Finite Group Databasebeta

← #26226685

Subgroups — #26226685

Order 12032.

754 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689150C211yesmaximal, characteristic
2#20245997C211yesmaximal, characteristic
3#20244156C211yesmaximal, characteristic, complemented
4#20244156C211yesmaximal, characteristic, complemented
5#20244154C211yesmaximal, characteristic, complemented
6#20244154C211yesmaximal, characteristic, complemented
7#20244151C211yesmaximal, characteristic, complemented
8#20244149C211yesmaximal, characteristic, complemented
9#20244146C211yesmaximal, characteristic, complemented
10#20244144C211yesmaximal, characteristic, complemented
11#20243419C211yesmaximal, characteristic
12#20243419C211yesmaximal, characteristic
13#20243415C211yesmaximal, characteristic
14#20243411C211yesmaximal, characteristic
15C47×#2314C211yesmaximal, characteristic
16C22×#10625329C2211yescharacteristic
17C23×C47⋊C8C2211yescharacteristic
18C2×#13093957C2211yescharacteristic
19C2×#13093957C2211yescharacteristic
20C2×#13093957C2211yescharacteristic
21C2×#13093957C2211yescharacteristic
22C2×#13093955C2211yescharacteristic
23C2×#13093955C2211yescharacteristic
24C2×#13093955C2211yescharacteristic
25C2×#13093955C2211yescharacteristic
26#15689150C418yescomplemented
27C2×C4⋊(C47⋊C8)C2211yescharacteristic
28C2×C4⋊(C47⋊C8)C2211yescharacteristic
29C2×C4⋊(C47⋊C8)C2211yescharacteristic
30C2×C4⋊(C47⋊C8)C2211yescharacteristic
31#15689016—21no—
32#15689016—21no—
33#15689009—21no—
34#15689009—21no—
35#15688993C2211yescharacteristic
36#15688989C2211yescharacteristic
37#15688983C2211yescharacteristic
38#15688983C2211yescharacteristic
39C4×C4⋊Dic47C2211yescharacteristic
40C4×C4⋊Dic47C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688885C2211yescharacteristic
47#15688885C2211yescharacteristic
48#15688885C2211yescharacteristic
49C2×#117×C47C2211yescharacteristic
50C47×#388—21no—
51C47×#388—21no—
52C47×#387—21no—
53C47×#387—21no—
54C47×#384C2211yescharacteristic
55C47×#382C2211yescharacteristic
56C47×#380C2211yescharacteristic
57C47×#380C2211yescharacteristic
58C4×C4⋊C4×C47C2211yescharacteristic
59C4×C4⋊C4×C47C2211yescharacteristic
60C2×#10625329—41no—
61C2×#10625329—41no—
62C2×#10625329C2×C414yes—
63C22×C47⋊C8—41no—
64C22×C47⋊C8—41no—
65C22×C47⋊C8C2×C414yes—
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68C22×C47⋊C8C2311yescharacteristic
69C22×C47⋊C8C2311yescharacteristic
70C2×C4⋊Dic47—21no—
71C2×C4⋊Dic47—21no—
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×C22⋊Dic47—21no—
77C2×C22⋊Dic47—21no—
78C2×C4×Dic47—21no—
79C2×C4×Dic47—21no—
80#13093957—41no—
81#13093957—41no—
82#13093957—41no—
83#13093957—41no—
84#13093955—41no—
85#13093955—41no—
86#13093955—41no—
87#13093955—41no—
88C4⋊(C47⋊C8)C2×C414yes—
89C4⋊(C47⋊C8)C2×C414yes—
90C4⋊(C47⋊C8)C2×C414yes—
91C4⋊(C47⋊C8)C2×C414yes—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C23×C4×C47C2311yescharacteristic
99#117×C47C2×C414yes—
100#117×C47Q814yes—
101C2×C4⋊C4×C47—21no—
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47D411yescharacteristic
104C2×C4⋊C4×C47D411yescharacteristic
105C2×C4⋊C4×C47C2311yescharacteristic
106C2×C4⋊C4×C47C2311yescharacteristic
