Finite Group Databasebeta

← #31919862

Subgroups — #31919862

Order 18176.

1854 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556093C211yesmaximal, characteristic
2C2×#17985903C212yesmaximal
3C2×#17985868C211yesmaximal, characteristic
4C2×#17985850C211yesmaximal, characteristic
5#22586594C211yesmaximal, characteristic
6#22586593C211yesmaximal, characteristic
7#22586590C211yesmaximal, characteristic
8Dic71⋊#95C212yesmaximal
9Dic71⋊#95C212yesmaximal
10Dic71⋊#95C212yesmaximal
11C71×#1441C211yesmaximal, characteristic
12C23×Dic142C2211yescharacteristic
13C2×#14556093—24no—
14C2×#14556093—24no—
15C2×#14556093C418yescomplemented
16C22×C4⋊Dic71C2211yescharacteristic
17C22×Dic71⋊C4C2212yes—
18C22×Dic71⋊C4C2212yes—
19C22×Dic71⋊C4C2211yescharacteristic
20C22×Dic71⋊C4C2211yescharacteristic
21C22×Dic71⋊C4C2212yes—
22C22×C4×Dic71C2212yes—
23C22×C4×Dic71C2211yescharacteristic
24C22×C4×Dic71C2211yescharacteristic
25C22×C4×Dic71C2211yescharacteristic
26#17985903—28no—
27#17985868—24no—
28#17985850—24no—
29C2×C71⋊#95C2212yes—
30C2×C71⋊#95C2212yes—
31C2×C71⋊#95C2212yes—
32C2×C71⋊#95C2211yescharacteristic
33C2×C71⋊#95C2212yes—
34C2×C71⋊#95C2212yes—
35C2×C71⋊#95C2212yes—
36C22×C4⋊C4×C71C2211yescharacteristic
37C22×C4⋊C4×C71C2211yescharacteristic
38C2×#95×C71C2212yes—
39C2×#95×C71C2211yescharacteristic
40C2×#95×C71C2212yes—
41C22×Dic142—22no—
42C22×Dic142—22no—
43C22×Dic142—22no—
44C22×Dic142—22no—
45C22×Dic142C2×C414yes—
46C22×Dic142D414yes—
47C22×Dic142Q814yes—
48C23×Dic71C2312yes—
49C23×Dic71C2311yescharacteristic
50C23×Dic71C2312yes—
51C23×Dic71C2311yescharacteristic
52C23×Dic71C2311yescharacteristic
53C23×Dic71C2311yescharacteristic
54#14556093—232no—
55C2×C4⋊Dic71—22no—
56C2×C4⋊Dic71—42no—
57C2×Dic71⋊C4—24no—
58C2×Dic71⋊C4—22no—
59C2×Dic71⋊C4—22no—
60C2×Dic71⋊C4—24no—
61C2×Dic71⋊C4—24no—
62C2×Dic71⋊C4—24no—
63C2×Dic71⋊C4—24no—
64C2×Dic71⋊C4—24no—
65C2×Dic71⋊C4—24no—
66C2×Dic71⋊C4—24no—
67C2×Dic71⋊C4—22no—
68C2×Dic71⋊C4—22no—
69C2×Dic71⋊C4C2×C414yes—
70C2×Dic71⋊C4C2×C414yes—
71C2×Dic71⋊C4Q818yescomplemented
72C2×C4×Dic71—22no—
73C2×C4×Dic71—22no—
74C2×C4×Dic71—22no—
75C2×C4×Dic71—22no—
76C2×C4×Dic71—24no—
77C2×C4×Dic71—24no—
78C2×C4×Dic71—24no—
79C2×C4×Dic71C2×C414yes—
80C2×C4×Dic71C2×C414yes—
81C2×C4×Dic71C2×C414yes—
82C2×C4×Dic71D414yes—
83C2×C4×Dic71D414yes—
84C71⋊#95—24no—
85C71⋊#95—24no—
86C71⋊#95—22no—
87C71⋊#95—24no—
88C71⋊#95—42no—
89C71⋊#95—42no—
90C71⋊#95—42no—
91C23×C4×C71C2312yes—
92C23×C4×C71C2311yescharacteristic
93C23×C4×C71C2312yes—
94C23×C4×C71C2311yescharacteristic
95C23×C4×C71C2311yescharacteristic
96C2×C4⋊C4×C71—22no—
97C2×C4⋊C4×C71—22no—
98C2×C4⋊C4×C71—22no—
99C2×C4⋊C4×C71—42no—
