Finite Group Databasebeta

← #32030498

Subgroups — #32030498

Order 18176.

1338 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986327C211yesmaximal, characteristic, complemented
2C22×#14556052C211yesmaximal, characteristic, complemented
3#22592648C211yesmaximal, characteristic
4#22589583C212yesmaximal, complemented
5#22589533C212yesmaximal, complemented
6#22588764C211yesmaximal, characteristic, complemented
7#22588633C211yesmaximal, characteristic
8#22588608C212yesmaximal, complemented
9#22588275C211yesmaximal, characteristic, complemented
10#22588275C212yesmaximal, complemented
11C2×D4×Q8×C71C211yesmaximal, characteristic
12C23×C71⋊C8C2211yescharacteristic
13C2×#14556109—21no—
14C22×C4⋊Dic71C2211yescharacteristic
15#17986327—24no—
16C2×#14556059C2212yes—
17C2×#14556052—24no—
18C2×#14556052C2212yes—
19C2×#14556052C2212yes—
20C2×#14556052C2212yes—
21C2×#14556052C2212yes—
22C2×#14556052C2212yes—
23C2×#14556052C2212yes—
24C2×C22⋊(C71⋊C8)C2211yescharacteristic
25C2×C22⋊(C71⋊C8)C2212yes—
26#17985929—21no—
27#17985924—21no—
28#17985897C2212yes—
29#17985860—21no—
30#17985859C2211yescharacteristic
31#17985859C2212yes—
32#17985761C2212yes—
33#17985754C2212yes—
34C23×Q8×C71C2211yescharacteristic
35C23×Q8×C71C2211yescharacteristic
36D4×Q8×C71—24no—
37D4×Q8×C71—24no—
38C2×C4⋊Q8×C71—21no—
39C2×C4⋊Q8×C71C2211yescharacteristic
40C2×C22⋊Q8×C71—21no—
41C2×C22⋊Q8×C71—21no—
42C2×C22⋊Q8×C71C2211yescharacteristic
43C2×C22⋊Q8×C71C2211yescharacteristic
44C2×C4×Q8×C71C2211yescharacteristic
45C2×C4×D4×C71—21no—
46C2×C4×D4×C71C2211yescharacteristic
47C23×Dic71—21no—
48C22×C71⋊C8—22no—
49C22×C71⋊C8—42no—
50C22×C71⋊C8C2312yes—
51C22×C71⋊C8C2312yes—
52#14556109—44no—
53C2×C4⋊Dic71—22no—
54C2×C4⋊Dic71—22no—
55C2×C4⋊Dic71—21no—
56C2×C4⋊Dic71C2312yes—
57C2×C4⋊Dic71C2312yes—
58C2×Dic71⋊C4—22no—
59C2×C22⋊Dic71—21no—
60C2×C22⋊Dic71—22no—
61#14556059—24no—
62#14556052—24no—
63#14556052—24no—
64#14556052—24no—
65#14556052—24no—
66#14556052—42no—
67#14556052—42no—
68C22⋊(C71⋊C8)—22no—
69C22⋊(C71⋊C8)—42no—
70C71⋊#95—22no—
71C71⋊#95—21no—
72C71⋊#95—22no—
73C22×Q8×C71—22no—
74C22×Q8×C71—22no—
75C22×Q8×C71—21no—
76C22×Q8×C71—21no—
77C22×Q8×C71—22no—
78C22×Q8×C71—24no—
79C22×Q8×C71—24no—
80C22×Q8×C71—42no—
81C22×Q8×C71—42no—
82C22×Q8×C71D414yes—
83C22×Q8×C71C2311yescharacteristic
84C22×Q8×C71C2311yescharacteristic
85C22×Q8×C71C2311yescharacteristic
86C22×D4×C71D411yescharacteristic
87C23×C4×C71—21no—
88C23×C4×C71—21no—
89C23×C4×C71C2311yescharacteristic
90C23×C4×C71C2311yescharacteristic
91C4⋊Q8×C71—28no—
