Finite Group Databasebeta

← #32056201

Subgroups — #32056201

Order 18176.

838 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986405C211yesmaximal, characteristic, complemented
2C22×#14556061C211yesmaximal, characteristic
3#22592064C211yesmaximal, characteristic
4#22589821C211yesmaximal, characteristic, complemented
5#22589812C211yesmaximal, characteristic, complemented
6#22589678C211yesmaximal, characteristic, complemented
7#22589665C211yesmaximal, characteristic, complemented
8#22589080C211yesmaximal, characteristic, complemented
9#22589075C211yesmaximal, characteristic, complemented
10#22588533C211yesmaximal, characteristic
11#22588527C211yesmaximal, characteristic
12#22588506C211yesmaximal, characteristic, complemented
13#22588363C211yesmaximal, characteristic
14#22588118C211yesmaximal, characteristic
15C71×#2440C211yesmaximal, characteristic
16C23×C71⋊C8C2211yescharacteristic
17C2×#14556120C2211yescharacteristic
18C22×C4⋊Dic71C2211yescharacteristic
19#17986405—24no—
20C2×#14556061—24no—
21C2×#14556061C2211yescharacteristic
22C2×#14556061C2211yescharacteristic
23C2×#14556061C2211yescharacteristic
24C2×#14556061C2211yescharacteristic
25C2×#14556061C2211yescharacteristic
26C2×#14556061C2211yescharacteristic
27C2×#14556059C2211yescharacteristic
28C2×#14556059C2211yescharacteristic
29C2×#14556059C2211yescharacteristic
30C2×#14556059C2211yescharacteristic
31C2×C22⋊(C71⋊C8)C2211yescharacteristic
32C2×#14556039C2211yescharacteristic
33C2×#14556039C2211yescharacteristic
34#17985941—21no—
35#17985933—21no—
36#17985897C2211yescharacteristic
37#17985897C2211yescharacteristic
38#17985871—21no—
39#17985865C2211yescharacteristic
40#17985859C2211yescharacteristic
41#17985857—21no—
42#17985761C2211yescharacteristic
43#17985761C2211yescharacteristic
44#17985761C2211yescharacteristic
45#17985761C2211yescharacteristic
46#17985761C2211yescharacteristic
47#17985761C2211yescharacteristic
48#17985754C2211yescharacteristic
49#17985754C2211yescharacteristic
50C2×C4⋊Q8×C71C2211yescharacteristic
51C2×C22⋊Q8×C71—21no—
52C22×C4⋊C4×C71C2211yescharacteristic
53#398×C71—21no—
54#395×C71—21no—
55#389×C71C2211yescharacteristic
56#389×C71C2211yescharacteristic
57#385×C71—21no—
58#384×C71C2211yescharacteristic
59#380×C71C2211yescharacteristic
60C4×C22⋊C4×C71C2211yescharacteristic
61C23×Dic71—21no—
62C22×C71⋊C8—22no—
63C22×C71⋊C8—22no—
64C22×C71⋊C8Q814yescomplemented
65C22×C71⋊C8C2311yescharacteristic
66C22×C71⋊C8C2311yescharacteristic
67C22×C71⋊C8C2311yescharacteristic
68C22×C71⋊C8C2311yescharacteristic
69#14556120—22no—
70#14556120—41no—
71C2×C4⋊Dic71—22no—
72C2×C4⋊Dic71—21no—
73C2×C4⋊Dic71—41no—
74C2×C4⋊Dic71C2311yescharacteristic
75C2×C4⋊Dic71C2311yescharacteristic
76C2×C4⋊Dic71C2311yescharacteristic
77C2×C4⋊Dic71C2311yescharacteristic
78C2×Dic71⋊C4—21no—
79C2×Dic71⋊C4—21no—
80C2×Dic71⋊C4—21no—
81C2×Dic71⋊C4—21no—
82C2×C22⋊Dic71—21no—
83#14556061—22no—
84#14556061—22no—
85#14556061—22no—
86#14556061—22no—
87#14556061—41no—
88#14556061D414yes—
89#14556059—22no—
90#14556059—22no—
91#14556059—22no—
92#14556059—41no—
93C22⋊(C71⋊C8)Q814yescomplemented
94#14556039—22no—
95#14556039—22no—
96C71⋊#95—21no—
97C71⋊#95—21no—
98C71⋊#95—21no—
99C71⋊#95—21no—
100C71⋊#95—21no—
101C22×Q8×C71—21no—
102C23×C4×C71—21no—
103C23×C4×C71C2311yescharacteristic
104C4⋊Q8×C71—22no—
105C4⋊Q8×C71—41no—
106C22⋊Q8×C71—44no—
107C2×C4⋊C4×C71—22no—
