Finite Group Databasebeta

← #31928741

Subgroups — #31928741

Order 18176.

2734 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556104C211yesmaximal, characteristic
2C22×C42×D71C211yesmaximal, characteristic
3C2×#17985906C211yesmaximal, characteristic, complemented
4C2×#17985856C211yesmaximal, characteristic
5#22586571C214yesmaximal, complemented
6#22586570C211yesmaximal, characteristic, complemented
7#22586569C211yesmaximal, characteristic, complemented
8(C2×D71)⋊#95C214yesmaximal
9C2×#391×C71C211yesmaximal, characteristic, complemented
10C23×C4×D71C2212yes—
11C23×C4×D71C2211yescharacteristic
12C2×#14556104—24no—
13C2×#14556104—24no—
14C2×#14556104C418yescomplemented
15C22×C4⋊Dic71C2211yescharacteristic
16C22×C4⋊Dic71C2211yescharacteristic
17C22×Dic71⋊C4C2212yes—
18C22×(C2×D71)⋊C4C2212yes—
19C22×(C2×D71)⋊C4C2212yes—
20C22×C4×Dic71C2212yes—
21C22×C4×Dic71C2211yescharacteristic
22C2×C42×D71—24no—
23C2×C42×D71—24no—
24C2×C42×D71—24no—
25#17985906—24no—
26#17985856—24no—
27C2×C71⋊#95C2214yes—
28C2×C71⋊#95C2214yes—
29C2×C71⋊#95C2214yes—
30C2×C71⋊#95C2212yes—
31C23×Q8×C71C2211yescharacteristic
32C22×C4⋊C4×C71C2211yescharacteristic
33C22×C42×C71C2211yescharacteristic
34#391×C71—24no—
35C2×#95×C71C2214yes—
36C24×D71C2311yescharacteristic
37C23×Dic71C2312yes—
38C23×Dic71C2312yes—
39C23×Dic71C2311yescharacteristic
40C23×Dic71C2312yes—
41C22×C4×D71—24no—
42C22×C4×D71—24no—
43C22×C4×D71—24no—
44C22×C4×D71—24no—
45C22×C4×D71—22no—
46C22×C4×D71—24no—
47C22×C4×D71—22no—
48C22×C4×D71—22no—
49C22×C4×D71—22no—
50C22×C4×D71—24no—
51C22×C4×D71C2×C414yes—
52C22×C4×D71D414yes—
53C22×C4×D71D414yes—
54C22×C4×D71Q818yescomplemented
55#14556104—232no—
56C2×C4⋊Dic71—22no—
57C2×C4⋊Dic71—22no—
58C2×C4⋊Dic71—22no—
59C2×C4⋊Dic71—22no—
60C2×C4⋊Dic71—22no—
61C2×C4⋊Dic71C2×C414yescomplemented
62C2×Dic71⋊C4—24no—
63C2×Dic71⋊C4—24no—
64C2×Dic71⋊C4C2×C418yescomplemented
65C2×(C2×D71)⋊C4—24no—
66C2×(C2×D71)⋊C4—24no—
67C2×(C2×D71)⋊C4—24no—
68C2×(C2×D71)⋊C4—28no—
69C2×(C2×D71)⋊C4C2×C418yes—
70C2×C4×Dic71—24no—
71C2×C4×Dic71—24no—
72C2×C4×Dic71—22no—
73C2×C4×Dic71—24no—
74C2×C4×Dic71—22no—
75C2×C4×Dic71D414yes—
76C42×D71—232no—
77C71⋊#95—24no—
78C71⋊#95—28no—
79C71⋊#95—28no—
80C71⋊#95—28no—
81C22×Q8×C71—22no—
82C22×Q8×C71—22no—
83C22×Q8×C71—22no—
84C22×Q8×C71—22no—
85C22×Q8×C71—22no—
86C22×Q8×C71—22no—
87C22×Q8×C71C2×C414yescomplemented
88C23×C4×C71C2312yes—
89C23×C4×C71C2312yes—
90C23×C4×C71C2312yes—
