Finite Group Databasebeta

← #31928745

Subgroups — #31928745

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×#14556108C211yesmaximal, characteristic
2C22×C42×D71C211yesmaximal, characteristic
3C2×#17985905C211yesmaximal, characteristic, complemented
4C2×#17985904C211yesmaximal, characteristic, complemented
5C2×#17985856C211yesmaximal, characteristic
6#22586574C214yesmaximal, complemented
7#22586569C211yesmaximal, characteristic, complemented
8(C2×D71)⋊#95C214yesmaximal
9C71×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×Dic142C2211yescharacteristic
11C23×C4×D71C2212yes—
12C23×C4×D71C2211yescharacteristic
13C2×#14556108—24no—
14C2×#14556108—24no—
15C2×#14556108C418yescomplemented
16C22×C4⋊Dic71C2211yescharacteristic
17C22×C4⋊Dic71C2212yes—
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#17985905—24no—
26#17985904—24no—
27#17985856—24no—
28C2×C71⋊#95C2214yes—
29C2×C71⋊#95C2214yes—
30C2×C71⋊#95C2214yes—
31C2×C71⋊#95C2212yes—
32C22×C4⋊C4×C71C2211yescharacteristic
33C22×C4⋊C4×C71C2211yescharacteristic
34C22×C42×C71C2211yescharacteristic
35C2×#95×C71C2214yes—
36C24×D71C2311yescharacteristic
37C22×Dic142—22no—
38C22×Dic142—22no—
39C22×Dic142—22no—
40C22×Dic142—22no—
41C22×Dic142—22no—
42C22×Dic142—22no—
43C22×Dic142C2×C414yescomplemented
44C23×Dic71C2312yes—
45C23×Dic71C2312yes—
46C23×Dic71C2311yescharacteristic
47C23×Dic71C2312yes—
48C22×C4×D71—24no—
49C22×C4×D71—24no—
50C22×C4×D71—24no—
51C22×C4×D71—24no—
52C22×C4×D71—22no—
53C22×C4×D71—24no—
54C22×C4×D71—22no—
55C22×C4×D71—24no—
56C22×C4×D71—22no—
57C22×C4×D71—22no—
58C22×C4×D71C2×C414yes—
59C22×C4×D71D414yes—
60C22×C4×D71D414yes—
61C22×C4×D71Q818yescomplemented
62#14556108—232no—
63C2×C4⋊Dic71—24no—
64C2×C4⋊Dic71—24no—
65C2×C4⋊Dic71—22no—
66C2×C4⋊Dic71—22no—
67C2×C4⋊Dic71—22no—
68C2×C4⋊Dic71C2×C418yescomplemented
69C2×(C2×D71)⋊C4—24no—
70C2×(C2×D71)⋊C4—24no—
71C2×(C2×D71)⋊C4—24no—
72C2×(C2×D71)⋊C4—28no—
73C2×(C2×D71)⋊C4C2×C418yes—
74C2×C4×Dic71—24no—
75C2×C4×Dic71—24no—
76C2×C4×Dic71—22no—
77C2×C4×Dic71—24no—
78C2×C4×Dic71—22no—
79C2×C4×Dic71D414yes—
80C42×D71—232no—
81C71⋊#95—24no—
82C71⋊#95—28no—
83C71⋊#95—28no—
84C71⋊#95—28no—
85C23×C4×C71C2312yes—
86C23×C4×C71C2312yes—
87C23×C4×C71C2312yes—
88C23×C4×C71C2311yescharacteristic
89C2×C4⋊C4×C71—22no—
90C2×C4⋊C4×C71—22no—
91C2×C4⋊C4×C71—22no—
92C2×C4⋊C4×C71—22no—
93C2×C4⋊C4×C71—22no—
94C2×C4⋊C4×C71C2×C414yescomplemented
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—22no—
103C23×D71—21no—
104C23×D71—21no—
105C23×D71C22×C412yes—
106C23×D71C2×D412yes—
107C23×D71C2×D412yes—
108C23×D71C2×Q814yes—
109C23×D71#4114yes—
110C2×Dic142—28no—
111C2×Dic142—28no—
112C2×Dic142—28no—
113C2×Dic142—44no—
114C2×Dic142—44no—
115C2×Dic142—44no—
116C2×Dic142—44no—
117C22×Dic71—21no—
118C22×Dic71—22no—
119C22×Dic71—21no—
120C22×Dic71—22no—
121C22×Dic71—22no—
122C22×Dic71—22no—
123C22×Dic71—21no—
124C22×Dic71—21no—
125C22×Dic71—22no—
126C22×Dic71—22no—
127C22×Dic71—22no—
128C22×Dic71—22no—
129C22×Dic71—22no—
130C22×Dic71—22no—
131C22×Dic71—22no—
132C22×Dic71—22no—
133C22×Dic71—24no—
134C22×Dic71—24no—
135C22×Dic71—22no—
136C22×Dic71C22×C414yes—
137C22×Dic71C22×C412yes—
138C22×Dic71C2×D412yes—
139C22×Dic71C2×D412yes—
140C22×Dic71C2×D414yes—
141C22×Dic71C2×Q814yes—
142C22×Dic71#4114yes—
143C2×C4×D71—216no—
144C2×C4×D71—216no—
145C2×C4×D71—216no—
146C2×C4×D71—216no—
147C2×C4×D71—216no—
148C2×C4×D71—28no—
149C2×C4×D71—28no—
150C2×C4×D71—28no—
