Finite Group Databasebeta

← #31924472

Subgroups — #31924472

Order 18176.

1454 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985911C211yesmaximal, characteristic
2C2×#17985905C211yesmaximal, characteristic, complemented
3#22586610C211yesmaximal, characteristic, complemented
4#22586600C212yesmaximal, complemented
5#22586596C211yesmaximal, characteristic, complemented
6#22586587C212yesmaximal
7#22586585C211yesmaximal, characteristic
8#22586577C212yesmaximal
9#22586572C212yesmaximal, complemented
10C4×C71⋊#95C211yesmaximal, characteristic, complemented
11C2×#388×C71C211yesmaximal, characteristic
12C23×Dic142C2211yescharacteristic
13C22×C4⋊Dic71C2211yescharacteristic
14C22×C22⋊Dic71C2212yes—
15C22×C22⋊Dic71C2212yes—
16C22×C4×Dic71C2212yes—
17#17985911C418yescomplemented
18#17985905—24no—
19C2×C71⋊#95C2212yes—
20C2×C71⋊#95C2212yes—
21C2×C71⋊#95C2211yescharacteristic
22C2×C71⋊#95C2212yes—
23C2×C71⋊#95C2212yes—
24C2×C71⋊#95C2212yes—
25C2×C71⋊#95C2212yes—
26C2×C71⋊#95C2212yes—
27C2×C71⋊#95C2212yes—
28C2×C71⋊#95C2211yescharacteristic
29C2×C71⋊#95C2212yes—
30C22×C4⋊C4×C71C2211yescharacteristic
31C22×C22⋊C4×C71C2211yescharacteristic
32C22×C22⋊C4×C71C2211yescharacteristic
33C22×C22⋊C4×C71C2211yescharacteristic
34C22×C42×C71C2211yescharacteristic
35#388×C71—24no—
36C2×#95×C71C2211yescharacteristic
37C2×#95×C71C2211yescharacteristic
38C22×Dic142—22no—
39C22×Dic142—22no—
40C22×Dic142—42no—
41C22×Dic142—42no—
42C22×Dic142C2×C414yes—
43C23×Dic71C2312yes—
44C23×Dic71C2312yes—
45C23×Dic71C2312yes—
46C23×Dic71C2312yes—
47C2×C4⋊Dic71—22no—
48C2×C4⋊Dic71—42no—
49C2×C22⋊Dic71—24no—
50C2×C22⋊Dic71—42no—
51C2×C22⋊Dic71—42no—
52C2×C22⋊Dic71—42no—
53C2×C22⋊Dic71—42no—
54C2×C22⋊Dic71C2×C418yes—
55C2×C4×Dic71—24no—
56C2×C4×Dic71—42no—
57C2×C4×Dic71—42no—
58C71⋊#95—24no—
59C71⋊#95—24no—
60C71⋊#95—24no—
61C71⋊#95—24no—
62C71⋊#95—22no—
63C71⋊#95—42no—
64C71⋊#95—42no—
65C71⋊#95—42no—
66C71⋊#95—42no—
67C71⋊#95C2×C418yes—
68C71⋊#95C2×C414yes—
69C25×C71C2311yescharacteristic
70C23×C4×C71C2311yescharacteristic
71C23×C4×C71C2311yescharacteristic
72C23×C4×C71C2311yescharacteristic
73C23×C4×C71C2311yescharacteristic
74C23×C4×C71C2311yescharacteristic
75C23×C4×C71C2311yescharacteristic
76C2×C4⋊C4×C71—22no—
77C2×C4⋊C4×C71—24no—
78C2×C22⋊C4×C71—22no—
79C2×C22⋊C4×C71—22no—
80C2×C22⋊C4×C71—22no—
81C2×C22⋊C4×C71—22no—
82C2×C22⋊C4×C71—24no—
83C2×C22⋊C4×C71—42no—
84C2×C22⋊C4×C71C2×C414yes—
85C2×C42×C71—42no—
86C2×C42×C71D414yes—
87#95×C71—22no—
88#95×C71—22no—
89C2×Dic142—28no—
90C2×Dic142—48no—
91C2×Dic142—44no—
92C2×Dic142—44no—
93C2×Dic142—48no—
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×Dic71—22no—
117C22×Dic71—22no—
118C22×Dic71—22no—
119C22×Dic71C22×C414yes—
120C22×Dic71C22×C414yes—
121C22×Dic71#4114yes—
122C4⋊Dic71—44no—
123C22⋊Dic71—48no—
124C22⋊Dic71—48no—
125C4×Dic71—48no—
126C24×C71—21no—
127C24×C71—22no—
128C24×C71—21no—
129C24×C71—22no—
130C24×C71—22no—
131C24×C71—22no—
132C24×C71—21no—
133C24×C71—21no—
134C24×C71—21no—
135C24×C71—21no—
136C24×C71C22×C412yes—
137C24×C71C2411yescharacteristic
138C22×C4×C71—22no—
139C22×C4×C71—22no—
140C22×C4×C71—22no—
141C22×C4×C71—21no—
