Finite Group Databasebeta

← #26187213

Subgroups — #26187213

Order 12032.

602 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253425C211yesmaximal, characteristic
2Q8×#10625324C211yesmaximal, characteristic
3#20252336C211yesmaximal, characteristic
4#20242284C211yesmaximal, characteristic, complemented
5#20242283C211yesmaximal, characteristic, complemented
6#20242240C211yesmaximal, characteristic, complemented
7#20242235C211yesmaximal, characteristic, complemented
8#20242235C211yesmaximal, characteristic, complemented
9Q16×C47⋊C8C211yesmaximal, characteristic, complemented
10C8×#10625337C211yesmaximal, characteristic, complemented
11#20242161C211yesmaximal, characteristic
12#20241841C211yesmaximal, characteristic
13Q8⋊C8×D47C211yesmaximal, characteristic
14#20241667C211yesmaximal, characteristic
15C47×#1584C211yesmaximal, characteristic, complemented
16#15689677C2211yescharacteristic
17C4×Q8×D47C2211yescharacteristic
18#15689460C411yescharacteristic
19#15689457C411yescharacteristic
20#15689417C411yescharacteristic
21#15689416C411yescharacteristic
22#15689332C411yescharacteristic
23#15689331C411yescharacteristic
24#15689280C411yescharacteristic
25#15689277C411yescharacteristic
26#15689253—21no—
27#15689253C2211yescharacteristic
28#15689251—21no—
29#15689250—21no—
30#15689250—21no—
31#15689250C2211yescharacteristic
32#15689250C2211yescharacteristic
33Q8×C47⋊C8C2211yescharacteristic
34Q8×C47⋊C8C2211yescharacteristic
35C8×Dic94C2211yescharacteristic
36C4×Dic188C2211yescharacteristic
37C4×#10625337C2211yescharacteristic
38C4×#10625337C2211yescharacteristic
39#15689174C2211yescharacteristic
40#15689148—21no—
41#15689145—21no—
42#15689145C2211yescharacteristic
43C4⋊C8×D47C2211yescharacteristic
44#15689080—21no—
45C4×#10625324—21no—
46C4×#10625324C2211yescharacteristic
47C4×C8×D47C2211yescharacteristic
48#15688876C2211yescharacteristic
49#15688875C2211yescharacteristic
50#15688875C2211yescharacteristic
51#15688855C2211yescharacteristic
52#15688855C2211yescharacteristic
53Q8⋊(C47⋊C8)C2211yescharacteristic
54Q8⋊(C47⋊C8)C2211yescharacteristic
55#15688853C2211yescharacteristic
56#15688853C2211yescharacteristic
57#15688845C2211yescharacteristic
58C8×C47⋊C8C2211yescharacteristic
59C8×C47⋊C8C2211yescharacteristic
60C47×#446C2211yescharacteristic
61C8×Q8×C47C2211yescharacteristic
62C4×Q16×C47C2211yescharacteristic
63C47×#335C2211yescharacteristic
64C47×Q8⋊C8C2211yescharacteristic
65C47×Q8⋊C8C2211yescharacteristic
66C47×#322C2211yescharacteristic
67C2×Q8×D47C2×C411yescharacteristic
68C2×Dic188C2×C411yescharacteristic
69C2×#10625337C2×C411yescharacteristic
70C2×#10625337C2×C411yescharacteristic
71C2×#10625324—21no—
72C2×#10625324D411yescharacteristic
73C2×#10625324D411yescharacteristic
