Finite Group Databasebeta

← #32030509

Subgroups — #32030509

Order 18176.

1318 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986332C211yesmaximal, characteristic, complemented
2C22×#14556052C211yesmaximal, characteristic, complemented
3#22592657C211yesmaximal, characteristic
4#22589583C212yesmaximal, complemented
5#22589540C212yesmaximal, complemented
6#22588764C211yesmaximal, characteristic, complemented
7#22588633C211yesmaximal, characteristic
8#22588594C212yesmaximal, complemented
9#22588275C211yesmaximal, characteristic, complemented
10#22588275C212yesmaximal, complemented
11C2×D4×Q8×C71C211yesmaximal, characteristic
12C23×C71⋊C8C2211yescharacteristic
13C22×C4⋊Dic71C2211yescharacteristic
14#17986332—24no—
15C2×#14556061C2212yes—
16C2×#14556052—24no—
17C2×#14556052C2212yes—
18C2×#14556052C2212yes—
19C2×#14556052C2212yes—
20C2×#14556052C2212yes—
21C2×#14556052C2212yes—
22C2×#14556052C2212yes—
23C2×C22⋊(C71⋊C8)C2211yescharacteristic
24C2×C22⋊(C71⋊C8)C2212yes—
25#17985938—21no—
26#17985935—21no—
27#17985897C2212yes—
28#17985879—21no—
29#17985871—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—
52C2×C4⋊Dic71—22no—
53C2×C4⋊Dic71—22no—
54C2×C4⋊Dic71—21no—
55C2×C4⋊Dic71C2312yes—
56C2×C4⋊Dic71C2312yes—
57C2×Dic71⋊C4—22no—
58C2×C22⋊Dic71—21no—
59C2×C22⋊Dic71—22no—
60#14556061—24no—
61#14556052—24no—
62#14556052—24no—
63#14556052—24no—
64#14556052—24no—
65#14556052—42no—
66#14556052—42no—
67C22⋊(C71⋊C8)—22no—
68C22⋊(C71⋊C8)—42no—
69C71⋊#95—22no—
70C71⋊#95—21no—
71C71⋊#95—22no—
72C22×Q8×C71—22no—
73C22×Q8×C71—22no—
74C22×Q8×C71—21no—
75C22×Q8×C71—21no—
76C22×Q8×C71—22no—
77C22×Q8×C71—24no—
78C22×Q8×C71—24no—
79C22×Q8×C71—42no—
80C22×Q8×C71—42no—
81C22×Q8×C71D414yes—
82C22×Q8×C71C2311yescharacteristic
83C22×Q8×C71C2311yescharacteristic
84C22×Q8×C71C2311yescharacteristic
85C22×D4×C71D411yescharacteristic
86C23×C4×C71—21no—
87C23×C4×C71—21no—
88C23×C4×C71C2311yescharacteristic
89C23×C4×C71C2311yescharacteristic
90C4⋊Q8×C71—28no—
91C4⋊Q8×C71—22no—
92C4⋊Q8×C71—22no—
93C22⋊Q8×C71—28no—
94C22⋊Q8×C71—28no—
95C22⋊Q8×C71—22no—
96C22⋊Q8×C71—22no—
97C22⋊Q8×C71—22no—
98C22⋊Q8×C71D414yes—
99C4×Q8×C71—22no—
100C4×Q8×C71—22no—
101C4×D4×C71—28no—
102C4×D4×C71—22no—
103C4×D4×C71—22no—
104C2×C4⋊C4×C71—21no—
105C2×C4⋊C4×C71—21no—
106C2×C4⋊C4×C71—21no—
107C2×C4⋊C4×C71—21no—
108C2×C4⋊C4×C71—21no—
109C2×C4⋊C4×C71D411yescharacteristic
110C2×C4⋊C4×C71C2311yescharacteristic
111C2×C22⋊C4×C71—21no—
112C2×C22⋊C4×C71—21no—
113C2×C22⋊C4×C71D411yescharacteristic
114C2×C22⋊C4×C71D411yescharacteristic
115C2×C42×C71—21no—
116C2×C42×C71C2311yescharacteristic
117C22×Dic71—22no—
118C22×Dic71—22no—
119C22×Dic71—42no—
120C22×Dic71—82no—
121C2×C71⋊C8—22no—
122C2×C71⋊C8—22no—
123C2×C71⋊C8—22no—
124C2×C71⋊C8—22no—
125C2×C71⋊C8—22no—
126C2×C71⋊C8—22no—
127C2×C71⋊C8—44no—
128C4⋊Dic71—22no—
129C4⋊Dic71—22no—
130C4⋊Dic71—22no—
