Finite Group Databasebeta

← #32005222

Subgroups — #32005222

Order 18176.

876 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986289C211yesmaximal, characteristic
2#22591088C211yesmaximal, characteristic, complemented
3#22589292C211yesmaximal, characteristic, complemented
4#22589108C211yesmaximal, characteristic, complemented
5#22588925C211yesmaximal, characteristic, complemented
6#22588918C211yesmaximal, characteristic, complemented
7#22588601C211yesmaximal, characteristic
8#22588587C211yesmaximal, characteristic
9#22588350C211yesmaximal, characteristic
10#22588238C211yesmaximal, characteristic, complemented
11C4×#14556059C211yesmaximal, characteristic, complemented
12#22588197C211yesmaximal, characteristic, complemented
13#22588165C211yesmaximal, characteristic
14C4×#14556049C211yesmaximal, characteristic
15C71×#2275C211yesmaximal, characteristic
16C2×#14556128C2211yescharacteristic
17#17986289C418yescomplemented
18C2×#14556061C2211yescharacteristic
19C2×#14556059C2211yescharacteristic
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×C4×C71⋊C8C2211yescharacteristic
30#17985903—21no—
31#17985903—21no—
32#17985897C2211yescharacteristic
33#17985897C2211yescharacteristic
34#17985882—21no—
35#17985882—21no—
36#17985865C2211yescharacteristic
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
52C4×C4⋊C4×C71C2211yescharacteristic
53C4×C4⋊C4×C71C2211yescharacteristic
54C4×C4⋊C4×C71C2211yescharacteristic
55C4×C22⋊C4×C71—21no—
56C4×C22⋊C4×C71—21no—
57C4×C22⋊C4×C71—21no—
58#376×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
67C2×C4⋊Dic71—21no—
68C2×C4⋊Dic71C2311yescharacteristic
69C2×C4⋊Dic71C2311yescharacteristic
70C2×C4⋊Dic71C2311yescharacteristic
71C2×C4⋊Dic71C2311yescharacteristic
72C2×Dic71⋊C4—21no—
73C2×Dic71⋊C4—21no—
74C2×C4×Dic71—21no—
75C2×C4×Dic71—21no—
76#14556061—22no—
77#14556059—22no—
78#14556059C2×C414yescomplemented
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—
87C4×C71⋊C8—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×C71Q814yescomplemented
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×C71C2311yescharacteristic
114C2×C42×C71C2311yescharacteristic
115C2×C42×C71C2311yescharacteristic
116#95×C71—21no—
117#95×C71—21no—
118#95×C71—21no—
119#95×C71D411yescharacteristic
120#95×C71D411yescharacteristic
121#95×C71D411yescharacteristic
122#95×C71D411yescharacteristic
123C2×Dic142—24no—
124C2×Dic142—81no—
125C2×Dic142—81no—
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—24no—
143C4⋊Dic71—21no—
144C4⋊Dic71—21no—
145C4⋊Dic71—21no—
146C4⋊Dic71—21no—
147C4⋊Dic71—21no—
148C4⋊Dic71—21no—
149C4⋊Dic71C22×C412yes—
150C4⋊Dic71C22×C412yes—
151Dic71⋊C4—81no—
152Dic71⋊C4—81no—
153C4×Dic71—81no—
154C4×Dic71—81no—
155C24×C71—21no—
156C2×D4×C71—21no—
157C2×D4×C71—21no—
158C2×D4×C71—21no—
159C2×D4×C71C22×C412yes—
160C2×D4×C71C2×Q812yes—
161C2×D4×C71#4112yes—
162C22×C4×C71—21no—
163C22×C4×C71—21no—
164C22×C4×C71—21no—
165C22×C4×C71—24no—
