Finite Group Databasebeta

← #26187730

Subgroups — #26187730

Order 12032.

654 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252408C211yesmaximal, characteristic, complemented
2#20252402C211yesmaximal, characteristic, complemented
3#20242279C212yesmaximal, complemented
4#20242239C212yesmaximal, complemented
5#20242218C212yesmaximal, complemented
6SD16×C47⋊C8C212yesmaximal, complemented
7#20242151C211yesmaximal, characteristic, complemented
8#20241822C211yesmaximal, characteristic, complemented
9#20241822C211yesmaximal, characteristic, complemented
10C8×#10625329C211yesmaximal, characteristic, complemented
11C4×#135×C47C211yesmaximal, characteristic
12#15689254C2212yes—
13Q8×C47⋊C8C2212yes—
14#15689201—22no—
15#15689201C2212yes—
16#15689195—22no—
17D4×C47⋊C8C2212yes—
18#15689180—21no—
19#15689180—21no—
20#15689180C2211yescharacteristic
21#15689180C2211yescharacteristic
22#15689176C2211yescharacteristic
23C4×#10625329—21no—
24C4×#10625329C2211yescharacteristic
25C2×#13093935—21no—
26#15688876C2212yes—
27#15688874C2212yes—
28Q8⋊(C47⋊C8)C2212yes—
29Q8⋊(C47⋊C8)C2212yes—
30D4⋊(C47⋊C8)C2212yes—
31D4⋊(C47⋊C8)C2212yes—
32#15688845C2212yes—
33C8×C47⋊C8C2212yes—
34C2×#135×C47C411yescharacteristic
35C4×#41×C47C2211yescharacteristic
36C4×#41×C47C2211yescharacteristic
37C4×Q16×C47C411yescharacteristic
38C4×Q16×C47C2211yescharacteristic
39C4×SD16×C47C411yescharacteristic
40C4×SD16×C47C411yescharacteristic
41C4×SD16×C47C2211yescharacteristic
42C4×SD16×C47C2211yescharacteristic
43C4×D8×C47C411yescharacteristic
44C4×D8×C47C2211yescharacteristic
45C47×#427C411yescharacteristic
46C47×#416C411yescharacteristic
47C47×#416C411yescharacteristic
48C2×C4×C8×C47C2211yescharacteristic
49C2×#10625329—22no—
50C2×#10625329D412yes—
51C22×C47⋊C8—22no—
52C4⋊(C47⋊C8)—21no—
53C4⋊(C47⋊C8)—22no—
54C4⋊(C47⋊C8)—22no—
55C4⋊(C47⋊C8)C2312yes—
56C4⋊(C47⋊C8)C2312yes—
57C22⋊(C47⋊C8)—21no—
58C22⋊(C47⋊C8)—21no—
59C22⋊(C47⋊C8)—21no—
60C22⋊(C47⋊C8)—22no—
61C22⋊(C47⋊C8)—22no—
62#13093935—21no—
63#13093935—21no—
64#13093935—22no—
65#13093935D411yescharacteristic
66#13093935D411yescharacteristic
67C4×C47⋊C8—22no—
68C4×C47⋊C8C2312yes—
69C4×C47⋊C8C2312yes—
70C2×#41×C47C2×C411yescharacteristic
71C2×#41×C47C2×C411yescharacteristic
72#135×C47—24no—
73C2×Q16×C47C2×C411yescharacteristic
74C2×SD16×C47C2×C411yescharacteristic
75C2×SD16×C47C2×C411yescharacteristic
76C2×D8×C47C2×C411yescharacteristic
77C22×C8×C47C2×C411yescharacteristic
78C22×C8×C47C2×C411yescharacteristic
79C4×Q8×C47C2×C411yescharacteristic
80C4×Q8×C47C2×C411yescharacteristic
81C4×Q8×C47C2311yescharacteristic
82C4×Q8×C47C2311yescharacteristic
83C4×D4×C47—21no—
84C4×D4×C47—21no—
85C4×D4×C47—21no—
86C4×D4×C47—21no—
87C4×D4×C47C2×C411yescharacteristic
88C4×D4×C47C2×C411yescharacteristic
89C4×D4×C47C2311yescharacteristic
90C4×D4×C47C2311yescharacteristic
91#117×C47—21no—
92#117×C47—21no—
93#117×C47C2×C411yescharacteristic
94#117×C47C2×C411yescharacteristic
95C2×C42×C47—21no—
96C2×C42×C47—21no—
97C2×C42×C47C2311yescharacteristic
98#107×C47C2×C411yescharacteristic
99#107×C47C2×C411yescharacteristic
100#106×C47C2×C411yescharacteristic
