Finite Group Databasebeta

← #31918591

Subgroups — #31918591

Order 18176.

2294 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556086C211yesmaximal, characteristic
2C2×#17985846C212yesmaximal
3C2×#17985844C211yesmaximal, characteristic, complemented
4C2×C42×Dic71C211yesmaximal, characteristic, complemented
5#22586578C211yesmaximal, characteristic
6#22586576C212yesmaximal
7#22586576C211yesmaximal, characteristic
8#22586574C212yesmaximal, complemented
9#22586572C211yesmaximal, characteristic, complemented
10#95×Dic71C212yesmaximal, complemented
11C71×#1436C211yesmaximal, characteristic, complemented
12C23×C4×D71C2211yescharacteristic
13C2×#14556086—24no—
14C2×#14556086—24no—
15C2×#14556086C418yescomplemented
16C22×C4⋊Dic71C2211yescharacteristic
17C22×(C2×D71)⋊C4C2211yescharacteristic
18C22×(C2×D71)⋊C4C2211yescharacteristic
19C22×(C2×D71)⋊C4C2212yes—
20C22×(C2×D71)⋊C4C2212yes—
21C22×C4×Dic71C2211yescharacteristic
22C22×C4×Dic71C2212yes—
23C22×C4×Dic71C2211yescharacteristic
24C22×C4×Dic71C2212yes—
25C22×C4×Dic71C2212yes—
26C22×C4×Dic71C2211yescharacteristic
27#17985846C4116yescomplemented
28#17985844—24no—
29C2×C71⋊#95C2212yes—
30C2×C71⋊#95C2212yes—
31C2×C71⋊#95C2212yes—
32C2×C71⋊#95C2211yescharacteristic
33C2×C71⋊#95C2212yes—
34C2×C71⋊#95C2212yes—
35C42×Dic71—24no—
36C22×C4⋊C4×C71C2211yescharacteristic
37C22×C42×C71C2211yescharacteristic
38C2×#95×C71C2212yes—
39C2×#95×C71C2211yescharacteristic
40C2×#95×C71C2212yes—
41C24×D71C2311yescharacteristic
42C23×Dic71C2312yes—
43C23×Dic71C2311yescharacteristic
44C23×Dic71C2311yescharacteristic
45C23×Dic71C2312yes—
46C23×Dic71C2311yescharacteristic
47C22×C4×D71—22no—
48C22×C4×D71—22no—
49C22×C4×D71—24no—
50C22×C4×D71C2×C418yes—
51C22×C4×D71C2×C414yes—
52#14556086—232no—
53C2×C4⋊Dic71—22no—
54C2×C4⋊Dic71—22no—
55C2×C4⋊Dic71C2×C414yes—
56C2×(C2×D71)⋊C4—22no—
57C2×(C2×D71)⋊C4—22no—
58C2×(C2×D71)⋊C4—24no—
59C2×(C2×D71)⋊C4—24no—
60C2×(C2×D71)⋊C4—22no—
61C2×(C2×D71)⋊C4—22no—
62C2×(C2×D71)⋊C4—24no—
63C2×(C2×D71)⋊C4—24no—
64C2×(C2×D71)⋊C4C2×C418yes—
65C2×(C2×D71)⋊C4C2×C418yes—
66C2×(C2×D71)⋊C4C2×C414yes—
67C2×(C2×D71)⋊C4C2×C414yes—
68C2×C4×Dic71—24no—
69C2×C4×Dic71—24no—
70C2×C4×Dic71—24no—
71C2×C4×Dic71—22no—
72C2×C4×Dic71—22no—
73C2×C4×Dic71—24no—
74C2×C4×Dic71—24no—
75C2×C4×Dic71—24no—
76C2×C4×Dic71—24no—
77C2×C4×Dic71—22no—
78C2×C4×Dic71—24no—
79C2×C4×Dic71C2×C418yes—
80C2×C4×Dic71C2×C418yes—
81C2×C4×Dic71C2×C414yes—
82C2×C4×Dic71C2×C414yes—
83C2×C4×Dic71D414yes—
84C2×C4×Dic71D414yes—
85C71⋊#95—24no—
86C71⋊#95—24no—
87C71⋊#95—24no—
88C71⋊#95—22no—
89C71⋊#95—24no—
90C71⋊#95C2×C418yes—
91C23×C4×C71C2311yescharacteristic
92C23×C4×C71C2312yes—
93C23×C4×C71C2311yescharacteristic
94C23×C4×C71C2311yescharacteristic
95C23×C4×C71C2312yes—
96C2×C4⋊C4×C71—22no—
97C2×C4⋊C4×C71—22no—
