Finite Group Databasebeta

← #32004695

Subgroups — #32004695

Order 18176.

876 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×Dic71C211yesmaximal, characteristic
2#117×Dic71C211yesmaximal, characteristic
3#22588983C211yesmaximal, characteristic, complemented
4#22588976C211yesmaximal, characteristic, complemented
5#22588934C211yesmaximal, characteristic, complemented
6#22588927C211yesmaximal, characteristic, complemented
7#22588653C211yesmaximal, characteristic
8#22588541C211yesmaximal, characteristic
9#22588435C211yesmaximal, characteristic
10#22588240C211yesmaximal, characteristic, complemented
11#22588239C211yesmaximal, characteristic, complemented
12#22588172C211yesmaximal, characteristic, complemented
13#22588172C211yesmaximal, characteristic, complemented
14#22588104C211yesmaximal, characteristic
15C2×#485×C71C211yesmaximal, characteristic
16C2×Q8×Dic71C2211yescharacteristic
17C2×#14556087C2211yescharacteristic
18C22×C4×Dic71C2211yescharacteristic
19C2×#14556062C2211yescharacteristic
20C2×#14556060C2211yescharacteristic
21C2×#14556052C2211yescharacteristic
22C2×#14556052C2211yescharacteristic
23C2×C22⋊(C71⋊C8)C2211yescharacteristic
24C2×#14556038C2211yescharacteristic
25C2×#14556038C2211yescharacteristic
26#17985906C2211yescharacteristic
27#17985898C2211yescharacteristic
28#17985885—21no—
29#17985874—21no—
30#17985859C2211yescharacteristic
31C4⋊C4×Dic71—21no—
32C4⋊C4×Dic71C2211yescharacteristic
33C4⋊C4×Dic71C2211yescharacteristic
34C4⋊C4×Dic71C2211yescharacteristic
35C4⋊C4×Dic71C2211yescharacteristic
36C4⋊C4×Dic71C2211yescharacteristic
37C22⋊C4×Dic71—21no—
38C22⋊C4×Dic71—21no—
39C22⋊C4×Dic71—21no—
40#17985845—21no—
41C42×Dic71—21no—
42#17985762C2211yescharacteristic
43#17985762C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985761C2211yescharacteristic
48#17985761C2211yescharacteristic
49#17985761C2211yescharacteristic
50#17985753C2211yescharacteristic
51#17985753C2211yescharacteristic
52C2×C22⋊Q8×C71C2211yescharacteristic
53C2×#117×C71C2211yescharacteristic
54#485×C71C418yescomplemented
55C2×#107×C71C2211yescharacteristic
56C2×#106×C71C2211yescharacteristic
57C2×#103×C71C2211yescharacteristic
58C2×#103×C71C2211yescharacteristic
59C2×C22⋊C8×C71C2211yescharacteristic
60C23×Dic71D411yescharacteristic
61C23×Dic71D411yescharacteristic
62C22×C71⋊C8C2311yescharacteristic
63C22×C71⋊C8C2311yescharacteristic
64Q8×Dic71—22no—
65Q8×Dic71—22no—
66#14556087—22no—
67#14556087—22no—
68C2×C4⋊Dic71—21no—
69C2×C4⋊Dic71—21no—
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C4⋊Dic71C2311yescharacteristic
72C2×C4⋊Dic71C2311yescharacteristic
73C2×C4⋊Dic71C2311yescharacteristic
74C2×C22⋊Dic71—21no—
75C2×C22⋊Dic71—21no—
76C2×C4×Dic71—22no—
77C2×C4×Dic71—22no—
78C2×C4×Dic71—21no—
79C2×C4×Dic71—21no—
80C2×C4×Dic71—21no—
81C2×C4×Dic71—21no—
82C2×C4×Dic71—21no—
83C2×C4×Dic71—21no—
84C2×C4×Dic71—21no—
85C2×C4×Dic71D411yescharacteristic
