Finite Group Databasebeta

← #31998334

Subgroups — #31998334

Order 18176.

1422 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986482C211yesmaximal, characteristic
2#22596930C211yesmaximal, characteristic, complemented
3#22596859C211yesmaximal, characteristic
4#22590963C211yesmaximal, characteristic, complemented
5#22588575C214yesmaximal, complemented
6#22588568C214yesmaximal, complemented
7C2×#17985767C211yesmaximal, characteristic, complemented
8#22588050C211yesmaximal, characteristic, complemented
9C2×#421×C71C211yesmaximal, characteristic
10C22×#11765849C2211yescharacteristic
11#17986482—22no—
12#17986482—22no—
13#17986482—22no—
14#17986482C2214yes—
15C2×Q8×Dic71—21no—
16C2×D4×Dic71—21no—
17C2×D4×Dic71—22no—
18C2×#14556087—21no—
19C2×#14556087C2211yescharacteristic
20C2×#14556087C2211yescharacteristic
21C22×C4⋊Dic71—21no—
22C22×C4⋊Dic71C2211yescharacteristic
23#17986041—21no—
24#17986041—21no—
25#17986041C2212yescomplemented
26#17986041C2212yescomplemented
27#17986037—22no—
28#17986008—21no—
29#17986004—21no—
30#17986000C2212yes—
31#17986000C2212yes—
32#17985995—22no—
33#17985898—22no—
34#17985874—22no—
35#17985869C2212yescomplemented
36#17985865C2212yescomplemented
37#17985767C2214yescomplemented
38#17985759C2214yescomplemented
39C22×#41×C71C2211yescharacteristic
40C22×M16×C71C2211yescharacteristic
41C2×#117×C71C2211yescharacteristic
42#421×C71C418yescomplemented
43C2×#104×C71C2214yes—
44C23×Dic71—22no—
45C23×Dic71D412yes—
46C2×#11765849—21no—
47C2×#11765849—22no—
48C2×#11765849—22no—
49C2×#11765849C2312yes—
50C22×C71⋊C8—21no—
51Q8×Dic71—24no—
52Q8×Dic71—24no—
53D4×Dic71—28no—
54D4×Dic71—24no—
55D4×Dic71—28no—
56D4×Dic71—24no—
57#14556087—24no—
58#14556087—24no—
59#14556087—21no—
60#14556087—21no—
61#14556087—21no—
62#14556087—21no—
63#14556087—21no—
64#14556087—21no—
65#14556087D412yes—
66#14556087C2312yes—
67C2×C4⋊Dic71—22no—
68C2×C4⋊Dic71—24no—
69C2×C4⋊Dic71—24no—
70C2×C4⋊Dic71—24no—
71C2×C4⋊Dic71—21no—
72C2×C4⋊Dic71—21no—
73C2×C4⋊Dic71—21no—
74C2×C4⋊Dic71—21no—
75C2×C4⋊Dic71D412yes—
76C2×C4⋊Dic71D411yescharacteristic
77C2×C4⋊Dic71D411yescharacteristic
78C2×C4⋊Dic71C2311yescharacteristic
79C2×C4⋊Dic71C2311yescharacteristic
80C2×C4⋊Dic71C2312yes—
81C2×C22⋊Dic71—22no—
82C2×C22⋊Dic71—21no—
83C2×C22⋊Dic71—21no—
84C2×C4×Dic71—22no—
85C2×C4×Dic71—21no—
86C2×C4×Dic71—21no—
87#14556062—22no—
88#14556062—22no—
89#14556060—22no—
90#14556060—22no—
91#14556052—22no—
92#14556052—22no—
93#14556049—22no—
94#14556049—22no—
95C71⋊#95—24no—
96C2×#41×C71—21no—
97C2×#41×C71—21no—
98C2×#41×C71—22no—
99C2×#41×C71—22no—
100C2×#41×C71C2×C414yes—
101C2×#41×C71D412yes—
102C2×#41×C71Q814yes—
103C2×#41×C71C2312yes—
104C22×Q8×C71D411yescharacteristic
105C22×D4×C71—21no—
106C22×D4×C71D411yescharacteristic
107C23×C4×C71—21no—
108C23×C4×C71C2311yescharacteristic
109C2×M16×C71—21no—
110C2×M16×C71—22no—
111C2×M16×C71C2×C414yescomplemented
112C2×M16×C71C2312yes—
113C22×C8×C71—21no—
114#117×C71—22no—
115#117×C71C2×C414yescomplemented
116C2×C4⋊C4×C71—21no—
117C2×C22⋊C4×C71—21no—
118C2×C42×C71C2312yes—
119#104×C71C2×C4116yescomplemented
120C22×Dic71—21no—
121C22×Dic71—22no—
122C22×Dic71—21no—
123C22×Dic71—28no—
124C22×Dic71—22no—
125C22×Dic71—21no—
126C22×Dic71—21no—
127C22×Dic71—82no—
128C22×Dic71—84no—
129C22×Dic71C2×D414yes—
130#11765849—22no—
131#11765849—22no—
132#11765849—42no—
133C2×C71⋊C8—22no—
