Finite Group Databasebeta

← #26186313

Subgroups — #26186313

Order 12032.

624 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253457C211yesmaximal, characteristic
2C8×#10625354C211yesmaximal, characteristic
3#20252317C211yesmaximal, characteristic, complemented
4#20242239C211yesmaximal, characteristic, complemented
5#20242235C211yesmaximal, characteristic, complemented
6#20242227C211yesmaximal, characteristic, complemented
7#20242223C211yesmaximal, characteristic, complemented
8#20242205C211yesmaximal, characteristic, complemented
9#20242202C211yesmaximal, characteristic, complemented
10#20242198C211yesmaximal, characteristic, complemented
11#20242175C211yesmaximal, characteristic, complemented
12#20241826C211yesmaximal, characteristic
13#20241807C211yesmaximal, characteristic, complemented
14#20241676C211yesmaximal, characteristic, complemented
15C47×#1579C211yesmaximal, characteristic, complemented
16#15689677C2211yescharacteristic
17C4×#10625354C2211yescharacteristic
18#15689466C411yescharacteristic
19#15689462C411yescharacteristic
20#15689406C411yescharacteristic
21#15689402C411yescharacteristic
22#15689331C411yescharacteristic
23#15689327C411yescharacteristic
24#15689288C411yescharacteristic
25#15689284C411yescharacteristic
26#15689254C2211yescharacteristic
27#15689252—21no—
28#15689250C2211yescharacteristic
29Q8×C47⋊C8C2211yescharacteristic
30C8×Dic94—21no—
31C8×Dic94C2211yescharacteristic
32C4×#10625337C2211yescharacteristic
33C4×#10625325C2211yescharacteristic
34C4×#10625336C2211yescharacteristic
35C8×D188—21no—
36C8×D188C2211yescharacteristic
37#15689188—21no—
38#15689188C2211yescharacteristic
39#15689175—21no—
40#15689175C2211yescharacteristic
41#15689174C2211yescharacteristic
42#15689148—21no—
43#15689145C2211yescharacteristic
44#15689077—21no—
45C4×#10625324C2211yescharacteristic
46C4×C8×D47—21no—
47C4×C8×D47C2211yescharacteristic
48#15688876C2211yescharacteristic
49#15688875C2211yescharacteristic
50#15688873C2211yescharacteristic
51#15688855C2211yescharacteristic
52Q8⋊(C47⋊C8)C2211yescharacteristic
53Q8⋊(C47⋊C8)C2211yescharacteristic
54#15688853C2211yescharacteristic
55#15688852C2211yescharacteristic
56#15688850C2211yescharacteristic
57#15688845C2211yescharacteristic
58#15688845C2211yescharacteristic
59#15688845C2211yescharacteristic
60C8×Q8×C47C2211yescharacteristic
61C8×Q8×C47C2211yescharacteristic
62C4×Q16×C47C2211yescharacteristic
63C47×#334C2211yescharacteristic
64C47×Q8⋊C8C2211yescharacteristic
65C47×Q8⋊C8C2211yescharacteristic
66C47×#322C2211yescharacteristic
67C2×#10625354C2×C411yescharacteristic
68C2×#10625337C2×C411yescharacteristic
69C2×#10625325C2×C411yescharacteristic
70C2×#10625336C2×C411yescharacteristic
71C2×#10625324—21no—
72C2×#10625324—21no—
73C2×C8×D47—21no—
74C2×C8×D47—21no—
75C2×C8×D47D411yescharacteristic
76C2×C8×D47D411yescharacteristic
77#13094023C2×C411yescharacteristic
78#13094022C2×C411yescharacteristic
79#13094014—21no—
80#13094004C2×C411yescharacteristic
81#13094002—21no—
82#13094000C2×C411yescharacteristic
83Q8×Dic47C2×C411yescharacteristic
84C4×Dic94—21no—
85C4×Dic94C2311yescharacteristic
86C4×D188—21no—
87C4×D188C2311yescharacteristic
88#13093986—21no—
89#13093986C2×C411yescharacteristic
90#13093982—21no—
91#13093982C2×C411yescharacteristic
92#13093981—21no—
93C42×D47—21no—
94C42×D47C2311yescharacteristic
95#13093957C2×C411yescharacteristic
