Finite Group Databasebeta

← #31967516

Subgroups — #31967516

Order 18176.

648 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
1#22598389C211yesmaximal, characteristic
2#22597491C211yesmaximal, characteristic, complemented
3#22597363C211yesmaximal, characteristic, complemented
4#22587206C211yesmaximal, characteristic, complemented
5#22587199C211yesmaximal, characteristic, complemented
6#22587178C211yesmaximal, characteristic, complemented
7#22587171C211yesmaximal, characteristic, complemented
8#22587165C211yesmaximal, characteristic, complemented
9#22587154C211yesmaximal, characteristic, complemented
10C8×#11765854C211yesmaximal, characteristic, complemented
11C8×D568C211yesmaximal, characteristic, complemented
12#22587129C211yesmaximal, characteristic, complemented
13#22586766C211yesmaximal, characteristic, complemented
14#22586749C211yesmaximal, characteristic, complemented
15C71×#1454C211yesmaximal, characteristic, complemented
16#17986533C2211yescharacteristic
17C2×C4×D284C2211yescharacteristic
18#17986335C411yescharacteristic
19#17986318C411yescharacteristic
20#17986300C411yescharacteristic
21#17986300C411yescharacteristic
22#17986215C411yescharacteristic
23#17986215C411yescharacteristic
24#17986152C411yescharacteristic
25#17986135C411yescharacteristic
26#17986129C2211yescharacteristic, complemented
27C8×Dic142C2211yescharacteristic, complemented
28C4×#11765854C2211yescharacteristic
29C4×#11765854C2211yescharacteristic
30C4×D568C2211yescharacteristic
31C4×D568C2211yescharacteristic
32#17986078C2211yescharacteristic, complemented
33#17986074—21no—
34#17986073—21no—
35C8×D284C2211yescharacteristic
36C8×D284C2211yescharacteristic
37C8×D284C2211yescharacteristic
38C8×D284C2211yescharacteristic
39C8×D284C2211yescharacteristic, complemented
40#17986064—21no—
41#17986064—21no—
42#17986059—21no—
43#17986052C2211yescharacteristic
44#17986024—21no—
45#17986023C2211yescharacteristic
46C22⋊C8×D71—21no—
47C22⋊C8×D71—21no—
48#17985752C2211yescharacteristic, complemented
49#17985752C2211yescharacteristic, complemented
50#17985750C2211yescharacteristic, complemented
51#17985750C2211yescharacteristic, complemented
52#17985731C2211yescharacteristic, complemented
53#17985731C2211yescharacteristic, complemented
54#17985728C2211yescharacteristic
55#17985728C2211yescharacteristic
56#17985728C2211yescharacteristic
57#17985728C2211yescharacteristic
58#17985728C2211yescharacteristic, complemented
59#17985728C2211yescharacteristic, complemented
60#431×C71C2211yescharacteristic
61#431×C71C2211yescharacteristic
62#431×C71C2211yescharacteristic
63#322×C71C2211yescharacteristic, complemented
64#322×C71C2211yescharacteristic, complemented
65#322×C71C2211yescharacteristic, complemented
66C82×C71C2211yescharacteristic, complemented
67C22×D284C2×C411yescharacteristic
68C22×C4×D71—21no—
69C2×#11765854C2×C411yescharacteristic
70C2×#11765854C2×C411yescharacteristic
71C2×D568C2×C411yescharacteristic
72C2×D568C2×C411yescharacteristic
73C2×#11765849—21no—
74C2×#11765848—21no—
75C2×C8×D71—21no—
76C2×C8×D71—21no—
77C2×C8×D71—21no—
78C2×C8×D71—21no—
79C2×C8×D71—21no—
80#14556122—21no—
81#14556120C2×C411yescharacteristic
82#14556116C2×C411yescharacteristic
83#14556111—21no—
84#14556107C2×C411yescharacteristic
85#14556102C2×C411yescharacteristic
86C4×Dic142C2311yescharacteristic
87C4×D284—21no—
88C4×D284—21no—
89C4×D284C2×C411yescharacteristic
90C4×D284C2×C411yescharacteristic
91C4×D284C2311yescharacteristic
92C4×D284C2311yescharacteristic
93C4×D284C2311yescharacteristic
94C4×C22⋊D71—21no—
95#14556085—21no—
96C2×C4⋊Dic71C2×C411yescharacteristic
97C2×(C2×D71)⋊C4—21no—
98#14556062C2×C411yescharacteristic
