Finite Group Databasebeta

← #32016839

Subgroups — #32016839

Order 18176.

894 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986099C211yesmaximal, characteristic
2#22591303C211yesmaximal, characteristic, complemented
3#22589029C211yesmaximal, characteristic, complemented
4#22589015C211yesmaximal, characteristic, complemented
5#22588929C211yesmaximal, characteristic, complemented
6#22588908C211yesmaximal, characteristic, complemented
7#22588674C211yesmaximal, characteristic
8#22588674C211yesmaximal, characteristic
9#22588475C211yesmaximal, characteristic, complemented
10#22588350C211yesmaximal, characteristic
11#22588342C211yesmaximal, characteristic, complemented
12#22588215C211yesmaximal, characteristic, complemented
13#22588165C211yesmaximal, characteristic
14C4×#14556049C211yesmaximal, characteristic
15D4×C4⋊C4×C71C211yesmaximal, characteristic
16C22×#11765853C2211yescharacteristic
17C2×C4×Dic142C2211yescharacteristic
18#17986099C418yescomplemented
19C2×#14556061C2211yescharacteristic
20C2×C4⋊(C71⋊C8)C2211yescharacteristic
21C2×#14556051C2211yescharacteristic
22C2×#14556051C2211yescharacteristic
23C2×#14556051C2211yescharacteristic
24C2×#14556049C2211yescharacteristic
25C2×#14556049C2211yescharacteristic
26C2×#14556049C2211yescharacteristic
27C2×#14556039C2211yescharacteristic
28C2×#14556039C2211yescharacteristic
29C2×C4×C71⋊C8C2211yescharacteristic
30#17985905C2211yescharacteristic
31#17985905C2211yescharacteristic
32#17985880—21no—
33#17985880—21no—
34#17985866—21no—
35#17985865C2211yescharacteristic
36C4×C4⋊Dic71C2211yescharacteristic
37C4×C4⋊Dic71C2211yescharacteristic
38#17985844—21no—
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985754C2211yescharacteristic
48#17985754C2211yescharacteristic
49C2×C4×D4×C71C2211yescharacteristic
50C2×C4×D4×C71C2211yescharacteristic
51C2×C4×D4×C71C2211yescharacteristic
52C22×C4⋊C4×C71—21no—
53#389×C71C2211yescharacteristic
54#389×C71C2211yescharacteristic
55#385×C71—21no—
56#385×C71—21no—
57#384×C71C2211yescharacteristic
58#380×C71—21no—
59C4×C4⋊C4×C71C2211yescharacteristic
60C22×Dic142C2311yescharacteristic
61C2×#11765853—22no—
62C2×#11765853—22no—
63C2×#11765853C2×C414yes—
64C22×C71⋊C8C2311yescharacteristic
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67C22×C71⋊C8C2311yescharacteristic
68C4×Dic142—22no—
69C4×Dic142C2×C414yescomplemented
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C4⋊Dic71C2311yescharacteristic
72C2×C4⋊Dic71C2311yescharacteristic
73C2×Dic71⋊C4—21no—
74C2×C4×Dic71—21no—
75C2×C4×Dic71—21no—
76C2×C4×Dic71—21no—
77#14556061C2×C414yescomplemented
78C4⋊(C71⋊C8)—22no—
79#14556051—41no—
80#14556051—41no—
81#14556051C2×C414yescomplemented
82#14556049—22no—
83#14556049—22no—
84#14556049C2×C414yes—
85#14556039—41no—
