Finite Group Databasebeta

← #31994865

Subgroups — #31994865

Order 18176.

2158 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556136C211yesmaximal, characteristic
2C2×#17986480C211yesmaximal, characteristic
3C2×#17986479C211yesmaximal, characteristic
4#22588060C211yesmaximal, characteristic, complemented
5#22588058C212yesmaximal, complemented
6#22588058C212yesmaximal, complemented
7#22588056C211yesmaximal, characteristic, complemented
8#22588055C212yesmaximal, complemented
9C2×#17985761C211yesmaximal, characteristic, complemented
10#340×D71C212yesmaximal, complemented
11C71×#1736C211yesmaximal, characteristic, complemented
12C22×#11765865C2211yescharacteristic
13C2×#14556136—24no—
14C2×#14556136—24no—
15C2×#14556136C418yescomplemented
16C22×#11765848C2212yes—
17C23×C71⋊C8C2211yescharacteristic
18C22×C8×D71C2212yes—
19#17986480—24no—
20#17986480C418yescomplemented
21#17986479—24no—
22#17986479C418yescomplemented
23C2×C4×Dic142—21no—
24C2×#14556093—21no—
25C2×C4×D284—21no—
26C2×#14556090—21no—
27C2×C4×C22⋊D71—22no—
28C2×(C22⋊D71)⋊C4—22no—
29C2×#14556087C2211yescharacteristic
30C2×#14556086C2212yes—
31C2×#14556086C2212yes—
32C22×C4⋊Dic71C2211yescharacteristic
33C2×C4⋊C4×D71C2212yes—
34C2×C4⋊C4×D71C2212yes—
35#17985762C2212yes—
36#17985762C2212yes—
37#17985761C2211yescharacteristic
38#17985761C2212yes—
39#17985761C2212yes—
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43C22×M16×C71C2211yescharacteristic
44C2×#117×C71C2211yescharacteristic
45C22×C4⋊C4×C71C2211yescharacteristic
46#340×C71C2212yes—
47#340×C71C2212yes—
48C2×#11765865C2×C414yes—
49C2×#11765865C2×C414yes—
50C2×#11765865C2×C414yes—
51C2×#11765865D414yes—
52C2×#11765865D414yes—
53C2×#11765865Q814yes—
54C22×Dic142D411yescharacteristic
55C22×D284D411yescharacteristic
56C22×C22⋊D71D412yes—
57C23×Dic71—21no—
58C22×C4×D71—22no—
59C22×C4×D71—22no—
60C22×C4×D71C2312yes—
61#14556136—232no—
62C2×#11765848—24no—
63C2×#11765848—24no—
64C2×#11765848C2×C418yescomplemented
65C22×C71⋊C8—22no—
66C22×C71⋊C8—22no—
67C22×C71⋊C8C2×C414yescomplemented
68C22×C71⋊C8C2311yescharacteristic
69C22×C71⋊C8C2311yescharacteristic
70C2×C8×D71—24no—
71C2×C8×D71—24no—
72C2×C8×D71C2×C418yescomplemented
73C4×Dic142—28no—
74#14556093—28no—
75C4×D284—28no—
76#14556090—28no—
77C4×C22⋊D71—216no—
78(C22⋊D71)⋊C4—216no—
79#14556087—22no—
80#14556087C2×C414yescomplemented
81#14556086—24no—
82#14556086—24no—
83#14556086C2×C418yescomplemented
84#14556086C2×C418yescomplemented
85C2×C4⋊Dic71—21no—
