Finite Group Databasebeta

← #32019416

Subgroups — #32019416

Order 18176.

878 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986289C211yesmaximal, characteristic
2#22591398C211yesmaximal, characteristic, complemented
3#22589312C211yesmaximal, characteristic, complemented
4#22589106C211yesmaximal, characteristic, complemented
5#22589080C211yesmaximal, characteristic, complemented
6#22589065C211yesmaximal, characteristic, complemented
7#22588922C211yesmaximal, characteristic, complemented
8#22588915C211yesmaximal, characteristic, complemented
9#22588601C211yesmaximal, characteristic
10#22588587C211yesmaximal, characteristic
11#22588215C211yesmaximal, characteristic, complemented
12#22588165C211yesmaximal, characteristic
13#22588165C211yesmaximal, characteristic
14C4×#14556049C211yesmaximal, characteristic
15C71×#2349C211yesmaximal, characteristic
16C2×#14556128C2211yescharacteristic
17C2×#14556093—21no—
18#17986289C418yescomplemented
19C2×#14556061C2211yescharacteristic
20C2×#14556059C2211yescharacteristic
21C2×#14556059C2211yescharacteristic
22C2×#14556059C2211yescharacteristic
23C2×C4⋊(C71⋊C8)C2211yescharacteristic
24C2×#14556049C2211yescharacteristic
25C2×#14556049C2211yescharacteristic
26C2×#14556049C2211yescharacteristic
27C2×#14556049C2211yescharacteristic
28C2×#14556039C2211yescharacteristic
29C2×#14556039C2211yescharacteristic
30C2×C4×C71⋊C8C2211yescharacteristic
31#17985904—21no—
32#17985897C2211yescharacteristic
33#17985897C2211yescharacteristic
34#17985884—21no—
35#17985884—21no—
36C4×C4⋊Dic71C2211yescharacteristic
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
52#389×C71C2211yescharacteristic
53#389×C71C2211yescharacteristic
54#385×C71—21no—
55#385×C71—21no—
56#382×C71C2211yescharacteristic
57#380×C71—21no—
58C4×C4⋊C4×C71C2211yescharacteristic
59C2×#95×C71—21no—
60C22×Dic142—21no—
61C22×C71⋊C8C2311yescharacteristic
62C22×C71⋊C8C2311yescharacteristic
63C22×C71⋊C8C2311yescharacteristic
64C22×C71⋊C8C2311yescharacteristic
65#14556128—22no—
66#14556128C2×C414yescomplemented
67#14556093—44no—
68C2×C4⋊Dic71—21no—
69C2×C4⋊Dic71C2311yescharacteristic
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C4⋊Dic71C2311yescharacteristic
72C2×C4⋊Dic71C2311yescharacteristic
73C2×Dic71⋊C4—21no—
74C2×Dic71⋊C4—21no—
75C2×C4×Dic71—21no—
76C2×C4×Dic71—21no—
77C2×C4×Dic71—21no—
78#14556061—41no—
79#14556059—41no—
80#14556059C2×C414yescomplemented
81#14556059C2×C414yescomplemented
82C4⋊(C71⋊C8)C2×C414yescomplemented
83#14556049—22no—
84#14556049C2×C414yes—
85#14556049C2×C414yes—
86#14556049Q814yescomplemented
87#14556039—22no—
88#14556039—41no—
89C4×C71⋊C8—41no—
90C71⋊#95—21no—
91C71⋊#95—21no—
92C71⋊#95—21no—
93C71⋊#95—21no—
94C71⋊#95—21no—
95C22×D4×C71C2311yescharacteristic
96C23×C4×C71—21no—
97C23×C4×C71—21no—
98C23×C4×C71—21no—
99C4×D4×C71—22no—
100C4×D4×C71—22no—
101C4×D4×C71—22no—
102C4×D4×C71—41no—
103C4×D4×C71—41no—
104C4×D4×C71C2×C414yes—
105C2×C4⋊C4×C71—21no—
106C2×C4⋊C4×C71—21no—
107C2×C4⋊C4×C71D411yescharacteristic
108C2×C4⋊C4×C71D411yescharacteristic
109C2×C4⋊C4×C71C2311yescharacteristic
110C2×C4⋊C4×C71C2311yescharacteristic
111C2×C4⋊C4×C71C2311yescharacteristic
112C2×C22⋊C4×C71—21no—
113C2×C22⋊C4×C71—21no—
114C2×C22⋊C4×C71—21no—
