Finite Group Databasebeta

← #31998341

Subgroups — #31998341

Order 18176.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986042C211yesmaximal, characteristic, complemented
2C2×#17986016C211yesmaximal, characteristic
3C2×#17986000C211yesmaximal, characteristic
4#22590960C211yesmaximal, characteristic, complemented
5#22588573C214yesmaximal, complemented
6#22588566C214yesmaximal, complemented
7#22588079C211yesmaximal, characteristic, complemented
8#22588049C211yesmaximal, characteristic, complemented
9C71×#3428C211yesmaximal, characteristic
10C22×#11765849C2211yescharacteristic
11C22×#11765849C2211yescharacteristic
12C2×#14556087C2211yescharacteristic
13C22×C4⋊Dic71C2211yescharacteristic
14#17986042—22no—
15#17986042C2214yescomplemented
16C2×#14556061—21no—
17C2×#14556059—21no—
18#17986016C418yescomplemented
19C2×#14556055C2214yes—
20#17986000—22no—
21#17986000C2214yes—
22C2×#14556052—21no—
23C2×#14556049—21no—
24#17985897—22no—
25#17985871—22no—
26#17985868C2212yescomplemented
27#17985864C2212yescomplemented
28#17985768C2212yescomplemented
29#17985768C2212yescomplemented
30#17985758C2214yescomplemented
31C22×#41×C71C2211yescharacteristic
32#520×C71—21no—
33#520×C71—21no—
34#520×C71C2212yes—
35#520×C71C2212yes—
36#519×C71—21no—
37#518×C71—21no—
38#518×C71—22no—
39C4×#41×C71—22no—
40C4×#41×C71—22no—
41C2×#117×C71—22no—
42C2×#117×C71C2211yescharacteristic
43C2×#117×C71C2211yescharacteristic
44C23×Dic71—21no—
45C2×#11765849—21no—
46C2×#11765849—21no—
47C2×#11765849—22no—
48C2×#11765849—22no—
49C2×#11765849—22no—
50C2×#11765849C2×C414yescomplemented
51C2×#11765849C2312yes—
52C2×#11765849C2312yes—
53C22×C71⋊C8—21no—
54C22×C71⋊C8—21no—
55#14556087—22no—
56#14556087C2×C414yescomplemented
57C2×C4⋊Dic71—21no—
58C2×C4⋊Dic71—21no—
59C2×C4⋊Dic71—21no—
60C2×C4⋊Dic71—21no—
61C2×C4⋊Dic71—21no—
62C2×C4⋊Dic71—21no—
63C2×C4⋊Dic71C2312yes—
64C2×Dic71⋊C4—24no—
65C2×C22⋊Dic71—21no—
66C2×C4×Dic71C2312yes—
67#14556061—24no—
68#14556059—24no—
69#14556055C2×C4116yescomplemented
70#14556052—24no—
71#14556049—24no—
72C71⋊#95—24no—
73C2×#41×C71—21no—
74C2×#41×C71—21no—
75C2×#41×C71—22no—
76C2×#41×C71—22no—
77C2×#41×C71C2×C414yes—
78C2×#41×C71D412yes—
79C2×#41×C71Q814yes—
80C2×#41×C71C2312yes—
81C22×Q8×C71D411yescharacteristic
82C22×D4×C71—21no—
83C22×D4×C71D411yescharacteristic
84C23×C4×C71—21no—
85C23×C4×C71C2311yescharacteristic
86C4×Q8×C71—22no—
87C4×Q8×C71—22no—
88C4×Q8×C71—22no—
89C4×Q8×C71—22no—
90C4×D4×C71—24no—
91C4×D4×C71—24no—
92C4×D4×C71—22no—
93C4×D4×C71—22no—
94C4×D4×C71—22no—
95C4×D4×C71—22no—
96C4×D4×C71—24no—
97C4×D4×C71—24no—
98#117×C71—24no—
99#117×C71—24no—
100#117×C71—21no—
101#117×C71—21no—
102#117×C71—21no—
103#117×C71—21no—
104#117×C71—21no—
105#117×C71—21no—
106#117×C71—22no—
107#117×C71—22no—
108#117×C71D412yes—
109#117×C71C2312yes—
110C2×C4⋊C4×C71—22no—
111C2×C4⋊C4×C71—22no—