108C2×C4⋊C4×C71—21no—
109C2×C4⋊C4×C71—21no—
110C2×C4⋊C4×C71—21no—
111C2×C4⋊C4×C71—21no—
112C2×C4⋊C4×C71—21no—
113C2×C4⋊C4×C71—21no—
114C2×C4⋊C4×C71—41no—
115C2×C4⋊C4×C71C2311yescharacteristic
116C2×C4⋊C4×C71C2311yescharacteristic
117C2×C4⋊C4×C71C2311yescharacteristic
118C2×C4⋊C4×C71C2311yescharacteristic
119C2×C22⋊C4×C71—21no—
120C2×C22⋊C4×C71D411yescharacteristic
121C2×C22⋊C4×C71D411yescharacteristic
122C2×C42×C71C2311yescharacteristic
123C2×C42×C71C2311yescharacteristic
124#95×C71—21no—
125#95×C71—21no—
126#95×C71—21no—
127#95×C71D411yescharacteristic
128#95×C71D411yescharacteristic
129C22×Dic71—21no—
130C22×Dic71—21no—
131C22×Dic71—21no—
132C22×Dic71—21no—
133C22×Dic71—42no—
134C22×Dic71—42no—
135C22×Dic71—81no—
136C2×C71⋊C8—21no—
137C2×C71⋊C8—21no—
138C2×C71⋊C8—21no—
139C2×C71⋊C8—21no—
140C2×C71⋊C8—21no—
141C2×C71⋊C8—21no—
142C2×C71⋊C8—28no—
143C2×C71⋊C8C2×D412yes—
144C2×C71⋊C8C2×Q812yes—
145C2×C71⋊C8C2×Q812yes—
146C2×C71⋊C8C2×Q812yes—
147C2×C71⋊C8C2×Q812yes—
148C2×C71⋊C8#4112yes—
149C4⋊Dic71—21no—
150C4⋊Dic71—21no—
151C4⋊Dic71—21no—
152C4⋊Dic71—21no—
153C4⋊Dic71—21no—
154C4⋊Dic71—21no—
155C4⋊Dic71—21no—
156C4⋊Dic71—42no—
157C4⋊Dic71C2×D412yes—
158Dic71⋊C4—42no—
159Dic71⋊C4—42no—
160Dic71⋊C4—81no—
161Dic71⋊C4—81no—
162C22⋊Dic71—42no—
163C24×C71C2×D411yescharacteristic
164C2×Q8×C71—42no—
165C2×Q8×C71—42no—
166C2×Q8×C71—81no—
167C22×C4×C71—21no—
168C22×C4×C71—21no—
169C22×C4×C71—21no—
170C22×C4×C71—21no—
171C22×C4×C71—21no—
172C22×C4×C71—21no—
173C22×C4×C71—21no—
174C22×C4×C71—21no—
175C22×C4×C71—42no—
176C22×C4×C71—42no—
177C22×C4×C71—81no—
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2×D411yescharacteristic
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71C2×Q812yes—
184C22×C4×C71C2×Q812yes—
185C22×C4×C71C2411yescharacteristic
186C4⋊C4×C71—21no—
187C4⋊C4×C71—21no—
188C4⋊C4×C71—21no—
189C4⋊C4×C71—21no—
190C4⋊C4×C71—21no—
191C4⋊C4×C71—21no—
192C4⋊C4×C71—21no—
193C4⋊C4×C71—42no—
194C4⋊C4×C71—42no—
195C4⋊C4×C71—42no—
196C4⋊C4×C71—42no—
197C4⋊C4×C71—42no—
198C4⋊C4×C71—81no—
199C4⋊C4×C71C2×D412yes—
200C22⋊C4×C71—22no—
201C22⋊C4×C71—22no—
202C22⋊C4×C71—42no—
203C42×C71—21no—
204C42×C71—21no—
205C42×C71—21no—
206C42×C71—21no—
207C23×C71—21no—
208C23×C71—21no—
209C23×C71—21no—
210C23×C71—21no—
211C23×C71C4⋊Q812yes—
212C23×C71AGL(1,ℤ/8)12yes—
213C23×C71#13712yes—
214C23×C71C22×D411yescharacteristic
215Q8×C71—84no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—22no—
219C2×C4×C71—22no—
220C2×C4×C71—22no—
221C2×C4×C71—22no—
222C2×C4×C71—21no—
223C2×C4×C71—21no—
224C2×C4×C71—22no—
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—21no—
234C2×C4×C71—21no—
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—41no—
244C2×C4×C71—41no—
245C2×C4×C71—41no—
246C2×C4×C71—84no—
247C2×C4×C71C4⋊Q812yes—
248C2×C4×C71C22×D411yescharacteristic
249C2×C4×C71C22×Q811yescharacteristic
250C2×C4×C71C22×Q811yescharacteristic
251C2×C4×C71C2×#4111yescharacteristic
252C2×C4×C71H(5,𝔽2)11yescharacteristic
253C2×C4×C712−1+411yescharacteristic
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—
267C71⋊C8—22no—
268C71⋊C8—22no—
269C71⋊C8—22no—
270C71⋊C8C22⋊Q814yescomplemented
271C22×C71—22no—
272C22×C71—22no—
273C22×C71—22no—
274C22×C71—22no—
275C22×C71—22no—
276C22×C71—22no—
277C22×C71#50114yescomplemented
278C22×C71C2×C4⋊Q811yescharacteristic
279C22×C71D4211yescharacteristic
280C22×C71#54711yescharacteristic
