Finite Group Databasebeta

← #26284290

Subgroups — #26284290

Order 12032.

680 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246482C211yesmaximal, characteristic, complemented
2#20244879C211yesmaximal, characteristic, complemented
3#20244624C211yesmaximal, characteristic
4#20244539C211yesmaximal, characteristic, complemented
5#20244534C211yesmaximal, characteristic
6#20244321C211yesmaximal, characteristic
7#20244088C211yesmaximal, characteristic
8#20244026C211yesmaximal, characteristic, complemented
9#20244005C211yesmaximal, characteristic
10#20244005C211yesmaximal, characteristic
11#20243977C211yesmaximal, characteristic, complemented
12#20243954C211yesmaximal, characteristic, complemented
13#20243642C211yesmaximal, characteristic, complemented
14#20243218C211yesmaximal, characteristic, complemented
15C47×#2448C211yesmaximal, characteristic
16C2×#13094016C2211yescharacteristic
17C2×#13093957C2211yescharacteristic
18C2×#13093957C2211yescharacteristic
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×#13093948C2211yescharacteristic
21C2×#13093948C2211yescharacteristic
22C2×#13093948C2211yescharacteristic
23C2×#13093948C2211yescharacteristic
24C2×#13093945C2211yescharacteristic
25C2×#13093945C2211yescharacteristic
26C2×#13093945C2211yescharacteristic
27C2×#13093945C2211yescharacteristic
28C2×#13093935C2211yescharacteristic
29#15689021C2211yescharacteristic
30#15689021C2211yescharacteristic
31#15689018—21no—
32#15689005—21no—
33#15689004—21no—
34C4×C4⋊Dic47C2211yescharacteristic
35C4×C4⋊Dic47C2211yescharacteristic
36C4×C4⋊Dic47C2211yescharacteristic
37#15688974—21no—
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688878C2211yescharacteristic
45#15688878C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48C2×#124×C47C2211yescharacteristic
49C2×C4×Q8×C47C2211yescharacteristic
50C47×#399—21no—
51C47×#397—21no—
52C47×#391C2211yescharacteristic
53C47×#390C2211yescharacteristic
54C47×#390C2211yescharacteristic
55C47×#389C2211yescharacteristic
56C47×#388—21no—
57C47×#388—21no—
58C4×C4⋊C4×C47C2211yescharacteristic
59C22×C47⋊C8C2311yescharacteristic
60C22×C47⋊C8C2311yescharacteristic
61C22×C47⋊C8C2311yescharacteristic
62C22×C47⋊C8C2311yescharacteristic
63#13094016—22no—
64#13094016—22no—
65C2×C4⋊Dic47C2311yescharacteristic
66C2×C4⋊Dic47C2311yescharacteristic
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×Dic47⋊C4—21no—
70C2×Dic47⋊C4—21no—
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C2×C4×Dic47—21no—
74C2×C4×Dic47—21no—
75C2×C4×Dic47—21no—
76#13093957—22no—
77#13093957—41no—
78C4⋊(C47⋊C8)—41no—
79#13093948—22no—
80#13093948—22no—
81#13093948—22no—
82#13093948—22no—
83#13093945—22no—
84#13093945—22no—
85#13093945—22no—
86#13093945—41no—
87#13093935—41no—
88C47⋊#95—21no—
89C47⋊#95—21no—
90C47⋊#95—21no—
91C47⋊#95—21no—
92C47⋊#95—21no—
93C22×Q8×C47C2311yescharacteristic
94C22×D4×C47C2311yescharacteristic
95#124×C47—22no—
96#124×C47—41no—
97C4×Q8×C47—22no—
98C4×Q8×C47—41no—
99C2×C4⋊C4×C47—21no—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47D411yescharacteristic
102C2×C4⋊C4×C47D411yescharacteristic
103C2×C4⋊C4×C47C2311yescharacteristic
