Finite Group Databasebeta

← #31997463

Subgroups — #31997463

Order 18176.

1370 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D71C211yesmaximal, characteristic
2C2×#17986051C211yesmaximal, characteristic
3#22590745C211yesmaximal, characteristic
4#22588429C212yesmaximal
5#22588346C211yesmaximal, characteristic, complemented
6#22588345C211yesmaximal, characteristic, complemented
7#22588342C211yesmaximal, characteristic, complemented
8#22588273C212yesmaximal
9#103×Dic71C212yesmaximal, complemented
10#22588159C212yesmaximal, complemented
11C71×#1803C211yesmaximal, characteristic, complemented
12C22×Q8×D71C2211yescharacteristic
13C2×Q8×Dic71C2211yescharacteristic
14C2×C4×Dic142—21no—
15C2×C4×Dic142C2211yescharacteristic
16C2×#14556093—21no—
17C2×#14556093C2211yescharacteristic
18C4×Q8×D71—24no—
19C4×Q8×D71C418yescomplemented
20C2×#14556086—21no—
21C2×C4⋊C4×D71—21no—
22C2×C4⋊C4×D71C2211yescharacteristic
23C2×C42×D71—21no—
24C2×C42×D71C2211yescharacteristic
25#17986051—24no—
26C2×C4⋊(C71⋊C8)C2211yescharacteristic
27C2×#14556053C2212yes—
28C2×#14556052C2212yes—
29C2×#14556051C2212yes—
30C2×(C2×D71)⋊C8C2212yes—
31C2×C8×Dic71C2212yes—
32#17985869C2211yescharacteristic
33#17985864—21no—
34#17985856—21no—
35#17985856C2212yes—
36C4⋊C4×Dic71—21no—
37C4⋊C4×Dic71C2212yes—
38#17985762C2212yes—
39#17985761C2212yes—
40#17985761C2212yes—
41C2×C4×Q8×C71C2211yescharacteristic
42C2×C4⋊C8×C71C2211yescharacteristic
43C2×#103×C71C2212yes—
44#384×C71C2211yescharacteristic
45#340×C71C2212yes—
46C22×Dic142—21no—
47C22×Dic142C2311yescharacteristic
48C2×Q8×D71—22no—
49C2×Q8×D71—22no—
50C2×Q8×D71—22no—
51C2×Q8×D71—22no—
52C2×Q8×D71—22no—
53C2×Q8×D71C2×C414yes—
54C22×C4×D71—21no—
55C22×C4×D71—21no—
56C22×C4×D71D411yescharacteristic
57C22×C4×D71D411yescharacteristic
58C22×C4×D71C2311yescharacteristic
59C22×C71⋊C8C2312yes—
60Q8×Dic71—22no—
61Q8×Dic71C2×C414yes—
62C4×Dic142—28no—
63C4×Dic142—22no—
64C4×Dic142C2×C414yes—
65#14556093—28no—
66#14556093—22no—
67#14556093C2×C414yes—
68#14556086—44no—
69C2×C4⋊Dic71—21no—
70C2×C4⋊Dic71—21no—
71C2×C4⋊Dic71—21no—
72C2×C4⋊Dic71C2312yes—
73C2×C4⋊Dic71C2311yescharacteristic
74C2×Dic71⋊C4—21no—
75C2×Dic71⋊C4—21no—
76C2×Dic71⋊C4—21no—
77C4⋊C4×D71—28no—
78C4⋊C4×D71—22no—
79C4⋊C4×D71C2×C414yes—
80C2×(C2×D71)⋊C4—22no—
81C2×C4×Dic71—21no—
82C2×C4×Dic71—21no—
83C2×C4×Dic71—22no—
84C2×C4×Dic71—21no—
85C2×C4×Dic71D411yescharacteristic
86C2×C4×Dic71D411yescharacteristic
87C2×C4×Dic71C2311yescharacteristic
88C42×D71—28no—
89C42×D71C2×C414yes—
90C42×D71D414yes—
91C4⋊(C71⋊C8)—22no—
92#14556053—24no—
93#14556052—24no—
94#14556051—24no—
95(C2×D71)⋊C8—24no—
96C8×Dic71—24no—
97C71⋊#95—22no—
98C71⋊#95—22no—
99C22×Q8×C71C2311yescharacteristic
100C22×C8×C71C2312yes—
