Finite Group Databasebeta

← #31924412

Subgroups — #31924412

Order 18176.

1558 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985935C211yesmaximal, characteristic
2C2×#17985897C211yesmaximal, characteristic, complemented
3#22586609C211yesmaximal, characteristic, complemented
4#22586596C211yesmaximal, characteristic, complemented
5#22586591C212yesmaximal, complemented
6#22586585C211yesmaximal, characteristic
7#22586583C212yesmaximal
8#22586573C212yesmaximal, complemented
9#22586567C211yesmaximal, characteristic, complemented
10#22586563C212yesmaximal
11C2×C4×C22⋊C4×C71C211yesmaximal, characteristic
12C22×C4⋊Dic71C2211yescharacteristic
13C22×C4⋊Dic71C2211yescharacteristic
14C22×Dic71⋊C4C2212yes—
15C22×Dic71⋊C4C2212yes—
16C22×C22⋊Dic71C2212yes—
17C22×C22⋊Dic71C2212yes—
18#17985935C418yescomplemented
19#17985897—24no—
20C2×C71⋊#95C2212yes—
21C2×C71⋊#95C2212yes—
22C2×C71⋊#95C2211yescharacteristic
23C2×C71⋊#95C2212yes—
24C2×C71⋊#95C2212yes—
25C2×C71⋊#95C2212yes—
26C2×C71⋊#95C2212yes—
27C2×C71⋊#95C2212yes—
28C2×C71⋊#95C2212yes—
29C2×C71⋊#95C2211yescharacteristic
30C24×C4×C71C2211yescharacteristic
31C22×C22⋊C4×C71C2211yescharacteristic
32C22×C22⋊C4×C71C2211yescharacteristic
33C22×C42×C71C2211yescharacteristic
34C22×C42×C71C2211yescharacteristic
35C4×C22⋊C4×C71—24no—
36C2×#95×C71C2211yescharacteristic
37C2×#95×C71C2211yescharacteristic
38C23×Dic71C2312yes—
39C23×Dic71C2312yes—
40C23×Dic71C2312yes—
41C23×Dic71C2312yes—
42C2×C4⋊Dic71—22no—
43C2×C4⋊Dic71—22no—
44C2×C4⋊Dic71—22no—
45C2×C4⋊Dic71—42no—
46C2×C4⋊Dic71C2×C414yescomplemented
47C2×Dic71⋊C4—24no—
48C2×Dic71⋊C4—42no—
49C2×Dic71⋊C4—42no—
50C2×Dic71⋊C4—42no—
51C2×Dic71⋊C4—42no—
52C2×Dic71⋊C4C2×C418yescomplemented
53C2×C22⋊Dic71—24no—
54C2×C22⋊Dic71—42no—
55C2×C22⋊Dic71—42no—
56C2×C22⋊Dic71—42no—
57C2×C22⋊Dic71—42no—
58C2×C22⋊Dic71C2×C418yes—
59C71⋊#95—24no—
60C71⋊#95—24no—
61C71⋊#95—22no—
62C71⋊#95—24no—
63C71⋊#95—42no—
64C71⋊#95—42no—
65C71⋊#95—42no—
66C71⋊#95—42no—
67C71⋊#95—42no—
68C71⋊#95C2×C414yescomplemented
69C25×C71C2311yescharacteristic
70C23×C4×C71—22no—
71C23×C4×C71—22no—
72C23×C4×C71—42no—
73C23×C4×C71—42no—
74C23×C4×C71Q814yes—
75C23×C4×C71C2311yescharacteristic
76C23×C4×C71C2311yescharacteristic
77C23×C4×C71C2311yescharacteristic
78C23×C4×C71C2311yescharacteristic
79C23×C4×C71C2311yescharacteristic
80C23×C4×C71C2311yescharacteristic
81C2×C22⋊C4×C71—22no—
82C2×C22⋊C4×C71—22no—
83C2×C22⋊C4×C71—22no—
84C2×C22⋊C4×C71—22no—
85C2×C22⋊C4×C71—22no—
86C2×C22⋊C4×C71C2×C414yes—
87C2×C42×C71—22no—
88C2×C42×C71—42no—
89C2×C42×C71—42no—
90C2×C42×C71D414yes—
91#95×C71—22no—
92#95×C71—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×Dic71—22no—
116C22×Dic71—22no—
117C22×Dic71—22no—
118C22×Dic71—22no—
119C22×Dic71C22×C414yes—
120C22×Dic71C22×C414yes—
121C4⋊Dic71—28no—
122C4⋊Dic71—44no—
123Dic71⋊C4—48no—
124Dic71⋊C4—48no—
125C22⋊Dic71—48no—
126C22⋊Dic71—48no—
127C24×C71—21no—
128C24×C71—21no—
129C24×C71—21no—
130C24×C71—21no—
131C24×C71—22no—
132C24×C71—22no—
133C24×C71—22no—
134C24×C71—22no—
135C24×C71C22×C412yes—
136C24×C71C2×Q812yes—
137C24×C71#4112yes—
138C24×C71C2411yescharacteristic
139C22×C4×C71—22no—
140C22×C4×C71—22no—
141C22×C4×C71—22no—
142C22×C4×C71—22no—
