Finite Group Databasebeta

← #32005132

Subgroups — #32005132

Order 18176.

936 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986299C211yesmaximal, characteristic
2C4×#14556120C211yesmaximal, characteristic, complemented
3#22589297C211yesmaximal, characteristic, complemented
4#22589108C211yesmaximal, characteristic, complemented
5#22588925C211yesmaximal, characteristic, complemented
6#22588911C211yesmaximal, characteristic, complemented
7#22588601C211yesmaximal, characteristic
8#22588587C211yesmaximal, characteristic
9#22588350C211yesmaximal, characteristic
10#22588238C211yesmaximal, characteristic, complemented
11#22588238C211yesmaximal, characteristic, complemented
12#22588193C211yesmaximal, characteristic, complemented
13#22588165C211yesmaximal, characteristic
14#22588165C211yesmaximal, characteristic
15C42×D4×C71C211yesmaximal, characteristic
16C2×#14556120C2211yescharacteristic
17#17986299C418yescomplemented
18C2×#14556061C2211yescharacteristic
19C2×#14556061C2211yescharacteristic
20C2×#14556059C2211yescharacteristic
21C2×#14556059C2211yescharacteristic
22C2×C4⋊(C71⋊C8)C2211yescharacteristic
23C2×C4⋊(C71⋊C8)C2211yescharacteristic
24C2×#14556049C2211yescharacteristic
25C2×#14556049C2211yescharacteristic
26C2×#14556049C2211yescharacteristic
27C2×#14556049C2211yescharacteristic
28C2×#14556039C2211yescharacteristic
29C2×#14556039C2211yescharacteristic
30#17985897C2211yescharacteristic
31#17985897C2211yescharacteristic
32#17985883—21no—
33#17985883—21no—
34#17985865C2211yescharacteristic
35C4×Dic71⋊C4—21no—
36C4×Dic71⋊C4—21no—
37C4×C4⋊Dic71C2211yescharacteristic
38C4×C4⋊Dic71C2211yescharacteristic
39#17985761C2211yescharacteristic
40#17985761C2211yescharacteristic
41#17985761C2211yescharacteristic
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985754C2211yescharacteristic
48#17985754C2211yescharacteristic
49C2×C4×D4×C71C2211yescharacteristic
50C2×C4×D4×C71C2211yescharacteristic
51C2×C4×D4×C71C2211yescharacteristic
52C22×C42×C71—21no—
53C4×C4⋊C4×C71C2211yescharacteristic
54C4×C4⋊C4×C71C2211yescharacteristic
55C4×C4⋊C4×C71C2211yescharacteristic
56C4×C22⋊C4×C71—21no—
57C4×C22⋊C4×C71—21no—
58C4×C22⋊C4×C71—21no—
59C43×C71C2211yescharacteristic
60C22×C71⋊C8C2311yescharacteristic
61C22×C71⋊C8C2311yescharacteristic
62C22×C71⋊C8C2311yescharacteristic
63C22×C71⋊C8C2311yescharacteristic
64#14556120—22no—
65#14556120C2×C414yescomplemented
66C2×C4⋊Dic71C2311yescharacteristic
67C2×C4⋊Dic71C2311yescharacteristic
68C2×C4⋊Dic71C2311yescharacteristic
69C2×C4⋊Dic71C2311yescharacteristic
70C2×Dic71⋊C4—21no—
71C2×Dic71⋊C4—21no—
72C2×Dic71⋊C4—21no—
73C2×Dic71⋊C4—21no—
74C2×C4×Dic71—21no—
75C2×C4×Dic71—21no—
76#14556061—22no—
77#14556061C2×C414yescomplemented
78#14556059—22no—
79#14556059C2×C414yescomplemented
80C4⋊(C71⋊C8)—22no—