107C2×C4⋊C4×C47C2311yescharacteristic
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C22⋊C4×C47—21no—
110C2×C22⋊C4×C47—21no—
111C2×C22⋊C4×C47D411yescharacteristic
112C2×C22⋊C4×C47D411yescharacteristic
113C2×C42×C47—21no—
114C2×C42×C47—21no—
115C2×C42×C47C2311yescharacteristic
116C2×C42×C47C2311yescharacteristic
117#95×C47—21no—
118#95×C47—21no—
119#95×C47—21no—
120#95×C47—21no—
121#95×C47—21no—
122#95×C47—21no—
123C24×C47C2×D411yescharacteristic
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×C47—21no—
131C22×C4×C47—21no—
132C22×C4×C47C22×C412yes—
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×Q812yes—
139C22×C4×C47C2411yescharacteristic
140C22×Dic47—21no—
141C22×Dic47—21no—
142C22×Dic47—21no—
143C22×Dic47—21no—
144C4⋊C4×C47—21no—
145C4⋊C4×C47—21no—
146C4⋊C4×C47—21no—
147C4⋊C4×C47—21no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—21no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—81no—
153C4⋊C4×C47—81no—
154C4⋊C4×C47C4⋊C414yes—
155C4⋊C4×C47C4⋊C414yes—
156C22⋊C4×C47—81no—
157C22⋊C4×C47—81no—
158C22⋊C4×C47C4⋊C414yes—
159C22⋊C4×C47C4⋊C414yes—
160C42×C47—81no—
161C42×C47—81no—
162C42×C47C22×C412yes—
163C42×C47C22×C412yes—
164C42×C47C2×Q812yes—
165C42×C47C2×Q812yes—
166C22⋊Dic47—81no—
167C22⋊Dic47—81no—
168C4⋊Dic47—21no—
169C4⋊Dic47—21no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—81no—
177C4⋊Dic47—81no—
178C4×Dic47—81no—
179C4×Dic47—81no—
180#10625329—44no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8—21no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8—44no—
190C2×C47⋊C8C22×C412yes—
191C2×C47⋊C8C22×C412yes—
192C2×C47⋊C8C22×C412yes—
193C2×C47⋊C8C22×C412yes—
194C23×C47—21no—
195C23×C47—21no—
196C23×C47—21no—
197C23×C47—21no—
198C23×C47—21no—
199C23×C47—21no—
200C23×C47C2×C4⋊C412yes—
201C23×C47C22×D411yescharacteristic
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—21no—
207C2×C4×C47—21no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—22no—
213C2×C4×C47—22no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—41no—
217C2×C4×C47—41no—
218C2×C4×C47—41no—
219C2×C4×C47—41no—
220C2×C4×C47—41no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47C2×C4⋊C412yes—
229C2×C4×C47C2×C4⋊C412yes—
230C2×C4×C47C2×C4⋊C412yes—
231C2×C4×C47C2×C4⋊C412yes—
232C2×C4×C47C2×C4⋊C412yes—
233C2×C4×C47C23×C411yescharacteristic
234C2×C4×C47C22×Q811yescharacteristic
235C2×C4×C47H(5,𝔽2)11yescharacteristic
236C2×C4×C47H(5,𝔽2)11yescharacteristic
237C2×C4×C472−1+411yescharacteristic
238C2×C4×C472−1+411yescharacteristic
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C47⋊C8—22no—
252C47⋊C8—22no—
253C47⋊C8—22no—
254C47⋊C8—22no—
255#21966—471nomaximal, Sylow
256C22×C47—22no—
257C22×C47—22no—
258C22×C47—22no—
259C22×C47—41no—
260C22×C47—41no—
261C22×C47—41no—
262C22×C47—41no—
263C22×C47C22×C4⋊C411yescharacteristic
264C22×C47#53711yescharacteristic
265C22×C47#53711yescharacteristic
266C22×C47#57611yescharacteristic
267C22×C47#57711yescharacteristic
268C22×C47#57711yescharacteristic
269C22×C47#57811yescharacteristic
270C4×C47—22no—
271C4×C47—22no—
272C4×C47—22no—
273C4×C47—22no—
274C4×C47—22no—
275C4×C47—81no—
276C4×C47—81no—
277C4×C47—81no—