134C2×C71⋊C8—42no—
135C4⋊Dic71—28no—
136C4⋊Dic71—22no—
137C4⋊Dic71—21no—
138C4⋊Dic71—22no—
139C4⋊Dic71—21no—
140C4⋊Dic71—22no—
141C4⋊Dic71—22no—
142C4⋊Dic71—21no—
143C4⋊Dic71—21no—
144C4⋊Dic71—21no—
145C4⋊Dic71—21no—
146C4⋊Dic71—84no—
147C4⋊Dic71C2×D412yes—
148C4⋊Dic71C2×D412yes—
149C22⋊Dic71—28no—
150C22⋊Dic71—22no—
151C22⋊Dic71—22no—
152C22⋊Dic71—41no—
153C22⋊Dic71—41no—
154C4×Dic71—28no—
155C4×Dic71—22no—
156C4×Dic71—22no—
157C4×Dic71—41no—
158C4×Dic71—41no—
159C24×C71—21no—
160C24×C71C2×D411yescharacteristic
161#41×C71—24no—
162#41×C71—24no—
163#41×C71—24no—
164#41×C71—24no—
165#41×C71—24no—
166#41×C71—22no—
167#41×C71—22no—
168#41×C71C4⋊C418yescomplemented
169#41×C71C22×C414yes—
170#41×C71C2×Q814yes—
171C2×Q8×C71—21no—
172C2×Q8×C71—22no—
173C2×Q8×C71C4⋊C414yescomplemented
174C2×Q8×C71C2×D412yes—
175C2×D4×C71—24no—
176C2×D4×C71—22no—
177C2×D4×C71—24no—
178C2×D4×C71—22no—
179C2×D4×C71—21no—
180C2×D4×C71—22no—
181C2×D4×C71C4⋊C414yescomplemented
182C2×D4×C71C2×D412yes—
183C22×C4×C71—21no—
184C22×C4×C71—24no—
185C22×C4×C71—22no—
186C22×C4×C71—24no—
187C22×C4×C71—22no—
188C22×C4×C71—22no—
189C22×C4×C71C22×C412yes—
190C22×C4×C71C2×D411yescharacteristic
191C22×C4×C71C2×D411yescharacteristic
192C22×C4×C71C2×D411yescharacteristic
193C22×C4×C71C2×D411yescharacteristic
194C22×C4×C71C2×D411yescharacteristic
195C22×C4×C71C2×Q812yes—
196C22×C4×C71#4112yes—
197C22×C4×C71#4112yes—
198C22×C4×C71C2411yescharacteristic
199M16×C71—24no—
200M16×C71—22no—
201M16×C71C22×C414yescomplemented
202C2×C8×C71—24no—
203C2×C8×C71—22no—
204C4⋊C4×C71—24no—
205C22⋊C4×C71—24no—
206C42×C71—22no—
207C42×C71C22×C414yescomplemented
208C23×C71—22no—
209C23×C71—24no—
210C23×C71—41no—
211C23×C71—41no—
212C23×C71—42no—
213C23×C71—42no—
214C23×C71C4×D412yes—
215C23×C71#12011yescharacteristic
216C23×C71#12011yescharacteristic
217C23×C71C22⋊Q812yes—
218C23×C71C22×D411yescharacteristic
219Q8×C71—22no—
220Q8×C71—24no—
221Q8×C71C2×C4⋊C414yes—
222D4×C71—24no—
223D4×C71—24no—
224D4×C71—28no—
225D4×C71—22no—
226D4×C71—24no—
227D4×C71C2×C4⋊C414yes—
228C2×C4×C71—22no—
229C2×C4×C71—24no—
230C2×C4×C71—24no—
231C2×C4×C71—24no—
232C2×C4×C71—28no—
233C2×C4×C71—24no—
234C2×C4×C71—42no—
235C2×C4×C71—42no—
236C2×C4×C71C2×C4⋊C412yes—
237C2×C4×C71C2×C4⋊C412yes—
238C2×C4×C71C4×D412yes—
239C2×C4×C71C4×D412yes—
240C2×C4×C71C4×D412yes—
241C2×C4×C71#12011yescharacteristic
242C2×C4×C71#12011yescharacteristic
243C2×C4×C71C22⋊Q812yes—
244C2×C4×C71C22⋊Q812yes—
245C2×C4×C71C22⋊Q812yes—
246C2×C4×C71#12312yes—
247C2×C4×C71#12312yes—
248C2×C4×C71C23×C411yescharacteristic
249C2×C4×C71C22×D411yescharacteristic
250C2×C4×C71C22×D411yescharacteristic
251C2×C4×C71C22×Q811yescharacteristic
252C2×C4×C71C2×#4111yescharacteristic
253C2×C4×C71C2×#4111yescharacteristic
254C2×Dic71—22no—
255C2×Dic71—24no—
256C2×Dic71—22no—
257C2×Dic71—42no—
258C2×Dic71—44no—
259C2×Dic71—42no—
260C2×Dic71—44no—
261C2×Dic71—42no—
262C2×Dic71—42no—
263C2×Dic71—41no—
264C2×Dic71—41no—
265C2×Dic71—82no—
266C2×Dic71—88no—
267C8×C71—24no—
268C71⋊C8—42no—
269C22×C71—21no—
270C22×C71—21no—
271C22×C71—21no—
272C22×C71—21no—
273C22×C71—21no—
274C22×C71—21no—
275C22×C71—24no—
276C22×C71—42no—
277C22×C71—42no—
278C22×C71—48no—
279C22×C71—42no—
280C22×C71—42no—
281C22×C71#38512yes—
282C22×C71#38512yes—
283C22×C71C2×C4×D411yescharacteristic
284C22×C71C2×#12011yescharacteristic
285C22×C71C2×C22⋊Q811yescharacteristic
