Finite Group Databasebeta

← #32016884

Subgroups — #32016884

Order 18176.

866 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
1C2×#17986097C211yesmaximal, characteristic
2#22591287C211yesmaximal, characteristic
3#22589036C211yesmaximal, characteristic
4#22589022C211yesmaximal, characteristic
5#22588929C211yesmaximal, characteristic, complemented
6#22588908C211yesmaximal, characteristic, complemented
7#22588695C211yesmaximal, characteristic
8#22588695C211yesmaximal, characteristic
9#22588473C211yesmaximal, characteristic, complemented
10#22588347C211yesmaximal, characteristic
11#22588345C211yesmaximal, characteristic, complemented
12#22588215C211yesmaximal, characteristic, complemented
13#22588170C211yesmaximal, characteristic, complemented
14C4×#14556052C211yesmaximal, characteristic, complemented
15C71×#2351C211yesmaximal, characteristic, complemented
16C22×#11765852C2211yescharacteristic
17C2×C4×D284C2211yescharacteristic
18C2×#14556086—21no—
19#17986097C418yescomplemented
20C2×#14556061C2211yescharacteristic
21C2×C4⋊(C71⋊C8)C2211yescharacteristic
22C2×#14556052C2211yescharacteristic
23C2×#14556052C2211yescharacteristic
24C2×#14556052C2211yescharacteristic
25C2×#14556048C2211yescharacteristic
26C2×#14556048C2211yescharacteristic
27C2×#14556048C2211yescharacteristic
28C2×#14556039C2211yescharacteristic
29C2×#14556039C2211yescharacteristic
30C2×C4×C71⋊C8C2211yescharacteristic
31#17985901C2211yescharacteristic
32#17985901C2211yescharacteristic
33#17985891—21no—
34#17985891—21no—
35#17985869C2211yescharacteristic
36#17985856—21no—
37C4×C4⋊Dic71C2211yescharacteristic
38C4×C4⋊Dic71C2211yescharacteristic
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985754C2211yescharacteristic
48#17985754C2211yescharacteristic
49C2×C4×Q8×C71C2211yescharacteristic
50C2×C4×Q8×C71C2211yescharacteristic
51C2×C4×Q8×C71C2211yescharacteristic
52#389×C71—21no—
53#389×C71—21no—
54#389×C71C2211yescharacteristic
55#389×C71C2211yescharacteristic
56#382×C71—21no—
57#382×C71C2211yescharacteristic
58C4×C4⋊C4×C71—21no—
59C4×C4⋊C4×C71C2211yescharacteristic
60C22×D284C2311yescharacteristic
61C22×C4×D71—21no—
62C2×#11765852—22no—
63C2×#11765852—22no—
64C2×#11765852C2×C414yes—
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67C22×C71⋊C8C2311yescharacteristic
68C22×C71⋊C8C2311yescharacteristic
69C4×D284—22no—
70C4×D284C2×C414yes—
71#14556086—44no—
72C2×C4⋊Dic71—21no—
73C2×C4⋊Dic71—21no—
74C2×C4⋊Dic71C2311yescharacteristic
75C2×C4⋊Dic71C2311yescharacteristic
76C2×C4⋊Dic71C2311yescharacteristic
77C2×(C2×D71)⋊C4—21no—
78C2×(C2×D71)⋊C4—21no—
79C2×(C2×D71)⋊C4—21no—
80C2×(C2×D71)⋊C4—21no—
81C2×(C2×D71)⋊C4—21no—
82C2×C4×Dic71—21no—
83C2×C4×Dic71—21no—
84#14556061C2×C414yes—
85C4⋊(C71⋊C8)—22no—
86#14556052—22no—
87#14556052—22no—
88#14556052C2×C414yes—
89#14556048—41no—
90#14556048—41no—
91#14556048C2×C414yes—
92#14556039—41no—
93#14556039C2×C414yes—
94C4×C71⋊C8—41no—
95C71⋊#95—21no—
96C71⋊#95—21no—
97C22×Q8×C71C2311yescharacteristic
98C4×Q8×C71—22no—
99C4×Q8×C71—22no—
100C4×Q8×C71—22no—
101C4×Q8×C71—41no—
102C4×Q8×C71—41no—
103C4×Q8×C71C2×C414yes—
104C2×C4⋊C4×C71—21no—
105C2×C4⋊C4×C71—21no—
106C2×C4⋊C4×C71—21no—
107C2×C4⋊C4×C71—21no—
108C2×C4⋊C4×C71—21no—
109C2×C4⋊C4×C71D411yescharacteristic
110C2×C4⋊C4×C71D411yescharacteristic
111C2×C4⋊C4×C71C2311yescharacteristic
112C2×C4⋊C4×C71C2311yescharacteristic