101#106×C47C2×C411yescharacteristic
102#103×C47C2×C411yescharacteristic
103#103×C47C2×C411yescharacteristic
104#103×C47C2×C411yescharacteristic
105#103×C47C2×C411yescharacteristic
106#102×C47C2×C411yescharacteristic
107#102×C47C2×C411yescharacteristic
108#102×C47C2×C411yescharacteristic
109#102×C47C2×C411yescharacteristic
110C4×C8×C47C2×C411yescharacteristic
111C4×C8×C47C2×C411yescharacteristic
112C4×C8×C47C2311yescharacteristic
113C4×C8×C47C2311yescharacteristic
114#41×C47—22no—
115#41×C47—22no—
116#41×C47—41no—
117#41×C47—41no—
118C2×Q8×C47C22×C411yescharacteristic
119C2×Q8×C47C22×C411yescharacteristic
120C2×D4×C47—21no—
121C2×D4×C47—21no—
122C2×D4×C47C22×C411yescharacteristic
123C2×D4×C47C22×C411yescharacteristic
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—21no—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47C22×C411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47#4111yescharacteristic
133Q16×C47—22no—
134SD16×C47—22no—
135SD16×C47M1614yescomplemented
136D8×C47—22no—
137C2×C8×C47—22no—
138C2×C8×C47—41no—
139C2×C8×C47C22×C411yescharacteristic
140C2×C8×C47C22×C411yescharacteristic
141C2×C8×C47C22×C411yescharacteristic
142C2×C8×C47C22×C411yescharacteristic
143C4⋊C4×C47—21no—
144C4⋊C4×C47—21no—
145C4⋊C4×C47—21no—
146C4⋊C4×C47—21no—
147C4⋊C4×C47C22×C411yescharacteristic
148C4⋊C4×C47C22×C411yescharacteristic
149C4⋊C4×C47C22×C411yescharacteristic
150C4⋊C4×C47C22×C411yescharacteristic
151C22⋊C4×C47—21no—
152C22⋊C4×C47—21no—
153C22⋊C4×C47—21no—
154C22⋊C4×C47—21no—
155C22⋊C4×C47—21no—
156C22⋊C4×C47—21no—
157C42×C47—21no—
158C42×C47—21no—
159C42×C47—21no—
160C42×C47—21no—
161C42×C47—21no—
162C42×C47—21no—
163C42×C47C22×C411yescharacteristic
164C42×C47C2×D411yescharacteristic
165C42×C47#4111yescharacteristic
166C42×C47C2411yescharacteristic
167#10625329—24no—
168#10625329—44no—
169C2×C47⋊C8—22no—
170C2×C47⋊C8—22no—
171C2×C47⋊C8—22no—
172C2×C47⋊C8—44no—
173C2×C47⋊C8C2×D412yes—
174C2×C47⋊C8C2×D412yes—
175C23×C47—21no—
176C23×C47—21no—
177C23×C47C4×D411yescharacteristic
178Q8×C47—21no—
179Q8×C47—21no—
180Q8×C47—21no—
181Q8×C47C2×M1612yes—
182D4×C47—21no—
183D4×C47—21no—
184D4×C47—21no—
185D4×C47—42no—
186D4×C47—42no—
187D4×C47C2×M1612yes—
188C2×C4×C47—21no—
189C2×C4×C47—21no—
190C2×C4×C47—21no—
191C2×C4×C47—22no—
192C2×C4×C47—21no—
193C2×C4×C47—21no—
194C2×C4×C47—21no—
195C2×C4×C47—21no—
196C2×C4×C47—21no—
197C2×C4×C47—41no—
198C2×C4×C47—42no—
199C2×C4×C47—42no—
200C2×C4×C47—42no—
201C2×C4×C47—42no—
202C2×C4×C47—81no—
203C2×C4×C47—81no—
204C2×C4×C47C4×D411yescharacteristic
205C2×C4×C47C4×D411yescharacteristic
206C2×C4×C47C4×D411yescharacteristic
207C2×C4×C47C23×C411yescharacteristic
208C2×C4×C47C22×D411yescharacteristic
209C2×C4×C47C2×#4111yescharacteristic
210C8×C47—21no—
211C8×C47—21no—
212C8×C47—21no—
213C8×C47C2×M1612yes—
214C47⋊C8—22no—
215C47⋊C8—42no—
216C47⋊C8—42no—
217C47⋊C8—42no—
218C47⋊C8#13514yescomplemented
219#18037—471nomaximal, Sylow
220C22×C47—21no—
221C22×C47—22no—
222C22×C47—21no—
223C22×C47—21no—
224C22×C47—41no—
225C22×C47—41no—
226C22×C47—41no—
227C22×C47—41no—
228C22×C47—41no—
229C22×C47C2×C4×D411yescharacteristic