86C2×C4⋊Dic71—22no—
87C2×C4⋊Dic71C2×C414yescomplemented
88C2×C4⋊Dic71C2311yescharacteristic
89C2×C4⋊Dic71C2312yes—
90C2×C4⋊Dic71C2311yescharacteristic
91C2×Dic71⋊C4—22no—
92C2×Dic71⋊C4—22no—
93C4⋊C4×D71—24no—
94C4⋊C4×D71—24no—
95C4⋊C4×D71C2×C418yescomplemented
96C4⋊C4×D71C2×C418yescomplemented
97C2×C22⋊Dic71—21no—
98C2×(C2×D71)⋊C4—22no—
99C2×(C2×D71)⋊C4—22no—
100C2×C4×Dic71—22no—
101C2×C4×Dic71—22no—
102C2×C4×Dic71—21no—
103C23×C4×C71—21no—
104C23×C4×C71C2311yescharacteristic
105C2×M16×C71—22no—
106C2×M16×C71—22no—
107C2×M16×C71C2×C414yescomplemented
108C22×C8×C71C2312yes—
109#117×C71—22no—
110#117×C71C2×C414yescomplemented
111C2×C4⋊C4×C71—21no—
112C2×C4⋊C4×C71—22no—
113C2×C4⋊C4×C71C2×C414yescomplemented
114C2×C4⋊C4×C71C2312yes—
115C2×C4⋊C4×C71C2311yescharacteristic
116C2×C4⋊C4×C71C2311yescharacteristic
117C2×C22⋊C4×C71—21no—
118C2×C42×C71—21no—
119C23×D71C2×D412yes—
120#11765865C42116yes—
121#11765865C22⋊C4116yes—
122#11765865C4⋊C4116yes—
123#11765865C4⋊C4116yes—
124C2×Dic142C22⋊C414yes—
125C2×Dic142C22⋊C414yes—
126C2×Dic142C4⋊C414yes—
127C2×D284C22⋊C414yes—
128C2×D284C22⋊C414yes—
129C2×D284C4⋊C414yes—
130C2×C22⋊D71C22⋊C418yes—
131C2×C22⋊D71C22⋊C418yes—
132C2×C22⋊D71C4⋊C418yes—
133C22×Dic71—21no—
134C22×Dic71—22no—
135C22×Dic71—21no—
136C22×Dic71—24no—
137C22×Dic71—81no—
138C22×Dic71—81no—
139C22×Dic71C2×D412yes—
140C2×C4×D71—28no—
141C2×C4×D71—28no—
142C2×C4×D71—82no—
143C2×C4×D71—82no—
144C2×C4×D71—82no—
145C2×C4×D71—82no—
146C2×C4×D71C22×C414yes—
147C2×C4×D71C22×C414yes—
148C2×C4×D71C22×C414yes—
149C2×C4×D71C2×D414yes—
150C2×C4×D71C2×D414yes—
151C2×C4×D71C2×Q814yes—
152#11765848—216no—
153C2×C71⋊C8—28no—
154C2×C71⋊C8—21no—
155C2×C71⋊C8—21no—
156C2×C71⋊C8—21no—
157C2×C71⋊C8—21no—
158C2×C71⋊C8C22×C412yes—
159C2×C71⋊C8C22×C412yes—
160C8×D71—216no—
161C4⋊Dic71—24no—
162C4⋊Dic71—21no—
163C4⋊Dic71—22no—
164C4⋊Dic71—21no—
165C4⋊Dic71C22×C412yes—
166C4⋊Dic71C22×C414yes—
167C4⋊Dic71C22×C412yes—
168Dic71⋊C4—28no—
169Dic71⋊C4—28no—
170C22⋊Dic71—24no—
171(C2×D71)⋊C4—28no—
172(C2×D71)⋊C4—28no—
173C4×Dic71—24no—
174C4×Dic71—28no—
175C4×Dic71—28no—
176C24×C71C2×D411yescharacteristic
177C22×C4×C71—21no—
178C22×C4×C71—22no—
179C22×C4×C71—21no—
180C22×C4×C71—24no—
181C22×C4×C71—81no—
182C22×C4×C71—81no—
183C22×C4×C71C22×C412yes—
184C22×C4×C71C22×C412yes—
185C22×C4×C71C22×C412yes—
186C22×C4×C71C2×D411yescharacteristic
187C22×C4×C71C2×D412yes—
188C22×C4×C71C2×D412yes—
189C22×C4×C71C2×Q812yes—
190C22×C4×C71C2411yescharacteristic
191M16×C71—28no—
192C2×C8×C71—22no—