112C2×C4⋊C4×C71D411yescharacteristic
113C2×C4⋊C4×C71D411yescharacteristic
114C2×C4⋊C4×C71C2311yescharacteristic
115C2×C4⋊C4×C71C2311yescharacteristic
116C2×C22⋊C4×C71—24no—
117C2×C22⋊C4×C71—21no—
118C2×C22⋊C4×C71D412yes—
119C2×C42×C71—24no—
120C2×C42×C71—21no—
121C2×C42×C71D412yes—
122C22×Dic71—22no—
123C22×Dic71—22no—
124C22×Dic71—41no—
125C22×Dic71—41no—
126C22×Dic71—82no—
127#11765849—24no—
128#11765849—22no—
129#11765849—22no—
130#11765849—22no—
131#11765849—42no—
132#11765849C22×C414yes—
133C2×C71⋊C8—24no—
134C2×C71⋊C8—22no—
135C2×C71⋊C8—22no—
136C2×C71⋊C8—42no—
137C4⋊Dic71—24no—
138C4⋊Dic71—22no—
139C4⋊Dic71—22no—
140C4⋊Dic71—41no—
141C4⋊Dic71—41no—
142Dic71⋊C4—84no—
143C22⋊Dic71—24no—
144C4×Dic71—22no—
145C4×Dic71C22×C414yes—
146C24×C71—21no—
147C24×C71C2×D411yescharacteristic
148#41×C71—24no—
149#41×C71—24no—
150#41×C71—24no—
151#41×C71—24no—
152#41×C71—24no—
153#41×C71—22no—
154#41×C71—22no—
155#41×C71C4⋊C418yescomplemented
156#41×C71C22×C414yes—
157#41×C71C2×Q814yes—
158C2×Q8×C71—21no—
159C2×Q8×C71—22no—
160C2×Q8×C71C4⋊C414yescomplemented
161C2×Q8×C71C2×D412yes—
162C2×D4×C71—24no—
163C2×D4×C71—22no—
164C2×D4×C71—24no—
165C2×D4×C71—22no—
166C2×D4×C71—21no—
167C2×D4×C71—22no—
168C2×D4×C71C4⋊C414yescomplemented
169C2×D4×C71C2×D412yes—
170C22×C4×C71—21no—
171C22×C4×C71—24no—
172C22×C4×C71—22no—
173C22×C4×C71—24no—
174C22×C4×C71—22no—
175C22×C4×C71—24no—
176C22×C4×C71—24no—
177C22×C4×C71—21no—
178C22×C4×C71—21no—
179C22×C4×C71C22×C412yes—
180C22×C4×C71C2×D412yes—
181C22×C4×C71C2×D412yes—
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71C2×D411yescharacteristic
184C22×C4×C71C2×D411yescharacteristic
185C22×C4×C71C2×D411yescharacteristic
186C22×C4×C71C2×D411yescharacteristic
187C22×C4×C71C2×Q812yes—
188C22×C4×C71#4112yes—
189C22×C4×C71#4112yes—
190C22×C4×C71C2411yescharacteristic
191C4⋊C4×C71—24no—
192C4⋊C4×C71—24no—
193C4⋊C4×C71—21no—
194C4⋊C4×C71—21no—
195C4⋊C4×C71—21no—
196C4⋊C4×C71—21no—
197C4⋊C4×C71—21no—
198C4⋊C4×C71—21no—
199C4⋊C4×C71C2×D412yes—
200C4⋊C4×C71C2×D412yes—
201C22⋊C4×C71—24no—
202C22⋊C4×C71—24no—
203C22⋊C4×C71—22no—
204C22⋊C4×C71—22no—
205C22⋊C4×C71—22no—
206C22⋊C4×C71—22no—
207C42×C71—24no—
208C42×C71—24no—
209C42×C71—22no—
210C42×C71—22no—
211C42×C71—22no—
212C42×C71—22no—
213C23×C71—22no—
214C23×C71—24no—
215C23×C71—41no—
216C23×C71—41no—
217C23×C71—81no—
218C23×C71—81no—
219C23×C71C4×D412yes—
220C23×C71#12011yescharacteristic
221C23×C71#12011yescharacteristic
222C23×C71C22⋊Q812yes—
223C23×C71C22×D411yescharacteristic
224Q8×C71—22no—
225Q8×C71—24no—
226Q8×C71C2×C4⋊C414yes—
227D4×C71—24no—
228D4×C71—24no—
229D4×C71—28no—
230D4×C71—22no—
231D4×C71—24no—
232D4×C71C2×C4⋊C414yes—
233C2×C4×C71—22no—
234C2×C4×C71—24no—
235C2×C4×C71—24no—
236C2×C4×C71—24no—
237C2×C4×C71—28no—
238C2×C4×C71—22no—
239C2×C4×C71—22no—
240C2×C4×C71—22no—
241C2×C4×C71—22no—