100C2×C4⋊C4×C71C2×C414yes—
101#95×C71—22no—
102#95×C71—24no—
103#95×C71—42no—
104C2×Dic142—28no—
105C2×Dic142—28no—
106C2×Dic142—28no—
107C2×Dic142—28no—
108C2×Dic142—28no—
109C2×Dic142—28no—
110C2×Dic142C4⋊C4116yescomplemented
111C22×Dic71—21no—
112C22×Dic71—21no—
113C22×Dic71—21no—
114C22×Dic71—21no—
115C22×Dic71—22no—
116C22×Dic71—22no—
117C22×Dic71—21no—
118C22×Dic71—21no—
119C22×Dic71—22no—
120C22×Dic71—22no—
121C22×Dic71—22no—
122C22×Dic71—22no—
123C22×Dic71—22no—
124C22×Dic71—22no—
125C22×Dic71—22no—
126C22×Dic71—22no—
127C22×Dic71—22no—
128C22×Dic71—22no—
129C22×Dic71—22no—
130C22×Dic71—22no—
131C22×Dic71—22no—
132C22×Dic71C22×C412yes—
133C22×Dic71C22×C412yes—
134C22×Dic71C22×C412yes—
135C22×Dic71C22×C412yes—
136C22×Dic71C2×D412yes—
137C22×Dic71C2×D412yes—
138C22×Dic71C2×D412yes—
139C22×Dic71C2×D412yes—
140C22×Dic71C2×D412yes—
141C22×Dic71C2×D412yes—
142C22×Dic71C2×Q814yes—
143C22×Dic71C2×Q812yes—
144C22×Dic71C2×Q812yes—
145C22×Dic71C2×Q814yes—
146C22×Dic71#4112yes—
147C22×Dic71#4112yes—
148C22×Dic71#4114yes—
149C4⋊Dic71—44no—
150Dic71⋊C4—28no—
151Dic71⋊C4—28no—
152Dic71⋊C4—216no—
153Dic71⋊C4—48no—
154Dic71⋊C4—48no—
155C4×Dic71—28no—
156C4×Dic71—28no—
157C4×Dic71—48no—
158C4×Dic71C22⋊C4116yes—
159C24×C71C2411yescharacteristic
160C22×C4×C71—21no—
161C22×C4×C71—21no—
162C22×C4×C71—21no—
163C22×C4×C71—21no—
164C22×C4×C71—22no—
165C22×C4×C71—22no—
166C22×C4×C71—22no—
167C22×C4×C71—22no—
168C22×C4×C71—21no—
169C22×C4×C71—21no—
170C22×C4×C71—22no—
171C22×C4×C71—22no—
172C22×C4×C71—22no—
173C22×C4×C71—22no—
174C22×C4×C71—22no—
175C22×C4×C71—22no—
176C22×C4×C71—22no—
177C22×C4×C71—22no—
178C22×C4×C71—22no—
179C22×C4×C71—22no—
180C22×C4×C71—22no—
181C22×C4×C71—22no—
182C22×C4×C71—22no—
183C22×C4×C71C22×C412yes—
184C22×C4×C71C22×C412yes—
185C22×C4×C71C22×C412yes—
186C22×C4×C71C2×D412yes—
187C22×C4×C71C2×D412yes—
188C22×C4×C71C2×D412yes—
189C22×C4×C71C2×Q814yes—
190C22×C4×C71C2×Q812yes—
191C4⋊C4×C71—28no—
192C4⋊C4×C71—44no—
193C23×C71C23×C411yescharacteristic
194C23×C71C22×D411yescharacteristic
195C23×C71C22×D411yescharacteristic
196C23×C71C22×D411yescharacteristic
197C23×C71C22×Q811yescharacteristic
198C23×C71C22×Q812yes—
199C23×C71C2×#4111yescharacteristic
200C23×C71C2×#4111yescharacteristic
201C23×C71C2×#4112yes—
202C23×C71H(5,𝔽2)12yes—
203C23×C712−1+412yes—
204C2×C4×C71—22no—
205C2×C4×C71—22no—
206C2×C4×C71—24no—
207C2×C4×C71—22no—
208C2×C4×C71—22no—
209C2×C4×C71—22no—
210C2×C4×C71—22no—
211C2×C4×C71—22no—
212C2×C4×C71—22no—
213C2×C4×C71—24no—
214C2×C4×C71—22no—
215C2×C4×C71—24no—
216C2×C4×C71—24no—
217C2×C4×C71—24no—
218C2×C4×C71—24no—
219C2×C4×C71—42no—
220C2×C4×C71—41no—
221C2×C4×C71—41no—
222C2×C4×C71—42no—