96#13093956C2×C411yescharacteristic
97#13093955C2×C411yescharacteristic
98C4⋊(C47⋊C8)—21no—
99C4⋊(C47⋊C8)—21no—
100C4⋊(C47⋊C8)C2311yescharacteristic
101C4⋊(C47⋊C8)C2311yescharacteristic
102Dic47⋊C8—21no—
103Dic47⋊C8—21no—
104Dic47⋊C8—21no—
105Dic47⋊C8C2×C411yescharacteristic
106Dic47⋊C8C2×C411yescharacteristic
107#13093949C2×C411yescharacteristic
108#13093948C2×C411yescharacteristic
109#13093948C2×C411yescharacteristic
110#13093947C2×C411yescharacteristic
111#13093946C2×C411yescharacteristic
112#13093944C2×C411yescharacteristic
113(C2×D47)⋊C8—21no—
114(C2×D47)⋊C8—21no—
115(C2×D47)⋊C8—21no—
116(C2×D47)⋊C8C2×C411yescharacteristic
117(C2×D47)⋊C8C2×C411yescharacteristic
118#13093935—21no—
119#13093934—21no—
120#13093934—21no—
121C4×C47⋊C8—21no—
122C4×C47⋊C8C2311yescharacteristic
123C4×C47⋊C8C2311yescharacteristic
124C4×C47⋊C8C2311yescharacteristic
125C8×Dic47—21no—
126C8×Dic47—21no—
127C8×Dic47D411yescharacteristic
128C8×Dic47D411yescharacteristic
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
139C4×C8×C47—21no—
140C4×C8×C47—21no—
141C4×C8×C47C2311yescharacteristic
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144C2×Q8×C47C22×C411yescharacteristic
145C2×Q8×C47C22×C411yescharacteristic
146#10625354—22no—
147#10625354—22no—
148C2×D188—21no—
149C2×D188C22×C411yescharacteristic
150C2×C4×D47—21no—
151C2×C4×D47—21no—
152C2×C4×D47—21no—
153C2×C4×D47C22×C411yescharacteristic
154C2×C4×D47C2×D411yescharacteristic
155C2×C4×D47#4111yescharacteristic
156C2×Dic94—21no—
157C2×Dic94C22×C411yescharacteristic
158Q16×C47—22no—
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47C22×C411yescharacteristic
163C2×C8×C47C22×C411yescharacteristic
164C2×C8×C47C2×D411yescharacteristic
165C2×C8×C47C2×D411yescharacteristic
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47C22×C411yescharacteristic
169C4⋊C4×C47C22×C411yescharacteristic
170C42×C47—21no—
171C42×C47—21no—
172C42×C47C2411yescharacteristic
173#10625337—22no—
174#10625336—22no—
175(C2×D47)⋊C4—21no—
176(C2×D47)⋊C4—21no—
177(C2×D47)⋊C4—21no—
178(C2×D47)⋊C4—21no—
179(C2×D47)⋊C4—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47C22×C411yescharacteristic
183C4⋊Dic47C22×C411yescharacteristic
184Dic47⋊C4—21no—
185Dic47⋊C4—21no—
186Dic47⋊C4—21no—
187Dic47⋊C4—21no—
188Dic47⋊C4—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
202#10625325—22no—
203#10625324—42no—
204#10625324—42no—
205C8×D47—22no—
206C8×D47—41no—
207C8×D47—42no—
208C8×D47—42no—
209C22×D47—21no—
210C22×D47C4×D411yescharacteristic
211Q8×C47—21no—
212Q8×C47—21no—
213Q8×C47—21no—
214Q8×C47#13112yes—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47C23×C411yescharacteristic
220C2×C4×C47C22×D411yescharacteristic
221C2×C4×C47C2×#4111yescharacteristic
222C2×Dic47—21no—
223C2×Dic47—21no—
224C2×Dic47—21no—
225C2×Dic47C4×D411yescharacteristic
226C2×Dic47C4×D411yescharacteristic
227C2×Dic47C4×D411yescharacteristic
228D188—24no—
229D188—21no—
230D188#13112yes—
231C4×D47—24no—
232C4×D47—21no—
233C4×D47—21no—
234C4×D47—21no—
235C4×D47—21no—
236C4×D47—22no—
237C4×D47—41no—
238C4×D47—41no—
239C4×D47—42no—
240Dic94—21no—
241Dic94—21no—
242Dic94—42no—
243C8×C47—21no—
244C8×C47—21no—
245C8×C47—21no—
246C8×C47—21no—
247C8×C47—41no—
248C8×C47—41no—
249C8×C47—41no—
250C47⋊C8—21no—
251C47⋊C8—21no—
252C47⋊C8—21no—
253C47⋊C8—41no—
254C47⋊C8—41no—
255C47⋊C8—41no—
256C47⋊C8#13712yes—