142C22×C4×C71—21no—
143C22×C4×C71—21no—
144C22×C4×C71—21no—
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—21no—
152C22×C4×C71—21no—
153C22×C4×C71—22no—
154C22×C4×C71—22no—
155C22×C4×C71—22no—
156C22×C4×C71—22no—
157C22×C4×C71—21no—
158C22×C4×C71—21no—
159C22×C4×C71—21no—
160C22×C4×C71—21no—
161C22×C4×C71C22×C412yes—
162C22×C4×C71C22×C412yes—
163C22×C4×C71C2×D412yes—
164C22×C4×C71C2×D412yes—
165C22×C4×C71C2×D412yes—
166C22×C4×C71#4114yes—
167C22×C4×C71#4112yes—
168C4⋊C4×C71—44no—
169C22⋊C4×C71—28no—
170C22⋊C4×C71—44no—
171C22⋊C4×C71—44no—
172C42×C71—44no—
173C23×C71—22no—
174C23×C71—24no—
175C23×C71—24no—
176C23×C71—22no—
177C23×C71—22no—
178C23×C71—41no—
179C23×C71—42no—
180C23×C71—41no—
181C23×C71—41no—
182C23×C71—41no—
183C23×C71—41no—
184C23×C71—41no—
185C23×C71—42no—
186C23×C71—42no—
187C23×C71—42no—
188C23×C71—42no—
189C23×C71—42no—
190C23×C71—41no—
191C23×C71—42no—
192C23×C71—42no—
193C23×C71—42no—
194C23×C71—42no—
195C23×C71—41no—
196C23×C71C23×C411yescharacteristic
197C23×C71C22×D411yescharacteristic
198C23×C71C2×#4112yes—
199C23×C71C2×#4112yes—
200C23×C71C2×#4111yescharacteristic
201C23×C71H(5,𝔽2)12yes—
202C23×C71H(5,𝔽2)12yes—
203C23×C71H(5,𝔽2)11yescharacteristic
204C23×C712−1+411yescharacteristic
205C23×C712−1+411yescharacteristic
206C23×C712−1+411yescharacteristic
207C2×C4×C71—24no—
208C2×C4×C71—22no—
209C2×C4×C71—22no—
210C2×C4×C71—22no—
211C2×C4×C71—22no—
212C2×C4×C71—24no—
213C2×C4×C71—24no—
214C2×C4×C71—24no—
215C2×C4×C71—22no—
216C2×C4×C71—24no—
217C2×C4×C71—22no—
218C2×C4×C71—22no—
219C2×C4×C71—22no—
220C2×C4×C71—41no—
221C2×C4×C71—42no—
222C2×C4×C71—41no—
223C2×C4×C71—42no—
224C2×C4×C71—42no—
225C2×C4×C71—41no—
226C2×C4×C71—41no—
227C2×C4×C71—41no—
228C2×C4×C71—41no—
229C2×C4×C71—41no—
230C2×C4×C71—41no—
231C2×C4×C71—42no—
232C2×C4×C71—42no—
233C2×C4×C71—42no—
234C2×C4×C71—42no—
235C2×C4×C71C4×D414yes—
236C2×C4×C71#12414yes—
237C2×Dic71—24no—
238C2×Dic71—24no—
239C2×Dic71—24no—
240C2×Dic71—24no—
241C2×Dic71—24no—
242C2×Dic71—24no—
243C2×Dic71—24no—
244C2×Dic71—24no—
245C2×Dic71—24no—
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×Dic71—42no—
257C2×Dic71—42no—
258C2×Dic71—42no—
259C2×Dic71—42no—
260C2×Dic71—42no—
261C2×Dic71—42no—
262C2×Dic71—42no—
263C2×Dic71—42no—
264C2×Dic71—42no—
265Dic142—416no—
266C22×C71—42no—
267C22×C71—42no—
268C22×C71—44no—
269C22×C71—44no—
270C22×C71—42no—
271C22×C71—81no—
272C22×C71—81no—
273C22×C71—82no—
274C22×C71—82no—
275C22×C71—81no—
276C22×C71—81no—
277C22×C71C2×C4×D411yescharacteristic
278C22×C71C4×#4112yes—
279C22×C71#51812yes—
280C22×C71#51911yescharacteristic
281C22×C71#52011yescharacteristic
282C22×C71C2×#12411yescharacteristic
283C22×C71#52912yes—
284C22×C71#53211yescharacteristic
285C22×C71#53411yescharacteristic
286C22×C71#53511yescharacteristic
287C22×C71#53611yescharacteristic
288C22×C71#53711yescharacteristic
289C22×C71#53812yes—
290C22×C71#53812yes—
291C22×C71#53812yes—
292C22×C71#53912yes—
293C22×C71#53912yes—
294C22×C71#54111yescharacteristic
295C22×C71#54212yes—
296C22×C71#54212yes—
297C22×C71#54212yes—
298C22×C71#54212yes—
299C22×C71#54211yescharacteristic
300C4×C71—24no—
301C4×C71—24no—