223C2×C4×C71—42no—
224C2×C4×C71—42no—
225C2×C4×C71—42no—
226C2×C4×C71—42no—
227C2×C4×C71—42no—
228C2×C4×C71—42no—
229C2×C4×C71—42no—
230C2×C4×C71—42no—
231C2×C4×C71—42no—
232C2×C4×C71—42no—
233C2×C4×C71C2×C22⋊C414yes—
234C2×C4×C71C2×C4⋊C414yes—
235C2×Dic71—22no—
236C2×Dic71—22no—
237C2×Dic71—22no—
238C2×Dic71—22no—
239C2×Dic71—22no—
240C2×Dic71—22no—
241C2×Dic71—22no—
242C2×Dic71—24no—
243C2×Dic71—24no—
244C2×Dic71—24no—
245C2×Dic71—24no—
246C2×Dic71—42no—
247C2×Dic71—41no—
248C2×Dic71—41no—
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×Dic71C2×C22⋊C414yes—
261C2×Dic71C2×C22⋊C414yes—
262C2×Dic71C2×C4⋊C414yes—
263C2×Dic71C2×C4⋊C414yes—
264C2×Dic71#11714yes—
265C2×Dic71C4×Q818yes—
266C2×Dic71#12114yes—
267C2×Dic71#12114yes—
268C2×Dic71C22⋊Q814yes—
269C2×Dic71C22⋊Q814yes—
270C2×Dic71C22⋊Q818yes—
271C2×Dic71#12414yes—
272C2×Dic71C4⋊Q814yes—
273Dic142—232no—
274C22×C71C22×C22⋊C411yescharacteristic
275C22×C71C22×C4⋊C411yescharacteristic
276C22×C71C2×#11711yescharacteristic
277C22×C71C2×C4×Q812yes—
278C22×C71#52012yes—
279C22×C71C2×#12111yescharacteristic
280C22×C71C2×#12111yescharacteristic
281C22×C71C2×C22⋊Q811yescharacteristic
282C22×C71C2×C22⋊Q811yescharacteristic
283C22×C71C2×C22⋊Q812yes—
284C22×C71C2×#12411yescharacteristic
285C22×C71C2×C4⋊Q811yescharacteristic
286C22×C71#53511yescharacteristic
287C22×C71#53611yescharacteristic
288C22×C71#53712yes—
289C22×C71D4×Q812yes—
290C22×C71#55012yes—
291C22×C71#55412yes—
292C22×C71#55612yes—
293C22×C71#55712yes—
294C22×C71#55712yes—
295C22×C71Q8212yes—
296C22×C71#55912yes—
297C4×C71—24no—
298C4×C71—24no—
299C4×C71—42no—
300C4×C71—44no—
301C4×C71—82no—
302Dic71—24no—
303Dic71—42no—
304Dic71—44no—
305Dic71—82no—
306Dic71#38018yes—
307Dic71#39118yescomplemented
308#13926—711nomaximal, Sylow
309C2×C71C2×#38011yescharacteristic
310C2×C71C2×#39111yescharacteristic
311C2×C71#231111yescharacteristic
312C2×C71Q8×C4⋊C412yes—
313C2×C71#235512yes—
314C2×C71#253812yes—
315C2×C71#254611yescharacteristic
316C2×C71#254811yescharacteristic
317C2×C71#256711yescharacteristic
318C2×C71#256811yescharacteristic
319C2×C71#258212yes—
320C22×C4×Q8—711no—
321C2×#391—712no—
322C2×#384—711no—
323C2×C4×C4⋊C4—711no—
324#1441—711no—
325#1441—711no—
326#1441—711no—
327#1441D7111yescharacteristic
328C4⋊#95—712no—
329C4⋊#95—712no—
330C4⋊#95—712no—
331C71#1392611yescharacteristic, complemented, Sylow
332C23×Q8—711no—
333C2×C4×Q8—718no—
334C2×C4×Q8—1424no—
335C2×C4×Q8—1424no—
336C22×C4⋊C4—711no—
337C22×C4⋊C4—711no—
338C22×C4⋊C4—711no—
339C22×C4⋊C4—712no—
340C22×C4⋊C4—712no—
341C22×C4⋊C4—712no—
342C22×C4⋊C4C2×D7111yescharacteristic
343C22×C4⋊C4C2×D7111yescharacteristic
344C22×C42—711no—