143C22×C4×C71—21no—
144C22×C4×C71—21no—
145C22×C4×C71—21no—
146C22×C4×C71—21no—
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—22no—
154C22×C4×C71—22no—
155C22×C4×C71—22no—
156C22×C4×C71—22no—
157C22×C4×C71—22no—
158C22×C4×C71—22no—
159C22×C4×C71—28no—
160C22×C4×C71—48no—
161C22×C4×C71—44no—
162C22×C4×C71—44no—
163C22×C4×C71—48no—
164C22×C4×C71C22×C412yes—
165C22×C4×C71C22×C412yes—
166C22×C4×C71C2×D412yes—
167C22×C4×C71C2×D412yes—
168C22×C4×C71C2×D412yes—
169C22×C4×C71C2×Q812yes—
170C22×C4×C71C2×Q812yes—
171C22×C4×C71#4112yes—
172C22×C4×C71#4112yes—
173C22×C4×C71#4112yes—
174C22⋊C4×C71—28no—
175C22⋊C4×C71—28no—
176C42×C71—44no—
177C42×C71—44no—
178C23×C71—22no—
179C23×C71—22no—
180C23×C71—24no—
181C23×C71—24no—
182C23×C71—22no—
183C23×C71—22no—
184C23×C71—22no—
185C23×C71—22no—
186C23×C71—24no—
187C23×C71—24no—
188C23×C71—24no—
189C23×C71—24no—
190C23×C71—41no—
191C23×C71—41no—
192C23×C71—41no—
193C23×C71—41no—
194C23×C71—42no—
195C23×C71—42no—
196C23×C71—42no—
197C23×C71—42no—
198C23×C71—42no—
199C23×C71—42no—
200C23×C71C4×Q814yes—
201C23×C71C23×C411yescharacteristic
202C23×C71C22×D411yescharacteristic
203C23×C71C22×Q811yescharacteristic
204C23×C71C2×#4111yescharacteristic
205C23×C71C2×#4111yescharacteristic
206C23×C71C2×#4111yescharacteristic
207C23×C71C2×#4111yescharacteristic
208C23×C71H(5,𝔽2)12yes—
209C23×C71H(5,𝔽2)12yes—
210C23×C71H(5,𝔽2)12yes—
211C23×C712−1+412yes—
212C2×C4×C71—24no—
213C2×C4×C71—22no—
214C2×C4×C71—22no—
215C2×C4×C71—22no—
216C2×C4×C71—22no—
217C2×C4×C71—24no—
218C2×C4×C71—24no—
219C2×C4×C71—24no—
220C2×C4×C71—22no—
221C2×C4×C71—24no—
222C2×C4×C71—22no—
223C2×C4×C71—22no—
224C2×C4×C71—22no—
225C2×C4×C71—22no—
226C2×C4×C71—22no—
227C2×C4×C71—24no—
228C2×C4×C71—24no—
229C2×C4×C71—416no—
230C2×C4×C71—41no—
231C2×C4×C71—41no—
232C2×C4×C71—41no—
233C2×C4×C71—41no—
234C2×C4×C71—41no—
235C2×C4×C71—41no—
236C2×C4×C71—42no—
237C2×C4×C71—42no—
238C2×C4×C71—42no—
239C2×C4×C71—42no—
240C2×C4×C71—42no—
241C2×C4×C71C4×D414yes—
242C2×C4×C71C22⋊Q814yes—
243C2×Dic71—24no—
244C2×Dic71—24no—
245C2×Dic71—24no—
246C2×Dic71—24no—
247C2×Dic71—24no—
248C2×Dic71—24no—
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—
265C2×Dic71—42no—
266C2×Dic71—42no—
267C2×Dic71—42no—
268C2×Dic71—42no—
269C2×Dic71—42no—
270C2×Dic71—42no—
271C22×C71—28no—
272C22×C71—24no—
273C22×C71—42no—
274C22×C71—42no—
275C22×C71—44no—
276C22×C71—42no—
277C22×C71—42no—
278C22×C71—42no—
279C22×C71—42no—
280C22×C71—44no—
281C22×C71—44no—
282C22×C71C2×C4×D411yescharacteristic
283C22×C71C2×C4×Q811yescharacteristic
284C22×C71C4×#4111yescharacteristic
285C22×C71#51812yes—
286C22×C71#52012yes—
287C22×C71C2×C22⋊Q811yescharacteristic
288C22×C71#52511yescharacteristic
289C22×C71#52911yescharacteristic
290C22×C71#53311yescharacteristic
291C22×C71#53411yescharacteristic
292C22×C71#53511yescharacteristic
293C22×C71#53511yescharacteristic
294C22×C71#53611yescharacteristic
295C22×C71#53812yes—
296C22×C71#53812yes—
297C22×C71#53912yes—
298C22×C71#53912yes—
299C22×C71#53912yes—
300C22×C71#54012yes—
301C22×C71#54012yes—
302C22×C71#54212yes—
303C22×C71#54312yes—
304C22×C71#54412yes—
305C4×C71—24no—
306C4×C71—24no—
307C4×C71—42no—
308C4×C71—42no—
