Finite Group Databasebeta

← #31919488

Subgroups — #31919488

Order 18176.

1342 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985865C211yesmaximal, characteristic
2C2×C4⋊C4×Dic71C211yesmaximal, characteristic
3C2×C4×Dic71⋊C4C211yesmaximal, characteristic
4#22586606C212yesmaximal
5#22586601C212yesmaximal
6#22586571C212yesmaximal
7#22586569C211yesmaximal, characteristic
8#22586569C211yesmaximal, characteristic
9#22586569C211yesmaximal, characteristic
10#22586566C212yesmaximal
11C71×C4⋊#95C211yesmaximal, characteristic
12C22×C4⋊Dic71C2211yescharacteristic
13C22×C4⋊Dic71C2211yescharacteristic
14C22×Dic71⋊C4C2211yescharacteristic
15C22×Dic71⋊C4C2211yescharacteristic
16C22×C4×Dic71C2212yes—
17C22×C4×Dic71C2211yescharacteristic
18C22×C4×Dic71C2211yescharacteristic
19#17985865—24no—
20C4⋊C4×Dic71—24no—
21C4×Dic71⋊C4C418yescomplemented
22C2×C71⋊#95C2212yes—
23C2×C71⋊#95C2212yes—
24C2×C71⋊#95C2212yes—
25C2×C71⋊#95C2212yes—
26C2×C71⋊#95C2212yes—
27C2×C71⋊#95C2212yes—
28C2×C71⋊#95C2212yes—
29C2×C71⋊#95C2212yes—
30C2×C71⋊#95C2212yes—
31C2×C71⋊#95C2211yescharacteristic
32C2×C71⋊#95C2211yescharacteristic
33C22×C4⋊C4×C71C2211yescharacteristic
34C22×C4⋊C4×C71C2211yescharacteristic
35C22×C42×C71C2211yescharacteristic
36C2×#95×C71C2212yes—
37C2×#95×C71C2212yes—
38C23×Dic71C2312yes—
39C23×Dic71C2311yescharacteristic
40C23×Dic71C2311yescharacteristic
41C23×Dic71C2312yes—
42C23×Dic71C2311yescharacteristic
43C23×Dic71C2311yescharacteristic
44C2×C4⋊Dic71—22no—
45C2×C4⋊Dic71—22no—
46C2×C4⋊Dic71—22no—
47C2×C4⋊Dic71—22no—
48C2×C4⋊Dic71—22no—
49C2×C4⋊Dic71D414yes—
50C2×Dic71⋊C4—22no—
51C2×Dic71⋊C4—22no—
52C2×Dic71⋊C4—22no—
53C2×Dic71⋊C4—22no—
54C2×Dic71⋊C4C2×C414yes—
55C2×Dic71⋊C4C2×C414yes—
56C2×C4×Dic71—24no—
57C2×C4×Dic71—24no—
58C2×C4×Dic71—22no—
59C2×C4×Dic71—22no—
60C2×C4×Dic71—22no—
61C2×C4×Dic71—22no—
62C2×C4×Dic71—42no—
63C2×C4×Dic71C2×C414yes—
64C2×C4×Dic71C2×C414yes—
65C71⋊#95—24no—
66C71⋊#95—24no—
67C71⋊#95—24no—
68C71⋊#95—24no—
69C71⋊#95—24no—
70C71⋊#95—24no—
71C71⋊#95—42no—
72C71⋊#95—42no—
73C71⋊#95—42no—
74C71⋊#95C2×C414yes—
75C71⋊#95C2×C414yes—
76C23×C4×C71C2312yes—
77C23×C4×C71C2311yescharacteristic
78C23×C4×C71C2311yescharacteristic
79C23×C4×C71C2312yes—
80C23×C4×C71C2311yescharacteristic
81C2×C4⋊C4×C71—22no—
82C2×C4⋊C4×C71—22no—
83C2×C4⋊C4×C71—22no—
84C2×C4⋊C4×C71—22no—
85C2×C4⋊C4×C71—22no—
86C2×C4⋊C4×C71Q814yes—
87C2×C42×C71—22no—
88C2×C42×C71—22no—
89C2×C42×C71C2×C414yes—
90#95×C71—42no—
91#95×C71—42no—
92C22×Dic71—21no—
93C22×Dic71—21no—
94C22×Dic71—21no—
95C22×Dic71—21no—
96C22×Dic71—21no—
97C22×Dic71—21no—
98C22×Dic71—21no—
99C22×Dic71—21no—
100C22×Dic71—21no—
101C22×Dic71—22no—
102C22×Dic71—22no—
103C22×Dic71—21no—
104C22×Dic71—21no—
105C22×Dic71—21no—
106C22×Dic71—21no—
107C22×Dic71—21no—
108C22×Dic71—22no—
109C22×Dic71—22no—
110C22×Dic71—21no—
111C22×Dic71—21no—
112C22×Dic71—21no—
113C22×Dic71—22no—
114C22×Dic71—21no—
115C22×Dic71—21no—
116C22×Dic71—21no—
117C22×Dic71—22no—
