Finite Group Databasebeta

← #31998427

Subgroups — #31998427

Order 18176.

1550 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11765865C211yesmaximal, characteristic
2C2×#17986039C211yesmaximal, characteristic, complemented
3C2×#17985998C211yesmaximal, characteristic
4C2×#17985998C211yesmaximal, characteristic
5#22588607C212yesmaximal, complemented
6#22588600C212yesmaximal, complemented
7#22588593C212yesmaximal, complemented
8#22588586C212yesmaximal, complemented
9#22588059C211yesmaximal, characteristic, complemented
10#22588034C211yesmaximal, characteristic, complemented
11C2×#389×C71C211yesmaximal, characteristic, complemented
12C22×#11765865C2211yescharacteristic
13C22×#11765849C2211yescharacteristic
14C22×#11765849C2211yescharacteristic
15C4×#11765865—24no—
16C4×#11765865—24no—
17C2×C4×Dic142C2211yescharacteristic
18C2×C4×Dic142C2211yescharacteristic
19C2×C4×D284C2211yescharacteristic
20C2×C4×D284C2211yescharacteristic
21C2×C4×C22⋊D71—21no—
22C2×C4×C22⋊D71—21no—
23C2×#14556087C2211yescharacteristic
24C2×#14556085—21no—
25C2×C42×D71—21no—
26#17986039—24no—
27C2×#14556061C2212yes—
28C2×#14556059C2212yes—
29#17985998—24no—
30#17985998C418yescomplemented
31C2×#14556051C2212yes—
32C2×#14556051C2212yes—
33C2×#14556048C2212yes—
34C2×#14556048C2212yes—
35#17985761C2212yes—
36#17985761C2212yes—
37#17985754C2212yes—
38#17985754C2212yes—
39C22×C4⋊C4×C71—21no—
40C22×C4⋊C4×C71C2211yescharacteristic
41C22×C4⋊C4×C71C2211yescharacteristic
42C22×C42×C71C2211yescharacteristic
43#389×C71—21no—
44#389×C71—21no—
45#389×C71C2212yes—
46#389×C71C2212yes—
47C2×#95×C71—21no—
48C2×#11765865—22no—
49C2×#11765865—22no—
50C2×#11765865—22no—
51C2×#11765865C2×C414yes—
52C2×#11765865D414yes—
53C2×#11765865Q814yes—
54C22×Dic142C2311yescharacteristic
55C22×D284C2311yescharacteristic
56C22×C22⋊D71—21no—
57C22×C4×D71—21no—
58C22×C4×D71—21no—
59C22×C4×D71—21no—
60C2×#11765849—22no—
61C2×#11765849—22no—
62C2×#11765849—22no—
63C2×#11765849—22no—
64C2×#11765849—22no—
65C2×#11765849C2×C414yescomplemented
66C22×C71⋊C8C2312yes—
67C22×C71⋊C8C2312yes—
68C4×Dic142—22no—
69C4×Dic142—22no—
70C4×Dic142—22no—
71C4×Dic142—22no—
72C4×D284—22no—
73C4×D284—22no—
74C4×D284—22no—
75C4×D284—22no—
76C4×C22⋊D71—28no—
77C4×C22⋊D71—28no—
78#14556087—22no—
79#14556087D414yes—
80#14556085—28no—
81C2×C4⋊Dic71C2311yescharacteristic
82C2×C4⋊Dic71C2311yescharacteristic
83C2×Dic71⋊C4—21no—
84C2×Dic71⋊C4—21no—
85C2×C22⋊Dic71—21no—
86C2×(C2×D71)⋊C4—21no—
87C2×(C2×D71)⋊C4—21no—
88C2×C4×Dic71—21no—
89C2×C4×Dic71—21no—
90C2×C4×Dic71—21no—
91C42×D71—28no—
92#14556061—24no—
93#14556059—24no—
94#14556051—24no—
95#14556051C2×C418yescomplemented
96#14556048—24no—
97#14556048C2×C418yescomplemented
98C23×C4×C71—21no—
99C23×C4×C71—21no—
100C23×C4×C71D411yescharacteristic
101C23×C4×C71D411yescharacteristic
102C23×C4×C71C2311yescharacteristic
103C2×C4⋊C4×C71—21no—
104C2×C4⋊C4×C71—22no—
105C2×C4⋊C4×C71—22no—
106C2×C4⋊C4×C71—22no—
107C2×C4⋊C4×C71—22no—
108C2×C4⋊C4×C71—22no—
109C2×C4⋊C4×C71—21no—
110C2×C4⋊C4×C71—44no—