92C4⋊Q8×C71—22no—
93C4⋊Q8×C71—22no—
94C22⋊Q8×C71—28no—
95C22⋊Q8×C71—28no—
96C22⋊Q8×C71—22no—
97C22⋊Q8×C71—22no—
98C22⋊Q8×C71—22no—
99C22⋊Q8×C71D414yes—
100C4×Q8×C71—22no—
101C4×Q8×C71—22no—
102C4×D4×C71—28no—
103C4×D4×C71—22no—
104C4×D4×C71—22no—
105C2×C4⋊C4×C71—21no—
106C2×C4⋊C4×C71—21no—
107C2×C4⋊C4×C71—21no—
108C2×C4⋊C4×C71—21no—
109C2×C4⋊C4×C71—21no—
110C2×C4⋊C4×C71D411yescharacteristic
111C2×C4⋊C4×C71C2311yescharacteristic
112C2×C22⋊C4×C71—21no—
113C2×C22⋊C4×C71—21no—
114C2×C22⋊C4×C71D411yescharacteristic
115C2×C22⋊C4×C71D411yescharacteristic
116C2×C42×C71—21no—
117C2×C42×C71C2311yescharacteristic
118C22×Dic71—22no—
119C22×Dic71—22no—
120C22×Dic71—42no—
121C22×Dic71—42no—
122C22×Dic71—42no—
123C2×C71⋊C8—22no—
124C2×C71⋊C8—22no—
125C2×C71⋊C8—22no—
126C2×C71⋊C8—22no—
127C2×C71⋊C8—22no—
128C2×C71⋊C8—22no—
129C2×C71⋊C8—44no—
130C4⋊Dic71—22no—
131C4⋊Dic71—22no—
132C4⋊Dic71—22no—
133C4⋊Dic71—22no—
134C4⋊Dic71—42no—
135Dic71⋊C4—44no—
136C22⋊Dic71—42no—
137C22⋊Dic71—44no—
138C24×C71C2×D411yescharacteristic
139C24×C71C2×D411yescharacteristic
140C2×Q8×C71—24no—
141C2×Q8×C71—24no—
142C2×Q8×C71—21no—
143C2×Q8×C71—21no—
144C2×Q8×C71—21no—
145C2×Q8×C71—22no—
146C2×Q8×C71—28no—
147C2×Q8×C71—22no—
148C2×Q8×C71—22no—
149C2×Q8×C71—22no—
150C2×Q8×C71—22no—
151C2×Q8×C71—44no—
152C2×Q8×C71—44no—
153C2×Q8×C71—44no—
154C2×Q8×C71—48no—
155C2×Q8×C71—48no—
156C2×Q8×C71—44no—
157C2×Q8×C71—44no—
158C2×Q8×C71C2×D412yes—
159C2×Q8×C71C2×D412yes—
160C2×Q8×C71C2×D412yes—
161C2×Q8×C71#4114yes—
162C2×D4×C71—22no—
163C2×D4×C71—22no—
164C2×D4×C71SD1614yes—
165C22×C4×C71—21no—
166C22×C4×C71—22no—
167C22×C4×C71—22no—
168C22×C4×C71—24no—
169C22×C4×C71—24no—
170C22×C4×C71—21no—
171C22×C4×C71—21no—
172C22×C4×C71—21no—
173C22×C4×C71—21no—
174C22×C4×C71—21no—
175C22×C4×C71—22no—
176C22×C4×C71—22no—
177C22×C4×C71—44no—
178C22×C4×C71—44no—
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2×D411yescharacteristic
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71C2×D412yes—
183C22×C4×C71C2×D412yes—
184C22×C4×C71C2×D411yescharacteristic
185C22×C4×C71C2411yescharacteristic
186C4⋊C4×C71—24no—
187C4⋊C4×C71—24no—
188C4⋊C4×C71—24no—
189C4⋊C4×C71—24no—
190C4⋊C4×C71—24no—
191C4⋊C4×C71—22no—
192C4⋊C4×C71—21no—
193C4⋊C4×C71C2×D412yes—
194C22⋊C4×C71—24no—
195C22⋊C4×C71—24no—
196C22⋊C4×C71—22no—
197C22⋊C4×C71—22no—
198C42×C71—24no—
199C42×C71—21no—
200C42×C71—21no—
201C23×C71—22no—
202C23×C71—22no—
203C23×C71—21no—
204C23×C71—22no—
205C23×C71—21no—
206C23×C71—22no—
207C23×C71#12012yes—
208C23×C71C2×SD1612yes—
209C23×C71C2×SD1612yes—
