Finite Group Databasebeta

← #32033527

Subgroups — #32033527

Order 18176.

1150 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986286C211yesmaximal, characteristic, complemented
2C2×#17986276C211yesmaximal, characteristic, complemented
3C4×#14556102C211yesmaximal, characteristic
4#22589675C212yesmaximal, complemented
5#22589483C212yesmaximal, complemented
6#22589293C211yesmaximal, characteristic, complemented
7#22589289C211yesmaximal, characteristic, complemented
8#22588662C212yesmaximal
9#22588539C212yesmaximal, complemented
10#22588320C211yesmaximal, characteristic, complemented
11C2×#533×C71C211yesmaximal, characteristic, complemented
12C2×#14556102C2211yescharacteristic
13C2×C4×D284C2211yescharacteristic
14C2×C4×C22⋊D71—21no—
15#17986286—24no—
16#17986276—24no—
17C2×#14556061C2212yes—
18C2×#14556059C2212yes—
19C2×C4⋊(C71⋊C8)C2211yescharacteristic
20C2×C4⋊(C71⋊C8)C2211yescharacteristic
21C2×#14556048C2212yes—
22C2×#14556048C2212yes—
23C2×C22⋊(C71⋊C8)C2212yes—
24#17985901C2211yescharacteristic
25#17985897C2211yescharacteristic
26#17985890—21no—
27#17985878—21no—
28#17985859C2211yescharacteristic
29C4×C4⋊Dic71C2211yescharacteristic
30C4×(C2×D71)⋊C4—21no—
31#17985761C2212yes—
32#17985761C2212yes—
33#17985761C2212yes—
34#17985761C2212yes—
35#17985754C2212yes—
36#533×C71—24no—
37#533×C71—24no—
38C2×C4⋊Q8×C71C2211yescharacteristic
39C2×C4⋊Q8×C71C2211yescharacteristic
40C2×#125×C71C2212yes—
41C2×C22⋊Q8×C71—21no—
42C2×C22⋊Q8×C71—21no—
43C2×C4×Q8×C71—21no—
44C2×C4×Q8×C71—21no—
45C2×#117×C71C2212yes—
46C22×C42×C71C2211yescharacteristic
47C22×D284C2311yescharacteristic
48C22×C22⋊D71—21no—
49C22×C4×D71—21no—
50C22×C71⋊C8C2312yes—
51C22×C71⋊C8C2312yes—
52#14556102—22no—
53#14556102Q814yescomplemented
54C4×D284—22no—
55C4×D284—22no—
56C4×C22⋊D71—44no—
57C2×C4⋊Dic71C2311yescharacteristic
58C2×C4⋊Dic71C2311yescharacteristic
59C2×C4⋊Dic71C2311yescharacteristic
60C2×Dic71⋊C4—21no—
61C2×C22⋊Dic71—21no—
62C2×(C2×D71)⋊C4—21no—
63C2×(C2×D71)⋊C4—21no—
64C2×(C2×D71)⋊C4—21no—
65C2×(C2×D71)⋊C4—21no—
66C2×C4×Dic71—21no—
67#14556061—24no—
68#14556059—24no—
69C4⋊(C71⋊C8)—22no—
70C4⋊(C71⋊C8)—22no—
71#14556048—24no—
72#14556048—24no—
73C22⋊(C71⋊C8)—24no—
74C71⋊#95—21no—
75C71⋊#95—21no—
76C71⋊#95—21no—
77C22×Q8×C71—21no—
78C22×Q8×C71—21no—
79C23×C4×C71D411yescharacteristic
80C23×C4×C71D411yescharacteristic
81C23×C4×C71C2311yescharacteristic
82C4⋊Q8×C71—22no—
83C4⋊Q8×C71—22no—
84C4⋊Q8×C71D414yescomplemented
85C4⋊Q8×C71D414yescomplemented
86#125×C71—24no—
87#125×C71—24no—
88C22⋊Q8×C71—28no—
89C22⋊Q8×C71—28no—
90C4×Q8×C71—28no—
91C4×Q8×C71—28no—
92#117×C71—24no—
93#117×C71—24no—
