Finite Group Databasebeta

← #31998452

Subgroups — #31998452

Order 18176.

1458 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986480C211yesmaximal, characteristic
2C2×#17986001C211yesmaximal, characteristic
3C2×#17985993C211yesmaximal, characteristic
4#22590956C211yesmaximal, characteristic, complemented
5#22588605C212yesmaximal, complemented
6#22588598C212yesmaximal, complemented
7#22588591C212yesmaximal, complemented
8#22588584C212yesmaximal, complemented
9C2×#17985761C211yesmaximal, characteristic, complemented
10#22588038C211yesmaximal, characteristic, complemented
11C2×#428×C71C211yesmaximal, characteristic, complemented
12C22×#11765865C2211yescharacteristic
13C23×C71⋊C8C2211yescharacteristic
14#17986480—24no—
15#17986480—24no—
16C2×#14556093C2212yes—
17C2×#14556090C2212yes—
18C2×C4×C22⋊D71—22no—
19C2×#14556087C2211yescharacteristic
20C2×#14556086—21no—
21C22×C4⋊Dic71C2211yescharacteristic
22C2×C4⋊C4×D71—21no—
23#17986001C418yescomplemented
24#17985993—24no—
25C2×#14556053C2212yes—
26C2×#14556051C2211yescharacteristic
27C2×#14556051C2211yescharacteristic
28C2×#14556050C2212yes—
29C2×#14556048C2211yescharacteristic
30C2×#14556048C2211yescharacteristic
31#17985897—21no—
32#17985897—21no—
33#17985895C2212yes—
34#17985895C2212yes—
35#17985857—22no—
36#17985761C2212yes—
37#17985761C2212yes—
38#17985753C2212yes—
39#17985753C2212yes—
40C22×M16×C71C2211yescharacteristic
41C2×#117×C71C2211yescharacteristic
42C22×C4⋊C4×C71C2211yescharacteristic
43#428×C71—24no—
44C2×#107×C71C2212yes—
45C2×#106×C71C2212yes—
46C2×#11765865—22no—
47C2×#11765865—22no—
48C2×#11765865—22no—
49C2×#11765865C2×C414yes—
50C2×#11765865D414yes—
51C2×#11765865Q814yes—
52C22×Dic142C2311yescharacteristic
53C22×D284C2311yescharacteristic
54C22×C22⋊D71—21no—
55C23×Dic71—21no—
56C22×C4×D71—22no—
57C22×C4×D71—21no—
58C22×C71⋊C8—22no—
59C22×C71⋊C8—22no—
60C22×C71⋊C8—22no—
61C22×C71⋊C8C2311yescharacteristic
62C22×C71⋊C8C2311yescharacteristic
63#14556093—24no—
64#14556093—24no—
65#14556090—24no—
66#14556090—24no—
67C4×C22⋊D71—216no—
68#14556087—22no—
69#14556087—22no—
70#14556086—28no—
71C2×C4⋊Dic71—21no—
72C2×C4⋊Dic71—22no—
73C2×C4⋊Dic71C2×C414yescomplemented
74C2×C4⋊Dic71D411yescharacteristic
75C2×C4⋊Dic71D411yescharacteristic
76C2×C4⋊Dic71C2312yes—
77C2×Dic71⋊C4—22no—
78C2×Dic71⋊C4—22no—
79C2×Dic71⋊C4—22no—
80C4⋊C4×D71—28no—
81C2×C22⋊Dic71D412yes—
82C2×(C2×D71)⋊C4—22no—
83C2×C4×Dic71—22no—
84C2×C4×Dic71C2312yes—
85#14556053C2×C418yescomplemented
86#14556051—22no—
87#14556051—22no—
88#14556050C2×C418yescomplemented
89#14556048—22no—
90#14556048—22no—
91C71⋊#95—22no—
92C71⋊#95—22no—
93C23×C4×C71—21no—
94C23×C4×C71C2311yescharacteristic
95C2×M16×C71—22no—
96C2×M16×C71—22no—