131C4⋊Dic71—22no—
132C4⋊Dic71—42no—
133Dic71⋊C4—82no—
134C22⋊Dic71—42no—
135C22⋊Dic71—82no—
136C24×C71C2×D411yescharacteristic
137C24×C71C2×D411yescharacteristic
138C2×Q8×C71—24no—
139C2×Q8×C71—24no—
140C2×Q8×C71—21no—
141C2×Q8×C71—21no—
142C2×Q8×C71—21no—
143C2×Q8×C71—22no—
144C2×Q8×C71—28no—
145C2×Q8×C71—22no—
146C2×Q8×C71—22no—
147C2×Q8×C71—22no—
148C2×Q8×C71—22no—
149C2×Q8×C71—44no—
150C2×Q8×C71—44no—
151C2×Q8×C71—44no—
152C2×Q8×C71—48no—
153C2×Q8×C71—48no—
154C2×Q8×C71—44no—
155C2×Q8×C71—44no—
156C2×Q8×C71C2×D412yes—
157C2×Q8×C71C2×D412yes—
158C2×Q8×C71C2×D412yes—
159C2×Q8×C71#4114yes—
160C2×D4×C71—22no—
161C2×D4×C71—22no—
162C2×D4×C71Q1614yes—
163C22×C4×C71—21no—
164C22×C4×C71—22no—
165C22×C4×C71—22no—
166C22×C4×C71—24no—
167C22×C4×C71—24no—
168C22×C4×C71—21no—
169C22×C4×C71—21no—
170C22×C4×C71—21no—
171C22×C4×C71—21no—
172C22×C4×C71—21no—
173C22×C4×C71—22no—
174C22×C4×C71—22no—
175C22×C4×C71—44no—
176C22×C4×C71—44no—
177C22×C4×C71C2×D411yescharacteristic
178C22×C4×C71C2×D411yescharacteristic
179C22×C4×C71C2×D411yescharacteristic
180C22×C4×C71C2×D412yes—
181C22×C4×C71C2×D412yes—
182C22×C4×C71C2×D411yescharacteristic
183C22×C4×C71C2411yescharacteristic
184C4⋊C4×C71—24no—
185C4⋊C4×C71—24no—
186C4⋊C4×C71—24no—
187C4⋊C4×C71—24no—
188C4⋊C4×C71—24no—
189C4⋊C4×C71—22no—
190C4⋊C4×C71—21no—
191C4⋊C4×C71C2×D412yes—
192C22⋊C4×C71—24no—
193C22⋊C4×C71—24no—
194C22⋊C4×C71—22no—
195C22⋊C4×C71—22no—
196C42×C71—24no—
197C42×C71—21no—
198C42×C71—21no—
199C23×C71—22no—
200C23×C71—22no—
201C23×C71—21no—
202C23×C71—22no—
203C23×C71—21no—
204C23×C71—22no—
205C23×C71#12012yes—
206C23×C71C2×Q1612yes—
207C23×C71C2×Q1612yes—
208C23×C71#13712yes—
209C23×C71C22×D411yescharacteristic
210Q8×C71—22no—
211Q8×C71—24no—
212Q8×C71—22no—
213Q8×C71—22no—
214Q8×C71—24no—
215Q8×C71—48no—
216Q8×C71—48no—
217Q8×C71—41no—
218Q8×C71—41no—
219Q8×C71—42no—
220Q8×C71#12314yes—
221D4×C71—28no—
222C2×C4×C71—21no—
223C2×C4×C71—22no—
224C2×C4×C71—22no—
225C2×C4×C71—24no—
226C2×C4×C71—21no—
227C2×C4×C71—22no—
228C2×C4×C71—22no—
229C2×C4×C71—24no—
230C2×C4×C71—21no—
231C2×C4×C71—22no—
232C2×C4×C71—22no—
233C2×C4×C71—22no—
234C2×C4×C71—21no—
235C2×C4×C71—22no—
236C2×C4×C71—22no—
237C2×C4×C71—48no—
238C2×C4×C71—48no—
239C2×C4×C71—42no—
240C2×C4×C71—42no—
241C2×C4×C71—42no—
242C2×C4×C71—41no—
243C2×C4×C71—41no—
244C2×C4×C71—42no—
245C2×C4×C71#12012yes—
246C2×C4×C71C2×Q1612yes—
247C2×C4×C71C22×D411yescharacteristic
248C2×C4×C71C22×D411yescharacteristic
249C2×C4×C71C2×#4112yes—
250C2×C4×C71H(5,𝔽2)12yes—
251C2×Dic71—42no—
252C2×Dic71—42no—
253C2×Dic71—42no—
254C2×Dic71—42no—
255C2×Dic71—42no—
256C2×Dic71—42no—
257C2×Dic71—84no—
258C71⋊C8—42no—
259C71⋊C8—42no—
260C22×C71—22no—
261C22×C71—22no—
262C22×C71—24no—
263C22×C71—24no—
264C22×C71—22no—
265C22×C71—24no—
266C22×C71—41no—
267C22×C71—41no—
268C22×C71—42no—
269C22×C71#45114yes—
270C22×C71C2×#12011yescharacteristic