86#14556039C2×C414yescomplemented
87C4×C71⋊C8—41no—
88C71⋊#95—21no—
89C71⋊#95—21no—
90C71⋊#95—21no—
91C71⋊#95—21no—
92C71⋊#95—21no—
93C71⋊#95—21no—
94C71⋊#95—21no—
95C71⋊#95—21no—
96C22×D4×C71C2311yescharacteristic
97C23×C4×C71—21no—
98C23×C4×C71—21no—
99C23×C4×C71—21no—
100C4×D4×C71—22no—
101C4×D4×C71—22no—
102C4×D4×C71—22no—
103C4×D4×C71—41no—
104C4×D4×C71—41no—
105C4×D4×C71C2×C414yes—
106C2×C4⋊C4×C71—21no—
107C2×C4⋊C4×C71—21no—
108C2×C4⋊C4×C71—21no—
109C2×C4⋊C4×C71—21no—
110C2×C4⋊C4×C71—44no—
111C2×C4⋊C4×C71D411yescharacteristic
112C2×C4⋊C4×C71D411yescharacteristic
113C2×C4⋊C4×C71C2311yescharacteristic
114C2×C4⋊C4×C71C2311yescharacteristic
115C2×C4⋊C4×C71C2311yescharacteristic
116C2×C22⋊C4×C71—21no—
117C2×C22⋊C4×C71—21no—
118C2×C22⋊C4×C71—21no—
119C2×C42×C71C2311yescharacteristic
120C2×C42×C71C2311yescharacteristic
121C2×C42×C71C2311yescharacteristic
122#95×C71—21no—
123#95×C71—21no—
124#95×C71D411yescharacteristic
125#95×C71D411yescharacteristic
126C2×Dic142—21no—
127C2×Dic142—21no—
128C2×Dic142—21no—
129C2×Dic142—21no—
130C2×Dic142—21no—
131C2×Dic142C22×C412yes—
132C22×Dic71—21no—
133C22×Dic71—21no—
134C22×Dic71—21no—
135C22×Dic71—21no—
136#11765853—28no—
137C2×C71⋊C8—21no—
138C2×C71⋊C8—21no—
139C2×C71⋊C8—21no—
140C2×C71⋊C8—21no—
141C2×C71⋊C8—21no—
142C2×C71⋊C8—21no—
143C2×C71⋊C8—21no—
144C2×C71⋊C8—21no—
145C2×C71⋊C8—21no—
146C2×C71⋊C8—21no—
147C2×C71⋊C8C22×C412yes—
148C2×C71⋊C8C22×C412yes—
149C4⋊Dic71—21no—
150C4⋊Dic71—21no—
151C4⋊Dic71—21no—
152C4⋊Dic71—21no—
153C4⋊Dic71—21no—
154C4⋊Dic71C22×C412yes—
155Dic71⋊C4—24no—
156C4×Dic71—24no—
157C4×Dic71—42no—
158C4×Dic71—42no—
159C24×C71—21no—
160C2×D4×C71—21no—
161C2×D4×C71—21no—
162C2×D4×C71—21no—
163C2×D4×C71C22×C412yes—
164C2×D4×C71#4112yes—
165C2×D4×C71#4112yes—
166C22×C4×C71—21no—
167C22×C4×C71—24no—
168C22×C4×C71—21no—
169C22×C4×C71—21no—
170C22×C4×C71—42no—
171C22×C4×C71—42no—
172C22×C4×C71—42no—
173C22×C4×C71—42no—
174C22×C4×C71—42no—
175C22×C4×C71—42no—
176C22×C4×C71—81no—
177C22×C4×C71—81no—
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2×D411yescharacteristic
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71C2×D411yescharacteristic
184C22×C4×C71C2411yescharacteristic
185C4⋊C4×C71—22no—
186C4⋊C4×C71—21no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71—21no—
190C4⋊C4×C71—21no—
191C4⋊C4×C71—41no—
192C4⋊C4×C71—42no—
193C4⋊C4×C71—42no—
194C4⋊C4×C71—42no—
195C4⋊C4×C71—81no—
196C4⋊C4×C71C22×C412yes—
197C22⋊C4×C71—24no—
198C22⋊C4×C71—42no—
199C22⋊C4×C71—42no—
200C42×C71—21no—
201C42×C71—21no—
202C42×C71—21no—
203C42×C71—21no—
204C42×C71C22×C412yes—
205C42×C71#4112yes—
206C23×C71—22no—
207C23×C71—22no—
208C23×C71—22no—
209C23×C71—41no—
210C23×C71—41no—
211C23×C71—41no—
212C23×C71—41no—
213C23×C71C22×D411yescharacteristic
214D4×C71—22no—
215D4×C71—22no—