99#14556062C2×C411yescharacteristic
100#14556060C2×C411yescharacteristic
101#14556060C2×C411yescharacteristic
102C4⋊(C71⋊C8)—21no—
103C4⋊(C71⋊C8)C2311yescharacteristic
104C4⋊(C71⋊C8)C2311yescharacteristic
105C4⋊(C71⋊C8)C2311yescharacteristic
106C4⋊(C71⋊C8)C2311yescharacteristic
107Dic71⋊C8—21no—
108Dic71⋊C8—21no—
109#14556053C2×C411yescharacteristic
110#14556053C2×C411yescharacteristic
111#14556050C2×C411yescharacteristic
112#14556050C2×C411yescharacteristic
113#14556050C2×C411yescharacteristic
114#14556050C2×C411yescharacteristic
115#14556050C2×C411yescharacteristic
116#14556050C2×C411yescharacteristic
117C22⋊(C71⋊C8)—21no—
118(C2×D71)⋊C8—21no—
119(C2×D71)⋊C8—21no—
120(C2×D71)⋊C8—21no—
121(C2×D71)⋊C8—21no—
122(C2×D71)⋊C8—21no—
123(C2×D71)⋊C8—21no—
124#14556038—21no—
125C4×C71⋊C8—21no—
126C8×Dic71—21no—
127C2×C42×C71C2311yescharacteristic
128C4⋊C8×C71—21no—
129C4⋊C8×C71C2×C411yescharacteristic
130C4⋊C8×C71C2×C411yescharacteristic
131C4⋊C8×C71D411yescharacteristic
132C4⋊C8×C71D411yescharacteristic
133C22⋊C8×C71—21no—
134C22⋊C8×C71C2×C411yescharacteristic
135C22⋊C8×C71C2×C411yescharacteristic
136C22⋊C8×C71D411yescharacteristic
137C22⋊C8×C71D411yescharacteristic
138C4×C8×C71C2311yescharacteristic
139C4×C8×C71C2311yescharacteristic
140C4×C8×C71C2311yescharacteristic
141C4×C8×C71C2311yescharacteristic
142C4×C8×C71C2311yescharacteristic
143C4×C8×C71C2311yescharacteristic
144C23×D71—21no—
145C2×Dic142C22×C411yescharacteristic
146C2×D284—21no—
147C2×D284—22no—
148C2×D284—22no—
149C2×D284C22×C411yescharacteristic
150C2×D284C22×C411yescharacteristic
151C2×D284C22×C411yescharacteristic
152C2×C22⋊D71—21no—
153C22×Dic71—21no—
154C2×C4×D71—21no—
155C2×C4×D71—21no—
156C2×C4×D71—21no—
157C2×C4×D71—21no—
158C2×C4×D71—21no—
159C2×C4×D71—81no—
160C2×C4×D71—81no—
161#11765854—22no—
162#11765854—22no—
163D568—22no—
164D568—22no—
165#11765849—81no—
166#11765848—81no—
167C2×C71⋊C8—21no—
168C2×C71⋊C8—21no—
169C2×C71⋊C8—21no—
170C2×C71⋊C8—21no—
171C8×D71—81no—
172C8×D71—81no—
173C8×D71—81no—
174C8×D71—81no—
175C8×D71—81no—
176C4⋊Dic71—21no—
177C4⋊Dic71C22×C411yescharacteristic
178C4⋊Dic71C22×C411yescharacteristic
179C4⋊Dic71C22×C411yescharacteristic
180C4⋊Dic71C22×C411yescharacteristic
181Dic71⋊C4—21no—
182Dic71⋊C4—21no—
183Dic71⋊C4—21no—
184Dic71⋊C4—21no—
185C22⋊Dic71—21no—
186(C2×D71)⋊C4—21no—
187(C2×D71)⋊C4—21no—
188(C2×D71)⋊C4—21no—
189(C2×D71)⋊C4—21no—
190(C2×D71)⋊C4—21no—
191(C2×D71)⋊C4—21no—
192(C2×D71)⋊C4—21no—
193(C2×D71)⋊C4—21no—
194C4×Dic71—21no—
195C22×C4×C71C22×C411yescharacteristic
196C22×C4×C71C2×D411yescharacteristic
197C22×C4×C71#4111yescharacteristic
198C2×C8×C71—21no—
199C2×C8×C71—21no—
200C2×C8×C71C22×C411yescharacteristic
201C2×C8×C71C22×C411yescharacteristic
202C2×C8×C71C22×C411yescharacteristic
203C2×C8×C71C22×C411yescharacteristic
204C2×C8×C71C2×D411yescharacteristic
205C2×C8×C71C2×D411yescharacteristic
206C2×C8×C71C2×D411yescharacteristic
207C2×C8×C71C2×D411yescharacteristic
208C42×C71C22×C411yescharacteristic
209C42×C71C2×D411yescharacteristic
210C42×C71#4111yescharacteristic
211C42×C71C2411yescharacteristic
212C23×C71C4×D411yescharacteristic
213C22×D71—21no—
214C22×D71—21no—
215C22×D71—24no—
216C22×D71—21no—
217C22×D71—81no—
218C22×D71—81no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—41no—
224C2×C4×C71C4×D411yescharacteristic
225C2×C4×C71C4×D411yescharacteristic
226C2×C4×C71C4×D411yescharacteristic
227C2×C4×C71C23×C411yescharacteristic
228C2×C4×C71C22×D411yescharacteristic
229C2×C4×C71C2×#4111yescharacteristic
230C22⋊D71—81no—
231C2×Dic71—21no—
232C2×Dic71—21no—
233C2×Dic71—21no—
234C2×Dic71—21no—
235C2×Dic71—21no—
236C2×Dic71—81no—
237D284—24no—
238D284—21no—
239D284—21no—