166C22×C4×C71—24no—
167C22×C4×C71—24no—
168C22×C4×C71—81no—
169C22×C4×C71—81no—
170C22×C4×C71—81no—
171C22×C4×C71—81no—
172C22×C4×C71—81no—
173C22×C4×C71—81no—
174C22×C4×C71C2×D411yescharacteristic
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71C2×D411yescharacteristic
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2411yescharacteristic
181C4⋊C4×C71—21no—
182C4⋊C4×C71—21no—
183C4⋊C4×C71—21no—
184C4⋊C4×C71C22×C412yes—
185C4⋊C4×C71C2×Q812yes—
186C4⋊C4×C71#4112yes—
187C22⋊C4×C71—24no—
188C22⋊C4×C71—24no—
189C22⋊C4×C71—24no—
190C42×C71—21no—
191C42×C71—21no—
192C42×C71—81no—
193C42×C71—81no—
194C42×C71—81no—
195C42×C71C22×C412yes—
196C42×C71C2×Q812yes—
197C42×C71#4112yes—
198C42×C71#4112yes—
199C23×C71—22no—
200C23×C71—22no—
201C23×C71—22no—
202C23×C71—41no—
203C23×C71—41no—
204C23×C71—41no—
205C23×C71—41no—
206C23×C71C22×D411yescharacteristic
207D4×C71—22no—
208D4×C71—22no—
209D4×C71—22no—
210D4×C71C4×Q814yes—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—21no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—22no—
220C2×C4×C71—22no—
221C2×C4×C71—21no—
222C2×C4×C71—21no—
223C2×C4×C71—22no—
224C2×C4×C71—21no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—41no—
229C2×C4×C71—41no—
230C2×C4×C71—41no—
231C2×C4×C71—41no—
232C2×C4×C71—41no—
233C2×C4×C71—41no—
234C2×C4×C71—84no—
235C2×C4×C71—84no—
236C2×C4×C71—84no—
237C2×C4×C71C4×D412yes—
238C2×C4×C71C4×D412yes—
239C2×C4×C71C22⋊Q812yes—
240C2×C4×C71C22⋊Q812yes—
241C2×C4×C71C23×C411yescharacteristic
242C2×C4×C71C22×Q811yescharacteristic
243C2×C4×C71C2×#4111yescharacteristic
244C2×C4×C71C2×#4111yescharacteristic
245C2×C4×C71C2×#4111yescharacteristic
246C2×C4×C71C2×#4111yescharacteristic
247C2×Dic71—22no—
248C2×Dic71—22no—
249C2×Dic71—41no—
250C2×Dic71—41no—
251C2×Dic71—41no—
252C2×Dic71—41no—
253C2×Dic71—41no—
254C2×Dic71—41no—
255C2×Dic71—41no—
256C2×Dic71—41no—
257C2×Dic71—41no—
258C2×Dic71—41no—
259Dic142—84no—
260C71⋊C8—22no—
261C71⋊C8—22no—
262C71⋊C8—41no—
263C71⋊C8—41no—
264C22×C71—24no—
265C22×C71—42no—
266C22×C71—42no—
267C22×C71—42no—
268C22×C71—42no—
269C22×C71—42no—
270C22×C71—42no—
271C22×C71C2×C4×D411yescharacteristic
272C22×C71C2×C22⋊Q811yescharacteristic
273C22×C71#52511yescharacteristic
274C22×C71#57611yescharacteristic
275C22×C71#57711yescharacteristic
276C22×C71#57711yescharacteristic
277C22×C71#57711yescharacteristic
278C4×C71—22no—
279C4×C71—22no—
280C4×C71—22no—
281C4×C71—22no—
282C4×C71—42no—
283C4×C71—41no—
284C4×C71—42no—
285C4×C71—41no—
286C4×C71—42no—
287C4×C71C2×C4×Q811yescharacteristic
288C4×C71C4×#4111yescharacteristic
289C4×C71#52911yescharacteristic
290C4×C71#53311yescharacteristic
291Dic71—42no—
292Dic71—42no—
293Dic71—81no—
294Dic71—81no—
295#21623—711nomaximal, Sylow
296C2×C71—44no—
297C2×C71C4×C22⋊Q811yescharacteristic
298C2×C71#294611yescharacteristic
299C2×C71#294911yescharacteristic
300C2×C71#308111yescharacteristic
301C2×C71#308411yescharacteristic