113C2×C4⋊C4×C71C2311yescharacteristic
114C2×C42×C71—21no—
115C2×C42×C71—21no—
116C2×C42×C71—21no—
117C2×C42×C71C2311yescharacteristic
118C2×C42×C71C2311yescharacteristic
119C2×C42×C71C2311yescharacteristic
120#95×C71—21no—
121#95×C71—21no—
122#95×C71—21no—
123#95×C71—21no—
124#95×C71D411yescharacteristic
125#95×C71D411yescharacteristic
126C23×D71—21no—
127C2×D284—21no—
128C2×D284—21no—
129C2×D284—21no—
130C2×D284—21no—
131C2×D284—21no—
132C2×D284C22×C412yes—
133C22×Dic71—21no—
134C22×Dic71—21no—
135C22×Dic71—21no—
136C2×C4×D71—24no—
137C2×C4×D71—42no—
138C2×C4×D71—42no—
139#11765852—28no—
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⋊C8—21no—
148C2×C71⋊C8—21no—
149C2×C71⋊C8—21no—
150C2×C71⋊C8C22×C412yes—
151C2×C71⋊C8C22×C412yes—
152C4⋊Dic71—21no—
153C4⋊Dic71—21no—
154C4⋊Dic71—21no—
155C4⋊Dic71—21no—
156C4⋊Dic71—21no—
157C4⋊Dic71—42no—
158C4⋊Dic71—81no—
159C4⋊Dic71C22×C412yes—
160(C2×D71)⋊C4—24no—
161(C2×D71)⋊C4—42no—
162(C2×D71)⋊C4—42no—
163(C2×D71)⋊C4—81no—
164(C2×D71)⋊C4—81no—
165C4×Dic71—42no—
166C4×Dic71—42no—
167C2×Q8×C71—21no—
168C2×Q8×C71—21no—
169C2×Q8×C71—21no—
170C2×Q8×C71C22×C412yes—
171C2×Q8×C71#4112yes—
172C2×Q8×C71#4112yes—
173C22×C4×C71—21no—
174C22×C4×C71—21no—
175C22×C4×C71—21no—
176C22×C4×C71—21no—
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2×D411yescharacteristic
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71C2411yescharacteristic
184C4⋊C4×C71—24no—
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—42no—
196C4⋊C4×C71C22×C412yes—
197C42×C71—24no—
198C42×C71—21no—
199C42×C71—21no—
200C42×C71—21no—
201C42×C71—21no—
202C42×C71—42no—
203C42×C71—42no—
204C42×C71C22×C412yes—
205C42×C71#4112yes—
206C23×C71C22×D411yescharacteristic
207C22×D71—22no—
208C22×D71—41no—
209C22×D71—41no—
210C22×D71—41no—
211C22×D71—41no—
212C22×D71—41no—
213C22×D71—41no—
214Q8×C71—22no—
215Q8×C71—22no—
216Q8×C71—22no—
217Q8×C71#11714yescomplemented
218C2×C4×C71—22no—
219C2×C4×C71—21no—
220C2×C4×C71—22no—
221C2×C4×C71—22no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—21no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—22no—
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—21no—
235C2×C4×C71—21no—
236C2×C4×C71—21no—
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—41no—
244C2×C4×C71—41no—
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—41no—
256C2×Dic71—41no—
257C2×Dic71—41no—
258C2×Dic71—41no—
259C2×Dic71—41no—
260C2×Dic71—41no—
261C2×Dic71—41no—
262C2×Dic71—41no—
263D284—22no—
264D284—22no—
265D284—41no—
266D284—41no—
267C4×D71—48no—
268C71⋊C8—22no—
269C71⋊C8—22no—
270C71⋊C8—41no—
271C71⋊C8—41no—
272C22×C71C2×C4×D411yescharacteristic
273C22×C71#54611yescharacteristic
274C22×C71#54811yescharacteristic
275C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C71#57511yescharacteristic
277C22×C71#57711yescharacteristic
278C22×C71#57811yescharacteristic
279C2×D71—42no—
280C2×D71—42no—
281C2×D71—42no—
282C2×D71—81no—
283C2×D71—81no—
284C2×D71—81no—
285C2×D71—81no—
286C4×C71—24no—
287C4×C71—22no—
288C4×C71—22no—
289C4×C71—22no—
290C4×C71—42no—
291C4×C71—41no—
292C4×C71—41no—
293C4×C71—41no—
294C4×C71—81no—
295C4×C71—81no—
296C4×C71C2×#11711yescharacteristic
297C4×C71#52011yescharacteristic
298C4×C71#55211yescharacteristic
299C4×C71#55311yescharacteristic
300Dic71—42no—
301Dic71—81no—
302Dic71—81no—