257#17916—471nomaximal, Sylow
258C22×C47C2×C4×D411yescharacteristic
259C2×D47—21no—
260C2×D47—22no—
261C2×D47—21no—
262C2×D47—21no—
263C2×D47—41no—
264C2×D47—41no—
265C4×C47—22no—
266C4×C47—21no—
267C4×C47—41no—
268C4×C47—41no—
269C4×C47—41no—
270C4×C47C2×#13111yescharacteristic
271C4×C47C2×#13111yescharacteristic
272C4×C47C2×#13711yescharacteristic
273C4×C47#57511yescharacteristic
274Dic47—21no—
275Dic47—21no—
276Dic47—41no—
277Dic47—41no—
278Dic47—41no—
279Dic47#44412yes—
280#3107—471no—
281C8×#41—471no—
282#2960—471no—
283#1584—471no—
284#1584—471no—
285#1582—471no—
286#1582—471no—
287#1579—471no—
288#1579D4711yescharacteristic, complemented
289SD16⋊C8—471no—
290SD16⋊C8—471no—
291#1574—471no—
292#1489—471no—
293#1485—471no—
294#1451—471no—
295C2×C47#292911yescharacteristic
296C2×C47C4×#13711yescharacteristic
297C2×C47C2×#44411yescharacteristic
298D47—41no—
299D47—42no—
300#533—471no—
301C4×#41—471no—
302#485—471no—
303#485—471no—
304#475—471no—
305#474—471no—
306#462—471no—
307#461—471no—
308#452—471no—
309#452—471no—
310#446—471no—
311#446—471no—
312#446—941no—
313C8×Q8—471no—
314C8×Q8—471no—
315C8×Q8—941no—
316C8×Q8C2×D4711yescharacteristic
317C8×Q8C2×D4711yescharacteristic
318C4×Q16—471no—
319C4×Q16C2×D4711yescharacteristic
320C4×SD16—471no—
321C4×SD16—471no—
322C8×D4—471no—
323C8×D4—471no—
324C8×D4—941no—
325C8×D4—941no—
326#431—471no—
327#431—471no—
328#431—941no—
329#424—941no—
330#423—471no—
331#405—941no—
332C4×M16—471no—
333C2×C4×C8—471no—
334C2×C4×C8—941no—
335#335—471no—
336#335—471no—
337#334—471no—
338#334C2×D4711yescharacteristic
339Q8⋊C8—471no—
340Q8⋊C8—471no—
341Q8⋊C8—471no—
342Q8⋊C8—471no—
343Q8⋊C8C2×D4711yescharacteristic
344Q8⋊C8C2×D4711yescharacteristic
345D4⋊C8—471no—
346D4⋊C8—471no—
347#322—471no—
348#322—471no—
349#322—471no—
350#322C2×D4711yescharacteristic
351C47#1791611yescharacteristic, complemented, Sylow
352C2×#41—471no—
353C2×Q16—471no—
354C2×Q16C4×D4711yescharacteristic
355C2×SD16—471no—
356C2×SD16—471no—
357C2×M16—941no—
358C2×M16—941no—
359C22×C8—471no—
360C22×C8—471no—
361C22×C8—941no—
362C22×C8—941no—
363C4⋊Q8—471no—
364C4⋊Q8—471no—
365#125—941no—
366C22⋊Q8—471no—
367C22⋊Q8—471no—
368C22⋊Q8—941no—
369C4×Q8—471no—
370C4×Q8—471no—
371C4×Q8—941no—
372C4×Q8C22×D4711yescharacteristic
373C4×Q8C22×D4711yescharacteristic
374C4×D4—471no—
375C4×D4—471no—
376C4×D4—941no—
377C4×D4—941no—
378#117—471no—
379#117—941no—
380#117—941no—
381C2×C42—471no—
382C2×C42—941no—
383#107—471no—
384#107C4×D4711yescharacteristic
385#106—471no—
386#106—471no—
387C4⋊C8—21no—
388C4⋊C8—21no—
389C4⋊C8—471no—
390C4⋊C8—471no—
391C4⋊C8—471no—
392C4⋊C8—471no—
393C4⋊C8—941no—
394C4⋊C8—941no—
395C4⋊C8—941no—
396C4⋊C8—941no—
397C4⋊C8—941no—
398C4⋊C8C22×D4711yescharacteristic
399C4⋊C8C22×D4711yescharacteristic
400#103—471no—
401#103—471no—
402#103—471no—
403#103—471no—
404#103C4×D4711yescharacteristic
405#103C4×D4711yescharacteristic
406#102—471no—
407#102—471no—
408C22⋊C8—471no—
409C22⋊C8—471no—
410C22⋊C8—941no—
411C22⋊C8—941no—
412C22⋊C8—941no—
413#97—941no—
414#97—941no—
415#97—941no—
416C4×C8—21no—
417C4×C8—21no—
418C4×C8—471no—
419C4×C8—471no—
420C4×C8—471no—
421C4×C8—471no—
422C4×C8—471no—
423C4×C8—941no—
424C4×C8—941no—
425C4×C8—941no—
426C4×C8C22×D4711yescharacteristic
427C4×C8C22×D4711yescharacteristic
428C4×C8C22×D4711yescharacteristic
429#41—942no—
430#41—942no—
431C2×Q8—471no—