91C23×C4×C71C2311yescharacteristic
92C2×C4⋊C4×C71—22no—
93C2×C4⋊C4×C71—22no—
94C2×C4⋊C4×C71—22no—
95C2×C42×C71—22no—
96C2×C42×C71—22no—
97C2×C42×C71D414yes—
98#95×C71—28no—
99C23×D71—21no—
100C23×D71—21no—
101C23×D71—22no—
102C23×D71—21no—
103C23×D71—21no—
104C23×D71—22no—
105C23×D71C22×C412yes—
106C23×D71C2×D412yes—
107C23×D71C2×D412yes—
108C23×D71C2×Q814yes—
109C23×D71#4114yes—
110C22×Dic71—21no—
111C22×Dic71—22no—
112C22×Dic71—21no—
113C22×Dic71—22no—
114C22×Dic71—22no—
115C22×Dic71—21no—
116C22×Dic71—21no—
117C22×Dic71—22no—
118C22×Dic71—22no—
119C22×Dic71—22no—
120C22×Dic71—24no—
121C22×Dic71—24no—
122C22×Dic71—22no—
123C22×Dic71—22no—
124C22×Dic71—24no—
125C22×Dic71—24no—
126C22×Dic71—22no—
127C22×Dic71C22×C414yes—
128C22×Dic71C22×C412yes—
129C22×Dic71C2×D412yes—
130C22×Dic71C2×D412yes—
131C22×Dic71C2×D414yes—
132C22×Dic71C2×Q814yes—
133C22×Dic71#4114yes—
134C2×C4×D71—216no—
135C2×C4×D71—216no—
136C2×C4×D71—216no—
137C2×C4×D71—216no—
138C2×C4×D71—216no—
139C2×C4×D71—28no—
140C2×C4×D71—28no—
141C2×C4×D71—28no—
142C2×C4×D71—28no—
143C2×C4×D71—28no—
144C2×C4×D71—216no—
145C2×C4×D71—28no—
146C2×C4×D71—216no—
147C2×C4×D71C22⋊C4116yes—
148C4⋊Dic71—28no—
149C4⋊Dic71—44no—
150Dic71⋊C4—216no—
151(C2×D71)⋊C4—216no—
152(C2×D71)⋊C4—48no—
153C4×Dic71—216no—
154C4×Dic71—28no—
155C24×C71C2411yescharacteristic
156C2×Q8×C71—28no—
157C2×Q8×C71—28no—
158C2×Q8×C71—28no—
159C2×Q8×C71—44no—
160C2×Q8×C71—44no—
161C2×Q8×C71—44no—
162C2×Q8×C71—44no—
163C22×C4×C71—22no—
164C22×C4×C71—22no—
165C22×C4×C71—22no—
166C22×C4×C71—22no—
167C22×C4×C71—22no—
168C22×C4×C71—22no—
169C22×C4×C71—22no—
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—24no—
177C22×C4×C71—24no—
178C22×C4×C71C22×C414yes—
179C22×C4×C71C22×C412yes—
180C22×C4×C71C2×D412yes—
181C22×C4×C71C2×D412yes—
182C22×C4×C71C2×D412yes—
183C22×C4×C71C2×D414yes—
184C22×C4×C71C2×D412yes—
185C22×C4×C71C2×Q814yes—
186C22×C4×C71#4112yes—
187C22×C4×C71#4112yes—
188C22×C4×C71#4114yes—
189C4⋊C4×C71—44no—
190C42×C71—28no—
191C23×C71C23×C411yescharacteristic
192C23×C71C22×D411yescharacteristic
193C23×C71C22×D411yescharacteristic
194C23×C71C22×D411yescharacteristic
195C23×C71C22×D411yescharacteristic
196C23×C71C22×Q812yes—
197C23×C71C2×#4112yes—
198C23×C71C2×#4112yes—
199C23×C71C2×#4111yescharacteristic
200C23×C71C2×#4111yescharacteristic
201C23×C71C2×#4112yes—
202C22×D71—22no—
203C22×D71—22no—
204C22×D71—24no—
205C22×D71—22no—
206C22×D71—22no—
207C22×D71—22no—
