Finite Group Databasebeta

← #26268160

Subgroups — #26268160

Order 12032.

774 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689424C211yesmaximal, characteristic
2#20248776C211yesmaximal, characteristic, complemented
3#20248746C211yesmaximal, characteristic
4#20244886C211yesmaximal, characteristic, complemented
5#20244881C211yesmaximal, characteristic, complemented
6#20244775C211yesmaximal, characteristic
7#20244761C211yesmaximal, characteristic
8#20244353C211yesmaximal, characteristic, complemented
9#20244158C211yesmaximal, characteristic, complemented
10#20244121C211yesmaximal, characteristic, complemented
11#20244116C211yesmaximal, characteristic, complemented
12#20244071C211yesmaximal, characteristic
13#20244071C211yesmaximal, characteristic
14C4×#13093946C211yesmaximal, characteristic
15C2×C8×Q8×C47C211yesmaximal, characteristic, complemented
16C2×#13094016C2211yescharacteristic
17C2×C4×D188C2211yescharacteristic
18#15689424—24no—
19C2×#13093958C2211yescharacteristic
20C2×#13093958C2211yescharacteristic
21C2×#13093956C2211yescharacteristic
22C2×#13093956C2211yescharacteristic
23C2×#13093946C2211yescharacteristic
24C2×#13093946C2211yescharacteristic
25C2×#13093946C2211yescharacteristic
26C2×#13093946C2211yescharacteristic
27#15689058—21no—
28#15689054—21no—
29#15689042—21no—
30#15689034—21no—
31#15689025C2211yescharacteristic
32#15689025C2211yescharacteristic
33#15689021C2211yescharacteristic
34#15689021C2211yescharacteristic
35#15689021C2211yescharacteristic
36#15689021C2211yescharacteristic
37#15689021C2211yescharacteristic
38#15689021C2211yescharacteristic
39#15689018—21no—
40#15689018—21no—
41#15688994—21no—
42#15688994—21no—
43#15688989C2211yescharacteristic
44C4×C4⋊Dic47C2211yescharacteristic
45#15688886C2211yescharacteristic
46#15688886C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688886C2211yescharacteristic
50#15688886C2211yescharacteristic
51#15688886C2211yescharacteristic
52#15688886C2211yescharacteristic
53C2×C4×Q8×C47C2211yescharacteristic
54C8×Q8×C47—24no—
55C2×C4⋊C8×C47C2211yescharacteristic
56C2×C4⋊C8×C47C2211yescharacteristic
57C2×C4⋊C8×C47C2211yescharacteristic
58C2×C4×C8×C47C2211yescharacteristic
59C2×C4×C8×C47C2211yescharacteristic
60C2×C4×C8×C47C2211yescharacteristic
61C22×D188C2311yescharacteristic
62C22×C4×D47—21no—
63#13094016—22no—
64#13094016—22no—
65C4×D188—22no—
66C4×D188—22no—
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×Dic47⋊C4—21no—
78C2×Dic47⋊C4—21no—
79C2×Dic47⋊C4—21no—
80C2×Dic47⋊C4—21no—
81C2×Dic47⋊C4—21no—
82C2×(C2×D47)⋊C4—21no—
83C2×(C2×D47)⋊C4—21no—
84C2×(C2×D47)⋊C4—21no—
85C2×(C2×D47)⋊C4—21no—
86C2×(C2×D47)⋊C4—21no—
87C2×C4×Dic47—21no—
88#13093958—22no—
89#13093958—22no—
90#13093956—22no—
91#13093956—22no—
92#13093946—22no—
93#13093946—22no—
94#13093946—22no—
95#13093946Q814yescomplemented
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C47⋊#95—21no—
104C47⋊#95—21no—
105C22×Q8×C47D411yescharacteristic
106C22×C8×C47C2311yescharacteristic
107C22×C8×C47C2311yescharacteristic
108C22×C8×C47C2311yescharacteristic
109C22×C8×C47C2311yescharacteristic
110C4×Q8×C47—22no—
111C4×Q8×C47—22no—
112C2×C4⋊C4×C47D411yescharacteristic
113C2×C4⋊C4×C47D411yescharacteristic
114C2×C4⋊C4×C47D411yescharacteristic
115C2×C42×C47C2311yescharacteristic
116C2×C42×C47C2311yescharacteristic