101C4×Q8×C71—22no—
102C4×Q8×C71C2×C414yes—
103C2×C4⋊C4×C71—21no—
104C2×C4⋊C4×C71—21no—
105C2×C4⋊C4×C71—21no—
106C2×C4⋊C4×C71C2312yes—
107C2×C4⋊C4×C71C2311yescharacteristic
108C2×C42×C71—21no—
109C2×C42×C71C2311yescharacteristic
110C4⋊C8×C71—22no—
111#103×C71—24no—
112C23×D71C2×D411yescharacteristic
113C2×Dic142—24no—
114C2×Dic142—24no—
115C2×Dic142—21no—
116C2×Dic142—21no—
117C2×Dic142—21no—
118C2×Dic142—24no—
119C2×Dic142C22×C412yes—
120C2×Dic142#4112yes—
121C2×Dic142#4112yes—
122Q8×D71—28no—
123Q8×D71—28no—
124Q8×D71—28no—
125Q8×D71—28no—
126C22×Dic71—21no—
127C22×Dic71—21no—
128C22×Dic71—22no—
129C22×Dic71C2×D411yescharacteristic
130C22×Dic71C2×D411yescharacteristic
131C22×Dic71C2×D411yescharacteristic
132C2×C4×D71—24no—
133C2×C4×D71—24no—
134C2×C4×D71—24no—
135C2×C4×D71—21no—
136C2×C4×D71—21no—
137C2×C4×D71—21no—
138C2×C4×D71—21no—
139C2×C4×D71—24no—
140C2×C4×D71—41no—
141C2×C4×D71—41no—
142C2×C4×D71—42no—
143C2×C4×D71—42no—
144C2×C4×D71C22⋊C414yes—
145C2×C4×D71C22⋊C414yes—
146C2×C4×D71SD1614yes—
147C2×C4×D71Q1614yes—
148C2×C4×D71C22×C412yes—
149C2×C4×D71C2×D412yes—
150C2×C71⋊C8—22no—
151C2×C71⋊C8—22no—
152C2×C71⋊C8—22no—
153C4⋊Dic71—21no—
154C4⋊Dic71—24no—
155C4⋊Dic71—22no—
156C4⋊Dic71—22no—
157C4⋊Dic71—42no—
158C4⋊Dic71—42no—
159C4⋊Dic71C22×C412yes—
160Dic71⋊C4—24no—
161Dic71⋊C4—24no—
162Dic71⋊C4—24no—
163(C2×D71)⋊C4—44no—
164C4×Dic71—24no—
165C4×Dic71—24no—
166C4×Dic71—24no—
167C4×Dic71—44no—
168C4×Dic71C22⋊C414yes—
169C4×Dic71C22⋊C414yes—
170C4×Dic71C22×C412yes—
171C4×Dic71C2×D412yes—
172C2×Q8×C71—21no—
173C2×Q8×C71—21no—
174C2×Q8×C71—21no—
175C2×Q8×C71C22×C412yes—
176C2×Q8×C71#4112yes—
177C2×Q8×C71#4112yes—
178C22×C4×C71—21no—
179C22×C4×C71—21no—
180C22×C4×C71—22no—
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71C2411yescharacteristic
184C2×C8×C71—22no—
185C2×C8×C71—22no—
186C2×C8×C71—22no—
187C4⋊C4×C71—24no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71—22no—
190C4⋊C4×C71—22no—
191C4⋊C4×C71—42no—
192C4⋊C4×C71—42no—
193C4⋊C4×C71C22×C412yes—
194C42×C71—24no—
195C42×C71C22×C412yes—
196C42×C71C2×D412yes—
197C23×C71C22×D411yescharacteristic
198C22×D71—21no—
199C22×D71—21no—
200C22×D71—21no—
201C22×D71—21no—
202C22×D71C2×C22⋊C412yes—
203C22×D71C2×SD1612yes—
204C22×D71C2×Q1612yes—
205Q8×C71—22no—
206Q8×C71—22no—
207Q8×C71—22no—
208Q8×C71#11714yescomplemented
209C2×C4×C71—22no—
210C2×C4×C71—22no—
211C2×C4×C71—22no—
212C2×C4×C71—21no—
213C2×C4×C71—21no—
214C2×C4×C71—22no—
215C2×C4×C71—41no—
216C2×C4×C71—41no—
217C2×C4×C71—42no—
218C2×C4×C71—42no—
219C2×C4×C71—42no—
220C2×C4×C71C2×C22⋊C412yes—
221C2×C4×C71C2×C22⋊C412yes—
222C2×C4×C71C2×SD1612yes—
223C2×C4×C71C2×Q1612yes—
224C2×C4×C71C23×C411yescharacteristic
225C2×C4×C71C22×D411yescharacteristic