242C2×C4×C71—42no—
243C2×C4×C71—42no—
244C2×C4×C71—81no—
245C2×C4×C71—81no—
246C2×C4×C71—84no—
247C2×C4×C71—84no—
248C2×C4×C71C2×C4⋊C412yes—
249C2×C4×C71C2×C4⋊C412yes—
250C2×C4×C71C4×D412yes—
251C2×C4×C71C4×D412yes—
252C2×C4×C71C4×D412yes—
253C2×C4×C71#12011yescharacteristic
254C2×C4×C71#12011yescharacteristic
255C2×C4×C71C22⋊Q812yes—
256C2×C4×C71C22⋊Q812yes—
257C2×C4×C71C22⋊Q812yes—
258C2×C4×C71#12312yes—
259C2×C4×C71#12312yes—
260C2×C4×C71C23×C411yescharacteristic
261C2×C4×C71C22×D411yescharacteristic
262C2×C4×C71C22×D411yescharacteristic
263C2×C4×C71C22×Q811yescharacteristic
264C2×C4×C71C2×#4111yescharacteristic
265C2×C4×C71C2×#4111yescharacteristic
266C2×Dic71—24no—
267C2×Dic71—42no—
268C2×Dic71—42no—
269C2×Dic71—44no—
270C2×Dic71—42no—
271C2×Dic71—82no—
272C2×Dic71—82no—
273C71⋊C8—24no—
274C71⋊C8—42no—
275C22×C71—21no—
276C22×C71—21no—
277C22×C71—21no—
278C22×C71—21no—
279C22×C71—21no—
280C22×C71—21no—
281C22×C71—24no—
282C22×C71—42no—
283C22×C71—42no—
284C22×C71—42no—
285C22×C71—42no—
286C22×C71—84no—
287C22×C71#38512yes—
288C22×C71#38512yes—
289C22×C71C2×C4×D411yescharacteristic
290C22×C71C2×#12011yescharacteristic
291C22×C71C2×C22⋊Q811yescharacteristic
292C4×C71—24no—
293C4×C71—42no—
294C4×C71—42no—
295C4×C71—42no—
296C4×C71—42no—
297C4×C71#38512yes—
298C4×C71#38512yes—
299C4×C71C22×C4⋊C411yescharacteristic
300C4×C71C2×C4×D411yescharacteristic
301C4×C71C2×C22⋊Q811yescharacteristic
302C4×C71C2×#12311yescharacteristic
303Dic71—44no—
304Dic71—82no—
305#20865—711nomaximal, Sylow
306C2×C71—21no—
307C2×C71—22no—
308C2×C71—21no—
309C2×C71—22no—
310C2×C71—44no—
311C2×C71C2×#38511yescharacteristic
312C2×C71#302111yescharacteristic
313C2×C71#302211yescharacteristic
314#3428D7111yescharacteristic
315C2×#428—711no—
316C2×#421—711no—
317C2×#417—711no—
318#2317—711no—
319#1860—714no—
320#1859—714no—
321#1742—711no—
322#1733—711no—
323C71#2086511yescharacteristic, complemented, Sylow
324C22×#41C2×D7111yescharacteristic
325C22×M16—711no—
326C22×M16—711no—
327#520—21no—
328#520—21no—
329#520C2×D7112yes—
330#520C2×D7112yes—
331#519—21no—
332#518—21no—
333#518—22no—
334C4×#41—22no—
335C4×#41—22no—
336C2×#117—22no—
337C2×#117—711no—
338C2×#117C2×D7111yescharacteristic
339C2×#117C2×D7111yescharacteristic
340C22×C4⋊C4—711no—
341#428—714no—
342#428—1422no—
343C2×#107—1421no—
344C2×#106—1421no—
345#421—718no—
346C2×#104—714no—
347#417—714no—
348#417—1422no—
349C2×#103—1421no—
350C2×#102—1421no—
351#389—1422no—
352#385—1422no—
353#384—712no—
354#382—712no—
355#343—712no—
356#343—712no—
357#339—714no—
358C2×#41—21no—
359C2×#41—21no—
360C2×#41—22no—
361C2×#41—22no—
362C2×#41Dic14214yes—
363C2×#41C4×D7114yes—
364C2×#41C22⋊D7112yes—
365C2×#41C22×D7112yes—
366C22×Q8C22⋊D7111yescharacteristic
367C22×D4—21no—
368C22×D4C22⋊D7111yescharacteristic
369C23×C4—21no—
370C23×C4—1421no—