118C22×Dic71—22no—
119C22×Dic71—22no—
120C22×Dic71—22no—
121C22×Dic71—22no—
122C22×Dic71—22no—
123C22×Dic71—22no—
124C22×Dic71—22no—
125C22×Dic71C22×C412yes—
126C22×Dic71C22×C412yes—
127C22×Dic71C22×C412yes—
128C22×Dic71C22×C412yes—
129C22×Dic71C2×D414yes—
130C22×Dic71#4112yes—
131C22×Dic71#4112yes—
132C22×Dic71#4112yes—
133C22×Dic71#4112yes—
134C4⋊Dic71—28no—
135C4⋊Dic71—44no—
136Dic71⋊C4—28no—
137Dic71⋊C4—28no—
138C4×Dic71—28no—
139C4×Dic71—28no—
140C4×Dic71—48no—
141C24×C71C2411yescharacteristic
142C22×C4×C71—21no—
143C22×C4×C71—21no—
144C22×C4×C71—22no—
145C22×C4×C71—21no—
146C22×C4×C71—21no—
147C22×C4×C71—22no—
148C22×C4×C71—22no—
149C22×C4×C71—22no—
150C22×C4×C71—21no—
151C22×C4×C71—21no—
152C22×C4×C71—21no—
153C22×C4×C71—21no—
154C22×C4×C71—21no—
155C22×C4×C71—21no—
156C22×C4×C71—22no—
157C22×C4×C71—22no—
158C22×C4×C71—21no—
159C22×C4×C71—21no—
160C22×C4×C71—22no—
161C22×C4×C71—22no—
162C22×C4×C71—22no—
163C22×C4×C71—22no—
164C22×C4×C71—22no—
165C22×C4×C71—22no—
166C22×C4×C71—22no—
167C22×C4×C71C22×C412yes—
168C22×C4×C71C22×C412yes—
169C22×C4×C71C22×C412yes—
170C22×C4×C71C2×D412yes—
171C22×C4×C71C2×Q814yes—
172C22×C4×C71C2×Q812yes—
173C22×C4×C71#4112yes—
174C22×C4×C71#4112yes—
175C22×C4×C71#4112yes—
176C22×C4×C71#4112yes—
177C4⋊C4×C71—28no—
178C4⋊C4×C71—44no—
179C42×C71—28no—
180C23×C71C23×C411yescharacteristic
181C23×C71C22×D411yescharacteristic
182C23×C71C22×Q811yescharacteristic
183C23×C71C2×#4111yescharacteristic
184C23×C71C2×#4111yescharacteristic
185C23×C71C2×#4111yescharacteristic
186C23×C71C2×#4111yescharacteristic
187C23×C71C2×#4111yescharacteristic
188C23×C71C2×#4111yescharacteristic
189C23×C71C2×#4111yescharacteristic
190C23×C71C2×#4111yescharacteristic
191C23×C71H(5,𝔽2)11yescharacteristic
192C23×C71H(5,𝔽2)11yescharacteristic
193C23×C712−1+411yescharacteristic
194C23×C712−1+411yescharacteristic
195C2×C4×C71—22no—
196C2×C4×C71—22no—
197C2×C4×C71—22no—
198C2×C4×C71—22no—
199C2×C4×C71—22no—
200C2×C4×C71—22no—
201C2×C4×C71—24no—
202C2×C4×C71—24no—
203C2×C4×C71—24no—
204C2×C4×C71—42no—
205C2×C4×C71—42no—
206C2×C4×C71—41no—
207C2×C4×C71—42no—
208C2×C4×C71—41no—
209C2×C4×C71—41no—
210C2×C4×C71—41no—
211C2×C4×C71—42no—
212C2×C4×C71—41no—
213C2×C4×C71—41no—
214C2×C4×C71—41no—
215C2×C4×C71—41no—
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#11714yes—
224C2×C4×C71C4×D414yes—
225C2×C4×C71C4×Q814yes—
226C2×C4×C71C22⋊Q814yes—
227C2×C4×C71#12314yes—
228C2×C4×C71#12514yes—
229C2×C4×C71#12614yes—
230C2×Dic71—22no—
231C2×Dic71—22no—
232C2×Dic71—22no—
233C2×Dic71—22no—
234C2×Dic71—22no—
235C2×Dic71—22no—
236C2×Dic71—22no—
237C2×Dic71—22no—
238C2×Dic71—22no—
239C2×Dic71—22no—
240C2×Dic71—22no—
241C2×Dic71—22no—
242C2×Dic71—22no—
243C2×Dic71—22no—
244C2×Dic71—22no—
245C2×Dic71—22no—
246C2×Dic71—22no—
247C2×Dic71—22no—
248C2×Dic71—24no—
249C2×Dic71—24no—
250C2×Dic71—24no—
251C2×Dic71—42no—
252C2×Dic71—41no—
253C2×Dic71—41no—
254C2×Dic71—41no—
255C2×Dic71—41no—
256C2×Dic71—42no—
257C2×Dic71—41no—