74C2×C8×D47—21no—
75C2×C8×D47—21no—
76#13094023C2×C411yescharacteristic
77#13094022C2×C411yescharacteristic
78#13094014—21no—
79#13094004C2×C411yescharacteristic
80#13094002—21no—
81#13094000C2×C411yescharacteristic
82Q8×Dic47C2×C411yescharacteristic
83C4×Dic94—21no—
84C4×Dic94—21no—
85C4×Dic94C2311yescharacteristic
86C4×Dic94C2311yescharacteristic
87#13093989—21no—
88#13093989C2×C411yescharacteristic
89#13093981—21no—
90C4⋊C4×D47—21no—
91C4⋊C4×D47C2×C411yescharacteristic
92C42×D47—21no—
93C42×D47C2311yescharacteristic
94#13093958C2×C411yescharacteristic
95#13093957C2×C411yescharacteristic
96#13093957C2×C411yescharacteristic
97C4⋊(C47⋊C8)—21no—
98C4⋊(C47⋊C8)—21no—
99C4⋊(C47⋊C8)C2311yescharacteristic
100C4⋊(C47⋊C8)C2311yescharacteristic
101Dic47⋊C8—21no—
102Dic47⋊C8—21no—
103Dic47⋊C8—21no—
104Dic47⋊C8—21no—
105Dic47⋊C8—21no—
106Dic47⋊C8—21no—
107Dic47⋊C8C2×C411yescharacteristic
108Dic47⋊C8C2×C411yescharacteristic
109#13093949C2×C411yescharacteristic
110#13093949C2×C411yescharacteristic
111#13093948C2×C411yescharacteristic
112#13093948C2×C411yescharacteristic
113#13093947C2×C411yescharacteristic
114#13093947C2×C411yescharacteristic
115(C2×D47)⋊C8C2×C411yescharacteristic
116(C2×D47)⋊C8C2×C411yescharacteristic
117#13093934—21no—
118#13093934—21no—
119#13093934—21no—
120#13093934D411yescharacteristic
121#13093934D411yescharacteristic
122C4×C47⋊C8—21no—
123C4×C47⋊C8—21no—
124C4×C47⋊C8C2311yescharacteristic
125C4×C47⋊C8C2311yescharacteristic
126C4×C47⋊C8C2311yescharacteristic
127C8×Dic47—21no—
128C8×Dic47—21no—
129C2×Q16×C47C2×C411yescharacteristic
130C4×Q8×C47C2311yescharacteristic
131C4×Q8×C47C2311yescharacteristic
132#107×C47C2×C411yescharacteristic
133C4⋊C8×C47—21no—
134C4⋊C8×C47—21no—
135C4⋊C8×C47C2311yescharacteristic
136C4⋊C8×C47C2311yescharacteristic
137#103×C47C2×C411yescharacteristic
138#103×C47C2×C411yescharacteristic
139#97×C47—21no—
140C4×C8×C47—21no—
141C4×C8×C47C2311yescharacteristic
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144C2×Q8×C47C22×C411yescharacteristic
145C2×Q8×C47C22×C411yescharacteristic
146Q8×D47—22no—
147Q8×D47—41no—
148C2×C4×D47—21no—
149C2×C4×D47—21no—
150C2×C4×D47C22×C411yescharacteristic
151C2×C4×D47C2×D411yescharacteristic
152C2×C4×D47#4111yescharacteristic
153C2×Dic94—21no—
154C2×Dic94—21no—
155C2×Dic94C22×C411yescharacteristic
156C2×Dic94C22×C411yescharacteristic
157Q16×C47—22no—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47C22×C411yescharacteristic
162C2×C8×C47C22×C411yescharacteristic
163C2×C8×C47C2×D411yescharacteristic
164C2×C8×C47C2×D411yescharacteristic
165C4⋊C4×C47—21no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47C22×C411yescharacteristic
168C4⋊C4×C47C22×C411yescharacteristic
169C42×C47—21no—