94C2×C4⋊C4×C71—22no—
95C2×C4⋊C4×C71—22no—
96C2×C4⋊C4×C71—21no—
97C2×C4⋊C4×C71—21no—
98C2×C4⋊C4×C71C2312yes—
99C2×C4⋊C4×C71C2312yes—
100C2×C22⋊C4×C71—22no—
101C2×C42×C71—22no—
102C2×C42×C71—22no—
103C2×C42×C71—22no—
104C2×C42×C71D411yescharacteristic
105C2×C42×C71D411yescharacteristic
106C2×C42×C71C2311yescharacteristic
107C2×C42×C71C2311yescharacteristic
108C23×D71—21no—
109C2×D284—21no—
110C2×D284—21no—
111C2×D284—21no—
112C2×D284—21no—
113C2×D284—21no—
114C2×D284C2×Q812yes—
115C2×C22⋊D71—24no—
116C2×C22⋊D71—42no—
117C2×C22⋊D71—42no—
118C22×Dic71—21no—
119C22×Dic71—21no—
120C22×Dic71—21no—
121C2×C4×D71—42no—
122C2×C4×D71—42no—
123C2×C4×D71—42no—
124C2×C71⋊C8—22no—
125C2×C71⋊C8—22no—
126C2×C71⋊C8—22no—
127C2×C71⋊C8—22no—
128C2×C71⋊C8—22no—
129C2×C71⋊C8—22no—
130C4⋊Dic71—21no—
131C4⋊Dic71—21no—
132C4⋊Dic71—21no—
133C4⋊Dic71—21no—
134C4⋊Dic71—21no—
135C4⋊Dic71C2×Q812yes—
136Dic71⋊C4—42no—
137C22⋊Dic71—42no—
138(C2×D71)⋊C4—24no—
139(C2×D71)⋊C4—42no—
140(C2×D71)⋊C4—42no—
141(C2×D71)⋊C4—42no—
142C4×Dic71—42no—
143C24×C71C2×D411yescharacteristic
144C2×Q8×C71—24no—
145C2×Q8×C71—24no—
146C2×Q8×C71—42no—
147C2×Q8×C71—42no—
148C2×Q8×C71—42no—
149C2×Q8×C71—42no—
150C22×C4×C71—21no—
151C22×C4×C71—21no—
152C22×C4×C71—21no—
153C22×C4×C71—21no—
154C22×C4×C71—22no—
155C22×C4×C71—22no—
156C22×C4×C71—22no—
157C22×C4×C71—22no—
158C22×C4×C71—22no—
159C22×C4×C71—22no—
160C22×C4×C71—41no—
161C22×C4×C71—41no—
162C22×C4×C71C2×D411yescharacteristic
163C22×C4×C71C2×D411yescharacteristic
164C22×C4×C71C2×D411yescharacteristic
165C22×C4×C71C2×D411yescharacteristic
166C22×C4×C71C2×D411yescharacteristic
167C22×C4×C71C2×Q812yes—
168C22×C4×C71#4112yes—
169C22×C4×C71C2411yescharacteristic
170C4⋊C4×C71—28no—
171C4⋊C4×C71—28no—
172C4⋊C4×C71—24no—
173C4⋊C4×C71—24no—
174C4⋊C4×C71—22no—
175C4⋊C4×C71—22no—
176C4⋊C4×C71C2×D414yes—
177C4⋊C4×C71C2×D414yes—
178C22⋊C4×C71—28no—
179C42×C71—28no—
180C42×C71—22no—
181C42×C71—22no—
182C42×C71—21no—
183C42×C71—21no—
184C42×C71C2×D412yes—
185C42×C71C2×D412yes—
186C23×C71—21no—
187C23×C71—21no—
188C23×C71—21no—
189C23×C71—21no—
190C23×C71—21no—
191C23×C71—21no—
192C23×C71C22⋊Q812yes—
193C23×C71C22×D411yescharacteristic
194C22×D71—22no—
195C22×D71—41no—
196C22×D71—41no—
197C22×D71—41no—
198C22×D71—41no—
199C22×D71—41no—
200C22×D71—41no—
201Q8×C71—48no—
202Q8×C71—48no—
203C2×C4×C71—21no—
204C2×C4×C71—21no—
205C2×C4×C71—21no—
206C2×C4×C71—21no—
207C2×C4×C71—22no—
208C2×C4×C71—22no—
209C2×C4×C71—22no—
210C2×C4×C71—22no—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—24no—
214C2×C4×C71—24no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—44no—