286C4×C71—24no—
287C4×C71—44no—
288C4×C71#38512yes—
289C4×C71#38512yes—
290C4×C71C22×C4⋊C411yescharacteristic
291C4×C71C2×C4×D411yescharacteristic
292C4×C71C2×C22⋊Q811yescharacteristic
293C4×C71C2×#12311yescharacteristic
294Dic71—44no—
295Dic71—42no—
296Dic71—42no—
297Dic71—81no—
298Dic71—81no—
299#20864—711nomaximal, Sylow
300C2×C71—21no—
301C2×C71—22no—
302C2×C71—21no—
303C2×C71—22no—
304C2×C71—44no—
305C2×C71C2×#38511yescharacteristic
306C2×C71#301911yescharacteristic
307C2×C71#302011yescharacteristic
308C2×#520—711no—
309#2911—711no—
310C2×#421D7111yescharacteristic
311#2896—711no—
312#2317—711no—
313#1860—714no—
314#1859—714no—
315C2×#343—711no—
316#1733—711no—
317C71#2086411yescharacteristic, complemented, Sylow
318C22×#41C2×D7111yescharacteristic
319C22×M16—711no—
320C22×M16C2×D7111yescharacteristic
321#520—714no—
322#520—1422no—
323#520—1422no—
324#520—1422no—
325C2×C4×Q8—1421no—
326C2×C4×D4—1422no—
327C2×C4×D4—1421no—
328C2×#117—711no—
329C2×#117—711no—
330C2×#117—1421no—
331C2×#117C2×D7111yescharacteristic
332C22×C4⋊C4—711no—
333C22×C4⋊C4—1421no—
334#428—712no—
335#428—712no—
336#428—1421no—
337#428—1421no—
338#427—1422no—
339#421Dic7118yescomplemented
340C2×#104C2×D7114yes—
341#419—1421no—
342#418—1421no—
343#417—712no—
344#417—712no—
345#416—1422no—
346#389—1422no—
347#385—1422no—
348#384—712no—
349#382—712no—
350#343—714no—
351#339—714no—
352C2×#41—21no—
353C2×#41—21no—
354C2×#41—22no—
355C2×#41—22no—
356C2×#41Dic14214yes—
357C2×#41D28412yes—
358C2×#41C2×Dic7114yes—
359C2×#41C22×D7112yes—
360C22×Q8D28411yescharacteristic
361C22×D4—21no—
362C22×D4D28411yescharacteristic
363C23×C4—21no—
364C23×C4—712no—
365C23×C4—1422no—
366C23×C4C22×D7111yescharacteristic
367C2×M16—21no—
368C2×M16—22no—
369C2×M16—712no—
370C2×M16—1422no—
371C2×M16—1422no—
372C2×M16—1421no—
373C2×M16C2×Dic7114yescomplemented
374C2×M16C22×D7112yes—
375C22×C8—21no—
376C22×C8—1421no—
377C4×Q8—1424no—
378C4×Q8—1424no—
379C4×D4—1428no—
380C4×D4—1428no—
381C4×D4—1424no—
382C4×D4—1424no—
383#117—22no—
384#117—712no—
385#117—712no—
386#117—1421no—
387#117—1421no—
388#117—1421no—
389#117—1421no—
390#117—1424no—
391#117—1424no—
392#117—1421no—
393#117—1421no—
394#117C2×Dic7114yescomplemented
395C2×C4⋊C4—21no—
396C2×C4⋊C4—711no—
397C2×C4⋊C4—711no—
398C2×C4⋊C4—712no—
399C2×C4⋊C4—712no—
400C2×C4⋊C4—711no—
401C2×C4⋊C4—711no—
402C2×C4⋊C4—1424no—
403C2×C4⋊C4—1424no—
404C2×C4⋊C4—1422no—
405C2×C4⋊C4—1424no—
406C2×C4⋊C4—1421no—
407C2×C4⋊C4—1421no—
408C2×C4⋊C4—1421no—
409C2×C4⋊C4—1421no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—1421no—
412C2×C22⋊C4—1422no—
413C2×C22⋊C4—1421no—
414C2×C42—1421no—
415C2×C42—1422no—
416C2×C42—1421no—
417C2×C42C22×D7112yes—
418#107—1422no—
419#107—1422no—
420#106—1422no—
421#106—1422no—
422#104C2×Dic71116yescomplemented
423#103—1422no—
424#103—1422no—
425#102—1422no—
426#102—1422no—
427#95—1424no—
428C24—21no—
429C24D4×D7111yescharacteristic
430#41—22no—
431#41—22no—
432#41—24no—
433#41—24no—
434#41—24no—
435#41—24no—
436#41—24no—
437#41C4⋊Dic7118yescomplemented
438#41C22×Dic7114yes—
439#41C2×Dic14214yes—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8C4⋊Dic7114yescomplemented
443C2×Q8C2×D28412yes—
444C2×D4—22no—
445C2×D4—24no—
446C2×D4—22no—
447C2×D4—24no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4C4⋊Dic7114yescomplemented
