Finite Group Databasebeta

← #31925954

Subgroups — #31925954

Order 18176.

1810 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556107C211yesmaximal, characteristic
2C2×#17985903C212yesmaximal, complemented
3#22586609C212yesmaximal, complemented
4#22586591C212yesmaximal, complemented
5#22586583C212yesmaximal
6#22586577C212yesmaximal
7#22586571C212yesmaximal, complemented
8#22586563C211yesmaximal, characteristic
9C71×C22⋊#95C211yesmaximal, characteristic
10C23×Dic142C2211yescharacteristic
11C2×#14556107—28no—
12C2×#14556107C418yescomplemented
13C22×C4⋊Dic71C2211yescharacteristic
14C22×Dic71⋊C4C2212yes—
15C22×Dic71⋊C4C2212yes—
16C22×C22⋊Dic71C2212yes—
17C22×C22⋊Dic71C2212yes—
18C22×C4×Dic71C2212yes—
19#17985903—28no—
20C2×C71⋊#95C2212yes—
21C2×C71⋊#95C2212yes—
22C2×C71⋊#95C2212yes—
23C2×C71⋊#95C2212yes—
24C2×C71⋊#95C2212yes—
25C2×C71⋊#95C2211yescharacteristic
26C2×C71⋊#95C2211yescharacteristic
27C2×C71⋊#95C2212yes—
28C2×C71⋊#95C2212yes—
29C24×C4×C71C2211yescharacteristic
30C22×C22⋊C4×C71C2211yescharacteristic
31C22×C22⋊C4×C71C2211yescharacteristic
32C2×#95×C71C2212yes—
33C2×#95×C71C2212yes—
34C22×Dic142—41no—
35C22×Dic142—42no—
36C22×Dic142—41no—
37C22×Dic142C2×C414yes—
38C22×Dic142D418yes—
39C23×Dic71C2312yes—
40C23×Dic71C2312yes—
41C23×Dic71C2312yes—
42C23×Dic71C2312yes—
43#14556107—232no—
44C2×C4⋊Dic71—24no—
45C2×C4⋊Dic71C2×C414yes—
46C2×Dic71⋊C4—24no—
47C2×Dic71⋊C4—24no—
48C2×Dic71⋊C4—24no—
49C2×Dic71⋊C4—42no—
50C2×Dic71⋊C4—42no—
51C2×Dic71⋊C4C2×C418yes—
52C2×C22⋊Dic71—24no—
53C2×C22⋊Dic71—24no—
54C2×C22⋊Dic71—24no—
55C2×C22⋊Dic71—24no—
56C2×C22⋊Dic71—42no—
57C2×C22⋊Dic71C2×C418yes—
58C2×C4×Dic71—24no—
59C2×C4×Dic71—42no—
60C2×C4×Dic71—42no—
61C71⋊#95—24no—
62C71⋊#95—24no—
63C71⋊#95—24no—
64C71⋊#95—24no—
65C71⋊#95—24no—
66C71⋊#95—42no—
67C71⋊#95—41no—
68C71⋊#95—41no—
69C71⋊#95—42no—
70C25×C71C2311yescharacteristic
71C23×C4×C71—24no—
72C23×C4×C71—42no—
73C23×C4×C71—41no—
74C23×C4×C71—41no—
75C23×C4×C71C2×C414yes—
76C23×C4×C71C2312yes—
77C23×C4×C71C2311yescharacteristic
78C23×C4×C71C2312yes—
79C23×C4×C71C2311yescharacteristic
80C2×C22⋊C4×C71—22no—
81C2×C22⋊C4×C71—41no—
82C2×C22⋊C4×C71—41no—
83C2×C22⋊C4×C71—41no—
84C2×C22⋊C4×C71—41no—
85C2×C22⋊C4×C71Q814yescomplemented
86#95×C71—24no—
87#95×C71—42no—
88C2×Dic142—44no—
89C2×Dic142—44no—
90C2×Dic142—48no—
91C2×Dic142—48no—
92C2×Dic142C22⋊C4116yescomplemented
93C22×Dic71—22no—
94C22×Dic71—22no—
95C22×Dic71—22no—
96C22×Dic71—22no—
97C22×Dic71—22no—
98C22×Dic71—22no—
99C22×Dic71—22no—
100C22×Dic71—22no—
101C22×Dic71—22no—
102C22×Dic71—22no—
103C22×Dic71—22no—
