Finite Group Databasebeta

← #32056125

Subgroups — #32056125

Order 18176.

894 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986403C211yesmaximal, characteristic, complemented
2C22×#14556061C211yesmaximal, characteristic
3#22592043C211yesmaximal, characteristic
4#22592036C211yesmaximal, characteristic
5#22589815C211yesmaximal, characteristic, complemented
6#22589814C211yesmaximal, characteristic, complemented
7#22589667C211yesmaximal, characteristic, complemented
8#22589666C211yesmaximal, characteristic, complemented
9#22589072C211yesmaximal, characteristic, complemented
10#22589072C211yesmaximal, characteristic, complemented
11#22588530C211yesmaximal, characteristic
12#22588498C211yesmaximal, characteristic, complemented
13#22588367C211yesmaximal, characteristic
14#22588114C211yesmaximal, characteristic
15C71×#1852C211yesmaximal, characteristic
16C23×C71⋊C8C2211yescharacteristic
17C2×#14556127C2211yescharacteristic
18C2×#14556127C2211yescharacteristic
19C2×#14556107—21no—
20C2×#14556105—21no—
21C22×C4⋊Dic71C2211yescharacteristic
22#17986403—24no—
23C2×#14556062C2211yescharacteristic
24C2×#14556062C2211yescharacteristic
25C2×#14556061—24no—
26C2×#14556061C2211yescharacteristic
27C2×#14556061C2211yescharacteristic
28C2×#14556061C2211yescharacteristic
29C2×#14556061C2211yescharacteristic
30C2×#14556061C2211yescharacteristic
31C2×#14556061C2211yescharacteristic
32C2×C8×Dic71C2211yescharacteristic
33C2×C8×Dic71C2211yescharacteristic
34#17985933—21no—
35#17985932—21no—
36#17985920—21no—
37#17985919—21no—
38#17985898C2211yescharacteristic
39#17985898C2211yescharacteristic
40#17985895C2211yescharacteristic
41#17985895C2211yescharacteristic
42#17985872—21no—
43#17985871—21no—
44#17985868C2211yescharacteristic
45#17985868C2211yescharacteristic
46#17985859C2211yescharacteristic
47C4×C22⋊Dic71C2211yescharacteristic
48#17985762C2211yescharacteristic
49#17985762C2211yescharacteristic
50#17985761C2211yescharacteristic
51#17985761C2211yescharacteristic
52#17985753C2211yescharacteristic
53#17985753C2211yescharacteristic
54C22×C4⋊C4×C71C2211yescharacteristic
55C2×#107×C71C2211yescharacteristic
56C2×#107×C71C2211yescharacteristic
57C2×C22⋊C8×C71C2211yescharacteristic
58#380×C71C2211yescharacteristic
59#340×C71C2211yescharacteristic
60#340×C71C2211yescharacteristic
61C22×Dic142—21no—
62C22×Dic142—21no—
63C23×Dic71—21no—
64C22×C71⋊C8—22no—
65C22×C71⋊C8—22no—
66C22×C71⋊C8Q814yescomplemented
67C22×C71⋊C8C2311yescharacteristic
68C22×C71⋊C8C2311yescharacteristic
69#14556127—22no—
70#14556127—22no—
71#14556127—41no—
72#14556127—41no—
73#14556107—44no—
74#14556105—44no—
75C2×C4⋊Dic71—22no—
76C2×C4⋊Dic71—21no—
77C2×C4⋊Dic71—21no—
78C2×C4⋊Dic71—21no—
79C2×C4⋊Dic71—41no—
80C2×C4⋊Dic71C2311yescharacteristic
81C2×C4⋊Dic71C2311yescharacteristic
82C2×C4⋊Dic71C2311yescharacteristic
83C2×C4⋊Dic71C2311yescharacteristic
84C2×Dic71⋊C4—21no—
85C2×Dic71⋊C4—21no—
86C2×Dic71⋊C4—21no—