151C2×C4×D71—28no—
152C2×C4×D71—28no—
153C2×C4×D71—216no—
154C2×C4×D71—28no—
155C2×C4×D71—216no—
156C2×C4×D71C22⋊C4116yes—
157C4⋊Dic71—216no—
158C4⋊Dic71—44no—
159(C2×D71)⋊C4—216no—
160(C2×D71)⋊C4—48no—
161C4×Dic71—216no—
162C4×Dic71—28no—
163C24×C71C2411yescharacteristic
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—24no—
171C22×C4×C71—24no—
172C22×C4×C71—22no—
173C22×C4×C71—22no—
174C22×C4×C71—24no—
175C22×C4×C71—22no—
176C22×C4×C71—24no—
177C22×C4×C71C22×C414yes—
178C22×C4×C71C22×C412yes—
179C22×C4×C71C2×D412yes—
180C22×C4×C71C2×D412yes—
181C22×C4×C71C2×D412yes—
182C22×C4×C71C2×D414yes—
183C22×C4×C71C2×D412yes—
184C22×C4×C71C2×Q814yes—
185C22×C4×C71#4114yes—
186C22×C4×C71#4112yes—
187C22×C4×C71#4112yes—
188C4⋊C4×C71—28no—
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×#4112yes—
200C23×C71C2×#4111yescharacteristic
201C23×C71C2×#4111yescharacteristic
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—
225C2×C4×C71—24no—
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—28no—
233C2×C4×C71—44no—
234C2×C4×C71—42no—
235C2×C4×C71—42no—
236C2×C4×C71—42no—
237C2×C4×C71—44no—
238C2×C4×C71—42no—
239C2×C4×C71—44no—
240C2×C4×C71—42no—
241C2×C4×C71C2×C22⋊C414yes—
242C2×C4×C71C4×D414yes—
243C2×C4×C71C4×D414yes—
244C2×C4×C71#12014yes—
245C2×C4×C71#12114yes—
246C2×C4×C71#12114yes—
247C2×C4×C71C22⋊Q818yescomplemented
248C2×C4×C71#12314yes—
249C2×Dic71—22no—
250C2×Dic71—22no—
251C2×Dic71—24no—
252C2×Dic71—22no—
253C2×Dic71—22no—
254C2×Dic71—22no—
255C2×Dic71—24no—
256C2×Dic71—24no—
257C2×Dic71—22no—
258C2×Dic71—24no—
259C2×Dic71—24no—
260C2×Dic71—24no—
261C2×Dic71—24no—
262C2×Dic71—24no—
263C2×Dic71—24no—
264C2×Dic71—42no—
265C2×Dic71—42no—
266C2×Dic71—42no—
267C2×Dic71—42no—
268C2×Dic71—42no—
269C2×Dic71—44no—
270C2×Dic71—42no—
271C2×Dic71—44no—
272C2×Dic71—42no—
273C2×Dic71C2×C22⋊C414yes—
274C2×Dic71C22⋊Q818yescomplemented
275C4×D71—264no—
276C4×D71—232no—
277Dic142—416no—
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×#391—711no—
337C2×#380—711no—
338#1436—714no—
339C4⋊#95—711no—
340C4⋊#95D7111yescharacteristic, complemented
341C22⋊#95—714no—
342C71#1470211yescharacteristic, complemented, Sylow
343C23×Q8—711no—
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⋊C4C2×D7111yescharacteristic
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—1424no—
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—714no—
371C22×Q8—1422no—
372C22×Q8—1422no—
373C22×Q8—1422no—
374C22×Q8—1422no—
375C22×Q8—1422no—
376C22×Q8—1422no—
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—1424no—
393C23×C4—1422no—
394C23×C4—1422no—
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—22no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—718no—
406C2×C4⋊C4—1422no—
407C2×C4⋊C4—1422no—
408C2×C4⋊C4—1422no—
409C2×C4⋊C4—1424no—
410C2×C4⋊C4—1424no—
411C2×C4⋊C4C4×D7114yescomplemented
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×C42D28414yes—
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—1422no—
441C24—1421no—
442C24—1421no—
443C24C23×D7111yescharacteristic
444C2×Q8—2844no—
445C2×Q8—2844no—
446C2×Q8—2844no—
447C2×Q8—2844no—
448C2×Q8—1428no—
449C2×Q8—1428no—
450C2×Q8—1428no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—712no—
465C22×C4—714no—
466C22×C4—712no—
467C22×C4—712no—
468C22×C4—714no—
469C22×C4—714no—
470C22×C4—7116no—
471C22×C4—714no—
472C22×C4—14216no—
473C22×C4—14216no—
474C22×C4—14216no—
475C22×C4—14216no—
476C22×C4—14216no—
