Finite Group Databasebeta

← #32008653

Subgroups — #32008653

Order 18176.

1296 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986132C211yesmaximal, characteristic
2C2×#17985901C211yesmaximal, characteristic, complemented
3#22589215C212yesmaximal, complemented
4#22589036C212yesmaximal, complemented
5#22588967C212yesmaximal, complemented
6#22588688C212yesmaximal, complemented
7#22588658C211yesmaximal, characteristic, complemented
8#22588532C211yesmaximal, characteristic, complemented
9#22588270C211yesmaximal, characteristic, complemented
10#22588102C211yesmaximal, characteristic, complemented
11C4×#121×C71C211yesmaximal, characteristic, complemented
12C23×D284C2211yescharacteristic
13C22×#11765850C2212yes—
14C22×C4⋊Dic71C2211yescharacteristic
15C22×(C2×D71)⋊C4—21no—
16C22×(C2×D71)⋊C4—21no—
17#17986132C418yescomplemented
18C2×#14556059C2212yescomplemented
19C2×#14556049C2212yescomplemented
20C2×#14556048C2212yes—
21C2×#14556048C2212yes—
22C2×#14556048C2212yes—
23C2×C22⋊(C71⋊C8)C2211yescharacteristic
24C2×C22⋊(C71⋊C8)C2211yescharacteristic
25C2×#14556039C2212yescomplemented
26#17985901—21no—
27#17985901—21no—
28#17985901C2212yes—
29#17985901C2212yes—
30#17985761C2212yescomplemented
31#17985761C2212yescomplemented
32#17985754C2212yescomplemented
33C2×#121×C71C2211yescharacteristic
34C2×C4×D4×C71—21no—
35C2×C4×D4×C71C2211yescharacteristic, complemented
36C22×C42×C71C2211yescharacteristic
37#390×C71C2211yescharacteristic, complemented
38#389×C71C2211yescharacteristic, complemented
39#388×C71—21no—
40#386×C71—21no—
41#380×C71C2211yescharacteristic
42C4×C4⋊C4×C71C2211yescharacteristic, complemented
43C4×C22⋊C4×C71—21no—
44C24×D71—21no—
45C22×D284—21no—
46C22×D284—22no—
47C22×D284—42no—
48C22×D284—42no—
49C22×D284C2×C414yescomplemented
50C22×D284C2312yes—
51C23×Dic71—21no—
52C2×#11765850—42no—
53C2×#11765850—42no—
54C2×#11765850C2×C418yescomplemented
55C22×C71⋊C8C2312yes—
56C22×C71⋊C8C2312yes—
57C2×C4⋊Dic71—21no—
58C2×C4⋊Dic71—42no—
59C2×C4⋊Dic71C2312yes—
60C2×(C2×D71)⋊C4—22no—
61C2×(C2×D71)⋊C4—22no—
62C2×(C2×D71)⋊C4—22no—
63C2×(C2×D71)⋊C4—22no—
64C2×(C2×D71)⋊C4—44no—
65C2×(C2×D71)⋊C4—44no—
66#14556059—42no—
67#14556049—42no—
68#14556048—42no—
69#14556048—42no—
70#14556048C2×C418yescomplemented
71C22⋊(C71⋊C8)—22no—
72C22⋊(C71⋊C8)C2×C414yescomplemented
73#14556039—42no—
74C22×D4×C71—21no—
75C22×D4×C71C2311yescharacteristic
76C23×C4×C71—21no—
77C23×C4×C71D411yescharacteristic
78C23×C4×C71D411yescharacteristic
79C23×C4×C71C2311yescharacteristic
80#121×C71—22no—
81#121×C71C2×C414yes—
82C4×D4×C71—42no—
83C4×D4×C71—44no—
84C2×C4⋊C4×C71—21no—
85C2×C4⋊C4×C71C2311yescharacteristic
86C2×C4⋊C4×C71C2311yescharacteristic
87C2×C4⋊C4×C71C2311yescharacteristic
88C2×C22⋊C4×C71—21no—