216D4×C71—22no—
217D4×C71#11714yes—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—22no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—21no—
229C2×C4×C71—21no—
230C2×C4×C71—21no—
231C2×C4×C71—21no—
232C2×C4×C71—21no—
233C2×C4×C71—21no—
234C2×C4×C71—48no—
235C2×C4×C71—41no—
236C2×C4×C71—41no—
237C2×C4×C71—41no—
238C2×C4×C71—41no—
239C2×C4×C71—41no—
240C2×C4×C71—41no—
241C2×C4×C71—41no—
242C2×C4×C71—41no—
243C2×C4×C71—84no—
244C2×C4×C71—84no—
245C2×C4×C71C4×D412yes—
246C2×C4×C71C4×D412yes—
247C2×C4×C71AGL(1,ℤ/8)12yes—
248C2×C4×C71C23×C411yescharacteristic
249C2×C4×C71C2×#4111yescharacteristic
250C2×C4×C71C2×#4111yescharacteristic
251C2×C4×C71C2×#4111yescharacteristic
252C2×C4×C71H(5,𝔽2)11yescharacteristic
253C2×C4×C712−1+411yescharacteristic
254C2×Dic71—22no—
255C2×Dic71—22no—
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—41no—
266Dic142—22no—
267Dic142—22no—
268Dic142—41no—
269Dic142—41no—
270C71⋊C8—22no—
271C71⋊C8—22no—
272C71⋊C8—41no—
273C71⋊C8—41no—
274C22×C71—24no—
275C22×C71—42no—
276C22×C71—42no—
277C22×C71—42no—
278C22×C71—42no—
279C22×C71—42no—
280C22×C71—42no—
281C22×C71C2×C4×D411yescharacteristic
282C22×C71#54611yescharacteristic
283C22×C71#54811yescharacteristic
284C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C71#57511yescharacteristic
286C22×C71#57711yescharacteristic
287C22×C71#57811yescharacteristic
288C4×C71—22no—
289C4×C71—22no—
290C4×C71—22no—
291C4×C71—42no—
292C4×C71—41no—
293C4×C71—41no—
294C4×C71—41no—
295C4×C71—81no—
296C4×C71—81no—
297C4×C71C2×#11711yescharacteristic
298C4×C71#52011yescharacteristic
299C4×C71#55211yescharacteristic
300C4×C71#55311yescharacteristic
301Dic71—42no—
302Dic71—42no—
303Dic71—81no—
304Dic71—81no—
305#22778—711nomaximal, Sylow
306C2×C71—44no—
307C2×C71#235211yescharacteristic
308C2×C71C4×AGL(1,ℤ/8)11yescharacteristic
309C2×C71#294711yescharacteristic
310C2×C71#331011yescharacteristic
311C2×C71#331511yescharacteristic
312C2×C71#336711yescharacteristic
313C2×C71#337611yescharacteristic
314C2×#440—711no—
315#2356—711no—
316D4×C4⋊C4D7111yescharacteristic
317#1935—711no—
318#1933—711no—
319#1921—711no—
320#1918—711no—
321#1877—711no—
322#1877—711no—
323#1836—711no—
324#1804—711no—
325#1803—711no—
326#1771—711no—
327#1762—711no—
328C4×#102—711no—
329C71#2277811yescharacteristic, complemented, Sylow
330C22×SD16—711no—
331C2×C4×Q8—711no—
332C2×C4×D4C2×D7111yescharacteristic
333C2×C4×D4C2×D7111yescharacteristic
334C2×C4×D4C2×D7111yescharacteristic
335C22×C4⋊C4—21no—
336#440—718no—
337C2×#107—711no—
338C2×C4⋊C8—711no—
339C2×#103—711no—
340C2×#103—711no—
341C2×#103—711no—
342C2×#102—711no—
343C2×#102—711no—
344C2×#102—711no—
345C2×#97—711no—
346C2×#97—711no—
347C2×C4×C8—711no—
348#391—711no—
349#391—711no—
350#389C2×D7111yescharacteristic
