Finite Group Databasebeta

← #32005573

Subgroups — #32005573

Order 18176.

826 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986361C211yesmaximal, characteristic
2#22591051C211yesmaximal, characteristic
3#22589302C211yesmaximal, characteristic, complemented
4#22589149C211yesmaximal, characteristic
5#22589036C211yesmaximal, characteristic
6#22589029C211yesmaximal, characteristic, complemented
7#22589022C211yesmaximal, characteristic
8#22589015C211yesmaximal, characteristic, complemented
9#22588473C211yesmaximal, characteristic, complemented
10#22588173C211yesmaximal, characteristic, complemented
11#22588168C211yesmaximal, characteristic
12C4×#14556051C211yesmaximal, characteristic, complemented
13C4×#14556048C211yesmaximal, characteristic
14#22588027C211yesmaximal, characteristic, complemented
15C71×#2331C211yesmaximal, characteristic, complemented
16C2×#14556116C2211yescharacteristic
17C2×C4×Dic142C2211yescharacteristic
18C2×C4×D284C2211yescharacteristic
19#17986361C418yescomplemented
20C2×#14556051C2211yescharacteristic
21C2×#14556051C2211yescharacteristic
22C2×#14556051C2211yescharacteristic
23C2×#14556051C2211yescharacteristic
24C2×#14556048C2211yescharacteristic
25C2×#14556048C2211yescharacteristic
26C2×#14556048C2211yescharacteristic
27C2×#14556048C2211yescharacteristic
28C2×#14556039C2211yescharacteristic
29C2×#14556039C2211yescharacteristic
30C2×#14556039C2211yescharacteristic
31C2×C4×C71⋊C8C2211yescharacteristic
32C2×C4×C71⋊C8C2211yescharacteristic
33C2×C4×C71⋊C8C2211yescharacteristic
34#17985905C2211yescharacteristic
35#17985901C2211yescharacteristic
36#17985891—21no—
37#17985891—21no—
38#17985873—21no—
39#17985873—21no—
40#17985869C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985761C2211yescharacteristic
48#17985761C2211yescharacteristic
49C2×C4⋊Q8×C71C2211yescharacteristic
50#391×C71—21no—
51#391×C71—21no—
52#389×C71C2211yescharacteristic
53#389×C71C2211yescharacteristic
54#387×C71—21no—
55#387×C71—21no—
56#382×C71C2211yescharacteristic
57C4×C4⋊C4×C71C2211yescharacteristic
58C4×C4⋊C4×C71C2211yescharacteristic
59#376×C71C2211yescharacteristic
60C22×Dic142C2311yescharacteristic
61C22×D284C2311yescharacteristic
62C22×C4×D71—21no—
63C22×C71⋊C8C2311yescharacteristic
64C22×C71⋊C8C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67#14556116—22no—
68#14556116C2×C414yes—
69C4×Dic142—22no—
70C4×Dic142—22no—
71C4×D284—22no—
72C4×D284—22no—
73C2×C4⋊Dic71—21no—
74C2×C4⋊Dic71—21no—
75C2×C4⋊Dic71C2311yescharacteristic
76C2×C4⋊Dic71C2311yescharacteristic
77C2×Dic71⋊C4—21no—
78C2×(C2×D71)⋊C4—21no—
79C2×(C2×D71)⋊C4—21no—
80C2×(C2×D71)⋊C4—21no—
81C2×(C2×D71)⋊C4—21no—
82C2×(C2×D71)⋊C4—21no—
83C2×C4×Dic71—21no—
84#14556051—22no—
85#14556051—22no—
86#14556051C2×C414yescomplemented
87#14556051C2×C414yescomplemented
88#14556048—22no—
89#14556048—22no—