281C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C71C2×#13711yescharacteristic
283C22×C71#57611yescharacteristic
284C22×C71#57811yescharacteristic
285C4×C71—22no—
286C4×C71—41no—
287C4×C71—41no—
288C4×C71—41no—
289C4×C71—42no—
290C4×C71—41no—
291C4×C71—81no—
292C4×C71—81no—
293C4×C71—81no—
294C4×C71C2×C22⋊Q811yescharacteristic
295C4×C71#54411yescharacteristic
296C4×C71D4×Q811yescharacteristic
297C4×C71#55411yescharacteristic
298Dic71—42no—
299Dic71—81no—
300Dic71—81no—
301Dic71—81no—
302#26071—711nomaximal, Sylow
303C2×C71—24no—
304C2×C71#244011yescharacteristic
305C2×C71C2×#50111yescharacteristic
306C2×C71#316411yescharacteristic
307C2×C71#326411yescharacteristic
308C2×C71#327011yescharacteristic
309C2×C71#334311yescharacteristic
310C2×C71#334611yescharacteristic
311C2×#501—711no—
312C22×#107—711no—
313#2442—711no—
314#2440D7111yescharacteristic
315#2085—711no—
316#2083—711no—
317#2059—711no—
318#2057—711no—
319#1944—711no—
320#1943—711no—
321#1853—711no—
322#1852—711no—
323#1847—711no—
324#1807—711no—
325#1752—711no—
326C71#2607111yescharacteristic, complemented, Sylow
327C23×C8—711no—
328C2×C4⋊Q8C2×D7111yescharacteristic
329C2×#125—711no—
330C2×C22⋊Q8—21no—
331C22×C4⋊C4—711no—
332C22×C4⋊C4C2×D7111yescharacteristic
333#501—1424no—
334C2×#107—711no—
335C2×#107—711no—
336C2×#107—711no—
337C2×#107—711no—
338C2×#107—711no—
339C2×#107—711no—
340C2×#107—1424no—
341C2×#106—711no—
342C2×#106—711no—
343C2×#106—711no—
344C2×#106—711no—
345C2×C22⋊C8—711no—
346C2×#97—711no—
347C2×#97—711no—
348#400—1421no—
349#398—21no—
350#398—1421no—
351#395—21no—
352#389—711no—
353#389—711no—
354#389C2×D7111yescharacteristic
355#389C2×D7111yescharacteristic
356#385—21no—
357#385—1421no—
358#384C2×D7111yescharacteristic
359#382—711no—
360#380—711no—
361#380—1421no—
362#380C2×D7111yescharacteristic
363C4×C22⋊C4C2×D7111yescharacteristic
364#340—711no—
365#340—711no—
366#340—711no—
367#340—711no—
368#340—711no—
369#340—711no—
370#336—711no—
371#336—711no—
372C22×Q8—21no—
373C23×C4—21no—
374C23×C4—1421no—
375C23×C4C22×D7111yescharacteristic
376C22×C8—711no—
377C22×C8—714no—
378C22×C8—711no—
379C22×C8—711no—
380C22×C8—711no—
381C22×C8—1422no—
382C22×C8—1422no—
383C4⋊Q8—22no—
384C4⋊Q8—41no—
385#125—2841no—
386#125—1422no—
387C22⋊Q8—44no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—41no—
396C2×C4⋊C4—2841no—
397C2×C4⋊C4—711no—
398C2×C4⋊C4—711no—
399C2×C4⋊C4—711no—
400C2×C4⋊C4—711no—
401C2×C4⋊C4—1421no—
402C2×C4⋊C4—1421no—
403C2×C4⋊C4—1421no—
404C2×C4⋊C4—1422no—
405C2×C4⋊C4—1421no—
406C2×C4⋊C4—1421no—
407C2×C4⋊C4C22×D7111yescharacteristic
408C2×C4⋊C4C22×D7111yescharacteristic
409C2×C4⋊C4C22×D7111yescharacteristic
410C2×C4⋊C4C22×D7111yescharacteristic
411C2×C22⋊C4—21no—
412C2×C22⋊C4—1421no—
413C2×C22⋊C4C22⋊D7111yescharacteristic
414C2×C22⋊C4C22⋊D7111yescharacteristic
415C2×C42C22×D7111yescharacteristic
416C2×C42C22×D7111yescharacteristic
417#107—2841no—
418#107—714no—
419#107—1422no—
420#107—1422no—
421#107—1422no—
422#107—1422no—
423#106—2841no—
424#106—1422no—
425#106—1422no—
426#106—1422no—
427C22⋊C8—714no—
428#97—1422no—
429#97—1422no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—1421no—
434#95—1421no—
435#95—1421no—
436#95—1421no—
437#95—1421no—
438#95C22⋊D7111yescharacteristic
439#95C22⋊D7111yescharacteristic
440C24C2×C22⋊D7111yescharacteristic