351#389C2×D7111yescharacteristic
352#387—1421no—
353#387—1421no—
354#385—21no—
355#385—21no—
356#384C2×D7111yescharacteristic
357#383—1421no—
358#382—711no—
359#380—21no—
360C4×C4⋊C4—711no—
361C4×C4⋊C4—711no—
362C4×C4⋊C4C2×D7111yescharacteristic
363#376—1421no—
364#340—711no—
365#340—711no—
366#340—711no—
367#340—711no—
368#340—711no—
369#340—711no—
370#340—711no—
371#340—711no—
372#336—711no—
373#336—711no—
374C22×Q8—711no—
375C22×D4C22×D7111yescharacteristic
376C23×C4—21no—
377C23×C4—21no—
378C23×C4—21no—
379C2×SD16—714no—
380C2×SD16—1422no—
381C2×SD16—1422no—
382C22×C8—711no—
383C22×C8—711no—
384C22×C8—711no—
385C22×C8—711no—
386C4×Q8—714no—
387C4×Q8—1422no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—41no—
392C4×D4—41no—
393C4×D4C4×D7114yes—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—44no—
399C2×C4⋊C4—711no—
400C2×C4⋊C4—711no—
401C2×C4⋊C4—711no—
402C2×C4⋊C4—1421no—
403C2×C4⋊C4C22⋊D7111yescharacteristic
404C2×C4⋊C4C22⋊D7111yescharacteristic
405C2×C4⋊C4C22×D7111yescharacteristic
406C2×C4⋊C4C22×D7111yescharacteristic
407C2×C4⋊C4C22×D7111yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C42—1421no—
412C2×C42—1421no—
413C2×C42—1421no—
414C2×C42C22×D7111yescharacteristic
415C2×C42C22×D7111yescharacteristic
416C2×C42C22×D7111yescharacteristic
417#107—714no—
418C4⋊C8—1422no—
419#103—2841no—
420#103—2841no—
421#103—714no—
422#102—714no—
423#102—1422no—
424#102—1422no—
425#97—2841no—
426#97—714no—
427C4×C8—2841no—
428#95—21no—
429#95—21no—
430#95—1421no—
431#95—1421no—
432#95—1421no—
433#95—1421no—
434#95—1421no—
435#95—1421no—
436#95—1421no—
437#95—1421no—
438#95C22⋊D7111yescharacteristic
439#95C22⋊D7111yescharacteristic
440C24—21no—
441C2×Q8—712no—
442C2×Q8—1421no—
443C2×Q8—1421no—
444C2×Q8—1421no—
445C2×Q8—1421no—
446C2×Q8—1421no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4C2×C4×D7112yes—
451C2×D4#1176586412yes—
452C2×D4#1176586612yes—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—81no—
464C22×C4—81no—
465C22×C4—1421no—
466C22×C4—1421no—
467C22×C4—1421no—
468C22×C4—1421no—
469C22×C4C2×C22⋊D7111yescharacteristic
470C22×C4C2×C22⋊D7111yescharacteristic
471C22×C4C2×C22⋊D7111yescharacteristic
472C22×C4C2×C22⋊D7111yescharacteristic
473C22×C4C2×C22⋊D7111yescharacteristic
474C22×C4C2×C22⋊D7111yescharacteristic
475C22×C4C23×D7111yescharacteristic
476SD16—1428no—
477C2×C8—712no—
478C2×C8—712no—
479C2×C8—1421no—
480C2×C8—1421no—
481C2×C8—1421no—
482C2×C8—1421no—
483C2×C8—1421no—
484C2×C8—1421no—
485C2×C8—1421no—
486C2×C8—1421no—
487C2×C8—1421no—
488C2×C8—1421no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—41no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—81no—
500C4⋊C4—712no—
501C4⋊C4—1421no—
502C4⋊C4—1421no—
503C4⋊C4—1421no—
504C4⋊C4—1421no—
505C4⋊C4—1421no—
506C4⋊C4—1424no—