81C4⋊(C71⋊C8)C2×C414yescomplemented
82#14556049—22no—
83#14556049—22no—
84#14556049C2×C414yes—
85#14556049C2×C414yes—
86#14556039—22no—
87#14556039—22no—
88C71⋊#95—21no—
89C71⋊#95—21no—
90C71⋊#95—21no—
91C71⋊#95—21no—
92C71⋊#95—21no—
93C71⋊#95—21no—
94C22×D4×C71C2311yescharacteristic
95C23×C4×C71—21no—
96C23×C4×C71—21no—
97C23×C4×C71—21no—
98C4×D4×C71—22no—
99C4×D4×C71—22no—
100C4×D4×C71—22no—
101C4×D4×C71—22no—
102C4×D4×C71C2×C414yes—
103C4×D4×C71Q814yes—
104C2×C4⋊C4×C71C2311yescharacteristic
105C2×C4⋊C4×C71C2311yescharacteristic
106C2×C4⋊C4×C71C2311yescharacteristic
107C2×C22⋊C4×C71—21no—
108C2×C22⋊C4×C71—21no—
109C2×C22⋊C4×C71—21no—
110C2×C42×C71—21no—
111C2×C42×C71—21no—
112C2×C42×C71—21no—
113C2×C42×C71—44no—
114C2×C42×C71D411yescharacteristic
115C2×C42×C71D411yescharacteristic
116C2×C42×C71D411yescharacteristic
117C2×C42×C71D411yescharacteristic
118C2×C42×C71C2311yescharacteristic
119C2×C42×C71C2311yescharacteristic
120C2×C42×C71C2311yescharacteristic
121#95×C71—21no—
122#95×C71—21no—
123#95×C71—21no—
124C22×Dic71—21no—
125C22×Dic71—21no—
126C22×Dic71—21no—
127C22×Dic71—21no—
128C2×C71⋊C8—21no—
129C2×C71⋊C8—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⋊C8C22×C412yes—
139C2×C71⋊C8C22×C412yes—
140C4⋊Dic71—21no—
141C4⋊Dic71—21no—
142C4⋊Dic71—21no—
143C4⋊Dic71—21no—
144C4⋊Dic71—21no—
145C4⋊Dic71—21no—
146C4⋊Dic71C22×C412yes—
147C4⋊Dic71C22×C412yes—
148Dic71⋊C4—24no—
149Dic71⋊C4—24no—
150Dic71⋊C4—42no—
151Dic71⋊C4—42no—
152C4×Dic71—42no—
153C4×Dic71—42no—
154C24×C71—21no—
155C2×D4×C71—21no—
156C2×D4×C71—21no—
157C2×D4×C71—21no—
158C2×D4×C71C22×C412yes—
159C2×D4×C71C2×Q812yes—
160C2×D4×C71#4112yes—
161C22×C4×C71—24no—
162C22×C4×C71—21no—
163C22×C4×C71—21no—
164C22×C4×C71—21no—
165C22×C4×C71—24no—
166C22×C4×C71—24no—
167C22×C4×C71—42no—
168C22×C4×C71—42no—
169C22×C4×C71—42no—
170C22×C4×C71—42no—
171C22×C4×C71—42no—
172C22×C4×C71—42no—
173C22×C4×C71C2×D411yescharacteristic
174C22×C4×C71C2×D411yescharacteristic
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71C2×D411yescharacteristic
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2411yescharacteristic
180C4⋊C4×C71—21no—
181C4⋊C4×C71—21no—
182C4⋊C4×C71—21no—
183C4⋊C4×C71C22×C412yes—
184C4⋊C4×C71C2×Q812yes—
185C4⋊C4×C71#4112yes—
186C22⋊C4×C71—24no—
187C22⋊C4×C71—24no—
188C22⋊C4×C71—24no—
189C42×C71—22no—
190C42×C71—21no—
191C42×C71—21no—
192C42×C71—41no—
193C42×C71—41no—
194C42×C71—41no—
195C42×C71—42no—
196C42×C71—42no—
197C42×C71—42no—
198C42×C71C22×C412yes—
199C42×C71C2×Q812yes—
200C42×C71#4112yes—
201C42×C71#4112yes—
202C23×C71—22no—
203C23×C71—22no—
204C23×C71—22no—
205C23×C71—41no—
206C23×C71—41no—
207C23×C71—41no—
208C23×C71—41no—
209C23×C71C22×D411yescharacteristic
210D4×C71—22no—
211D4×C71—22no—
212D4×C71—22no—