258C2×Dic71—41no—
259C2×Dic71—41no—
260C2×Dic71—41no—
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#11714yes—
271C2×Dic71#11714yes—
272C22×C71C2×#11711yescharacteristic
273C22×C71C2×#11711yescharacteristic
274C22×C71C2×#11711yescharacteristic
275C22×C71C2×C4×D411yescharacteristic
276C22×C71C2×C4×Q811yescharacteristic
277C22×C71#52011yescharacteristic
278C22×C71#52011yescharacteristic
279C22×C71C2×C22⋊Q811yescharacteristic
280C22×C71C2×#12311yescharacteristic
281C22×C71C2×#12511yescharacteristic
282C22×C71C2×#12611yescharacteristic
283C22×C71#52911yescharacteristic
284C22×C71#52911yescharacteristic
285C22×C71#52911yescharacteristic
286C22×C71#52911yescharacteristic
287C22×C71#54011yescharacteristic
288C22×C71#54111yescharacteristic
289C22×C71#54211yescharacteristic
290C22×C71#54211yescharacteristic
291C22×C71#54611yescharacteristic
292C22×C71#54711yescharacteristic
293C22×C71#54811yescharacteristic
294C22×C71#55011yescharacteristic
295C22×C71#55211yescharacteristic
296C22×C71#55311yescharacteristic
297C22×C71#55411yescharacteristic
298C22×C71#55411yescharacteristic
299C22×C71#55511yescharacteristic
300C22×C71#55611yescharacteristic
301C22×C71#55611yescharacteristic
302C22×C71#55611yescharacteristic
303C22×C71#55711yescharacteristic
304C22×C71#55711yescharacteristic
305C22×C71#55911yescharacteristic
306C22×C71#55911yescharacteristic
307C4×C71—42no—
308C4×C71—42no—
309C4×C71—44no—
310C4×C71—82no—
311C4×C71#38718yes—
312Dic71—24no—
313Dic71—24no—
314Dic71—42no—
315Dic71—42no—
316Dic71—44no—
317Dic71—82no—
318#13885—711nomaximal, Sylow
319C2×C71#228311yescharacteristic
320C2×C71C2×#38711yescharacteristic
321C2×C71#231411yescharacteristic
322C2×C71#235211yescharacteristic
323C2×C71#235411yescharacteristic
324C2×C71#235511yescharacteristic
325C2×C71#235511yescharacteristic
326C2×C71#244711yescharacteristic
327C2×C71#244811yescharacteristic
328C2×C71#245311yescharacteristic
329C2×C71#245411yescharacteristic
330C2×C71#250811yescharacteristic
331C2×C71#250911yescharacteristic
332C2×C71#251311yescharacteristic
333C2×C71#251411yescharacteristic
334C2×#382—711no—
335C2×C4×C4⋊C4—711no—
336C2×C4×C4⋊C4—711no—
337#1445—712no—
338#1443—712no—
339#1436—712no—
340C4⋊#95—711no—
341C4⋊#95—711no—
342C4⋊#95—711no—
343C4⋊#95D7111yescharacteristic
344#1434—712no—
345C71#1388511yescharacteristic, complemented, Sylow
346C22×C4⋊C4—711no—
347C22×C4⋊C4—711no—
348C22×C4⋊C4—711no—
349C22×C4⋊C4—711no—
350C22×C4⋊C4C2×D7111yescharacteristic
351C22×C4⋊C4C2×D7111yescharacteristic
352C22×C42—712no—
353C22×C42—711no—
354C22×C42—711no—
355C22×C42C2×D7111yescharacteristic
356#382—1424no—
357C4×C4⋊C4—718no—
358C4×C4⋊C4—1424no—
359C2×#95—712no—
360C2×#95—712no—
361C2×#95—712no—
362C2×#95—711no—
363C2×#95—711no—
364C2×#95—712no—
365C2×#95—712no—
366C2×#95—712no—
367C2×#95—712no—
368C2×#95—712no—
369C2×#95—712no—
370C2×#95C2×D7112yes—
371C2×#95C2×D7112yes—
372C23×C4—711no—
373C23×C4—711no—
374C23×C4—711no—
375C23×C4—712no—