432C2×Q8—941no—
433C2×Q8C2×C4×D4711yescharacteristic
434C2×Q8C2×C4×D4711yescharacteristic
435C2×D4—471no—
436C2×D4—941no—
437C22×C4—471no—
438C22×C4—471no—
439C22×C4—471no—
440C22×C4—941no—
441C22×C4—941no—
442C22×C4—941no—
443Q16—22no—
444Q16—942no—
445SD16—942no—
446SD16—942no—
447M16—1882no—
448M16—1882no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—471no—
453C2×C8—471no—
454C2×C8—471no—
455C2×C8—471no—
456C2×C8—942no—
457C2×C8—941no—
458C2×C8—941no—
459C2×C8—941no—
460C2×C8—1881no—
461C2×C8—1882no—
462C2×C8—1882no—
463C2×C8C2×C4×D4711yescharacteristic
464C2×C8C2×C4×D4711yescharacteristic
465C2×C8D4×D4711yescharacteristic
466C2×C8D4×D4711yescharacteristic
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—471no—
470C4⋊C4—471no—
471C4⋊C4—941no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4—941no—
475C4⋊C4—941no—
476C4⋊C4—941no—
477C4⋊C4—941no—
478C4⋊C4C2×C4×D4711yescharacteristic
479C4⋊C4C2×C4×D4711yescharacteristic
480C22⋊C4—941no—
481C22⋊C4—941no—
482C22⋊C4—941no—
483C22⋊C4—941no—
484C22⋊C4—941no—
485C42—21no—
486C42—21no—
487C42—471no—
488C42—471no—
489C42—471no—
490C42—941no—
491C42—941no—
492C42—941no—
493C42C23×D4711yescharacteristic
494C23—471no—
495C23—941no—
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—941no—
500Q8—941no—
501Q8—1882no—
502Q8#1309403012yes—
503D4—472no—
504D4—944no—
505D4—941no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—471no—
511C2×C4—471no—
512C2×C4—471no—
513C2×C4—942no—
514C2×C4—944no—
515C2×C4—941no—
516C2×C4—941no—
517C2×C4—941no—
518C2×C4—941no—
519C2×C4—941no—
520C2×C4—941no—
521C2×C4—941no—
522C2×C4—1882no—
523C2×C4—1881no—
524C2×C4—1881no—
525C2×C4C22×C4×D4711yescharacteristic
526C2×C4C2×D4×D4711yescharacteristic
527C2×C4#41×D4711yescharacteristic
528C8—21no—
529C8—21no—
530C8—21no—
531C8—21no—
532C8—41no—
533C8—41no—
534C8—41no—
535C8—472no—
536C8—941no—
537C8—941no—
538C8—941no—
539C8—1881no—
540C8—1881no—
541C8—1881no—
542C22—941no—
543C22—941no—
544C22—942no—
545C22—941no—
546C22—1881no—
547C22—1881no—
548C22C4×D4×D4711yescharacteristic
549C4—21no—
550C4—22no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—472no—
555C4—941no—
556C4—941no—
557C4—1881no—
558C4—1881no—
559C4—1881no—
560C4#131×D4711yescharacteristic
561C4C2×#1309403011yescharacteristic
562C4#137×D4711yescharacteristic
563C4#1568992211yescharacteristic
564C2—1881no—
565C2—1882no—
566C2#2025207711yescharacteristic
567C2#2025220411yescharacteristic
568C2#444×D4711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1579C2
socleC22×C47C2×C4×D4
prime radicalC2×#10625354C2×C4
cosocleC42×C47C24
hypercenter#1579D47
hypocenterC47#17916
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1579

Complements

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

NH ≅ G/Ntype
3. #20252317C2semidirect
4. #20242239C2semidirect
5. #20242235C2semidirect
6. #20242227C2semidirect
7. #20242223C2semidirect
8. #20242205C2semidirect
9. #20242202C2semidirect
10. #20242198C2semidirect
11. #20242175C2semidirect
13. #20241807C2semidirect
14. #20241676C2semidirect
15. C47×#1579C2semidirect
288. #1579D47semidirect
351. C47#17916semidirect

Maximal subgroups

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

subgroupindex
#202534572
C8×#106253542
#202523172
#202422392
#202422352
#202422272
#202422232
#202422052
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