226C2×C4×C71C2×#4111yescharacteristic
227C2×C4×C71C2×#4111yescharacteristic
228C2×C4×C71C2×#4111yescharacteristic
229C2×C4×C71C2×#4111yescharacteristic
230C2×Dic71—22no—
231C2×Dic71—22no—
232C2×Dic71—22no—
233C2×Dic71—21no—
234C2×Dic71—21no—
235C2×Dic71—21no—
236C2×Dic71—21no—
237C2×Dic71—22no—
238C2×Dic71—41no—
239C2×Dic71—41no—
240C2×Dic71—42no—
241C2×Dic71—42no—
242C2×Dic71—42no—
243C2×Dic71C2×C22⋊C412yes—
244C2×Dic71C2×C22⋊C412yes—
245C2×Dic71C2×C22⋊C412yes—
246C2×Dic71C2×SD1612yes—
247C2×Dic71C2×Q1612yes—
248C4×D71—216no—
249C4×D71—22no—
250C4×D71—22no—
251C4×D71—22no—
252C4×D71—22no—
253C4×D71—44no—
254C4×D71—48no—
255C4×D71#103116yescomplemented
256Dic142—216no—
257Dic142—22no—
258Dic142—22no—
259Dic142—22no—
260Dic142#11714yescomplemented
261C8×C71—42no—
262C71⋊C8—42no—
263C22×C71C22×C22⋊C411yescharacteristic
264C22×C71#52511yescharacteristic
265C22×C71#52511yescharacteristic
266C22×C71C22×SD1611yescharacteristic
267C22×C71C22×Q1611yescharacteristic
268C22×C71#57511yescharacteristic
269C22×C71#57511yescharacteristic
270C2×D71—22no—
271C2×D71—22no—
272C2×D71—22no—
273C2×D71—22no—
274C2×D71—22no—
275C2×D71—22no—
276C2×D71C2×#10314yes—
277C4×C71—22no—
278C4×C71—24no—
279C4×C71—42no—
280C4×C71—82no—
281C4×C71C2×#10314yes—
282C4×C71C2×#11711yescharacteristic
283C4×C71C2×#11711yescharacteristic
284C4×C71#52511yescharacteristic
285C4×C71#52511yescharacteristic
286Dic71—22no—
287Dic71—22no—
288Dic71—24no—
289Dic71—42no—
290Dic71—82no—
291Dic71C2×#10314yes—
292#20734—711nomaximal, Sylow
293C2×C71#228011yescharacteristic
294C2×C71C22×#10311yescharacteristic
295C2×C71#289611yescharacteristic
296C2×C71#310311yescharacteristic
297C2×C71#310311yescharacteristic
298C2×C71#310411yescharacteristic
299C2×C71#310411yescharacteristic
300D71—24no—
301C22×C4×Q8—711no—
302C2×#431—711no—
303#2285—711no—
304#1824—712no—
305#1803—711no—
306#1803—711no—
307#1803—711no—
308#1803D7111yescharacteristic, complemented
309#1787—712no—
310C4×#103—712no—
311C4×#103—712no—
312C71#2073411yescharacteristic, complemented, Sylow
313C23×Q8—711no—
314C2×C4×Q8—711no—
315C2×C4×Q8—718no—
316C2×C4×Q8—711no—
317C2×C4×Q8—711no—
318C2×C4×Q8—1421no—
319C2×C4×Q8—1421no—
320C2×C4×Q8—1424no—
321C2×C4×Q8C2×D7111yescharacteristic
322C2×#117—1421no—
323C22×C4⋊C4—711no—
324C22×C4⋊C4—1421no—
325C22×C42—711no—
326C22×C42—1421no—
327#431—1424no—
328C2×C4⋊C8—711no—
329C2×C4⋊C8C2×D7111yescharacteristic
330C2×#103—712no—
331C2×#103—712no—
332C2×#103—712no—
333C2×#103C2×D7112yes—
334C2×C22⋊C8—712no—
335C2×C4×C8—712no—
336#384—711no—
337#384C2×D7111yescharacteristic
338#382—1421no—
339#380—712no—
340#380—1421no—
341C4×C4⋊C4—712no—
342C4×C4⋊C4—1421no—
343#340—712no—
344#340—712no—
345#340—712no—
