Finite Group Databasebeta

← #31919436

Subgroups — #31919436

Order 18176.

2510 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D71C211yesmaximal, characteristic
2C2×#17985873C214yesmaximal
3C2×#17985869C211yesmaximal, characteristic, complemented
4C2×#17985868C211yesmaximal, characteristic, complemented
5C2×#17985856C211yesmaximal, characteristic
6C2×C4×(C2×D71)⋊C4C211yesmaximal, characteristic
7#22586604C214yesmaximal, complemented
8#22586569C211yesmaximal, characteristic, complemented
9C71×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C4×D71C2212yes—
11C23×C4×D71C2211yescharacteristic
12C22×C4⋊Dic71C2211yescharacteristic
13C22×C4⋊Dic71C2214yes—
14C22×C4⋊Dic71C2211yescharacteristic
15C22×Dic71⋊C4C2212yes—
16C22×Dic71⋊C4C2214yes—
17C2×C4⋊C4×D71—24no—
18C2×C4⋊C4×D71—24no—
19C2×C4⋊C4×D71—24no—
20C22×(C2×D71)⋊C4C2212yes—
21C22×(C2×D71)⋊C4C2212yes—
22C22×C4×Dic71C2211yescharacteristic
23#17985873—216no—
24#17985869—24no—
25#17985868—24no—
26#17985856—24no—
27C4×(C2×D71)⋊C4C418yescomplemented
28C2×C71⋊#95C2214yes—
29C2×C71⋊#95C2212yes—
30C2×C71⋊#95C2212yes—
31C22×C4⋊C4×C71C2211yescharacteristic
32C22×C4⋊C4×C71C2211yescharacteristic
33C22×C42×C71C2211yescharacteristic
34C2×#95×C71C2214yes—
35C24×D71C2311yescharacteristic
36C23×Dic71C2312yes—
37C23×Dic71C2312yes—
38C23×Dic71C2311yescharacteristic
39C23×Dic71C2312yes—
40C22×C4×D71—24no—
41C22×C4×D71—24no—
42C22×C4×D71—28no—
43C22×C4×D71—22no—
44C22×C4×D71—22no—
45C22×C4×D71—24no—
46C22×C4×D71—22no—
47C22×C4×D71—22no—
48C22×C4×D71—44no—
49C22×C4×D71C2×C414yes—
50C22×C4×D71D414yes—
51C22×C4×D71Q814yes—
52C2×C4⋊Dic71—28no—
53C2×C4⋊Dic71—22no—
54C2×C4⋊Dic71—22no—
55C2×C4⋊Dic71—22no—
56C2×C4⋊Dic71—22no—
57C2×C4⋊Dic71—22no—
58C2×C4⋊Dic71—28no—
59C2×C4⋊Dic71—44no—
60C2×C4⋊Dic71D414yes—
61C2×Dic71⋊C4—24no—
62C2×Dic71⋊C4—24no—
63C2×Dic71⋊C4—28no—
64C2×Dic71⋊C4—24no—
65C2×Dic71⋊C4—28no—
66C2×Dic71⋊C4—44no—
67C4⋊C4×D71—232no—
68C2×(C2×D71)⋊C4—24no—
69C2×(C2×D71)⋊C4—28no—
70C2×(C2×D71)⋊C4C2×C418yes—
71C2×(C2×D71)⋊C4D418yes—
72C2×(C2×D71)⋊C4Q818yes—
73C2×C4×Dic71—22no—
74C2×C4×Dic71—22no—
75C2×C4×Dic71C2×C414yescomplemented
76C71⋊#95—24no—
77C71⋊#95—28no—
78C71⋊#95C2×C418yescomplemented
79C23×C4×C71C2312yes—
80C23×C4×C71C2312yes—
81C23×C4×C71C2312yes—
82C23×C4×C71C2311yescharacteristic
83C2×C4⋊C4×C71—22no—
84C2×C4⋊C4×C71—22no—
85C2×C4⋊C4×C71—22no—
86C2×C4⋊C4×C71—22no—
87C2×C4⋊C4×C71—22no—
88C2×C4⋊C4×C71D414yes—
89C2×C42×C71—22no—
90C2×C42×C71—22no—
91C2×C42×C71C2×C414yescomplemented
92#95×C71—28no—
93C23×D71—21no—
94C23×D71—21no—
95C23×D71—21no—
96C23×D71—21no—
97C23×D71—22no—
98C23×D71—22no—
99C23×D71C22×C412yes—
100C23×D71C2×D412yes—
101C23×D71C2×D414yes—
102C23×D71C2×Q812yes—
103C23×D71C2×Q814yes—
104C22×Dic71—21no—
105C22×Dic71—21no—
106C22×Dic71—22no—
107C22×Dic71—21no—
108C22×Dic71—21no—
109C22×Dic71—22no—