87C2×Dic71⋊C4—21no—
88C2×Dic71⋊C4—21no—
89C2×Dic71⋊C4—21no—
90C2×Dic71⋊C4—21no—
91C2×Dic71⋊C4—21no—
92C2×C22⋊Dic71—21no—
93C2×C22⋊Dic71D411yescharacteristic
94C2×C22⋊Dic71D411yescharacteristic
95C2×C4×Dic71C2311yescharacteristic
96C2×C4×Dic71C2311yescharacteristic
97#14556062—22no—
98#14556062—41no—
99#14556061—22no—
100#14556061—22no—
101#14556061—22no—
102#14556061—22no—
103#14556061—22no—
104#14556061D414yes—
105C8×Dic71—22no—
106C8×Dic71—22no—
107C71⋊#95—21no—
108C71⋊#95—21no—
109C71⋊#95—21no—
110C71⋊#95—21no—
111C71⋊#95—21no—
112C71⋊#95D411yescharacteristic
113C71⋊#95D411yescharacteristic
114C23×C4×C71—21no—
115C23×C4×C71C2311yescharacteristic
116C22×C8×C71C2311yescharacteristic
117C22×C8×C71C2311yescharacteristic
118C2×C4⋊C4×C71—22no—
119C2×C4⋊C4×C71—21no—
120C2×C4⋊C4×C71—41no—
121C2×C4⋊C4×C71C2311yescharacteristic
122C2×C4⋊C4×C71C2311yescharacteristic
123C2×C4⋊C4×C71C2311yescharacteristic
124C2×C4⋊C4×C71C2311yescharacteristic
125C2×C22⋊C4×C71—21no—
126#107×C71—22no—
127#107×C71—41no—
128C22⋊C8×C71Q814yescomplemented
129#95×C71—21no—
130C2×Dic142—42no—
131C2×Dic142—42no—
132C2×Dic142—42no—
133C2×Dic142—42no—
134C2×Dic142—81no—
135C2×Dic142—81no—
136C22×Dic71—21no—
137C22×Dic71—21no—
138C22×Dic71—21no—
139C22×Dic71—21no—
140C22×Dic71—42no—
141C22×Dic71—42no—
142C22×Dic71—81no—
143C22×Dic71C2×D411yescharacteristic
144C22×Dic71C2×D411yescharacteristic
145C22×Dic71C2×D411yescharacteristic
146C22×Dic71C2×D411yescharacteristic
147C2×C71⋊C8—21no—
148C2×C71⋊C8—21no—
149C2×C71⋊C8—28no—
150C2×C71⋊C8C2×D412yes—
151C2×C71⋊C8C2×Q812yes—
152C2×C71⋊C8C2×Q812yes—
153C2×C71⋊C8#4112yes—
154C4⋊Dic71—21no—
155C4⋊Dic71—21no—
156C4⋊Dic71—21no—
157C4⋊Dic71—21no—
158C4⋊Dic71—21no—
159C4⋊Dic71—21no—
160C4⋊Dic71—21no—
161C4⋊Dic71—42no—
162C4⋊Dic71—42no—
163C4⋊Dic71—42no—
164C4⋊Dic71C2×D412yes—
165Dic71⋊C4—42no—
166Dic71⋊C4—42no—
167Dic71⋊C4—42no—
168Dic71⋊C4—42no—
169Dic71⋊C4—42no—
170Dic71⋊C4—42no—
171Dic71⋊C4—81no—
172Dic71⋊C4—81no—
173C22⋊Dic71—42no—
174C22⋊Dic71D814yes—
175C22⋊Dic71Q1614yes—
176C4×Dic71—21no—
177C4×Dic71—21no—
178C4×Dic71—21no—
179C4×Dic71—21no—
180C24×C71C2×D411yescharacteristic
181C22×C4×C71—21no—
182C22×C4×C71—21no—
183C22×C4×C71—21no—
184C22×C4×C71—21no—
185C22×C4×C71—21no—
186C22×C4×C71—21no—
187C22×C4×C71—21no—
188C22×C4×C71—21no—
189C22×C4×C71—42no—
190C22×C4×C71—42no—
191C22×C4×C71—81no—
192C22×C4×C71C2×D411yescharacteristic
193C22×C4×C71C2×Q812yes—
194C22×C4×C71C2×Q812yes—
195C22×C4×C71C2411yescharacteristic
196C2×C8×C71—21no—
197C2×C8×C71—21no—
198C2×C8×C71—21no—
199C2×C8×C71—21no—
200C2×C8×C71C2×Q812yes—
201C2×C8×C71C2×Q812yes—
202C4⋊C4×C71—21no—
203C4⋊C4×C71—21no—
204C4⋊C4×C71—21no—
205C4⋊C4×C71—21no—
206C4⋊C4×C71—21no—
207C4⋊C4×C71—21no—
208C4⋊C4×C71—21no—
209C4⋊C4×C71—42no—
210C4⋊C4×C71C2×D412yes—
211C22⋊C4×C71—42no—
212C23×C71—21no—
213C23×C71—21no—
214C23×C71—21no—
215C23×C71—21no—
216C23×C71C4⋊Q812yes—
217C23×C71C2×D812yes—