346#340C2×D7112yes—
347C22×Q8—714no—
348C22×Q8—711no—
349C22×Q8—1421no—
350C22×Q8—1422no—
351C22×Q8—1422no—
352C22×Q8—1422no—
353C22×Q8—1422no—
354C22×Q8—1422no—
355C22×Q8C22×D7111yescharacteristic
356C23×C4—711no—
357C23×C4—711no—
358C23×C4—711no—
359C23×C4—1421no—
360C23×C4—1421no—
361C22×C8—712no—
362C22×C8C22×D7112yes—
363C4×Q8—22no—
364C4×Q8—714no—
365C4×Q8—714no—
366C4×Q8—714no—
367C4×Q8—1428no—
368C4×Q8—1422no—
369C4×Q8—1428no—
370C4×Q8—1422no—
371C4×Q8—1422no—
372C4×Q8C4×D7114yes—
373#117—2844no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—711no—
378C2×C4⋊C4—714no—
379C2×C4⋊C4—712no—
380C2×C4⋊C4—1422no—
381C2×C4⋊C4—1428no—
382C2×C4⋊C4—1421no—
383C2×C4⋊C4—1421no—
384C2×C4⋊C4—1421no—
385C2×C4⋊C4—1421no—
386C2×C4⋊C4—1421no—
387C2×C4⋊C4—1421no—
388C2×C4⋊C4C22×D7112yes—
389C2×C4⋊C4C22×D7111yescharacteristic
390C2×C22⋊C4—1422no—
391C2×C42—21no—
392C2×C42—711no—
393C2×C42—711no—
394C2×C42—711no—
395C2×C42—714no—
396C2×C42—714no—
397C2×C42—1428no—
398C2×C42—1422no—
399C2×C42—1421no—
400C2×C42—1421no—
401C2×C42—1421no—
402C2×C42C22×D7111yescharacteristic
403C4⋊C8—22no—
404C4⋊C8—1422no—
405#103—24no—
406#103—1424no—
407#103—1424no—
408#103—1424no—
409C22⋊C8—1424no—
410C4×C8—1424no—
411#95—1422no—
412#95—1422no—
413C24—711no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—712no—
418C2×Q8—712no—
419C2×Q8—712no—
420C2×Q8—1424no—
421C2×Q8—1424no—
422C2×Q8—1424no—
423C2×Q8—1428no—
424C2×Q8—1428no—
425C2×Q8—1428no—
426C2×Q8—1428no—
427C2×Q8—1421no—
428C2×Q8—1421no—
429C2×Q8—1421no—
430C2×Q8C2×C4×D7112yes—
431C2×Q8#1176586412yes—
432C2×Q8#1176586612yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—2841no—
437C22×C4—2841no—
438C22×C4—2842no—
439C22×C4—2842no—
440C22×C4—714no—
441C22×C4—711no—
442C22×C4—714no—
443C22×C4—714no—
444C22×C4—714no—
445C22×C4—711no—
446C22×C4—711no—
447C22×C4—712no—
448C22×C4—712no—
449C22×C4—1424no—
450C22×C4—1424no—
451C22×C4—1424no—
452C22×C4—1424no—
453C22×C4—1422no—
454C22×C4—1421no—
455C22×C4—1421no—
456C22×C4—1421no—
457C22×C4—1421no—
458C22×C4—1421no—
459C22×C4—1421no—
460C22×C4D4×D7111yescharacteristic
461C22×C4D4×D7111yescharacteristic
462C22×C4C23×D7111yescharacteristic
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—1422no—
467C2×C8—1422no—
468C2×C8—1422no—
469C4⋊C4—21no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—2842no—
476C4⋊C4—2842no—
477C4⋊C4—712no—
478C4⋊C4—1424no—
479C4⋊C4—1424no—
480C4⋊C4—1424no—
481C4⋊C4—1422no—
482C4⋊C4—1422no—
483C4⋊C4—1421no—
484C4⋊C4—1424no—
485C4⋊C4C2×C4×D7112yes—
486C22⋊C4—2844no—
487C42—24no—
488C42—2844no—
489C42—712no—
490C42—712no—
491C42—714no—
492C42—714no—
493C42—1424no—
494C42—1424no—
495C42—1424no—
496C42C2×C4×D7112yes—
497C42D4×D7112yes—
498C23—712no—
