Finite Group Databasebeta

← #31996935

Subgroups — #31996935

Order 18176.

1450 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D71C211yesmaximal, characteristic
2C2×#17986021C211yesmaximal, characteristic
3C2×#17986006C212yesmaximal
4#22590745C211yesmaximal, characteristic
5#22588354C212yesmaximal, complemented
6#22588353C212yesmaximal, complemented
7#22588346C211yesmaximal, characteristic, complemented
8#22588345C211yesmaximal, characteristic, complemented
9#22588342C211yesmaximal, characteristic, complemented
10#22588058C212yesmaximal
11C71×#1803C211yesmaximal, characteristic, complemented
12C22×Q8×D71C2211yescharacteristic
13C22×#11765848C2212yes—
14C2×Q8×Dic71C2211yescharacteristic
15C2×C4×Dic142—21no—
16C2×C4×Dic142C2211yescharacteristic
17C2×#14556093—21no—
18C2×#14556093C2211yescharacteristic
19C4×Q8×D71—24no—
20C4×Q8×D71C418yescomplemented
21C2×#14556086C2212yes—
22C2×C4⋊C4×D71—21no—
23C2×C4⋊C4×D71C2211yescharacteristic
24C2×C42×D71—21no—
25C2×C42×D71C2211yescharacteristic
26#17986021—24no—
27C2×C4⋊(C71⋊C8)C2211yescharacteristic
28C2×Dic71⋊C8C2212yes—
29#17986006—28no—
30C2×#14556053C2212yes—
31C2×#14556052C2212yes—
32C2×#14556051C2212yes—
33#17985869C2211yescharacteristic
34#17985864C2212yes—
35#17985856—21no—
36#17985856—21no—
37C4⋊C4×Dic71—21no—
38C4⋊C4×Dic71—21no—
39#17985762C2212yes—
40#17985761C2212yes—
41#17985761C2212yes—
42C2×C4×Q8×C71C2211yescharacteristic
43C2×C4⋊C8×C71C2211yescharacteristic
44C2×#103×C71C2212yes—
45#384×C71C2211yescharacteristic
46#340×C71C2212yes—
47C22×Dic142—21no—
48C22×Dic142C2311yescharacteristic
49C2×Q8×D71—22no—
50C2×Q8×D71—22no—
51C2×Q8×D71—22no—
52C2×Q8×D71C2×C414yes—
53C2×Q8×D71D414yes—
54C2×Q8×D71Q814yes—
55C22×C4×D71—21no—
56C22×C4×D71—21no—
57C22×C4×D71D411yescharacteristic
58C22×C4×D71D411yescharacteristic
59C22×C4×D71C2311yescharacteristic
60C2×#11765848—24no—
61C2×#11765848—24no—
62C2×#11765848—24no—
63C22×C71⋊C8C2312yes—
64Q8×Dic71—22no—
65Q8×Dic71C2×C414yes—
66C4×Dic142—28no—
67C4×Dic142—22no—
68C4×Dic142C2×C414yes—
69#14556093—28no—
70#14556093—22no—
71#14556093C2×C414yes—
72#14556086—24no—
73#14556086—24no—
74C2×C4⋊Dic71—21no—
75C2×C4⋊Dic71—21no—
76C2×C4⋊Dic71—21no—
77C2×C4⋊Dic71C2312yes—
78C2×C4⋊Dic71C2311yescharacteristic
79C2×Dic71⋊C4—21no—
80C2×Dic71⋊C4—21no—
81C2×Dic71⋊C4—21no—
82C4⋊C4×D71—28no—
83C4⋊C4×D71—22no—
84C4⋊C4×D71C2×C414yes—
85C2×(C2×D71)⋊C4—22no—
86C2×C4×Dic71—21no—
87C2×C4×Dic71—21no—
88C2×C4×Dic71—22no—
89C2×C4×Dic71—21no—
90C2×C4×Dic71D411yescharacteristic
91C2×C4×Dic71D411yescharacteristic
92C2×C4×Dic71C2311yescharacteristic
93C42×D71—28no—
94C42×D71C2×C414yes—
95C42×D71D414yes—
96C4⋊(C71⋊C8)—22no—
97Dic71⋊C8—24no—
98#14556053—24no—
99#14556052—24no—
100#14556051—24no—
101C71⋊#95—22no—