170C42×C47—21no—
171C42×C47C2411yescharacteristic
172#10625337—22no—
173#10625337—41no—
174(C2×D47)⋊C4—21no—
175(C2×D47)⋊C4—21no—
176C4⋊Dic47—21no—
177C4⋊Dic47—21no—
178C4⋊Dic47C22×C411yescharacteristic
179C4⋊Dic47C22×C411yescharacteristic
180Dic47⋊C4—21no—
181Dic47⋊C4—21no—
182Dic47⋊C4—21no—
183Dic47⋊C4—21no—
184Dic47⋊C4—21no—
185Dic47⋊C4—21no—
186Dic47⋊C4—21no—
187Dic47⋊C4—21no—
188C4×Dic47—21no—
189C4×Dic47—21no—
190C4×Dic47—21no—
191C4×Dic47—21no—
192C4×Dic47C22×C411yescharacteristic
193C4×Dic47C2×D411yescharacteristic
194C4×Dic47#4111yescharacteristic
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8C22×C411yescharacteristic
199C2×C47⋊C8C22×C411yescharacteristic
200C2×C47⋊C8C2×D411yescharacteristic
201C2×C47⋊C8C2×D411yescharacteristic
202Dic188—41no—
203#10625324—22no—
204#10625324—42no—
205#10625324Q1614yescomplemented
206C8×D47—42no—
207C8×D47—42no—
208C22×D47C4×D411yescharacteristic
209Q8×C47—21no—
210Q8×C47—21no—
211Q8×C47—21no—
212Q8×C47#13112yes—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47C23×C411yescharacteristic
218C2×C4×C47C22×D411yescharacteristic
219C2×C4×C47C2×#4111yescharacteristic
220C2×Dic47—21no—
221C2×Dic47—21no—
222C2×Dic47—21no—
223C2×Dic47—21no—
224C2×Dic47C4×D411yescharacteristic
225C2×Dic47C4×D411yescharacteristic
226C2×Dic47C4×D411yescharacteristic
227C4×D47—21no—
228C4×D47—21no—
229C4×D47—41no—
230C4×D47—42no—
231C4×D47—81no—
232C4×D47C2×Q1612yes—
233C4×D47#13512yes—
234Dic94—21no—
235Dic94—21no—
236Dic94—21no—
237Dic94—21no—
238Dic94—42no—
239Dic94—81no—
240C8×C47—21no—
241C8×C47—21no—
242C8×C47—21no—
243C8×C47—41no—
244C8×C47—41no—
245C8×C47—41no—
246C8×C47C2×Q1612yes—
247C47⋊C8—22no—
248C47⋊C8—21no—
249C47⋊C8—21no—
250C47⋊C8—21no—
251C47⋊C8—41no—
252C47⋊C8—41no—
253C47⋊C8C2×Q1612yes—
254#17998—471nomaximal, Sylow
255C22×C47C2×C4×D411yescharacteristic
256C2×D47—21no—
257C2×D47—21no—
258C2×D47C4×Q1612yes—
259C4×C47—22no—
260C4×C47—21no—
261C4×C47—41no—
262C4×C47—41no—
263C4×C47—41no—
264C4×C47C2×#13111yescharacteristic
265C4×C47#56811yescharacteristic
266C4×C47C22×Q1611yescharacteristic
267C4×C47C2×#13511yescharacteristic
268Dic47—21no—
269Dic47—21no—
270Dic47—41no—
271Dic47—41no—
272Dic47—41no—
273Dic47—41no—
274Dic47C4×Q1612yes—
275#3104—471no—
276Q8×M16—471no—
277#2962—471no—
278#1591—471no—
279#1591—471no—
280#1584—471no—
281#1584—471no—
282#1584—471no—
283#1584D4711yescharacteristic, complemented
284C8×Q16—471no—
285C8×Q16—471no—
286#1571—471no—
287#1493—471no—
288C2×Q8⋊C8—471no—
289C8×M16—471no—
290C2×C47#293311yescharacteristic
291C2×C47C2×C4×Q1611yescharacteristic
292C2×C47#295611yescharacteristic
293D47—22no—
294#533—471no—
295C2×C4×Q8—471no—
296#484—471no—
297#484—471no—
298#477—471no—
299#477—471no—
300#462—471no—
301#462—471no—
