Finite Group Databasebeta

← #32003104

Subgroups — #32003104

Order 18176.

1868 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D284C211yesmaximal, characteristic
2C2×#17986149C211yesmaximal, characteristic
3C2×#17986097C212yesmaximal, complemented
4#22588677C212yesmaximal, complemented
5#22588648C211yesmaximal, characteristic, complemented
6#22588586C212yesmaximal, complemented
7#22588527C211yesmaximal, characteristic, complemented
8#22588270C211yesmaximal, characteristic
9#22588168C212yesmaximal, complemented
10#22588102C211yesmaximal, characteristic, complemented
11C4×C22⋊Q8×C71C211yesmaximal, characteristic, complemented
12C23×D284C2211yescharacteristic
13C23×C4×D71—21no—
14C22×#11765852C2212yes—
15C2×C4×D284—28no—
16C2×C4×D284—21no—
17C2×C4×D284—21no—
18C2×C4×D284C2212yes—
19C2×C4×D284C2212yes—
20C22×C4⋊Dic71C2211yescharacteristic
21C22×(C2×D71)⋊C4—21no—
22#17986149C418yescomplemented
23#17986097—28no—
24C2×#14556061C2212yescomplemented
25C2×#14556052C2212yescomplemented
26C2×#14556048C2212yes—
27C2×#14556048C2212yes—
28C2×#14556048C2212yes—
29C2×C22⋊(C71⋊C8)C2211yescharacteristic
30C2×C22⋊(C71⋊C8)C2211yescharacteristic
31C2×#14556039C2212yescomplemented
32#17985761C2212yescomplemented
33#17985761C2212yescomplemented
34#17985754C2212yescomplemented
35C2×C22⋊Q8×C71C2211yescharacteristic
36C2×C4×Q8×C71C2211yescharacteristic, complemented
37C22×C42×C71C2211yescharacteristic
38#391×C71C2211yescharacteristic, complemented
39#389×C71C2211yescharacteristic, complemented
40#387×C71—21no—
41#385×C71—21no—
42#380×C71C2211yescharacteristic
43C4×C4⋊C4×C71—21no—
44C4×C4⋊C4×C71C2211yescharacteristic, complemented
45C4×C22⋊C4×C71—21no—
46C24×D71—21no—
47C22×D284—24no—
48C22×D284—24no—
49C22×D284—21no—
50C22×D284—22no—
51C22×D284C2×C414yes—
52C22×D284C2312yes—
53C23×Dic71—21no—
54C22×C4×D71—28no—
55C22×C4×D71—22no—
56C22×C4×D71—22no—
57C22×C4×D71—44no—
58C22×C4×D71—44no—
59C2×#11765852—24no—
60C2×#11765852—24no—
61C2×#11765852C2×C418yescomplemented
62C22×C71⋊C8C2312yes—
63C22×C71⋊C8C2312yes—
64C4×D284—28no—
65C4×D284—28no—
66C4×D284—24no—
67C4×D284—24no—
68C4×D284—24no—
69C4×D284D418yes—
70C2×C4⋊Dic71—24no—
71C2×C4⋊Dic71—21no—
72C2×C4⋊Dic71C2312yes—
73C2×(C2×D71)⋊C4—28no—
74C2×(C2×D71)⋊C4—22no—
75C2×(C2×D71)⋊C4—22no—
76#14556061—24no—
77#14556052—24no—
78#14556048—24no—
79#14556048—24no—
80#14556048C2×C418yescomplemented
81C22⋊(C71⋊C8)—22no—
82C22⋊(C71⋊C8)C2×C414yescomplemented
83#14556039—24no—
84C22×Q8×C71C2311yescharacteristic
85C23×C4×C71D411yescharacteristic
86C23×C4×C71D411yescharacteristic
87C23×C4×C71C2311yescharacteristic
88C22⋊Q8×C71—22no—
89C22⋊Q8×C71C2×C414yescomplemented
90C4×Q8×C71—24no—
91C2×C4⋊C4×C71—21no—
92C2×C4⋊C4×C71—21no—
93C2×C4⋊C4×C71—21no—
94C2×C4⋊C4×C71C2311yescharacteristic
95C2×C4⋊C4×C71C2311yescharacteristic
96C2×C4⋊C4×C71C2311yescharacteristic