208C22×D71—24no—
209C22×D71—24no—
210C22×D71—22no—
211C22×D71—24no—
212C22×D71—24no—
213C22×D71—24no—
214C22×D71—24no—
215C22×D71—24no—
216C22×D71—22no—
217C22×D71—22no—
218C22×D71—24no—
219C22×D71—24no—
220C22×D71C2×C22⋊C414yes—
221C22×D71C4×Q818yes—
222C22×D71C22⋊Q818yes—
223C22×D71#12414yes—
224C22×D71C4⋊Q814yes—
225Q8×C71—416no—
226C2×C4×C71—24no—
227C2×C4×C71—24no—
228C2×C4×C71—24no—
229C2×C4×C71—24no—
230C2×C4×C71—24no—
231C2×C4×C71—24no—
232C2×C4×C71—24no—
233C2×C4×C71—24no—
234C2×C4×C71—24no—
235C2×C4×C71—42no—
236C2×C4×C71—42no—
237C2×C4×C71—42no—
238C2×C4×C71—42no—
239C2×C4×C71—42no—
240C2×C4×C71—44no—
241C2×C4×C71—42no—
242C2×C4×C71—44no—
243C2×C4×C71—42no—
244C2×C4×C71C2×C22⋊C414yes—
245C2×C4×C71C4×D414yes—
246C2×C4×C71C4×D414yes—
247C2×C4×C71#12014yes—
248C2×C4×C71#12114yes—
249C2×C4×C71#12114yes—
250C2×C4×C71C22⋊Q818yescomplemented
251C2×C4×C71#12314yes—
252C2×Dic71—22no—
253C2×Dic71—22no—
254C2×Dic71—24no—
255C2×Dic71—22no—
256C2×Dic71—22no—
257C2×Dic71—22no—
258C2×Dic71—24no—
259C2×Dic71—24no—
260C2×Dic71—22no—
261C2×Dic71—24no—
262C2×Dic71—24no—
263C2×Dic71—24no—
264C2×Dic71—24no—
265C2×Dic71—28no—
266C2×Dic71—44no—
267C2×Dic71—42no—
268C2×Dic71—42no—
269C2×Dic71—42no—
270C2×Dic71—44no—
271C2×Dic71—42no—
272C2×Dic71—44no—
273C2×Dic71—42no—
274C2×Dic71C2×C22⋊C414yes—
275C2×Dic71C22⋊Q818yescomplemented
276C4×D71—264no—
277C4×D71—232no—
278C22×C71C22×C22⋊C411yescharacteristic
279C22×C71C2×C4×D411yescharacteristic
280C22×C71C2×C4×D411yescharacteristic
281C22×C71C2×C4×Q812yes—
282C22×C71C4×#4112yes—
283C22×C71C2×#12011yescharacteristic
284C22×C71C2×#12111yescharacteristic
285C22×C71C2×#12111yescharacteristic
286C22×C71C2×C22⋊Q812yes—
287C22×C71C2×C22⋊Q812yes—
288C22×C71C2×C22⋊Q812yes—
289C22×C71C2×#12311yescharacteristic
290C22×C71#52512yes—
291C22×C71#52512yes—
292C22×C71#52512yes—
293C22×C71C2×#12411yescharacteristic
294C22×C71#52912yes—
295C22×C71#52912yes—
296C22×C71C2×C4⋊Q811yescharacteristic
297C22×C71#53211yescharacteristic
298C22×C71#53211yescharacteristic
299C22×C71#53312yes—
300C22×C71#53312yes—
301C2×D71—28no—
302C2×D71—24no—
303C2×D71—24no—
304C2×D71—28no—
305C2×D71—24no—
306C2×D71—24no—
307C2×D71—24no—
308C2×D71—24no—
309C2×D71—28no—
310C2×D71—28no—
311C2×D71#39118yescomplemented
312C4×C71—28no—
313C4×C71—44no—
314C4×C71—82no—
315C4×C71#38618yes—
316Dic71—28no—
317Dic71—24no—
318Dic71—44no—
319Dic71—82no—
320#14702—711nomaximal, Sylow
321C2×C71C2×#38611yescharacteristic