220C2×C4×C71—42no—
221C2×C4×C71—42no—
222C2×C4×C71—42no—
223C2×C4×C71—42no—
224C2×C4×C71—44no—
225C2×C4×C71#12012yes—
226C2×C4×C71#12012yes—
227C2×C4×C71C22⋊Q812yes—
228C2×C4×C71#13512yes—
229C2×C4×C71AGL(1,ℤ/8)12yes—
230C2×C4×C71C22×D411yescharacteristic
231C2×C4×C71C22×D411yescharacteristic
232C2×C4×C71C22×Q811yescharacteristic
233C2×C4×C71C2×#4111yescharacteristic
234C2×C4×C712−1+411yescharacteristic
235C2×C4×C712−1+411yescharacteristic
236C22⋊D71—48no—
237C2×Dic71—22no—
238C2×Dic71—41no—
239C2×Dic71—41no—
240C2×Dic71—41no—
241C2×Dic71—41no—
242C2×Dic71—41no—
243C2×Dic71—41no—
244C2×Dic71—41no—
245C2×Dic71—41no—
246D284—22no—
247D284—22no—
248D284—41no—
249D284—41no—
250C4×D71—84no—
251C71⋊C8—42no—
252C71⋊C8—42no—
253C22×C71—22no—
254C22×C71—22no—
255C22×C71—22no—
256C22×C71—41no—
257C22×C71—41no—
258C22×C71—41no—
259C22×C71—41no—
260C22×C71C2×#12011yescharacteristic
261C22×C71C2×C22⋊Q811yescharacteristic
262C22×C71#53611yescharacteristic
263C22×C71C2×#13511yescharacteristic
264C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C71#57511yescharacteristic
266C22×C71#57511yescharacteristic
267C2×D71—42no—
268C2×D71—42no—
269C2×D71—42no—
270C2×D71—81no—
271C2×D71—81no—
272C2×D71—81no—
273C2×D71—81no—
274C4×C71—22no—
275C4×C71—22no—
276C4×C71—41no—
277C4×C71—44no—
278C4×C71—41no—
279C4×C71—44no—
280C4×C71#44914yes—
281C4×C71#54811yescharacteristic
282C4×C71#54811yescharacteristic
283C4×C71D4×Q811yescharacteristic
284C4×C71D4×Q811yescharacteristic
285Dic71—42no—
286Dic71—81no—
287Dic71—81no—
288#24300—711nomaximal, Sylow
289C2×C71—42no—
290C2×C71#240911yescharacteristic
291C2×C71C2×#44911yescharacteristic
292C2×C71#300511yescharacteristic
293C2×C71#307611yescharacteristic
294C2×C71#307711yescharacteristic
295C2×C71#323311yescharacteristic
296C2×C71#323511yescharacteristic
297D71—82no—
298C2×#533D7111yescharacteristic, complemented
299C2×#476—711no—
300C2×#474—711no—
301C4×#121—711no—
302#2059—712no—
303#2025—712no—
304#1988—711no—
305#1987—711no—
306#1875—712no—
307#1854—712no—
308#1797—711no—
309C71#2430011yescharacteristic, complemented, Sylow
310#533—24no—
311#533—24no—
312C2×C4⋊Q8C2×D7111yescharacteristic
313C2×C4⋊Q8C2×D7111yescharacteristic
314C2×#125C2×D7112yes—
315C2×C22⋊Q8—21no—
316C2×C22⋊Q8—21no—
317C2×#121—711no—
318C2×C4×Q8—21no—
319C2×C4×Q8—21no—
320C2×C4×D4—711no—
321C2×C4×D4—1421no—
322C2×#117C2×D7112yes—
323C22×C42C2×D7111yescharacteristic
324#476—1424no—
325#474—1424no—
326C2×#107—712no—
327C2×#106—712no—
328C2×C4⋊C8—711no—
329C2×C4⋊C8—711no—
330C2×#102—712no—
331C2×#102—712no—
332C2×C22⋊C8—712no—
333#390—711no—
334#389—711no—
335#388—1421no—