371C23×C4C22×D7111yescharacteristic
372C2×M16—714no—
373C2×M16—712no—
374C2×M16—712no—
375C2×M16—1422no—
376C2×M16—1422no—
377C2×M16—1421no—
378C2×M16—1421no—
379C2×M16—1422no—
380C22×C8—1421no—
381C22×C8—1421no—
382C4×Q8—22no—
383C4×Q8—22no—
384C4×Q8—22no—
385C4×Q8—22no—
386C4×D4—24no—
387C4×D4—24no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—24no—
393C4×D4—24no—
394#117—24no—
395#117—24no—
396#117—21no—
397#117—21no—
398#117—21no—
399#117—21no—
400#117—21no—
401#117—21no—
402#117—22no—
403#117—22no—
404#117—714no—
405#117—1422no—
406#117C22⋊D7112yes—
407#117C22×D7112yes—
408C2×C4⋊C4—22no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—712no—
411C2×C4⋊C4—1421no—
412C2×C4⋊C4—1421no—
413C2×C4⋊C4—1421no—
414C2×C4⋊C4—1421no—
415C2×C4⋊C4—1421no—
416C2×C4⋊C4—1424no—
417C2×C4⋊C4—1421no—
418C2×C4⋊C4C22⋊D7111yescharacteristic
419C2×C4⋊C4C22⋊D7111yescharacteristic
420C2×C4⋊C4C22×D7111yescharacteristic
421C2×C4⋊C4C22×D7111yescharacteristic
422C2×C22⋊C4—24no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1421no—
425C2×C22⋊C4C22⋊D7112yes—
426C2×C42—24no—
427C2×C42—21no—
428C2×C42—712no—
429C2×C42C22⋊D7112yes—
430#107—1424no—
431#106—1424no—
432#104—7116no—
433#103—1424no—
434#102—1424no—
435#95—1424no—
436C24—21no—
437C24C2×C22⋊D7111yescharacteristic
438#41—22no—
439#41—22no—
440#41—24no—
441#41—24no—
442#41—24no—
443#41—24no—
444#41—24no—
445#41Dic71⋊C418yescomplemented
446#41C2×C4×D7114yes—
447#41C2×Dic14214yes—
448C2×Q8—21no—
449C2×Q8—22no—
450C2×Q8Dic71⋊C414yescomplemented
451C2×Q8C2×C22⋊D7112yes—
452C2×D4—22no—
453C2×D4—24no—
454C2×D4—22no—
455C2×D4—24no—
456C2×D4—21no—
457C2×D4—22no—
458C2×D4Dic71⋊C414yescomplemented
459C2×D4C2×C22⋊D7112yes—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—2841no—
470C22×C4—2841no—
471C22×C4—5682no—
472C22×C4—1422no—
473C22×C4—1422no—
474C22×C4C2×C4×D7112yes—
475C22×C4C2×C22⋊D7112yes—
476C22×C4C2×C22⋊D7112yes—
477C22×C4C2×C22⋊D7111yescharacteristic
478C22×C4C2×C22⋊D7111yescharacteristic
479C22×C4C2×C22⋊D7111yescharacteristic
480C22×C4C2×C22⋊D7111yescharacteristic
481C22×C4C2×C22⋊D7111yescharacteristic
482C22×C4C2×Dic14212yes—
483C22×C4#1176586512yes—
484C22×C4#1176586512yes—
485C22×C4C23×D7111yescharacteristic
486M16—2842no—
487M16—714no—
488M16—1422no—
489M16—1422no—
490M16—1422no—
491M16—1424no—
492C2×C8—2842no—
493C2×C8—1422no—
494C2×C8—1422no—
495C2×C8—1424no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—2841no—
505C4⋊C4—2841no—
506C4⋊C4—5684no—
507C4⋊C4—1422no—
508C4⋊C4—1422no—
509C4⋊C4—1424no—
510C4⋊C4C2×C22⋊D7112yes—
511C4⋊C4C2×C22⋊D7112yes—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—1424no—
519C42—24no—
520C42—24no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—714no—
526C42—1422no—
527C23—22no—
528C23—24no—
529C23—41no—
530C23—41no—
531C23—81no—
532C23—81no—
533C23C4×C22⋊D7112yes—
534C23#1455609711yescharacteristic
