Finite Group Databasebeta

← #31999218

Subgroups — #31999218

Order 18176.

1954 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986510C211yesmaximal, characteristic
2C2×#17986017C212yesmaximal, complemented
3C2×#17985963C211yesmaximal, characteristic, complemented
4#22588788C214yesmaximal, complemented
5#22588783C214yesmaximal, complemented
6#22588050C212yesmaximal, complemented
7C22×#117×C71C211yesmaximal, characteristic, complemented
8C22×#11765865C2211yescharacteristic
9C22×#11765849C2212yes—
10#17986510—22no—
11#17986510—21no—
12#17986510—21no—
13#17986510C414yescomplemented
14#17986510C2212yes—
15#17986510C2212yes—
16C2×#14556111—21no—
17C2×#14556107C412yescomplemented
18C2×#14556102C412yescomplemented
19C2×#14556097—21no—
20C2×C4×C22⋊D71—22no—
21C2×#14556087C2211yescharacteristic
22#17986017—28no—
23C2×#14556056C2218yescomplemented
24#17985963—22no—
25#17985963C2212yescomplemented
26#17985963C2212yescomplemented
27C2×C22⋊(C71⋊C8)—22no—
28#17985759C2218yescomplemented
29C24×C4×C71C2211yescharacteristic
30C2×#117×C71—21no—
31C2×#117×C71—21no—
32C2×#117×C71—28no—
33C2×#117×C71C2212yes—
34C2×#117×C71C2212yescomplemented
35C22×C4⋊C4×C71—21no—
36C22×C22⋊C4×C71—21no—
37C22×C42×C71C2212yescomplemented
38C2×#11765865—21no—
39C2×#11765865—21no—
40C2×#11765865—24no—
41C2×#11765865—24no—
42C2×#11765865C2×C412yescomplemented
43C2×#11765865C2312yes—
44C22×Dic142C2×C411yescharacteristic, complemented
45C22×D284C2×C411yescharacteristic, complemented
46C22×C22⋊D71—22no—
47C22×C22⋊D71—21no—
48C22×C4×D71—21no—
49C2×#11765849—22no—
50C2×#11765849—24no—
51C2×#11765849—24no—
52C2×#11765849C2314yes—
53C22×C71⋊C8—22no—
54#14556111—24no—
55#14556111—22no—
56#14556111—22no—
57#14556107—22no—
58#14556107—22no—
59#14556107C2×C414yes—
60#14556107C2×C414yescomplemented
61#14556102—22no—
62#14556102—22no—
63#14556102C2×C414yes—
64#14556102C2×C414yescomplemented
65#14556097—24no—
66#14556097—22no—
67#14556097—22no—
68C4×C22⋊D71—24no—
69C4×C22⋊D71—24no—
70C4×C22⋊D71—24no—
71C4×C22⋊D71—24no—
72#14556087—21no—
73#14556087—21no—
74#14556087C2×C412yescomplemented
75#14556087C2312yes—
76C2×C4⋊Dic71C2×C412yescomplemented
77C2×Dic71⋊C4—22no—
78C2×C22⋊Dic71—22no—
79C2×C22⋊Dic71—21no—
80C2×(C2×D71)⋊C4—22no—
81C2×C4×Dic71—21no—
82#14556056—216no—
83C22⋊(C71⋊C8)—24no—
84C22⋊(C71⋊C8)—24no—
85C25×C71D411yescharacteristic
86C23×C4×C71—24no—
87C23×C4×C71—24no—
88C23×C4×C71—22no—
89C23×C4×C71C2×C412yes—
90C23×C4×C71C2×C412yes—
91C23×C4×C71D411yescharacteristic
92C23×C4×C71D411yescharacteristic
93C23×C4×C71D411yescharacteristic
94C23×C4×C71D412yes—
95C23×C4×C71D412yes—
96C23×C4×C71C2311yescharacteristic
97C23×C4×C71C2311yescharacteristic
98C23×C4×C71C2311yescharacteristic
99#117×C71—28no—
100#117×C71—28no—
101#117×C71—24no—
102#117×C71—28no—
103#117×C71D418yescomplemented
104C2×C4⋊C4×C71—28no—
105C2×C4⋊C4×C71—22no—
106C2×C4⋊C4×C71—22no—
107C2×C22⋊C4×C71—28no—
108C2×C22⋊C4×C71—22no—
109C2×C22⋊C4×C71—22no—
110C2×C42×C71—22no—
111C2×C42×C71—28no—