240D284—21no—
241D284—21no—
242D284#13112yes—
243D284#13112yes—
244C4×D71—41no—
245C4×D71—41no—
246C4×D71—41no—
247C4×D71—41no—
248C4×D71—41no—
249C4×D71—41no—
250C4×D71—81no—
251C4×D71—81no—
252Dic142—21no—
253Dic142—21no—
254C8×C71—21no—
255C8×C71—21no—
256C8×C71—21no—
257C8×C71—21no—
258C8×C71—21no—
259C8×C71—21no—
260C8×C71—21no—
261C8×C71—21no—
262C8×C71—41no—
263C8×C71—41no—
264C71⋊C8—41no—
265C71⋊C8—41no—
266C71⋊C8—41no—
267C71⋊C8—41no—
268C22×C71—21no—
269C22×C71—21no—
270C22×C71—21no—
271C22×C71C2×C4×D411yescharacteristic
272C2×D71—22no—
273C2×D71—22no—
274C2×D71—41no—
275C2×D71—41no—
276C2×D71—41no—
277C2×D71—41no—
278C2×D71—41no—
279C2×D71—41no—
280C2×D71—41no—
281C2×D71—84no—
282C4×C71—21no—
283C4×C71—21no—
284C4×C71—21no—
285C4×C71—21no—
286C4×C71C2×#13111yescharacteristic
287C4×C71C2×#13111yescharacteristic
288C4×C71#57611yescharacteristic
289C4×C71#57711yescharacteristic
290Dic71—41no—
291Dic71—41no—
292Dic71—41no—
293Dic71—41no—
294Dic71—41no—
295#17918—711nomaximal, Sylow
296C2×C71—41no—
297C2×C71#292911yescharacteristic
298C2×C71#294611yescharacteristic
299C2×C71#295711yescharacteristic
300D71—42no—
301D71—42no—
302D71—81no—
303#3105—711no—
304#2993—711no—
305#2972—711no—
306#1586—711no—
307#1585—711no—
308#1582—711no—
309#1581—711no—
310#1580—711no—
311SD16⋊C8—711no—
312C8×SD16—711no—
313C8×D8—711no—
314#1574—711no—
315#1486—711no—
316#1482—711no—
317#1454D7111yescharacteristic, complemented
318C71#1791811yescharacteristic, complemented, Sylow
319#529—711no—
320C2×C4×D4—711no—
321#484—711no—
322#481—711no—
323#478—711no—
324#478—711no—
325#463—711no—
326#463—711no—
327#450—711no—
328#447—711no—
329#446—711no—
330C8×Q8—711no—
331C4×SD16—711no—
332C4×SD16—711no—
333C4×D8—711no—
334C4×D8—711no—
335#436—711no—
336#436—1421no—
337#435—1421no—
338C8×D4—711no—
339C8×D4—711no—
340C8×D4—711no—
341C8×D4—711no—
342C8×D4—711no—
343C8×D4—1421no—
344C8×D4—1421no—
345#433—1421no—
346#431—711no—
347#431C2×D7111yescharacteristic
348#431C2×D7111yescharacteristic
349#431C2×D7111yescharacteristic
350#424—1421no—
351#423—711no—
352C2×C22⋊C8—1421no—
353C2×C22⋊C8—1421no—
354#335—711no—
355#335—711no—
356#334—711no—
357#334—711no—
358Q8⋊C8—711no—
359Q8⋊C8—711no—
360D4⋊C8—711no—
361D4⋊C8—711no—
362D4⋊C8—711no—
363D4⋊C8—711no—
364D4⋊C8—711no—
365D4⋊C8—711no—
366#322C2×D7111yescharacteristic, complemented
367#322C2×D7111yescharacteristic, complemented
368#322C2×D7111yescharacteristic, complemented
369C82C2×D7111yescharacteristic, complemented
370C22×D4—711no—
371C23×C4—1421no—
372C2×SD16—711no—
373C2×SD16—711no—
374C2×D8—711no—
375C2×D8—711no—
376C2×M16—1421no—
377C2×M16—1421no—
378C22×C8—1421no—
379C22×C8—1421no—
380C22×C8—1421no—
381C22×C8—1421no—
382C22×C8—1421no—
383#126—1421no—
384#125—711no—
385#124—711no—
386#123—1421no—
387C22⋊Q8—711no—
388#121—711no—
389C4×Q8—711no—
390C4×D4—711no—
391C4×D4—711no—
392C4×D4—711no—
393C4×D4—711no—
394C4×D4—711no—
395C4×D4—1421no—
396C4×D4—1421no—
397C4×D4—1421no—
398#117—1421no—
399C2×C4⋊C4—711no—
400C2×C22⋊C4—1421no—
401C2×C42C22×D7111yescharacteristic
402#107—711no—
403#107—711no—
404#106—711no—
405#106—711no—
406C4⋊C8—21no—
407C4⋊C8—711no—
408C4⋊C8—711no—
409C4⋊C8—711no—
410C4⋊C8—711no—
411C4⋊C8—1421no—
412C4⋊C8—1421no—
413C4⋊C8—1421no—
414C4⋊C8C4×D7111yescharacteristic
415C4⋊C8C4×D7111yescharacteristic
416C4⋊C8D28411yescharacteristic
417C4⋊C8D28411yescharacteristic
418#103—711no—
419#103—711no—
420#102—711no—
421#102—711no—
422#102—711no—
423#102—711no—
424#102—711no—
425#102—711no—
426C22⋊C8—21no—
427C22⋊C8—1421no—
428C22⋊C8—1421no—
429C22⋊C8—1421no—
430C22⋊C8—1421no—
431C22⋊C8—1421no—
432C22⋊C8—1421no—
433C22⋊C8—1421no—
434C22⋊C8C4×D7111yescharacteristic