210C23×C71#13712yes—
211C23×C71C22×D411yescharacteristic
212Q8×C71—22no—
213Q8×C71—24no—
214Q8×C71—22no—
215Q8×C71—22no—
216Q8×C71—24no—
217Q8×C71—48no—
218Q8×C71—48no—
219Q8×C71—41no—
220Q8×C71—41no—
221Q8×C71—42no—
222Q8×C71#12314yescomplemented
223D4×C71—28no—
224C2×C4×C71—21no—
225C2×C4×C71—22no—
226C2×C4×C71—22no—
227C2×C4×C71—24no—
228C2×C4×C71—21no—
229C2×C4×C71—22no—
230C2×C4×C71—22no—
231C2×C4×C71—24no—
232C2×C4×C71—21no—
233C2×C4×C71—22no—
234C2×C4×C71—22no—
235C2×C4×C71—22no—
236C2×C4×C71—21no—
237C2×C4×C71—22no—
238C2×C4×C71—22no—
239C2×C4×C71—48no—
240C2×C4×C71—48no—
241C2×C4×C71—42no—
242C2×C4×C71—42no—
243C2×C4×C71—42no—
244C2×C4×C71—41no—
245C2×C4×C71—41no—
246C2×C4×C71—42no—
247C2×C4×C71#12012yes—
248C2×C4×C71C2×SD1612yes—
249C2×C4×C71C22×D411yescharacteristic
250C2×C4×C71C22×D411yescharacteristic
251C2×C4×C71C2×#4112yes—
252C2×C4×C71H(5,𝔽2)12yes—
253C2×Dic71—42no—
254C2×Dic71—42no—
255C2×Dic71—42no—
256C2×Dic71—42no—
257C2×Dic71—42no—
258C2×Dic71—42no—
259C2×Dic71—84no—
260C71⋊C8—42no—
261C71⋊C8—42no—
262C22×C71—22no—
263C22×C71—22no—
264C22×C71—24no—
265C22×C71—24no—
266C22×C71—22no—
267C22×C71—24no—
268C22×C71—41no—
269C22×C71—41no—
270C22×C71—42no—
271C22×C71#44814yes—
272C22×C71C2×#12011yescharacteristic
273C22×C71#54612yes—
274C22×C71C22×SD1611yescharacteristic
275C22×C71C2×#13711yescharacteristic
276C22×C71#57812yes—
277C4×C71—22no—
278C4×C71—24no—
279C4×C71—22no—
280C4×C71—41no—
281C4×C71—42no—
282C4×C71—42no—
283C4×C71—41no—
284C4×C71—42no—
285C4×C71C2×#12311yescharacteristic
286C4×C71#53411yescharacteristic
287C4×C71#54612yes—
288Dic71—82no—
289Dic71—82no—
290#24041—711nomaximal, Sylow
291C2×C71—24no—
292C2×C71—42no—
293C2×C71#240511yescharacteristic
294C2×C71C2×#44811yescharacteristic
295C2×C71#301111yescharacteristic
296C2×C71#329812yes—
297C2×C71#331212yes—
298C2×D4×Q8D7111yescharacteristic
299C2×#483—711no—
300C22×#103—711no—
301#2486—711no—
302#2043—712no—
303#2035—712no—
304#1892—711no—
305#1869—711no—
306#1865—712no—
307#1787—712no—
308#1787—711no—
309C71#2404111yescharacteristic, complemented, Sylow
310C23×Q8C2×D7111yescharacteristic
311C23×Q8C2×D7111yescharacteristic
312C23×C8—711no—
313D4×Q8—24no—
314D4×Q8—24no—
315C2×C4⋊Q8—21no—
316C2×C4⋊Q8C2×D7111yescharacteristic
317C2×#123—1421no—
318C2×C22⋊Q8—21no—
319C2×C22⋊Q8—21no—
320C2×C22⋊Q8C2×D7111yescharacteristic
321C2×C22⋊Q8C2×D7111yescharacteristic
322C2×C4×Q8C2×D7111yescharacteristic
323C2×C4×D4—21no—
324C2×C4×D4C2×D7111yescharacteristic
325C22×C4⋊C4—711no—
326#483—1424no—
327C2×#106—712no—