97C2×M16×C71C2×C414yescomplemented
98C22×C8×C71C2312yes—
99#117×C71—22no—
100#117×C71—22no—
101C2×C4⋊C4×C71—21no—
102C2×C4⋊C4×C71—22no—
103C2×C4⋊C4×C71D414yes—
104C2×C4⋊C4×C71C2311yescharacteristic
105C2×C4⋊C4×C71C2311yescharacteristic
106C2×C4⋊C4×C71C2312yes—
107C2×C22⋊C4×C71—21no—
108C2×C42×C71—21no—
109#107×C71—24no—
110#106×C71—24no—
111C23×D71—21no—
112#11765865—28no—
113#11765865—28no—
114#11765865—28no—
115#11765865C4⋊C4116yescomplemented
116C2×Dic142—21no—
117C2×Dic142—21no—
118C2×Dic142—21no—
119C2×Dic142C22×C412yes—
120C2×Dic142C2×D412yes—
121C2×Dic142C2×Q812yes—
122C2×D284—21no—
123C2×D284—21no—
124C2×D284—21no—
125C2×D284C22×C412yes—
126C2×D284C2×D412yes—
127C2×D284C2×Q812yes—
128C2×C22⋊D71—24no—
129C2×C22⋊D71—24no—
130C2×C22⋊D71—24no—
131C22×Dic71—21no—
132C22×Dic71—22no—
133C22×Dic71—24no—
134C22×Dic71—81no—
135C22×Dic71—81no—
136C22×Dic71C2×D412yes—
137C22×Dic71C2×D412yes—
138C2×C4×D71—24no—
139C2×C4×D71—24no—
140C2×C4×D71—24no—
141C2×C4×D71—28no—
142C2×C4×D71—82no—
143C2×C4×D71—82no—
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⋊C8—44no—
151C4⋊Dic71—24no—
152C4⋊Dic71—22no—
153C4⋊Dic71—22no—
154C4⋊Dic71—22no—
155C4⋊Dic71C22×C414yes—
156Dic71⋊C4—28no—
157Dic71⋊C4—82no—
158Dic71⋊C4—82no—
159C22⋊Dic71—24no—
160(C2×D71)⋊C4—28no—
161C4×Dic71—28no—
162C4×Dic71—22no—
163C4×Dic71—22no—
164C24×C71C2×D411yescharacteristic
165C22×C4×C71—22no—
166C22×C4×C71—24no—
167C22×C4×C71—21no—
168C22×C4×C71—21no—
169C22×C4×C71—81no—
170C22×C4×C71—81no—
171C22×C4×C71C22×C412yes—
172C22×C4×C71C2×D412yes—
173C22×C4×C71C2×D411yescharacteristic
174C22×C4×C71C2×D412yes—
175C22×C4×C71C2×Q812yes—
176C22×C4×C71#4112yes—
177C22×C4×C71#4112yes—
178C22×C4×C71C2411yescharacteristic
179M16×C71—28no—
180C2×C8×C71—22no—
181C2×C8×C71—22no—
182C2×C8×C71C22×C414yes—
183C4⋊C4×C71—24no—
184C4⋊C4×C71—22no—
185C4⋊C4×C71—21no—
186C4⋊C4×C71—21no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71C2×D414yes—
190C22⋊C4×C71—42no—
191C42×C71—42no—
192C23×C71—21no—
193C23×C71—21no—
194C23×C71—21no—
195C23×C71—21no—
196C23×C71C4×D412yes—
197C23×C71#12012yes—
198C23×C71C22⋊Q812yes—
199C23×C71C22×D411yescharacteristic
200C22×D71—22no—
201C22×D71—22no—
202C22×D71—22no—
203C22×D71—41no—
204C22×D71—41no—
205C22×D71—41no—
206C22×D71—41no—
207C2×C4×C71—24no—
208C2×C4×C71—42no—
209C2×C4×C71—42no—
210C2×C4×C71—41no—
211C2×C4×C71—41no—
212C2×C4×C71—41no—
213C2×C4×C71—41no—
214C2×C4×C71—41no—
215C2×C4×C71—41no—
216C2×C4×C71—84no—
217C2×C4×C71C2×C4⋊C414yes—
218C2×C4×C71C4×D414yes—
219C2×C4×C71C4×D412yes—
220C2×C4×C71#12012yes—
221C2×C4×C71C22⋊Q814yes—
222C2×C4×C71C22⋊Q812yes—
223C2×C4×C71#12314yes—
224C2×C4×C71C23×C411yescharacteristic
225C2×C4×C71C22×D411yescharacteristic
226C2×C4×C71C22×D411yescharacteristic