111C2×C4⋊C4×C71C2×C414yescomplemented
112C2×C4⋊C4×C71C2312yes—
113C2×C4⋊C4×C71C2312yes—
114C2×C42×C71—22no—
115C2×C42×C71—22no—
116C2×C42×C71D411yescharacteristic
117C2×C42×C71D411yescharacteristic
118C2×C42×C71C2311yescharacteristic
119C2×C42×C71C2311yescharacteristic
120#95×C71—22no—
121#95×C71—22no—
122C23×D71—21no—
123#11765865—28no—
124#11765865—28no—
125#11765865—28no—
126#11765865C4⋊C4116yescomplemented
127C2×Dic142—21no—
128C2×Dic142—21no—
129C2×Dic142—21no—
130C2×Dic142C22×C412yes—
131C2×Dic142C2×D412yes—
132C2×Dic142C2×Q812yes—
133C2×D284—21no—
134C2×D284—21no—
135C2×D284—21no—
136C2×D284C22×C412yes—
137C2×D284C2×D412yes—
138C2×D284C2×Q812yes—
139C2×C22⋊D71—24no—
140C2×C22⋊D71—24no—
141C2×C22⋊D71—24no—
142C22×Dic71—21no—
143C22×Dic71—21no—
144C22×Dic71—21no—
145C2×C4×D71—24no—
146C2×C4×D71—24no—
147C2×C4×D71—24no—
148C2×C4×D71—24no—
149C2×C4×D71—24no—
150C2×C4×D71—42no—
151C2×C4×D71—42no—
152C2×C4×D71—42no—
153C2×C4×D71—42no—
154#11765849—28no—
155#11765849—44no—
156C2×C71⋊C8—22no—
157C2×C71⋊C8—22no—
158C2×C71⋊C8—22no—
159C2×C71⋊C8—22no—
160C2×C71⋊C8—22no—
161C2×C71⋊C8C22×C414yes—
162C4⋊Dic71—21no—
163C4⋊Dic71—21no—
164C4⋊Dic71C2×D412yes—
165C4⋊Dic71C2×D412yes—
166Dic71⋊C4—24no—
167Dic71⋊C4—24no—
168C22⋊Dic71—24no—
169(C2×D71)⋊C4—24no—
170(C2×D71)⋊C4—24no—
171C4×Dic71—24no—
172C4×Dic71—24no—
173C4×Dic71—24no—
174C24×C71C2×D411yescharacteristic
175C22×C4×C71—24no—
176C22×C4×C71—22no—
177C22×C4×C71—22no—
178C22×C4×C71—22no—
179C22×C4×C71—22no—
180C22×C4×C71—22no—
181C22×C4×C71—22no—
182C22×C4×C71—41no—
183C22×C4×C71—41no—
184C22×C4×C71—42no—
185C22×C4×C71—42no—
186C22×C4×C71—42no—
187C22×C4×C71—42no—
188C22×C4×C71—42no—
189C22×C4×C71C22×C412yes—
190C22×C4×C71C2×D412yes—
191C22×C4×C71C2×D412yes—
192C22×C4×C71C2×D411yescharacteristic
193C22×C4×C71C2×D411yescharacteristic
194C22×C4×C71C2×D411yescharacteristic
195C22×C4×C71C2×D411yescharacteristic
196C22×C4×C71C2×D411yescharacteristic
197C22×C4×C71C2×Q812yes—
198C22×C4×C71#4112yes—
199C22×C4×C71#4112yes—
200C22×C4×C71C2411yescharacteristic
201C4⋊C4×C71—24no—
202C4⋊C4×C71—22no—
203C4⋊C4×C71—22no—
204C4⋊C4×C71—22no—
205C4⋊C4×C71—44no—
206C4⋊C4×C71—42no—
207C4⋊C4×C71—44no—
208C4⋊C4×C71C22×C414yes—
209C42×C71—22no—
210C42×C71—22no—
211C42×C71—21no—
212C42×C71—21no—
213C42×C71—21no—
214C42×C71—21no—
215C23×C71—21no—
216C23×C71—21no—
217C23×C71—21no—
218C23×C71—21no—
219C23×C71C4×D412yes—
220C23×C71#12012yes—
221C23×C71C22⋊Q812yes—
222C23×C71C22×D411yescharacteristic
223C22×D71—22no—
224C22×D71—22no—
225C22×D71—22no—
226C22×D71—41no—
227C22×D71—41no—
228C22×D71—41no—
229C22×D71—41no—
230C2×C4×C71—21no—
231C2×C4×C71—21no—
232C2×C4×C71—21no—
233C2×C4×C71—21no—
234C2×C4×C71—24no—
235C2×C4×C71—21no—
236C2×C4×C71—21no—
237C2×C4×C71—21no—
238C2×C4×C71—21no—
239C2×C4×C71—21no—
240C2×C4×C71—21no—
241C2×C4×C71—44no—
242C2×C4×C71—44no—
243C2×C4×C71—48no—
244C2×C4×C71—42no—
245C2×C4×C71—42no—
246C2×C4×C71—42no—
247C2×C4×C71—42no—
248C2×C4×C71—42no—
249C2×C4×C71—84no—
250C2×C4×C71C2×C4⋊C414yes—