302C4×C71—42no—
303C4×C71—42no—
304C4×C71—81no—
305C4×C71—81no—
306Dic71—44no—
307Dic71—44no—
308Dic71—44no—
309Dic71—82no—
310#14323—711nomaximal, Sylow
311C2×C71—82no—
312C2×C71C4×#12411yescharacteristic
313C2×C71#232111yescharacteristic
314C2×C71#232311yescharacteristic
315C2×C71#233212yes—
316C2×C71#233712yes—
317C2×C71#263911yescharacteristic
318C2×C71#264011yescharacteristic
319C2×C71#265012yes—
320C2×C71#265312yes—
321C2×C71#266211yescharacteristic
322C2×C71#266311yescharacteristic
323C2×#393—711no—
324C2×#391—711no—
325C2×#388D7111yescharacteristic
326#1446—711no—
327#1443—712no—
328#1442—711no—
329#1440—712no—
330#1439—711no—
331#1437—712no—
332#1436—712no—
333C4×#95—711no—
334C71#1432311yescharacteristic, complemented, Sylow
335C23×Q8—711no—
336C22×C4⋊C4—711no—
337C22×C4⋊C4C2×D7111yescharacteristic
338C22×C22⋊C4—712no—
339C22×C22⋊C4—712no—
340C22×C22⋊C4C2×D7111yescharacteristic
341C22×C22⋊C4C2×D7111yescharacteristic
342C22×C22⋊C4C2×D7111yescharacteristic
343C22×C42—712no—
344C22×C42C2×D7111yescharacteristic
345#393—718no—
346#391—1424no—
347#388—24no—
348C2×#95—712no—
349C2×#95—712no—
350C2×#95—712no—
351C2×#95—712no—
352C2×#95—711no—
353C2×#95—712no—
354C2×#95—711no—
355C2×#95—712no—
356C2×#95—712no—
357C2×#95—712no—
358C2×#95—712no—
359C2×#95C2×D7111yescharacteristic
360C2×#95C2×D7111yescharacteristic
361C25C22×D7111yescharacteristic
362C22×Q8—2842no—
363C22×Q8—2842no—
364C22×Q8—714no—
365C22×Q8—1422no—
366C22×Q8—1422no—
367C23×C4—712no—
368C23×C4—712no—
369C23×C4—712no—
370C23×C4—712no—
371C23×C4C22×D7111yescharacteristic
372C23×C4C22×D7111yescharacteristic
373C23×C4C22×D7111yescharacteristic
374C23×C4C22×D7111yescharacteristic
375C23×C4C22×D7111yescharacteristic
376C23×C4C22×D7111yescharacteristic
377C2×C4⋊C4—22no—
378C2×C4⋊C4—24no—
379C2×C4⋊C4—2842no—
380C2×C4⋊C4—1422no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—42no—
387C2×C22⋊C4—2842no—
388C2×C22⋊C4—2842no—
389C2×C22⋊C4—2842no—
390C2×C22⋊C4—2842no—
391C2×C22⋊C4—718no—
392C2×C22⋊C4—1424no—
393C2×C22⋊C4C4×D7114yes—
394C2×C42—42no—
395C2×C42—2842no—
396C2×C42—2842no—
397C2×C42—1424no—
398C2×C42C22⋊D7114yes—
399#95—22no—
400#95—22no—
401#95—2842no—
402#95—2842no—
403#95—2842no—
404#95—2842no—
405#95—714no—
406#95—718no—
407#95—1424no—
408#95—1424no—
409#95—1424no—
410#95—1424no—
411#95—1422no—
412C24—21no—
413C24—22no—
414C24—22no—
415C24—21no—
416C24—22no—
417C24—22no—
418C24—21no—
419C24—21no—
420C24—21no—
421C24—21no—
422C24C2×C4×D7112yes—
423C24C23×D7111yescharacteristic
424C2×Q8—2844no—
425C2×Q8—2848no—
426C2×Q8—2844no—
427C2×Q8—2848no—
428C2×Q8—1428no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—714no—
453C22×C4—714no—
454C22×C4—714no—
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×C4—1422no—
472C22×C4—1422no—
473C22×C4—1422no—
474C22×C4—1422no—
475C22×C4—1422no—
476C22×C4—1422no—
477C22×C4—1422no—
478C22×C4—1422no—
479C22×C4—1422no—
480C22×C4C2×C4×D7112yes—
481C22×C4C2×C4×D7112yes—
482C22×C4C2×C22⋊D7112yes—
483C22×C4C2×C22⋊D7112yes—
484C22×C4C2×C22⋊D7112yes—
485C22×C4#1176586414yes—
486C22×C4#1176586512yes—
487C4⋊C4—44no—
488C4⋊C4—2844no—