117C2×C42×C47C2311yescharacteristic
118C4⋊C8×C47—22no—
119C4⋊C8×C47—22no—
120C4⋊C8×C47D414yes—
121C4×C8×C47—22no—
122C4×C8×C47—22no—
123C4×C8×C47—22no—
124C23×D47—21no—
125C2×Q8×C47—22no—
126C2×Q8×C47—22no—
127C2×Q8×C47—22no—
128C22×C4×C47C2×D411yescharacteristic
129C22×C4×C47C2×D411yescharacteristic
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2411yescharacteristic
135C22×Dic47—21no—
136C22×Dic47—21no—
137C22×Dic47—21no—
138C22×Dic47—21no—
139C22×Dic47—21no—
140C22×Dic47—21no—
141C22×Dic47—21no—
142C2×D188—21no—
143C2×D188—21no—
144C2×D188—21no—
145C2×D188—21no—
146C2×D188—21no—
147C2×D188C2×Q812yes—
148C2×C4×D47—42no—
149C2×C4×D47—81no—
150C2×C4×D47—81no—
151C2×C8×C47—21no—
152C2×C8×C47—21no—
153C2×C8×C47—21no—
154C2×C8×C47—21no—
155C2×C8×C47—21no—
156C2×C8×C47—21no—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47—21no—
160C2×C8×C47C2×D412yes—
161C2×C8×C47C2×D412yes—
162C2×C8×C47C2×Q812yes—
163C4⋊C4×C47—22no—
164C4⋊C4×C47—22no—
165C4⋊C4×C47—22no—
166C42×C47—21no—
167C42×C47—21no—
168C42×C47—21no—
169C42×C47—21no—
170C42×C47C2×D412yes—
171C42×C47#4112yes—
172(C2×D47)⋊C4—42no—
173(C2×D47)⋊C4—42no—
174(C2×D47)⋊C4—42no—
175(C2×D47)⋊C4—81no—
176(C2×D47)⋊C4—81no—
177C4⋊Dic47—21no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—21no—
181C4⋊Dic47—21no—
182C4⋊Dic47—21no—
183C4⋊Dic47—21no—
184C4⋊Dic47—21no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47C2×Q812yes—
190C4⋊Dic47#4112yes—
191Dic47⋊C4—42no—
192Dic47⋊C4—42no—
193Dic47⋊C4—42no—
194Dic47⋊C4—42no—
195Dic47⋊C4—81no—
196Dic47⋊C4—81no—
197Dic47⋊C4—81no—
198Dic47⋊C4—81no—
199C4×Dic47—81no—
200C23×C47C22×D411yescharacteristic
201C22×D47—22no—
202C22×D47—41no—
203C22×D47—41no—
204C22×D47—41no—
205C22×D47—41no—
206C22×D47—41no—
207C22×D47—41no—
208Q8×C47—28no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
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—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47#12112yes—
224C2×C4×C47#12112yes—
225C2×C4×C47#13512yes—
226C2×C4×C47#13512yes—
227C2×C4×C47C22×D411yescharacteristic
228C2×C4×C47C22×Q811yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×C4×C472−1+411yescharacteristic
232C2×C4×C472−1+411yescharacteristic
233C2×Dic47—22no—
234C2×Dic47—22no—
235C2×Dic47—41no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—41no—
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—
251C2×Dic47—41no—
252C2×Dic47—41no—
253C2×Dic47—41no—
254D188—22no—
255D188—22no—
256D188—41no—
257D188—41no—
258C4×D47—84no—
259C8×C47—22no—
260C8×C47—22no—
261C8×C47—22no—
262C8×C47—41no—
263#25473—471nomaximal, Sylow
264C22×C47C2×#12111yescharacteristic
265C22×C47#54811yescharacteristic
266C22×C47D4×Q811yescharacteristic
267C22×C47C2×#13511yescharacteristic
268C22×C47C2×#13511yescharacteristic
269C22×C47#57611yescharacteristic
270C22×C47#57711yescharacteristic
271C2×D47—42no—
272C2×D47—42no—
273C2×D47—42no—
274C2×D47—81no—
275C2×D47—81no—
276C2×D47—81no—
277C2×D47—81no—
278C4×C47—22no—
279C4×C47—22no—
280C4×C47—22no—
281C4×C47—22no—
282C4×C47—41no—
283C4×C47—41no—
284C4×C47#53311yescharacteristic
285C4×C47#54111yescharacteristic
286C4×C47#54711yescharacteristic
287C4×C47D4×Q811yescharacteristic
288Dic47—42no—
289Dic47—42no—
290Dic47—81no—
291Dic47—81no—
292Dic47—81no—
293Dic47—81no—
294Dic47—81no—
