Finite Group Databasebeta

← #32004673

Subgroups — #32004673

Order 18176.

960 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986320C211yesmaximal, characteristic
2C4×#14556087C211yesmaximal, characteristic, complemented
3#22588964C211yesmaximal, characteristic, complemented
4#22588957C211yesmaximal, characteristic, complemented
5#22588915C211yesmaximal, characteristic, complemented
6#22588908C211yesmaximal, characteristic, complemented
7#22588663C211yesmaximal, characteristic
8#22588537C211yesmaximal, characteristic, complemented
9#22588432C211yesmaximal, characteristic, complemented
10#22588238C211yesmaximal, characteristic, complemented
11#22588238C211yesmaximal, characteristic, complemented
12#22588165C211yesmaximal, characteristic, complemented
13#22588165C211yesmaximal, characteristic, complemented
14#22588102C211yesmaximal, characteristic, complemented
15C4×#121×C71C211yesmaximal, characteristic
16C2×#14556087C2211yescharacteristic
17C2×#14556087C2211yescharacteristic
18#17986320C418yescomplemented
19C2×#14556061C2211yescharacteristic
20C2×#14556061C2211yescharacteristic
21C2×#14556059C2211yescharacteristic
22C2×#14556059C2211yescharacteristic
23C2×#14556049C2211yescharacteristic
24C2×#14556049C2211yescharacteristic
25C2×#14556049C2211yescharacteristic
26C2×#14556049C2211yescharacteristic
27C2×C22⋊(C71⋊C8)C2211yescharacteristic
28C2×C22⋊(C71⋊C8)C2211yescharacteristic
29C2×#14556039C2211yescharacteristic
30C2×#14556039C2211yescharacteristic
31C4×C4⋊Dic71C2211yescharacteristic
32C4×C4⋊Dic71C2211yescharacteristic
33C4×C4⋊Dic71C2211yescharacteristic
34C4×C4⋊Dic71C2211yescharacteristic
35C4×C22⋊Dic71—21no—
36C4×C22⋊Dic71—21no—
37#17985845—21no—
38C42×Dic71—21no—
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985754C2211yescharacteristic
48#17985754C2211yescharacteristic
49C2×#121×C71C2211yescharacteristic
50C2×C4×D4×C71—21no—
51C2×C4×D4×C71C2211yescharacteristic
52C22×C42×C71C2211yescharacteristic
53#390×C71C2211yescharacteristic
54#389×C71C2211yescharacteristic
55#388×C71—21no—
56#386×C71—21no—
57#380×C71C2211yescharacteristic
58C4×C4⋊C4×C71C2211yescharacteristic
59C4×C22⋊C4×C71—21no—
60C22×C71⋊C8C2311yescharacteristic
61C22×C71⋊C8C2311yescharacteristic
62C22×C71⋊C8C2311yescharacteristic
63C22×C71⋊C8C2311yescharacteristic
64#14556087—22no—
65#14556087—22no—
66#14556087—22no—
67#14556087C2×C414yescomplemented
68C2×C4⋊Dic71C2311yescharacteristic
69C2×C4⋊Dic71C2311yescharacteristic
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C4⋊Dic71C2311yescharacteristic
72C2×C22⋊Dic71—21no—
73C2×C22⋊Dic71—21no—
74C2×C4×Dic71—21no—
75C2×C4×Dic71—21no—
76C2×C4×Dic71—21no—
77C2×C4×Dic71—21no—
78C2×C4×Dic71—21no—
79C2×C4×Dic71—21no—
80#14556061—22no—
81#14556061C2×C414yescomplemented
82#14556059—22no—
83#14556059C2×C414yescomplemented
84#14556049—22no—
85#14556049—22no—
86#14556049C2×C414yescomplemented
87#14556049C2×C414yescomplemented
88C22⋊(C71⋊C8)—22no—