535C23#1455609711yescharacteristic
536C23#1455610712yes—
537C23C22×C22⋊D7111yescharacteristic
538Q8—24no—
539Q8—22no—
540Q8C2×Dic71⋊C414yes—
541D4—28no—
542D4—24no—
543D4—24no—
544D4—24no—
545D4—22no—
546D4C2×Dic71⋊C414yes—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—2842no—
563C2×C4—2842no—
564C2×C4—2844no—
565C2×C4—2842no—
566C2×C4—5682no—
567C2×C4—5682no—
568C2×C4—1424no—
569C2×C4C2×Dic71⋊C412yes—
570C2×C4C2×Dic71⋊C412yes—
571C2×C4C4×C22⋊D7112yes—
572C2×C4C4×C22⋊D7112yes—
573C2×C4C4×C22⋊D7112yes—
574C2×C4#1455609711yescharacteristic
575C2×C4#1455609711yescharacteristic
576C2×C4#1455610712yes—
577C2×C4#1455610712yes—
578C2×C4#1455610712yes—
579C2×C4#1455611112yes—
580C2×C4#1455611112yes—
581C2×C4C22×C4×D7111yescharacteristic
582C2×C4C22×C22⋊D7111yescharacteristic
583C2×C4C22×C22⋊D7111yescharacteristic
584C2×C4C22×Dic14211yescharacteristic
585C2×C4C2×#1176586511yescharacteristic
586C2×C4C2×#1176586511yescharacteristic
587C8—2842no—
588C8—1424no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—24no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—84no—
601C22#1798587212yes—
602C22#1798587212yes—
603C22C2×C4×C22⋊D7111yescharacteristic
604C22C2×#1455609711yescharacteristic
605C22C2×#1455610711yescharacteristic
606C4—24no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—2844no—
612C4—5682no—
613C4#1798587212yes—
614C4#1798587212yes—
615C4C22×Dic71⋊C411yescharacteristic
616C4C2×C4×C22⋊D7111yescharacteristic
617C4C2×#1455610711yescharacteristic
618C4C2×#1455611111yescharacteristic
619C2—21no—
620C2—22no—
621C2—22no—
622C2—21no—
623C2—44no—
624C2C2×#1798587211yescharacteristic
625C2#2259771211yescharacteristic
626C2#2259771911yescharacteristic
subgroupG/H
centerC22C2×#14556097
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#3428C2
socleC22×C71C2×#120
prime radicalC22×#41×C71C22
cosocleC22×C4×C71C24
hypercenter#3428D71
hypocenterC71#20865
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3428

Complements

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

NH ≅ G/Ntype
1. C2×#17986042C2semidirect
4. #22590960C2semidirect
5. #22588573C2semidirect
6. #22588566C2semidirect
7. #22588079C2semidirect
8. #22588049C2semidirect
15. #17986042C22semidirect
18. #17986016C4semidirect
26. #17985868C22semidirect
27. #17985864C22semidirect
28. #17985768C22semidirect
29. #17985768C22semidirect
30. #17985758C22semidirect
50. C2×#11765849C2×C4semidirect
56. #14556087C2×C4semidirect
69. #14556055C2×C4semidirect
155. #41×C71C4⋊C4semidirect
160. C2×Q8×C71C4⋊C4semidirect
168. C2×D4×C71C4⋊C4semidirect
323. C71#20865semidirect
445. #41Dic71⋊C4semidirect
450. C2×Q8Dic71⋊C4semidirect
458. C2×D4Dic71⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#179860422
C2×#179860162
C2×#179860002
#225909602
#225885732
#225885662
#225880792
#225880492
C71×#34282
#2086571