328C2×#103—712no—
329C2×#103—712no—
330C2×#103—712no—
331C2×#103—712no—
332C2×#103—712no—
333C2×#103—712no—
334C2×#103—1424no—
335C2×C22⋊C8—712no—
336C2×C22⋊C8—711no—
337#398—1421no—
338#397—1421no—
339#389—712no—
340#381—1421no—
341#380—711no—
342#380—712no—
343#340—712no—
344#336—712no—
345C22×Q8—22no—
346C22×Q8—22no—
347C22×Q8—21no—
348C22×Q8—21no—
349C22×Q8—22no—
350C22×Q8—24no—
351C22×Q8—24no—
352C22×Q8—42no—
353C22×Q8—42no—
354C22×Q8C22⋊D7114yes—
355C22×Q8C22×D7111yescharacteristic
356C22×Q8C22×D7111yescharacteristic
357C22×Q8C22×D7111yescharacteristic
358C22×D4C22⋊D7111yescharacteristic
359C23×C4—21no—
360C23×C4—21no—
361C23×C4—1421no—
362C23×C4C22×D7111yescharacteristic
363C23×C4C22×D7111yescharacteristic
364C22×C8—2842no—
365C22×C8—712no—
366C22×C8—712no—
367C22×C8—1422no—
368C4⋊Q8—28no—
369C4⋊Q8—22no—
370C4⋊Q8—22no—
371#123—2844no—
372C22⋊Q8—28no—
373C22⋊Q8—28no—
374C22⋊Q8—22no—
375C22⋊Q8—22no—
376C22⋊Q8—22no—
377C22⋊Q8C22⋊D7114yes—
378C4×Q8—22no—
379C4×Q8—22no—
380C4×D4—28no—
381C4×D4—22no—
382C4×D4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—712no—
389C2×C4⋊C4—712no—
390C2×C4⋊C4—1421no—
391C2×C4⋊C4—1422no—
392C2×C4⋊C4—1422no—
393C2×C4⋊C4—1422no—
394C2×C4⋊C4C22⋊D7111yescharacteristic
395C2×C4⋊C4C22×D7111yescharacteristic
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1421no—
399C2×C22⋊C4—1422no—
400C2×C22⋊C4C22⋊D7111yescharacteristic
401C2×C22⋊C4C22⋊D7111yescharacteristic
402C2×C42—21no—
403C2×C42C22×D7111yescharacteristic
404#106—1424no—
405#103—2842no—
406#103—2842no—
407#103—1424no—
408#103—1424no—
409#103—1424no—
410#103—1424no—
411C22⋊C8—2842no—
412C22⋊C8—1422no—
413#95—1422no—
414#95—1421no—
415#95—1422no—
416C24C2×C22⋊D7111yescharacteristic
417C24C2×C22⋊D7111yescharacteristic
418C2×Q8—24no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—28no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8—44no—
432C2×Q8—48no—
433C2×Q8—48no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8C2×C22⋊D7112yes—
437C2×Q8C2×C22⋊D7112yes—
438C2×Q8C2×C22⋊D7112yes—
439C2×Q8#1176586414yes—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4#1176585214yes—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—2842no—
458C22×C4—2842no—
459C22×C4—2842no—
460C22×C4—1422no—
461C22×C4—1422no—
462C22×C4C2×C22⋊D7111yescharacteristic
463C22×C4C2×C22⋊D7111yescharacteristic
464C22×C4C2×C22⋊D7112yes—
465C22×C4C2×C22⋊D7112yes—
466C22×C4C2×C22⋊D7111yescharacteristic
467C22×C4C2×C22⋊D7111yescharacteristic
468C22×C4C23×D7111yescharacteristic
469C2×C8—2844no—
470C2×C8—1422no—