89C2×C22⋊C4×C71—21no—
90C2×C22⋊C4×C71—21no—
91C2×C22⋊C4×C71—21no—
92C2×C22⋊C4×C71—21no—
93C2×C42×C71—21no—
94C2×C42×C71—42no—
95C2×C42×C71D411yescharacteristic
96C2×C42×C71D411yescharacteristic
97C2×C42×C71C2311yescharacteristic
98C2×C42×C71C2311yescharacteristic
99#95×C71—21no—
100#95×C71—21no—
101#95×C71—21no—
102C23×D71—24no—
103C23×D71—22no—
104C23×D71—42no—
105C23×D71—42no—
106C23×D71—42no—
107C23×D71—42no—
108C23×D71—42no—
109C23×D71—42no—
110C2×D284—22no—
111C2×D284—22no—
112C2×D284—22no—
113C2×D284—22no—
114C2×D284—24no—
115C2×D284—22no—
116C2×D284—48no—
117C2×D284—42no—
118C2×D284—42no—
119C2×D284—48no—
120C2×D284C22×C414yes—
121C22×Dic71—22no—
122C22×Dic71—42no—
123C22×Dic71—42no—
124C22×Dic71—42no—
125#11765850—48no—
126C2×C71⋊C8—22no—
127C2×C71⋊C8—22no—
128C2×C71⋊C8—22no—
129C2×C71⋊C8—22no—
130C2×C71⋊C8—22no—
131C2×C71⋊C8C22×C414yes—
132C4⋊Dic71—22no—
133C4⋊Dic71—22no—
134C4⋊Dic71—42no—
135(C2×D71)⋊C4—44no—
136(C2×D71)⋊C4—44no—
137(C2×D71)⋊C4—44no—
138(C2×D71)⋊C4—44no—
139C24×C71—21no—
140C24×C71C2×D411yescharacteristic
141C2×D4×C71—24no—
142C2×D4×C71—21no—
143C2×D4×C71—22no—
144C2×D4×C71—22no—
145C2×D4×C71—42no—
146C2×D4×C71—42no—
147C2×D4×C71C22×C412yes—
148C22×C4×C71—22no—
149C22×C4×C71—22no—
150C22×C4×C71—22no—
151C22×C4×C71—22no—
152C22×C4×C71—21no—
153C22×C4×C71—22no—
154C22×C4×C71—21no—
155C22×C4×C71—21no—
156C22×C4×C71—41no—
157C22×C4×C71—41no—
158C22×C4×C71—41no—
159C22×C4×C71—42no—
160C22×C4×C71—42no—
161C22×C4×C71—42no—
162C22×C4×C71C22×C412yes—
163C22×C4×C71C2×D411yescharacteristic
164C22×C4×C71C2×D411yescharacteristic
165C22×C4×C71C2×D411yescharacteristic
166C22×C4×C71C2×D411yescharacteristic
167C22×C4×C71C2×D411yescharacteristic
168C22×C4×C71#4112yes—
169C22×C4×C71C2411yescharacteristic
170C4⋊C4×C71—21no—
171C4⋊C4×C71—22no—
172C4⋊C4×C71—22no—
173C4⋊C4×C71—42no—
174C4⋊C4×C71C22×C412yes—
175C22⋊C4×C71—24no—
176C22⋊C4×C71—42no—
177C22⋊C4×C71—42no—
178C22⋊C4×C71—42no—
179C22⋊C4×C71—42no—
180C42×C71—22no—
181C42×C71—22no—
182C42×C71—41no—
183C42×C71—41no—
184C42×C71—42no—
185C23×C71—21no—
186C23×C71—21no—
187C23×C71—21no—
188C23×C71—21no—
189C23×C71—22no—
190C23×C71—41no—
191C23×C71—41no—
192C23×C71—41no—
193C23×C71—41no—
194C23×C71—41no—
195C23×C71—41no—
196C23×C71C4×D412yes—
197C23×C71AGL(1,ℤ/8)12yes—
198C23×C71AGL(1,ℤ/8)12yes—
199C23×C71C22×D411yescharacteristic
200C22×D71—24no—
201C22×D71—48no—
202C22×D71—48no—
203C22×D71—42no—
204C22×D71—42no—
205C22×D71—42no—
206C22×D71—42no—
207C22×D71—42no—
208C22×D71—42no—
209C22×D71—48no—
210C22×D71—84no—
211C22×D71—84no—
212C22×D71—84no—
213C22×D71—84no—
214D4×C71—24no—
215D4×C71—48no—
216D4×C71—42no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—24no—
222C2×C4×C71—24no—
223C2×C4×C71—22no—