218C23×C71C2×Q1612yes—
219C23×C71C22×D411yescharacteristic
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—22no—
223C2×C4×C71—22no—
224C2×C4×C71—22no—
225C2×C4×C71—22no—
226C2×C4×C71—22no—
227C2×C4×C71—41no—
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—41no—
235C2×C4×C71—41no—
236C2×C4×C71—41no—
237C2×C4×C71—41no—
238C2×C4×C71—84no—
239C2×C4×C71C4⋊Q812yes—
240C2×C4×C71C22×D411yescharacteristic
241C2×C4×C71C22×Q811yescharacteristic
242C2×C4×C71C22×Q811yescharacteristic
243C2×C4×C71C2×#4111yescharacteristic
244C2×C4×C71H(5,𝔽2)11yescharacteristic
245C2×C4×C712−1+411yescharacteristic
246C2×Dic71—21no—
247C2×Dic71—21no—
248C2×Dic71—21no—
249C2×Dic71—21no—
250C2×Dic71—21no—
251C2×Dic71—21no—
252C2×Dic71—21no—
253C2×Dic71—21no—
254C2×Dic71—22no—
255C2×Dic71—41no—
256C2×Dic71—41no—
257C2×Dic71—41no—
258C2×Dic71—41no—
259C2×Dic71—41no—
260C2×Dic71—41no—
261C2×Dic71—41no—
262C2×Dic71—41no—
263C2×Dic71—41no—
264C2×Dic71—41no—
265C2×Dic71—41no—
266C2×Dic71—84no—
267C2×Dic71C2×D812yes—
268C2×Dic71C2×D812yes—
269C2×Dic71C2×Q1612yes—
270C2×Dic71C2×Q1612yes—
271Dic142—84no—
272Dic142—84no—
273C8×C71—22no—
274C8×C71—22no—
275C71⋊C8—22no—
276C71⋊C8C22⋊Q814yescomplemented
277C22×C71—22no—
278C22×C71—22no—
279C22×C71—22no—
280C22×C71—22no—
281C22×C71—22no—
282C22×C71—22no—
283C22×C71#50014yescomplemented
284C22×C71C2×C4⋊Q811yescharacteristic
285C22×C71D4211yescharacteristic
286C22×C71#54711yescharacteristic
287C22×C71C22×D811yescharacteristic
288C22×C71C22×Q1611yescharacteristic
289C22×C71#57611yescharacteristic
290C22×C71#57811yescharacteristic
291C4×C71—22no—
292C4×C71—42no—
293C4×C71—81no—
294C4×C71—81no—
295C4×C71—81no—
296C4×C71C2×C22⋊Q811yescharacteristic
297C4×C71#54411yescharacteristic
298C4×C71D4×Q811yescharacteristic
299C4×C71#55411yescharacteristic
300Dic71—22no—
301Dic71—22no—
302Dic71—22no—
303Dic71—22no—
304Dic71—42no—
305Dic71—81no—
306Dic71—81no—
307Dic71—81no—
308#26065—711nomaximal, Sylow
309C2×C71—24no—
310C2×C71#244011yescharacteristic
311C2×C71C2×#50011yescharacteristic
312C2×C71#316411yescharacteristic
313C2×C71D4×D811yescharacteristic
314C2×C71D4×Q1611yescharacteristic
315C2×C71#333411yescharacteristic
316C2×C71#333811yescharacteristic
317C2×#500—711no—
318C22×#107—711no—
319#2440—711no—
320#2440—711no—
321#2083—711no—
322#2083—711no—
323#2057—711no—
324#2057—711no—
325#1942—711no—
326#1942—711no—
327#1852—711no—
328#1852D7111yescharacteristic
329#1845—711no—
330#1808—711no—
331C8×C22⋊C4—711no—
332C71#2606511yescharacteristic, complemented, Sylow
333C23×C8—711no—
334C2×C4⋊Q8—711no—
335C2×C4⋊Q8—711no—
336C2×C22⋊Q8—1421no—
337C2×C22⋊Q8—1421no—
338C22×C4⋊C4—711no—
339C22×C4⋊C4C2×D7111yescharacteristic
340#500—1424no—
341C2×#107—711no—
342C2×#107—711no—
343C2×#107—711no—
344C2×#107—711no—
345C2×#107—711no—
346C2×#107—711no—
347C2×#107—711no—
348C2×#107—711no—
349C2×#107—1424no—
350C2×#107C2×D7111yescharacteristic
351C2×#107C2×D7111yescharacteristic