251C2×C4×C71C4×D412yes—
252C2×C4×C71C4×D414yes—
253C2×C4×C71#12012yes—
254C2×C4×C71C22⋊Q812yes—
255C2×C4×C71C22⋊Q814yes—
256C2×C4×C71#12314yes—
257C2×C4×C71AGL(1,ℤ/8)12yes—
258C2×C4×C71#13712yes—
259C2×C4×C71C23×C411yescharacteristic
260C2×C4×C71C22×D411yescharacteristic
261C2×C4×C71C22×D411yescharacteristic
262C2×C4×C71C22×Q811yescharacteristic
263C2×C4×C71C2×#4111yescharacteristic
264C2×C4×C71C2×#4111yescharacteristic
265C22⋊D71—216no—
266C2×Dic71—22no—
267C2×Dic71—22no—
268C2×Dic71—22no—
269C2×Dic71—22no—
270C2×Dic71—22no—
271C2×Dic71—41no—
272C2×Dic71—41no—
273C2×Dic71—41no—
274C2×Dic71—41no—
275D284—22no—
276D284—22no—
277D284—22no—
278D284C2×C4⋊C414yescomplemented
279C4×D71—216no—
280C4×D71—48no—
281C4×D71—48no—
282Dic142—22no—
283Dic142—22no—
284Dic142—22no—
285Dic142C2×C4⋊C414yescomplemented
286C71⋊C8—24no—
287C71⋊C8—42no—
288C22×C71—22no—
289C22×C71—22no—
290C22×C71—22no—
291C22×C71—22no—
292C22×C71—22no—
293C22×C71—22no—
294C22×C71#38514yes—
295C22×C71C2×C4×D411yescharacteristic
296C22×C71C2×#12011yescharacteristic
297C22×C71C2×C22⋊Q811yescharacteristic
298C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C71C2×#13711yescharacteristic
300C22×C71#57511yescharacteristic
301C22×C71#57511yescharacteristic
302C2×D71—24no—
303C2×D71—42no—
304C2×D71—42no—
305C2×D71—42no—
306C2×D71—42no—
307C2×D71—42no—
308C2×D71—42no—
309C4×C71—22no—
310C4×C71—22no—
311C4×C71—44no—
312C4×C71—41no—
313C4×C71—41no—
314C4×C71—82no—
315C4×C71#38514yes—
316C4×C71C22×C4⋊C411yescharacteristic
317C4×C71C2×C4×D411yescharacteristic
318C4×C71C2×C22⋊Q811yescharacteristic
319C4×C71C2×#12311yescharacteristic
320Dic71—24no—
321Dic71—42no—
322Dic71—42no—
323#20875—711nomaximal, Sylow
324C2×C71—24no—
325C2×C71C2×#38511yescharacteristic
326C2×C71C4×AGL(1,ℤ/8)11yescharacteristic
327C2×C71C4×#13711yescharacteristic
328C2×C71#300611yescharacteristic
329C2×C71#300711yescharacteristic
330C2×C71#307911yescharacteristic
331C2×C71#308011yescharacteristic
332D71—44no—
333C2×C4×#41—711no—
334C2×#428—711no—
335C2×#417—711no—
336C2×#417—711no—
337C2×#389D7111yescharacteristic, complemented
338#1865—712no—
339#1864—712no—
340#1863—712no—
341#1862—712no—
342#1736—711no—
343#1728—711no—
344C71#2087511yescharacteristic, complemented, Sylow
345C22×#41—711no—
346C22×M16—711no—
347C22×M16—711no—
348C4×#41—1424no—
349C4×#41—1424no—
350C2×C4×Q8—711no—
351C2×C4×Q8—711no—
352C2×C4×D4—711no—
353C2×C4×D4—711no—
354C2×C4×D4—1421no—
355C2×C4×D4—1421no—
356C2×#117—711no—
357C2×#117—1421no—
358C22×C4⋊C4—21no—
359C22×C4⋊C4C2×D7111yescharacteristic
360C22×C4⋊C4C2×D7111yescharacteristic
361C22×C42—1421no—
362C22×C42C2×D7111yescharacteristic
363#428—1424no—
364C2×#107—712no—
365C2×#106—712no—
366#417—718no—
367#417—1424no—
368C2×#103—712no—
369C2×#103—712no—
370C2×#102—712no—
371C2×#102—712no—
372#389—21no—
373#389—21no—
374#389C2×D7112yes—
375#389C2×D7112yes—
376C2×#95—21no—
377#340—712no—
378#340—712no—
379#336—712no—
380#336—712no—
381C2×#41—714no—
382C2×#41—714no—
383C2×#41—714no—