193C2×C8×C71—22no—
194C2×C8×C71C22×C414yes—
195C4⋊C4×C71—24no—
196C4⋊C4×C71—22no—
197C4⋊C4×C71—21no—
198C4⋊C4×C71—21no—
199C4⋊C4×C71C22×C414yes—
200C4⋊C4×C71C22×C412yes—
201C4⋊C4×C71C22×C412yes—
202C22⋊C4×C71—24no—
203C42×C71—24no—
204C23×C71—21no—
205C23×C71—21no—
206C23×C71—21no—
207C23×C71—21no—
208C23×C71C2×C22⋊C412yes—
209C23×C71C2×C22⋊C412yes—
210C23×C71C2×C4⋊C412yes—
211C23×C71C22×D411yescharacteristic
212C22×D71—22no—
213C22×D71—22no—
214C22×D71—22no—
215C22×D71—22no—
216C22×D71C2×C22⋊C414yes—
217C22×D71C2×C22⋊C414yes—
218C22×D71C2×C4⋊C414yes—
219C2×C4×C71—22no—
220C2×C4×C71—22no—
221C2×C4×C71—24no—
222C2×C4×C71—42no—
223C2×C4×C71—41no—
224C2×C4×C71—41no—
225C2×C4×C71—42no—
226C2×C4×C71—41no—
227C2×C4×C71—41no—
228C2×C4×C71—84no—
229C2×C4×C71C2×C4214yes—
230C2×C4×C71C2×C22⋊C414yes—
231C2×C4×C71C2×C22⋊C412yes—
232C2×C4×C71C2×C22⋊C412yes—
233C2×C4×C71C2×C4⋊C414yes—
234C2×C4×C71C2×C4⋊C414yes—
235C2×C4×C71C2×C4⋊C412yes—
236C2×C4×C71C23×C411yescharacteristic
237C2×C4×C71C23×C411yescharacteristic
238C2×C4×C71C23×C411yescharacteristic
239C2×C4×C71C22×D411yescharacteristic
240C2×C4×C71C22×D411yescharacteristic
241C2×C4×C71C22×Q811yescharacteristic
242C22⋊D71#95132yes—
243C2×Dic71—22no—
244C2×Dic71—24no—
245C2×Dic71—22no—
246C2×Dic71—41no—
247C2×Dic71—42no—
248C2×Dic71—41no—
249C2×Dic71—42no—
250C2×Dic71—41no—
251C2×Dic71—41no—
252C2×Dic71—84no—
253C2×Dic71C2×C22⋊C414yes—
254C2×Dic71C2×C22⋊C414yes—
255C2×Dic71C2×C4⋊C414yes—
256D284#95116yes—
257C4×D71—88no—
258C4×D71—88no—
259C4×D71C2×C4218yes—
260C4×D71C2×C22⋊C418yes—
261C4×D71C2×C4⋊C418yes—
262C4×D71C2×C4⋊C418yes—
263Dic142#95116yes—
264C8×C71—24no—
265C71⋊C8—22no—
266C71⋊C8—22no—
267C22×C71—22no—
268C22×C71—22no—
269C22×C71—22no—
270C22×C71—22no—
271C22×C71—22no—
272C22×C71—22no—
273C22×C71C2×#9514yes—
274C22×C71C22×C22⋊C411yescharacteristic
275C22×C71C22×C22⋊C411yescharacteristic
276C22×C71C22×C4⋊C411yescharacteristic
277C22×C71#57611yescharacteristic
278C22×C71#57711yescharacteristic
279C22×C71#57711yescharacteristic
280C22×C71#57811yescharacteristic
281C2×D71—24no—
282C2×D71—24no—
283C2×D71—24no—
284C2×D71—24no—
285C2×D71—24no—
286C2×D71—24no—
287C2×D71C2×#9518yes—
288C4×C71—42no—
289C4×C71—44no—
290C4×C71—42no—
291C4×C71C2×#9514yes—
292C4×C71C22×C4211yescharacteristic
293C4×C71C22×C22⋊C411yescharacteristic
294C4×C71C22×C4⋊C411yescharacteristic
295C4×C71C22×C4⋊C411yescharacteristic
296Dic71—42no—
297Dic71—44no—
298Dic71—42no—
299Dic71C2×#9518yes—
300#20348—711nomaximal, Sylow