224C2×C4×C71—21no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—21no—
229C2×C4×C71—21no—
230C2×C4×C71—21no—
231C2×C4×C71—21no—
232C2×C4×C71—21no—
233C2×C4×C71—44no—
234C2×C4×C71—44no—
235C2×C4×C71—41no—
236C2×C4×C71—41no—
237C2×C4×C71—41no—
238C2×C4×C71—41no—
239C2×C4×C71—41no—
240C2×C4×C71—41no—
241C2×C4×C71—41no—
242C2×C4×C71—41no—
243C2×C4×C71—84no—
244C2×C4×C71C4×D412yes—
245C2×C4×C71AGL(1,ℤ/8)12yes—
246C2×C4×C71C23×C411yescharacteristic
247C2×C4×C71C2×#4111yescharacteristic
248C2×C4×C71H(5,𝔽2)12yes—
249C2×C4×C71H(5,𝔽2)12yes—
250C2×Dic71—42no—
251C2×Dic71—42no—
252C2×Dic71—42no—
253C2×Dic71—84no—
254D284—24no—
255D284—24no—
256D284—48no—
257D284—42no—
258D284—42no—
259C71⋊C8—24no—
260C71⋊C8—42no—
261C22×C71—22no—
262C22×C71—22no—
263C22×C71—22no—
264C22×C71—22no—
265C22×C71—22no—
266C22×C71—22no—
267C22×C71—42no—
268C22×C71—42no—
269C22×C71—42no—
270C22×C71—81no—
271C22×C71—81no—
272C22×C71—81no—
273C22×C71—81no—
274C22×C71#44214yes—
275C22×C71C2×C4×D411yescharacteristic
276C22×C71#53411yescharacteristic
277C22×C71#53511yescharacteristic
278C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C71#57511yescharacteristic
282C2×D71—44no—
283C2×D71—44no—
284C2×D71—44no—
285C2×D71—816no—
286C2×D71—82no—
287C2×D71—82no—
288C2×D71—82no—
289C2×D71—82no—
290C4×C71—22no—
291C4×C71—22no—
292C4×C71—41no—
293C4×C71—41no—
294C4×C71—41no—
295C4×C71—42no—
296C4×C71—81no—
297C4×C71—81no—
298C4×C71#51812yes—
299C4×C71#53812yes—
300Dic71—82no—
301#22046—711nomaximal, Sylow
302C2×C71—24no—
303C2×C71—82no—
304C2×C71#232111yescharacteristic
305C2×C71C2×#44211yescharacteristic
306C2×C71C4×AGL(1,ℤ/8)11yescharacteristic
307C2×C71#319211yescharacteristic
308C2×C71#319411yescharacteristic
309C2×C71#321311yescharacteristic
310C2×C71#321711yescharacteristic
311D71—84no—
312C2×#447—711no—
313C4×#121D7111yescharacteristic, complemented
314C2×#390—711no—
315#1972—712no—
316#1936—712no—
317#1926—712no—
318#1879—712no—
319#1874—711no—
320#1853—711no—
321#1786—711no—
322#1749—711no—
323C71#2204611yescharacteristic, complemented, Sylow
324C23×D4—711no—
325C22×D8—712no—
326C2×#121C2×D7111yescharacteristic
327C2×C4×D4—21no—
328C2×C4×D4C2×D7111yescharacteristic, complemented
329C22×C4⋊C4—711no—
330C22×C22⋊C4—1421no—
331C22×C22⋊C4—1421no—
332C22×C42C2×D7111yescharacteristic
333#447—718no—
334C2×#106—712no—
335C2×#102—712no—
336C2×#102—712no—
337C2×#102—712no—
338C2×#102—712no—
339C2×C22⋊C8—711no—
340C2×C22⋊C8—711no—
341C2×#97—712no—
342#390—712no—
343#390—712no—
344#390—1421no—
345#390—1421no—
346#390C2×D7111yescharacteristic, complemented
347#389C2×D7111yescharacteristic, complemented
348#388—21no—
349#386—21no—
350#380C2×D7111yescharacteristic
351C4×C4⋊C4C2×D7111yescharacteristic, complemented