98C2×C4⋊C4×C71C2×C414yes—
99C2×C42×C71—22no—
100C2×C42×C71—24no—
101#95×C71—22no—
102#95×C71—24no—
103#95×C71C2×C418yes—
104C23×D71—21no—
105C23×D71—21no—
106C23×D71—22no—
107C23×D71—22no—
108C23×D71—22no—
109C23×D71—22no—
110C23×D71—22no—
111C23×D71C22×C412yes—
112C23×D71C22×C414yes—
113C22×Dic71—22no—
114C22×Dic71—22no—
115C22×Dic71—22no—
116C22×Dic71—22no—
117C22×Dic71—22no—
118C22×Dic71—22no—
119C22×Dic71—22no—
120C22×Dic71—21no—
121C22×Dic71—21no—
122C22×Dic71—22no—
123C22×Dic71—22no—
124C22×Dic71—22no—
125C22×Dic71—22no—
126C22×Dic71—22no—
127C22×Dic71—22no—
128C22×Dic71—22no—
129C22×Dic71C22×C414yes—
130C22×Dic71C22×C412yes—
131C22×Dic71C22×C412yes—
132C22×Dic71C22×C414yes—
133C22×Dic71C22×C412yes—
134C22×Dic71C22×C414yes—
135C22×Dic71C2×D412yes—
136C22×Dic71C2×D412yes—
137C22×Dic71C2×D412yes—
138C22×Dic71C2×D412yes—
139C22×Dic71#4114yes—
140C22×Dic71#4114yes—
141C22×Dic71#4114yes—
142C2×C4×D71—28no—
143C2×C4×D71—28no—
144C2×C4×D71—216no—
145C2×C4×D71—216no—
146C2×C4×D71C42116yescomplemented
147C4⋊Dic71—28no—
148(C2×D71)⋊C4—28no—
149(C2×D71)⋊C4—216no—
150(C2×D71)⋊C4—28no—
151(C2×D71)⋊C4—216no—
152C4×Dic71—216no—
153C4×Dic71—28no—
154C4×Dic71—28no—
155C4×Dic71—216no—
156C4×Dic71—216no—
157C4×Dic71C22⋊C4116yescomplemented
158C24×C71C2411yescharacteristic
159C22×C4×C71—21no—
160C22×C4×C71—21no—
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×C71—22no—
168C22×C4×C71—22no—
169C22×C4×C71—21no—
170C22×C4×C71—21no—
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×C71C22×C414yes—
179C22×C4×C71C22×C412yes—
180C22×C4×C71C22×C414yes—
181C22×C4×C71C22×C412yes—
182C22×C4×C71C22×C412yes—
183C22×C4×C71C22×C414yes—
184C22×C4×C71C2×D412yes—
185C22×C4×C71C2×D412yes—
186C22×C4×C71#4114yes—
187C22×C4×C71#4114yes—
188C4⋊C4×C71—28no—
189C42×C71—28no—
190C23×C71C23×C411yescharacteristic
191C23×C71C23×C412yes—
192C23×C71C22×D411yescharacteristic
193C23×C71C22×D411yescharacteristic
194C23×C71C2×#4112yes—
195C23×C71C2×#4112yes—
196C23×C71C2×#4112yes—
197C23×C71C2×#4112yes—
198C23×C71C2×#4112yes—
199C22×D71—22no—
200C22×D71—24no—
201C22×D71—24no—
202C22×D71—22no—
203C22×D71—24no—
204C22×D71—24no—
205C22×D71—24no—
206C22×D71—24no—
207C22×D71—24no—
208C22×D71—24no—
209C22×D71—41no—
210C22×D71—41no—
211C22×D71—42no—
212C22×D71—42no—
213C22×D71—42no—
214C22×D71—42no—
215C22×D71—42no—
216C22×D71—42no—
217C22×D71—41no—
218C22×D71—41no—
219C22×D71C2×C4214yes—
220C2×C4×C71—22no—
221C2×C4×C71—22no—
222C2×C4×C71—24no—
223C2×C4×C71—24no—
224C2×C4×C71—24no—
225C2×C4×C71—24no—
226C2×C4×C71—24no—
227C2×C4×C71—22no—
228C2×C4×C71—24no—
229C2×C4×C71—22no—
230C2×C4×C71—24no—
231C2×C4×C71—24no—
232C2×C4×C71—24no—
233C2×C4×C71—24no—
234C2×C4×C71—22no—
235C2×C4×C71—24no—
236C2×C4×C71—24no—
237C2×C4×C71—24no—
238C2×C4×C71—24no—
239C2×C4×C71—41no—