104C2×C4⋊C4×C47C2311yescharacteristic
105C2×C22⋊C4×C47—21no—
106C2×C22⋊C4×C47—21no—
107C2×C22⋊C4×C47—21no—
108C2×C22⋊C4×C47—21no—
109C2×C22⋊C4×C47—21no—
110C2×C22⋊C4×C47—21no—
111C2×C42×C47—21no—
112C2×C42×C47C2311yescharacteristic
113C2×C42×C47C2311yescharacteristic
114C2×C42×C47C2311yescharacteristic
115#95×C47—21no—
116#95×C47—21no—
117#95×C47—21no—
118#95×C47D411yescharacteristic
119#95×C47D411yescharacteristic
120C24×C47—21no—
121C2×Q8×C47—21no—
122C2×Q8×C47—21no—
123C2×Q8×C47—21no—
124C2×Q8×C47—21no—
125C2×Q8×C47—21no—
126C2×Q8×C47#4112yes—
127C2×D4×C47—21no—
128C2×D4×C47—21no—
129C2×D4×C47—21no—
130C2×D4×C47—21no—
131C2×D4×C47—21no—
132C2×D4×C47—21no—
133C22×C4×C47—21no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2×D411yescharacteristic
138C22×C4×C47C2×D411yescharacteristic
139C22×C4×C47C2×D411yescharacteristic
140C22×C4×C47C2×D411yescharacteristic
141C22×C4×C47C2×D411yescharacteristic
142C22×C4×C47C2411yescharacteristic
143C22×Dic47—21no—
144C22×Dic47—21no—
145C22×Dic47—21no—
146C22×Dic47—21no—
147C4⋊C4×C47—21no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—21no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—41no—
152C4⋊C4×C47—41no—
153C4⋊C4×C47—42no—
154C4⋊C4×C47—81no—
155C22⋊C4×C47—42no—
156C22⋊C4×C47—42no—
157C22⋊C4×C47—42no—
158C22⋊C4×C47—42no—
159C22⋊C4×C47—81no—
160C22⋊C4×C47—81no—
161C42×C47—21no—
162C42×C47—21no—
163C42×C47—21no—
164C42×C47—21no—
165C42×C47—21no—
166C42×C47—21no—
167C42×C47—42no—
168C4⋊Dic47—21no—
169C4⋊Dic47—21no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47#4112yes—
174C4⋊Dic47#4112yes—
175C4⋊Dic47#4112yes—
176Dic47⋊C4—24no—
177Dic47⋊C4—24no—
178Dic47⋊C4—42no—
179Dic47⋊C4—81no—
180C4×Dic47—42no—
181C4×Dic47—81no—
182C4×Dic47—81no—
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—21no—
190C2×C47⋊C8—21no—
191C2×C47⋊C8—21no—
192C2×C47⋊C8—21no—
193C2×C47⋊C8—21no—
194C2×C47⋊C8—21no—
195C23×C47—41no—
196C23×C47—41no—
197C23×C47—41no—
198C23×C47—41no—
199C23×C47—41no—
200C23×C47—41no—
201C23×C47—41no—
202C23×C47C22×D411yescharacteristic
203Q8×C47—22no—
204Q8×C47—22no—
205Q8×C47—41no—
206Q8×C47—41no—
207D4×C47—41no—
208D4×C47—41no—
209D4×C47—41no—
210D4×C47—41no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—22no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47C2×#4111yescharacteristic
239C2×C4×C47C2×#4111yescharacteristic
240C2×C4×C47C2×#4111yescharacteristic
241C2×C4×C47C2×#4111yescharacteristic
242C2×C4×C47H(5,𝔽2)11yescharacteristic
243C2×C4×C472−1+411yescharacteristic
244C2×Dic47—22no—
245C2×Dic47—22no—
246C2×Dic47—22no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—41no—
250C2×Dic47—41no—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254C2×Dic47—41no—
255C2×Dic47—41no—
256C47⋊C8—41no—
257C47⋊C8—41no—
258C47⋊C8—41no—
259C47⋊C8—41no—
260#26719—471nomaximal, Sylow
261C22×C47—81no—
262C22×C47—81no—
263C22×C47—81no—
264C22×C47—81no—
265C22×C47—81no—
266C22×C47—81no—
267C22×C47—81no—
268C22×C47#52511yescharacteristic
269C22×C47#54611yescharacteristic
270C22×C47#54811yescharacteristic
271C22×C47#57511yescharacteristic
272C22×C47#57511yescharacteristic
273C22×C47#57611yescharacteristic
274C22×C47#57811yescharacteristic