213D4×C71C4×Q814yes—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—22no—
225C2×C4×C71—22no—
226C2×C4×C71—21no—
227C2×C4×C71—22no—
228C2×C4×C71—21no—
229C2×C4×C71—21no—
230C2×C4×C71—48no—
231C2×C4×C71—48no—
232C2×C4×C71—48no—
233C2×C4×C71—41no—
234C2×C4×C71—41no—
235C2×C4×C71—41no—
236C2×C4×C71—41no—
237C2×C4×C71—41no—
238C2×C4×C71—41no—
239C2×C4×C71C4×D412yes—
240C2×C4×C71C4×D412yes—
241C2×C4×C71C22⋊Q812yes—
242C2×C4×C71C22⋊Q812yes—
243C2×C4×C71AGL(1,ℤ/8)12yes—
244C2×C4×C71C23×C411yescharacteristic
245C2×C4×C71C22×Q811yescharacteristic
246C2×C4×C71C2×#4111yescharacteristic
247C2×C4×C71C2×#4111yescharacteristic
248C2×C4×C71C2×#4111yescharacteristic
249C2×C4×C71C2×#4111yescharacteristic
250C2×Dic71—22no—
251C2×Dic71—22no—
252C2×Dic71—41no—
253C2×Dic71—41no—
254C2×Dic71—41no—
255C2×Dic71—41no—
256C2×Dic71—41no—
257C2×Dic71—41no—
258C2×Dic71—41no—
259C2×Dic71—41no—
260C2×Dic71—41no—
261C2×Dic71—41no—
262C71⋊C8—22no—
263C71⋊C8—22no—
264C71⋊C8—41no—
265C71⋊C8—41no—
266C22×C71—24no—
267C22×C71—42no—
268C22×C71—42no—
269C22×C71—42no—
270C22×C71—42no—
271C22×C71—42no—
272C22×C71—42no—
273C22×C71C2×C4×D411yescharacteristic
274C22×C71C2×C22⋊Q811yescharacteristic
275C22×C71#52511yescharacteristic
276C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C71#57511yescharacteristic
278C22×C71#57511yescharacteristic
279C22×C71#57511yescharacteristic
280C4×C71—22no—
281C4×C71—22no—
282C4×C71—22no—
283C4×C71—22no—
284C4×C71—22no—
285C4×C71—42no—
286C4×C71—41no—
287C4×C71—42no—
288C4×C71—42no—
289C4×C71C2×C4×Q811yescharacteristic
290C4×C71C4×#4111yescharacteristic
291C4×C71#52911yescharacteristic
292C4×C71#53311yescharacteristic
293Dic71—42no—
294Dic71—42no—
295Dic71—81no—
296Dic71—81no—
297#21617—711nomaximal, Sylow
298C2×C71—44no—
299C2×C71C4×C22⋊Q811yescharacteristic
300C2×C71#294311yescharacteristic
301C2×C71C4×AGL(1,ℤ/8)11yescharacteristic
302C2×C71#307711yescharacteristic
303C2×C71#308011yescharacteristic
304C2×C71#311111yescharacteristic
305C2×C71#311411yescharacteristic
306C2×#478—711no—
307C4×#125—711no—
308C42×D4D7111yescharacteristic
309#1989—711no—
310#1949—711no—
311#1920—711no—
312#1918—711no—
313#1864—711no—
314#1862—711no—
315#1804—711no—
316#1776—711no—
317#1776—711no—
318#1767—711no—
319#1762—711no—
320#1762—711no—
321C71#2161711yescharacteristic, complemented, Sylow
322C2×#125—711no—
323C2×C4×D4C2×D7111yescharacteristic
324C2×C4×D4C2×D7111yescharacteristic
325C2×C4×D4C2×D7111yescharacteristic
326C22×C42—21no—
327#478—718no—
328C2×#107—711no—
329C2×#107—711no—
330C2×#106—711no—
331C2×#106—711no—
332C2×C4⋊C8—711no—
333C2×C4⋊C8—711no—
334C2×#102—711no—
335C2×#102—711no—
336C2×#102—711no—
337C2×#102—711no—
338C2×#97—711no—
339C2×#97—711no—
340#389—711no—
341#389—711no—
342#387—1421no—
343#387—1421no—
344#382—711no—
345C4×C4⋊C4—711no—
346C4×C4⋊C4—711no—