384C2×#41—1422no—
385C2×#41—1422no—
386C2×#41—1422no—
387C22×Q8—711no—
388C22×D4—711no—
389C22×D4—1421no—
390C23×C4—21no—
391C23×C4—21no—
392C23×C4—1421no—
393C23×C4—1421no—
394C23×C4—1421no—
395C23×C4C22⋊D7111yescharacteristic
396C23×C4C22⋊D7111yescharacteristic
397C23×C4C22×D7111yescharacteristic
398C2×M16—714no—
399C2×M16—1422no—
400C2×M16—1422no—
401C2×M16—1422no—
402C2×M16—1422no—
403C2×M16—1422no—
404C22×C8—712no—
405C22×C8—712no—
406C4×Q8—1422no—
407C4×Q8—1422no—
408C4×Q8—1422no—
409C4×Q8—1422no—
410C4×D4—1422no—
411C4×D4—1422no—
412C4×D4—1422no—
413C4×D4—1422no—
414C4×D4—1428no—
415C4×D4—1428no—
416#117—714no—
417#117—1428no—
418#117—1422no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—22no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4—22no—
423C2×C4⋊C4—22no—
424C2×C4⋊C4—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—44no—
427C2×C4⋊C4—711no—
428C2×C4⋊C4—711no—
429C2×C4⋊C4—1421no—
430C2×C4⋊C4—1421no—
431C2×C4⋊C4C4×D7114yescomplemented
432C2×C4⋊C4C22×D7112yes—
433C2×C4⋊C4C22×D7112yes—
434C2×C22⋊C4—1421no—
435C2×C22⋊C4—1421no—
436C2×C22⋊C4—1421no—
437C2×C42—22no—
438C2×C42—22no—
439C2×C42—1428no—
440C2×C42—1421no—
441C2×C42—1421no—
442C2×C42—1421no—
443C2×C42C22⋊D7111yescharacteristic
444C2×C42C22⋊D7111yescharacteristic
445C2×C42C22×D7111yescharacteristic
446C2×C42C22×D7111yescharacteristic
447#107—1424no—
448#106—1424no—
449#103—718no—
450#103—1424no—
451#102—718no—
452#102—1424no—
453#95—22no—
454#95—22no—
455C24—1421no—
456C24C2×C22⋊D7111yescharacteristic
457#41—7116no—
458#41—1428no—
459#41—1428no—
460#41—1428no—
461C2×Q8—712no—
462C2×Q8—712no—
463C2×Q8—712no—
464C2×Q8—1421no—
465C2×Q8—1421no—
466C2×Q8—1421no—
467C2×D4—712no—
468C2×D4—712no—
469C2×D4—712no—
470C2×D4—1424no—
471C2×D4—1424no—
472C2×D4—1424no—
473C2×D4—1421no—
474C2×D4—1421no—
475C2×D4—1421no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—2842no—
491C22×C4—2842no—
492C22×C4—2842no—
493C22×C4—2842no—
494C22×C4—1424no—
495C22×C4—1424no—
496C22×C4—1421no—
497C22×C4—1421no—
498C22×C4—1421no—
499C22×C4—1424no—
500C22×C4—1424no—
501C22×C4—1424no—
502C22×C4C2×C4×D7112yes—
503C22×C4D4×D7112yes—
504C22×C4D4×D7112yes—
505C22×C4C2×C22⋊D7111yescharacteristic
506C22×C4C2×C22⋊D7111yescharacteristic
507C22×C4C2×C22⋊D7111yescharacteristic
508C22×C4C2×C22⋊D7111yescharacteristic
509C22×C4C2×C22⋊D7111yescharacteristic
510C22×C4Q8×D7112yes—
511C22×C4#1176586512yes—
512C22×C4#1176586612yes—
513C22×C4C23×D7111yescharacteristic
514M16—2844no—
515M16—1428no—
516C2×C8—714no—
517C2×C8—1422no—
518C2×C8—1422no—
519C2×C8—1422no—
520C2×C8—1422no—
521C2×C8—1422no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—712no—
530C4⋊C4—712no—
531C4⋊C4—1424no—
532C4⋊C4—1424no—
533C4⋊C4—1421no—
534C4⋊C4—1421no—
535C4⋊C4C2×C4×D7114yes—
536C22⋊C4—1424no—
537C22⋊C4—1424no—
538C22⋊C4—1424no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—22no—
544C42—22no—
545C42—1424no—
546C42—1424no—
547C42—1424no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—2841no—
553C23—2841no—