278C4×C47—81no—
279C4×C47#51811yescharacteristic
280C4×C47#51911yescharacteristic
281C4×C47#54411yescharacteristic
282C4×C47#54411yescharacteristic
283Dic47—81no—
284Dic47—81no—
285Dic47—81no—
286Dic47—81no—
287C2×#424—471no—
288#2314—471no—
289#2314D4711yescharacteristic
290#1949—471no—
291#1949—471no—
292#1949—471no—
293#1949—471no—
294#1948—471no—
295#1948—471no—
296#1947—471no—
297#1947—471no—
298#1809—471no—
299#1809—471no—
300#1808—471no—
301#1807—471no—
302C2×C47—42no—
303C2×C47#231611yescharacteristic
304C2×C47#291211yescharacteristic
305C2×C47#291311yescharacteristic
306C2×C47#325711yescharacteristic
307C2×C47#325711yescharacteristic
308C2×C47#325811yescharacteristic
309C2×C47#325811yescharacteristic
310C22×M16—471no—
311C23×C8—471no—
312C2×#117C2×D4711yescharacteristic
313C2×#107—471no—
314C2×#107—471no—
315C2×#107—471no—
316C2×#107—471no—
317C2×#106—471no—
318C2×#106—471no—
319C2×#106—471no—
320C2×#106—471no—
321#424—478no—
322C2×C4⋊C8—471no—
323C2×C4⋊C8—471no—
324C2×C4⋊C8—471no—
325C2×C4⋊C8—471no—
326#388—21no—
327#388—21no—
328#388—941no—
329#388—941no—
330#387—21no—
331#387—21no—
332#387—941no—
333#387—941no—
334#384—471no—
335#384C2×D4711yescharacteristic
336#382—471no—
337#382C2×D4711yescharacteristic
338#380—471no—
339#380—471no—
340#380C2×D4711yescharacteristic
341#380C2×D4711yescharacteristic
342C4×C4⋊C4—471no—
343C4×C4⋊C4—471no—
344C4×C4⋊C4C2×D4711yescharacteristic
345C4×C4⋊C4C2×D4711yescharacteristic
346#340—471no—
347#340—471no—
348#340—471no—
349#340—471no—
350#340—471no—
351#340—471no—
352#340—471no—
353#340—471no—
354C47#2196611yescharacteristic, complemented, Sylow
355C23×C4C22×D4711yescharacteristic
356C2×M16—474no—
357C2×M16—1881no—
358C2×M16—1881no—
359C22×C8—471no—
360C22×C8—471no—
361C22×C8—471no—
362C22×C8—471no—
363C22×C8—474no—
364C22×C8—1881no—
365C22×C8—1881no—
366#117Dic9414yes—
367#117C4×D4714yes—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—471no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4C22⋊D4711yescharacteristic
377C2×C4⋊C4C22⋊D4711yescharacteristic
378C2×C4⋊C4C22×D4711yescharacteristic
379C2×C4⋊C4C22×D4711yescharacteristic
380C2×C4⋊C4C22×D4711yescharacteristic
381C2×C4⋊C4C22×D4711yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—941no—
385C2×C22⋊C4—941no—
386C2×C22⋊C4C22⋊D4711yescharacteristic
387C2×C22⋊C4C22⋊D4711yescharacteristic
388C2×C42—21no—
389C2×C42—21no—
390C2×C42—941no—
391C2×C42—941no—
392C2×C42C22×D4711yescharacteristic
393C2×C42C22×D4711yescharacteristic
394#107—1881no—
395#107—1881no—
396#107—1881no—
397#107—1881no—
398#106—1881no—
399#106—1881no—
400#106—1881no—
401#106—1881no—
402C4⋊C8—474no—
403C4⋊C8—474no—
404C4⋊C8—474no—
405C4⋊C8—474no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—941no—
413#95—941no—
414#95—941no—
415#95—941no—
416#95—941no—
417#95—941no—
418C24C2×C22⋊D4711yescharacteristic
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—941no—
428C22×C4—941no—
429C22×C4—941no—
430C22×C4—941no—
431C22×C4C2×Dic9412yes—
432C22×C4C2×C4×D4712yes—
433C22×C4C2×C22⋊D4711yescharacteristic
434C22×C4C2×C22⋊D4711yescharacteristic
435C22×C4C2×C22⋊D4711yescharacteristic
436C22×C4C2×C22⋊D4711yescharacteristic