322C2×C71C2×#39111yescharacteristic
323C2×C71#229711yescharacteristic
324C2×C71C4×C22⋊Q812yes—
325C2×C71C4×C22⋊Q812yes—
326C2×C71#284812yes—
327C2×C71#285011yescharacteristic
328C2×C71#285011yescharacteristic
329C2×C71#285512yes—
330C2×C71#285711yescharacteristic
331C2×C71#285711yescharacteristic
332D71—28no—
333C22×C22⋊Q8—711no—
334C23×C42—711no—
335C2×#391—711no—
336C2×#391D7111yescharacteristic, complemented
337C2×#380—711no—
338#1436—714no—
339C4⋊#95—711no—
340C4⋊#95—711no—
341C22⋊#95—714no—
342C71#1470211yescharacteristic, complemented, Sylow
343C23×Q8C2×D7111yescharacteristic
344C24×C4—711no—
345C24×C4—712no—
346C2×C22⋊Q8—718no—
347C2×C22⋊Q8—1424no—
348C2×C22⋊Q8—1424no—
349C22×C4⋊C4—712no—
350C22×C4⋊C4—711no—
351C22×C4⋊C4—711no—
352C22×C4⋊C4C2×D7111yescharacteristic
353C22×C22⋊C4—712no—
354C22×C22⋊C4—712no—
355C22×C42—711no—
356C22×C42—712no—
357C22×C42—1424no—
358C22×C42—1424no—
359C22×C42—1424no—
360C22×C42C2×D7111yescharacteristic
361#391—24no—
362#391—1424no—
363#380—1424no—
364C2×#95—714no—
365C2×#95—714no—
366C2×#95—714no—
367C2×#95—712no—
368C2×#95C2×D7114yes—
369C25—711no—
370C22×Q8—22no—
371C22×Q8—22no—
372C22×Q8—22no—
373C22×Q8—22no—
374C22×Q8—22no—
375C22×Q8—22no—
376C22×Q8C4×D7114yescomplemented
377C23×C4—711no—
378C23×C4—712no—
379C23×C4—714no—
380C23×C4—714no—
381C23×C4—714no—
382C23×C4—712no—
383C23×C4—718no—
384C23×C4—712no—
385C23×C4—1424no—
386C23×C4—1424no—
387C23×C4—1424no—
388C23×C4—1424no—
389C23×C4—1422no—
390C23×C4—1424no—
391C23×C4—1422no—
392C23×C4—1422no—
393C23×C4—1422no—
394C23×C4—1424no—
395C23×C4C22×D7112yes—
396C23×C4C22×D7112yes—
397C23×C4C22×D7112yes—
398C23×C4C22×D7111yescharacteristic
399C22⋊Q8—14232no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—718no—
404C2×C4⋊C4—714no—
405C2×C4⋊C4—1422no—
406C2×C4⋊C4—1422no—
407C2×C4⋊C4—1422no—
408C2×C4⋊C4—1424no—
409C2×C4⋊C4—1424no—
410C2×C4⋊C4—1422no—
411C2×C4⋊C4—1422no—
412C2×C22⋊C4—718no—
413C2×C22⋊C4—1424no—
414C2×C22⋊C4—1428no—
415C2×C22⋊C4—1424no—
416C2×C22⋊C4—1424no—
417C2×C42—22no—
418C2×C42—22no—
419C2×C42—714no—
420C2×C42—14232no—
421C2×C42—1424no—
422C2×C42—1424no—
423C2×C42—1422no—
424C2×C42—1424no—
425C2×C42—1422no—
426C2×C42C22⋊D7114yes—
427#95—28no—
428#95—1424no—
429#95—1428no—
430#95—1428no—
431#95—1428no—
432C24—712no—
433C24—712no—
434C24—712no—
435C24—714no—
436C24—714no—
437C24—1421no—
438C24—1421no—
439C24—1422no—
440C24—1421no—
441C24—1421no—
442C24—1422no—