352C4×C22⋊C4—21no—
353#340—712no—
354#340—712no—
355#336—712no—
356C25—1421no—
357C22×D4—21no—
358C22×D4—2842no—
359C22×D4—2842no—
360C22×D4—714no—
361C22×D4—712no—
362C22×D4—1421no—
363C22×D4—1422no—
364C22×D4C22×D7111yescharacteristic
365C23×C4—21no—
366C23×C4—1421no—
367C23×C4C22⋊D7111yescharacteristic
368C23×C4C22⋊D7111yescharacteristic
369C23×C4C22×D7111yescharacteristic
370C2×D8—2842no—
371C2×D8—2842no—
372C2×D8—718no—
373C22×C8—712no—
374C22×C8—712no—
375#121—22no—
376#121C4×D7114yes—
377C4×D4—42no—
378C4×D4—44no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—2842no—
381C2×C4⋊C4—712no—
382C2×C4⋊C4—1421no—
383C2×C4⋊C4C22×D7111yescharacteristic
384C2×C4⋊C4C22×D7111yescharacteristic
385C2×C4⋊C4C22×D7111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—2844no—
392C2×C22⋊C4—2844no—
393C2×C22⋊C4—1422no—
394C2×C22⋊C4—1422no—
395C2×C22⋊C4—1422no—
396C2×C22⋊C4—1422no—
397C2×C42—21no—
398C2×C42—42no—
399C2×C42C22⋊D7111yescharacteristic
400C2×C42C22⋊D7111yescharacteristic
401C2×C42C22×D7111yescharacteristic
402C2×C42C22×D7111yescharacteristic
403#106—2842no—
404#102—2842no—
405#102—2842no—
406#102—2842no—
407#102—718no—
408C22⋊C8—714no—
409C22⋊C8—1422no—
410#97—2842no—
411#95—21no—
412#95—21no—
413#95—21no—
414C24—21no—
415C24—2842no—
416C24—2842no—
417C24—2842no—
418C24—2842no—
419C24—2842no—
420C24—2842no—
421C24—1422no—
422C24—1424no—
423C24C2×C22⋊D7111yescharacteristic
424C2×D4—24no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—2842no—
431C2×D4—2842no—
432C2×D4—2848no—
433C2×D4—2848no—
434C2×D4—714no—
435C2×D4—1422no—
436C2×D4—1422no—
437C2×D4—1422no—
438C2×D4—1422no—
439C2×D4—1422no—
440C2×D4—1424no—
441C2×D4C2×C4×D7112yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—2842no—
457C22×C4—2842no—
458C22×C4—2842no—
459C22×C4—1422no—
460C22×C4C2×C4×D7112yes—
461C22×C4C2×C22⋊D7111yescharacteristic
462C22×C4C2×C22⋊D7111yescharacteristic
463C22×C4C2×C22⋊D7111yescharacteristic
464C22×C4C2×C22⋊D7111yescharacteristic
465C22×C4C2×C22⋊D7111yescharacteristic
466C22×C4#1176586512yes—
467C22×C4C23×D7111yescharacteristic
468D8—2848no—
469C2×C8—714no—
470C2×C8—1422no—
471C2×C8—1422no—
472C2×C8—1422no—
473C2×C8—1422no—
474C2×C8—1422no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—42no—
479C4⋊C4—2842no—
480C4⋊C4—1422no—
481C4⋊C4—1422no—
482C4⋊C4C2×C4×D7112yes—
483C22⋊C4—24no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—2844no—
489C22⋊C4—2844no—
490C22⋊C4—2844no—
491C22⋊C4—2844no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42—41no—
496C42—42no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—2848no—
509C23—2848no—
510C23—2842no—
511C23—2848no—
512C23—2842no—
513C23—2842no—
514C23—2842no—
515C23—2842no—
516C23—2842no—
517C23—5684no—
518C23—5684no—