112C2×C42×C71C2314yes—
113C23×D71—21no—
114#11765865—28no—
115#11765865—24no—
116#11765865—24no—
117#11765865—42no—
118#11765865—44no—
119#11765865—42no—
120C2×Dic142—24no—
121C2×Dic142—21no—
122C2×Dic142C22×C412yes—
123C2×D284—24no—
124C2×D284—21no—
125C2×D284C22×C412yes—
126C2×C22⋊D71—22no—
127C2×C22⋊D71—24no—
128C2×C22⋊D71—24no—
129C2×C22⋊D71—22no—
130C2×C22⋊D71—48no—
131C2×C22⋊D71—42no—
132C2×C22⋊D71—42no—
133C22×Dic71—21no—
134C22×Dic71—22no—
135C2×C4×D71—22no—
136C2×C4×D71—22no—
137C2×C4×D71—44no—
138#11765849—24no—
139#11765849—24no—
140#11765849—44no—
141C2×C71⋊C8—24no—
142C2×C71⋊C8—44no—
143C4⋊Dic71—22no—
144C4⋊Dic71C22×C414yes—
145Dic71⋊C4—24no—
146Dic71⋊C4—24no—
147C22⋊Dic71—22no—
148C22⋊Dic71—24no—
149C22⋊Dic71—24no—
150C22⋊Dic71—22no—
151(C2×D71)⋊C4—24no—
152(C2×D71)⋊C4—24no—
153C4×Dic71—22no—
154C4×Dic71—22no—
155C24×C71—22no—
156C24×C71—22no—
157C24×C71—44no—
158C24×C71C22⋊C412yes—
159C24×C71C22⋊C412yes—
160C24×C71C2×D411yescharacteristic
161C24×C71C2×D411yescharacteristic
162C24×C71C2×D411yescharacteristic
163C22×C4×C71—22no—
164C22×C4×C71—22no—
165C22×C4×C71—28no—
166C22×C4×C71—28no—
167C22×C4×C71—22no—
168C22×C4×C71—22no—
169C22×C4×C71—28no—
170C22×C4×C71—28no—
171C22×C4×C71—22no—
172C22×C4×C71—22no—
173C22×C4×C71—22no—
174C22×C4×C71—22no—
175C22×C4×C71—22no—
176C22×C4×C71—22no—
177C22×C4×C71—28no—
178C22×C4×C71—24no—
179C22×C4×C71—48no—
180C22×C4×C71C22⋊C412yes—
181C22×C4×C71C22⋊C412yes—
182C22×C4×C71C22⋊C412yes—
183C22×C4×C71C22⋊C412yes—
184C22×C4×C71C22⋊C412yes—
185C22×C4×C71C22⋊C412yes—
186C22×C4×C71C22×C412yes—
187C22×C4×C71C22×C412yes—
188C22×C4×C71C22×C411yescharacteristic
189C22×C4×C71C22×C411yescharacteristic
190C22×C4×C71C2×D411yescharacteristic
191C22×C4×C71C2×D411yescharacteristic
192C22×C4×C71C2×D411yescharacteristic
193C22×C4×C71C2×D414yes—
194C22×C4×C71C2×D412yes—
195C22×C4×C71C2×D412yes—
196C22×C4×C71C2×D411yescharacteristic
197C22×C4×C71C2×D411yescharacteristic
198C22×C4×C71#4114yes—
199C22×C4×C71C2411yescharacteristic
200C4⋊C4×C71—28no—
201C4⋊C4×C71—28no—
202C22⋊C4×C71—28no—
203C22⋊C4×C71—28no—
204C42×C71—28no—
205C42×C71—24no—
206C42×C71C2×D418yescomplemented
207C23×C71—21no—
208C23×C71—21no—
209C23×C71—24no—
210C23×C71—21no—
211C23×C71—21no—
212C23×C71—24no—
213C23×C71—24no—
214C23×C71—24no—
215C23×C71—24no—
216C23×C71—42no—
217C23×C71—42no—
218C23×C71—42no—
219C23×C71—42no—
220C23×C71—48no—
221C23×C71—48no—
222C23×C71C2×C22⋊C412yes—
223C23×C71C2×C22⋊C411yescharacteristic
224C23×C71C2×C22⋊C411yescharacteristic
225C23×C71C2×C22⋊C412yes—
226C23×C71#12012yes—
227C23×C71#12312yes—
228C23×C71C22×D411yescharacteristic
229C22×D71—22no—
230C22×D71—44no—
231C22×D71—42no—
232C2×C4×C71—21no—
233C2×C4×C71—21no—
234C2×C4×C71—21no—
235C2×C4×C71—21no—
236C2×C4×C71—24no—
237C2×C4×C71—24no—
238C2×C4×C71—24no—
239C2×C4×C71—24no—
240C2×C4×C71—24no—
241C2×C4×C71—24no—
242C2×C4×C71—24no—
243C2×C4×C71—24no—
244C2×C4×C71—24no—
245C2×C4×C71—24no—
246C2×C4×C71—28no—
247C2×C4×C71—416no—