240C2×C4×C71—41no—
241C2×C4×C71—42no—
242C2×C4×C71—42no—
243C2×C4×C71—42no—
244C2×C4×C71—42no—
245C2×C4×C71—42no—
246C2×C4×C71C2×C4214yes—
247C2×C4×C71C2×C22⋊C414yes—
248C2×C4×C71#11718yescomplemented
249C2×Dic71—24no—
250C2×Dic71—24no—
251C2×Dic71—24no—
252C2×Dic71—24no—
253C2×Dic71—24no—
254C2×Dic71—24no—
255C2×Dic71—24no—
256C2×Dic71—24no—
257C2×Dic71—22no—
258C2×Dic71—22no—
259C2×Dic71—22no—
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—41no—
269C2×Dic71—42no—
270C2×Dic71—41no—
271C2×Dic71C2×C4214yes—
272C2×Dic71C2×C22⋊C414yes—
273C2×Dic71C2×C22⋊C414yes—
274C2×Dic71#11714yes—
275C2×Dic71#11718yescomplemented
276C2×Dic71C4×D418yes—
277C2×Dic71C4×D418yes—
278C4×D71—232no—
279C22×C71C22×C4211yescharacteristic
280C22×C71C22×C22⋊C411yescharacteristic
281C22×C71C2×#11711yescharacteristic
282C22×C71C2×#11712yes—
283C22×C71C2×#11712yes—
284C22×C71C2×C4×D412yes—
285C22×C71C2×C4×D412yes—
286C22×C71C4×#4112yes—
287C22×C71C4×#4112yes—
288C22×C71C4×#4112yes—
289C22×C71C4×#4112yes—
290C22×C71#52511yescharacteristic
291C22×C71#52511yescharacteristic
292C22×C71#52512yes—
293C22×C71#52912yes—
294C22×C71#52912yes—
295C22×C71#52912yes—
296C22×C71#52912yes—
297C22×C71#53212yes—
298C22×C71#53211yescharacteristic
299C22×C71#53211yescharacteristic
300C2×D71—24no—
301C2×D71—28no—
302C2×D71—44no—
303C2×D71—44no—
304C2×D71—44no—
305C2×D71—44no—
306C2×D71—42no—
307C2×D71—42no—
308C2×D71—44no—
309C4×C71—24no—
310C4×C71—28no—
311C4×C71—24no—
312C4×C71—44no—
313C4×C71—42no—
314Dic71—28no—
315Dic71—24no—
316Dic71—44no—
317Dic71—42no—
318Dic71C4×C22⋊C418yes—
319#13761—711nomaximal, Sylow
320C2×C71C2×C4×C22⋊C411yescharacteristic
321C2×C71C4×#11712yes—
322C2×C71#229012yes—
323C2×C71#229012yes—
324C2×C71#229312yes—
325C2×C71#229312yes—
326C2×C71#229711yescharacteristic
327C2×C71#229711yescharacteristic
328C2×C71#229812yes—
329D71—44no—
330C22×#117—711no—
331C2×C4×C22⋊C4—712no—
332C2×#376—711no—
333C2×C43—711no—
334#1437—711no—
335#1437—711no—
336#1437—712no—
337#1436—712no—
338#1436—711no—
339#1436D7111yescharacteristic, complemented
340C4×#95—712no—
341C71#1376111yescharacteristic, complemented, Sylow
342C24×C4—711no—
343C2×#117—718no—
344C2×#117—1424no—
345C2×#117—1424no—
346C22×C4⋊C4—711no—
347C22×C4⋊C4C2×D7111yescharacteristic
348C22×C22⋊C4—711no—
349C22×C22⋊C4—711no—
350C22×C22⋊C4—712no—
351C22×C22⋊C4—712no—
352C22×C42—711no—
353C22×C42—711no—
354C22×C42—712no—
355C22×C42—712no—
356C22×C42—712no—
357C22×C42—711no—
358C22×C42C2×D7111yescharacteristic
359C4×C22⋊C4—7116no—
360#376—1424no—
361C2×#95—712no—
362C2×#95—712no—
363C2×#95—712no—
364C2×#95—712no—
365C2×#95—712no—
366C2×#95—711no—
367C2×#95C2×D7112yes—
368C2×#95C2×D7111yescharacteristic
369C2×#95C2×D7112yes—
370C43—1424no—
371C25—711no—
372C23×C4—711no—
373C23×C4—711no—