443C24C23×D7111yescharacteristic
444C2×Q8—28no—
445C2×Q8—28no—
446C2×Q8—28no—
447C2×Q8—44no—
448C2×Q8—44no—
449C2×Q8—44no—
450C2×Q8—44no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—712no—
467C22×C4—714no—
468C22×C4—712no—
469C22×C4—712no—
470C22×C4—714no—
471C22×C4—714no—
472C22×C4—7116no—
473C22×C4—714no—
474C22×C4—14216no—
475C22×C4—14216no—
476C22×C4—14216no—
477C22×C4—14216no—
478C22×C4—14216no—
479C22×C4—14216no—
480C22×C4—1422no—
481C22×C4—1422no—
482C22×C4—1424no—
483C22×C4—1422no—
484C22×C4—1424no—
485C22×C4—1422no—
486C22×C4—1422no—
487C22×C4—1424no—
488C22×C4—1424no—
489C22×C4—1421no—
490C22×C4—1422no—
491C22×C4—1421no—
492C22×C4—1422no—
493C22×C4—1422no—
494C22×C4—1421no—
495C22×C4—1421no—
496C22×C4—1422no—
497C22×C4—1428no—
498C22×C4—1428no—
499C22×C4—1428no—
500C22×C4—1428no—
501C22×C4—1428no—
502C22×C4—14216no—
503C22×C4—1428no—
504C22×C4C2×C4×D7114yes—
505C22×C4C2×C4×D7112yes—
506C22×C4D4×D7112yes—
507C22×C4D4×D7112yes—
508C22×C4D4×D7112yes—
509C22×C4C2×C22⋊D7114yes—
510C22×C4C2×C22⋊D7112yes—
511C22×C4Q8×D7114yes—
512C22×C4#1176586412yes—
513C22×C4#1176586512yes—
514C22×C4#1176586614yes—
515C4⋊C4—44no—
516C4⋊C4—2844no—
517C4⋊C4—14216no—
518C4⋊C4—1428no—
519C22⋊C4—2848no—
520C22⋊C4—14216no—
521C42—28no—
522C42—14216no—
523C42—1428no—
524C23—714no—
525C23—718no—
526C23—718no—
527C23—714no—
528C23—714no—
529C23—1422no—
530C23—1422no—
531C23—1424no—
532C23—1422no—
533C23—1422no—
534C23—1422no—
535C23—1424no—
536C23—1424no—
537C23—1422no—
538C23—1424no—
539C23—1424no—
540C23—1424no—
541C23—1424no—
542C23—1424no—
543C23—1422no—
544C23—1422no—
545C23—1424no—
546C23—1424no—
547C23C22×C4×D7111yescharacteristic
548C23C2×D4×D7111yescharacteristic
549C23C2×D4×D7111yescharacteristic
550C23C2×D4×D7111yescharacteristic
551C23C22×C22⋊D7111yescharacteristic
552C23C2×Q8×D7112yes—
553C23#41×D7112yes—
554C23#41×D7112yes—
555C23C2×#1176586411yescharacteristic
556C23C2×#1176586511yescharacteristic
557C23C2×#1176586612yes—
558Q8—416no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—2844no—
578C2×C4—2842no—
579C2×C4—2842no—
580C2×C4—2842no—
581C2×C4—2844no—
582C2×C4—2842no—
583C2×C4—2844no—
584C2×C4—2842no—
585C2×C4—714no—
586C2×C4—718no—
587C2×C4—14264no—
588C2×C4—1424no—
589C2×C4—1428no—
590C2×C4—1422no—
591C2×C4—1422no—
592C2×C4—1424no—
593C2×C4—1422no—
594C2×C4—1422no—
595C2×C4—1422no—
596C2×C4—1424no—
597C2×C4—1424no—
598C2×C4—1422no—
599C2×C4—1424no—
600C2×C4—1424no—