97C2×C22⋊C4×C71—21no—
98C2×C22⋊C4×C71—21no—
99C2×C42×C71—24no—
100C2×C42×C71—21no—
101C2×C42×C71—21no—
102C2×C42×C71D411yescharacteristic
103C2×C42×C71D411yescharacteristic
104C2×C42×C71C2311yescharacteristic
105C2×C42×C71C2311yescharacteristic
106#95×C71—21no—
107#95×C71—21no—
108#95×C71—21no—
109#95×C71—21no—
110#95×C71—21no—
111C23×D71—24no—
112C23×D71—24no—
113C23×D71—24no—
114C23×D71—22no—
115C23×D71—42no—
116C23×D71—42no—
117C23×D71—42no—
118C23×D71—42no—
119C2×D284—24no—
120C2×D284—216no—
121C2×D284—24no—
122C2×D284—24no—
123C2×D284—22no—
124C2×D284—22no—
125C2×D284—22no—
126C2×D284—216no—
127C2×D284C22×C414yes—
128C2×D284C2×D414yes—
129C2×D284#4114yes—
130C22×Dic71—24no—
131C22×Dic71—22no—
132C22×Dic71—42no—
133C22×Dic71—42no—
134C2×C4×D71—28no—
135C2×C4×D71—28no—
136C2×C4×D71—432no—
137C2×C4×D71—44no—
138C2×C4×D71—44no—
139C2×C4×D71—44no—
140C2×C4×D71—44no—
141#11765852—216no—
142C2×C71⋊C8—22no—
143C2×C71⋊C8—22no—
144C2×C71⋊C8—22no—
145C2×C71⋊C8—22no—
146C2×C71⋊C8—22no—
147C2×C71⋊C8C22×C414yescomplemented
148C4⋊Dic71—24no—
149C4⋊Dic71—22no—
150C4⋊Dic71C2×D414yes—
151(C2×D71)⋊C4—28no—
152(C2×D71)⋊C4—28no—
153C24×C71C2×D411yescharacteristic
154C2×Q8×C71—21no—
155C2×Q8×C71—22no—
156C2×Q8×C71—22no—
157C2×Q8×C71C22×C412yescomplemented
158C22×C4×C71—22no—
159C22×C4×C71—22no—
160C22×C4×C71—22no—
161C22×C4×C71—22no—
162C22×C4×C71—22no—
163C22×C4×C71—22no—
164C22×C4×C71—21no—
165C22×C4×C71—21no—
166C22×C4×C71—22no—
167C22×C4×C71—21no—
168C22×C4×C71—21no—
169C22×C4×C71—41no—
170C22×C4×C71C22×C412yes—
171C22×C4×C71C2×D411yescharacteristic
172C22×C4×C71C2×D411yescharacteristic
173C22×C4×C71C2×D411yescharacteristic
174C22×C4×C71C2×D411yescharacteristic
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71#4112yes—
177C22×C4×C71C2411yescharacteristic
178C4⋊C4×C71—24no—
179C4⋊C4×C71—21no—
180C4⋊C4×C71—22no—
181C4⋊C4×C71—22no—
182C4⋊C4×C71—42no—
183C4⋊C4×C71—42no—
184C4⋊C4×C71C22×C412yescomplemented
185C22⋊C4×C71—24no—
186C22⋊C4×C71—42no—
187C42×C71—22no—
188C42×C71—22no—
189C42×C71—22no—
190C42×C71—42no—
191C42×C71—42no—
192C42×C71C2×D414yes—
193C23×C71—21no—
194C23×C71—21no—
195C23×C71—21no—
196C23×C71—21no—
197C23×C71C4×D412yes—
198C23×C71AGL(1,ℤ/8)12yes—
199C23×C71#13712yes—
200C23×C71C22×D411yescharacteristic
201C22×D71—216no—
202C22×D71—24no—
203C22×D71—24no—
204C22×D71—24no—
205C22×D71—48no—
206C22×D71—48no—
207C22×D71—48no—
208C22×D71—48no—
209C22×D71—48no—
210C22×D71—42no—
211C22×D71—42no—
212C22×D71—42no—
213C22×D71—42no—
214C22×D71—48no—
215Q8×C71—24no—
216Q8×C71—42no—
217C2×C4×C71—28no—
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—21no—
225C2×C4×C71—21no—
226C2×C4×C71—24no—
227C2×C4×C71—24no—
228C2×C4×C71—22no—
229C2×C4×C71—22no—
230C2×C4×C71—21no—
231C2×C4×C71—21no—
232C2×C4×C71—21no—
233C2×C4×C71—44no—
234C2×C4×C71—41no—