499C23—712no—
500C23—712no—
501C23—1421no—
502C23—1421no—
503C23—1421no—
504C23—1421no—
505C23C2×D4×D7111yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—22no—
509Q8—714no—
510Q8—14216no—
511Q8—1422no—
512Q8—1422no—
513Q8—1422no—
514Q8#1455608614yescomplemented
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—2844no—
527C2×C4—2848no—
528C2×C4—2841no—
529C2×C4—2841no—
530C2×C4—2842no—
531C2×C4—2842no—
532C2×C4—2842no—
533C2×C4—712no—
534C2×C4—712no—
535C2×C4—712no—
536C2×C4—7116no—
537C2×C4—712no—
538C2×C4—712no—
539C2×C4—14216no—
540C2×C4—1422no—
541C2×C4—1422no—
542C2×C4—1421no—
543C2×C4—1422no—
544C2×C4—1421no—
545C2×C4—1422no—
546C2×C4—1421no—
547C2×C4—1421no—
548C2×C4—1422no—
549C2×C4—1422no—
550C2×C4—1422no—
551C2×C4—1422no—
552C2×C4C22⋊C4×D7112yes—
553C2×C4C22⋊C4×D7112yes—
554C2×C4SD16×D7112yes—
555C2×C4Q16×D7112yes—
556C2×C4C22×C4×D7111yescharacteristic
557C2×C4C2×D4×D7111yescharacteristic
558C2×C4#41×D7111yescharacteristic
559C2×C4#41×D7111yescharacteristic
560C2×C4C2×#1176586411yescharacteristic
561C2×C4C2×#1176586611yescharacteristic
562C8—42no—
563C8—2842no—
564C22—714no—
565C22—1422no—
566C22—1422no—
567C22—1422no—
568C22—1422no—
569C22—1422no—
570C22—1422no—
571C22C2×C22⋊C4×D7111yescharacteristic
572C22#1798650911yescharacteristic
573C22#1798651211yescharacteristic
574C22C2×SD16×D7111yescharacteristic
575C22C2×Q16×D7111yescharacteristic
576C22#1798679511yescharacteristic
577C22#1798679811yescharacteristic
578C4—24no—
579C4—22no—
580C4—42no—
581C4—82no—
582C4—2842no—
583C4—5682no—
584C4—714no—
585C4—1422no—
586C4—1424no—
587C4—1422no—
588C4#103×D7114yes—
589C4#117×D7111yescharacteristic
590C4C2×#1455608611yescharacteristic
591C4#1798650911yescharacteristic
592C4#1798651211yescharacteristic
593C2—1424no—
594C2#2259072211yescharacteristic
595C2C2×#103×D7111yescharacteristic
596C2#2259685611yescharacteristic
597C2#2259837011yescharacteristic
598C2#2259837311yescharacteristic
599C2#2259837811yescharacteristic
600C2#2259838111yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1803C2
socleC23×C71C22×D4
prime radicalC23×D71C2×D4
cosocleC22×C4×C71C24
hypercenter#1803D71
hypocenterC71#20734
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
5. #22588346C2semidirect
6. #22588345C2semidirect
7. #22588342C2semidirect
9. #103×Dic71C2semidirect
10. #22588159C2semidirect
11. C71×#1803C2semidirect
19. C4×Q8×D71C4semidirect
208. Q8×C71#117semidirect
255. C4×D71#103semidirect
260. Dic142#117semidirect
308. #1803D71semidirect
312. C71#20734semidirect
514. Q8#14556086semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D712
C2×#179860512
#225907452
#225884292
#225883462
#225883452
#225883422
#225882732
#103×Dic712
#225881592
C71×#18032
#2073471