437C22×C4C2×C22⋊D4711yescharacteristic
438C22×C4C23×D4711yescharacteristic
439M16—1884no—
440C2×C8—472no—
441C2×C8—472no—
442C2×C8—472no—
443C2×C8—472no—
444C2×C8—941no—
445C2×C8—941no—
446C2×C8—941no—
447C2×C8—941no—
448C2×C8—941no—
449C2×C8—941no—
450C2×C8—941no—
451C2×C8—941no—
452C2×C8—1884no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—81no—
462C4⋊C4—81no—
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—3761no—
472C4⋊C4—3761no—
473C4⋊C4Dic47⋊C414yes—
474C4⋊C4Dic47⋊C414yes—
475C22⋊C4—81no—
476C22⋊C4—81no—
477C22⋊C4—3761no—
478C22⋊C4—3761no—
479C22⋊C4Dic47⋊C414yes—
480C22⋊C4Dic47⋊C414yes—
481C42—81no—
482C42—81no—
483C42—3761no—
484C42—3761no—
485C42C2×Dic9412yes—
486C42C2×Dic9412yes—
487C42C2×C4×D4712yes—
488C42C2×C4×D4712yes—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23C2×Dic47⋊C412yes—
496C23C22×C22⋊D4711yescharacteristic
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—1881no—
524C2×C4—1881no—
525C2×C4—1881no—
526C2×C4—1881no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4—1881no—
532C2×C4—1881no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4C2×Dic47⋊C412yes—
536C2×C4C2×Dic47⋊C412yes—
537C2×C4C2×Dic47⋊C412yes—
538C2×C4C2×Dic47⋊C412yes—
539C2×C4C2×Dic47⋊C412yes—
540C2×C4C22×C4×D4711yescharacteristic
541C2×C4C22×Dic9411yescharacteristic
542C2×C4#1309407011yescharacteristic
543C2×C4#1309407011yescharacteristic
544C2×C4#1309407111yescharacteristic
545C2×C4#1309407111yescharacteristic
546C8—942no—
547C8—942no—
548C8—942no—
549C8—942no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C22×Dic47⋊C411yescharacteristic
558C22#1568970111yescharacteristic
559C22#1568970111yescharacteristic
560C22#1568992611yescharacteristic
561C22#1568993111yescharacteristic
562C22#1568993411yescharacteristic
563C22#1568993811yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—22no—
568C4—22no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—3761no—
574C4—3761no—
575C4—3761no—
576C4—3761no—
577C4#1568959711yescharacteristic
578C4#1568960011yescharacteristic
579C4#1568975111yescharacteristic
580C4#1568975111yescharacteristic
581C2—42no—
582C2#2024600411yescharacteristic
583C2#2025198111yescharacteristic
584C2#2025198611yescharacteristic
585C2#2025479311yescharacteristic
586C2#2025479911yescharacteristic
587C2#2025480811yescharacteristic
588C2#2025481411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2314C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
cosocleC22×C4×C47C24
hypercenter#2314D47
hypocenterC47#21966
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2314

Complements

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

NH ≅ G/Ntype
3. #20244156C2semidirect
4. #20244156C2semidirect
5. #20244154C2semidirect
6. #20244154C2semidirect
7. #20244151C2semidirect
8. #20244149C2semidirect
9. #20244146C2semidirect
10. #20244144C2semidirect
26. #15689150C4semidirect
354. C47#21966semidirect

Maximal subgroups

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

subgroupindex
C2×#156891502
#202459972
#202441562
#202441542
#202441512
#202441492
#202441462
#202441442
#202434192
#202434152
#202434112
C47×#23142
#2196647