104C22×Dic71—22no—
105C22×Dic71—22no—
106C22×Dic71—22no—
107C22×Dic71—22no—
108C22×Dic71—22no—
109C22×Dic71—22no—
110C22×Dic71—22no—
111C22×Dic71—22no—
112C22×Dic71—22no—
113C22×Dic71—22no—
114C22×Dic71—22no—
115C22×Dic71—22no—
116C22×Dic71C22×C414yes—
117C22×Dic71C22×C414yes—
118C22×Dic71C2×D414yes—
119C22×Dic71C2×D414yes—
120C22×Dic71#4114yes—
121C4⋊Dic71—28no—
122Dic71⋊C4—216no—
123Dic71⋊C4—48no—
124C22⋊Dic71—216no—
125C22⋊Dic71—48no—
126C4×Dic71—48no—
127C24×C71—22no—
128C24×C71—22no—
129C24×C71—21no—
130C24×C71—21no—
131C24×C71—21no—
132C24×C71—22no—
133C24×C71—22no—
134C24×C71—21no—
135C24×C71C22×C412yes—
136C24×C71C2×Q812yes—
137C24×C71#4112yes—
138C24×C71C2411yescharacteristic
139C22×C4×C71—21no—
140C22×C4×C71—21no—
141C22×C4×C71—22no—
142C22×C4×C71—22no—
143C22×C4×C71—22no—
144C22×C4×C71—22no—
145C22×C4×C71—22no—
146C22×C4×C71—22no—
147C22×C4×C71—22no—
148C22×C4×C71—22no—
149C22×C4×C71—21no—
150C22×C4×C71—21no—
151C22×C4×C71—22no—
152C22×C4×C71—28no—
153C22×C4×C71—22no—
154C22×C4×C71—22no—
155C22×C4×C71—22no—
156C22×C4×C71—22no—
157C22×C4×C71—22no—
158C22×C4×C71—22no—
159C22×C4×C71—22no—
160C22×C4×C71—44no—
161C22×C4×C71—44no—
162C22×C4×C71—48no—
163C22×C4×C71—48no—
164C22×C4×C71C22×C412yes—
165C22×C4×C71C22×C412yes—
166C22×C4×C71C2×D414yes—
167C22×C4×C71C2×Q814yes—
168C22⋊C4×C71—44no—
169C22⋊C4×C71—44no—
170C23×C71—22no—
171C23×C71—22no—
172C23×C71—22no—
173C23×C71—22no—
174C23×C71—22no—
175C23×C71—22no—
176C23×C71—22no—
177C23×C71—22no—
178C23×C71—22no—
179C23×C71—22no—
180C23×C71—24no—
181C23×C71—24no—
182C23×C71—24no—
183C23×C71—24no—
184C23×C71—42no—
185C23×C71—42no—
186C23×C71—42no—
187C23×C71—42no—
188C23×C71—42no—
189C23×C71—42no—
190C23×C71—42no—
191C23×C71—42no—
192C23×C71C4×Q814yes—
193C23×C71C23×C411yescharacteristic
194C23×C71C22×D412yes—
195C23×C71C22×Q811yescharacteristic
196C23×C71C2×#4112yes—
197C23×C71C2×#4111yescharacteristic
198C23×C71H(5,𝔽2)11yescharacteristic
199C23×C71H(5,𝔽2)11yescharacteristic
200C23×C71H(5,𝔽2)12yes—
201C23×C71H(5,𝔽2)11yescharacteristic
202C23×C712−1+412yes—
203C23×C712−1+411yescharacteristic
204C2×C4×C71—22no—
205C2×C4×C71—22no—
206C2×C4×C71—24no—
207C2×C4×C71—24no—
208C2×C4×C71—22no—
209C2×C4×C71—22no—
210C2×C4×C71—22no—
211C2×C4×C71—24no—
212C2×C4×C71—24no—
213C2×C4×C71—24no—
214C2×C4×C71—416no—
215C2×C4×C71—42no—
216C2×C4×C71—42no—
217C2×C4×C71—42no—
218C2×C4×C71—42no—
219C2×C4×C71—42no—
220C2×C4×C71—42no—
221C2×C4×C71—42no—
222C2×C4×C71—42no—
223C2×C4×C71—42no—
224C2×C4×C71—42no—
225C2×C4×C71—42no—
226C2×C4×C71—42no—
227C2×C4×C71—42no—
228C2×C4×C71C2×C22⋊C414yes—