86C2×C4×Dic71D411yescharacteristic
87C2×C4×Dic71C2311yescharacteristic
88C2×C4×Dic71C2311yescharacteristic
89#14556062—22no—
90#14556060—22no—
91#14556052—22no—
92#14556052—22no—
93C22⋊(C71⋊C8)—22no—
94#14556038—22no—
95#14556038—22no—
96C71⋊#95—21no—
97C71⋊#95—21no—
98C71⋊#95—21no—
99C71⋊#95—21no—
100C71⋊#95—21no—
101C71⋊#95—21no—
102C71⋊#95—21no—
103C71⋊#95—21no—
104C71⋊#95—21no—
105C22×Q8×C71C2311yescharacteristic
106C23×C4×C71C2311yescharacteristic
107C22×C8×C71C2311yescharacteristic
108C22×C8×C71C2311yescharacteristic
109C22⋊Q8×C71—22no—
110C22⋊Q8×C71C2×C414yes—
111#117×C71—22no—
112#117×C71C2×C414yes—
113C2×C4⋊C4×C71—21no—
114C2×C4⋊C4×C71C2311yescharacteristic
115C2×C4⋊C4×C71C2311yescharacteristic
116C2×C4⋊C4×C71C2311yescharacteristic
117C2×C22⋊C4×C71—21no—
118C2×C22⋊C4×C71—21no—
119C2×C42×C71—21no—
120#107×C71C2×C414yescomplemented
121#106×C71C2×C414yescomplemented
122#103×C71C2×C414yescomplemented
123#103×C71C2×C414yescomplemented
124C22⋊C8×C71C2×C414yes—
125C22×Dic71—21no—
126C22×Dic71—21no—
127C22×Dic71—22no—
128C22×Dic71—22no—
129C22×Dic71—22no—
130C22×Dic71—21no—
131C22×Dic71—21no—
132C22×Dic71—41no—
133C22×Dic71—41no—
134C22×Dic71—41no—
135C22×Dic71C2×D411yescharacteristic
136C22×Dic71C2×D411yescharacteristic
137C22×Dic71C2×D411yescharacteristic
138C22×Dic71C2×D411yescharacteristic
139C2×C71⋊C8—21no—
140C2×C71⋊C8—21no—
141C2×C71⋊C8—21no—
142C2×C71⋊C8—21no—
143C2×C71⋊C8—21no—
144C2×C71⋊C8—21no—
145C4⋊Dic71—21no—
146C4⋊Dic71—21no—
147C4⋊Dic71—21no—
148C4⋊Dic71—21no—
149C4⋊Dic71—21no—
150C4⋊Dic71—21no—
151C4⋊Dic71—42no—
152C4⋊Dic71—42no—
153C4⋊Dic71#4112yes—
154C4⋊Dic71#4112yes—
155C22⋊Dic71—24no—
156C22⋊Dic71—42no—
157C4×Dic71—24no—
158C4×Dic71—21no—
159C4×Dic71—21no—
160C4×Dic71—22no—
161C4×Dic71—22no—
162C4×Dic71—42no—
163C4×Dic71—42no—
164C4×Dic71—42no—
165C4×Dic71—42no—
166C4×Dic71—42no—
167C4×Dic71—42no—
168C4×Dic71#4112yes—
169C4×Dic71#4112yes—
170C24×C71C2×D411yescharacteristic
171C2×Q8×C71—21no—
172C2×Q8×C71—21no—
173C2×Q8×C71—21no—
174C2×Q8×C71—21no—
175C2×Q8×C71—21no—
176C2×Q8×C71C22×C412yes—
177C22×C4×C71—21no—
178C22×C4×C71—21no—
179C22×C4×C71—21no—
180C22×C4×C71—21no—
181C22×C4×C71—21no—
182C22×C4×C71—21no—
183C22×C4×C71—21no—
184C22×C4×C71—21no—
185C22×C4×C71C22×C412yes—
186C22×C4×C71C2×D411yescharacteristic
187C22×C4×C71#4112yes—
188C22×C4×C71C2411yescharacteristic
189C2×C8×C71—21no—
190C2×C8×C71—21no—
191C2×C8×C71—21no—
192C2×C8×C71—21no—
193C2×C8×C71C22×C412yes—
194C2×C8×C71C22×C412yes—
195C4⋊C4×C71—24no—
196C4⋊C4×C71—21no—
197C4⋊C4×C71—21no—
198C4⋊C4×C71—21no—
199C4⋊C4×C71C22×C412yes—
200C4⋊C4×C71C22×C412yes—
201C4⋊C4×C71C22×C412yes—
202C22⋊C4×C71—24no—
203C22⋊C4×C71—24no—
204C42×C71—24no—
205C23×C71—21no—
206C23×C71—21no—
207C23×C71—21no—
208C23×C71—21no—
209C23×C71—21no—
210C23×C71—21no—
211C23×C71C4×D412yes—
212C23×C71C22×D411yescharacteristic
213Q8×C71—22no—
214Q8×C71—22no—
215Q8×C71—41no—
216Q8×C71—41no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—22no—