435C22⋊C8C4×D7111yescharacteristic
436C22⋊C8D28411yescharacteristic
437C22⋊C8D28411yescharacteristic
438#97—1421no—
439C4×C8—1421no—
440C4×C8—1421no—
441C4×C8C22×D7111yescharacteristic
442C4×C8C22×D7111yescharacteristic
443C4×C8C22×D7111yescharacteristic
444C4×C8C22×D7111yescharacteristic
445C4×C8C22×D7111yescharacteristic
446C4×C8C22×D7111yescharacteristic
447C24—1421no—
448C2×Q8—711no—
449C2×D4—711no—
450C2×D4—711no—
451C2×D4—711no—
452C2×D4—1422no—
453C2×D4—1422no—
454C2×D4—1421no—
455C2×D4—1421no—
456C22×C4—5681no—
457C22×C4—5681no—
458C22×C4—1421no—
459C22×C4—1421no—
460C22×C4—1421no—
461C22×C4—1421no—
462C22×C4—1421no—
463C22×C4—1421no—
464C22×C4C2×C4×D7111yescharacteristic
465C22×C4C2×D28411yescharacteristic
466C22×C4#1176586511yescharacteristic
467SD16—1422no—
468SD16—1422no—
469D8—1422no—
470D8—1422no—
471M16—5681no—
472M16—5681no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—5681no—
476C2×C8—5681no—
477C2×C8—5681no—
478C2×C8—5681no—
479C2×C8—5681no—
480C2×C8—1421no—
481C2×C8—1421no—
482C2×C8—1421no—
483C2×C8—1421no—
484C2×C8C2×C4×D7111yescharacteristic
485C2×C8C2×C4×D7111yescharacteristic
486C2×C8C2×C4×D7111yescharacteristic
487C2×C8C2×C4×D7111yescharacteristic
488C2×C8C2×D28411yescharacteristic
489C2×C8C2×D28411yescharacteristic
490C2×C8C2×D28411yescharacteristic
491C2×C8C2×D28411yescharacteristic
492C4⋊C4—711no—
493C4⋊C4—711no—
494C4⋊C4—711no—
495C4⋊C4—711no—
496C4⋊C4—1421no—
497C4⋊C4—1421no—
498C4⋊C4—1421no—
499C4⋊C4—1421no—
500C4⋊C4—1421no—
501C22⋊C4—1421no—
502C22⋊C4—1421no—
503C22⋊C4—1421no—
504C22⋊C4—1421no—
505C22⋊C4—1421no—
506C22⋊C4—1421no—
507C22⋊C4—1421no—
508C22⋊C4—1421no—
509C22⋊C4—1421no—
510C42—1421no—
511C42C2×C4×D7111yescharacteristic
512C42C2×D28411yescharacteristic
513C42#1176586511yescharacteristic
514C42C23×D7111yescharacteristic
515C23—5681no—
516C23—5681no—
517C23—1424no—
518C23—1421no—
519C23—1421no—
520C23—1421no—
521C23C4×D28411yescharacteristic
522Q8—1421no—
523Q8—1421no—
524D4—5681no—
525D4—712no—
526D4—712no—
527D4—1424no—
528D4—1421no—
529D4—1421no—
530D4—1421no—
531D4—1421no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—2841no—
538C2×C4—2841no—
539C2×C4—2841no—
540C2×C4—2841no—
541C2×C4—2841no—
542C2×C4—2841no—
543C2×C4—5681no—
544C2×C4—5681no—
545C2×C4—5681no—
546C2×C4—1421no—
547C2×C4—1421no—
548C2×C4—1421no—
549C2×C4—1421no—
550C2×C4—1421no—
551C2×C4C4×D28411yescharacteristic
552C2×C4C4×D28411yescharacteristic
553C2×C4C4×D28411yescharacteristic
554C2×C4C22×C4×D7111yescharacteristic
555C2×C4C22×D28411yescharacteristic
556C2×C4C2×#1176586511yescharacteristic
557C8—21no—
558C8—21no—
559C8—21no—
560C8—21no—
561C8—21no—
562C8—21no—
563C8—21no—
564C8—21no—
565C8—41no—
566C8—41no—
567C8—2841no—
568C8—2841no—
569C8—2841no—
570C8—2841no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—2841no—
575C22—2841no—
576C22—2841no—
577C22—2841no—
578C22—2841no—
579C22—2841no—
580C22—2841no—
581C22—5684no—
582C22—1422no—
583C22—1422no—
584C22C2×C4×D28411yescharacteristic
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—2841no—
590C4—2841no—
591C4—2841no—
592C4—2841no—
593C4—2841no—
594C4#131×D7111yescharacteristic
595C4#131×D7111yescharacteristic
596C4#1798680411yescharacteristic
597C4#1798681111yescharacteristic
598C2—41no—
599C2—2842no—
600C2—2842no—
601C2—5681no—
602C2#2259703611yescharacteristic
603C2#2259717411yescharacteristic
604C2#2259725811yescharacteristic
subgroupG/H
centerC2×C4C22×D284
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#1454C2
socleC22×C71C2×C4×D4
prime radical#17986135C4
cosocleC42×C71C24
hypercenter#1454D71
hypocenterC71#17918
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1454