227C2×C4×C71C22×Q811yescharacteristic
228C2×C4×C71C2×#4111yescharacteristic
229C2×C4×C71C2×#4111yescharacteristic
230C22⋊D71—216no—
231C2×Dic71—22no—
232C2×Dic71—22no—
233C2×Dic71—22no—
234C2×Dic71—24no—
235C2×Dic71—22no—
236C2×Dic71—22no—
237C2×Dic71—22no—
238C2×Dic71—22no—
239C2×Dic71—22no—
240C2×Dic71—22no—
241C2×Dic71—42no—
242C2×Dic71—42no—
243C2×Dic71—84no—
244D284—22no—
245D284—22no—
246D284—22no—
247D284C2×C4⋊C414yes—
248C4×D71—216no—
249C4×D71—88no—
250Dic142—22no—
251Dic142—22no—
252Dic142—22no—
253Dic142C2×C4⋊C414yes—
254C8×C71—24no—
255C71⋊C8—41no—
256C71⋊C8—41no—
257C22×C71—22no—
258C22×C71—22no—
259C22×C71—22no—
260C22×C71—22no—
261C22×C71—22no—
262C22×C71—22no—
263C22×C71#38514yes—
264C22×C71C2×C4×D411yescharacteristic
265C22×C71C2×#12011yescharacteristic
266C22×C71C2×C22⋊Q811yescharacteristic
267C22×C71#57611yescharacteristic
268C22×C71#57711yescharacteristic
269C22×C71#57711yescharacteristic
270C22×C71#57811yescharacteristic
271C2×D71—24no—
272C2×D71—42no—
273C2×D71—42no—
274C2×D71—42no—
275C2×D71—42no—
276C2×D71—42no—
277C2×D71—42no—
278C4×C71—44no—
279C4×C71—81no—
280C4×C71—81no—
281C4×C71#38514yes—
282C4×C71C22×C4⋊C411yescharacteristic
283C4×C71C2×C4×D411yescharacteristic
284C4×C71C2×C22⋊Q811yescharacteristic
285C4×C71C2×#12311yescharacteristic
286Dic71—24no—
287Dic71—44no—
288Dic71—42no—
289Dic71—42no—
290#20878—711nomaximal, Sylow
291C2×C71—24no—
292C2×C71C2×#38511yescharacteristic
293C2×C71#294611yescharacteristic
294C2×C71#294711yescharacteristic
295C2×C71#301211yescharacteristic
296C2×C71#301311yescharacteristic
297C2×C71#308311yescharacteristic
298C2×C71#308411yescharacteristic
299D71—44no—
300C2×#520—711no—
301C2×#428D7111yescharacteristic, complemented
302C2×#417—711no—
303C2×#416—711no—
304#2316—711no—
305#1865—712no—
306#1864—712no—
307#1863—712no—
308#1862—712no—
309C2×#340—711no—
310#1729—711no—
311C71#2087811yescharacteristic, complemented, Sylow
312C22×#41—711no—
313C22×M16C2×D7111yescharacteristic
314C23×C8—711no—
315#520—1424no—
316#520—1424no—
317C2×C4×Q8—712no—
318C2×C4×D4—712no—
319C2×C4×D4—1422no—
320C2×#117—711no—
321C2×#117—1421no—
322C2×#117C2×D7111yescharacteristic
323C22×C4⋊C4—711no—
324C22×C4⋊C4—1421no—
325C22×C4⋊C4C2×D7111yescharacteristic
326#428—24no—
327C2×#107C2×D7112yes—
328C2×#106C2×D7112yes—
329#417—718no—
330#416—1424no—
331C2×#103—712no—
332C2×#103—711no—
333C2×#103—711no—
334C2×#102—712no—
335C2×#102—711no—
336C2×#102—711no—
337#389—712no—
338#389—712no—
339#389—1421no—
340#389—1421no—
341#380—1422no—
342#340—712no—
343#340—712no—
344#336—712no—
345#336—712no—
346C2×#41—714no—
347C2×#41—714no—
348C2×#41—714no—
349C2×#41—1422no—
350C2×#41—1422no—
351C2×#41—1422no—
352C22×Q8—711no—
353C22×D4—711no—