477C22×C4—14216no—
478C22×C4—1422no—
479C22×C4—1422no—
480C22×C4—1424no—
481C22×C4—1424no—
482C22×C4—1422no—
483C22×C4—1422no—
484C22×C4—1422no—
485C22×C4—1422no—
486C22×C4—1422no—
487C22×C4—1422no—
488C22×C4—1422no—
489C22×C4—1421no—
490C22×C4—1422no—
491C22×C4—1421no—
492C22×C4—1422no—
493C22×C4—1422no—
494C22×C4—1422no—
495C22×C4—1421no—
496C22×C4—1421no—
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×D28414yes—
510C22×C4C2×D28412yes—
511C22×C4Q8×D7114yes—
512C22×C4#1176586414yes—
513C22×C4#1176586512yes—
514C22×C4#1176586612yes—
515C4⋊C4—28no—
516C4⋊C4—44no—
517C4⋊C4—2844no—
518C4⋊C4—14216no—
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×D28411yescharacteristic
552C23C2×Q8×D7112yes—
553C23#41×D7112yes—
554C23#41×D7112yes—
555C23C2×#1176586412yes—
556C23C2×#1176586511yescharacteristic
557C23C2×#1176586611yescharacteristic
558Q8—28416no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—2842no—
576C2×C4—2842no—
577C2×C4—2842no—
578C2×C4—2842no—
579C2×C4—2842no—
580C2×C4—2844no—
581C2×C4—2842no—
582C2×C4—2844no—
583C2×C4—2842no—
584C2×C4—714no—
585C2×C4—718no—
586C2×C4—14264no—
587C2×C4—1424no—
588C2×C4—1424no—
589C2×C4—1424no—
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#1455609014yes—
605C2×C4C4×D28414yes—
606C2×C4#1455609814yes—
607C2×C4#1455610114yes—
608C2×C4#1455610314yes—
609C2×C4#1455610818yescomplemented
610C2×C4#1455611314yes—
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×#1455609011yescharacteristic
624C22C2×C4×D28411yescharacteristic
625C22C4×Q8×D7112yes—
626C22C4×#1176586412yes—
627C22C2×#1455609811yescharacteristic
628C22C2×#1455610111yescharacteristic
629C22C2×#1455610311yescharacteristic
630C22C22⋊Q8×D7112yes—
631C22C22⋊Q8×D7112yes—
632C22C2×#1455610812yes—
633C22C2×#1455611311yescharacteristic
634C22#1798650912yes—
635C22#1798650912yes—
636C22#1798651112yes—
637C22#124×D7111yescharacteristic
638C22#1798653112yes—
639C22#1798653412yes—
640C22C4⋊Q8×D7111yescharacteristic
641C22#1798654711yescharacteristic
642C22#1798654811yescharacteristic
643C22#1798655312yes—
644C22#1798655412yes—
645C4—28no—
646C4—44no—
647C4—82no—
648C4—2844no—
649C4—5682no—
650C4—1428no—
651C4—1424no—
652C4#1798587718yes—
653C2—1428no—
654C2C2×#1798587711yescharacteristic
655C2#391×D7111yescharacteristic
656C2#2259082711yescharacteristic
657C2#2259087012yes—
658C2C4×#1455610812yes—
659C2#2259662812yes—
660C2#2259664011yescharacteristic
661C2#2259664211yescharacteristic
662C2#2259667812yes—
663C2#2259668911yescharacteristic
664C2#2259669311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC71×C4⋊#95C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenterC4⋊#95D71
hypocenterC71#14702
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
3. C2×#17985905C2semidirect
4. C2×#17985904C2semidirect
6. #22586574C2semidirect
7. #22586569C2semidirect
9. C71×C4⋊#95C2semidirect
15. C2×#14556108C4semidirect
43. C22×Dic142C2×C4semidirect
61. C22×C4×D71Q8semidirect
68. C2×C4⋊Dic71C2×C4semidirect
94. C2×C4⋊C4×C71C2×C4semidirect
247. C2×C4×C71C22⋊Q8semidirect
274. C2×Dic71C22⋊Q8semidirect
311. C2×D71#391semidirect
340. C4⋊#95D71semidirect
342. C71#14702semidirect
411. C2×C4⋊C4C4×D71semidirect
609. C2×C4#14556108semidirect

Maximal subgroups

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

subgroupindex
C22×#145561082
C22×C42×D712
C2×#179859052
C2×#179859042
C2×#179858562
#225865742
#225865692
(C2×D71)⋊#952
C71×C4⋊#952
#1470271