275C4×C47—41no—
276C4×C47—41no—
277C4×C47—42no—
278C4×C47—41no—
279C4×C47—41no—
280C4×C47—41no—
281C4×C47—41no—
282C4×C47—81no—
283C4×C47—81no—
284C4×C47#52911yescharacteristic
285C4×C47#54211yescharacteristic
286C4×C47#55211yescharacteristic
287C4×C47#55911yescharacteristic
288Dic47—42no—
289Dic47—42no—
290Dic47—42no—
291Dic47—81no—
292#2448D4711yescharacteristic
293#2370—471no—
294#2086—471no—
295#2042—471no—
296#2026—471no—
297#2025—471no—
298#1983—471no—
299#1936—471no—
300#1928—471no—
301#1925—471no—
302#1925—471no—
303#1921—471no—
304#1917—471no—
305#1863—471no—
306#1763—471no—
307C2×C47—161no—
308C2×C47#245811yescharacteristic
309C2×C47#311111yescharacteristic
310C2×C47#312411yescharacteristic
311C2×C47#330911yescharacteristic
312C2×C47#332711yescharacteristic
313C2×C47#337011yescharacteristic
314C2×C47#337711yescharacteristic
315C2×#125—471no—
316C2×#124C2×D4711yescharacteristic
317C2×C4×Q8C2×D4711yescharacteristic
318C2×#107—471no—
319C2×#107—471no—
320C2×C4⋊C8—471no—
321C2×#103—471no—
322C2×#103—471no—
323C2×#103—471no—
324C2×#103—471no—
325C2×#102—471no—
326C2×#102—471no—
327C2×#102—471no—
328C2×#102—471no—
329C2×#97—471no—
330#399—21no—
331#397—21no—
332#391C2×D4711yescharacteristic
333#390C2×D4711yescharacteristic
334#390C2×D4711yescharacteristic
335#389—471no—
336#389—471no—
337#389—941no—
338#389C2×D4711yescharacteristic
339#388—21no—
340#388—21no—
341#387—941no—
342#387—941no—
343C4×C4⋊C4—471no—
344C4×C4⋊C4—471no—
345C4×C4⋊C4—471no—
346C4×C4⋊C4—941no—
347C4×C4⋊C4C2×D4711yescharacteristic
348#340—471no—
349#340—471no—
350#340—471no—
351#340—471no—
352#340—471no—
353#340—471no—
354#336—471no—
355#336—471no—
356#336—471no—
357#336—471no—
358C47#2671911yescharacteristic, complemented, Sylow
359C22×Q8C22×D4711yescharacteristic
360C22×D4C22×D4711yescharacteristic
361C22×C8—471no—
362C22×C8—471no—
363C22×C8—471no—
364C22×C8—471no—
365#125—942no—
366#125—942no—
367#124—22no—
368#124—41no—
369C4×Q8—22no—
370C4×Q8—41no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—471no—
374C2×C4⋊C4—471no—
375C2×C4⋊C4—471no—
376C2×C4⋊C4—471no—
377C2×C4⋊C4—941no—
378C2×C4⋊C4—941no—
379C2×C4⋊C4—941no—
380C2×C4⋊C4—941no—
381C2×C4⋊C4C22⋊D4711yescharacteristic
382C2×C4⋊C4C22⋊D4711yescharacteristic
383C2×C4⋊C4C22×D4711yescharacteristic
384C2×C4⋊C4C22×D4711yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C42—21no—
392C2×C42—941no—
393C2×C42—941no—
394C2×C42—941no—
395C2×C42C22×D4711yescharacteristic
396C2×C42C22×D4711yescharacteristic
397C2×C42C22×D4711yescharacteristic
398#107—942no—
399#107—1881no—
400C4⋊C8—1881no—
401#103—942no—
402#103—942no—
403#103—942no—
404#103—942no—
405#102—942no—
406#102—942no—
407#102—942no—
408#102—1881no—
409#97—1881no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—941no—
414#95—941no—
415#95—941no—
416#95—941no—
417#95—941no—
418#95C22⋊D4711yescharacteristic
419#95C22⋊D4711yescharacteristic
420C24—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8#1062535212yes—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—941no—
437C22×C4—941no—
438C22×C4—941no—
439C22×C4—941no—
440C22×C4C2×C22⋊D4711yescharacteristic
441C22×C4C2×C22⋊D4711yescharacteristic