110C22×Dic71—22no—
111C22×Dic71—22no—
112C22×Dic71—24no—
113C22×Dic71—22no—
114C22×Dic71—22no—
115C22×Dic71—24no—
116C22×Dic71—22no—
117C22×Dic71—24no—
118C22×Dic71—22no—
119C22×Dic71—22no—
120C22×Dic71—24no—
121C22×Dic71C22×C412yes—
122C22×Dic71C22×C414yes—
123C22×Dic71C2×D414yes—
124C22×Dic71C2×D412yes—
125C22×Dic71C2×D414yes—
126C22×Dic71C2×Q812yes—
127C22×Dic71C2×Q814yes—
128C2×C4×D71—28no—
129C2×C4×D71—28no—
130C2×C4×D71—28no—
131C2×C4×D71—28no—
132C2×C4×D71—28no—
133C2×C4×D71—28no—
134C2×C4×D71—216no—
135C2×C4×D71—416no—
136C2×C4×D71—416no—
137C2×C4×D71—48no—
138C2×C4×D71—48no—
139C2×C4×D71C4⋊C4116yescomplemented
140C4⋊Dic71—28no—
141C4⋊Dic71—416no—
142C4⋊Dic71—44no—
143Dic71⋊C4—216no—
144Dic71⋊C4—416no—
145(C2×D71)⋊C4—48no—
146(C2×D71)⋊C4C4⋊C4132yescomplemented
147C4×Dic71—28no—
148C24×C71C2411yescharacteristic
149C22×C4×C71—22no—
150C22×C4×C71—22no—
151C22×C4×C71—22no—
152C22×C4×C71—22no—
153C22×C4×C71—24no—
154C22×C4×C71—22no—
155C22×C4×C71—22no—
156C22×C4×C71—24no—
157C22×C4×C71—24no—
158C22×C4×C71—22no—
159C22×C4×C71—22no—
160C22×C4×C71—22no—
161C22×C4×C71—24no—
162C22×C4×C71C22×C414yes—
163C22×C4×C71C22×C412yes—
164C22×C4×C71C2×D414yes—
165C22×C4×C71C2×D412yes—
166C22×C4×C71C2×D412yes—
167C22×C4×C71C2×D414yes—
168C22×C4×C71C2×D412yes—
169C22×C4×C71C2×Q814yes—
170C22×C4×C71C2×Q812yes—
171C22×C4×C71#4112yes—
172C22×C4×C71#4112yes—
173C4⋊C4×C71—28no—
174C4⋊C4×C71—44no—
175C42×C71—28no—
176C23×C71C23×C411yescharacteristic
177C23×C71C22×D411yescharacteristic
178C23×C71C22×D412yes—
179C23×C71C22×D411yescharacteristic
180C23×C71C22×D411yescharacteristic
181C23×C71C22×Q811yescharacteristic
182C23×C71C22×Q812yes—
183C23×C71C2×#4111yescharacteristic
184C23×C71C2×#4111yescharacteristic
185C23×C71H(5,𝔽2)12yes—
186C23×C712−1+412yes—
187C22×D71—22no—
188C22×D71—22no—
189C22×D71—24no—
190C22×D71—22no—
191C22×D71—22no—
192C22×D71—22no—
193C22×D71—22no—
194C22×D71—22no—
195C22×D71—22no—
196C22×D71—24no—
197C22×D71—24no—
198C22×D71—24no—
199C22×D71—24no—
200C22×D71—24no—
201C22×D71—24no—
202C22×D71—24no—
203C22×D71—24no—
204C22×D71—24no—
205C22×D71C2×C4⋊C418yes—
206C22×D71C2×C4⋊C414yes—
207C22×D71#12714yes—
208C22×D71C4⋊Q814yes—
209C22×D71C4⋊Q818yes—
210C2×C4×C71—24no—
211C2×C4×C71—24no—
212C2×C4×C71—24no—
213C2×C4×C71—24no—
214C2×C4×C71—24no—
215C2×C4×C71—24no—
216C2×C4×C71—24no—
217C2×C4×C71—28no—
218C2×C4×C71—42no—
219C2×C4×C71—42no—
220C2×C4×C71—44no—
221C2×C4×C71—42no—
222C2×C4×C71—42no—
223C2×C4×C71—42no—
224C2×C4×C71—44no—
225C2×C4×C71—44no—
226C2×C4×C71C2×C4⋊C414yes—
227C2×C4×C71C2×C4⋊C418yescomplemented
228C2×C4×C71C4×D414yes—
229C2×C4×C71C4×D414yes—
230C2×C4×C71#12014yes—
231C2×C4×C71C22⋊Q814yes—
232C2×C4×C71C22⋊Q814yes—
233C2×C4×C71#12314yes—
234C2×Dic71—24no—
235C2×Dic71—22no—
236C2×Dic71—22no—
237C2×Dic71—24no—
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—24no—