248C2×C4×C71—48no—
249C2×C4×C71C2×C22⋊C412yes—
250C2×C4×C71C2×C22⋊C412yes—
251C2×C4×C71C2×C22⋊C411yescharacteristic
252C2×C4×C71C2×C22⋊C411yescharacteristic
253C2×C4×C71C4×D414yes—
254C2×C4×C71C4×D414yes—
255C2×C4×C71#12012yes—
256C2×C4×C71#12114yes—
257C2×C4×C71#12114yes—
258C2×C4×C71#12312yes—
259C2×C4×C71C23×C411yescharacteristic
260C2×C4×C71C22×D412yes—
261C2×C4×C71C22×D411yescharacteristic
262C2×C4×C71C2×#4112yes—
263C22⋊D71—48no—
264C22⋊D71—42no—
265C22⋊D71—42no—
266C22⋊D71—44no—
267C22⋊D71—48no—
268C2×Dic71—24no—
269C2×Dic71—22no—
270C2×Dic71—44no—
271C2×Dic71—42no—
272D284—24no—
273D284—42no—
274C4×D71—44no—
275C4×D71—44no—
276Dic142—24no—
277Dic142—42no—
278C71⋊C8—44no—
279C22×C71—24no—
280C22×C71—24no—
281C22×C71—22no—
282C22×C71—22no—
283C22×C71—22no—
284C22×C71—22no—
285C22×C71—22no—
286C22×C71—22no—
287C22×C71—22no—
288C22×C71—22no—
289C22×C71—42no—
290C22×C71—42no—
291C22×C71—48no—
292C22×C71—48no—
293C22×C71—42no—
294C22×C71—42no—
295C22×C71#38612yes—
296C22×C71#38612yes—
297C22×C71#42114yescomplemented
298C22×C71C22×C22⋊C411yescharacteristic
299C22×C71C2×#12011yescharacteristic
300C22×C71C2×#12311yescharacteristic
301C2×D71—44no—
302C2×D71—42no—
303C2×D71—84no—
304C4×C71—22no—
305C4×C71—22no—
306C4×C71—48no—
307C4×C71#38612yes—
308C4×C71#38612yes—
309C4×C71C2×C4×D412yes—
310C4×C71C2×#12112yes—
311Dic71—44no—
312Dic71—42no—
313#20979—711nomaximal, Sylow
314C2×C71—21no—
315C2×C71—22no—
316C2×C71—21no—
317C2×C71—24no—
318C2×C71—22no—
319C2×C71—42no—
320C2×C71C2×#38611yescharacteristic
321C2×C71C2×#42112yes—
322D71—82no—
323C2×#525—711no—
324C22×#117D7111yescharacteristic, complemented
325C2×#421—712no—
326C2×#407—711no—
327#1897—714no—
328#1896—714no—
329#1733—712no—
330C71#2097911yescharacteristic, complemented, Sylow
331C22×#41—711no—
332C24×C4C2×D7111yescharacteristic
333C22×M16—712no—
334#525—714no—
335#525—712no—
336#525—712no—
337#525—1422no—
338#525—1421no—
339#525—1421no—
340C2×#123—1421no—
341C2×C22⋊Q8—712no—
342C2×#121—712no—
343C2×#120—1421no—
344C2×C4×D4—1422no—
345C2×#117—21no—
346C2×#117—21no—
347C2×#117—28no—
348C2×#117—711no—
349C2×#117C2×D7112yes—
350C2×#117C2×D7112yescomplemented
351C22×C4⋊C4—21no—
352C22×C22⋊C4—21no—
353C22×C42C2×D7112yescomplemented
354#421—1428no—
355C2×#104—718no—
356#407—712no—
357#407—712no—
358#407—1422no—
359C2×C22⋊C8—1422no—
360#339—718no—
361C25C22⋊D7111yescharacteristic
362C2×#41—712no—
363C2×#41—712no—
364C2×#41—1424no—
365C2×#41—1424no—
366C2×#41—1421no—
367C2×#41—1421no—
368C22×Q8—711no—
369C22×D4—711no—
370C22×D4—1422no—
371C22×D4—1421no—
372C23×C4—24no—
373C23×C4—24no—
374C23×C4—22no—
375C23×C4—1421no—
376C23×C4C4×D7112yes—
377C23×C4C4×D7112yes—
378C23×C4D28412yes—
379C23×C4D28412yes—
380C23×C4C22⋊D7111yescharacteristic
381C23×C4C22⋊D7111yescharacteristic
382C23×C4C22⋊D7111yescharacteristic
383C23×C4C22×D7111yescharacteristic
384C23×C4C22×D7111yescharacteristic
385C23×C4C22×D7111yescharacteristic
386C2×M16—714no—
387C2×M16—1424no—
388C2×M16—1424no—