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—41no—
244C2×C4×C71C4×D412yes—
245C2×C4×C71#12012yes—
246C2×C4×C71#12012yes—
247C2×C4×C71AGL(1,ℤ/8)12yes—
248C2×C4×C71C23×C411yescharacteristic
249C2×C4×C71C22×D412yes—
250C2×C4×C71C2×#4112yes—
251C2×C4×C71C2×#4111yescharacteristic
252C2×Dic71—24no—
253C2×Dic71—48no—
254C2×Dic71—42no—
255C2×Dic71—42no—
256D284—216no—
257D284—24no—
258D284—24no—
259D284—24no—
260D284C4×D418yes—
261C4×D71—416no—
262C4×D71—416no—
263C71⋊C8—24no—
264C71⋊C8—42no—
265C22×C71—22no—
266C22×C71—22no—
267C22×C71—22no—
268C22×C71—22no—
269C22×C71—22no—
270C22×C71—22no—
271C22×C71#44014yescomplemented
272C22×C71C2×C4×D411yescharacteristic
273C22×C71C2×#12011yescharacteristic
274C22×C71#52511yescharacteristic
275C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C71C2×#13711yescharacteristic
278C22×C71#57511yescharacteristic
279C2×D71—28no—
280C2×D71—432no—
281C2×D71—44no—
282C2×D71—44no—
283C2×D71—44no—
284C2×D71—44no—
285C2×D71—44no—
286C2×D71—44no—
287C4×C71—22no—
288C4×C71—22no—
289C4×C71—22no—
290C4×C71—22no—
291C4×C71—42no—
292C4×C71—41no—
293C4×C71—42no—
294C4×C71—81no—
295C4×C71—81no—
296C4×C71C2×C4×D412yes—
297C4×C71#52512yes—
298Dic71—44no—
299#21453—711nomaximal, Sylow
300C2×C71—24no—
301C2×C71C4×#12011yescharacteristic
302C2×C71C2×#44011yescharacteristic
303C2×C71C4×AGL(1,ℤ/8)11yescharacteristic
304C2×C71#300311yescharacteristic
305C2×C71#300711yescharacteristic
306C2×C71#311011yescharacteristic
307C2×C71#311411yescharacteristic
308D71—48no—
309C22×C4×D4—711no—
310C2×#450—711no—
311C2×#440—712no—
312C4×C22⋊Q8D7111yescharacteristic, complemented
313#1877—712no—
314#1872—711no—
315#1862—712no—
316#1852—711no—
317#1786—711no—
318#1762—712no—
319#1749—711no—
320C71#2145311yescharacteristic, complemented, Sylow
321C23×D4—711no—
322C24×C4—1421no—
323C22×SD16—712no—
324C2×C22⋊Q8C2×D7111yescharacteristic
325C2×C4×Q8C2×D7111yescharacteristic, complemented
326C2×C4×D4—712no—
327C2×C4×D4—712no—
328C2×C4×D4—1428no—
329C2×C4×D4—1421no—
330C2×C4×D4—1421no—
331C22×C4⋊C4—711no—
332C22×C22⋊C4—1421no—
333C22×C42C2×D7111yescharacteristic
334#450—718no—
335#440—1428no—
336C2×#107—712no—
337C2×#103—712no—
338C2×#102—712no—
339C2×#102—712no—
340C2×#102—712no—
341C2×C22⋊C8—711no—
342C2×C22⋊C8—711no—
343C2×#97—712no—
344#391C2×D7111yescharacteristic, complemented
345#389C2×D7111yescharacteristic, complemented
346#387—21no—
347#385—21no—
348#380C2×D7111yescharacteristic
349C4×C4⋊C4—21no—
350C4×C4⋊C4C2×D7111yescharacteristic, complemented
351C4×C22⋊C4—21no—
352#340—712no—
353#340—712no—
354#336—712no—
355C25—1421no—
356C22×Q8C22×D7111yescharacteristic
357C22×D4—714no—
358C22×D4—712no—
359C22×D4—1424no—
360C22×D4—1424no—
361C22×D4—1421no—
362C22×D4—1422no—
363C23×C4—2844no—
364C23×C4—2844no—
365C23×C4—1428no—
366C23×C4—1421no—
367C23×C4—1422no—
368C23×C4—1422no—
369C23×C4C22⋊D7111yescharacteristic