519C23—5684no—
520C23—5684no—
521C23—1424no—
522C23C4×C22⋊D7112yes—
523C23#1455615312yes—
524C23#1455615412yes—
525C23C22×C22⋊D7111yescharacteristic
526D4—24no—
527D4—48no—
528D4—42no—
529D4—2848no—
530D4—2842no—
531D4—2842no—
532D4—1424no—
533D4—1424no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—84no—
561C2×C4—2842no—
562C2×C4—2842no—
563C2×C4—2842no—
564C2×C4—5684no—
565C2×C4C4×C22⋊D7112yes—
566C2×C4#1455615312yes—
567C2×C4C22×C4×D7111yescharacteristic
568C2×C4C2×#1176586511yescharacteristic
569C2×C4#1455617312yes—
570C2×C4#1455617412yes—
571C8—2842no—
572C8—1424no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—2844no—
587C22—2844no—
588C22—2844no—
589C22—56816no—
590C22—5682no—
591C22—5682no—
592C22—5682no—
593C22—5682no—
594C22#1798610914yes—
595C22C2×C4×C22⋊D7111yescharacteristic
596C22#1798655811yescharacteristic
597C22#1798656311yescharacteristic
598C22C2×#1455615311yescharacteristic
599C22C2×#1455615311yescharacteristic
600C22C2×#1455615411yescharacteristic
601C22#1798679711yescharacteristic
602C4—22no—
603C4—22no—
604C4—42no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—81no—
609C4—81no—
610C4—5682no—
611C4#1798647112yes—
612C4#1798658212yes—
613C2—24no—
614C2—82no—
615C2—5684no—
616C2#2259099411yescharacteristic
617C2C2×#1798610911yescharacteristic
618C2C4×#1455615311yescharacteristic
619C2#2259913711yescharacteristic
620C2#2259915011yescharacteristic
621C2#2259932311yescharacteristic
622C2#2259935511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC4×#121×C71C2
socleC23×C71C22×D4
prime radicalC2×#17986132C2
cosocleC22×C4×C71C24
hypercenterC4×#121D71
hypocenterC71#22046
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. C2×#17985901C2semidirect
3. #22589215C2semidirect
4. #22589036C2semidirect
5. #22588967C2semidirect
6. #22588688C2semidirect
7. #22588658C2semidirect
8. #22588532C2semidirect
9. #22588270C2semidirect
10. #22588102C2semidirect
11. C4×#121×C71C2semidirect
17. #17986132C4semidirect
18. C2×#14556059C22semidirect
19. C2×#14556049C22semidirect
25. C2×#14556039C22semidirect
30. #17985761C22semidirect
31. #17985761C22semidirect
32. #17985754C22semidirect
35. C2×C4×D4×C71C22semidirect
37. #390×C71C22semidirect
38. #389×C71C22semidirect
42. C4×C4⋊C4×C71C22semidirect
49. C22×D284C2×C4semidirect
54. C2×#11765850C2×C4semidirect
70. #14556048C2×C4semidirect
72. C22⋊(C71⋊C8)C2×C4semidirect
313. C4×#121D71semidirect
323. C71#22046semidirect
328. C2×C4×D4C2×D71semidirect
346. #390C2×D71semidirect
347. #389C2×D71semidirect
351. C4×C4⋊C4C2×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179861322
C2×#179859012
#225892152
#225890362
#225889672
#225886882
#225886582
#225885322
#225882702
#225881022
C4×#121×C712
#2204671