376C23×C4—711no—
377C23×C4—712no—
378C23×C4C22×D7112yes—
379C23×C4C22×D7111yescharacteristic
380C23×C4C22×D7111yescharacteristic
381C23×C4C22×D7112yes—
382C23×C4C22×D7111yescharacteristic
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—714no—
389C2×C4⋊C4—714no—
390C2×C4⋊C4—714no—
391C2×C4⋊C4—1422no—
392C2×C4⋊C4—1422no—
393C2×C4⋊C4—1422no—
394C2×C4⋊C4—1422no—
395C2×C4⋊C4—1422no—
396C2×C4⋊C4—1422no—
397C2×C4⋊C4—1422no—
398C2×C4⋊C4—1422no—
399C2×C4⋊C4—1422no—
400C2×C4⋊C4Dic14214yes—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—2842no—
404C2×C42—714no—
405C2×C42—714no—
406C2×C42—1424no—
407C2×C42—1424no—
408C2×C42—1422no—
409C2×C42—1422no—
410C2×C42—1422no—
411C2×C42—1422no—
412C2×C42C4×D7114yes—
413#95—42no—
414#95—42no—
415#95—2842no—
416#95—2842no—
417#95—2842no—
418#95—714no—
419#95—714no—
420#95—1424no—
421#95—1424no—
422#95—1424no—
423#95—1424no—
424#95—1424no—
425#95—1424no—
426C24C23×D7111yescharacteristic
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
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—712no—
453C22×C4—712no—
454C22×C4—712no—
455C22×C4—712no—
456C22×C4—712no—
457C22×C4—712no—
458C22×C4—712no—
459C22×C4—712no—
460C22×C4—714no—
461C22×C4—1422no—
462C22×C4—1422no—
463C22×C4—1422no—
464C22×C4—1422no—
465C22×C4—1422no—
466C22×C4—1422no—
467C22×C4—1422no—
468C22×C4—1421no—
469C22×C4—1422no—
470C22×C4—1422no—
471C22×C4—1421no—
472C22×C4—1421no—
473C22×C4—1421no—
474C22×C4—1421no—
475C22×C4—1422no—
476C22×C4—1421no—
477C22×C4—1422no—
478C22×C4—1421no—
479C22×C4—1421no—
480C22×C4—1421no—
481C22×C4—1422no—
482C22×C4—1421no—
483C22×C4—1421no—
484C22×C4—1422no—
485C22×C4—1421no—
486C22×C4—1421no—
487C22×C4—1421no—
488C22×C4—1421no—
489C22×C4—1421no—
490C22×C4—1421no—
491C22×C4—1421no—
492C22×C4—1421no—
493C22×C4—1421no—
494C22×C4C2×C4×D7112yes—
495C22×C4C2×C4×D7112yes—
496C22×C4C2×C4×D7112yes—
497C22×C4D4×D7112yes—
498C22×C4C2×Dic14214yes—
499C22×C4C2×Dic14212yes—
500C22×C4#1176586412yes—
501C22×C4#1176586412yes—
502C22×C4#1176586512yes—
503C22×C4#1176586612yes—
504C4⋊C4—28no—
505C4⋊C4—44no—
506C4⋊C4—2844no—
507C4⋊C4—1428no—
508C4⋊C4—1428no—
509C4⋊C4—1428no—
510C42—28no—
511C42—2848no—
512C42—1428no—
513C42—1428no—
514C23C22×C4×D7111yescharacteristic
515C23C2×D4×D7111yescharacteristic
516C23C22×Dic14211yescharacteristic
517C23#41×D7111yescharacteristic
518C23#41×D7111yescharacteristic
519C23#41×D7111yescharacteristic
520C23#41×D7111yescharacteristic
521C23C2×#1176586411yescharacteristic
522C23C2×#1176586411yescharacteristic
523C23C2×#1176586511yescharacteristic
524C23C2×#1176586611yescharacteristic
525C23#1455617311yescharacteristic
526C23#1455617311yescharacteristic
527C23#1455617611yescharacteristic
528C23#1455617611yescharacteristic
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—2841no—
558C2×C4—2841no—
559C2×C4—2842no—
560C2×C4—2841no—
561C2×C4—2841no—
562C2×C4—2841no—
563C2×C4—2841no—
564C2×C4—2841no—
565C2×C4—2841no—