451C2×D4C2×D28412yes—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—5682no—
459C22×C4—5684no—
460C22×C4—714no—
461C22×C4—1421no—
462C22×C4—1421no—
463C22×C4—1421no—
464C22×C4—1421no—
465C22×C4—1422no—
466C22×C4—1428no—
467C22×C4—1422no—
468C22×C4C22×Dic7112yes—
469C22×C4D4×D7111yescharacteristic
470C22×C4D4×D7111yescharacteristic
471C22×C4D4×D7111yescharacteristic
472C22×C4C2×D28411yescharacteristic
473C22×C4C2×D28411yescharacteristic
474C22×C4C2×Dic14212yes—
475C22×C4#1176586412yes—
476C22×C4#1176586412yes—
477C22×C4C23×D7111yescharacteristic
478M16—22no—
479M16—24no—
480M16—2842no—
481M16—1422no—
482M16—1422no—
483M16C22×Dic7114yescomplemented
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—2842no—
487C2×C8—1422no—
488C4⋊C4—24no—
489C4⋊C4—5684no—
490C4⋊C4—712no—
491C4⋊C4—712no—
492C4⋊C4—1428no—
493C4⋊C4—1421no—
494C4⋊C4—1421no—
495C4⋊C4—1421no—
496C4⋊C4—1421no—
497C4⋊C4—1422no—
498C4⋊C4—1422no—
499C4⋊C4—1421no—
500C4⋊C4—1422no—
501C4⋊C4—1422no—
502C4⋊C4—1421no—
503C22⋊C4—24no—
504C22⋊C4—2841no—
505C22⋊C4—2841no—
506C22⋊C4—1428no—
507C22⋊C4—1422no—
508C22⋊C4—1422no—
509C42—22no—
510C42—2841no—
511C42—2841no—
512C42—1428no—
513C42—1422no—
514C42—1422no—
515C42C22×Dic7114yescomplemented
516C23—24no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23D4×Dic7112yes—
523C23#1455609811yescharacteristic
524C23#1455609811yescharacteristic
525C23#1455610512yes—
526C23C2×D4×D7111yescharacteristic
527Q8—24no—
528Q8—22no—
529Q8C2×C4⋊Dic7114yes—
530D4—28no—
531D4—24no—
532D4—24no—
533D4—24no—
534D4—22no—
535D4C2×C4⋊Dic7114yes—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—2844no—
545C2×C4—2842no—
546C2×C4—2842no—
547C2×C4—2844no—
548C2×C4—2842no—
549C2×C4—2842no—
550C2×C4—2841no—
551C2×C4—2841no—
552C2×C4—5682no—
553C2×C4—5688no—
554C2×C4—1422no—
555C2×C4—1422no—
556C2×C4—1424no—
557C2×C4C2×C4⋊Dic7112yes—
558C2×C4C2×C4⋊Dic7112yes—
559C2×C4D4×Dic7112yes—
560C2×C4D4×Dic7112yes—
561C2×C4D4×Dic7112yes—
562C2×C4#1455609811yescharacteristic
563C2×C4#1455609811yescharacteristic
564C2×C4#1455610512yes—
565C2×C4#1455610512yes—
566C2×C4#1455610512yes—
567C2×C4#1455611212yes—
568C2×C4#1455611212yes—
569C2×C4C23×Dic7111yescharacteristic
570C2×C4C2×D4×D7111yescharacteristic
571C2×C4C22×D28411yescharacteristic
572C2×C4C22×Dic14211yescharacteristic
573C2×C4C2×#1176586411yescharacteristic
574C2×C4C2×#1176586411yescharacteristic
575C8—24no—
576C8—2842no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—24no—
584C22—42no—
585C22—42no—
586C22—48no—
587C22—42no—
588C22—42no—
589C22#1798587412yes—
590C22#1798587412yes—
591C22C2×D4×Dic7111yescharacteristic
592C22C2×#1455609811yescharacteristic
593C22C2×#1455610511yescharacteristic
594C4—24no—
595C4—44no—
596C4—2844no—
597C4—2842no—
598C4—2842no—
599C4—5681no—
600C4—5681no—
601C4#1798587412yes—
602C4#1798587412yes—
603C4C22×C4⋊Dic7111yescharacteristic
604C4C2×D4×Dic7111yescharacteristic
605C4C2×#1455610511yescharacteristic
606C4C2×#1455611211yescharacteristic
607C2—21no—
608C2—22no—
609C2—22no—
610C2—21no—
611C2—44no—
612C2C2×#1798587411yescharacteristic
613C2#2259769911yescharacteristic
614C2#2259770611yescharacteristic
subgroupG/H
centerC22C2×#14556098
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#421×C71C2
socleC22×C71C2×#120
prime radicalC22×#41×C71C22
cosocleC22×C4×C71C24
hypercenterC2×#421D71
hypocenterC71#20864
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#421