230C4×C47—21no—
231C4×C47—22no—
232C4×C47—21no—
233C4×C47—21no—
234C4×C47—21no—
235C4×C47—41no—
236C4×C47—41no—
237C4×C47—41no—
238C4×C47—41no—
239C4×C47—41no—
240C4×C47C22×M1611yescharacteristic
241C4×C47#56811yescharacteristic
242C4×C47C2×#13512yes—
243#2971—471no—
244#2970—471no—
245C4×#135D4711yescharacteristic
246#1590—472no—
247#1584—472no—
248#1581—472no—
249C8×SD16—472no—
250#1568—471no—
251#1488—471no—
252#1488—471no—
253C8×M16—471no—
254C2×C47—41no—
255C2×C47—42no—
256C2×C47—81no—
257C2×C47D4×M1611yescharacteristic
258C2×C47C4×#13511yescharacteristic
259C2×C47#295711yescharacteristic
260C2×#135Dic4711yescharacteristic
261C4×#41C2×D4711yescharacteristic
262C4×#41C2×D4711yescharacteristic
263#446—472no—
264C8×Q8—472no—
265C4×Q16Dic4711yescharacteristic
266C4×Q16C2×D4711yescharacteristic
267C4×SD16Dic4711yescharacteristic
268C4×SD16Dic4711yescharacteristic
269C4×SD16C2×D4711yescharacteristic
270C4×SD16C2×D4711yescharacteristic
271C4×D8Dic4711yescharacteristic
272C4×D8C2×D4711yescharacteristic
273#436—472no—
274#436—942no—
275#435—942no—
276C8×D4—472no—
277#432—471no—
278#432—471no—
279#432—941no—
280#432—941no—
281#431—471no—
282#427Dic4711yescharacteristic
283#416Dic4711yescharacteristic
284#416Dic4711yescharacteristic
285C4×M16—471no—
286C4×M16—941no—
287C2×#97—941no—
288C2×C4×C8C2×D4711yescharacteristic
289#335—472no—
290#334—472no—
291Q8⋊C8—472no—
292Q8⋊C8—472no—
293D4⋊C8—472no—
294D4⋊C8—472no—
295#322—472no—
296C82—472no—
297C47#1803711yescharacteristic, complemented, Sylow
298C2×#41C2×Dic4711yescharacteristic
299C2×#41C2×Dic4711yescharacteristic
300#135—24no—
301C2×Q16C2×Dic4711yescharacteristic
302C2×SD16C2×Dic4711yescharacteristic
303C2×SD16C2×Dic4711yescharacteristic
304C2×D8C2×Dic4711yescharacteristic
305C2×M16—472no—
306C2×M16—942no—
307C22×C8—942no—
308C22×C8C2×Dic4711yescharacteristic
309C22×C8C2×Dic4711yescharacteristic
310C4×Q8C2×Dic4711yescharacteristic
311C4×Q8C2×Dic4711yescharacteristic
312C4×Q8C22×D4711yescharacteristic
313C4×Q8C22×D4711yescharacteristic
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4C2×Dic4711yescharacteristic
319C4×D4C2×Dic4711yescharacteristic
320C4×D4C22×D4711yescharacteristic
321C4×D4C22×D4711yescharacteristic
322#117—21no—
323#117—21no—
324#117C2×Dic4711yescharacteristic
325#117C2×Dic4711yescharacteristic
326C2×C42—21no—
327C2×C42—21no—
328C2×C42C22×D4711yescharacteristic
329#107C2×Dic4711yescharacteristic
330#107C2×Dic4711yescharacteristic
331#106C2×Dic4711yescharacteristic
332#106C2×Dic4711yescharacteristic
333C4⋊C8—472no—
334C4⋊C8—472no—
335C4⋊C8—942no—
336C4⋊C8—942no—
337C4⋊C8—941no—
338#103C2×Dic4711yescharacteristic
339#103C2×Dic4711yescharacteristic
340#103C2×Dic4711yescharacteristic
341#103C2×Dic4711yescharacteristic
342#102C2×Dic4711yescharacteristic
343#102C2×Dic4711yescharacteristic
344#102C2×Dic4711yescharacteristic
345#102C2×Dic4711yescharacteristic
346C22⋊C8—942no—
347C22⋊C8—942no—
348C22⋊C8—941no—
349C22⋊C8—941no—
350C22⋊C8—941no—
351#97—471no—
352#97—471no—
353#97—942no—
354#97—941no—
355#97—941no—
356C4×C8—472no—
357C4×C8—472no—
358C4×C8—942no—