229C2×Dic71—24no—
230C2×Dic71—24no—
231C2×Dic71—24no—
232C2×Dic71—24no—
233C2×Dic71—24no—
234C2×Dic71—24no—
235C2×Dic71—24no—
236C2×Dic71—24no—
237C2×Dic71—24no—
238C2×Dic71—24no—
239C2×Dic71—24no—
240C2×Dic71—24no—
241C2×Dic71—42no—
242C2×Dic71—42no—
243C2×Dic71—42no—
244C2×Dic71—42no—
245C2×Dic71—42no—
246C2×Dic71—42no—
247C2×Dic71—42no—
248C2×Dic71—42no—
249C2×Dic71—42no—
250C2×Dic71—42no—
251C2×Dic71—42no—
252C2×Dic71—42no—
253C2×Dic71—42no—
254C2×Dic71—42no—
255C2×Dic71—42no—
256C2×Dic71C2×C22⋊C418yes—
257Dic142—416no—
258C22×C71—24no—
259C22×C71—24no—
260C22×C71—24no—
261C22×C71—42no—
262C22×C71—42no—
263C22×C71—42no—
264C22×C71—42no—
265C22×C71—44no—
266C22×C71—44no—
267C22×C71—44no—
268C22×C71—44no—
269C22×C71C22×C22⋊C411yescharacteristic
270C22×C71C2×C4×Q811yescharacteristic
271C22×C71C4×#4111yescharacteristic
272C22×C71#51811yescharacteristic
273C22×C71#51811yescharacteristic
274C22×C71#51911yescharacteristic
275C22×C71#52011yescharacteristic
276C22×C71#53412yes—
277C22×C71#53412yes—
278C22×C71#53811yescharacteristic
279C22×C71#53912yes—
280C22×C71#53911yescharacteristic
281C22×C71#54212yes—
282C22×C71#54311yescharacteristic
283C22×C71#54411yescharacteristic
284C22×C71#54812yes—
285C22×C71#54812yes—
286C22×C71D4×Q812yes—
287C22×C71#55012yes—
288C22×C71#55112yes—
289C22×C71#55412yes—
290C22×C71#56212yes—
291C22×C71#56312yes—
292C4×C71—24no—
293C4×C71—42no—
294C4×C71—44no—
295C4×C71—82no—
296Dic71—28no—
297Dic71—44no—
298Dic71—44no—
299Dic71—82no—
300#14449—711nomaximal, Sylow
301C2×C71—44no—
302C2×C71#230811yescharacteristic
303C2×C71#232911yescharacteristic
304C2×C71#233211yescharacteristic
305C2×C71Q8×C22⋊C411yescharacteristic
306C2×C71#234111yescharacteristic
307C2×C71#236811yescharacteristic
308C2×C71#237811yescharacteristic
309C2×C71#272812yes—
310C2×C71#272912yes—
311C2×C71#273012yes—
312C2×C71#273212yes—
313C22×C22⋊Q8—711no—
314C2×#391—712no—
315#1445—712no—
316#1441—712no—
317#1439—712no—
318#1437—712no—
319#1436—712no—
320C22⋊#95—711no—
321C22⋊#95D7111yescharacteristic
322C71#1444911yescharacteristic, complemented, Sylow
323C23×Q8—711no—
324C24×C4C2×D7111yescharacteristic
325C2×C22⋊Q8—718no—
326C2×C22⋊Q8—1428no—
327C22×C4⋊C4—712no—
328C22×C4⋊C4—712no—
329C22×C4⋊C4—711no—
330C22×C22⋊C4—712no—
331C22×C22⋊C4—712no—
332C22×C22⋊C4C2×D7111yescharacteristic
333C22×C22⋊C4C2×D7111yescharacteristic
334C22×C42—712no—
335#391—1428no—
336C2×#95—711no—
337C2×#95—711no—
338C2×#95—712no—
339C2×#95—712no—
340C2×#95—712no—
341C2×#95—712no—
342C2×#95—712no—
343C2×#95—712no—
344C2×#95—712no—
345C2×#95C2×D7112yes—
346C2×#95C2×D7112yes—