302#451—471no—
303#451—471no—
304#446—471no—
305#446—471no—
306#446—471no—
307#446—941no—
308#446—941no—
309#446—941no—
310#446—941no—
311#446C2×D4711yescharacteristic
312C8×Q8—471no—
313C8×Q8—471no—
314C8×Q8—471no—
315C8×Q8C2×D4711yescharacteristic
316C4×Q16—471no—
317C4×Q16—471no—
318C4×Q16—471no—
319C4×Q16C2×D4711yescharacteristic
320#431—471no—
321#424—941no—
322#423—471no—
323#423—941no—
324C2×C4⋊C8—471no—
325#405—941no—
326C4×M16—471no—
327C4×M16—941no—
328C2×C4×C8—471no—
329#335—471no—
330#335—471no—
331#335—471no—
332#335C2×D4711yescharacteristic
333Q8⋊C8—471no—
334Q8⋊C8—471no—
335Q8⋊C8—471no—
336Q8⋊C8—471no—
337Q8⋊C8—471no—
338Q8⋊C8—471no—
339Q8⋊C8C2×D4711yescharacteristic
340Q8⋊C8C2×D4711yescharacteristic
341#322—471no—
342#322C2×D4711yescharacteristic
343C82—471no—
344C82—471no—
345C47#1799811yescharacteristic, complemented, Sylow
346C22×Q8—471no—
347C2×Q16—471no—
348C2×Q16—471no—
349C2×Q16—471no—
350C2×Q16C4×D4711yescharacteristic
351C2×M16—471no—
352C2×M16—471no—
353C2×M16—941no—
354C22×C8—941no—
355C22×C8—941no—
356C4⋊Q8—471no—
357C4⋊Q8—471no—
358#125—941no—
359C22⋊Q8—471no—
360C22⋊Q8—471no—
361C22⋊Q8—941no—
362C4×Q8—471no—
363C4×Q8—471no—
364C4×Q8—471no—
365C4×Q8—471no—
366C4×Q8—941no—
367C4×Q8—941no—
368C4×Q8—941no—
369C4×Q8C22×D4711yescharacteristic
370C4×Q8C22×D4711yescharacteristic
371#117—941no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—941no—
374C2×C42—471no—
375C2×C42—941no—
376#107—471no—
377#107—471no—
378#107—471no—
379#107C4×D4711yescharacteristic
380C4⋊C8—21no—
381C4⋊C8—21no—
382C4⋊C8—471no—
383C4⋊C8—471no—
384C4⋊C8—471no—
385C4⋊C8—471no—
386C4⋊C8—941no—
387C4⋊C8—941no—
388C4⋊C8—941no—
389C4⋊C8—941no—
390C4⋊C8—941no—
391C4⋊C8—941no—
392C4⋊C8—941no—
393C4⋊C8—941no—
394C4⋊C8C22×D4711yescharacteristic
395C4⋊C8C22×D4711yescharacteristic
396#103—471no—
397#103—471no—
398#103—471no—
399#103—471no—
400#103—471no—
401#103—471no—
402#103C4×D4711yescharacteristic
403#103C4×D4711yescharacteristic
404C22⋊C8—471no—
405C22⋊C8—471no—
406#97—21no—
407#97—471no—
408#97—471no—
409#97—941no—
410#97—941no—
411#97—941no—
412C4×C8—21no—
413C4×C8—471no—
414C4×C8—471no—
415C4×C8—471no—
416C4×C8—941no—
417C4×C8—941no—
418C4×C8—941no—
419C4×C8—941no—
420C4×C8C22×D4711yescharacteristic
421C4×C8C22×D4711yescharacteristic
422C4×C8C22×D4711yescharacteristic
423C2×Q8—471no—
424C2×Q8—471no—
425C2×Q8—941no—
426C2×Q8—941no—
427C2×Q8—942no—
428C2×Q8—1881no—
429C2×Q8C2×C4×D4711yescharacteristic
430C2×Q8C2×C4×D4711yescharacteristic
431C22×C4—471no—
432C22×C4—471no—
433C22×C4—471no—
434C22×C4—941no—
435C22×C4—941no—
436Q16—22no—
437Q16—942no—
438Q16—1881no—
439Q16—1881no—
440M16—474no—
441M16—942no—
442M16—1882no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—471no—