247C2×Dic71—28no—
248C2×Dic71—42no—
249C2×Dic71—42no—
250C2×Dic71—44no—
251C2×Dic71—42no—
252C2×Dic71—42no—
253C2×Dic71—42no—
254C2×Dic71—44no—
255C2×Dic71—44no—
256C2×Dic71C2×C4⋊C414yes—
257C2×Dic71C2×C4⋊C418yescomplemented
258C4×D71—232no—
259C4×D71—432no—
260C22×C71C22×C4⋊C411yescharacteristic
261C22×C71C22×C4⋊C412yes—
262C22×C71C2×C4×D411yescharacteristic
263C22×C71C2×C4×D411yescharacteristic
264C22×C71#52012yes—
265C22×C71C2×#12011yescharacteristic
266C22×C71C2×C22⋊Q811yescharacteristic
267C22×C71C2×C22⋊Q811yescharacteristic
268C22×C71C2×#12311yescharacteristic
269C22×C71C2×#12711yescharacteristic
270C22×C71C2×C4⋊Q811yescharacteristic
271C22×C71C2×C4⋊Q812yes—
272C22×C71#53511yescharacteristic
273C22×C71#53611yescharacteristic
274C22×C71#54412yes—
275C22×C71D4212yes—
276C22×C71D4212yes—
277C22×C71#54712yes—
278C22×C71#54712yes—
279C22×C71D4×Q812yes—
280C22×C71D4×Q812yes—
281C22×C71#55412yes—
282C22×C71#55412yes—
283C2×D71—24no—
284C2×D71—28no—
285C2×D71—28no—
286C2×D71—24no—
287C2×D71—24no—
288C2×D71—28no—
289C2×D71—28no—
290C2×D71—24no—
291C2×D71—24no—
292C2×D71—24no—
293C2×D71#38418yescomplemented
294C4×C71—28no—
295C4×C71—44no—
296C4×C71—82no—
297C4×C71#38518yescomplemented
298Dic71—28no—
299Dic71—24no—
300Dic71—44no—
301Dic71—82no—
302#13880—711nomaximal, Sylow
303C2×C71C2×#38411yescharacteristic
304C2×C71C2×#38511yescharacteristic
305C2×C71#231711yescharacteristic
306C2×C71D4×C4⋊C412yes—
307C2×C71D4×C4⋊C412yes—
308C2×C71#243311yescharacteristic
309C2×C71#243411yescharacteristic
310C2×C71#244012yes—
311C2×C71#244012yes—
312C2×C71#250111yescharacteristic
313C2×C71#250211yescharacteristic
314D71—28no—
315C23×C4⋊C4—711no—
316C2×#385—714no—
317C2×#384—711no—
318C2×#384—711no—
319C2×#380—711no—
320C2×C4×C22⋊C4—711no—
321#1444—714no—
322C4⋊#95—711no—
323C4⋊#95D7111yescharacteristic, complemented
324C71#1388011yescharacteristic, complemented, Sylow
325C24×C4—712no—
326C24×C4—711no—
327C22×C4⋊C4—714no—
328C22×C4⋊C4—714no—
329C22×C4⋊C4—711no—
330C22×C4⋊C4—712no—
331C22×C4⋊C4—711no—
332C22×C4⋊C4—1424no—
333C22×C4⋊C4—1424no—
334C22×C4⋊C4—1424no—
335C22×C4⋊C4C2×D7111yescharacteristic
336C22×C4⋊C4C2×D7111yescharacteristic
337C22×C22⋊C4—712no—
338C22×C22⋊C4—712no—
339C22×C42—711no—
340C22×C42C2×D7111yescharacteristic
341#385—14216no—
342#384—1424no—
343#384—1424no—
344#380—1424no—
345C4×C22⋊C4—718no—
346C2×#95—712no—
347C2×#95—714no—
348C2×#95—712no—
349C2×#95C2×D7114yes—
350C25—711no—
351C23×C4—2844no—
352C23×C4—712no—
353C23×C4—711no—
354C23×C4—714no—
355C23×C4—714no—
356C23×C4—714no—
357C23×C4—712no—
358C23×C4—712no—
359C23×C4—1428no—
360C23×C4—1424no—
361C23×C4—1424no—
362C23×C4—1424no—
363C23×C4—1422no—
364C23×C4—1422no—
365C23×C4—1422no—
366C23×C4—1422no—
367C23×C4C22×D7112yes—
368C23×C4C22×D7112yes—
369C23×C4C22×D7112yes—
370C23×C4C22×D7111yescharacteristic