302C2×C71#311811yescharacteristic
303C2×C71#312511yescharacteristic
304C2×#476—711no—
305#2331—711no—
306#2275D7111yescharacteristic
307#1988—711no—
308#1949—711no—
309#1920—711no—
310#1919—711no—
311#1864—711no—
312#1862—711no—
313#1804—711no—
314#1776—711no—
315C4×#106—711no—
316#1768—711no—
317#1762—711no—
318C4×#102—711no—
319C71#2162311yescharacteristic, complemented, Sylow
320C2×C4⋊Q8—711no—
321C2×C4×D4C2×D7111yescharacteristic
322C2×C4×D4C2×D7111yescharacteristic
323C2×C4×D4C2×D7111yescharacteristic
324#476—718no—
325C2×#107—711no—
326C2×#106—711no—
327C2×#106—711no—
328C2×#106—711no—
329C2×C4⋊C8—711no—
330C2×C4⋊C8—711no—
331C2×#102—711no—
332C2×#102—711no—
333C2×#102—711no—
334C2×#102—711no—
335C2×#97—711no—
336C2×C4×C8—711no—
337#391—1421no—
338#391—1421no—
339#389—711no—
340#389—711no—
341#387—1421no—
342#387—1421no—
343#382—711no—
344C4×C4⋊C4—711no—
345C4×C4⋊C4—711no—
346C4×C4⋊C4C2×D7111yescharacteristic
347C4×C4⋊C4C2×D7111yescharacteristic
348C4×C4⋊C4C2×D7111yescharacteristic
349C4×C22⋊C4—21no—
350C4×C22⋊C4—21no—
351C4×C22⋊C4—21no—
352#376C2×D7111yescharacteristic
353C2×#95—21no—
354#340—711no—
355#340—711no—
356#340—711no—
357#340—711no—
358#340—711no—
359#340—711no—
360#340—711no—
361#340—711no—
362#336—711no—
363#336—711no—
364C22×Q8—1421no—
365C22×D4C22×D7111yescharacteristic
366C23×C4—21no—
367C23×C4—21no—
368C23×C4—21no—
369C22×C8—711no—
370C22×C8—711no—
371C22×C8—711no—
372C22×C8—711no—
373C4⋊Q8—714no—
374C4⋊Q8—1422no—
375C4×D4—22no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4Dic14214yescomplemented
380C4×D4C4×D7114yes—
381C2×C4⋊C4—711no—
382C2×C4⋊C4—711no—
383C2×C4⋊C4—711no—
384C2×C4⋊C4—711no—
385C2×C4⋊C4—1421no—
386C2×C4⋊C4—1421no—
387C2×C4⋊C4—1421no—
388C2×C4⋊C4C22×D7111yescharacteristic
389C2×C4⋊C4C22×D7111yescharacteristic
390C2×C4⋊C4C22×D7111yescharacteristic
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—1421no—
398C2×C42—1421no—
399C2×C42C22×D7111yescharacteristic
400C2×C42C22×D7111yescharacteristic
401C2×C42C22×D7111yescharacteristic
402#107—1422no—
403#106—714no—
404#106—714no—
405#106—1422no—
406C4⋊C8—714no—
407C4⋊C8—1422no—
408#102—714no—
409#102—714no—
410#102—1422no—
411#102—1422no—
412#97—1422no—
413C4×C8—1422no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—1421no—
418#95—1421no—
419#95—1421no—
420#95—1421no—
421#95—1421no—
422#95—1421no—
423#95C22⋊D7111yescharacteristic
424#95C22⋊D7111yescharacteristic
425#95C22⋊D7111yescharacteristic
426#95C22⋊D7111yescharacteristic
427C24—21no—
428C2×Q8—5681no—
429C2×Q8—5681no—
430C2×Q8—1424no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4C2×C4×D7112yes—
435C2×D4C2×Dic14212yes—
436C2×D4#1176586512yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—81no—
446C22×C4—81no—
447C22×C4—81no—
448C22×C4—81no—
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—5681no—
476C4⋊C4—5681no—
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⋊C4C2×C4×D7112yes—
487C4⋊C4C2×Dic14212yes—