359C4×C8C2×Dic4711yescharacteristic
360C4×C8C2×Dic4711yescharacteristic
361C4×C8C22×D4711yescharacteristic
362C4×C8C22×D4711yescharacteristic
363#41—22no—
364#41—22no—
365#41—41no—
366#41—41no—
367C2×Q8C22×Dic4711yescharacteristic
368C2×Q8C22×Dic4711yescharacteristic
369C2×D4—21no—
370C2×D4—21no—
371C2×D4C22×Dic4711yescharacteristic
372C2×D4C22×Dic4711yescharacteristic
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4D4×D4711yescharacteristic
380C22×C4#1062535211yescharacteristic
381C22×C4C22×Dic4711yescharacteristic
382Q16—22no—
383SD16—22no—
384SD16#1062532914yescomplemented
385D8—22no—
386M16—944no—
387M16—1884no—
388C2×C8—22no—
389C2×C8—41no—
390C2×C8—472no—
391C2×C8—472no—
392C2×C8—942no—
393C2×C8—942no—
394C2×C8—942no—
395C2×C8—1884no—
396C2×C8C22×Dic4711yescharacteristic
397C2×C8C22×Dic4711yescharacteristic
398C2×C8C22×Dic4711yescharacteristic
399C2×C8C22×Dic4711yescharacteristic
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4C22×Dic4711yescharacteristic
405C4⋊C4C22×Dic4711yescharacteristic
406C4⋊C4C22×Dic4711yescharacteristic
407C4⋊C4C22×Dic4711yescharacteristic
408C22⋊C4—21no—
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42D4×D4711yescharacteristic
421C42#1062535211yescharacteristic
422C42C22×Dic4711yescharacteristic
423C42C23×D4711yescharacteristic
424C23—21no—
425C23—21no—
426C23D4×Dic4711yescharacteristic
427Q8—21no—
428Q8—21no—
429Q8—21no—
430Q8C2×#1062532912yes—
431D4—21no—
432D4—21no—
433D4—21no—
434D4—42no—
435D4—42no—
436D4C2×#1062532912yes—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4D4×Dic4711yescharacteristic
454C2×C4D4×Dic4711yescharacteristic
455C2×C4D4×Dic4711yescharacteristic
456C2×C4C23×Dic4711yescharacteristic
457C2×C4C2×D4×D4711yescharacteristic
458C2×C4C2×#1062535211yescharacteristic
459C8—21no—
460C8—21no—
461C8—21no—
462C8—474no—
463C8—942no—
464C8—1882no—
465C8—1882no—
466C8—1882no—
467C8C2×#1062532912yes—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22C2×D4×Dic4711yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—22no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4C22×#1062532911yescharacteristic
489C4#1568989011yescharacteristic
490C4#135×D4712yes—
491C2—41no—
492C2—42no—
493C2—81no—
494C2D4×#1062532911yescharacteristic
495C2#135×Dic4711yescharacteristic
496C2#2025230511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC4×#135×C47C2
socleC22×C47C2×C4×D4
prime radicalC2×#135×C47C4
cosocleC42×C47C24
hypercenterC4×#135D47
hypocenterC47#18037
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
1. #20252408C2semidirect
2. #20252402C2semidirect
3. #20242279C2semidirect
4. #20242239C2semidirect
5. #20242218C2semidirect
6. SD16×C47⋊C8C2semidirect
7. #20242151C2semidirect
8. #20241822C2semidirect
9. #20241822C2semidirect
10. C8×#10625329C2semidirect
135. SD16×C47M16semidirect
218. C47⋊C8#135semidirect
297. C47#18037semidirect
384. SD16#10625329semidirect

Maximal subgroups

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

subgroupindex
#202524082
#202524022
#202422792
#202422392
#202422182
SD16×C47⋊C82
#202421512
#202418222
C8×#106253292
C4×#135×C472
#1803747