354C22×D4—1421no—
355C23×C4—21no—
356C23×C4—1422no—
357C23×C4—1421no—
358C23×C4—1421no—
359C23×C4C22×D7111yescharacteristic
360C2×M16—22no—
361C2×M16—22no—
362C2×M16C4×D7114yescomplemented
363C22×C8—711no—
364C22×C8—711no—
365C22×C8—1422no—
366C22×C8—1422no—
367C22×C8—1422no—
368C22×C8C22×D7112yes—
369C4×Q8—1424no—
370C4×Q8—1424no—
371C4×D4—1424no—
372C4×D4—14216no—
373C4×D4—1424no—
374#117—22no—
375#117—22no—
376#117—1428no—
377#117—1422no—
378#117—1422no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—714no—
382C2×C4⋊C4—712no—
383C2×C4⋊C4—711no—
384C2×C4⋊C4—711no—
385C2×C4⋊C4—1428no—
386C2×C4⋊C4—1422no—
387C2×C4⋊C4—1422no—
388C2×C4⋊C4—1422no—
389C2×C4⋊C4—1421no—
390C2×C4⋊C4—1422no—
391C2×C4⋊C4C22⋊D7114yes—
392C2×C4⋊C4C22×D7111yescharacteristic
393C2×C4⋊C4C22×D7111yescharacteristic
394C2×C4⋊C4C22×D7112yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—712no—
397C2×C22⋊C4—1422no—
398C2×C42—21no—
399C2×C42—712no—
400C2×C42—1422no—
401#107—24no—
402#106—24no—
403#103—718no—
404#103—1422no—
405#103—1422no—
406#102—718no—
407#102—1422no—
408#102—1422no—
409#95—1422no—
410#95—1422no—
411C24—1421no—
412C24D4×D7111yescharacteristic
413#41—7116no—
414#41—1428no—
415#41—1428no—
416#41—1428no—
417C2×Q8—712no—
418C2×Q8—712no—
419C2×Q8—712no—
420C2×Q8—1421no—
421C2×Q8—1421no—
422C2×Q8—1421no—
423C2×D4—712no—
424C2×D4—712no—
425C2×D4—712no—
426C2×D4—1424no—
427C2×D4—1424no—
428C2×D4—1424no—
429C2×D4—1421no—
430C2×D4—1421no—
431C2×D4—1421no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—5682no—
439C22×C4—5682no—
440C22×C4—5681no—
441C22×C4—5681no—
442C22×C4—712no—
443C22×C4—712no—
444C22×C4—1428no—
445C22×C4—1422no—
446C22×C4—1424no—
447C22×C4—1421no—
448C22×C4—1424no—
449C22×C4—1424no—
450C22×C4—1424no—
451C22×C4C2×C4×D7112yes—
452C22×C4D4×D7112yes—
453C22×C4D4×D7111yescharacteristic
454C22×C4C2×C22⋊D7112yes—
455C22×C4Q8×D7112yes—
456C22×C4#1176586512yes—
457C22×C4#1176586612yes—
458C22×C4C23×D7111yescharacteristic
459M16—28no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—2844no—
463C2×C8—1421no—
464C2×C8—1421no—
465C2×C8—1421no—
466C2×C8—1421no—
467C2×C8—1421no—
468C2×C8—1421no—
469C2×C8C2×C4×D7114yes—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—5682no—
477C4⋊C4—5682no—
478C4⋊C4—714no—
479C4⋊C4—1428no—
480C4⋊C4—1422no—
481C4⋊C4—1424no—
482C4⋊C4—1422no—
483C4⋊C4—1422no—
484C4⋊C4C2×C22⋊D7114yes—
485C22⋊C4—42no—
486C22⋊C4—1428no—
487C22⋊C4—1424no—
488C42—42no—
489C42—1428no—
490C42—1422no—
491C42—1422no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—2841no—
497C23—2841no—
498C23—2841no—
499C23—2841no—
500C23—1422no—
501C23—1422no—
502C23—1422no—
503C23#1455609012yes—
504C23#1455609612yes—
505C23#1455610612yes—
506C23C2×D4×D7111yescharacteristic