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—2844no—
377C2×C4⋊C4—2844no—
378C2×C4⋊C4—714no—
379C2×C4⋊C4—14232no—
380C2×C4⋊C4—1428no—
381C2×C4⋊C4—1428no—
382C2×C4⋊C4—1422no—
383C2×C4⋊C4—1422no—
384C2×C4⋊C4—1428no—
385C2×C4⋊C4—1428no—
386C2×C4⋊C4—1422no—
387C2×C4⋊C4—1424no—
388C2×C4⋊C4—1424no—
389C2×C4⋊C4—1422no—
390C2×C4⋊C4—1424no—
391C2×C4⋊C4—1422no—
392C2×C4⋊C4D28414yes—
393C2×C22⋊C4—718no—
394C2×C22⋊C4—718no—
395C2×C22⋊C4—718no—
396C2×C22⋊C4—1424no—
397C2×C22⋊C4—1428no—
398C2×C42—22no—
399C2×C42—22no—
400C2×C42—714no—
401C2×C42—1422no—
402C2×C42—1422no—
403C2×C42C4×D7114yescomplemented
404#95—28no—
405#95—718no—
406#95—1424no—
407#95—1428no—
408C24—712no—
409C24—714no—
410C24—712no—
411C24—714no—
412C24—712no—
413C24—1421no—
414C24—1421no—
415C24—1421no—
416C24—1421no—
417C24—1422no—
418C24—1422no—
419C24C23×D7111yescharacteristic
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—28416no—
434C22×C4—2848no—
435C22×C4—2848no—
436C22×C4—28416no—
437C22×C4—712no—
438C22×C4—714no—
439C22×C4—714no—
440C22×C4—712no—
441C22×C4—714no—
442C22×C4—712no—
443C22×C4—7116no—
444C22×C4—714no—
445C22×C4—14216no—
446C22×C4—1422no—
447C22×C4—1424no—
448C22×C4—1422no—
449C22×C4—1422no—
450C22×C4—1424no—
451C22×C4—1424no—
452C22×C4—1422no—
453C22×C4—1422no—
454C22×C4—1424no—
455C22×C4—1421no—
456C22×C4—1422no—
457C22×C4—1421no—
458C22×C4—1421no—
459C22×C4—1422no—
460C22×C4—1421no—
461C22×C4—1422no—
462C22×C4—1422no—
463C22×C4—1428no—
464C22×C4—1428no—
465C22×C4—1428no—
466C22×C4—1428no—
467C22×C4—1428no—
468C22×C4—1428no—
469C22×C4C2×C4×D7114yes—
470C22×C4C2×C4×D7112yes—
471C22×C4D4×D7114yes—
472C22×C4D4×D7112yes—
473C22×C4D4×D7112yes—
474C22×C4C2×D28414yes—
475C22×C4C2×D28412yes—
476C22×C4Q8×D7114yes—
477C22×C4Q8×D7112yes—
478C22×C4#1176586412yes—
479C22×C4#1176586512yes—
480C4⋊C4—28no—
481C4⋊C4—44no—
482C4⋊C4—28416no—
483C4⋊C4—28416no—
484C4⋊C4—2844no—
485C4⋊C4—14216no—
486C4⋊C4—1428no—
487C22⋊C4—2848no—
488C22⋊C4—7132no—
489C42—28no—
490C42—1428no—
491C23—718no—
492C23—714no—
493C23—714no—
494C23—718no—
495C23—714no—
496C23—1422no—
497C23—1422no—
498C23—1424no—
499C23—1424no—
500C23—1424no—
501C23—1424no—
502C23—1424no—
503C23—1424no—
504C23—1424no—
505C23—1424no—
506C23—1424no—
507C23—1422no—
508C23—1422no—
509C23—1424no—
510C23—1422no—
511C23—1422no—
512C23—1422no—
513C23—1422no—
514C23C22×C4×D7111yescharacteristic
515C23C2×D4×D7111yescharacteristic
516C23C2×D4×D7111yescharacteristic
517C23C2×D4×D7112yes—
518C23C22×D28411yescharacteristic
519C23C2×Q8×D7111yescharacteristic
520C23C2×Q8×D7112yes—
521C23C2×#1176586411yescharacteristic
522C23C2×#1176586511yescharacteristic
523C23#1455617412yes—
524C23#1455617512yes—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—28432no—
542C2×C4—2842no—
543C2×C4—2844no—
544C2×C4—2842no—
545C2×C4—2842no—