352C2×C22⋊C8C2×D7111yescharacteristic
353C2×C4×C8—711no—
354C2×C4×C8—711no—
355#398—1421no—
356#398—1421no—
357#395—1421no—
358#395—1421no—
359#389—711no—
360#389—711no—
361#389—711no—
362#389—711no—
363#385—1421no—
364#385—1421no—
365#384—711no—
366#384—711no—
367#380—711no—
368#380C2×D7111yescharacteristic
369C4×C22⋊C4—711no—
370#340—711no—
371#340—711no—
372#340—711no—
373#340—711no—
374#340C2×D7111yescharacteristic
375#340C2×D7111yescharacteristic
376#336—711no—
377#336—711no—
378C22×Q8—1421no—
379C22×Q8—1421no—
380C23×C4—21no—
381C23×C4—1421no—
382C23×C4C22×D7111yescharacteristic
383C22×C8—711no—
384C22×C8—714no—
385C22×C8—711no—
386C22×C8—1422no—
387C22×C8—1422no—
388C22×C8C22×D7111yescharacteristic
389C22×C8C22×D7111yescharacteristic
390C4⋊Q8—2841no—
391C4⋊Q8—2841no—
392C4⋊Q8—1422no—
393C4⋊Q8—1422no—
394C22⋊Q8—2844no—
395C22⋊Q8—2844no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—41no—
399C2×C4⋊C4—2841no—
400C2×C4⋊C4—711no—
401C2×C4⋊C4—711no—
402C2×C4⋊C4—711no—
403C2×C4⋊C4—711no—
404C2×C4⋊C4—1421no—
405C2×C4⋊C4—1421no—
406C2×C4⋊C4—1421no—
407C2×C4⋊C4—1421no—
408C2×C4⋊C4—1421no—
409C2×C4⋊C4—1421no—
410C2×C4⋊C4—1421no—
411C2×C4⋊C4—1421no—
412C2×C4⋊C4—1421no—
413C2×C4⋊C4—1422no—
414C2×C4⋊C4—1421no—
415C2×C4⋊C4—1421no—
416C2×C4⋊C4C22×D7111yescharacteristic
417C2×C4⋊C4C22×D7111yescharacteristic
418C2×C4⋊C4C22×D7111yescharacteristic
419C2×C4⋊C4C22×D7111yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—711no—
422C2×C22⋊C4—711no—
423C2×C22⋊C4—1421no—
424C2×C42—711no—
425C2×C42—711no—
426#107—22no—
427#107—41no—
428#107—2841no—
429#107—714no—
430#107—1422no—
431#107—1422no—
432#107—1422no—
433#107—1422no—
434#107—1422no—
435#107—1422no—
436C22⋊C8Dic14214yescomplemented
437C4×C8—1422no—
438C4×C8—1422no—
439#95—21no—
440#95—711no—
441#95—711no—
442#95—1421no—
443#95—1421no—
444#95—1421no—
445#95—1421no—
446#95—1421no—
447C24D4×D7111yescharacteristic
448C2×Q8—2842no—
449C2×Q8—2842no—
450C2×Q8—2842no—
451C2×Q8—2842no—
452C2×Q8—5681no—
453C2×Q8—5681no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—81no—
465C22×C4—2842no—
466C22×C4—2842no—
467C22×C4—5681no—
468C22×C4—711no—
469C22×C4—711no—
470C22×C4—711no—
471C22×C4—711no—
472C22×C4—1421no—
473C22×C4—1421no—
474C22×C4—1421no—
475C22×C4—1421no—
476C22×C4D4×D7111yescharacteristic
477C22×C4Q8×D7112yes—
478C22×C4C2×Dic14212yes—
479C22×C4C23×D7111yescharacteristic
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—712no—
485C2×C8—712no—
486C2×C8—712no—
487C2×C8—712no—
488C2×C8—1428no—
489C2×C8—1421no—
490C2×C8—1421no—
491C2×C8C2×Dic14212yes—
492C2×C8C2×Dic14212yes—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—42no—
501C4⋊C4—2842no—
502C4⋊C4—2842no—
503C4⋊C4—2842no—
504C4⋊C4—2842no—
505C4⋊C4—2842no—
506C4⋊C4—2842no—
507C4⋊C4—2842no—
508C4⋊C4—2842no—
509C4⋊C4—2842no—
510C4⋊C4—5681no—
511C4⋊C4—5681no—
512C4⋊C4—712no—
513C4⋊C4—1421no—
514C4⋊C4—1421no—
515C4⋊C4—1421no—