220C2×C4×C71—22no—
221C2×C4×C71—22no—
222C2×C4×C71—22no—
223C2×C4×C71—22no—
224C2×C4×C71—22no—
225C2×C4×C71—22no—
226C2×C4×C71—22no—
227C2×C4×C71—41no—
228C2×C4×C71—41no—
229C2×C4×C71—41no—
230C2×C4×C71—41no—
231C2×C4×C71—41no—
232C2×C4×C71—41no—
233C2×C4×C71—41no—
234C2×C4×C71—41no—
235C2×C4×C71C4×D412yes—
236C2×C4×C71C23×C411yescharacteristic
237C2×C4×C71C2×#4111yescharacteristic
238C2×C4×C71C2×#4111yescharacteristic
239C2×C4×C71C2×#4111yescharacteristic
240C2×C4×C71C2×#4111yescharacteristic
241C2×C4×C71C2×#4111yescharacteristic
242C2×Dic71—21no—
243C2×Dic71—21no—
244C2×Dic71—21no—
245C2×Dic71—21no—
246C2×Dic71—21no—
247C2×Dic71—21no—
248C2×Dic71—21no—
249C2×Dic71—22no—
250C2×Dic71—22no—
251C2×Dic71—21no—
252C2×Dic71—21no—
253C2×Dic71—44no—
254C2×Dic71—44no—
255C2×Dic71—41no—
256C2×Dic71—41no—
257C2×Dic71—41no—
258C2×Dic71—41no—
259C2×Dic71—41no—
260C2×Dic71—41no—
261C2×Dic71—41no—
262C2×Dic71—41no—
263C2×Dic71—41no—
264C2×Dic71—41no—
265C2×Dic71#12312yes—
266C2×Dic71#12312yes—
267C2×Dic71#13712yes—
268C8×C71—22no—
269C8×C71—22no—
270C71⋊C8—41no—
271C71⋊C8—41no—
272C22×C71—22no—
273C22×C71—22no—
274C22×C71—22no—
275C22×C71—41no—
276C22×C71—41no—
277C22×C71—41no—
278C22×C71—41no—
279C22×C71C2×C4×D411yescharacteristic
280C22×C71C2×#12311yescharacteristic
281C22×C71#52511yescharacteristic
282C22×C71C2×#13711yescharacteristic
283C22×C71#57511yescharacteristic
284C22×C71#57511yescharacteristic
285C22×C71#57511yescharacteristic
286C4×C71—22no—
287C4×C71—42no—
288C4×C71—42no—
289C4×C71—42no—
290C4×C71—42no—
291C4×C71C4×#4111yescharacteristic
292C4×C71C4×#4111yescharacteristic
293C4×C71#52911yescharacteristic
294C4×C71#52911yescharacteristic
295Dic71—22no—
296Dic71—22no—
297Dic71—22no—
298Dic71—41no—
299Dic71—42no—
300Dic71—42no—
301Dic71—81no—
302Dic71—81no—
303#21583—711nomaximal, Sylow
304C2×C71—42no—
305C2×C71C4×#12311yescharacteristic
306C2×C71#294311yescharacteristic
307C2×C71C4×#13711yescharacteristic
308C2×C71#309311yescharacteristic
309C2×C71#309511yescharacteristic
310C2×C71#311111yescharacteristic
311C2×C71#311611yescharacteristic
312C2×#485D7111yescharacteristic
313C4×C22⋊Q8—711no—
314C4×#117—711no—
315#1928—711no—
316#1927—711no—
317#1921—711no—
318#1920—711no—
319#1873—711no—
320#1854—711no—
321#1825—711no—
322#1776—711no—
323#1776—711no—
324#1763—711no—
325#1763—711no—
326#1749—711no—
327C71#2158311yescharacteristic, complemented, Sylow
328C2×C22⋊Q8C2×D7111yescharacteristic
329C2×C4×Q8—711no—
330C2×#117—711no—
331C2×#117C2×D7111yescharacteristic
332C22×C42—711no—
333#485Dic7118yescomplemented
334C2×#107—711no—
335C2×#107C2×D7111yescharacteristic
336C2×#106—711no—
337C2×#106C2×D7111yescharacteristic
338C2×#103—711no—
339C2×#103—711no—
340C2×#103C2×D7111yescharacteristic
341C2×#103C2×D7111yescharacteristic
342C2×C22⋊C8—711no—
343C2×C22⋊C8C2×D7111yescharacteristic
344C2×#97—711no—
345C2×#97—711no—
346#391—711no—