546C2×C4—2844no—
547C2×C4—2844no—
548C2×C4—2842no—
549C2×C4—2842no—
550C2×C4—718no—
551C2×C4—714no—
552C2×C4—1428no—
553C2×C4—1424no—
554C2×C4—1424no—
555C2×C4—1424no—
556C2×C4—1424no—
557C2×C4—1424no—
558C2×C4—1422no—
559C2×C4—1424no—
560C2×C4—1422no—
561C2×C4—1424no—
562C2×C4—1422no—
563C2×C4—1422no—
564C2×C4—1422no—
565C2×C4—1422no—
566C2×C4—14232no—
567C2×C4C4⋊C4×D7114yes—
568C2×C4C4⋊C4×D7118yescomplemented
569C2×C4(C22⋊D71)⋊C414yes—
570C2×C4C4×D28414yes—
571C2×C4#1455609814yes—
572C2×C4#1455610614yes—
573C2×C4#1455610814yes—
574C2×C4#1455611014yes—
575C22—718no—
576C22—1428no—
577C22—1428no—
578C22—1424no—
579C22—1428no—
580C22—1424no—
581C22—1428no—
582C22—1424no—
583C22—1424no—
584C22—1424no—
585C22—1424no—
586C22C2×C4⋊C4×D7112yes—
587C22C2×C4⋊C4×D7111yescharacteristic
588C22C2×(C22⋊D71)⋊C411yescharacteristic
589C22C2×C4×D28411yescharacteristic
590C22#1798647912yes—
591C22C2×#1455609811yescharacteristic
592C22C2×#1455610611yescharacteristic
593C22C2×#1455610811yescharacteristic
594C22C2×#1455611011yescharacteristic
595C22#127×D7111yescharacteristic
596C22C4⋊Q8×D7112yes—
597C22C4⋊Q8×D7111yescharacteristic
598C22#1798656711yescharacteristic
599C22#1798656911yescharacteristic
600C22#1798662912yes—
601C22#1798663312yes—
602C22D4×D28412yes—
603C22#1798664912yes—
604C22#1798665412yes—
605C22#1798666212yes—
606C22Q8×D28412yes—
607C22#1798670712yes—
608C22#1798670912yes—
609C4—28no—
610C4—44no—
611C4—82no—
612C4—2844no—
613C4—5682no—
614C4—1428no—
615C4—1424no—
616C4#1798587318yescomplemented
617C2—1428no—
618C2#384×D7111yescharacteristic
619C2C2×#1798587311yescharacteristic
620C2#2259096511yescharacteristic
621C2#2259123612yes—
622C2C4⋊C4×D28412yes—
623C2#2259197611yescharacteristic
624C2#2259198311yescharacteristic
625C2#2259204012yes—
626C2#2259204412yes—
627C2#2259281011yescharacteristic
628C2#2259281711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D71
commutatorC23×C71C23×C4
FittingC71×C4⋊#95C2
socle, cosocleC24×C71C24
prime radicalC24×D71C23
hypercenterC4⋊#95D71
hypocenterC71#13880
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
3. C2×#17985869C2semidirect
4. C2×#17985868C2semidirect
7. #22586604C2semidirect
8. #22586569C2semidirect
9. C71×C4⋊#95C2semidirect
27. C4×(C2×D71)⋊C4C4semidirect
75. C2×C4×Dic71C2×C4semidirect
78. C71⋊#95C2×C4semidirect
91. C2×C42×C71C2×C4semidirect
139. C2×C4×D71C4⋊C4semidirect
146. (C2×D71)⋊C4C4⋊C4semidirect
227. C2×C4×C71C2×C4⋊C4semidirect
257. C2×Dic71C2×C4⋊C4semidirect
293. C2×D71#384semidirect
297. C4×C71#385semidirect
323. C4⋊#95D71semidirect
324. C71#13880semidirect
403. C2×C42C4×D71semidirect
568. C2×C4C4⋊C4×D71semidirect
616. C4#17985873semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D712
C2×#179858732
C2×#179858692
C2×#179858682
C2×#179858562
C2×C4×(C2×D71)⋊C42
#225866042
#225865692
C71×C4⋊#952
#1388071