89C22⋊(C71⋊C8)C2×C414yescomplemented
90#14556039—22no—
91#14556039—22no—
92C71⋊#95—21no—
93C71⋊#95—21no—
94C71⋊#95—21no—
95C71⋊#95—21no—
96C22×D4×C71—21no—
97C22×D4×C71C2311yescharacteristic
98C23×C4×C71—21no—
99C23×C4×C71D411yescharacteristic
100C23×C4×C71D411yescharacteristic
101C23×C4×C71C2311yescharacteristic
102#121×C71—22no—
103#121×C71C2×C414yes—
104C4×D4×C71—22no—
105C4×D4×C71—22no—
106C4×D4×C71—44no—
107C2×C4⋊C4×C71—21no—
108C2×C4⋊C4×C71C2311yescharacteristic
109C2×C4⋊C4×C71C2311yescharacteristic
110C2×C4⋊C4×C71C2311yescharacteristic
111C2×C22⋊C4×C71—21no—
112C2×C22⋊C4×C71—21no—
113C2×C22⋊C4×C71—21no—
114C2×C22⋊C4×C71—21no—
115C2×C22⋊C4×C71—21no—
116C2×C42×C71—22no—
117C2×C42×C71—22no—
118C2×C42×C71—21no—
119C2×C42×C71D411yescharacteristic
120C2×C42×C71D411yescharacteristic
121C2×C42×C71C2311yescharacteristic
122C2×C42×C71C2311yescharacteristic
123#95×C71—21no—
124#95×C71—21no—
125#95×C71—21no—
126C22×Dic71—21no—
127C22×Dic71—21no—
128C22×Dic71—21no—
129C22×Dic71—21no—
130C2×C71⋊C8—21no—
131C2×C71⋊C8—21no—
132C2×C71⋊C8—21no—
133C2×C71⋊C8—21no—
134C2×C71⋊C8—21no—
135C2×C71⋊C8—21no—
136C2×C71⋊C8—21no—
137C2×C71⋊C8—21no—
138C2×C71⋊C8—21no—
139C2×C71⋊C8—21no—
140C2×C71⋊C8C22×C412yes—
141C2×C71⋊C8C22×C412yes—
142C4⋊Dic71—21no—
143C4⋊Dic71—21no—
144C4⋊Dic71—21no—
145C4⋊Dic71—21no—
146C4⋊Dic71C22×C412yes—
147C4⋊Dic71C22×C412yes—
148C4⋊Dic71#4112yes—
149C4⋊Dic71#4112yes—
150C22⋊Dic71—24no—
151C22⋊Dic71—24no—
152C4×Dic71—24no—
153C4×Dic71—24no—
154C4×Dic71—42no—
155C4×Dic71—42no—
156C4×Dic71—42no—
157C4×Dic71—42no—
158C24×C71—21no—
159C24×C71C2×D411yescharacteristic
160C2×D4×C71—24no—
161C2×D4×C71—21no—
162C2×D4×C71—21no—
163C2×D4×C71—21no—
164C2×D4×C71—21no—
165C2×D4×C71—21no—
166C2×D4×C71—42no—
167C2×D4×C71—42no—
168C2×D4×C71C22×C412yes—
169C22×C4×C71—21no—
170C22×C4×C71—21no—
171C22×C4×C71—21no—
172C22×C4×C71—21no—
173C22×C4×C71—21no—
174C22×C4×C71—22no—
175C22×C4×C71—22no—
176C22×C4×C71—22no—
177C22×C4×C71—21no—
178C22×C4×C71—21no—
179C22×C4×C71—41no—
180C22×C4×C71—41no—
181C22×C4×C71—41no—
182C22×C4×C71—42no—
183C22×C4×C71—42no—
184C22×C4×C71—42no—
185C22×C4×C71C22×C412yes—
186C22×C4×C71C2×D411yescharacteristic
187C22×C4×C71C2×D411yescharacteristic
188C22×C4×C71C2×D411yescharacteristic
189C22×C4×C71C2×D411yescharacteristic
190C22×C4×C71C2×D411yescharacteristic
191C22×C4×C71#4112yes—
192C22×C4×C71C2411yescharacteristic
193C4⋊C4×C71—21no—
194C4⋊C4×C71—21no—
195C4⋊C4×C71—21no—
196C4⋊C4×C71—21no—
197C4⋊C4×C71—21no—
198C4⋊C4×C71—42no—
199C4⋊C4×C71C22×C412yes—
200C22⋊C4×C71—24no—
201C22⋊C4×C71—42no—
202C22⋊C4×C71—42no—
203C22⋊C4×C71—42no—
204C22⋊C4×C71—42no—
205C42×C71—22no—
206C42×C71—22no—
207C42×C71—21no—
208C42×C71—21no—
209C42×C71—42no—
210C42×C71#4112yes—
211C42×C71#4112yes—
212C23×C71—21no—
213C23×C71—21no—
214C23×C71—21no—
215C23×C71—21no—
216C23×C71—21no—
217C23×C71—21no—
218C23×C71—22no—
219C23×C71—41no—
220C23×C71—41no—
221C23×C71—41no—
222C23×C71—41no—
223C23×C71—41no—