347#389—711no—
348#387—1421no—
349#385—1421no—
350#380—711no—
351C4×C4⋊C4—711no—
352C4×C4⋊C4—711no—
353C4×C4⋊C4—711no—
354C4×C4⋊C4—711no—
355C4×C4⋊C4—711no—
356C4×C4⋊C4—1421no—
357C4×C22⋊C4—1421no—
358C4×C22⋊C4—1421no—
359C4×C22⋊C4—1421no—
360#376—1421no—
361C43—1421no—
362#340—711no—
363#340—711no—
364#340—711no—
365#340—711no—
366#340—711no—
367#340—711no—
368#340—711no—
369#340—711no—
370#336—711no—
371#336—711no—
372C22×Q8C22×D7111yescharacteristic
373C23×C4—711no—
374C23×C4—711no—
375C23×C4C22×D7111yescharacteristic
376C22×C8—711no—
377C22×C8—711no—
378C22×C8C22×D7111yescharacteristic
379C22×C8C22×D7111yescharacteristic
380C22⋊Q8—22no—
381C22⋊Q8C2×Dic7114yes—
382C4×Q8—1422no—
383C4×Q8—1422no—
384#117—22no—
385#117—1422no—
386#117—1422no—
387#117C2×Dic7114yes—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—711no—
390C2×C4⋊C4—711no—
391C2×C4⋊C4—711no—
392C2×C4⋊C4—711no—
393C2×C4⋊C4—1421no—
394C2×C4⋊C4—1421no—
395C2×C4⋊C4C22×D7111yescharacteristic
396C2×C4⋊C4C22×D7111yescharacteristic
397C2×C4⋊C4C22×D7111yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—1421no—
401C2×C22⋊C4—1421no—
402C2×C42—21no—
403C2×C42—711no—
404C2×C42—711no—
405C2×C42—711no—
406C2×C42—711no—
407C2×C42—1421no—
408C2×C42—1421no—
409C2×C42—1421no—
410C2×C42—1421no—
411C2×C42—1421no—
412C2×C42—1421no—
413C2×C42—1422no—
414C2×C42—1422no—
415C2×C42—1421no—
416#107—1422no—
417#107C2×Dic7114yescomplemented
418#106—1422no—
419#106C2×Dic7114yescomplemented
420#103—1422no—
421#103—1422no—
422#103C2×Dic7114yescomplemented
423#103C2×Dic7114yescomplemented
424C22⋊C8—1422no—
425C22⋊C8C2×Dic7114yes—
426#97—1422no—
427#97—1422no—
428#95—1421no—
429#95—1421no—
430#95—1421no—
431#95—1421no—
432#95—1421no—
433#95—1421no—
434#95—1421no—
435#95—1421no—
436#95—1421no—
437C24D4×D7111yescharacteristic
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8C22×Dic7112yes—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—2841no—
453C22×C4—2841no—
454C22×C4—2841no—
455C22×C4—711no—
456C22×C4—711no—
457C22×C4—711no—
458C22×C4—711no—
459C22×C4—1421no—
460C22×C4—1421no—
461C22×C4—1422no—
462C22×C4—1422no—
463C22×C4—1422no—
464C22×C4—1421no—
465C22×C4—1421no—
466C22×C4C22×Dic7112yes—
467C22×C4D4×D7111yescharacteristic
468C22×C4#1176586412yes—
469C22×C4C23×D7111yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—1421no—
475C2×C8—1421no—
476C2×C8—1421no—
477C2×C8—1421no—
478C2×C8—1421no—
479C2×C8—1421no—
480C2×C8C22×Dic7112yes—
481C2×C8C22×Dic7112yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—24no—
486C4⋊C4—2842no—
487C4⋊C4—2842no—
488C4⋊C4—712no—
489C4⋊C4—712no—
490C4⋊C4—1421no—
491C4⋊C4—1421no—
492C4⋊C4—1421no—
493C4⋊C4—1421no—
494C4⋊C4—1421no—
495C4⋊C4—1421no—
496C4⋊C4C22×Dic7112yes—
497C4⋊C4C22×Dic7112yes—
498C4⋊C4C22×Dic7112yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—2842no—
502C22⋊C4—1424no—
503C42—24no—
504C42—2842no—
505C42—2842no—
506C42—2842no—
507C42—2842no—