601C2×C4—1424no—
602C2×C4—14232no—
603C2×C4C22⋊C4×D7114yes—
604C2×C4(C22⋊D71)⋊C414yes—
605C2×C4C4×C22⋊D7114yes—
606C2×C4#1455609614yes—
607C2×C4#1455609914yes—
608C2×C4#1455610114yes—
609C2×C4#1455610418yescomplemented
610C2×C4#1455611014yes—
611C22—718no—
612C22—1428no—
613C22—1424no—
614C22—1424no—
615C22—1424no—
616C22—1428no—
617C22—1424no—
618C22—1424no—
619C22—1424no—
620C22—1428no—
621C22—1428no—
622C22C2×C22⋊C4×D7111yescharacteristic
623C22C2×(C22⋊D71)⋊C411yescharacteristic
624C22C2×C4×C22⋊D7111yescharacteristic
625C22C4×Q8×D7112yes—
626C22C4×#1176586612yes—
627C22C2×#1455609611yescharacteristic
628C22C2×#1455609911yescharacteristic
629C22C2×#1455610111yescharacteristic
630C22C22⋊Q8×D7112yes—
631C22C22⋊Q8×D7112yes—
632C22C2×#1455610412yes—
633C22C2×#1455611011yescharacteristic
634C22#1798650612yes—
635C22#1798651212yes—
636C22#1798651212yes—
637C22#124×D7111yescharacteristic
638C22#1798652812yes—
639C22#1798653612yes—
640C22C4⋊Q8×D7111yescharacteristic
641C22#1798654811yescharacteristic
642C22#1798655011yescharacteristic
643C22#1798655312yes—
644C22#1798655412yes—
645C4—28no—
646C4—44no—
647C4—82no—
648C4—2844no—
649C4—5682no—
650C4—1428no—
651C4—1424no—
652C4#1798587518yes—
653C2—1428no—
654C2C2×#1798587511yescharacteristic
655C2#391×D7111yescharacteristic
656C2#2259082911yescharacteristic
657C2C4×#1455610412yes—
658C2#2259087812yes—
659C2#2259662412yes—
660C2#2259663611yescharacteristic
661C2#2259664411yescharacteristic
662C2#2259668112yes—
663C2#2259668711yescharacteristic
664C2#2259668911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC2×#391×C71C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenterC2×#391D71
hypocenterC71#14702
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
3. C2×#17985906C2semidirect
5. #22586571C2semidirect
6. #22586570C2semidirect
7. #22586569C2semidirect
9. C2×#391×C71C2semidirect
14. C2×#14556104C4semidirect
54. C22×C4×D71Q8semidirect
61. C2×C4⋊Dic71C2×C4semidirect
64. C2×Dic71⋊C4C2×C4semidirect
87. C22×Q8×C71C2×C4semidirect
250. C2×C4×C71C22⋊Q8semidirect
275. C2×Dic71C22⋊Q8semidirect
311. C2×D71#391semidirect
336. C2×#391D71semidirect
342. C71#14702semidirect
376. C22×Q8C4×D71semidirect
609. C2×C4#14556104semidirect

Maximal subgroups

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

subgroupindex
C22×#145561042
C22×C42×D712
C2×#179859062
C2×#179858562
#225865712
#225865702
#225865692
(C2×D71)⋊#952
C2×#391×C712
#1470271