224C23×C71—41no—
225C23×C71C4×D412yes—
226C23×C71C22×D411yescharacteristic
227D4×C71—22no—
228D4×C71—22no—
229D4×C71—48no—
230D4×C71—41no—
231D4×C71—41no—
232C2×C4×C71—21no—
233C2×C4×C71—21no—
234C2×C4×C71—21no—
235C2×C4×C71—21no—
236C2×C4×C71—21no—
237C2×C4×C71—21no—
238C2×C4×C71—21no—
239C2×C4×C71—21no—
240C2×C4×C71—22no—
241C2×C4×C71—22no—
242C2×C4×C71—22no—
243C2×C4×C71—22no—
244C2×C4×C71—22no—
245C2×C4×C71—21no—
246C2×C4×C71—21no—
247C2×C4×C71—21no—
248C2×C4×C71—44no—
249C2×C4×C71—44no—
250C2×C4×C71—41no—
251C2×C4×C71—41no—
252C2×C4×C71—41no—
253C2×C4×C71—41no—
254C2×C4×C71—41no—
255C2×C4×C71—41no—
256C2×C4×C71—41no—
257C2×C4×C71—41no—
258C2×C4×C71—84no—
259C2×C4×C71C4×D412yes—
260C2×C4×C71#12312yes—
261C2×C4×C71#12312yes—
262C2×C4×C71AGL(1,ℤ/8)12yes—
263C2×C4×C71C23×C411yescharacteristic
264C2×C4×C71C2×#4111yescharacteristic
265C2×C4×C71C2×#4111yescharacteristic
266C2×C4×C71C2×#4111yescharacteristic
267C2×C4×C71C2×#4111yescharacteristic
268C2×C4×C71C2×#4111yescharacteristic
269C2×Dic71—22no—
270C2×Dic71—22no—
271C2×Dic71—22no—
272C2×Dic71—22no—
273C2×Dic71—41no—
274C2×Dic71—41no—
275C2×Dic71—41no—
276C2×Dic71—41no—
277C2×Dic71—41no—
278C2×Dic71—41no—
279C2×Dic71—41no—
280C2×Dic71—41no—
281C71⋊C8—22no—
282C71⋊C8—22no—
283C71⋊C8—41no—
284C71⋊C8—41no—
285C22×C71—22no—
286C22×C71—22no—
287C22×C71—22no—
288C22×C71—41no—
289C22×C71—41no—
290C22×C71—41no—
291C22×C71—41no—
292C22×C71—42no—
293C22×C71—42no—
294C22×C71—42no—
295C22×C71—81no—
296C22×C71—81no—
297C22×C71—81no—
298C22×C71—81no—
299C22×C71C2×C4×D411yescharacteristic
300C22×C71C2×#12311yescharacteristic
301C22×C71#52511yescharacteristic
302C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
303C22×C71#57511yescharacteristic
304C22×C71#57511yescharacteristic
305C22×C71#57511yescharacteristic
306C4×C71—22no—
307C4×C71—22no—
308C4×C71—22no—
309C4×C71—22no—
310C4×C71—41no—
311C4×C71—42no—
312C4×C71—81no—
313C4×C71—81no—
314C4×C71C4×#4111yescharacteristic
315C4×C71C4×#4111yescharacteristic
316C4×C71#52911yescharacteristic
317C4×C71#52911yescharacteristic
318Dic71—42no—
319Dic71—42no—
320Dic71—42no—
321Dic71—42no—
322#21582—711nomaximal, Sylow
323C2×C71—42no—
324C2×C71—82no—
325C2×C71C4×#12311yescharacteristic
326C2×C71#294311yescharacteristic
327C2×C71C4×AGL(1,ℤ/8)11yescharacteristic
328C2×C71#309311yescharacteristic
329C2×C71#309411yescharacteristic
330C2×C71#311111yescharacteristic
331C2×C71#311511yescharacteristic
332C2×#482—711no—
333C4×#121D7111yescharacteristic
334C4×#117—711no—
335#1926—711no—
336#1925—711no—
337#1919—711no—
338#1918—711no—
339#1875—711no—
340#1854—711no—
341#1825—711no—
342#1776—711no—
343#1776—711no—
344#1762—711no—
345#1762—711no—
346#1749—711no—
347C71#2158211yescharacteristic, complemented, Sylow
348C2×#121C2×D7111yescharacteristic
349C2×C4×D4—21no—
350C2×C4×D4C2×D7111yescharacteristic
351C2×#117—711no—
352C2×#117—711no—
353C22×C42C2×D7111yescharacteristic
354#482—718no—
355C2×#107—711no—