370C23×C4C22⋊D7111yescharacteristic
371C23×C4C22×D7111yescharacteristic
372C2×SD16—718no—
373C2×SD16—1424no—
374C2×SD16—1424no—
375C22×C8—712no—
376C22×C8—712no—
377C22⋊Q8—22no—
378C22⋊Q8C4×D7114yescomplemented
379C4×Q8—24no—
380C4×D4—718no—
381C4×D4—1424no—
382C4×D4—1424no—
383C4×D4—1428no—
384C4×D4—1428no—
385C4×D4—1424no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—712no—
390C2×C4⋊C4—1424no—
391C2×C4⋊C4—1421no—
392C2×C4⋊C4C22×D7111yescharacteristic
393C2×C4⋊C4C22×D7111yescharacteristic
394C2×C4⋊C4C22×D7111yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—1428no—
398C2×C22⋊C4—1422no—
399C2×C22⋊C4—1422no—
400C2×C42—24no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42C22⋊D7111yescharacteristic
404C2×C42C22⋊D7111yescharacteristic
405C2×C42C22×D7111yescharacteristic
406C2×C42C22×D7111yescharacteristic
407#107—1424no—
408#103—1424no—
409#102—718no—
410#102—1424no—
411#102—1424no—
412C22⋊C8—714no—
413C22⋊C8—1422no—
414#97—1424no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—21no—
419#95—21no—
420C24—2842no—
421C24—2842no—
422C24—2842no—
423C24—2842no—
424C24—1424no—
425C24—1424no—
426C24—1422no—
427C24—1424no—
428C24C2×C22⋊D7111yescharacteristic
429C2×Q8—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8C2×C4×D7112yescomplemented
433C2×D4—714no—
434C2×D4—714no—
435C2×D4—714no—
436C2×D4—14216no—
437C2×D4—1424no—
438C2×D4—1424no—
439C2×D4—14216no—
440C2×D4—1422no—
441C2×D4—1422no—
442C2×D4—1422no—
443C2×D4—1424no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—41no—
456C22×C4—28432no—
457C22×C4—2844no—
458C22×C4—2844no—
459C22×C4—2844no—
460C22×C4—2842no—
461C22×C4—2842no—
462C22×C4—2844no—
463C22×C4—1428no—
464C22×C4—1424no—
465C22×C4—1422no—
466C22×C4—1428no—
467C22×C4C2×C4×D7112yes—
468C22×C4C2×C22⋊D7111yescharacteristic
469C22×C4C2×C22⋊D7111yescharacteristic
470C22×C4C2×C22⋊D7111yescharacteristic
471C22×C4C2×C22⋊D7111yescharacteristic
472C22×C4C2×C22⋊D7111yescharacteristic
473C22×C4#1176586512yes—
474C22×C4C23×D7111yescharacteristic
475SD16—14216no—
476C2×C8—714no—
477C2×C8—1422no—
478C2×C8—1422no—
479C2×C8—1422no—
480C2×C8—1422no—
481C2×C8—1422no—
482C4⋊C4—21no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—714no—
489C4⋊C4—1424no—
490C4⋊C4—1422no—
491C4⋊C4C2×C4×D7112yescomplemented
492C22⋊C4—24no—
493C22⋊C4—42no—
494C22⋊C4—1428no—
495C22⋊C4—1428no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—42no—
500C42—42no—
501C42D4×D7114yes—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—2848no—
507C23—2848no—
508C23—2848no—
509C23—2848no—
510C23—2848no—
511C23—2848no—
512C23—2842no—
513C23—2842no—
514C23—2842no—
515C23—2842no—
516C23—14216no—
517C23—1424no—
518C23—1424no—
519C23—1424no—
520C23C4×C22⋊D7112yes—
521C23#1455615312yes—
522C23#1455615912yes—
523C23C22×C22⋊D7111yescharacteristic
524Q8—24no—
525Q8—42no—