295C2×#478—471no—
296C2×C8×Q8D4711yescharacteristic, complemented
297#2584—471no—
298#2580—471no—
299#2088—471no—
300#2087—471no—
301#2067—471no—
302#2065—471no—
303#1991—471no—
304#1949—471no—
305#1942—471no—
306#1941—471no—
307#1934—471no—
308#1934—471no—
309C4×#102—471no—
310C2×C47#242911yescharacteristic
311C2×C47#305011yescharacteristic
312C2×C47#306211yescharacteristic
313C2×C47#336111yescharacteristic
314C2×C47#336511yescharacteristic
315C2×C47#338011yescharacteristic
316C2×C47#338111yescharacteristic
317D47—82no—
318C2×#125—471no—
319C2×C4×Q8C2×D4711yescharacteristic
320C2×C4×D4—471no—
321#478—944no—
322C8×Q8—24no—
323C2×#107—471no—
324C2×#107—471no—
325C2×#106—471no—
326C2×#106—471no—
327C2×C4⋊C8C2×D4711yescharacteristic
328C2×C4⋊C8C2×D4711yescharacteristic
329C2×C4⋊C8C2×D4711yescharacteristic
330C2×#102—471no—
331C2×#102—471no—
332C2×#102—471no—
333C2×#102—471no—
334C2×C4×C8C2×D4711yescharacteristic
335C2×C4×C8C2×D4711yescharacteristic
336C2×C4×C8C2×D4711yescharacteristic
337#399—941no—
338#398—941no—
339#395—941no—
340#393—941no—
341#390—471no—
342#390—471no—
343#389—471no—
344#389—471no—
345#389—471no—
346#389—471no—
347#389—471no—
348#389—471no—
349#389—941no—
350#389—941no—
351#385—941no—
352#385—941no—
353#382—471no—
354C4×C4⋊C4—471no—
355#340—471no—
356#340—471no—
357#340—471no—
358#340—471no—
359#340—471no—
360#340—471no—
361#340—471no—
362#340—471no—
363C47#2547311yescharacteristic, complemented, Sylow
364C22×Q8D18811yescharacteristic
365C22×D4—471no—
366C23×C4—941no—
367C22×C8C22×D4711yescharacteristic
368C22×C8C22×D4711yescharacteristic
369C22×C8C22×D4711yescharacteristic
370C22×C8C22×D4711yescharacteristic
371#125—942no—
372#125—942no—
373C4×Q8—22no—
374C4×Q8—22no—
375C4×D4—942no—
376C4×D4—942no—
377C2×C4⋊C4—471no—
378C2×C4⋊C4—471no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—471no—
381C2×C4⋊C4—471no—
382C2×C4⋊C4—471no—
383C2×C4⋊C4—471no—
384C2×C4⋊C4—941no—
385C2×C4⋊C4—941no—
386C2×C4⋊C4—941no—
387C2×C4⋊C4—941no—
388C2×C4⋊C4—941no—
389C2×C4⋊C4—941no—
390C2×C4⋊C4—941no—
391C2×C4⋊C4—941no—
392C2×C4⋊C4D18811yescharacteristic
393C2×C4⋊C4D18811yescharacteristic
394C2×C4⋊C4D18811yescharacteristic
395C2×C22⋊C4—941no—
396C2×C22⋊C4—941no—
397C2×C22⋊C4—941no—
398C2×C22⋊C4—941no—
399C2×C22⋊C4—941no—
400C2×C42—941no—
401C2×C42C22×D4711yescharacteristic
402C2×C42C22×D4711yescharacteristic
403C2×C42C22×D4711yescharacteristic
404#107—942no—
405#107—942no—
406#106—942no—
407#106—942no—
408C4⋊C8—22no—
409C4⋊C8—22no—
410C4⋊C8C22⋊D4714yes—
411#102—474no—
412#102—942no—
413#102—942no—
414#102—942no—
415C4×C8—22no—
416C4×C8—22no—
417C4×C8—22no—
418#95—941no—
419#95—941no—
420#95—941no—
421#95—941no—
422#95—941no—
423#95—941no—
424#95—941no—
425#95—941no—
426#95—941no—
427C24—941no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×D4—472no—
432C2×D4—941no—
433C2×D4—941no—
434C2×D4—941no—
435C2×D4—941no—
436C2×D4—941no—
437C22×C4—941no—
438C22×C4—941no—
439C22×C4—941no—
440C22×C4—941no—
441C22×C4—941no—
442C22×C4—941no—
443C22×C4—941no—
444C22×C4—3761no—
445C22×C4—3761no—
446C22×C4—1882no—
447C22×C4C2×D18811yescharacteristic
448C22×C4C2×D18811yescharacteristic
449C22×C4C2×D18811yescharacteristic
450C22×C4C2×D18811yescharacteristic
451C22×C4C2×D18811yescharacteristic
452C22×C4C2×D18811yescharacteristic