303#22781—711nomaximal, Sylow
304C2×C71#235211yescharacteristic
305C2×C71C4×AGL(1,ℤ/8)11yescharacteristic
306C2×C71#294711yescharacteristic
307C2×C71#330811yescharacteristic
308C2×C71#332111yescharacteristic
309C2×C71#336811yescharacteristic
310C2×C71#337711yescharacteristic
311D71—82no—
312C2×#440—711no—
313#2354—711no—
314#2351D7111yescharacteristic, complemented
315#1936—711no—
316#1934—711no—
317#1921—711no—
318#1918—711no—
319#1880—711no—
320#1880—711no—
321#1836—711no—
322#1804—711no—
323#1803—711no—
324#1771—711no—
325#1763—711no—
326C4×#103—711no—
327C71#2278111yescharacteristic, complemented, Sylow
328C22×SD16—711no—
329C2×C4×Q8C2×D7111yescharacteristic
330C2×C4×Q8C2×D7111yescharacteristic
331C2×C4×Q8C2×D7111yescharacteristic
332C2×C4×D4—711no—
333C2×#117—1421no—
334#440—718no—
335C2×#107—711no—
336C2×C4⋊C8—711no—
337C2×#103—711no—
338C2×#103—711no—
339C2×#103—711no—
340C2×#102—711no—
341C2×#102—711no—
342C2×#102—711no—
343C2×#97—711no—
344C2×#97—711no—
345C2×C4×C8—711no—
346#390—711no—
347#390—711no—
348#389—21no—
349#389—21no—
350#389C2×D7111yescharacteristic
351#389C2×D7111yescharacteristic
352#388—1421no—
353#388—1421no—
354#384—711no—
355#382—21no—
356#382C2×D7111yescharacteristic
357#380—1421no—
358C4×C4⋊C4—21no—
359C4×C4⋊C4—711no—
360C4×C4⋊C4—711no—
361C4×C4⋊C4C2×D7111yescharacteristic
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
373C22×D4—711no—
374C23×C4—1421no—
375C2×SD16—714no—
376C2×SD16—1422no—
377C2×SD16—1422no—
378C22×C8—711no—
379C22×C8—711no—
380C22×C8—711no—
381C22×C8—711no—
382C4×Q8—22no—
383C4×Q8—22no—
384C4×Q8—22no—
385C4×Q8—41no—
386C4×Q8—41no—
387C4×Q8C4×D7114yes—
388C4×D4—714no—
389C4×D4—1422no—
390#117—2844no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—711no—
397C2×C4⋊C4—711no—
398C2×C4⋊C4—711no—
399C2×C4⋊C4—1421no—
400C2×C4⋊C4—1421no—
401C2×C4⋊C4C22⋊D7111yescharacteristic
402C2×C4⋊C4C22⋊D7111yescharacteristic
403C2×C4⋊C4C22×D7111yescharacteristic
404C2×C4⋊C4C22×D7111yescharacteristic
405C2×C4⋊C4C22×D7111yescharacteristic
406C2×C22⋊C4—1421no—
407C2×C22⋊C4—1421no—
408C2×C22⋊C4—1421no—
409C2×C22⋊C4—1421no—
410C2×C22⋊C4—1421no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—21no—
414C2×C42—1421no—
415C2×C42—1421no—
416C2×C42C22×D7111yescharacteristic
417C2×C42C22×D7111yescharacteristic
418C2×C42C22×D7111yescharacteristic
419#107—714no—
420C4⋊C8—1422no—
421#103—714no—
422#103—1422no—
423#103—1422no—
424#102—2841no—
425#102—2841no—
426#102—714no—
427#97—2841no—
428#97—714no—
429C4×C8—2841no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—1421no—
435#95—1421no—
436#95C22⋊D7111yescharacteristic
437#95C22⋊D7111yescharacteristic
438C24—1421no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×C4×D7112yes—
443C2×Q8#1176586412yes—
444C2×Q8#1176586612yes—
445C2×D4—712no—
446C2×D4—1421no—
447C2×D4—1421no—
448C2×D4—1421no—
449C2×D4—1421no—
450C2×D4—1421no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—2842no—
456C22×C4—2842no—
457C22×C4—1421no—
458C22×C4—1421no—
459C22×C4—1421no—
460C22×C4—1424no—
461C22×C4C2×C22⋊D7111yescharacteristic
462C22×C4C2×C22⋊D7111yescharacteristic
463C22×C4C2×C22⋊D7111yescharacteristic
464C22×C4C2×C22⋊D7111yescharacteristic
465C22×C4C2×C22⋊D7111yescharacteristic
466C22×C4C2×C22⋊D7111yescharacteristic
467C22×C4C23×D7111yescharacteristic
468SD16—1428no—
469C2×C8—712no—
470C2×C8—712no—