301C2×C71—24no—
302C2×C71C22×#9511yescharacteristic
303C2×C71#289711yescharacteristic
304C2×C71#289711yescharacteristic
305C2×C71#289811yescharacteristic
306C2×C71#289811yescharacteristic
307C2×C71#291211yescharacteristic
308C2×C71#291311yescharacteristic
309D71—28no—
310C22×#131—711no—
311C2×#520—711no—
312C2×#520—711no—
313#1736—712no—
314#1736—712no—
315#1736—711no—
316#1736D7111yescharacteristic, complemented
317#1735—712no—
318#1735—711no—
319C2×#340—712no—
320C2×#340—711no—
321C71#2034811yescharacteristic, complemented, Sylow
322C22×#41—711no—
323C2×#131—718no—
324C2×#131—1424no—
325C2×#131—1424no—
326C22×M16—712no—
327C22×M16C2×D7111yescharacteristic
328C23×C8—711no—
329C23×C8—712no—
330#520—718no—
331#520—718no—
332#520—1424no—
333#520—1424no—
334C2×C4×Q8—1421no—
335C2×C4×Q8—1421no—
336C2×C4×D4—1421no—
337C2×C4×D4—1422no—
338C2×C4×D4—1421no—
339C2×C4×D4—1422no—
340C2×#117—711no—
341C2×#117—712no—
342C2×#117—712no—
343C2×#117C2×D7111yescharacteristic
344C22×C4⋊C4—711no—
345C22×C4⋊C4—712no—
346C22×C4⋊C4—712no—
347C22×C4⋊C4C2×D7111yescharacteristic
348#340—712no—
349#340—712no—
350#340—711no—
351#340—712no—
352#340—712no—
353#340—711no—
354#340—711no—
355#340—711no—
356#340C2×D7112yes—
357#340C2×D7112yes—
358C2×#41—714no—
359C2×#41—714no—
360C2×#41—714no—
361C2×#41—714no—
362C2×#41—714no—
363C2×#41—714no—
364C22×Q8—711no—
365C22×D4—712no—
366C22×D4—711no—
367C23×C4—21no—
368C23×C4—712no—
369C23×C4—1422no—
370C23×C4—1422no—
371C23×C4—1421no—
372C23×C4C22×D7111yescharacteristic
373#131—14232no—
374C2×M16—22no—
375C2×M16—22no—
376C2×M16—718no—
377C2×M16—1424no—
378C2×M16—1424no—
379C2×M16C4×D7114yescomplemented
380C22×C8—714no—
381C22×C8—711no—
382C22×C8—711no—
383C22×C8—718no—
384C22×C8—1424no—
385C22×C8—1424no—
386C22×C8—1422no—
387C22×C8—1422no—
388C22×C8C22×D7112yes—
389C4×Q8—1428no—
390C4×Q8—1428no—
391C4×D4—1428no—
392C4×D4—14216no—
393C4×D4—1428no—
394C4×D4—14216no—
395#117—22no—
396#117—714no—
397#117—718no—
398#117—718no—
399#117—1424no—
400#117—1424no—
401#117—1422no—
402#117C4×D7114yescomplemented
403C2×C4⋊C4—21no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—714no—
406C2×C4⋊C4—712no—
407C2×C4⋊C4—711no—
408C2×C4⋊C4—711no—
409C2×C4⋊C4—718no—
410C2×C4⋊C4—718no—
411C2×C4⋊C4—1424no—
412C2×C4⋊C4—1424no—
413C2×C4⋊C4—1422no—
414C2×C4⋊C4—1422no—
415C2×C4⋊C4—1421no—
416C2×C4⋊C4—1422no—
417C2×C4⋊C4C4×D7114yescomplemented
418C2×C4⋊C4C22×D7111yescharacteristic
419C2×C4⋊C4C22×D7112yes—