115C2×C42×C71C2311yescharacteristic
116C2×C42×C71C2311yescharacteristic
117C2×C42×C71C2311yescharacteristic
118#95×C71—21no—
119#95×C71—21no—
120#95×C71—21no—
121#95×C71—21no—
122#95×C71D411yescharacteristic
123#95×C71D411yescharacteristic
124C2×Dic142—24no—
125C2×Dic142—42no—
126C2×Dic142—42no—
127C22×Dic71—21no—
128C22×Dic71—21no—
129C22×Dic71—21no—
130C22×Dic71—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—
142C2×C71⋊C8C2×Q812yes—
143C4⋊Dic71—24no—
144C4⋊Dic71—21no—
145C4⋊Dic71—21no—
146C4⋊Dic71—21no—
147C4⋊Dic71—21no—
148C4⋊Dic71—21no—
149C4⋊Dic71C22×C412yes—
150C4⋊Dic71C22×C412yes—
151C4⋊Dic71C2×Q812yes—
152Dic71⋊C4—42no—
153Dic71⋊C4—42no—
154C4×Dic71—42no—
155C4×Dic71—42no—
156C4×Dic71—42no—
157C24×C71—21no—
158C2×D4×C71—21no—
159C2×D4×C71—21no—
160C2×D4×C71—21no—
161C2×D4×C71C22×C412yes—
162C2×D4×C71C2×Q812yes—
163C2×D4×C71#4112yes—
164C22×C4×C71—21no—
165C22×C4×C71—24no—
166C22×C4×C71—21no—
167C22×C4×C71—21no—
168C22×C4×C71—42no—
169C22×C4×C71—42no—
170C22×C4×C71—42no—
171C22×C4×C71—42no—
172C22×C4×C71—81no—
173C22×C4×C71—81no—
174C22×C4×C71—81no—
175C22×C4×C71—81no—
176C22×C4×C71C2×D411yescharacteristic
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2×D411yescharacteristic
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2411yescharacteristic
183C4⋊C4×C71—22no—
184C4⋊C4×C71—21no—
185C4⋊C4×C71—21no—
186C4⋊C4×C71—21no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71—41no—
190C4⋊C4×C71—81no—
191C4⋊C4×C71—81no—
192C4⋊C4×C71C22×C412yes—
193C22⋊C4×C71—24no—
194C22⋊C4×C71—42no—
195C22⋊C4×C71—42no—
196C42×C71—21no—
197C42×C71—21no—
198C42×C71—21no—
199C42×C71—21no—
200C42×C71C22×C412yes—
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×Q814yescomplemented
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—22no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
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—41no—
231C2×C4×C71—41no—
232C2×C4×C71—41no—
233C2×C4×C71—41no—
234C2×C4×C71—41no—
235C2×C4×C71—41no—
236C2×C4×C71—41no—
237C2×C4×C71—41no—
238C2×C4×C71—84no—
239C2×C4×C71—84no—
240C2×C4×C71—84no—
241C2×C4×C71C4×D412yes—
242C2×C4×C71C4×D412yes—
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×C71H(5,𝔽2)11yescharacteristic
249C2×C4×C712−1+411yescharacteristic
250C2×Dic71—22no—
251C2×Dic71—22no—
252C2×Dic71—22no—
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—
262Dic142—48no—
263C71⋊C8—22no—
264C71⋊C8—22no—
265C71⋊C8—22no—
266C71⋊C8—41no—
267C22×C71—24no—
268C22×C71—42no—
269C22×C71—42no—
270C22×C71—42no—
271C22×C71—42no—
272C22×C71—42no—
273C22×C71—42no—
274C22×C71C2×C4×D411yescharacteristic
275C22×C71#54611yescharacteristic
276C22×C71D4×Q811yescharacteristic
277C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C71#57511yescharacteristic
279C22×C71#57711yescharacteristic
280C22×C71#57711yescharacteristic
281C4×C71—22no—
282C4×C71—22no—
283C4×C71—22no—
284C4×C71—42no—
285C4×C71—41no—
286C4×C71—41no—
287C4×C71—41no—
288C4×C71—81no—
289C4×C71—81no—
290C4×C71C2×C4×Q811yescharacteristic
291C4×C71#52011yescharacteristic
292C4×C71#55211yescharacteristic
293C4×C71#55411yescharacteristic
294Dic71—42no—
295Dic71—42no—
296Dic71—42no—
297Dic71—81no—
298#22975—711nomaximal, Sylow
299C2×C71—44no—
300C2×C71#236011yescharacteristic