489C22⋊C4—28no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—2848no—
493C22⋊C4—2848no—
494C42—44no—
495C42—2848no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—22no—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—41no—
518C23—41no—
519C23C22×C4×D7111yescharacteristic
520C23C22×C22⋊D7111yescharacteristic
521C23#41×D7112yes—
522C23C2×#1176586412yes—
523C23C2×#1176586511yescharacteristic
524C23#1455617312yes—
525C23#1455617412yes—
526C23#1455617411yescharacteristic
527C23#1455617511yescharacteristic
528C23#1455617511yescharacteristic
529C23#1455617511yescharacteristic
530Q8—28416no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—2842no—
560C2×C4—2842no—
561C2×C4—2842no—
562C2×C4—2842no—
563C2×C4—2842no—
564C2×C4—2842no—
565C2×C4—2842no—
566C2×C4—2842no—
567C2×C4—2842no—
568C2×C4—2842no—
569C2×C4—2842no—
570C2×C4—2842no—
571C2×C4—2842no—
572C2×C4—2842no—
573C2×C4—2842no—
574C2×C4—2842no—
575C2×C4—2842no—
576C2×C4—2842no—
577C2×C4—2842no—
578C2×C4—1424no—
579C2×C4—1424no—
580C2×C4—1424no—
581C2×C4—1424no—
582C2×C4—1424no—
583C2×C4—1424no—
584C2×C4—1424no—
585C2×C4—1424no—
586C2×C4—1424no—
587C2×C4C4×C22⋊D7114yes—
588C2×C4#1455611514yes—
589C22—42no—
590C22—42no—
591C22—44no—
592C22—42no—
593C22—44no—
594C22—82no—
595C22—82no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22C2×C4×C22⋊D7111yescharacteristic
601C22C4×#1176586412yes—
602C22#1798647112yes—
603C22#1798647611yescharacteristic
604C22#1798647911yescharacteristic
605C22C2×#1455611511yescharacteristic
606C22#1798653112yes—
607C22#1798654511yescharacteristic
608C22#1798655811yescharacteristic
609C22#1798656311yescharacteristic
610C22#1798657011yescharacteristic
611C22#1798657711yescharacteristic
612C22#1798658212yes—
613C22#1798658612yes—
614C22#1798658812yes—
615C22#1798659212yes—
616C22#1798659412yes—
617C22#1798661111yescharacteristic
618C22#1798661512yes—
619C22#1798661712yes—
620C22#1798661812yes—
621C22#1798661812yes—
622C22#1798662011yescharacteristic
623C4—24no—
624C4—24no—
625C4—42no—
626C4—42no—
627C4—81no—
628C4—81no—
629C4—2844no—
630C4—2844no—
631C4—2844no—
632C4—5682no—
633C2—82no—
634C2C4×#1455611511yescharacteristic
635C2#2259099411yescharacteristic
636C2#2259102211yescharacteristic
637C2#2259109012yes—
638C2#2259114712yes—
639C2#2259427311yescharacteristic
640C2#2259428011yescharacteristic
641C2#2259436212yes—
642C2#2259440712yes—
643C2#2259448011yescharacteristic
644C2#2259449011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC2×#388×C71C2
socle, cosocleC24×C71C24
prime radicalC25×C71C23
hypercenterC2×#388D71
hypocenterC71#14323
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#388

Complements

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

NH ≅ G/Ntype
2. C2×#17985905C2semidirect
3. #22586610C2semidirect
4. #22586600C2semidirect
5. #22586596C2semidirect
9. #22586572C2semidirect
10. C4×C71⋊#95C2semidirect
17. #17985911C4semidirect
334. C71#14323semidirect

Maximal subgroups

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

subgroupindex
C2×#179859112
C2×#179859052
#225866102
#225866002
#225865962
#225865872
#225865852
#225865772
#225865722
C4×C71⋊#952
C2×#388×C712
#1432371