Finite Group Databasebeta

← #31925384

Subgroups — #31925384

Order 18176.

1750 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985898C212yesmaximal, complemented
2C2×#17985892C212yesmaximal
3C2×#17985859C212yesmaximal
4#22586605C212yesmaximal, complemented
5#22586593C212yesmaximal, complemented
6#22586585C211yesmaximal, characteristic
7#22586585C211yesmaximal, characteristic
8#22586574C212yesmaximal, complemented
9C2×#385×C71C211yesmaximal, characteristic
10C22×C4⋊Dic71C2212yes—
11C22×C4⋊Dic71C2211yescharacteristic
12C22×C4⋊Dic71C2212yes—
13C22×C4⋊Dic71C2212yes—
14C22×C4⋊Dic71C2211yescharacteristic
15C22×C22⋊Dic71C2212yes—
16C22×C22⋊Dic71C2212yes—
17C22×C4×Dic71C2212yes—
18#17985898—28no—
19#17985892—28no—
20#17985859—28no—
21C2×C71⋊#95C2212yes—
22C2×C71⋊#95C2212yes—
23C2×C71⋊#95C2212yes—
24C2×C71⋊#95C2212yes—
25C2×C71⋊#95C2211yescharacteristic
26C2×C71⋊#95C2212yes—
27C2×C71⋊#95C2211yescharacteristic
28C2×C71⋊#95C2212yes—
29C24×C4×C71C2211yescharacteristic
30C22×C4⋊C4×C71C2212yes—
31C22×C22⋊C4×C71C2211yescharacteristic
32C22×C22⋊C4×C71C2211yescharacteristic
33#385×C71C418yescomplemented
34C2×#95×C71C2212yes—
35C23×Dic71C2312yes—
36C23×Dic71C2312yes—
37C23×Dic71C2312yes—
38C23×Dic71C2312yes—
39C2×C4⋊Dic71—24no—
40C2×C4⋊Dic71—24no—
41C2×C4⋊Dic71—24no—
42C2×C4⋊Dic71—24no—
43C2×C4⋊Dic71—24no—
44C2×C4⋊Dic71—24no—
45C2×C4⋊Dic71—24no—
46C2×C4⋊Dic71—42no—
47C2×C4⋊Dic71—42no—
48C2×C4⋊Dic71—42no—
49C2×C4⋊Dic71—41no—
50C2×C4⋊Dic71—41no—
51C2×C4⋊Dic71—42no—
52C2×C22⋊Dic71—24no—
53C2×C22⋊Dic71—24no—
54C2×C22⋊Dic71—24no—
55C2×C22⋊Dic71—24no—
56C2×C22⋊Dic71—42no—
57C2×C22⋊Dic71—42no—
58C2×C4×Dic71—24no—
59C2×C4×Dic71—24no—
60C2×C4×Dic71—42no—
61C71⋊#95—24no—
62C71⋊#95—24no—
63C71⋊#95—24no—
64C71⋊#95—24no—
65C71⋊#95—41no—
66C71⋊#95—41no—
67C71⋊#95—42no—
68C71⋊#95D418yes—
69C25×C71C2311yescharacteristic
70C23×C4×C71—24no—
71C23×C4×C71—24no—
72C23×C4×C71—41no—
73C23×C4×C71—41no—
74C23×C4×C71C2×C414yes—
75C23×C4×C71C2311yescharacteristic
76C23×C4×C71C2311yescharacteristic
77C23×C4×C71C2312yes—
78C23×C4×C71C2312yes—
79C2×C4⋊C4×C71—24no—
80C2×C4⋊C4×C71—24no—
81C2×C4⋊C4×C71C2×C418yes—
82C2×C22⋊C4×C71—41no—
83C2×C22⋊C4×C71—41no—
84C2×C22⋊C4×C71C2×C414yes—
85C2×C22⋊C4×C71C2×C414yes—
86C2×C22⋊C4×C71D414yes—
87C2×C22⋊C4×C71Q814yes—
88#95×C71C2×C418yes—
89C22×Dic71—22no—
90C22×Dic71—22no—
91C22×Dic71—22no—
92C22×Dic71—22no—
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×Dic71C2×D414yes—
116C22×Dic71C2×D414yes—
117C4⋊Dic71—44no—
118C4⋊Dic71—48no—
119C4⋊Dic71—48no—
120C4⋊Dic71—48no—
121C4⋊Dic71—44no—
122C22⋊Dic71—48no—
123C22⋊Dic71—48no—
124C4×Dic71—48no—
125C24×C71—22no—
126C24×C71—22no—
127C24×C71—21no—
128C24×C71—21no—
129C24×C71—22no—
130C24×C71—22no—
131C24×C71—21no—
132C24×C71—21no—
133C24×C71C22×C412yes—
134C24×C71C2×D412yes—
135C24×C71C2×Q812yes—
136C24×C71C2411yescharacteristic
137C22×C4×C71—22no—
138C22×C4×C71—22no—
139C22×C4×C71—22no—
140C22×C4×C71—22no—
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—21no—