301C2×C71C4×AGL(1,ℤ/8)11yescharacteristic
302C2×C71#294911yescharacteristic
303C2×C71#330611yescharacteristic
304C2×C71#331111yescharacteristic
305C2×C71#338411yescharacteristic
306C2×C71#338811yescharacteristic
307C2×#476—711no—
308#2365—711no—
309#2349D7111yescharacteristic
310#1993—711no—
311#1949—711no—
312#1944—711no—
313#1941—711no—
314#1920—711no—
315#1919—711no—
316#1864—711no—
317#1862—711no—
318#1771—711no—
319#1762—711no—
320#1762—711no—
321C4×#102—711no—
322C71#2297511yescharacteristic, complemented, Sylow
323C2×C4⋊Q8—711no—
324C2×C4×Q8—1421no—
325C2×C4×D4C2×D7111yescharacteristic
326C2×C4×D4C2×D7111yescharacteristic
327C2×C4×D4C2×D7111yescharacteristic
328#476—718no—
329C2×#107—711no—
330C2×#106—711no—
331C2×#106—711no—
332C2×#106—711no—
333C2×C4⋊C8—711no—
334C2×#102—711no—
335C2×#102—711no—
336C2×#102—711no—
337C2×#102—711no—
338C2×#97—711no—
339C2×#97—711no—
340C2×C4×C8—711no—
341#391—1421no—
342#389—711no—
343#389—711no—
344#389C2×D7111yescharacteristic
345#389C2×D7111yescharacteristic
346#387—1421no—
347#387—1421no—
348#385—21no—
349#385—21no—
350#382C2×D7111yescharacteristic
351#380—21no—
352C4×C4⋊C4—711no—
353C4×C4⋊C4—711no—
354C4×C4⋊C4—711no—
355C4×C4⋊C4C2×D7111yescharacteristic
356C2×#95—21no—
357#340—711no—
358#340—711no—
359#340—711no—
360#340—711no—
361#340—711no—
362#340—711no—
363#340—711no—
364#340—711no—
365#336—711no—
366#336—711no—
367C22×Q8—1421no—
368C22×D4C22×D7111yescharacteristic
369C23×C4—21no—
370C23×C4—21no—
371C23×C4—21no—
372C22×C8—711no—
373C22×C8—711no—
374C22×C8—711no—
375C22×C8—711no—
376C4⋊Q8—714no—
377C4⋊Q8—1422no—
378C4×Q8—2844no—
379C4×D4—22no—
380C4×D4—22no—
381C4×D4—22no—
382C4×D4—41no—
383C4×D4—41no—
384C4×D4C4×D7114yes—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—711no—
388C2×C4⋊C4—711no—
389C2×C4⋊C4—711no—
390C2×C4⋊C4—711no—
391C2×C4⋊C4—1421no—
392C2×C4⋊C4—1421no—
393C2×C4⋊C4—1421no—
394C2×C4⋊C4C22⋊D7111yescharacteristic
395C2×C4⋊C4C22⋊D7111yescharacteristic
396C2×C4⋊C4C22×D7111yescharacteristic
397C2×C4⋊C4C22×D7111yescharacteristic
398C2×C4⋊C4C22×D7111yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C42—1421no—
403C2×C42—1421no—
404C2×C42—1421no—
405C2×C42C22×D7111yescharacteristic
406C2×C42C22×D7111yescharacteristic
407C2×C42C22×D7111yescharacteristic
408#107—2841no—
409#106—2841no—
410#106—714no—
411#106—714no—
412C4⋊C8—714no—
413#102—714no—
414#102—714no—
415#102—714no—
416#102—1422no—
417#97—2841no—
418#97—1422no—
419C4×C8—2841no—
420#95—21no—
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#95C22⋊D7111yescharacteristic
430#95C22⋊D7111yescharacteristic
431C24—21no—
432C2×Q8—2842no—
433C2×Q8—2842no—
434C2×Q8—1424no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4C2×C4×D7112yes—
439C2×D4Q8×D7112yes—
440C2×D4#1176586412yes—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—81no—
452C22×C4—81no—
453C22×C4—1421no—
454C22×C4—1421no—
455C22×C4—1421no—
456C22×C4—1421no—
457C22×C4C2×C22⋊D7111yescharacteristic
458C22×C4C2×C22⋊D7111yescharacteristic
459C22×C4C2×C22⋊D7111yescharacteristic
460C22×C4C2×C22⋊D7111yescharacteristic
461C22×C4C2×C22⋊D7111yescharacteristic
462C22×C4C2×C22⋊D7111yescharacteristic
463C22×C4C23×D7111yescharacteristic
464C2×C8—712no—
465C2×C8—712no—
466C2×C8—712no—
467C2×C8—1421no—