309C4×C71—42no—
310C4×C71—42no—
311Dic71—44no—
312Dic71—44no—
313Dic71—82no—
314Dic71—82no—
315#14317—711nomaximal, Sylow
316C2×C71—44no—
317C2×C71C4×C22⋊Q811yescharacteristic
318C2×C71#232211yescharacteristic
319C2×C71#232811yescharacteristic
320C2×C71#232912yes—
321C2×C71#233512yes—
322C2×C71#262611yescharacteristic
323C2×C71#262811yescharacteristic
324C2×C71#263311yescharacteristic
325C2×C71#263511yescharacteristic
326C2×C71#264912yes—
327C2×C71#266712yes—
328C2×#399—711no—
329C2×#389—711no—
330C2×C4×C22⋊C4D7111yescharacteristic
331#1445—711no—
332#1442—711no—
333#1441—712no—
334#1439—712no—
335#1439—711no—
336#1436—712no—
337#1434—711no—
338C22⋊#95—712no—
339C71#1431711yescharacteristic, complemented, Sylow
340C24×C4C2×D7111yescharacteristic
341C22×C4⋊C4—712no—
342C22×C4⋊C4—712no—
343C22×C4⋊C4—711no—
344C22×C4⋊C4—711no—
345C22×C22⋊C4—712no—
346C22×C22⋊C4—712no—
347C22×C22⋊C4C2×D7111yescharacteristic
348C22×C22⋊C4C2×D7111yescharacteristic
349C22×C42C2×D7111yescharacteristic
350C22×C42C2×D7111yescharacteristic
351#399—718no—
352#389—1424no—
353C4×C22⋊C4—24no—
354C2×#95—712no—
355C2×#95—712no—
356C2×#95—712no—
357C2×#95—712no—
358C2×#95—712no—
359C2×#95—711no—
360C2×#95—712no—
361C2×#95—712no—
362C2×#95—711no—
363C2×#95—712no—
364C2×#95C2×D7111yescharacteristic
365C2×#95C2×D7111yescharacteristic
366C25C22×D7111yescharacteristic
367C23×C4—22no—
368C23×C4—22no—
369C23×C4—42no—
370C23×C4—42no—
371C23×C4—712no—
372C23×C4—712no—
373C23×C4—712no—
374C23×C4—712no—
375C23×C4Dic14214yes—
376C23×C4C22×D7111yescharacteristic
377C23×C4C22×D7111yescharacteristic
378C23×C4C22×D7111yescharacteristic
379C23×C4C22×D7111yescharacteristic
380C23×C4C22×D7111yescharacteristic
381C23×C4C22×D7111yescharacteristic
382C2×C4⋊C4—2842no—
383C2×C4⋊C4—2842no—
384C2×C4⋊C4—2842no—
385C2×C4⋊C4—2842no—
386C2×C4⋊C4—2842no—
387C2×C4⋊C4—718no—
388C2×C4⋊C4—714no—
389C2×C4⋊C4—1424no—
390C2×C4⋊C4—1422no—
391C2×C4⋊C4—1422no—
392C2×C4⋊C4—1422no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—22no—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—2842no—
399C2×C22⋊C4—2842no—
400C2×C22⋊C4—2842no—
401C2×C22⋊C4—2842no—
402C2×C22⋊C4—718no—
403C2×C22⋊C4—1424no—
404C2×C22⋊C4C4×D7114yes—
405C2×C42—22no—
406C2×C42—42no—
407C2×C42—42no—
408C2×C42C22⋊D7114yes—
409#95—22no—
410#95—22no—
411#95—2842no—
412#95—2842no—
413#95—2842no—
414#95—2842no—
415#95—2842no—
416#95—714no—
417#95—1424no—
418#95—1422no—
419#95—1424no—
420#95—1424no—
421C24—21no—
422C24—21no—
423C24—21no—
424C24—21no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24C2×C4×D7112yes—
430C24C2×Dic14212yes—
431C24#1176586512yes—
432C24C23×D7111yescharacteristic
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—28no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—44no—
458C22×C4—714no—
459C22×C4—714no—
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×C4—1422no—
481C22×C4—1422no—
482C22×C4—1422no—
483C22×C4—1422no—
484C22×C4—1422no—
485C22×C4—1422no—
486C22×C4C2×C4×D7112yes—
487C22×C4C2×C4×D7112yes—
488C22×C4C2×C22⋊D7112yes—
489C22×C4C2×C22⋊D7112yes—
490C22×C4C2×C22⋊D7112yes—
491C22×C4C2×Dic14212yes—
492C22×C4C2×Dic14212yes—
493C22×C4#1176586512yes—
494C22×C4#1176586512yes—
