—
384C4⋊C4—941no—
385C4⋊C4—941no—
386C4⋊C4—941no—
387C4⋊C4—941no—
388C4⋊C4—941no—
389C4⋊C4—1881no—
390C4⋊C4—1881no—
391C4⋊C4C2×D18812yes—
392C22⋊C4—1881no—
393C22⋊C4—1881no—
394C22⋊C4—1881no—
395C42—22no—
396C42—22no—
397C42—1881no—
398C42C2×D18811yescharacteristic
399C42#1062535011yescharacteristic
400C42C2×C22⋊D4711yescharacteristic
401C42C23×D4711yescharacteristic
402C23—1881no—
403C23#1309399811yescharacteristic
404Q8—24no—
405Q8—22no—
406Q8—3761no—
407Q8—1881no—
408Q8—1881no—
409Q8—1881no—
410Q8—1881no—
411Q8C2×#1062532514yes—
412D4—3761no—
413D4—3762no—
414D4—1881no—
415D4—1881no—
416D4—1881no—
417D4—1881no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—3762no—
426C2×C4—1881no—
427C2×C4—1881no—
428C2×C4—1881no—
429C2×C4#1309399811yescharacteristic
430C2×C4#1309399811yescharacteristic
431C2×C4#1309399811yescharacteristic
432C2×C4C2×#1062532512yes—
433C2×C4#1309404612yes—
434C2×C4C22×C22⋊D4711yescharacteristic
435C2×C4C22×D18811yescharacteristic
436C2×C4C2×#1062535011yescharacteristic
437C8—21no—
438C8—21no—
439C8—21no—
440C8—42no—
441C8—1882no—
442C8—1882no—
443C8#1309405512yescomplemented
444C22—21no—
445C22—21no—
446C22—3761no—
447C22—3761no—
448C22—3761no—
449C22#1568935212yes—
450C22C2×#1309399811yescharacteristic
451C4—24no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—42no—
456C4—3761no—
457C4—3761no—
458C4—3761no—
459C4#1568935212yes—
460C4C22×#1062532511yescharacteristic
461C4C2×#1309404611yescharacteristic
462C4C2×#1309405511yescharacteristic
463C4#1568992111yescharacteristic
464C2—22no—
465C2—7521no—
466C2#2025286011yescharacteristic
467C2C2×#1568935211yescharacteristic
468C2#2025310811yescharacteristic
subgroupG/H
centerC22C2×#13093998
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C8×Q8×C47C2
socleC22×C47C2×#121
prime radical#15689673C22
hypercenterC2×C8×Q8D47
hypocenterC47#19260
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. #20253193C2semidirect
4. #20242818C2semidirect
5. #20242804C2semidirect
6. #20242770C2semidirect
7. #20242736C2semidirect
8. #20242144C2semidirect
9. #20242076C2semidirect
11. C2×C8×Q8×C47C2semidirect
112. C2×Q8×C47SD16semidirect
206. C8×C47#137semidirect
233. C2×C8×Q8D47semidirect
283. C47#19260semidirect
352. C2×Q8#10625325semidirect
443. C8#13094055semidirect

Maximal subgroups

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

subgroupindex
#202539472
#202531932
#202528602
#202428182
#202428042
#202427702
#202427362
#202421442
#202420762
#202420632
C2×C8×Q8×C472
#1926047