507Q8—714no—
508Q8—1422no—
509Q8—1422no—
510Q8—1422no—
511D4—714no—
512D4—14216no—
513D4—1422no—
514D4—1422no—
515D4—1422no—
516C2×C4—24no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—84no—
526C2×C4—2842no—
527C2×C4—2842no—
528C2×C4—5688no—
529C2×C4—5684no—
530C2×C4—1424no—
531C2×C4—1422no—
532C2×C4—1422no—
533C2×C4—1422no—
534C2×C4—1422no—
535C2×C4—1422no—
536C2×C4—1422no—
537C2×C4—1422no—
538C2×C4—1422no—
539C2×C4—1422no—
540C2×C4—14216no—
541C2×C4C4⋊C4×D7114yes—
542C2×C4C4×C22⋊D7114yes—
543C2×C4#1455609012yes—
544C2×C4#1455609612yes—
545C2×C4#1455610414yes—
546C2×C4#1455610612yes—
547C2×C4#1455611314yes—
548C2×C4C22×C4×D7111yescharacteristic
549C2×C4C2×D4×D7111yescharacteristic
550C2×C4C22×C22⋊D7111yescharacteristic
551C2×C4C2×Q8×D7111yescharacteristic
552C2×C4C2×#1176586511yescharacteristic
553C2×C4C2×#1176586611yescharacteristic
554C8—24no—
555C8—2841no—
556C8—2841no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—2842no—
564C22—2842no—
565C22—2842no—
566C22—2842no—
567C22—2842no—
568C22—2842no—
569C22—1424no—
570C22#1798587014yes—
571C22C2×#1455609011yescharacteristic
572C22C2×#1455609611yescharacteristic
573C22C2×#1455610611yescharacteristic
574C22#1798680311yescharacteristic
575C22#1798680811yescharacteristic
576C22#1798680911yescharacteristic
577C22#1798681511yescharacteristic
578C4—44no—
579C4—81no—
580C4—81no—
581C4—2842no—
582C4—2842no—
583C4—2844no—
584C4—1424no—
585C4#1798587014yes—
586C4C2×C4⋊C4×D7111yescharacteristic
587C4C2×C4×C22⋊D7111yescharacteristic
588C4C2×#1455610411yescharacteristic
589C4C2×#1455611311yescharacteristic
590C2—24no—
591C2—2844no—
592C2C2×#1798587011yescharacteristic
593C2#2259717011yescharacteristic
594C2#2259718111yescharacteristic
595C2#2259764311yescharacteristic
596C2#2259765411yescharacteristic
597C2#2259819011yescharacteristic
598C2#2259820111yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#428×C71C2
socleC23×C71C22×D4
prime radicalC22×#11765865C22
cosocleC22×C4×C71C24
hypercenterC2×#428D71
hypocenterC71#20878
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. #22590956C2semidirect
5. #22588605C2semidirect
6. #22588598C2semidirect
7. #22588591C2semidirect
8. #22588584C2semidirect
9. C2×#17985761C2semidirect
10. #22588038C2semidirect
11. C2×#428×C71C2semidirect
23. #17986001C4semidirect
73. C2×C4⋊Dic71C2×C4semidirect
85. #14556053C2×C4semidirect
88. #14556050C2×C4semidirect
97. C2×M16×C71C2×C4semidirect
115. #11765865C4⋊C4semidirect
301. C2×#428D71semidirect
311. C71#20878semidirect
362. C2×M16C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179864802
C2×#179860012
C2×#179859932
#225909562
#225886052
#225885982
#225885912
#225885842
C2×#179857612
#225880382
C2×#428×C712
#2087871