148C22×C4×C71—21no—
149C22×C4×C71—21no—
150C22×C4×C71—21no—
151C22×C4×C71—22no—
152C22×C4×C71—22no—
153C22×C4×C71—216no—
154C22×C4×C71—44no—
155C22×C4×C71—44no—
156C22×C4×C71—48no—
157C22×C4×C71—44no—
158C22×C4×C71C22×C412yes—
159C22×C4×C71C22×C412yes—
160C22×C4×C71C22×C414yes—
161C22×C4×C71C22×C414yes—
162C22×C4×C71C2×D414yes—
163C22×C4×C71C2×D414yes—
164C22×C4×C71C2×Q814yes—
165C22×C4×C71#4114yes—
166C4⋊C4×C71—216no—
167C22⋊C4×C71—44no—
168C22⋊C4×C71C4⋊C4116yescomplemented
169C23×C71—24no—
170C23×C71—22no—
171C23×C71—22no—
172C23×C71—24no—
173C23×C71—22no—
174C23×C71—22no—
175C23×C71—22no—
176C23×C71—22no—
177C23×C71—24no—
178C23×C71—24no—
179C23×C71—24no—
180C23×C71—24no—
181C23×C71—42no—
182C23×C71—42no—
183C23×C71—42no—
184C23×C71—42no—
185C23×C71—41no—
186C23×C71—42no—
187C23×C71—41no—
188C23×C71—41no—
189C23×C71—42no—
190C23×C71—41no—
191C23×C71C2×C4⋊C414yes—
192C23×C71C23×C411yescharacteristic
193C23×C71C22×D412yes—
194C23×C71C22×D411yescharacteristic
195C23×C71C22×Q811yescharacteristic
196C23×C71C2×#4112yes—
197C23×C71H(5,𝔽2)11yescharacteristic
198C23×C71H(5,𝔽2)11yescharacteristic
199C23×C71H(5,𝔽2)12yes—
200C23×C712−1+412yes—
201C23×C712−1+411yescharacteristic
202C23×C712−1+411yescharacteristic
203C2×C4×C71—24no—
204C2×C4×C71—24no—
205C2×C4×C71—24no—
206C2×C4×C71—24no—
207C2×C4×C71—24no—
208C2×C4×C71—24no—
209C2×C4×C71—24no—
210C2×C4×C71—24no—
211C2×C4×C71—24no—
212C2×C4×C71—24no—
213C2×C4×C71—24no—
214C2×C4×C71—24no—
215C2×C4×C71—24no—
216C2×C4×C71—22no—
217C2×C4×C71—22no—
218C2×C4×C71—24no—
219C2×C4×C71—416no—
220C2×C4×C71—41no—
221C2×C4×C71—42no—
222C2×C4×C71—41no—
223C2×C4×C71—41no—
224C2×C4×C71—42no—
225C2×C4×C71—41no—
226C2×C4×C71C2×C4⋊C418yes—
227C2×C4×C71C4×D418yes—
228C2×Dic71—24no—
229C2×Dic71—24no—
230C2×Dic71—24no—
231C2×Dic71—24no—
232C2×Dic71—24no—
233C2×Dic71—24no—
234C2×Dic71—42no—
235C2×Dic71—42no—
236C2×Dic71—42no—
237C2×Dic71—42no—
238C2×Dic71—42no—
239C2×Dic71—42no—
240C2×Dic71—42no—
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—
256C22×C71—24no—
257C22×C71—28no—
258C22×C71—44no—
259C22×C71—42no—
260C22×C71—42no—
261C22×C71—44no—
262C22×C71—42no—
263C22×C71—42no—
264C22×C71—44no—
265C22×C71—42no—
266C22×C71—42no—
267C22×C71C22×C4⋊C411yescharacteristic
268C22×C71C2×C4×D412yes—
269C22×C71#51811yescharacteristic
270C22×C71#51911yescharacteristic
271C22×C71#52012yes—
272C22×C71#53612yes—
273C22×C71#53712yes—
274C22×C71#53711yescharacteristic
275C22×C71#53711yescharacteristic
276C22×C71#54112yes—
277C22×C71#54212yes—
278C22×C71#54411yescharacteristic
279C22×C71#54411yescharacteristic
280C22×C71D4211yescharacteristic
281C22×C71D4212yes—
282C22×C71#54612yes—
283C22×C71#54712yes—
284C22×C71D4×Q812yes—
285C22×C71#55111yescharacteristic
286C22×C71#55412yes—
287C22×C71#56011yescharacteristic
288C22×C71#56212yes—
289C22×C71#56311yescharacteristic
290C4×C71—28no—
291C4×C71—28no—
292C4×C71—42no—
293C4×C71—42no—
294Dic71—44no—
295Dic71—44no—
296Dic71—82no—
297Dic71—82no—