345C22×C42—711no—
346C22×C42—711no—
347C22×C42—712no—
348#391—1428no—
349#384—1424no—
350C4×C4⋊C4—1424no—
351C2×#95—712no—
352C2×#95—712no—
353C2×#95—712no—
354C2×#95—712no—
355C2×#95—712no—
356C2×#95—712no—
357C2×#95—711no—
358C2×#95C2×D7112yes—
359C2×#95C2×D7111yescharacteristic
360C2×#95C2×D7112yes—
361C22×Q8—714no—
362C22×Q8—714no—
363C22×Q8—714no—
364C22×Q8—1422no—
365C22×Q8—1422no—
366C22×Q8—1422no—
367C22×Q8—1422no—
368C23×C4—711no—
369C23×C4—711no—
370C23×C4—711no—
371C23×C4—711no—
372C23×C4—712no—
373C23×C4—712no—
374C23×C4C22×D7112yes—
375C23×C4C22×D7111yescharacteristic
376C23×C4C22×D7112yes—
377C23×C4C22×D7111yescharacteristic
378C23×C4C22×D7111yescharacteristic
379C4×Q8—14232no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—42no—
384C2×C4⋊C4—2842no—
385C2×C4⋊C4—714no—
386C2×C4⋊C4—714no—
387C2×C4⋊C4—718no—
388C2×C4⋊C4—1424no—
389C2×C4⋊C4—1424no—
390C2×C4⋊C4—1424no—
391C2×C4⋊C4—1424no—
392C2×C4⋊C4—1424no—
393C2×C4⋊C4—1424no—
394C2×C4⋊C4—1422no—
395C2×C4⋊C4—1422no—
396C2×C4⋊C4—1422no—
397C2×C4⋊C4—1424no—
398C2×C4⋊C4—1424no—
399C2×C4⋊C4—1422no—
400C2×C4⋊C4—1422no—
401C2×C4⋊C4C4×D7114yes—
402C2×C42—714no—
403C2×C42—714no—
404C2×C42—714no—
405C2×C42—714no—
406C2×C42—714no—
407C2×C42—1424no—
408C2×C42—1424no—
409C2×C42—1424no—
410C2×C42—1422no—
411C2×C42—1422no—
412C2×C42—1422no—
413C2×C42—1422no—
414#95—22no—
415#95—24no—
416#95—42no—
417#95—2842no—
418#95—2842no—
419#95—2842no—
420#95—1424no—
421#95—1424no—
422#95—1422no—
423#95—1424no—
424C24C23×D7111yescharacteristic
425C2×Q8—7116no—
426C2×Q8—1428no—
427C2×Q8—1428no—
428C2×Q8—1428no—
429C2×Q8—1428no—
430C2×Q8—1428no—
431C2×Q8—1428no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
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—22no—
454C22×C4—22no—
455C22×C4—712no—
456C22×C4—712no—
457C22×C4—712no—
458C22×C4—712no—
459C22×C4—712no—
460C22×C4—712no—
461C22×C4—712no—
462C22×C4—714no—
463C22×C4—712no—
464C22×C4—712no—
465C22×C4—714no—
466C22×C4—712no—
467C22×C4—712no—
468C22×C4—712no—
469C22×C4—712no—
470C22×C4—712no—
471C22×C4—714no—
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—1421no—
482C22×C4—1421no—
483C22×C4—1422no—
484C22×C4—1422no—
485C22×C4—1422no—
486C22×C4—1422no—
487C22×C4—1422no—
488C22×C4—1422no—
489C22×C4—1421no—
490C22×C4—1421no—
491C22×C4—1421no—
492C22×C4—1421no—
493C22×C4C2×C4×D7112yes—
494C22×C4C2×C4×D7112yes—
495C22×C4C2×C4×D7112yes—
496C22×C4D4×D7112yes—
497C22×C4D4×D7112yes—
498C22×C4D4×D7112yes—
499C22×C4Q8×D7114yes—
500C22×C4Q8×D7112yes—
501C4⋊C4—28no—
502C4⋊C4—44no—
503C4⋊C4—2844no—
504C4⋊C4—2848no—
505C4⋊C4—2848no—
506C4⋊C4—14216no—
507C4⋊C4—1428no—
508C4⋊C4—1428no—
509C42—2848no—
510C42—7116no—