347C25C22×D7111yescharacteristic
348C22×Q8—2841no—
349C22×Q8—2842no—
350C22×Q8—2841no—
351C22×Q8—714no—
352C22×Q8—718no—
353C23×C4—24no—
354C23×C4—42no—
355C23×C4—41no—
356C23×C4—41no—
357C23×C4—712no—
358C23×C4—712no—
359C23×C4—712no—
360C23×C4—712no—
361C23×C4C4×D7114yes—
362C23×C4C22×D7112yes—
363C23×C4C22×D7111yescharacteristic
364C23×C4C22×D7112yes—
365C23×C4C22×D7111yescharacteristic
366C22⋊Q8—14232no—
367C2×C4⋊C4—2842no—
368C2×C4⋊C4—2842no—
369C2×C4⋊C4—718no—
370C2×C4⋊C4—714no—
371C2×C4⋊C4—1424no—
372C2×C4⋊C4—1424no—
373C2×C4⋊C4—1424no—
374C2×C4⋊C4—1424no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—41no—
377C2×C22⋊C4—41no—
378C2×C22⋊C4—41no—
379C2×C22⋊C4—41no—
380C2×C22⋊C4—2842no—
381C2×C22⋊C4—718no—
382C2×C22⋊C4—1424no—
383C2×C22⋊C4—1424no—
384C2×C22⋊C4—1424no—
385C2×C22⋊C4—1424no—
386C2×C22⋊C4Dic14214yescomplemented
387C2×C42—2842no—
388C2×C42—2842no—
389C2×C42—1424no—
390#95—24no—
391#95—42no—
392#95—2842no—
393#95—2841no—
394#95—2841no—
395#95—2842no—
396#95—1424no—
397#95—1424no—
398#95—1424no—
399#95—1424no—
400#95—1424no—
401C24—22no—
402C24—22no—
403C24—21no—
404C24—21no—
405C24—21no—
406C24—22no—
407C24—22no—
408C24—21no—
409C24C2×C4×D7112yes—
410C24C2×Dic14212yes—
411C24#1176586512yes—
412C24C23×D7111yescharacteristic
413C2×Q8—2844no—
414C2×Q8—2844no—
415C2×Q8—2848no—
416C2×Q8—2848no—
417C2×Q8—7116no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—28no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—714no—
444C22×C4—714no—
445C22×C4—714no—
446C22×C4—714no—
447C22×C4—714no—
448C22×C4—1422no—
449C22×C4—1422no—
450C22×C4—1422no—
451C22×C4—1422no—
452C22×C4—1422no—
453C22×C4—1422no—
454C22×C4—1422no—
455C22×C4—1422no—
456C22×C4—1422no—
457C22×C4—1422no—
458C22×C4—1422no—
459C22×C4—1422no—
460C22×C4—1422no—
461C22×C4—1422no—
462C22×C4—1422no—
463C22×C4—1422no—
464C22×C4—1422no—
465C22×C4—1422no—
466C22×C4—1422no—
467C22×C4—1422no—
468C22×C4—1422no—
469C22×C4—1422no—
470C22×C4—1422no—
471C22×C4C2×C4×D7112yes—
472C22×C4C2×C4×D7112yes—
473C22×C4D4×D7114yes—
474C22×C4C2×Dic14214yes—
475C4⋊C4—2848no—
476C4⋊C4—14216no—
477C4⋊C4—1428no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—2848no—
481C22⋊C4—14216no—
482C42—2848no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23C4×Dic14214yes—
506C23C22×C4×D7111yescharacteristic
507C23C2×D4×D7112yes—
508C23C22×Dic14211yescharacteristic
509C23#41×D7112yes—
510C23C2×#1176586511yescharacteristic
511C23#1455617311yescharacteristic
512C23#1455617311yescharacteristic
513C23#1455617311yescharacteristic
514C23#1455617312yes—
515C23#1455617512yes—