374C23×C4—712no—
375C23×C4—714no—
376C23×C4—718no—
377C23×C4—711no—
378C23×C4—712no—
379C23×C4—1422no—
380C23×C4—1422no—
381C23×C4—1424no—
382C23×C4C22×D7111yescharacteristic
383C23×C4C22×D7112yes—
384C23×C4C22×D7111yescharacteristic
385C23×C4C22×D7111yescharacteristic
386C23×C4C22×D7112yes—
387#117—14232no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—714no—
391C2×C4⋊C4—1422no—
392C2×C4⋊C4—1422no—
393C2×C4⋊C4C4×D7114yes—
394C2×C22⋊C4—714no—
395C2×C22⋊C4—714no—
396C2×C22⋊C4—718no—
397C2×C22⋊C4—718no—
398C2×C22⋊C4—1422no—
399C2×C22⋊C4—1422no—
400C2×C22⋊C4—1424no—
401C2×C22⋊C4—1424no—
402C2×C22⋊C4—1422no—
403C2×C22⋊C4—1422no—
404C2×C22⋊C4—1424no—
405C2×C22⋊C4—1424no—
406C2×C42—22no—
407C2×C42—24no—
408C2×C42—714no—
409C2×C42—714no—
410C2×C42—714no—
411C2×C42—714no—
412C2×C42—718no—
413C2×C42—718no—
414C2×C42—1422no—
415C2×C42—1422no—
416C2×C42—1424no—
417C2×C42—1424no—
418C2×C42—1424no—
419C2×C42—1424no—
420C2×C42—1424no—
421C2×C42—1424no—
422C2×C42—1424no—
423C2×C42—1422no—
424C2×C42—1424no—
425#95—22no—
426#95—24no—
427#95—718no—
428#95—1424no—
429#95—1424no—
430#95—1424no—
431#95—1422no—
432#95—1424no—
433#95C4×D7118yes—
434C24—712no—
435C24—714no—
436C24—1421no—
437C24—1421no—
438C24—1422no—
439C24—1422no—
440C24—1422no—
441C24—1422no—
442C24—1422no—
443C24C23×D7111yescharacteristic
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—712no—
464C22×C4—714no—
465C22×C4—712no—
466C22×C4—712no—
467C22×C4—714no—
468C22×C4—714no—
469C22×C4—712no—
470C22×C4—712no—
471C22×C4—712no—
472C22×C4—714no—
473C22×C4—714no—
474C22×C4—7116no—
475C22×C4—714no—
476C22×C4—712no—
477C22×C4—1428no—
478C22×C4—1428no—
479C22×C4—14216no—
480C22×C4—14216no—
481C22×C4—1422no—
482C22×C4—1421no—
483C22×C4—1421no—
484C22×C4—1422no—
485C22×C4—1422no—
486C22×C4—1422no—
487C22×C4—1422no—
488C22×C4—1422no—
489C22×C4—1422no—
490C22×C4—1422no—
491C22×C4—1422no—
492C22×C4—1422no—
493C22×C4—1422no—
494C22×C4—1422no—
495C22×C4—1422no—
496C22×C4—1422no—
497C22×C4C2×C4×D7114yes—
498C22×C4C2×C4×D7112yes—
499C22×C4C2×C4×D7114yes—
500C22×C4C2×C4×D7112yes—
501C22×C4C2×C4×D7112yes—
502C22×C4C2×C4×D7114yes—
503C22×C4D4×D7112yes—
504C22×C4D4×D7112yes—
505C22×C4#1176586414yes—
506C22×C4#1176586614yes—
507C4⋊C4—28no—
508C4⋊C4—1428no—
509C22⋊C4—14216no—
510C22⋊C4—1428no—
511C22⋊C4—1428no—
512C22⋊C4—14216no—
513C42—28no—
514C42—7116no—
515C42—14216no—
516C42—14216no—
517C42—1428no—
518C42—1428no—
519C42—14216no—
520C23—2841no—
521C23—2841no—
522C23—2842no—
523C23—2842no—
524C23—2842no—
525C23—2842no—
526C23—2842no—
527C23—2842no—
528C23—2841no—
529C23—2841no—
530C23—714no—
531C23—1422no—
532C23—1424no—
533C23—1422no—
534C23—1424no—
535C23—1424no—
536C23—1424no—
537C23—1424no—
538C23—1424no—
539C23—1424no—
540C23—1424no—
541C23C22×C4×D7111yescharacteristic