102C71⋊#95—22no—
103C22×Q8×C71C2311yescharacteristic
104C22×C8×C71C2312yes—
105C4×Q8×C71—22no—
106C4×Q8×C71C2×C414yes—
107C2×C4⋊C4×C71—21no—
108C2×C4⋊C4×C71—21no—
109C2×C4⋊C4×C71—21no—
110C2×C4⋊C4×C71C2312yes—
111C2×C4⋊C4×C71C2311yescharacteristic
112C2×C42×C71—21no—
113C2×C42×C71C2311yescharacteristic
114C4⋊C8×C71—22no—
115#103×C71—24no—
116C23×D71C2×D411yescharacteristic
117C2×Dic142—21no—
118C2×Dic142—21no—
119C2×Dic142—21no—
120C2×Dic142—24no—
121C2×Dic142—24no—
122C2×Dic142—24no—
123C2×Dic142C22×C412yes—
124C2×Dic142C2×D412yes—
125C2×Dic142C2×Q812yes—
126Q8×D71—28no—
127Q8×D71—28no—
128Q8×D71—28no—
129Q8×D71C4⋊C4116yescomplemented
130C22×Dic71—21no—
131C22×Dic71—21no—
132C22×Dic71—22no—
133C22×Dic71C2×D411yescharacteristic
134C22×Dic71C2×D411yescharacteristic
135C22×Dic71C2×D411yescharacteristic
136C2×C4×D71—24no—
137C2×C4×D71—24no—
138C2×C4×D71—24no—
139C2×C4×D71—22no—
140C2×C4×D71—22no—
141C2×C4×D71—24no—
142C2×C4×D71—41no—
143C2×C4×D71—41no—
144C2×C4×D71—42no—
145C2×C4×D71—42no—
146C2×C4×D71C22⋊C414yes—
147C2×C4×D71C22⋊C414yes—
148C2×C4×D71C22×C412yes—
149C2×C4×D71C2×D412yes—
150C2×C4×D71C2×D412yes—
151C2×C4×D71C2×Q812yes—
152C2×C4×D71#4112yes—
153C2×C4×D71#4112yes—
154#11765848—48no—
155C2×C71⋊C8—22no—
156C2×C71⋊C8—22no—
157C2×C71⋊C8—22no—
158C4⋊Dic71—21no—
159C4⋊Dic71—24no—
160C4⋊Dic71—22no—
161C4⋊Dic71—22no—
162C4⋊Dic71—42no—
163C4⋊Dic71—42no—
164C4⋊Dic71C22×C412yes—
165Dic71⋊C4—24no—
166Dic71⋊C4—24no—
167Dic71⋊C4—24no—
168(C2×D71)⋊C4—44no—
169C4×Dic71—24no—
170C4×Dic71—24no—
171C4×Dic71—24no—
172C4×Dic71—44no—
173C4×Dic71C22⋊C414yes—
174C4×Dic71C22⋊C414yes—
175C4×Dic71C22×C412yes—
176C4×Dic71C2×D412yes—
177C2×Q8×C71—21no—
178C2×Q8×C71—21no—
179C2×Q8×C71—21no—
180C2×Q8×C71C22×C412yes—
181C2×Q8×C71C2×D412yes—
182C2×Q8×C71C2×Q812yes—
183C22×C4×C71—21no—
184C22×C4×C71—21no—
185C22×C4×C71—22no—
186C22×C4×C71C2×D411yescharacteristic
187C22×C4×C71C2×D411yescharacteristic
188C22×C4×C71C2411yescharacteristic
189C2×C8×C71—22no—
190C2×C8×C71—22no—
191C2×C8×C71—22no—
192C4⋊C4×C71—24no—
193C4⋊C4×C71—21no—
194C4⋊C4×C71—22no—
195C4⋊C4×C71—22no—
196C4⋊C4×C71—42no—
197C4⋊C4×C71—42no—
198C4⋊C4×C71C22×C412yes—
199C42×C71—24no—
200C42×C71C22×C412yes—
201C42×C71C2×D412yes—
202C23×C71C22×D411yescharacteristic
203C22×D71—21no—
204C22×D71—21no—
205C22×D71—21no—
206C22×D71—21no—
207C22×D71C2×C22⋊C412yes—
208C22×D71#12112yes—
209C22×D71C22⋊Q812yes—
210Q8×C71—22no—
211Q8×C71—22no—
212Q8×C71—22no—
213Q8×C71C2×C4⋊C414yes—
214C2×C4×C71—22no—
215C2×C4×C71—22no—
216C2×C4×C71—22no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—22no—
220C2×C4×C71—41no—
221C2×C4×C71—41no—
222C2×C4×C71—42no—
223C2×C4×C71—42no—
224C2×C4×C71—42no—
225C2×C4×C71C2×C22⋊C412yes—
226C2×C4×C71C2×C22⋊C412yes—
227C2×C4×C71#13712yes—