Complements

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

NH ≅ G/Ntype
2. #22597491C2semidirect
3. #22597363C2semidirect
4. #22587206C2semidirect
5. #22587199C2semidirect
6. #22587178C2semidirect
7. #22587171C2semidirect
8. #22587165C2semidirect
9. #22587154C2semidirect
10. C8×#11765854C2semidirect
11. C8×D568C2semidirect
12. #22587129C2semidirect
13. #22586766C2semidirect
14. #22586749C2semidirect
15. C71×#1454C2semidirect
26. #17986129C22semidirect
27. C8×Dic142C22semidirect
32. #17986078C22semidirect
39. C8×D284C22semidirect
48. #17985752C22semidirect
49. #17985752C22semidirect
50. #17985750C22semidirect
51. #17985750C22semidirect
52. #17985731C22semidirect
53. #17985731C22semidirect
58. #17985728C22semidirect
59. #17985728C22semidirect
63. #322×C71C22semidirect
64. #322×C71C22semidirect
65. #322×C71C22semidirect
66. C82×C71C22semidirect
317. #1454D71semidirect
318. C71#17918semidirect
366. #322C2×D71semidirect
367. #322C2×D71semidirect
368. #322C2×D71semidirect
369. C82C2×D71semidirect

Maximal subgroups

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

subgroupindex
#225983892
#225974912
#225973632
#225872062
#225871992
#225871782
#225871712
#225871652
#225871542
C8×#117658542
C8×D5682
#225871292
#225867662
#225867492
C71×#14542
#1791871