526D4—718no—
527D4—14216no—
528D4—1424no—
529D4—1424no—
530D4—1424no—
531C2×C4—28no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—44no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—28416no—
559C2×C4—28416no—
560C2×C4—2848no—
561C2×C4—2842no—
562C2×C4—2842no—
563C2×C4—1424no—
564C2×C4C4×C22⋊D7112yes—
565C2×C4#1455609612yes—
566C2×C4#1455609612yes—
567C2×C4#1455615312yes—
568C2×C4C22×C4×D7111yescharacteristic
569C2×C4C2×D4×D7112yes—
570C2×C4#41×D7112yes—
571C2×C4C2×#1176586511yescharacteristic
572C8—2842no—
573C8—1424no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—28432no—
581C22—2844no—
582C22—2844no—
583C22—2844no—
584C22—2844no—
585C22—2844no—
586C22—2844no—
587C22—1428no—
588C22#1798609714yescomplemented
589C22C2×C4×C22⋊D7111yescharacteristic
590C22C2×#1455609611yescharacteristic
591C22#1798650611yescharacteristic
592C22C2×#1455615311yescharacteristic
593C22C2×#1455615311yescharacteristic
594C22C2×#1455615911yescharacteristic
595C22#1798679711yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—81no—
604C4—81no—
605C4—2844no—
606C4C4×D4×D7112yes—
607C4#1798650812yes—
608C2—24no—
609C2—2848no—
610C2C4×#1455609611yescharacteristic
611C2C2×#1798609711yescharacteristic
612C2C4×#1455615311yescharacteristic
613C2#2259756611yescharacteristic
614C2#2259759811yescharacteristic
615C2#2259843511yescharacteristic
616C2#2259847411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC4×C22⋊Q8×C71C2
socleC23×C71C22×D4
prime radicalC23×D284C22
cosocleC22×C4×C71C24
hypercenterC4×C22⋊Q8D71
hypocenterC71#21453
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#17986097C2semidirect
4. #22588677C2semidirect
5. #22588648C2semidirect
6. #22588586C2semidirect
7. #22588527C2semidirect
9. #22588168C2semidirect
10. #22588102C2semidirect
11. C4×C22⋊Q8×C71C2semidirect
22. #17986149C4semidirect
24. C2×#14556061C22semidirect
25. C2×#14556052C22semidirect
31. C2×#14556039C22semidirect
32. #17985761C22semidirect
33. #17985761C22semidirect
34. #17985754C22semidirect
36. C2×C4×Q8×C71C22semidirect
38. #391×C71C22semidirect
39. #389×C71C22semidirect
44. C4×C4⋊C4×C71C22semidirect
61. C2×#11765852C2×C4semidirect
80. #14556048C2×C4semidirect
82. C22⋊(C71⋊C8)C2×C4semidirect
89. C22⋊Q8×C71C2×C4semidirect
147. C2×C71⋊C8C22×C4semidirect
157. C2×Q8×C71C22×C4semidirect
184. C4⋊C4×C71C22×C4semidirect
271. C22×C71#440semidirect
312. C4×C22⋊Q8D71semidirect
320. C71#21453semidirect
325. C2×C4×Q8C2×D71semidirect
344. #391C2×D71semidirect
345. #389C2×D71semidirect
350. C4×C4⋊C4C2×D71semidirect
378. C22⋊Q8C4×D71semidirect
432. C2×Q8C2×C4×D71semidirect
491. C4⋊C4C2×C4×D71semidirect
588. C22#17986097semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D2842
C2×#179861492
C2×#179860972
#225886772
#225886482
#225885862
#225885272
#225882702
#225881682
#225881022
C4×C22⋊Q8×C712
#2145371