90#14556048C2×C414yes—
91#14556048C2×C414yes—
92#14556039—22no—
93#14556039—22no—
94#14556039C2×C414yescomplemented
95C4×C71⋊C8—22no—
96C4×C71⋊C8—22no—
97C4×C71⋊C8—22no—
98C71⋊#95—21no—
99C71⋊#95—21no—
100C22×Q8×C71—21no—
101C4⋊Q8×C71—22no—
102C4⋊Q8×C71C2×C414yescomplemented
103C2×C4⋊C4×C71—21no—
104C2×C4⋊C4×C71—21no—
105C2×C4⋊C4×C71—21no—
106C2×C4⋊C4×C71C2311yescharacteristic
107C2×C4⋊C4×C71C2311yescharacteristic
108C2×C4⋊C4×C71C2311yescharacteristic
109C2×C4⋊C4×C71C2311yescharacteristic
110C2×C42×C71—21no—
111C2×C42×C71—21no—
112C2×C42×C71C2311yescharacteristic
113C2×C42×C71C2311yescharacteristic
114C2×C42×C71C2311yescharacteristic
115#95×C71—21no—
116#95×C71—21no—
117#95×C71—21no—
118#95×C71—21no—
119#95×C71—21no—
120#95×C71—21no—
121#95×C71D411yescharacteristic
122#95×C71D411yescharacteristic
123#95×C71D411yescharacteristic
124#95×C71D411yescharacteristic
125C23×D71—21no—
126C2×Dic142—21no—
127C2×Dic142—21no—
128C2×Dic142—21no—
129C2×Dic142—21no—
130C2×Dic142—21no—
131C2×Dic142C22×C412yes—
132C2×D284—21no—
133C2×D284—21no—
134C2×D284—21no—
135C2×D284—21no—
136C2×D284—21no—
137C2×D284C22×C412yes—
138C22×Dic71—21no—
139C22×Dic71—21no—
140C22×Dic71—21no—
141C2×C4×D71—42no—
142C2×C4×D71—81no—
143C2×C4×D71—81no—
144C2×C71⋊C8—21no—
145C2×C71⋊C8—21no—
146C2×C71⋊C8—21no—
147C2×C71⋊C8—21no—
148C2×C71⋊C8—21no—
149C2×C71⋊C8—21no—
150C2×C71⋊C8—21no—
151C2×C71⋊C8—21no—
152C2×C71⋊C8C22×C412yes—
153C2×C71⋊C8C22×C412yes—
154C2×C71⋊C8#4112yes—
155C2×C71⋊C8#4112yes—
156C4⋊Dic71—21no—
157C4⋊Dic71—21no—
158C4⋊Dic71—21no—
159C4⋊Dic71—21no—
160C4⋊Dic71—81no—
161C4⋊Dic71—81no—
162Dic71⋊C4—42no—
163(C2×D71)⋊C4—24no—
164(C2×D71)⋊C4—24no—
165(C2×D71)⋊C4—42no—
166(C2×D71)⋊C4—81no—
167(C2×D71)⋊C4—81no—
168C4×Dic71—42no—
169C2×Q8×C71—24no—
170C2×Q8×C71—81no—
171C2×Q8×C71—81no—
172C22×C4×C71—21no—
173C22×C4×C71—21no—
174C22×C4×C71—21no—
175C22×C4×C71—21no—
176C22×C4×C71C2×D411yescharacteristic
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2×D411yescharacteristic
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2411yescharacteristic
183C4⋊C4×C71—24no—
184C4⋊C4×C71—21no—
185C4⋊C4×C71—21no—
186C4⋊C4×C71—21no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71—21no—
190C4⋊C4×C71—81no—
191C4⋊C4×C71—81no—
192C4⋊C4×C71C22×C412yes—
193C4⋊C4×C71C22×C412yes—
194C42×C71—21no—
195C42×C71—21no—
196C42×C71—81no—
197C42×C71—81no—
198C42×C71C22×C412yes—
199C42×C71#4112yes—
200C42×C71#4112yes—
201C42×C71#4112yes—
202C23×C71C22×D411yescharacteristic
203C22×D71—22no—
204C22×D71—41no—
205C22×D71—41no—
206C22×D71—41no—
207C22×D71—41no—
208C22×D71—41no—
209C22×D71—41no—
210Q8×C71—84no—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—21no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—22no—
220C2×C4×C71—22no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—21no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—41no—