507C4⋊C4C2×C4×D7112yes—
508C22⋊C4—24no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—2842no—
516C42—2842no—
517C42—1424no—
518C42C2×C4×D7112yes—
519C42#1176586512yes—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23C22×C22⋊D7111yescharacteristic
528Q8—2841no—
529Q8—2841no—
530Q8—1422no—
531Q8—1422no—
532D4—22no—
533D4—22no—
534D4—22no—
535D4#1455608614yes—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—48no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—2841no—
564C2×C4—2841no—
565C2×C4—2841no—
566C2×C4—2841no—
567C2×C4—2841no—
568C2×C4—2841no—
569C2×C4—2841no—
570C2×C4—2841no—
571C2×C4—2841no—
572C2×C4—2841no—
573C2×C4—1422no—
574C2×C4—1422no—
575C2×C4C4×C22⋊D7112yes—
576C2×C4C4×C22⋊D7112yes—
577C2×C4#1455615412yes—
578C2×C4C22×C4×D7111yescharacteristic
579C2×C4C2×#1176586411yescharacteristic
580C2×C4C2×#1176586511yescharacteristic
581C2×C4C2×#1176586611yescharacteristic
582C2×C4#1455617311yescharacteristic
583C2×C4#1455617611yescharacteristic
584C8—2841no—
585C8—2841no—
586C8—1422no—
587C8—1422no—
588C22—24no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22C2×C4×C22⋊D7111yescharacteristic
596C22#1798664311yescharacteristic
597C22#1798666011yescharacteristic
598C22C2×#1455615411yescharacteristic
599C22#1798679711yescharacteristic
600C22#1798681011yescharacteristic
601C22#1798681411yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—42no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—81no—
610C4—81no—
611C4—2842no—
612C4—2842no—
613C4—5681no—
614C4—5681no—
615C4C2×#1455608611yescharacteristic
616C4#1798648011yescharacteristic
617C4#1798668411yescharacteristic
618C4#1798669711yescharacteristic
619C2—44no—
620C2#2259126511yescharacteristic
621C2C4×#1455615411yescharacteristic
622C2#2259717911yescharacteristic
623C2#2260036111yescharacteristic
624C2#2260043611yescharacteristic
625C2#2260105811yescharacteristic
626C2#2260116711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingD4×C4⋊C4×C71C2
socleC23×C71C22×D4
prime radicalC22×D4×C71C23
cosocleC22×C4×C71C24
hypercenterD4×C4⋊C4D71
hypocenterC71#22778
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #22591303C2semidirect
3. #22589029C2semidirect
4. #22589015C2semidirect
5. #22588929C2semidirect
6. #22588908C2semidirect
9. #22588475C2semidirect
11. #22588342C2semidirect
12. #22588215C2semidirect
18. #17986099C4semidirect
69. C4×Dic142C2×C4semidirect
77. #14556061C2×C4semidirect
81. #14556051C2×C4semidirect
86. #14556039C2×C4semidirect
329. C71#22778semidirect

Maximal subgroups

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

subgroupindex
C2×#179860992
#225913032
#225890292
#225890152
#225889292
#225889082
#225886742
#225884752
#225883502
#225883422
#225882152
#225881652
C4×#145560492
D4×C4⋊C4×C712
#2277871