554C23—2841no—
555C23—2841no—
556C23—1422no—
557C23—1422no—
558C23—1422no—
559C23C4×C22⋊D7112yes—
560C23#1455609612yes—
561C23#1455610412yes—
562C23C22×C22⋊D7111yescharacteristic
563Q8—714no—
564Q8—1422no—
565Q8—1422no—
566Q8—1422no—
567D4—714no—
568D4—14216no—
569D4—1422no—
570D4—1422no—
571D4—1422no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—24no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—84no—
592C2×C4—2848no—
593C2×C4—2848no—
594C2×C4—2841no—
595C2×C4—2841no—
596C2×C4—2841no—
597C2×C4—2841no—
598C2×C4—1422no—
599C2×C4—1422no—
600C2×C4—1422no—
601C2×C4—1422no—
602C2×C4—1422no—
603C2×C4—14216no—
604C2×C4C4⋊C4×D7114yes—
605C2×C4C4×C22⋊D7112yes—
606C2×C4#1455609014yes—
607C2×C4#1455609612yes—
608C2×C4#1455610412yes—
609C2×C4#1455610614yes—
610C2×C4#1455611314yes—
611C2×C4#1455615412yes—
612C2×C4#1455615912yes—
613C2×C4C22×C4×D7111yescharacteristic
614C2×C4C2×D4×D7111yescharacteristic
615C2×C4C2×D4×D7111yescharacteristic
616C2×C4C2×Q8×D7111yescharacteristic
617C2×C4C2×#1176586511yescharacteristic
618C2×C4C2×#1176586611yescharacteristic
619C8—2842no—
620C8—1424no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—2842no—
628C22—2842no—
629C22—2842no—
630C22—2842no—
631C22—2842no—
632C22—2842no—
633C22—1424no—
634C22#1798587014yes—
635C22C2×C4×C22⋊D7111yescharacteristic
636C22C2×#1455609611yescharacteristic
637C22C2×#1455610411yescharacteristic
638C22C2×#1455615411yescharacteristic
639C22C2×#1455615911yescharacteristic
640C22#1798679711yescharacteristic
641C22#1798679711yescharacteristic
642C4—22no—
643C4—22no—
644C4—44no—
645C4—41no—
646C4—41no—
647C4—82no—
648C4—2842no—
649C4—2842no—
650C4—1424no—
651C4#1798587014yes—
652C4C2×C4⋊C4×D7111yescharacteristic
653C4C2×#1455609011yescharacteristic
654C4C2×#1455610611yescharacteristic
655C4C2×#1455611311yescharacteristic
656C2—24no—
657C2—2844no—
658C2C2×#1798587011yescharacteristic
659C2C4×#1455615411yescharacteristic
660C2C4×#1455615911yescharacteristic
661C2#2259759011yescharacteristic
662C2#2259759811yescharacteristic
663C2#2259814911yescharacteristic
664C2#2259815811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC2×#389×C71C2
socleC23×C71C22×D4
prime radicalC22×#11765865C22
cosocleC22×C4×C71C24
hypercenterC2×#389D71
hypocenterC71#20875
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
2. C2×#17986039C2semidirect
5. #22588607C2semidirect
6. #22588600C2semidirect
7. #22588593C2semidirect
8. #22588586C2semidirect
9. #22588059C2semidirect
10. #22588034C2semidirect
11. C2×#389×C71C2semidirect
30. #17985998C4semidirect
65. C2×#11765849C2×C4semidirect
95. #14556051C2×C4semidirect
97. #14556048C2×C4semidirect
111. C2×C4⋊C4×C71C2×C4semidirect
126. #11765865C4⋊C4semidirect
278. D284C2×C4⋊C4semidirect
285. Dic142C2×C4⋊C4semidirect
337. C2×#389D71semidirect
344. C71#20875semidirect
431. C2×C4⋊C4C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117658652
C2×#179860392
C2×#179859982
#225886072
#225886002
#225885932
#225885862
#225880592
#225880342
C2×#389×C712
#2087571