228C2×C4×C71—41no—
229C2×C4×C71—41no—
230C2×C4×C71—41no—
231C2×C4×C71—41no—
232C2×C4×C71—41no—
233C2×C4×C71—41no—
234C2×C4×C71—41no—
235C2×C4×C71—41no—
236C2×C4×C71—41no—
237C2×C4×C71C4×D412yes—
238C2×C4×C71C4×D412yes—
239C2×C4×C71#12412yes—
240C2×C4×C71#12412yes—
241C2×C4×C71C23×C411yescharacteristic
242C2×C4×C71C2×#4111yescharacteristic
243C2×C4×C71C2×#4111yescharacteristic
244C2×C4×C71C2×#4111yescharacteristic
245C2×C4×C71C2×#4111yescharacteristic
246C2×C4×C71C2×#4111yescharacteristic
247C2×Dic71—22no—
248C2×Dic71—41no—
249C2×Dic71—41no—
250C2×Dic71—41no—
251C2×Dic71—41no—
252C2×Dic71—41no—
253C2×Dic71—41no—
254C2×Dic71—41no—
255C2×Dic71—41no—
256D284—22no—
257D284—22no—
258D284—41no—
259D284—41no—
260C4×D71—84no—
261Dic142—22no—
262Dic142—22no—
263Dic142—41no—
264Dic142—41no—
265C71⋊C8—22no—
266C71⋊C8—22no—
267C71⋊C8—22no—
268C71⋊C8—22no—
269C22×C71C2×C4×D411yescharacteristic
270C22×C71C2×#12411yescharacteristic
271C22×C71#53211yescharacteristic
272C22×C71#57611yescharacteristic
273C22×C71#57711yescharacteristic
274C22×C71#57711yescharacteristic
275C22×C71#57811yescharacteristic
276C2×D71—42no—
277C2×D71—42no—
278C2×D71—42no—
279C2×D71—81no—
280C2×D71—81no—
281C2×D71—81no—
282C2×D71—81no—
283C4×C71—22no—
284C4×C71—22no—
285C4×C71—22no—
286C4×C71—22no—
287C4×C71—41no—
288C4×C71—41no—
289C4×C71—42no—
290C4×C71—42no—
291C4×C71—81no—
292C4×C71—81no—
293C4×C71C4×#4111yescharacteristic
294C4×C71C4×#4111yescharacteristic
295C4×C71#52911yescharacteristic
296C4×C71#52911yescharacteristic
297Dic71—42no—
298Dic71—81no—
299Dic71—81no—
300#21654—711nomaximal, Sylow
301C2×C71C4×#12411yescharacteristic
302C2×C71#294611yescharacteristic
303C2×C71#294711yescharacteristic
304C2×C71#313611yescharacteristic
305C2×C71#313711yescharacteristic
306C2×C71#317311yescharacteristic
307C2×C71#317511yescharacteristic
308D71—82no—
309C2×#489—711no—
310#2331D7111yescharacteristic, complemented
311#2326—711no—
312#1990—711no—
313#1958—711no—
314#1936—711no—
315#1935—711no—
316#1934—711no—
317#1933—711no—
318#1836—711no—
319#1763—711no—
320#1762—711no—
321C4×#103—711no—
322C4×#102—711no—
323#1726—711no—
324C71#2165411yescharacteristic, complemented, Sylow
325C2×C4⋊Q8C2×D7111yescharacteristic
326C2×#124—711no—
327C2×C4×Q8—711no—
328C2×C4×D4—711no—
329#489—718no—
330C2×#103—711no—
331C2×#103—711no—
332C2×#103—711no—
333C2×#103—711no—
334C2×#102—711no—
335C2×#102—711no—
336C2×#102—711no—
337C2×#102—711no—
338C2×#97—711no—
339C2×#97—711no—
340C2×#97—711no—
341C2×C4×C8—711no—
342C2×C4×C8—711no—
343C2×C4×C8—711no—
344#391—21no—
345#391—21no—
346#391—711no—
347#390—711no—
348#389C2×D7111yescharacteristic
349#389C2×D7111yescharacteristic
350#388—1421no—
351#388—1421no—
352#387—21no—
353#387—21no—
354#385—1421no—
355#385—1421no—
356#384—711no—