516C4⋊C4—1421no—
517C4⋊C4—1421no—
518C4⋊C4—1421no—
519C4⋊C4—1421no—
520C4⋊C4D4×D7112yes—
521C22⋊C4—42no—
522C22⋊C4—2842no—
523C22⋊C4—714no—
524C22⋊C4—714no—
525C42—1421no—
526C42—1421no—
527C42—1421no—
528C42—1421no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23#1455612712yes—
534C23D8×D7112yes—
535C23Q16×D7112yes—
536C23C2×D4×D7111yescharacteristic
537Q8—5684no—
538Q8—5684no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—84no—
558C2×C4—2841no—
559C2×C4—2841no—
560C2×C4—2841no—
561C2×C4—2841no—
562C2×C4—2841no—
563C2×C4—2841no—
564C2×C4—2841no—
565C2×C4—2841no—
566C2×C4—2841no—
567C2×C4—2841no—
568C2×C4—2841no—
569C2×C4—5684no—
570C2×C4—712no—
571C2×C4—712no—
572C2×C4—712no—
573C2×C4—712no—
574C2×C4—1422no—
575C2×C4—1421no—
576C2×C4—1421no—
577C2×C4—1421no—
578C2×C4—1421no—
579C2×C4—1421no—
580C2×C4—1421no—
581C2×C4—1421no—
582C2×C4—1421no—
583C2×C4#1455612712yes—
584C2×C4C2×D4×D7111yescharacteristic
585C2×C4C2×Q8×D7111yescharacteristic
586C2×C4C22×Dic14211yescharacteristic
587C2×C4#41×D7111yescharacteristic
588C2×C4#1455617311yescharacteristic
589C2×C4#1455617511yescharacteristic
590C8—22no—
591C8—22no—
592C8—714no—
593C8—1422no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22#1798640314yescomplemented
601C22C2×#1455612711yescharacteristic
602C22#1798663111yescharacteristic
603C22#1798664711yescharacteristic
604C22C2×D8×D7111yescharacteristic
605C22C2×Q16×D7111yescharacteristic
606C22#1798680311yescharacteristic
607C22#1798681511yescharacteristic
608C4—22no—
609C4—42no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—2842no—
614C4—5681no—
615C4—5681no—
616C4—5681no—
617C4—1422no—
618C4—1422no—
619C4—1422no—
620C4—1422no—
621C4C22⋊Q8×D7111yescharacteristic
622C4#1798662811yescharacteristic
623C4D4×Dic14211yescharacteristic
624C4#1798670811yescharacteristic
625C2—24no—
626C2#2259204111yescharacteristic
627C2C2×#1798640311yescharacteristic
628C2#2259887811yescharacteristic
629C2#2260001011yescharacteristic
630C2#2260008511yescharacteristic
631C2#2260068211yescharacteristic
632C2#2260072711yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#1852C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
hypercenter#1852D71
hypocenterC71#26065
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
1. C2×#17986403C2semidirect
5. #22589815C2semidirect
6. #22589814C2semidirect
7. #22589667C2semidirect
8. #22589666C2semidirect
9. #22589072C2semidirect
10. #22589072C2semidirect
12. #22588498C2semidirect
66. C22×C71⋊C8Q8semidirect
128. C22⋊C8×C71Q8semidirect
276. C71⋊C8C22⋊Q8semidirect
283. C22×C71#500semidirect
332. C71#26065semidirect
436. C22⋊C8Dic142semidirect
600. C22#17986403semidirect

Maximal subgroups

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

subgroupindex
C2×#179864032
C22×#145560612
#225920432
#225920362
#225898152
#225898142
#225896672
#225896662
#225890722
#225885302
#225884982
#225883672
#225881142
C71×#18522
#2606571