488C4⋊C4#1176586512yes—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C42—21no—
493C42—21no—
494C42—81no—
495C42—81no—
496C42—81no—
497C42—5681no—
498C42—5681no—
499C42C2×C4×D7112yes—
500C42C2×Dic14212yes—
501C42#1176586512yes—
502C42#1176586512yes—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C22×C22⋊D7111yescharacteristic
511Q8—5684no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4C4×Dic14214yes—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4—2841no—
543C2×C4—2841no—
544C2×C4—2841no—
545C2×C4—2841no—
546C2×C4—2841no—
547C2×C4—2841no—
548C2×C4—2841no—
549C2×C4—2841no—
550C2×C4—2841no—
551C2×C4—2841no—
552C2×C4—1422no—
553C2×C4—1422no—
554C2×C4C4×C22⋊D7112yes—
555C2×C4C4×C22⋊D7112yes—
556C2×C4#1455610712yes—
557C2×C4#1455610712yes—
558C2×C4C22×C4×D7111yescharacteristic
559C2×C4C22×Dic14211yescharacteristic
560C2×C4C2×#1176586511yescharacteristic
561C2×C4C2×#1176586511yescharacteristic
562C2×C4C2×#1176586511yescharacteristic
563C2×C4C2×#1176586511yescharacteristic
564C8—2841no—
565C8—2841no—
566C8—1422no—
567C8—1422no—
568C22—24no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22C2×C4×C22⋊D7111yescharacteristic
576C22C2×#1455610711yescharacteristic
577C22#1798651011yescharacteristic
578C22#1798680211yescharacteristic
579C22#1798680711yescharacteristic
580C22#1798681011yescharacteristic
581C22#1798681011yescharacteristic
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—42no—
587C4—42no—
588C4—41no—
589C4—42no—
590C4—41no—
591C4—2842no—
592C4—2842no—
593C4—5681no—
594C4—5681no—
595C4C2×C4×Dic14211yescharacteristic
596C4C4×#1176586511yescharacteristic
597C4#1798653311yescharacteristic
598C4#1798655611yescharacteristic
599C2—44no—
600C2C4×#1455610711yescharacteristic
601C2#2259716811yescharacteristic
602C2#2259719711yescharacteristic
603C2#2259817111yescharacteristic
604C2#2259820211yescharacteristic
605C2#2259852411yescharacteristic
606C2#2259861711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2275C2
socleC23×C71C22×D4
prime radicalC22×D4×C71C23
cosocleC22×C4×C71C24
hypercenter#2275D71
hypocenterC71#21623
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ #376 ◁ #2275

Complements

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

NH ≅ G/Ntype
2. #22591088C2semidirect
3. #22589292C2semidirect
4. #22589108C2semidirect
5. #22588925C2semidirect
6. #22588918C2semidirect
10. #22588238C2semidirect
11. C4×#14556059C2semidirect
12. #22588197C2semidirect
17. #17986289C4semidirect
66. #14556128C2×C4semidirect
78. #14556059C2×C4semidirect
79. #14556059C2×C4semidirect
81. C4⋊(C71⋊C8)C2×C4semidirect
103. C4×D4×C71Q8semidirect
319. C71#21623semidirect
379. C4×D4Dic142semidirect

Maximal subgroups

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

subgroupindex
C2×#179862892
#225910882
#225892922
#225891082
#225889252
#225889182
#225886012
#225885872
#225883502
#225882382
C4×#145560592
#225881972
#225881652
C4×#145560492
C71×#22752
#2162371