511C42—1428no—
512C42—1428no—
513C23C22×C4×D7111yescharacteristic
514C23C2×D4×D7111yescharacteristic
515C23C2×D4×D7111yescharacteristic
516C23C2×D4×D7111yescharacteristic
517C23C2×Q8×D7112yes—
518C23C2×Q8×D7111yescharacteristic
519C23#41×D7112yes—
520C23#41×D7111yescharacteristic
521C23#41×D7111yescharacteristic
522C23#1455617312yes—
523C23#1455617512yes—
524Q8—14232no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—2841no—
555C2×C4—2841no—
556C2×C4—2842no—
557C2×C4—2842no—
558C2×C4—2842no—
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—714no—
569C2×C4—714no—
570C2×C4—714no—
571C2×C4—714no—
572C2×C4—718no—
573C2×C4—714no—
574C2×C4—714no—
575C2×C4—718no—
576C2×C4—714no—
577C2×C4—714no—
578C2×C4—714no—
579C2×C4—714no—
580C2×C4—714no—
581C2×C4—1424no—
582C2×C4—1422no—
583C2×C4—1424no—
584C2×C4—1424no—
585C2×C4—1424no—
586C2×C4—1422no—
587C2×C4—1422no—
588C2×C4—1422no—
589C2×C4—1422no—
590C2×C4—1422no—
591C2×C4—1422no—
592C2×C4C22⋊C4×D7114yes—
593C2×C4C4⋊C4×D7114yes—
594C22C2×C22⋊C4×D7111yescharacteristic
595C22C2×C4⋊C4×D7111yescharacteristic
596C22#117×D7111yescharacteristic
597C22C4×Q8×D7112yes—
598C22#1798647812yes—
599C22#121×D7111yescharacteristic
600C22#121×D7111yescharacteristic
601C22C22⋊Q8×D7111yescharacteristic
602C22C22⋊Q8×D7112yes—
603C22C22⋊Q8×D7111yescharacteristic
604C22#124×D7111yescharacteristic
605C22C4⋊Q8×D7111yescharacteristic
606C22#1798656411yescharacteristic
607C22#1798657111yescharacteristic
608C22#1798657612yes—
609C22#1798666212yes—
610C22#1798666712yes—
611C22#1798670312yes—
612C22#1798671612yes—
613C22#1798672612yes—
614C22#1798672612yes—
615C22#1798672912yes—
616C22#1798673312yes—
617C4—24no—
618C4—24no—
619C4—42no—
620C4—44no—
621C4—82no—
622C4—2842no—
623C4—2844no—
624C4—5682no—
625C4—718no—
626C4—718no—
627C4—1424no—
628C2#380×D7111yescharacteristic
629C2#391×D7111yescharacteristic
630C2#2259092711yescharacteristic
631C2#2259125412yes—
632C2#2259129412yes—
633C2#2259325712yes—
634C2#2259332411yescharacteristic
635C2#2259333811yescharacteristic
636C2#2259354411yescharacteristic
637C2#2259355111yescharacteristic
638C2#2259370912yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC71×#1441C2
socle, prime radical, cosocleC24×C71C24
hypercenter#1441D71
hypocenterC71#13926
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
15. C2×#14556093C4semidirect
71. C2×Dic71⋊C4Q8semidirect
110. C2×Dic142C4⋊C4semidirect
307. Dic71#391semidirect
331. C71#13926semidirect

Maximal subgroups

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

subgroupindex
C22×#145560932
C2×#179859032
C2×#179858682
C2×#179858502
#225865942
#225865932
#225865902
Dic71⋊#952
C71×#14412
#1392671