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—714no—
577C2×C4—714no—
578C2×C4—1424no—
579C2×C4—1424no—
580C2×C4—1422no—
581C2×C4—1422no—
582C2×C4—1422no—
583C2×C4—1422no—
584C2×C4—1422no—
585C2×C4—1422no—
586C2×C4—1424no—
587C2×C4—1422no—
588C2×C4—1422no—
589C2×C4—1422no—
590C2×C4—1422no—
591C2×C4—1422no—
592C2×C4—1422no—
593C2×C4—1422no—
594C2×C4—1422no—
595C2×C4—1422no—
596C2×C4—1422no—
597C2×C4—1422no—
598C2×C4—1422no—
599C2×C4#1455608614yes—
600C2×C4(C22⋊D71)⋊C414yes—
601C2×C4C4×Dic14214yes—
602C2×C4#1455610514yes—
603C2×C4#1455611314yes—
604C2×C4#1455611914yes—
605C2×C4#1455612114yes—
606C22#117×D7111yescharacteristic
607C22#117×D7111yescharacteristic
608C22C2×#1455608611yescharacteristic
609C22C2×(C22⋊D71)⋊C411yescharacteristic
610C22C2×C4×Dic14211yescharacteristic
611C22#1798648011yescharacteristic
612C22#1798648011yescharacteristic
613C22C2×#1455610511yescharacteristic
614C22C2×#1455611311yescharacteristic
615C22C2×#1455611911yescharacteristic
616C22C2×#1455612111yescharacteristic
617C22#1798652811yescharacteristic
618C22#1798652811yescharacteristic
619C22#1798653411yescharacteristic
620C22#1798653411yescharacteristic
621C22#1798660311yescharacteristic
622C22#1798661011yescharacteristic
623C22#1798662311yescharacteristic
624C22#1798662311yescharacteristic
625C22#1798664611yescharacteristic
626C22#1798665111yescharacteristic
627C22#1798665911yescharacteristic
628C22#1798666711yescharacteristic
629C22#1798668311yescharacteristic
630C22#1798669611yescharacteristic
631C22#1798670811yescharacteristic
632C22#1798670811yescharacteristic
633C22#1798671311yescharacteristic
634C22#1798672011yescharacteristic
635C22#1798672111yescharacteristic
636C22#1798672111yescharacteristic
637C22#1798672511yescharacteristic
638C22#1798672511yescharacteristic
639C22#1798673211yescharacteristic
640C22#1798673211yescharacteristic
641C4—42no—
642C4—42no—
643C4—44no—
644C4—82no—
645C4—2842no—
646C4—2844no—
647C4—2842no—
648C4—5682no—
649C4—1424no—
650C4—1424no—
651C4#1798588618yes—
652C2#2259073311yescharacteristic
653C2C2×#1798588611yescharacteristic
654C2#2259094811yescharacteristic
655C2#2259126211yescharacteristic
656C2#2259128411yescharacteristic
657C2#2259129811yescharacteristic
658C2#2259129811yescharacteristic
659C2#2259212911yescharacteristic
660C2#2259214011yescharacteristic
661C2#2259220311yescharacteristic
662C2#2259221411yescharacteristic
663C2#2259288011yescharacteristic
664C2#2259289111yescharacteristic
665C2#2259293911yescharacteristic
666C2#2259295011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC71×C4⋊#95C2
socle, prime radical, cosocleC24×C71C24
hypercenterC4⋊#95D71
hypocenterC71#13885
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
21. C4×Dic71⋊C4C4semidirect
345. C71#13885semidirect

Maximal subgroups

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

subgroupindex
C2×#179858652
C2×C4⋊C4×Dic712
C2×C4×Dic71⋊C42
#225866062
#225866012
#225865712
#225865692
#225865662
C71×C4⋊#952
#1388571