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—81no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—81no—
455C22×C4—2842no—
456C22×C4—2842no—
457C22×C4—5681no—
458C22×C4—1421no—
459C22×C4—1421no—
460C22×C4—1421no—
461C22×C4—1421no—
462C22×C4C2×C22⋊D7111yescharacteristic
463C22×C4C2×C22⋊D7111yescharacteristic
464C22×C4C2×C22⋊D7111yescharacteristic
465C22×C4C2×C22⋊D7111yescharacteristic
466C22×C4C2×C22⋊D7111yescharacteristic
467C22×C4Q8×D7112yes—
468C22×C4Q8×D7112yes—
469C22×C4C23×D7111yescharacteristic
470C2×C8—712no—
471C2×C8—712no—
472C2×C8—712no—
473C2×C8—712no—
474C2×C8—712no—
475C2×C8—712no—
476C2×C8—1428no—
477C2×C8—1421no—
478C2×C8—1421no—
479C2×C8—1421no—
480C2×C8—1421no—
481C2×C8—1421no—
482C2×C8—1421no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4—2842no—
497C4⋊C4—2842no—
498C4⋊C4—2842no—
499C4⋊C4—5681no—
500C4⋊C4—5681no—
501C4⋊C4—712no—
502C4⋊C4—1421no—
503C4⋊C4—1421no—
504C4⋊C4—1421no—
505C4⋊C4—1421no—
506C4⋊C4—1421no—
507C4⋊C4—1421no—
508C4⋊C4—1421no—
509C4⋊C4D4×D7112yes—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—42no—
513C22⋊C4—2842no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23#1455612612yes—
523C23#1455615412yes—
524C23#1455615912yes—
525C23C22×C22⋊D7111yescharacteristic
526Q8—84no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
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—1422no—
571C2×C4#1455612612yes—
572C2×C4C2×D4×D7111yescharacteristic
573C2×C4C2×Q8×D7111yescharacteristic
574C2×C4C2×Q8×D7111yescharacteristic
575C2×C4#41×D7111yescharacteristic
576C2×C4#1455617311yescharacteristic
577C2×C4#1455617511yescharacteristic
578C8—714no—
579C8—1422no—
580C8—1422no—
581C8—1422no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22#1798640514yescomplemented
589C22C2×#1455612611yescharacteristic
590C22D4×C22⋊D7111yescharacteristic
591C22#1798664811yescharacteristic
592C22C2×#1455615411yescharacteristic
593C22C2×#1455615911yescharacteristic
594C22#1798680211yescharacteristic
595C22#1798681411yescharacteristic
596C4—22no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—42no—
601C4—41no—
602C4—81no—
603C4—81no—
604C4—81no—
605C4—2842no—
606C4—5681no—
607C4—5681no—
608C4—5681no—
609C4C22⋊Q8×D7111yescharacteristic
610C4#1798662811yescharacteristic
611C4#1798666211yescharacteristic
612C4#1798670311yescharacteristic
613C2—24no—
614C2#2259203911yescharacteristic
615C2C2×#1798640511yescharacteristic
616C2#2259887411yescharacteristic
617C2#2259981611yescharacteristic
618C2#2259991111yescharacteristic
619C2#2260077411yescharacteristic
620C2#2260081911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC71×#2440C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
hypercenter#2440D71
hypocenterC71#26071
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
1. C2×#17986405C2semidirect
4. #22589821C2semidirect
5. #22589812C2semidirect
6. #22589678C2semidirect
7. #22589665C2semidirect
8. #22589080C2semidirect
9. #22589075C2semidirect
12. #22588506C2semidirect
64. C22×C71⋊C8Q8semidirect
93. C22⋊(C71⋊C8)Q8semidirect
270. C71⋊C8C22⋊Q8semidirect
277. C22×C71#501semidirect
326. C71#26071semidirect
588. C22#17986405semidirect

Maximal subgroups

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

subgroupindex
C2×#179864052
C22×#145560612
#225920642
#225898212
#225898122
#225896782
#225896652
#225890802
#225890752
#225885332
#225885272
#225885062
#225883632
#225881182
C71×#24402
#2607171