357#382C2×D7111yescharacteristic
358C4×C4⋊C4C2×D7111yescharacteristic
359C4×C4⋊C4C2×D7111yescharacteristic
360#376C2×D7111yescharacteristic
361#340—711no—
362#340—711no—
363#340—711no—
364#340—711no—
365#340—711no—
366#340—711no—
367#340—711no—
368#340—711no—
369C22×Q8—21no—
370C22×Q8—711no—
371C22×D4—711no—
372C23×C4—1421no—
373C22×C8—711no—
374C22×C8—711no—
375C22×C8—711no—
376C22×C8—711no—
377C4⋊Q8—22no—
378C4⋊Q8C4×D7114yescomplemented
379#124—714no—
380#124—1422no—
381C4×Q8—1422no—
382C4×Q8—1422no—
383C4×D4—1422no—
384C4×D4—1422no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—711no—
389C2×C4⋊C4—711no—
390C2×C4⋊C4—1421no—
391C2×C4⋊C4—1421no—
392C2×C4⋊C4—1421no—
393C2×C4⋊C4C22×D7111yescharacteristic
394C2×C4⋊C4C22×D7111yescharacteristic
395C2×C4⋊C4C22×D7111yescharacteristic
396C2×C4⋊C4C22×D7111yescharacteristic
397C2×C22⋊C4—1421no—
398C2×C22⋊C4—1421no—
399C2×C22⋊C4—1421no—
400C2×C22⋊C4—1421no—
401C2×C22⋊C4—1421no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42—1421no—
405C2×C42C22×D7111yescharacteristic
406C2×C42C22×D7111yescharacteristic
407C2×C42C22×D7111yescharacteristic
408#103—714no—
409#103—714no—
410#103—1422no—
411#103—1422no—
412#102—714no—
413#102—714no—
414#102—1422no—
415#102—1422no—
416#97—714no—
417#97—1422no—
418#97—1422no—
419C4×C8—1422no—
420C4×C8—1422no—
421C4×C8—1422no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—1421no—
429#95—1421no—
430#95C22⋊D7111yescharacteristic
431#95C22⋊D7111yescharacteristic
432#95C22⋊D7111yescharacteristic
433#95C22⋊D7111yescharacteristic
434C24—1421no—
435C2×Q8—24no—
436C2×Q8—81no—
437C2×Q8—81no—
438C2×Q8—712no—
439C2×Q8—1421no—
440C2×Q8—1421no—
441C2×Q8—1421no—
442C2×Q8—1421no—
443C2×Q8—1421no—
444C2×D4—712no—
445C2×D4—1421no—
446C2×D4—1421no—
447C2×D4—1421no—
448C2×D4—1421no—
449C2×D4—1421no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—2842no—
455C22×C4—5681no—
456C22×C4—5681no—
457C22×C4—1421no—
458C22×C4—1421no—
459C22×C4—1421no—
460C22×C4C2×C22⋊D7111yescharacteristic
461C22×C4C2×C22⋊D7111yescharacteristic
462C22×C4C2×C22⋊D7111yescharacteristic
463C22×C4C2×C22⋊D7111yescharacteristic
464C22×C4C2×C22⋊D7111yescharacteristic
465C22×C4C2×C22⋊D7111yescharacteristic
466C22×C4C23×D7111yescharacteristic
467C2×C8—712no—
468C2×C8—712no—
469C2×C8—712no—
470C2×C8—712no—
471C2×C8—1421no—
472C2×C8—1421no—
473C2×C8—1421no—
474C2×C8—1421no—
475C2×C8—1421no—
476C2×C8—1421no—
477C2×C8—1421no—
478C2×C8—1421no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—81no—
487C4⋊C4—81no—
488C4⋊C4—2842no—
489C4⋊C4—5681no—
490C4⋊C4—5681no—
491C4⋊C4—1421no—
492C4⋊C4—1421no—
493C4⋊C4—1421no—
494C4⋊C4—1421no—
495C4⋊C4C2×C4×D7112yes—
496C4⋊C4C2×C4×D7112yes—
497C22⋊C4—2842no—
498C22⋊C4—5681no—
499C22⋊C4—5681no—
500C22⋊C4—1424no—
501C22⋊C4—1424no—
502C42—21no—
503C42—21no—
504C42—81no—
505C42—81no—
506C42—2842no—