356C2×#107—711no—
357C2×#106—711no—
358C2×#106—711no—
359C2×#102—711no—
360C2×#102—711no—
361C2×#102—711no—
362C2×#102—711no—
363C2×C22⋊C8—711no—
364C2×C22⋊C8—711no—
365C2×#97—711no—
366C2×#97—711no—
367#390C2×D7111yescharacteristic
368#389C2×D7111yescharacteristic
369#388—21no—
370#386—21no—
371#380C2×D7111yescharacteristic
372C4×C4⋊C4—711no—
373C4×C4⋊C4—711no—
374C4×C4⋊C4—711no—
375C4×C4⋊C4—711no—
376C4×C4⋊C4C2×D7111yescharacteristic
377C4×C22⋊C4—21no—
378C4×C22⋊C4—1421no—
379C4×C22⋊C4—1421no—
380#376—1421no—
381C43—1421no—
382#340—711no—
383#340—711no—
384#340—711no—
385#340—711no—
386#340—711no—
387#340—711no—
388#340—711no—
389#340—711no—
390#336—711no—
391#336—711no—
392C22×D4—21no—
393C22×D4C22×D7111yescharacteristic
394C23×C4—21no—
395C23×C4C22⋊D7111yescharacteristic
396C23×C4C22⋊D7111yescharacteristic
397C23×C4C22×D7111yescharacteristic
398C22×C8—711no—
399C22×C8—711no—
400C22×C8—711no—
401C22×C8—711no—
402#121—22no—
403#121C4×D7114yes—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—44no—
407#117—714no—
408#117—1422no—
409#117—1422no—
410#117—1422no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—711no—
413C2×C4⋊C4—711no—
414C2×C4⋊C4—711no—
415C2×C4⋊C4—711no—
416C2×C4⋊C4C22×D7111yescharacteristic
417C2×C4⋊C4C22×D7111yescharacteristic
418C2×C4⋊C4C22×D7111yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—1421no—
425C2×C22⋊C4—1421no—
426C2×C42—22no—
427C2×C42—22no—
428C2×C42—21no—
429C2×C42—1421no—
430C2×C42—1421no—
431C2×C42—1421no—
432C2×C42—1421no—
433C2×C42—1421no—
434C2×C42—1421no—
435C2×C42C22⋊D7111yescharacteristic
436C2×C42C22⋊D7111yescharacteristic
437C2×C42C22×D7111yescharacteristic
438C2×C42C22×D7111yescharacteristic
439#107—714no—
440#107—1422no—
441#106—714no—
442#106—1422no—
443#102—714no—
444#102—714no—
445#102—1422no—
446#102—1422no—
447C22⋊C8—714no—
448C22⋊C8—1422no—
449#97—1422no—
450#97—1422no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—1421no—
455#95—1421no—
456#95—1421no—
457#95—1421no—
458C24—21no—
459C24C2×C22⋊D7111yescharacteristic
460C2×D4—24no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4C2×C4×D7112yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—1421no—
486C22×C4—1421no—
487C22×C4—1421no—
488C22×C4—1421no—
489C22×C4C2×C4×D7112yes—
490C22×C4C2×C22⋊D7111yescharacteristic
491C22×C4C2×C22⋊D7111yescharacteristic
492C22×C4C2×C22⋊D7111yescharacteristic
493C22×C4C2×C22⋊D7111yescharacteristic
494C22×C4C2×C22⋊D7111yescharacteristic
495C22×C4#1176586512yes—
496C22×C4C23×D7111yescharacteristic
497C2×C8—712no—
498C2×C8—712no—
499C2×C8—1421no—
500C2×C8—1421no—
501C2×C8—1421no—
502C2×C8—1421no—
503C2×C8—1421no—
504C2×C8—1421no—
505C2×C8—1421no—
506C2×C8—1421no—
507C2×C8—1421no—
508C2×C8—1421no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—42no—
515C4⋊C4—712no—
516C4⋊C4—712no—
517C4⋊C4—712no—
518C4⋊C4—712no—
519C4⋊C4—1421no—
520C4⋊C4—1421no—
521C4⋊C4—1421no—
522C4⋊C4—1421no—
523C4⋊C4C2×C4×D7112yes—