495C22×C4#1176586512yes—
496C4⋊C4—2848no—
497C4⋊C4—2844no—
498C4⋊C4—2848no—
499C4⋊C4—1428no—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—2848no—
503C22⋊C4—2848no—
504C42—44no—
505C42—44no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—41no—
528C23C4×Dic14214yes—
529C23C22×C4×D7111yescharacteristic
530C23C22×C22⋊D7111yescharacteristic
531C23C22×Dic14211yescharacteristic
532C23C2×#1176586511yescharacteristic
533C23C2×#1176586511yescharacteristic
534C23C2×#1176586511yescharacteristic
535C23C2×#1176586511yescharacteristic
536C23#1455617312yes—
537C23#1455617312yes—
538C23#1455617412yes—
539C23#1455617512yes—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—416no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
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—2842no—
579C2×C4—2842no—
580C2×C4—2842no—
581C2×C4—2842no—
582C2×C4—2842no—
583C2×C4—2842no—
584C2×C4—2842no—
585C2×C4—2842no—
586C2×C4—2842no—
587C2×C4—2842no—
588C2×C4—2842no—
589C2×C4—2842no—
590C2×C4—2842no—
591C2×C4—1424no—
592C2×C4—1424no—
593C2×C4—1424no—
594C2×C4—1424no—
595C2×C4—1424no—
596C2×C4—1424no—
597C2×C4C4×C22⋊D7114yes—
598C2×C4#1455610714yes—
599C22—28no—
600C22—24no—
601C22—42no—
602C22—44no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—44no—
607C22—42no—
608C22—42no—
609C22—44no—
610C22C2×C4×C22⋊D7111yescharacteristic
611C22C2×C4×Dic14211yescharacteristic
612C22C4×#1176586511yescharacteristic
613C22#1798647112yes—
614C22#1798647812yes—
615C22C2×#1455610711yescharacteristic
616C22#1798651011yescharacteristic
617C22#1798653311yescharacteristic
618C22#1798655611yescharacteristic
619C22#1798655811yescharacteristic
620C22#1798656311yescharacteristic
621C22#1798656611yescharacteristic
622C22#1798657011yescharacteristic
623C22#1798658212yes—
624C22#1798658712yes—
625C22#1798659112yes—
626C22#1798659112yes—
627C22#1798659112yes—
628C22#1798660012yes—
629C22#1798660012yes—
630C22#1798662012yes—
631C22#1798662512yes—
632C22#1798662712yes—
633C4—24no—
634C4—24no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—2844no—
640C4—2844no—
641C4—5682no—
642C4—5682no—
643C2—44no—
644C2C4×#1455610711yescharacteristic
645C2#2259100511yescharacteristic
646C2#2259106211yescharacteristic
647C2#2259106812yes—
648C2#2259113212yes—
649C2#2259414511yescharacteristic
650C2#2259416111yescharacteristic
651C2#2259420811yescharacteristic
652C2#2259422411yescharacteristic
653C2#2259433712yes—
654C2#2259453412yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC2×C4×C22⋊C4×C71C2
socle, cosocleC24×C71C24
prime radicalC25×C71C23
hypercenterC2×C4×C22⋊C4D71
hypocenterC71#14317
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#17985897C2semidirect
3. #22586609C2semidirect
4. #22586596C2semidirect
5. #22586591C2semidirect
8. #22586573C2semidirect
9. #22586567C2semidirect
18. #17985935C4semidirect
46. C2×C4⋊Dic71C2×C4semidirect
52. C2×Dic71⋊C4C2×C4semidirect
68. C71⋊#95C2×C4semidirect
339. C71#14317semidirect

Maximal subgroups

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

subgroupindex
C2×#179859352
C2×#179858972
#225866092
#225865962
#225865912
#225865852
#225865832
#225865732
#225865672
#225865632
C2×C4×C22⋊C4×C712
#1431771