468C2×C8—1421no—
469C2×C8—1421no—
470C2×C8—1421no—
471C2×C8—1421no—
472C2×C8—1421no—
473C2×C8—1421no—
474C2×C8—1421no—
475C2×C8—1421no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—41no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—2842no—
486C4⋊C4—2842no—
487C4⋊C4—712no—
488C4⋊C4—712no—
489C4⋊C4—712no—
490C4⋊C4—1421no—
491C4⋊C4—1421no—
492C4⋊C4—1421no—
493C4⋊C4—1421no—
494C4⋊C4—1421no—
495C4⋊C4—1424no—
496C4⋊C4C2×C4×D7112yes—
497C22⋊C4—24no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—2842no—
505C42—2842no—
506C42—2842no—
507C42C2×C4×D7112yes—
508C42#1176586512yes—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23C22×C22⋊D7111yescharacteristic
517Q8—2848no—
518D4—22no—
519D4—22no—
520D4—22no—
521D4#1455609314yescomplemented
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—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4—2841no—
550C2×C4—2841no—
551C2×C4—2841no—
552C2×C4—2841no—
553C2×C4—2841no—
554C2×C4—2841no—
555C2×C4—2841no—
556C2×C4—2841no—
557C2×C4—2841no—
558C2×C4—1422no—
559C2×C4—1422no—
560C2×C4—1422no—
561C2×C4C4×C22⋊D7112yes—
562C2×C4C4×C22⋊D7112yes—
563C2×C4#1455615312yes—
564C2×C4C22×C4×D7111yescharacteristic
565C2×C4C2×Q8×D7111yescharacteristic
566C2×C4C2×#1176586411yescharacteristic
567C2×C4C2×#1176586511yescharacteristic
568C2×C4#1455617311yescharacteristic
569C2×C4#1455617611yescharacteristic
570C8—2841no—
571C8—1422no—
572C8—1422no—
573C8—1422no—
574C22—24no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22C2×C4×C22⋊D7111yescharacteristic
582C22#1798664311yescharacteristic
583C22Q8×C22⋊D7111yescharacteristic
584C22C2×#1455615311yescharacteristic
585C22#1798679711yescharacteristic
586C22#1798681011yescharacteristic
587C22#1798681011yescharacteristic
588C4—22no—
589C4—22no—
590C4—22no—
591C4—42no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—81no—
596C4—81no—
597C4—2842no—
598C4—2842no—
599C4—2842no—
600C4—5681no—
601C4C2×#1455609311yescharacteristic
602C4#1798648011yescharacteristic
603C4#1798668411yescharacteristic
604C4#1798670411yescharacteristic
605C2—44no—
606C2#2259134511yescharacteristic
607C2C4×#1455615311yescharacteristic
608C2#2259719711yescharacteristic
609C2#2260030111yescharacteristic
610C2#2260037611yescharacteristic
611C2#2260123911yescharacteristic
612C2#2260128511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2349C2
socleC23×C71C22×D4
prime radicalC22×D4×C71C23
cosocleC22×C4×C71C24
hypercenter#2349D71
hypocenterC71#22975
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
2. #22591398C2semidirect
3. #22589312C2semidirect
4. #22589106C2semidirect
5. #22589080C2semidirect
6. #22589065C2semidirect
7. #22588922C2semidirect
8. #22588915C2semidirect
11. #22588215C2semidirect
18. #17986289C4semidirect
66. #14556128C2×C4semidirect
80. #14556059C2×C4semidirect
81. #14556059C2×C4semidirect
82. C4⋊(C71⋊C8)C2×C4semidirect
86. #14556049Q8semidirect
213. D4×C71C4×Q8semidirect
322. C71#22975semidirect
521. D4#14556093semidirect

Maximal subgroups

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

subgroupindex
C2×#179862892
#225913982
#225893122
#225891062
#225890802
#225890652
#225889222
#225889152
#225886012
#225885872
#225882152
#225881652
C4×#145560492
C71×#23492
#2297571