336#386—1421no—
337#380—711no—
338C4×C4⋊C4—711no—
339C4×C22⋊C4—1421no—
340#340—712no—
341#340—712no—
342#340—712no—
343#340—712no—
344#336—712no—
345C22×Q8—21no—
346C22×Q8—21no—
347C22×D4—711no—
348C22×D4—1421no—
349C23×C4—1421no—
350C23×C4C22⋊D7111yescharacteristic
351C23×C4C22⋊D7111yescharacteristic
352C23×C4C22×D7111yescharacteristic
353C22×C8—712no—
354C22×C8—712no—
355C4⋊Q8—22no—
356C4⋊Q8—22no—
357C4⋊Q8C22⋊D7114yescomplemented
358C4⋊Q8C22⋊D7114yescomplemented
359#125—24no—
360#125—24no—
361C22⋊Q8—28no—
362C22⋊Q8—28no—
363#121—714no—
364#121—1422no—
365C4×Q8—28no—
366C4×Q8—28no—
367C4×D4—2844no—
368C4×D4—1422no—
369C4×D4—1422no—
370#117—24no—
371#117—24no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—711no—
377C2×C4⋊C4—711no—
378C2×C4⋊C4—711no—
379C2×C4⋊C4—1421no—
380C2×C4⋊C4C22×D7112yes—
381C2×C4⋊C4C22×D7112yes—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—1421no—
384C2×C22⋊C4—1421no—
385C2×C22⋊C4—1421no—
386C2×C22⋊C4—1421no—
387C2×C22⋊C4—1421no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—1421no—
392C2×C42C22⋊D7111yescharacteristic
393C2×C42C22⋊D7111yescharacteristic
394C2×C42C22×D7111yescharacteristic
395C2×C42C22×D7111yescharacteristic
396#107—1424no—
397#106—1424no—
398C4⋊C8—1422no—
399C4⋊C8—1422no—
400#102—1424no—
401#102—1424no—
402C22⋊C8—1424no—
403#95—1421no—
404#95—1421no—
405#95—1421no—
406C24—1421no—
407C24C2×C22⋊D7111yescharacteristic
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×D4—2842no—
415C2×D4—2842no—
416C2×D4—712no—
417C2×D4—1424no—
418C2×D4—1421no—
419C2×D4—1421no—
420C2×D4—1421no—
421C2×D4—1421no—
422C2×D4—1421no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—2842no—
436C22×C4—2842no—
437C22×C4—2842no—
438C22×C4—1421no—
439C22×C4—1421no—
440C22×C4—1421no—
441C22×C4C2×C22⋊D7111yescharacteristic
442C22×C4C2×C22⋊D7111yescharacteristic
443C22×C4C2×C22⋊D7111yescharacteristic
444C22×C4C2×C22⋊D7111yescharacteristic
445C22×C4C2×C22⋊D7111yescharacteristic
446C22×C4Q8×D7112yes—
447C22×C4#1176586612yes—
448C22×C4C23×D7111yescharacteristic
449C2×C8—1422no—
450C2×C8—1422no—
451C2×C8—1422no—
452C2×C8—1422no—
453C2×C8—1422no—
454C2×C8—1422no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—2842no—
462C4⋊C4—712no—
463C4⋊C4—1421no—
464C4⋊C4—1421no—
465C4⋊C4—1421no—
466C4⋊C4—1421no—
467C4⋊C4—1421no—
468C4⋊C4C2×C22⋊D7114yes—
469C4⋊C4C2×C22⋊D7114yes—
470C22⋊C4—28no—
471C22⋊C4—2842no—
472C22⋊C4—2842no—
473C22⋊C4—2842no—
474C22⋊C4—2842no—
475C22⋊C4—1424no—
476C42—28no—
477C42—21no—
478C42—21no—
479C42—22no—
480C42—22no—
481C42—2842no—
482C42C2×C22⋊D7112yes—
483C42C2×C22⋊D7112yes—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—2841no—