Complements

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

NH ≅ G/Ntype
2. #22596930C2semidirect
4. #22590963C2semidirect
5. #22588575C2semidirect
6. #22588568C2semidirect
7. C2×#17985767C2semidirect
8. #22588050C2semidirect
25. #17986041C22semidirect
26. #17986041C22semidirect
35. #17985869C22semidirect
36. #17985865C22semidirect
37. #17985767C22semidirect
38. #17985759C22semidirect
42. #421×C71C4semidirect
111. C2×M16×C71C2×C4semidirect
115. #117×C71C2×C4semidirect
119. #104×C71C2×C4semidirect
168. #41×C71C4⋊C4semidirect
173. C2×Q8×C71C4⋊C4semidirect
181. C2×D4×C71C4⋊C4semidirect
201. M16×C71C22×C4semidirect
207. C42×C71C22×C4semidirect
317. C71#20864semidirect
339. #421Dic71semidirect
373. C2×M16C2×Dic71semidirect
394. #117C2×Dic71semidirect
422. #104C2×Dic71semidirect
437. #41C4⋊Dic71semidirect
442. C2×Q8C4⋊Dic71semidirect
450. C2×D4C4⋊Dic71semidirect
483. M16C22×Dic71semidirect
515. C42C22×Dic71semidirect

Maximal subgroups

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

subgroupindex
C2×#179864822
#225969302
#225968592
#225909632
#225885752
#225885682
C2×#179857672
#225880502
C2×#421×C712
#2086471