507C42C2×C4×D7112yes—
508C42#1176586512yes—
509C42#1176586612yes—
510C42#1176586612yes—
511C23—2841no—
512C23—2841no—
513C23—2841no—
514C23—2841no—
515C23—2841no—
516C23—2841no—
517C23—1422no—
518C23C22×C22⋊D7111yescharacteristic
519Q8—84no—
520Q8—2841no—
521Q8—2841no—
522Q8—1422no—
523Q8—1422no—
524D4—2841no—
525D4—2841no—
526D4—1422no—
527D4—1422no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—2841no—
555C2×C4—2841no—
556C2×C4—2841no—
557C2×C4—2841no—
558C2×C4—2841no—
559C2×C4—2841no—
560C2×C4—2841no—
561C2×C4—2841no—
562C2×C4—5684no—
563C2×C4—1422no—
564C2×C4C4×C22⋊D7112yes—
565C2×C4C4×C22⋊D7112yes—
566C2×C4#1455611712yes—
567C2×C4#1455611712yes—
568C2×C4C22×C4×D7111yescharacteristic
569C2×C4#41×D7111yescharacteristic
570C2×C4#41×D7111yescharacteristic
571C2×C4C2×#1176586511yescharacteristic
572C2×C4C2×#1176586611yescharacteristic
573C2×C4C2×#1176586611yescharacteristic
574C8—1422no—
575C8—1422no—
576C8—1422no—
577C8—1422no—
578C22—2842no—
579C22—2842no—
580C22—2842no—
581C22—5681no—
582C22—5681no—
583C22—5681no—
584C22—5681no—
585C22C2×C4×C22⋊D7111yescharacteristic
586C22C2×#1455611711yescharacteristic
587C22#1798654511yescharacteristic
588C22#1798680211yescharacteristic
589C22#1798680711yescharacteristic
590C22#1798681011yescharacteristic
591C22#1798681411yescharacteristic
592C4—22no—
593C4—22no—
594C4—22no—
595C4—22no—
596C4—41no—
597C4—42no—
598C4—42no—
599C4—41no—
600C4—81no—
601C4—81no—
602C4—2842no—
603C4—5681no—
604C4—5681no—
605C4C4×#1176586611yescharacteristic
606C4C4×#1176586611yescharacteristic
607C4#1798653611yescharacteristic
608C4#1798653611yescharacteristic
609C2—5682no—
610C2C4×#1455611711yescharacteristic
611C2#2259717211yescharacteristic
612C2#2259718311yescharacteristic
613C2#2259869911yescharacteristic
614C2#2259871011yescharacteristic
615C2#2259895111yescharacteristic
616C2#2259897311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2331C2
socleC23×C71C22×D4
prime radicalC22×D284C23
cosocleC22×C4×C71C24
hypercenter#2331D71
hypocenterC71#21654
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ #376 ◁ #2331

Complements

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

NH ≅ G/Ntype
3. #22589302C2semidirect
6. #22589029C2semidirect
8. #22589015C2semidirect
9. #22588473C2semidirect
10. #22588173C2semidirect
12. C4×#14556051C2semidirect
14. #22588027C2semidirect
15. C71×#2331C2semidirect
19. #17986361C4semidirect
86. #14556051C2×C4semidirect
87. #14556051C2×C4semidirect
94. #14556039C2×C4semidirect
102. C4⋊Q8×C71C2×C4semidirect
310. #2331D71semidirect
324. C71#21654semidirect
378. C4⋊Q8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179863612
#225910512
#225893022
#225891492
#225890362
#225890292
#225890222
#225890152
#225884732
#225881732
#225881682
C4×#145560512
C4×#145560482
#225880272
C71×#23312
#2165471