491C23—2841no—
492C23—2841no—
493C23—2841no—
494C23—2841no—
495C23—2841no—
496C23—1422no—
497C23#1455610412yes—
498C23C22×C22⋊D7111yescharacteristic
499Q8—48no—
500Q8—48no—
501D4—2848no—
502D4—2841no—
503D4—2841no—
504D4—1422no—
505D4—1422no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—2841no—
529C2×C4—2841no—
530C2×C4—2841no—
531C2×C4—2841no—
532C2×C4—2841no—
533C2×C4—2841no—
534C2×C4—2841no—
535C2×C4—2841no—
536C2×C4—5684no—
537C2×C4—1422no—
538C2×C4#1455609712yes—
539C2×C4#1455609712yes—
540C2×C4#1455610412yes—
541C2×C4#1455614812yes—
542C2×C4#1455615312yes—
543C2×C4C22×C22⋊D7111yescharacteristic
544C2×C4C22×C22⋊D7111yescharacteristic
545C2×C4C2×Q8×D7111yescharacteristic
546C2×C4C2×#1176586611yescharacteristic
547C2×C4#1455617611yescharacteristic
548C2×C4#1455617611yescharacteristic
549C8—2842no—
550C8—2842no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—2842no—
559C22—2842no—
560C22—2842no—
561C22—5681no—
562C22—5681no—
563C22—5681no—
564C22—5681no—
565C22C2×#1455609711yescharacteristic
566C22C2×#1455610411yescharacteristic
567C22#1798657311yescharacteristic
568C22C2×#1455614811yescharacteristic
569C22C2×#1455615311yescharacteristic
570C22#1798679711yescharacteristic
571C22#1798679711yescharacteristic
572C4—22no—
573C4—22no—
574C4—41no—
575C4—44no—
576C4—41no—
577C4—44no—
578C4—2842no—
579C4—5681no—
580C4—5681no—
581C4#1798614414yes—
582C4#1798666011yescharacteristic
583C4#1798666011yescharacteristic
584C4Q8×C22⋊D7111yescharacteristic
585C4Q8×C22⋊D7111yescharacteristic
586C2—42no—
587C2—5682no—
588C2#2259169911yescharacteristic
589C2C2×#1798614411yescharacteristic
590C2#2259758211yescharacteristic
591C2#2259812211yescharacteristic
592C2#2259813311yescharacteristic
593C2#2259951911yescharacteristic
594C2#2259953811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC2×#533×C71C2
socleC23×C71C22×D4
prime radicalC2×#14556102C22
hypercenterC2×#533D71
hypocenterC71#24300
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×#17986286C2semidirect
2. C2×#17986276C2semidirect
4. #22589675C2semidirect
5. #22589483C2semidirect
6. #22589293C2semidirect
7. #22589289C2semidirect
9. #22588539C2semidirect
10. #22588320C2semidirect
11. C2×#533×C71C2semidirect
53. #14556102Q8semidirect
84. C4⋊Q8×C71D4semidirect
85. C4⋊Q8×C71D4semidirect
298. C2×#533D71semidirect
309. C71#24300semidirect
357. C4⋊Q8C22⋊D71semidirect
358. C4⋊Q8C22⋊D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179862862
C2×#179862762
C4×#145561022
#225896752
#225894832
#225892932
#225892892
#225886622
#225885392
#225883202
C2×#533×C712
#2430071