298#14402—711nomaximal, Sylow
299C2×C71—44no—
300C2×C71#231611yescharacteristic
301C2×C71#232312yes—
302C2×C71D4×C4⋊C412yes—
303C2×C71#235811yescharacteristic
304C2×C71#237611yescharacteristic
305C2×C71#267212yes—
306C2×C71#267511yescharacteristic
307C2×C71#267711yescharacteristic
308C2×C71#270412yes—
309C2×C71#272211yescharacteristic
310C2×C71#274211yescharacteristic
311C2×#389—712no—
312C2×#388—712no—
313C2×#385D7111yescharacteristic
314C2×#380—712no—
315#1444—712no—
316#1441—712no—
317#1439—711no—
318#1439—711no—
319#1436—712no—
320C71#1440211yescharacteristic, complemented, Sylow
321C24×C4C2×D7111yescharacteristic
322C22×C4⋊C4—711no—
323C22×C4⋊C4—712no—
324C22×C4⋊C4—711no—
325C22×C4⋊C4—712no—
326C22×C4⋊C4—712no—
327C22×C4⋊C4C2×D7112yes—
328C22×C22⋊C4—712no—
329C22×C22⋊C4—712no—
330C22×C22⋊C4C2×D7111yescharacteristic
331C22×C22⋊C4C2×D7111yescharacteristic
332C22×C42—712no—
333#389—1428no—
334#388—1428no—
335#385Dic7118yescomplemented
336#380—1428no—
337C2×#95—711no—
338C2×#95—712no—
339C2×#95—711no—
340C2×#95—712no—
341C2×#95—712no—
342C2×#95—712no—
343C2×#95—712no—
344C2×#95—712no—
345C2×#95C2×D7112yes—
346C25C22×D7111yescharacteristic
347C23×C4—24no—
348C23×C4—24no—
349C23×C4—41no—
350C23×C4—41no—
351C23×C4—712no—
352C23×C4—712no—
353C23×C4—712no—
354C23×C4—712no—
355C23×C4C2×Dic7114yes—
356C23×C4C22×D7111yescharacteristic
357C23×C4C22×D7111yescharacteristic
358C23×C4C22×D7112yes—
359C23×C4C22×D7112yes—
360C2×C4⋊C4—24no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—2842no—
363C2×C4⋊C4—2842no—
364C2×C4⋊C4—2842no—
365C2×C4⋊C4—2841no—
366C2×C4⋊C4—2842no—
367C2×C4⋊C4—2841no—
368C2×C4⋊C4—1424no—
369C2×C4⋊C4—1424no—
370C2×C4⋊C4—1424no—
371C2×C4⋊C4—1424no—
372C2×C4⋊C4—1424no—
373C2×C4⋊C4—1424no—
374C2×C4⋊C4—1424no—
375C2×C4⋊C4C2×Dic7118yes—
376C2×C22⋊C4—41no—
377C2×C22⋊C4—41no—
378C2×C22⋊C4—2842no—
379C2×C22⋊C4—2842no—
380C2×C22⋊C4—1424no—
381C2×C22⋊C4—1424no—
382C2×C22⋊C4—1424no—
383C2×C22⋊C4—1424no—
384C2×C22⋊C4Dic14214yes—
385C2×C22⋊C4D28414yes—
386C2×C22⋊C4C2×Dic7114yes—
387C2×C22⋊C4C2×Dic7114yes—
388C2×C42—2842no—
389C2×C42—1424no—
390C2×C42—1424no—
391#95—2841no—
392#95—2841no—
393#95—2842no—
394#95—718no—
395#95—1424no—
396#95—1424no—
397#95—1424no—
398#95—1424no—
399#95C2×Dic7118yes—
400C24—22no—
401C24—22no—
402C24—21no—
403C24—21no—
404C24—22no—
405C24—22no—
406C24—21no—
407C24—21no—
408C24C22×Dic7112yes—
409C24C2×D28412yes—
410C24C2×Dic14212yes—
411C24C23×D7111yescharacteristic
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—216no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—714no—
434C22×C4—714no—
435C22×C4—1422no—
436C22×C4—1422no—
437C22×C4—1422no—
438C22×C4—1422no—
439C22×C4—1422no—
440C22×C4—1422no—
441C22×C4—1422no—
442C22×C4—1422no—
443C22×C4—1422no—
444C22×C4—1422no—
445C22×C4—1422no—
446C22×C4—1422no—
447C22×C4—1422no—
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×C4C22×Dic7112yes—
462C22×C4C22×Dic7112yes—
463C22×C4C22×Dic7114yes—
464C22×C4C22×Dic7114yes—