471C2×C8—1422no—
472C2×C8—1422no—
473C2×C8—1422no—
474C2×C8—1422no—
475C2×C8—1422no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—2842no—
484C4⋊C4—2844no—
485C4⋊C4—1422no—
486C4⋊C4—1422no—
487C4⋊C4—1422no—
488C4⋊C4—1422no—
489C4⋊C4C2×C22⋊D7112yes—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—2842no—
495C22⋊C4—2844no—
496C42—24no—
497C42—21no—
498C42—21no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—21no—
503C23—21no—
504C23—22no—
505C23#1455609712yes—
506C23C2×#1176585212yes—
507C23C2×#1176585212yes—
508C23#1455615912yes—
509C23C22×C22⋊D7111yescharacteristic
510Q8—24no—
511Q8—22no—
512Q8—24no—
513Q8—22no—
514Q8—22no—
515Q8—48no—
516Q8—48no—
517Q8—42no—
518Q8—41no—
519Q8—41no—
520Q8#1455610914yescomplemented
521D4—28no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—2842no—
546C2×C4—2842no—
547C2×C4—2842no—
548C2×C4—2842no—
549C2×C4—2842no—
550C2×C4—2842no—
551C2×C4—5684no—
552C2×C4#1455609712yes—
553C2×C4C2×#1176585212yes—
554C2×C4C22×C22⋊D7111yescharacteristic
555C2×C4C22×C22⋊D7111yescharacteristic
556C2×C4C2×#1176586412yes—
557C2×C4#1455617312yes—
558C8—2842no—
559C8—2842no—
560C22—22no—
561C22—24no—
562C22—22no—
563C22—24no—
564C22—24no—
565C22—22no—
566C22—41no—
567C22—41no—
568C22—42no—
569C22#1798613814yes—
570C22C2×#1455609711yescharacteristic
571C22#1798664312yes—
572C22C22×#1176585211yescharacteristic
573C22C2×#1455615911yescharacteristic
574C22#1798681412yes—
575C4—22no—
576C4—22no—
577C4—24no—
578C4—41no—
579C4—42no—
580C4—41no—
581C4—42no—
582C4—42no—
583C4—5682no—
584C4—5682no—
585C4C2×#1455610911yescharacteristic
586C4#1798655811yescharacteristic
587C4#1798664312yes—
588C2—24no—
589C2—42no—
590C2#2259166311yescharacteristic
591C2C2×#1798613811yescharacteristic
592C2#2259763511yescharacteristic
593C2#2260020512yes—
594C2#2260039112yes—
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC2×D4×Q8×C71C2
socleC23×C71C22×D4
prime radicalC22×D4×C71D4
hypercenterC2×D4×Q8D71
hypocenterC71#24041
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#17986327C2semidirect
2. C22×#14556052C2semidirect
4. #22589583C2semidirect
5. #22589533C2semidirect
6. #22588764C2semidirect
8. #22588608C2semidirect
9. #22588275C2semidirect
10. #22588275C2semidirect
222. Q8×C71#123semidirect
309. C71#24041semidirect
520. Q8#14556109semidirect

Maximal subgroups

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

subgroupindex
C2×#179863272
C22×#145560522
#225926482
#225895832
#225895332
#225887642
#225886332
#225886082
#225882752
C2×D4×Q8×C712
#2404171