516C23#1455617611yescharacteristic
517Q8—28416no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—416no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—2842no—
543C2×C4—2842no—
544C2×C4—2842no—
545C2×C4—2842no—
546C2×C4—2842no—
547C2×C4—2842no—
548C2×C4—2842no—
549C2×C4—2842no—
550C2×C4—2842no—
551C2×C4—2842no—
552C2×C4—2842no—
553C2×C4—2842no—
554C2×C4—2842no—
555C2×C4—2842no—
556C2×C4—2842no—
557C2×C4—718no—
558C2×C4—1424no—
559C2×C4—1424no—
560C2×C4—1424no—
561C2×C4—1424no—
562C2×C4—1424no—
563C2×C4—1424no—
564C2×C4—1424no—
565C2×C4—1424no—
566C2×C4—1424no—
567C2×C4—1424no—
568C2×C4—1424no—
569C2×C4—1424no—
570C2×C4C22⋊C4×D7114yes—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—44no—
575C22—42no—
576C22—44no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—44no—
581C22—44no—
582C22C2×C22⋊C4×D7111yescharacteristic
583C22C2×C4×Dic14211yescharacteristic
584C22#1798646711yescharacteristic
585C22#1798647211yescharacteristic
586C22#1798647211yescharacteristic
587C22#1798647611yescharacteristic
588C22#1798648011yescharacteristic
589C22#1798655912yes—
590C22#1798655912yes—
591C22#1798658711yescharacteristic
592C22#1798659212yes—
593C22#1798659611yescharacteristic
594C22#1798661712yes—
595C22#1798662511yescharacteristic
596C22#1798663011yescharacteristic
597C22#1798665812yes—
598C22#1798665912yes—
599C22D4×Dic14212yes—
600C22#1798666712yes—
601C22#1798667812yes—
602C22#1798670812yes—
603C22#1798674412yes—
604C22#1798675012yes—
605C4—24no—
606C4—44no—
607C4—42no—
608C4—82no—
609C4—2844no—
610C4—2844no—
611C4—5682no—
612C4—1428no—
613C2—44no—
614C2#2259091611yescharacteristic
615C2#2259107311yescharacteristic
616C2#2259109311yescharacteristic
617C2C22⋊C4×Dic14211yescharacteristic
618C2#2259117711yescharacteristic
619C2#2259142311yescharacteristic
620C2#2259150311yescharacteristic
621C2#2259531312yes—
622C2#2259532612yes—
623C2#2259534012yes—
624C2#2259536012yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC71×C22⋊#95C2
socle, cosocleC24×C71C24
prime radicalC25×C71C23
hypercenterC22⋊#95D71
hypocenterC71#14449
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#17985903C2semidirect
3. #22586609C2semidirect
4. #22586591C2semidirect
7. #22586571C2semidirect
12. C2×#14556107C4semidirect
85. C2×C22⋊C4×C71Q8semidirect
92. C2×Dic142C22⋊C4semidirect
322. C71#14449semidirect
386. C2×C22⋊C4Dic142semidirect

Maximal subgroups

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

subgroupindex
C22×#145561072
C2×#179859032
#225866092
#225865912
#225865832
#225865772
#225865712
#225865632
C71×C22⋊#952
#1444971