508C42—2842no—
509C42—2842no—
510C42—712no—
511C42—712no—
512C42—1422no—
513C42—1422no—
514C42—1424no—
515C42—1421no—
516C42—1421no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23D4×Dic7112yes—
524C23C2×D4×D7111yescharacteristic
525Q8—22no—
526Q8—22no—
527Q8—41no—
528Q8—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—2841no—
548C2×C4—2841no—
549C2×C4—2841no—
550C2×C4—2841no—
551C2×C4—2844no—
552C2×C4—2841no—
553C2×C4—2841no—
554C2×C4—2841no—
555C2×C4—2841no—
556C2×C4—2841no—
557C2×C4—2841no—
558C2×C4—2844no—
559C2×C4—712no—
560C2×C4—712no—
561C2×C4—712no—
562C2×C4—1421no—
563C2×C4—1422no—
564C2×C4—1422no—
565C2×C4—1421no—
566C2×C4—1421no—
567C2×C4—1421no—
568C2×C4—1421no—
569C2×C4—1421no—
570C2×C4—1421no—
571C2×C4—1421no—
572C2×C4—1421no—
573C2×C4D4×Dic7112yes—
574C2×C4C23×Dic7111yescharacteristic
575C2×C4#41×D7111yescharacteristic
576C2×C4#41×D7111yescharacteristic
577C2×C4#41×D7111yescharacteristic
578C2×C4#41×D7111yescharacteristic
579C2×C4C2×#1176586411yescharacteristic
580C8—22no—
581C8—22no—
582C8—2841no—
583C8—2841no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22C2×D4×Dic7111yescharacteristic
592C22#123×D7111yescharacteristic
593C22#1798650711yescharacteristic
594C22#137×D7111yescharacteristic
595C22#1798679611yescharacteristic
596C22#1798679611yescharacteristic
597C22#1798679811yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—2842no—
604C4—2842no—
605C4—2841no—
606C4—5681no—
607C4—5681no—
608C4—1422no—
609C4—1422no—
610C4—1422no—
611C4#41×Dic7111yescharacteristic
612C4#41×Dic7111yescharacteristic
613C4#1798652611yescharacteristic
614C4#1798652611yescharacteristic
615C2—42no—
616C2#123×Dic7111yescharacteristic
617C2#2259714111yescharacteristic
618C2#137×Dic7111yescharacteristic
619C2#2259827211yescharacteristic
620C2#2259828711yescharacteristic
621C2#2259845211yescharacteristic
622C2#2259849711yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#485×C71C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
cosocleC22×C4×C71C24
hypercenterC2×#485D71
hypocenterC71#21583
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#485

Complements

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

NH ≅ G/Ntype
3. #22588983C2semidirect
4. #22588976C2semidirect
5. #22588934C2semidirect
6. #22588927C2semidirect
10. #22588240C2semidirect
11. #22588239C2semidirect
12. #22588172C2semidirect
13. #22588172C2semidirect
54. #485×C71C4semidirect
120. #107×C71C2×C4semidirect
121. #106×C71C2×C4semidirect
122. #103×C71C2×C4semidirect
123. #103×C71C2×C4semidirect
327. C71#21583semidirect
333. #485Dic71semidirect
417. #107C2×Dic71semidirect
419. #106C2×Dic71semidirect
422. #103C2×Dic71semidirect
423. #103C2×Dic71semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×Dic712
#117×Dic712
#225889832
#225889762
#225889342
#225889272
#225886532
#225885412
#225884352
#225882402
#225882392
#225881722
#225881042
C2×#485×C712
#2158371