389C2×M16—1422no—
390C22×C8—1422no—
391#123—1424no—
392#123—1422no—
393#123—1422no—
394C22⋊Q8—714no—
395C22⋊Q8—714no—
396C22⋊Q8—1422no—
397C22⋊Q8—1422no—
398#121—714no—
399#121—714no—
400#121—1422no—
401#121—1422no—
402#120—1424no—
403#120—1422no—
404#120—1422no—
405C4×D4—1424no—
406C4×D4—1424no—
407C4×D4—1424no—
408C4×D4—1424no—
409#117—28no—
410#117—28no—
411#117—24no—
412#117—28no—
413#117—712no—
414#117—712no—
415#117—1421no—
416#117—1421no—
417#117C22⋊D7118yescomplemented
418C2×C4⋊C4—28no—
419C2×C4⋊C4—22no—
420C2×C4⋊C4—22no—
421C2×C4⋊C4—712no—
422C2×C4⋊C4—1422no—
423C2×C22⋊C4—28no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—1422no—
427C2×C22⋊C4—1421no—
428C2×C22⋊C4—1422no—
429C2×C42—22no—
430C2×C42—28no—
431C2×C42—1421no—
432C2×C42C22×D7114yes—
433#104—14216no—
434C22⋊C8—1424no—
435C22⋊C8—1424no—
436C24—22no—
437C24—22no—
438C24—44no—
439C24—1421no—
440C24(C2×D71)⋊C412yes—
441C24(C2×D71)⋊C412yes—
442C24C2×C22⋊D7111yescharacteristic
443C24C2×C22⋊D7111yescharacteristic
444C24C2×C22⋊D7111yescharacteristic
445#41—2842no—
446#41—2844no—
447#41—2842no—
448#41—1424no—
449#41—1424no—
450#41—1428no—
451C2×Q8—712no—
452C2×Q8—1424no—
453C2×Q8—1421no—
454C2×D4—2848no—
455C2×D4—2842no—
456C2×D4—2842no—
457C2×D4—712no—
458C2×D4—1422no—
459C2×D4—1424no—
460C2×D4—1424no—
461C2×D4—1422no—
462C2×D4—1421no—
463C2×D4—1424no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—28no—
471C22×C4—28no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—28no—
475C22×C4—28no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—28no—
479C22×C4—24no—
480C22×C4—48no—
481C22×C4—2844no—
482C22×C4—1422no—
483C22×C4—1421no—
484C22×C4—1422no—
485C22×C4—1422no—
486C22×C4(C2×D71)⋊C412yes—
487C22×C4(C2×D71)⋊C412yes—
488C22×C4(C2×D71)⋊C412yes—
489C22×C4(C2×D71)⋊C412yes—
490C22×C4(C2×D71)⋊C412yes—
491C22×C4(C2×D71)⋊C412yes—
492C22×C4C2×C4×D7112yes—
493C22×C4C2×C4×D7112yes—
494C22×C4C2×C4×D7111yescharacteristic
495C22×C4C2×C4×D7111yescharacteristic
496C22×C4C2×C22⋊D7111yescharacteristic
497C22×C4C2×C22⋊D7111yescharacteristic
498C22×C4C2×C22⋊D7114yes—
499C22×C4C2×C22⋊D7111yescharacteristic
500C22×C4C2×D28412yes—
501C22×C4C2×D28412yes—
502C22×C4C2×D28411yescharacteristic
503C22×C4C2×D28411yescharacteristic
504C22×C4#1176586514yes—
505C22×C4C23×D7111yescharacteristic
506M16—2844no—
507M16—1424no—
508M16—1424no—
509C2×C8—2844no—
510C2×C8—1424no—
511C4⋊C4—28no—
512C4⋊C4—28no—
513C4⋊C4—714no—
514C4⋊C4—1424no—
515C4⋊C4—1424no—
516C4⋊C4—1422no—
517C22⋊C4—28no—
518C22⋊C4—28no—
519C22⋊C4—1422no—
520C22⋊C4—1422no—
521C22⋊C4—1424no—
522C22⋊C4—1424no—
523C22⋊C4—1424no—
524C22⋊C4—1424no—
525C42—28no—
526C42—24no—
527C42—1422no—
528C42—1422no—
529C42C2×C22⋊D7118yescomplemented
530C23—21no—
531C23—21no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—21no—
536C23—21no—
537C23—24no—
538C23—24no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—48no—
544C23—48no—
545C23—2844no—
546C23—2842no—
547C23—1422no—
548C23C2×(C2×D71)⋊C412yes—