228C2×C4×C71#13712yes—
229C2×C4×C71C23×C411yescharacteristic
230C2×C4×C71C22×D411yescharacteristic
231C2×C4×C71C22×D411yescharacteristic
232C2×C4×C71C22×Q811yescharacteristic
233C2×C4×C71C2×#4111yescharacteristic
234C2×C4×C71C2×#4111yescharacteristic
235C2×Dic71—22no—
236C2×Dic71—22no—
237C2×Dic71—22no—
238C2×Dic71—21no—
239C2×Dic71—21no—
240C2×Dic71—21no—
241C2×Dic71—21no—
242C2×Dic71—22no—
243C2×Dic71—41no—
244C2×Dic71—41no—
245C2×Dic71—42no—
246C2×Dic71—42no—
247C2×Dic71—42no—
248C2×Dic71C2×C22⋊C412yes—
249C2×Dic71C2×C22⋊C412yes—
250C2×Dic71C2×C22⋊C412yes—
251C2×Dic71#12112yes—
252C2×Dic71C22⋊Q812yes—
253C4×D71—216no—
254C4×D71—28no—
255C4×D71—44no—
256C4×D71—48no—
257C4×D71C2×C4⋊C414yes—
258C4×D71#11714yes—
259C4×D71#12114yes—
260C4×D71C22⋊Q814yes—
261Dic142—22no—
262Dic142—22no—
263Dic142—22no—
264Dic142—216no—
265Dic142C2×C4⋊C414yes—
266C8×C71—42no—
267C71⋊C8—42no—
268C22×C71C22×C22⋊C411yescharacteristic
269C22×C71C2×#12111yescharacteristic
270C22×C71C2×C22⋊Q811yescharacteristic
271C22×C71C2×#13711yescharacteristic
272C22×C71C2×#13711yescharacteristic
273C22×C71#57511yescharacteristic
274C22×C71#57511yescharacteristic
275C2×D71—22no—
276C2×D71—22no—
277C2×D71—22no—
278C2×D71—22no—
279C2×D71—22no—
280C2×D71—22no—
281C2×D71#38014yes—
282C4×C71—22no—
283C4×C71—24no—
284C4×C71—42no—
285C4×C71—82no—
286C4×C71#41914yescomplemented
287C4×C71C22×C4⋊C411yescharacteristic
288C4×C71C2×#11711yescharacteristic
289C4×C71C2×#12111yescharacteristic
290C4×C71C2×C22⋊Q811yescharacteristic
291Dic71—22no—
292Dic71—22no—
293Dic71—24no—
294Dic71—42no—
295Dic71—82no—
296Dic71#38014yes—
297#20647—711nomaximal, Sylow
298C2×C71C2×#38011yescharacteristic
299C2×C71C2×#41911yescharacteristic
300C2×C71#289611yescharacteristic
301C2×C71#303711yescharacteristic
302C2×C71#303711yescharacteristic
303C2×C71#307811yescharacteristic
304C2×C71#307811yescharacteristic
305D71—24no—
306C22×C4×Q8—711no—
307C2×#423—711no—
308C2×#419—712no—
309#2285—711no—
310#1805—712no—
311#1805—712no—
312#1803—711no—
313#1803—711no—
314#1803—711no—
315#1803D7111yescharacteristic, complemented
316#1736—712no—
317C71#2064711yescharacteristic, complemented, Sylow
318C23×Q8—711no—
319C22×M16—712no—
320C2×C4×Q8—711no—
321C2×C4×Q8—718no—
322C2×C4×Q8—711no—
323C2×C4×Q8—711no—
324C2×C4×Q8—1421no—
325C2×C4×Q8—1421no—
326C2×C4×Q8—1424no—
327C2×C4×Q8C2×D7111yescharacteristic
328C2×#117—712no—
329C22×C4⋊C4—711no—
330C22×C4⋊C4—1421no—
331C22×C42—711no—
332C22×C42—1421no—
333#423—1424no—
334C2×C4⋊C8—712no—
335C2×C4⋊C8—711no—
336C2×C4⋊C8C2×D7111yescharacteristic
337#419—1428no—
338C2×#103—712no—
339C2×#103—712no—
340C2×#103—712no—
341C2×#103C2×D7112yes—
342#384—711no—
343#384C2×D7111yescharacteristic
344#382—712no—
345#380—1421no—