347C4×C4⋊C4—1421no—
348C4×C4⋊C4—1421no—
349C4×C4⋊C4C2×D7111yescharacteristic
350C4×C4⋊C4C2×D7111yescharacteristic
351C4×C4⋊C4C2×D7111yescharacteristic
352C4×C22⋊C4—21no—
353C4×C22⋊C4—21no—
354C4×C22⋊C4—21no—
355C43C2×D7111yescharacteristic
356#340—711no—
357#340—711no—
358#340—711no—
359#340—711no—
360#340—711no—
361#340—711no—
362#340—711no—
363#340—711no—
364#336—711no—
365#336—711no—
366C22×D4C22×D7111yescharacteristic
367C23×C4—21no—
368C23×C4—21no—
369C23×C4—21no—
370C22×C8—711no—
371C22×C8—711no—
372C22×C8—711no—
373C22×C8—711no—
374#125—714no—
375#125—1422no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4Dic14214yes—
381C4×D4C4×D7114yes—
382C2×C4⋊C4—711no—
383C2×C4⋊C4—711no—
384C2×C4⋊C4—711no—
385C2×C4⋊C4—711no—
386C2×C4⋊C4—1421no—
387C2×C4⋊C4—1421no—
388C2×C4⋊C4—1421no—
389C2×C4⋊C4—1421no—
390C2×C4⋊C4C22×D7111yescharacteristic
391C2×C4⋊C4C22×D7111yescharacteristic
392C2×C4⋊C4C22×D7111yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42—44no—
400C2×C42—1421no—
401C2×C42—1421no—
402C2×C42C22⋊D7111yescharacteristic
403C2×C42C22⋊D7111yescharacteristic
404C2×C42C22⋊D7111yescharacteristic
405C2×C42C22⋊D7111yescharacteristic
406C2×C42C22×D7111yescharacteristic
407C2×C42C22×D7111yescharacteristic
408C2×C42C22×D7111yescharacteristic
409#107—714no—
410#107—1422no—
411#106—714no—
412#106—1422no—
413C4⋊C8—714no—
414C4⋊C8—1422no—
415#102—714no—
416#102—714no—
417#102—1422no—
418#102—1422no—
419#97—1422no—
420#97—1422no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—1421no—
425#95—1421no—
426#95—1421no—
427#95—1421no—
428#95—1421no—
429#95—1421no—
430C24—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4C2×C4×D7112yes—
435C2×D4C2×Dic14212yes—
436C2×D4#1176586512yes—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—1421no—
450C22×C4—1421no—
451C22×C4—1421no—
452C22×C4—1421no—
453C22×C4C2×C22⋊D7111yescharacteristic
454C22×C4C2×C22⋊D7111yescharacteristic
455C22×C4C2×C22⋊D7111yescharacteristic
456C22×C4C2×C22⋊D7111yescharacteristic
457C22×C4C2×C22⋊D7111yescharacteristic
458C22×C4C2×C22⋊D7111yescharacteristic
459C22×C4C23×D7111yescharacteristic
460C2×C8—712no—
461C2×C8—712no—
462C2×C8—1421no—
463C2×C8—1421no—
464C2×C8—1421no—
465C2×C8—1421no—
466C2×C8—1421no—
467C2×C8—1421no—
468C2×C8—1421no—
469C2×C8—1421no—
470C2×C8—1421no—
471C2×C8—1421no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—2842no—
476C4⋊C4—2842no—
477C4⋊C4—712no—
478C4⋊C4—712no—
479C4⋊C4—1421no—
480C4⋊C4—1421no—
481C4⋊C4—1421no—
482C4⋊C4—1421no—
483C4⋊C4—1421no—
484C4⋊C4—1421no—
485C4⋊C4—1424no—
486C4⋊C4—1424no—
487C4⋊C4C2×C4×D7112yes—
488C4⋊C4C2×Dic14212yes—
489C4⋊C4#1176586512yes—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C42—21no—
494C42—21no—
495C42—22no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—42no—
500C42—42no—