471C2×C8—1421no—
472C2×C8—1421no—
473C2×C8—1421no—
474C2×C8—1421no—
475C2×C8—1421no—
476C2×C8—1421no—
477C2×C8—1421no—
478C2×C8—1421no—
479C2×C8—1421no—
480C2×C8—1421no—
481C4⋊C4—21no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—41no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—2842no—
494C4⋊C4—5681no—
495C4⋊C4—712no—
496C4⋊C4—1421no—
497C4⋊C4—1421no—
498C4⋊C4—1421no—
499C4⋊C4—1421no—
500C4⋊C4—1421no—
501C4⋊C4C2×C4×D7112yes—
502C22⋊C4—2842no—
503C22⋊C4—2842no—
504C22⋊C4—5681no—
505C22⋊C4—5681no—
506C22⋊C4—1424no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—24no—
512C42—42no—
513C42—42no—
514C42—2842no—
515C42—2842no—
516C42C2×C4×D7112yes—
517C42#1176586512yes—
518C23—2841no—
519C23—2841no—
520C23—2841no—
521C23—2841no—
522C23—2841no—
523C23—2841no—
524C23—1422no—
525C23C22×C22⋊D7111yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—22no—
529Q8#1455608614yescomplemented
530D4—2841no—
531D4—2841no—
532D4—1422no—
533D4—1422no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
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—2848no—
562C2×C4—2841no—
563C2×C4—2841no—
564C2×C4—2841no—
565C2×C4—2841no—
566C2×C4—2841no—
567C2×C4—2841no—
568C2×C4—2841no—
569C2×C4—2841no—
570C2×C4—1422no—
571C2×C4C4×C22⋊D7112yes—
572C2×C4C4×C22⋊D7112yes—
573C2×C4#1455615312yes—
574C2×C4C22×C4×D7111yescharacteristic
575C2×C4C2×#1176586411yescharacteristic
576C2×C4C2×#1176586511yescharacteristic
577C2×C4C2×#1176586611yescharacteristic
578C2×C4#1455617311yescharacteristic
579C2×C4#1455617611yescharacteristic
580C8—2841no—
581C8—2841no—
582C8—1422no—
583C8—1422no—
584C22—2842no—
585C22—2842no—
586C22—2842no—
587C22—5681no—
588C22—5681no—
589C22—5681no—
590C22—5681no—
591C22C2×C4×C22⋊D7111yescharacteristic
592C22#1798664311yescharacteristic
593C22#1798666011yescharacteristic
594C22C2×#1455615311yescharacteristic
595C22#1798679711yescharacteristic
596C22#1798680711yescharacteristic
597C22#1798681411yescharacteristic
598C4—22no—
599C4—24no—
600C4—22no—
601C4—22no—
602C4—42no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—81no—
607C4—81no—
608C4—2842no—
609C4—5681no—
610C4—5681no—
611C4C2×#1455608611yescharacteristic
612C4#1798648011yescharacteristic
613C4#1798668411yescharacteristic
614C4#1798669711yescharacteristic
615C2—5682no—
616C2#2259126511yescharacteristic
617C2C4×#1455615311yescharacteristic
618C2#2259718311yescharacteristic
619C2#2260033111yescharacteristic
620C2#2260052611yescharacteristic
621C2#2260106911yescharacteristic
622C2#2260118211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2351C2
socleC23×C71C22×D4
prime radicalC22×D284C23
cosocleC22×C4×C71C24
hypercenter#2351D71
hypocenterC71#22781
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
5. #22588929C2semidirect
6. #22588908C2semidirect
9. #22588473C2semidirect
11. #22588345C2semidirect
12. #22588215C2semidirect
13. #22588170C2semidirect
14. C4×#14556052C2semidirect
15. C71×#2351C2semidirect
19. #17986097C4semidirect
217. Q8×C71#117semidirect
314. #2351D71semidirect
327. C71#22781semidirect
529. Q8#14556086semidirect

Maximal subgroups

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

subgroupindex
C2×#179860972
#225912872
#225890362
#225890222
#225889292
#225889082
#225886952
#225884732
#225883472
#225883452
#225882152
#225881702
C4×#145560522
C71×#23512
#2278171