524C22⋊C4—24no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—1424no—
530C22⋊C4—1424no—
531C42—21no—
532C42—21no—
533C42—22no—
534C42—22no—
535C42—42no—
536C42—2842no—
537C42—2842no—
538C42—2842no—
539C42—2842no—
540C42—1424no—
541C42—1424no—
542C42#1176586412yes—
543C42#1176586412yes—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C4×C22⋊D7112yes—
558C23C22×C22⋊D7111yescharacteristic
559D4—22no—
560D4—22no—
561D4—48no—
562D4—41no—
563D4—41no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—84no—
591C2×C4—2841no—
592C2×C4—2841no—
593C2×C4—2841no—
594C2×C4—2841no—
595C2×C4—2841no—
596C2×C4—2841no—
597C2×C4—2841no—
598C2×C4—2841no—
599C2×C4—1422no—
600C2×C4—1422no—
601C2×C4—1422no—
602C2×C4—1422no—
603C2×C4C4×C22⋊D7112yes—
604C2×C4#1455610912yes—
605C2×C4#1455610912yes—
606C2×C4#1455615412yes—
607C2×C4C22×C4×D7111yescharacteristic
608C2×C4#41×D7111yescharacteristic
609C2×C4#41×D7111yescharacteristic
610C2×C4C2×#1176586411yescharacteristic
611C2×C4C2×#1176586411yescharacteristic
612C2×C4C2×#1176586511yescharacteristic
613C8—2841no—
614C8—2841no—
615C8—1422no—
616C8—1422no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22C2×C4×C22⋊D7111yescharacteristic
632C22C2×#1455610911yescharacteristic
633C22#1798650611yescharacteristic
634C22C2×#1455615411yescharacteristic
635C22#1798679711yescharacteristic
636C22#1798679711yescharacteristic
637C22#1798679711yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—42no—
643C4—41no—
644C4—81no—
645C4—81no—
646C4—2842no—
647C4—2842no—
648C4—2842no—
649C4—2842no—
650C4C4×#1176586411yescharacteristic
651C4C4×#1176586411yescharacteristic
652C4#1798652911yescharacteristic
653C4#1798652911yescharacteristic
654C2—42no—
655C2—82no—
656C2C4×#1455610911yescharacteristic
657C2#2259713811yescharacteristic
658C2C4×#1455615411yescharacteristic
659C2#2259826911yescharacteristic
660C2#2259827611yescharacteristic
661C2#2259844311yescharacteristic
662C2#2259848511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC4×#121×C71C2
socleC23×C71C22×D4
prime radicalC2×#121×C71C22
cosocleC22×C4×C71C24
hypercenterC4×#121D71
hypocenterC71#21582
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
2. C4×#14556087C2semidirect
3. #22588964C2semidirect
4. #22588957C2semidirect
5. #22588915C2semidirect
6. #22588908C2semidirect
8. #22588537C2semidirect
9. #22588432C2semidirect
10. #22588238C2semidirect
11. #22588238C2semidirect
12. #22588165C2semidirect
13. #22588165C2semidirect
14. #22588102C2semidirect
18. #17986320C4semidirect
67. #14556087C2×C4semidirect
81. #14556061C2×C4semidirect
83. #14556059C2×C4semidirect
86. #14556049C2×C4semidirect
87. #14556049C2×C4semidirect
89. C22⋊(C71⋊C8)C2×C4semidirect
347. C71#21582semidirect

Maximal subgroups

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

subgroupindex
C2×#179863202
C4×#145560872
#225889642
#225889572
#225889152
#225889082
#225886632
#225885372
#225884322
#225882382
#225881652
#225881022
C4×#121×C712
#2158271