442C22×C4C2×C22⋊D4711yescharacteristic
443C22×C4C2×C22⋊D4711yescharacteristic
444C22×C4C2×C22⋊D4711yescharacteristic
445C22×C4C2×C22⋊D4711yescharacteristic
446C22×C4C23×D4711yescharacteristic
447C2×C8—941no—
448C2×C8—941no—
449C2×C8—941no—
450C2×C8—941no—
451C2×C8—941no—
452C2×C8—941no—
453C2×C8—941no—
454C2×C8—941no—
455C2×C8—941no—
456C2×C8—941no—
457C2×C8—941no—
458C2×C8—941no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—42no—
466C4⋊C4—81no—
467C4⋊C4—472no—
468C4⋊C4—472no—
469C4⋊C4—472no—
470C4⋊C4—944no—
471C4⋊C4—944no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4—941no—
475C4⋊C4—941no—
476C4⋊C4—941no—
477C4⋊C4—3761no—
478C4⋊C4—1882no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—81no—
484C22⋊C4—81no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—42no—
492C42—3761no—
493C42—3761no—
494C42—1882no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23C22×C22⋊D4711yescharacteristic
503Q8—22no—
504Q8—22no—
505Q8—41no—
506Q8—41no—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—41no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—942no—
539C2×C4—942no—
540C2×C4—942no—
541C2×C4—1881no—
542C2×C4—1881no—
543C2×C4—1881no—
544C2×C4—1881no—
545C2×C4—1881no—
546C2×C4—1881no—
547C2×C4—1881no—
548C2×C4—1881no—
549C2×C4—1881no—
550C2×C4#41×D4711yescharacteristic
551C2×C4#41×D4711yescharacteristic
552C2×C4#41×D4711yescharacteristic
553C2×C4C2×#1062535211yescharacteristic
554C2×C4#1309406911yescharacteristic
555C2×C4#1309407111yescharacteristic
556C8—1881no—
557C8—1881no—
558C8—1881no—
559C8—1881no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22#1568963011yescharacteristic
568C22#1568976711yescharacteristic
569C22#1568978011yescharacteristic
570C22#1568992111yescharacteristic
571C22#1568992111yescharacteristic
572C22#1568992611yescharacteristic
573C22#1568993811yescharacteristic
574C4—42no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—81no—
582C4—81no—
583C4—3761no—
584C4—1882no—
585C4—1882no—
586C4—1882no—
587C4#1568965011yescharacteristic
588C4#1568974111yescharacteristic
589C4#1568981011yescharacteristic
590C4#1568985711yescharacteristic
591C2—161no—
592C2#2024730611yescharacteristic
593C2#2025349111yescharacteristic
594C2#2025364411yescharacteristic
595C2#2025539411yescharacteristic
596C2#2025565611yescharacteristic
597C2#2025613511yescharacteristic
598C2#2025622411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2448C2
socleC23×C47C22×D4
prime radicalC22×D4×C47C23
hypercenter#2448D47
hypocenterC47#26719
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2448

Complements

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

NH ≅ G/Ntype
1. #20246482C2semidirect
2. #20244879C2semidirect
4. #20244539C2semidirect
8. #20244026C2semidirect
11. #20243977C2semidirect
12. #20243954C2semidirect
13. #20243642C2semidirect
14. #20243218C2semidirect
358. C47#26719semidirect

Maximal subgroups

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

subgroupindex
#202464822
#202448792
#202446242
#202445392
#202445342
#202443212
#202440882
#202440262
#202440052
#202439772
#202439542
#202436422
#202432182
C47×#24482
#2671947