549C23C2×(C2×D71)⋊C411yescharacteristic
550C23C2×(C2×D71)⋊C411yescharacteristic
551C23C2×(C2×D71)⋊C412yes—
552C23#1455609712yes—
553C23#1455611112yes—
554C23C22×C22⋊D7111yescharacteristic
555Q8—2842no—
556Q8—1424no—
557D4—2848no—
558D4—2844no—
559D4—2842no—
560D4—2842no—
561D4—2848no—
562D4—2842no—
563D4—1424no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—416no—
580C2×C4—48no—
581C2×C4—2844no—
582C2×C4—2842no—
583C2×C4—2844no—
584C2×C4—2844no—
585C2×C4—1424no—
586C2×C4—1422no—
587C2×C4C2×(C2×D71)⋊C412yes—
588C2×C4C2×(C2×D71)⋊C412yes—
589C2×C4C2×(C2×D71)⋊C411yescharacteristic
590C2×C4C2×(C2×D71)⋊C411yescharacteristic
591C2×C4C4×C22⋊D7114yes—
592C2×C4C4×C22⋊D7114yes—
593C2×C4#1455609712yes—
594C2×C4#1455610214yes—
595C2×C4#1455610214yes—
596C2×C4#1455611112yes—
597C2×C4C22×C4×D7111yescharacteristic
598C2×C4C22×C22⋊D7112yes—
599C2×C4C22×D28411yescharacteristic
600C2×C4C2×#1176586512yes—
601C8—2844no—
602C22—24no—
603C22—24no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—42no—
613C22—42no—
614C22—48no—
615C22—48no—
616C22—42no—
617C22—42no—
618C22—2844no—
619C22—2842no—
620C22—5684no—
621C22#1798587812yes—
622C22#1798587812yes—
623C22#1798601714yescomplemented
624C22C22×(C2×D71)⋊C411yescharacteristic
625C22C2×#1455609711yescharacteristic
626C22C2×#1455611111yescharacteristic
627C4—22no—
628C4—22no—
629C4—48no—
630C4—2844no—
631C4—2842no—
632C4#1798587812yes—
633C4#1798587812yes—
634C4C2×C4×C22⋊D7112yes—
635C4C2×#1455610212yes—
636C2—21no—
637C2—21no—
638C2—24no—
639C2—22no—
640C2—22no—
641C2—42no—
642C2—5682no—
643C2C2×#1798587811yescharacteristic
644C2C2×#1798601712yes—
subgroupG/H
centerC2×C4C22×D284
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC22×#117×C71C2
socleC22×C71C22×C22⋊C4
prime radicalC2×#17986510C2
cosocleC22×C4×C71C24
hypercenterC22×#117D71
hypocenterC71#20979
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. C2×#17986017C2semidirect
3. C2×#17985963C2semidirect
4. #22588788C2semidirect
5. #22588783C2semidirect
6. #22588050C2semidirect
7. C22×#117×C71C2semidirect
13. #17986510C4semidirect
17. C2×#14556107C4semidirect
18. C2×#14556102C4semidirect
23. C2×#14556056C22semidirect
25. #17985963C22semidirect
26. #17985963C22semidirect
28. #17985759C22semidirect
34. C2×#117×C71C22semidirect
37. C22×C42×C71C22semidirect
42. C2×#11765865C2×C4semidirect
44. C22×Dic142C2×C4semidirect
45. C22×D284C2×C4semidirect
60. #14556107C2×C4semidirect
64. #14556102C2×C4semidirect
74. #14556087C2×C4semidirect
76. C2×C4⋊Dic71C2×C4semidirect
103. #117×C71D4semidirect
206. C42×C71C2×D4semidirect
297. C22×C71#421semidirect
324. C22×#117D71semidirect
330. C71#20979semidirect
350. C2×#117C2×D71semidirect
353. C22×C42C2×D71semidirect
417. #117C22⋊D71semidirect
529. C42C2×C22⋊D71semidirect
623. C22#17986017semidirect

Maximal subgroups

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

subgroupindex
C2×#179865102
C2×#179860172
C2×#179859632
#225887882
#225887832
#225880502
C22×#117×C712
#2097971