346#380—1421no—
347C4×C4⋊C4—1421no—
348C4×C4⋊C4—1421no—
349#340—712no—
350#340—712no—
351#340—712no—
352#340C2×D7112yes—
353C22×Q8—714no—
354C22×Q8—714no—
355C22×Q8—714no—
356C22×Q8—711no—
357C22×Q8—1421no—
358C22×Q8—1422no—
359C22×Q8—1422no—
360C22×Q8—1422no—
361C22×Q8C22×D7111yescharacteristic
362C23×C4—711no—
363C23×C4—711no—
364C23×C4—711no—
365C23×C4—1421no—
366C23×C4—1421no—
367C2×M16—1424no—
368C2×M16—1424no—
369C2×M16—1424no—
370C22×C8—712no—
371C22×C8C22×D7112yes—
372C4×Q8—22no—
373C4×Q8—714no—
374C4×Q8—714no—
375C4×Q8—714no—
376C4×Q8—1428no—
377C4×Q8—1422no—
378C4×Q8—1428no—
379C4×Q8—1422no—
380C4×Q8—1422no—
381C4×Q8C4×D7114yes—
382#117—1424no—
383#117—1424no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—711no—
388C2×C4⋊C4—714no—
389C2×C4⋊C4—712no—
390C2×C4⋊C4—1422no—
391C2×C4⋊C4—1428no—
392C2×C4⋊C4—1421no—
393C2×C4⋊C4—1421no—
394C2×C4⋊C4—1421no—
395C2×C4⋊C4—1421no—
396C2×C4⋊C4—1421no—
397C2×C4⋊C4—1421no—
398C2×C4⋊C4C22×D7112yes—
399C2×C4⋊C4C22×D7111yescharacteristic
400C2×C22⋊C4—1422no—
401C2×C42—21no—
402C2×C42—711no—
403C2×C42—711no—
404C2×C42—711no—
405C2×C42—714no—
406C2×C42—714no—
407C2×C42—1428no—
408C2×C42—1422no—
409C2×C42—1421no—
410C2×C42—1421no—
411C2×C42—1421no—
412C2×C42C22×D7111yescharacteristic
413C4⋊C8—22no—
414C4⋊C8—1424no—
415C4⋊C8—1422no—
416#103—24no—
417#103—1424no—
418#103—1424no—
419#103—1424no—
420#95—1422no—
421#95—1422no—
422C24—711no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—7116no—
427C2×Q8—712no—
428C2×Q8—712no—
429C2×Q8—712no—
430C2×Q8—1424no—
431C2×Q8—1424no—
432C2×Q8—1424no—
433C2×Q8—1428no—
434C2×Q8—1428no—
435C2×Q8—1428no—
436C2×Q8—1421no—
437C2×Q8—1421no—
438C2×Q8—1421no—
439C2×Q8C2×C4×D7112yes—
440C2×Q8D4×D7112yes—
441C2×Q8Q8×D7112yes—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—2841no—
446C22×C4—2841no—
447C22×C4—2842no—
448C22×C4—2842no—
449C22×C4—714no—
450C22×C4—711no—
451C22×C4—712no—
452C22×C4—712no—
453C22×C4—712no—
454C22×C4—712no—
455C22×C4—712no—
456C22×C4—712no—
457C22×C4—711no—
458C22×C4—711no—
459C22×C4—714no—
460C22×C4—1424no—
461C22×C4—1424no—
462C22×C4—1424no—
463C22×C4—1424no—
464C22×C4—1422no—
465C22×C4—1421no—
466C22×C4—1421no—
467C22×C4—1422no—
468C22×C4—1422no—
469C22×C4D4×D7111yescharacteristic
470C22×C4D4×D7111yescharacteristic
471C22×C4C23×D7111yescharacteristic
472M16—2848no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—1422no—
477C2×C8—1422no—
478C2×C8—1422no—
479C4⋊C4—21no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—2842no—
486C4⋊C4—2842no—
487C4⋊C4—712no—
488C4⋊C4—1424no—
489C4⋊C4—1424no—
490C4⋊C4—1424no—
491C4⋊C4—1422no—
492C4⋊C4—1422no—
493C4⋊C4—1421no—
494C4⋊C4—1424no—
495C4⋊C4C2×C4×D7112yes—
496C22⋊C4—2844no—
497C42—24no—
498C42—2844no—
499C42—712no—