447C2×C8—471no—
448C2×C8—471no—
449C2×C8—471no—
450C2×C8—941no—
451C2×C8—941no—
452C2×C8—941no—
453C2×C8—1882no—
454C2×C8—1882no—
455C2×C8C2×C4×D4711yescharacteristic
456C2×C8C2×C4×D4711yescharacteristic
457C2×C8D4×D4711yescharacteristic
458C2×C8D4×D4711yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—471no—
462C4⋊C4—471no—
463C4⋊C4—941no—
464C4⋊C4—941no—
465C4⋊C4—941no—
466C4⋊C4—941no—
467C4⋊C4—941no—
468C4⋊C4—941no—
469C4⋊C4—941no—
470C4⋊C4—941no—
471C4⋊C4—941no—
472C4⋊C4—941no—
473C4⋊C4C2×C4×D4711yescharacteristic
474C4⋊C4C2×C4×D4711yescharacteristic
475C22⋊C4—941no—
476C22⋊C4—941no—
477C42—21no—
478C42—21no—
479C42—471no—
480C42—471no—
481C42—471no—
482C42—941no—
483C42—941no—
484C42—941no—
485C42—941no—
486C42C23×D4711yescharacteristic
487C23—471no—
488Q8—21no—
489Q8—21no—
490Q8—21no—
491Q8—941no—
492Q8—941no—
493Q8—941no—
494Q8—941no—
495Q8—3761no—
496Q8—1882no—
497Q8#1309403212yes—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—471no—
503C2×C4—472no—
504C2×C4—472no—
505C2×C4—471no—
506C2×C4—471no—
507C2×C4—941no—
508C2×C4—941no—
509C2×C4—941no—
510C2×C4—941no—
511C2×C4—941no—
512C2×C4—941no—
513C2×C4—3761no—
514C2×C4—1881no—
515C2×C4—1882no—
516C2×C4C22×C4×D4711yescharacteristic
517C2×C4C2×D4×D4711yescharacteristic
518C2×C4#41×D4711yescharacteristic
519C8—21no—
520C8—21no—
521C8—21no—
522C8—41no—
523C8—41no—
524C8—41no—
525C8—472no—
526C8—942no—
527C8—941no—
528C8—941no—
529C8—941no—
530C8—1881no—
531C8—1881no—
532C8Q16×D4712yes—
533C22—472no—
534C22—941no—
535C22—941no—
536C22C4×D4×D4711yescharacteristic
537C4—21no—
538C4—22no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—472no—
543C4—941no—
544C4—941no—
545C4—1881no—
546C4—1881no—
547C4—1881no—
548C4—1881no—
549C4C2×#1309403211yescharacteristic
550C4#1568988811yescharacteristic
551C4C2×Q16×D4711yescharacteristic
552C4#135×D4711yescharacteristic
553C2—942no—
554C2#2025211411yescharacteristic
555C2C4×Q16×D4711yescharacteristic
556C2#2025229611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1584C2
socleC22×C47C2×C4×D4
prime radicalC22×D47C4×D4
cosocleC42×C47C24
hypercenter#1584D47
hypocenterC47#17998
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
4. #20242284C2semidirect
5. #20242283C2semidirect
6. #20242240C2semidirect
7. #20242235C2semidirect
8. #20242235C2semidirect
9. Q16×C47⋊C8C2semidirect
10. C8×#10625337C2semidirect
15. C47×#1584C2semidirect
205. #10625324Q16semidirect
283. #1584D47semidirect
345. C47#17998semidirect

Maximal subgroups

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

subgroupindex
#202534252
Q8×#106253242
#202523362
#202422842
#202422832
#202422402
#202422352
Q16×C47⋊C82
C8×#106253372
#202421612
#202418412
Q8⋊C8×D472
#202416672
C47×#15842
#1799847