420C2×C4⋊C4C22×D7111yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—1422no—
423C2×C22⋊C4—1422no—
424C2×C22⋊C4—1421no—
425C2×C42—21no—
426C2×C42—1422no—
427C2×C42—1422no—
428C2×C42—1421no—
429C24—712no—
430C24D4×D7111yescharacteristic
431#41—7116no—
432#41—7116no—
433#41—7116no—
434#41—7116no—
435C2×Q8—714no—
436C2×Q8—714no—
437C2×Q8—714no—
438C2×D4—718no—
439C2×D4—718no—
440C2×D4—718no—
441C2×D4—714no—
442C2×D4—714no—
443C2×D4—714no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4—5682no—
451C22×C4—5682no—
452C22×C4—5682no—
453C22×C4—5682no—
454C22×C4—5681no—
455C22×C4—5681no—
456C22×C4—712no—
457C22×C4—714no—
458C22×C4—714no—
459C22×C4—714no—
460C22×C4—714no—
461C22×C4—714no—
462C22×C4—714no—
463C22×C4—1428no—
464C22×C4—1428no—
465C22×C4—1421no—
466C22×C4—1422no—
467C22×C4—1421no—
468C22×C4—1424no—
469C22×C4C2×C4×D7112yes—
470C22×C4C2×C4×D7112yes—
471C22×C4C2×C4×D7112yes—
472C22×C4D4×D7112yes—
473C22×C4D4×D7112yes—
474C22×C4D4×D7111yescharacteristic
475C22×C4Q8×D7112yes—
476C22×C4C23×D7111yescharacteristic
477M16—28no—
478M16—14216no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—712no—
482C2×C8—712no—
483C2×C8—14216no—
484C2×C8—1421no—
485C2×C8—1421no—
486C2×C8—1421no—
487C2×C8—1421no—
488C2×C8—1428no—
489C2×C8C2×C4×D7114yes—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—24no—
494C4⋊C4—712no—
495C4⋊C4—714no—
496C4⋊C4—712no—
497C4⋊C4—1428no—
498C4⋊C4—1428no—
499C4⋊C4—1421no—
500C4⋊C4—1422no—
501C4⋊C4—1421no—
502C4⋊C4—1424no—
503C4⋊C4C2×C4×D7112yes—
504C4⋊C4C2×C4×D7114yes—
505C4⋊C4C2×C4×D7112yes—
506C22⋊C4—24no—
507C22⋊C4—1428no—
508C22⋊C4—1428no—
509C22⋊C4—1424no—
510C42—24no—
511C42—1428no—
512C42—1428no—
513C42—1424no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—714no—
519C23—714no—
520C23—714no—
521C23—1422no—
522C23—1422no—
523C23—1422no—
524C23—1422no—
525C23C22⋊C4×D7112yes—
526C23C22⋊C4×D7112yes—
527C23C4⋊C4×D7112yes—
528C23C2×D4×D7111yescharacteristic
529Q8—7116no—
530D4—7132no—
531D4—7116no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—84no—
542C2×C4—2842no—
543C2×C4—2841no—
544C2×C4—2842no—
545C2×C4—2841no—
546C2×C4—2841no—
547C2×C4—2841no—
548C2×C4—5688no—
549C2×C4—5688no—
550C2×C4—5684no—
551C2×C4—714no—
552C2×C4—714no—
553C2×C4—714no—
554C2×C4—718no—
555C2×C4—718no—
556C2×C4—718no—
557C2×C4—718no—
558C2×C4—1422no—
559C2×C4—1424no—
560C2×C4—1422no—
561C2×C4C42×D7114yes—
562C2×C4C22⋊C4×D7112yes—
563C2×C4C22⋊C4×D7112yes—
564C2×C4C22⋊C4×D7114yes—
565C2×C4C4⋊C4×D7112yes—
566C2×C4C4⋊C4×D7114yes—
567C2×C4C4⋊C4×D7114yes—