500C42—712no—
501C42—714no—
502C42—714no—
503C42—1424no—
504C42—1424no—
505C42—1424no—
506C42C2×C4×D7112yes—
507C42D4×D7112yes—
508C23—712no—
509C23—712no—
510C23—712no—
511C23—1421no—
512C23—1421no—
513C23—1421no—
514C23—1421no—
515C23C2×D4×D7111yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—22no—
519Q8—714no—
520Q8—14216no—
521Q8—1422no—
522Q8—1422no—
523Q8—1422no—
524Q8C4⋊C4×D7114yes—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—2844no—
537C2×C4—2848no—
538C2×C4—2841no—
539C2×C4—2841no—
540C2×C4—2842no—
541C2×C4—2842no—
542C2×C4—2842no—
543C2×C4—712no—
544C2×C4—712no—
545C2×C4—712no—
546C2×C4—714no—
547C2×C4—714no—
548C2×C4—714no—
549C2×C4—714no—
550C2×C4—712no—
551C2×C4—712no—
552C2×C4—14216no—
553C2×C4—1422no—
554C2×C4—1422no—
555C2×C4—1422no—
556C2×C4—1422no—
557C2×C4—1421no—
558C2×C4—1421no—
559C2×C4—1421no—
560C2×C4—1421no—
561C2×C4—1428no—
562C2×C4C22⋊C4×D7112yes—
563C2×C4C22⋊C4×D7112yes—
564C2×C4#1455615712yes—
565C2×C4#1455616012yes—
566C2×C4C22×C4×D7111yescharacteristic
567C2×C4C2×D4×D7111yescharacteristic
568C2×C4C2×D4×D7111yescharacteristic
569C2×C4C2×Q8×D7111yescharacteristic
570C2×C4#41×D7111yescharacteristic
571C2×C4#41×D7111yescharacteristic
572C8—42no—
573C8—2842no—
574C22—714no—
575C22—1422no—
576C22—1422no—
577C22—1422no—
578C22—1422no—
579C22—1422no—
580C22—1422no—
581C22C2×C22⋊C4×D7111yescharacteristic
582C22#121×D7111yescharacteristic
583C22C22⋊Q8×D7111yescharacteristic
584C22C2×#1455615711yescharacteristic
585C22C2×#1455616011yescharacteristic
586C22#1798679511yescharacteristic
587C22#1798679811yescharacteristic
588C4—24no—
589C4—22no—
590C4—42no—
591C4—82no—
592C4—2842no—
593C4—5682no—
594C4—714no—
595C4—1422no—
596C4—1424no—
597C4—1422no—
598C4#1798600614yescomplemented
599C4C2×C4⋊C4×D7111yescharacteristic
600C4#117×D7111yescharacteristic
601C4#121×D7111yescharacteristic
602C4C22⋊Q8×D7111yescharacteristic
603C2—1424no—
604C2#380×D7111yescharacteristic
605C2C2×#1798600611yescharacteristic
606C2#2259685611yescharacteristic
607C2#2259780711yescharacteristic
608C2#2259781311yescharacteristic
609C2#2259814011yescharacteristic
610C2#2259814611yescharacteristic
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#20647
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. #22588354C2semidirect
6. #22588353C2semidirect
7. #22588346C2semidirect
8. #22588345C2semidirect
9. #22588342C2semidirect
11. C71×#1803C2semidirect
20. C4×Q8×D71C4semidirect
129. Q8×D71C4⋊C4semidirect
286. C4×C71#419semidirect
315. #1803D71semidirect
317. C71#20647semidirect
598. C4#17986006semidirect

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×#179860212
C2×#179860062
#225907452
#225883542
#225883532
#225883462
#225883452
#225883422
#225880582
C71×#18032
#2064771