501C42—42no—
502C42—2842no—
503C42—2842no—
504C42C2×C4×D7112yes—
505C42C2×Dic14212yes—
506C42#1176586512yes—
507C42#1176586512yes—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C22×C22⋊D7111yescharacteristic
516D4—22no—
517D4—22no—
518D4—22no—
519D4C4×Dic14214yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—48no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—2841no—
546C2×C4—2841no—
547C2×C4—2841no—
548C2×C4—2841no—
549C2×C4—2841no—
550C2×C4—2841no—
551C2×C4—2841no—
552C2×C4—2841no—
553C2×C4—2841no—
554C2×C4—2841no—
555C2×C4—1422no—
556C2×C4—1422no—
557C2×C4C4×C22⋊D7112yes—
558C2×C4C4×C22⋊D7112yes—
559C2×C4#1455610712yes—
560C2×C4#1455610712yes—
561C2×C4#1455615412yes—
562C2×C4C22×C4×D7111yescharacteristic
563C2×C4C22×Dic14211yescharacteristic
564C2×C4C2×#1176586511yescharacteristic
565C2×C4C2×#1176586511yescharacteristic
566C2×C4C2×#1176586511yescharacteristic
567C2×C4C2×#1176586511yescharacteristic
568C8—2841no—
569C8—2841no—
570C8—1422no—
571C8—1422no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C2×C4×C22⋊D7111yescharacteristic
580C22C2×#1455610711yescharacteristic
581C22#1798651011yescharacteristic
582C22C2×#1455615411yescharacteristic
583C22#1798679711yescharacteristic
584C22#1798679711yescharacteristic
585C22#1798679711yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—22no—
590C4—22no—
591C4—42no—
592C4—41no—
593C4—42no—
594C4—42no—
595C4—2842no—
596C4—2842no—
597C4—5681no—
598C4—5681no—
599C4C2×C4×Dic14211yescharacteristic
600C4C4×#1176586511yescharacteristic
601C4#1798653311yescharacteristic
602C4#1798655611yescharacteristic
603C2—44no—
604C2C4×#1455610711yescharacteristic
605C2#2259713811yescharacteristic
606C2C4×#1455615411yescharacteristic
607C2#2259813711yescharacteristic
608C2#2259816211yescharacteristic
609C2#2259844911yescharacteristic
610C2#2259847811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC42×D4×C71C2
socleC23×C71C22×D4
prime radicalC22×D4×C71C23
cosocleC22×C4×C71C24
hypercenterC42×D4D71
hypocenterC71#21617
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C43 ◁ C42×D4

Complements

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

NH ≅ G/Ntype
2. C4×#14556120C2semidirect
3. #22589297C2semidirect
4. #22589108C2semidirect
5. #22588925C2semidirect
6. #22588911C2semidirect
10. #22588238C2semidirect
11. #22588238C2semidirect
12. #22588193C2semidirect
17. #17986299C4semidirect
65. #14556120C2×C4semidirect
77. #14556061C2×C4semidirect
79. #14556059C2×C4semidirect
81. C4⋊(C71⋊C8)C2×C4semidirect
321. C71#21617semidirect

Maximal subgroups

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

subgroupindex
C2×#179862992
C4×#145561202
#225892972
#225891082
#225889252
#225889112
#225886012
#225885872
#225883502
#225882382
#225881932
#225881652
C42×D4×C712
#2161771