453C22×C4C23×D4711yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8Q8×D4712yes—
464C2×C8C2×C22⋊D4712yes—
465C2×C8C2×C22⋊D4712yes—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—472no—
470C4⋊C4—472no—
471C4⋊C4—941no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4—941no—
475C4⋊C4—941no—
476C4⋊C4—941no—
477C4⋊C4—941no—
478C4⋊C4—941no—
479C4⋊C4—941no—
480C4⋊C4—941no—
481C4⋊C4—941no—
482C4⋊C4—941no—
483C4⋊C4—3761no—
484C4⋊C4—3761no—
485C4⋊C4—3761no—
486C4⋊C4—3761no—
487C4⋊C4—1882no—
488C4⋊C4—1882no—
489C4⋊C4—1882no—
490C4⋊C4—1882no—
491C22⋊C4—3761no—
492C22⋊C4—3761no—
493C22⋊C4—1882no—
494C22⋊C4—1882no—
495C22⋊C4—1882no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—3761no—
501C42#1062535012yes—
502C42C2×C22⋊D4712yes—
503C23—942no—
504C23—1881no—
505C23—1881no—
506C23—1881no—
507C23—1881no—
508C23—1881no—
509C23—1881no—
510C23C22×D18811yescharacteristic
511Q8—28no—
512D4—942no—
513D4—942no—
514D4—1881no—
515D4—1881no—
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—942no—
531C2×C4—942no—
532C2×C4—3764no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4—1881no—
536C2×C4—1881no—
537C2×C4—1881no—
538C2×C4—1881no—
539C2×C4—1881no—
540C2×C4—1881no—
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—1881no—
551C2×C4—1881no—
552C2×C4#1309399812yes—
553C2×C4#1309399812yes—
554C2×C4#1309404612yes—
555C2×C4#1309404612yes—
556C2×C4C22×C22⋊D4711yescharacteristic
557C2×C4C2×Q8×D4711yescharacteristic
558C2×C4#41×D4711yescharacteristic
559C2×C4C2×#1062535011yescharacteristic
560C2×C4#1309407111yescharacteristic
561C2×C4#1309407211yescharacteristic
562C8—22no—
563C8—22no—
564C8—22no—
565C8—41no—
566C22—3761no—
567C22—3761no—
568C22—3761no—
569C22—3761no—
570C22—1882no—
571C22—1882no—
572C22—1882no—
573C22C2×#1309399811yescharacteristic
574C22#1568978511yescharacteristic
575C22Q8×D18811yescharacteristic
576C22C2×#1309404611yescharacteristic
577C22C2×#1309404611yescharacteristic
578C22#1568992811yescharacteristic
579C22#1568993511yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—3761no—
587C4—3761no—
588C4—3761no—
589C4—3761no—
590C4—3761no—
591C4—1882no—
592C4—1882no—
593C4#1568967711yescharacteristic
594C4#1568973111yescharacteristic
595C4#1568977211yescharacteristic
596C4Q8×C22⋊D4711yescharacteristic
597C2—3762no—
598C2#2024698111yescharacteristic
599C2#2025296911yescharacteristic
600C2#2025306811yescharacteristic
601C2#2025605011yescharacteristic
602C2#2025608611yescharacteristic
603C2#2025625611yescharacteristic
604C2#2025626511yescharacteristic
subgroupG/H
centerC23C22×D188
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C8×Q8×C47C2
socleC23×C47C22×D4
prime radicalC22×D188C23
hypercenterC2×C8×Q8D47
hypocenterC47#25473
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. #20248776C2semidirect
4. #20244886C2semidirect
5. #20244881C2semidirect
8. #20244353C2semidirect
9. #20244158C2semidirect
10. #20244121C2semidirect
11. #20244116C2semidirect
15. C2×C8×Q8×C47C2semidirect
95. #13093946Q8semidirect
296. C2×C8×Q8D47semidirect
363. C47#25473semidirect

Maximal subgroups

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

subgroupindex
C2×#156894242
#202487762
#202487462
#202448862
#202448812
#202447752
#202447612
#202443532
#202441582
#202441212
#202441162
#202440712
C4×#130939462
C2×C8×Q8×C472
#2547347