568C2×C4C22×C4×D7111yescharacteristic
569C2×C4C22×C4×D7111yescharacteristic
570C2×C4C22×C4×D7111yescharacteristic
571C2×C4C2×D4×D7111yescharacteristic
572C2×C4C2×D4×D7111yescharacteristic
573C2×C4C2×Q8×D7111yescharacteristic
574C8—24no—
575C8—1422no—
576C8—1422no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—718no—
584C22—1424no—
585C22—1424no—
586C22—1424no—
587C22—1424no—
588C22—1424no—
589C22—1424no—
590C22#95×D7114yes—
591C22C2×C22⋊C4×D7111yescharacteristic
592C22C2×C22⋊C4×D7111yescharacteristic
593C22C2×C4⋊C4×D7111yescharacteristic
594C22#1798680311yescharacteristic
595C22#1798680811yescharacteristic
596C22#1798680911yescharacteristic
597C22#1798681511yescharacteristic
598C4—42no—
599C4—44no—
600C4—42no—
601C4—2844no—
602C4—2842no—
603C4—2842no—
604C4—718no—
605C4#95×D7114yes—
606C4C2×C42×D7111yescharacteristic
607C4C2×C22⋊C4×D7111yescharacteristic
608C4C2×C4⋊C4×D7111yescharacteristic
609C4C2×C4⋊C4×D7111yescharacteristic
610C2—24no—
611C2—1428no—
612C2C2×#95×D7111yescharacteristic
613C2#2259686411yescharacteristic
614C2#2259686511yescharacteristic
615C2#2259687111yescharacteristic
616C2#2259687211yescharacteristic
617C2#2259693411yescharacteristic
618C2#2259693911yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC4×C71C22×C42
FrattiniC22×C4C23×D71
FittingC71×#1736C2
socleC23×C71C22×D4
prime radicalC22×#11765865C22
cosocleC22×C4×C71C24
hypercenter#1736D71
hypocenterC71#20348
LCSG ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. #22588060C2semidirect
5. #22588058C2semidirect
6. #22588058C2semidirect
7. #22588056C2semidirect
8. #22588055C2semidirect
9. C2×#17985761C2semidirect
10. #340×D71C2semidirect
11. C71×#1736C2semidirect
15. C2×#14556136C4semidirect
20. #17986480C4semidirect
22. #17986479C4semidirect
64. C2×#11765848C2×C4semidirect
67. C22×C71⋊C8C2×C4semidirect
72. C2×C8×D71C2×C4semidirect
80. #14556087C2×C4semidirect
83. #14556086C2×C4semidirect
84. #14556086C2×C4semidirect
87. C2×C4⋊Dic71C2×C4semidirect
95. C4⋊C4×D71C2×C4semidirect
96. C4⋊C4×D71C2×C4semidirect
107. C2×M16×C71C2×C4semidirect
110. #117×C71C2×C4semidirect
113. C2×C4⋊C4×C71C2×C4semidirect
316. #1736D71semidirect
321. C71#20348semidirect
379. C2×M16C4×D71semidirect
402. #117C4×D71semidirect
417. C2×C4⋊C4C4×D71semidirect

Maximal subgroups

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

subgroupindex
C22×#145561362
C2×#179864802
C2×#179864792
#225880602
#225880582
#225880562
#225880552
C2×#179857612
#340×D712
C71×#17362
#2034871