465C22×C4D4×D7114yes—
466C22×C4C2×D28414yes—
467C22×C4C2×Dic14214yes—
468C22×C4#1176586414yes—
469C4⋊C4—216no—
470C4⋊C4—2848no—
471C4⋊C4—2848no—
472C4⋊C4—2848no—
473C4⋊C4—2844no—
474C4⋊C4—2844no—
475C22⋊C4—44no—
476C22⋊C4—2848no—
477C22⋊C4—2848no—
478C22⋊C4C4⋊Dic71116yescomplemented
479C42—2848no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—41no—
502C23C2×C4⋊Dic7114yes—
503C23C23×Dic7111yescharacteristic
504C23C2×D4×D7112yes—
505C23C22×D28411yescharacteristic
506C23C22×Dic14211yescharacteristic
507C23C2×#1176586412yes—
508C23#1455617311yescharacteristic
509C23#1455617311yescharacteristic
510C23#1455617412yes—
511C23#1455617512yes—
512C23#1455617611yescharacteristic
513C23#1455617611yescharacteristic
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—416no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—2842no—
538C2×C4—2842no—
539C2×C4—2842no—
540C2×C4—2842no—
541C2×C4—2842no—
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—2842no—
558C2×C4—2842no—
559C2×C4—1424no—
560C2×C4—1424no—
561C2×C4—1424no—
562C2×C4—1424no—
563C2×C4—1424no—
564C2×C4—1424no—
565C2×C4C2×C4⋊Dic7118yes—
566C2×C4D4×Dic7118yes—
567C22—28no—
568C22—24no—
569C22—42no—
570C22—44no—
571C22—44no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22C22×C4⋊Dic7111yescharacteristic
579C22C2×D4×Dic7112yes—
580C22#1798647411yescharacteristic
581C22#1798647711yescharacteristic
582C22#1798648212yes—
583C22#1798657212yes—
584C22#1798657912yes—
585C22#1798658011yescharacteristic
586C22#1798658011yescharacteristic
587C22#1798660912yes—
588C22#1798662212yes—
589C22#1798663011yescharacteristic
590C22#1798663011yescharacteristic
591C22#1798663111yescharacteristic
592C22D4×D28412yes—
593C22#1798663612yes—
594C22#1798665312yes—
595C22D4×Dic14212yes—
596C22#1798668011yescharacteristic
597C22#1798670612yes—
598C22#1798673911yescharacteristic
599C22#1798674612yes—
600C22#1798675211yescharacteristic
601C4—28no—
602C4—28no—
603C4—42no—
604C4—42no—
605C4—2844no—
606C4—2844no—
607C4—5682no—
608C4—5682no—
609C2—44no—
610C2#2259095811yescharacteristic
611C2#2259102412yes—
612C2D4×C4⋊Dic7112yes—
613C2#2259132411yescharacteristic
614C2#2259148811yescharacteristic
615C2#2259459412yes—
616C2#2259463111yescharacteristic
617C2#2259465511yescharacteristic
618C2#2259500212yes—
619C2#2259524611yescharacteristic
620C2#2259546911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC2×#385×C71C2
socle, cosocleC24×C71C24
prime radicalC25×C71C23
hypercenterC2×#385D71
hypocenterC71#14402
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
1. C2×#17985898C2semidirect
4. #22586605C2semidirect
5. #22586593C2semidirect
8. #22586574C2semidirect
33. #385×C71C4semidirect
168. C22⋊C4×C71C4⋊C4semidirect
320. C71#14402semidirect
335. #385Dic71semidirect
478. C22⋊C4C4⋊Dic71semidirect

Maximal subgroups

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

subgroupindex
C2×#179858982
C2×#179858922
C2×#179858592
#225866052
#225865932
#225865852
#225865742
C2×#385×C712
#1440271