542C23C22×C4×D7112yes—
543C23C2×D4×D7111yescharacteristic
544C23C2×D4×D7111yescharacteristic
545C23#41×D7112yes—
546C23#41×D7112yes—
547C23#41×D7112yes—
548C23C2×#1176586412yes—
549C23C2×#1176586612yes—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—2842no—
577C2×C4—2842no—
578C2×C4—2842no—
579C2×C4—2842no—
580C2×C4—2841no—
581C2×C4—2842no—
582C2×C4—2841no—
583C2×C4—714no—
584C2×C4—714no—
585C2×C4—714no—
586C2×C4—718no—
587C2×C4—718no—
588C2×C4—714no—
589C2×C4—718no—
590C2×C4—14232no—
591C2×C4—1422no—
592C2×C4—1424no—
593C2×C4—1424no—
594C2×C4—1424no—
595C2×C4—1424no—
596C2×C4—1424no—
597C2×C4—1424no—
598C2×C4—1424no—
599C2×C4—1424no—
600C2×C4—1424no—
601C2×C4—1424no—
602C2×C4—1424no—
603C2×C4—1424no—
604C2×C4—1422no—
605C2×C4—1422no—
606C2×C4C42×D7114yes—
607C2×C4C22⋊C4×D7114yes—
608C2×C4#1455608618yescomplemented
609C22—2844no—
610C22—2844no—
611C22—2844no—
612C22—2844no—
613C22—2842no—
614C22—2842no—
615C22—2844no—
616C22—1424no—
617C22—1428no—
618C22C2×C42×D7111yescharacteristic
619C22C2×C22⋊C4×D7111yescharacteristic
620C22#117×D7112yes—
621C22#117×D7111yescharacteristic
622C22C2×#1455608612yes—
623C22C4×D4×D7112yes—
624C22C4×D4×D7112yes—
625C22C4×#1176586412yes—
626C22C4×#1176586412yes—
627C22C4×#1176586612yes—
628C22C4×#1176586612yes—
629C22#1798650511yescharacteristic
630C22#1798650711yescharacteristic
631C22#1798651212yes—
632C22#1798652612yes—
633C22#1798653212yes—
634C22#1798653412yes—
635C22#1798653512yes—
636C22#1798654412yes—
637C22#1798654811yescharacteristic
638C22#1798655011yescharacteristic
639C4—28no—
640C4—24no—
641C4—24no—
642C4—44no—
643C4—42no—
644C4—2844no—
645C4—2842no—
646C4—718no—
647C4—1428no—
648C4—1424no—
649C2—2844no—
650C2C4×C22⋊C4×D7111yescharacteristic
651C2C4×#1455608612yes—
652C2#2259077612yes—
653C2#2259077812yes—
654C2#2259079812yes—
655C2#2259080012yes—
656C2#2259082511yescharacteristic
657C2#2259082911yescharacteristic
658C2#2259083212yes—
subgroupG/H
center, FrattiniC24C23×D71
commutatorC22×C71C22×C42
FittingC71×#1436C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenter#1436D71
hypocenterC71#13761
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
3. C2×#17985844C2semidirect
4. C2×C42×Dic71C2semidirect
8. #22586574C2semidirect
9. #22586572C2semidirect
10. #95×Dic71C2semidirect
11. C71×#1436C2semidirect
15. C2×#14556086C4semidirect
27. #17985846C4semidirect
146. C2×C4×D71C42semidirect
157. C4×Dic71C22⋊C4semidirect
248. C2×C4×C71#117semidirect
275. C2×Dic71#117semidirect
339. #1436D71semidirect
341. C71#13761semidirect
608. C2×C4#14556086semidirect

Maximal subgroups

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

subgroupindex
C22×#145560862
C2×#179858462
C2×#179858442
C2×C42×Dic712
#225865782
#225865762
#225865742
#225865722
#95×Dic712
C71×#14362
#1376171