271C22×C71#54612yes—
272C22×C71C22×Q1611yescharacteristic
273C22×C71C2×#13711yescharacteristic
274C22×C71#57712yes—
275C4×C71—22no—
276C4×C71—24no—
277C4×C71—22no—
278C4×C71—41no—
279C4×C71—42no—
280C4×C71—42no—
281C4×C71—41no—
282C4×C71—42no—
283C4×C71C2×#12311yescharacteristic
284C4×C71#53411yescharacteristic
285C4×C71#54612yes—
286Dic71—82no—
287Dic71—82no—
288#24042—711nomaximal, Sylow
289C2×C71—24no—
290C2×C71—42no—
291C2×C71#240511yescharacteristic
292C2×C71C2×#45111yescharacteristic
293C2×C71#301011yescharacteristic
294C2×C71#329912yes—
295C2×C71#332012yes—
296C2×D4×Q8D7111yescharacteristic
297C2×#484—711no—
298C22×#103—711no—
299#2487—711no—
300#2043—712no—
301#2036—712no—
302#1892—711no—
303#1869—711no—
304#1863—712no—
305#1787—712no—
306#1787—711no—
307C71#2404211yescharacteristic, complemented, Sylow
308C23×Q8C2×D7111yescharacteristic
309C23×Q8C2×D7111yescharacteristic
310C23×C8—711no—
311D4×Q8—24no—
312D4×Q8—24no—
313C2×C4⋊Q8—21no—
314C2×C4⋊Q8C2×D7111yescharacteristic
315C2×C22⋊Q8—21no—
316C2×C22⋊Q8—21no—
317C2×C22⋊Q8C2×D7111yescharacteristic
318C2×C22⋊Q8C2×D7111yescharacteristic
319C2×C4×Q8C2×D7111yescharacteristic
320C2×C4×D4—21no—
321C2×C4×D4C2×D7111yescharacteristic
322C22×C4⋊C4—711no—
323#484—1424no—
324C2×#107—712no—
325C2×#103—712no—
326C2×#103—712no—
327C2×#103—712no—
328C2×#103—712no—
329C2×#103—712no—
330C2×#103—712no—
331C2×#103—1424no—
332C2×C22⋊C8—712no—
333C2×C22⋊C8—711no—
334#400—1421no—
335#399—1421no—
336#389—712no—
337#386—1421no—
338#385—1421no—
339#380—711no—
340#380—712no—
341#340—712no—
342#336—712no—
343C22×Q8—22no—
344C22×Q8—22no—
345C22×Q8—21no—
346C22×Q8—21no—
347C22×Q8—22no—
348C22×Q8—24no—
349C22×Q8—24no—
350C22×Q8—42no—
351C22×Q8—42no—
352C22×Q8C22⋊D7114yes—
353C22×Q8C22×D7111yescharacteristic
354C22×Q8C22×D7111yescharacteristic
355C22×Q8C22×D7111yescharacteristic
356C22×D4C22⋊D7111yescharacteristic
357C23×C4—21no—
358C23×C4—21no—
359C23×C4—1421no—
360C23×C4C22×D7111yescharacteristic
361C23×C4C22×D7111yescharacteristic
362C22×C8—2842no—
363C22×C8—712no—
364C22×C8—712no—
365C22×C8—1422no—
366C4⋊Q8—28no—
367C4⋊Q8—22no—
368C4⋊Q8—22no—
369C22⋊Q8—28no—
370C22⋊Q8—28no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374C22⋊Q8C22⋊D7114yes—
375C4×Q8—22no—
376C4×Q8—22no—
377C4×D4—28no—
378C4×D4—22no—
379C4×D4—22no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—712no—
386C2×C4⋊C4—712no—
387C2×C4⋊C4—1421no—
388C2×C4⋊C4—1422no—
389C2×C4⋊C4—1422no—
390C2×C4⋊C4—1422no—
391C2×C4⋊C4C22⋊D7111yescharacteristic
392C2×C4⋊C4C22×D7111yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—1421no—
396C2×C22⋊C4—1422no—
397C2×C22⋊C4C22⋊D7111yescharacteristic
398C2×C22⋊C4C22⋊D7111yescharacteristic
399C2×C42—21no—
400C2×C42C22×D7111yescharacteristic
401#107—1424no—
402#103—2842no—
403#103—2842no—
404#103—1424no—
405#103—1424no—
406#103—1424no—
407#103—1424no—
408C22⋊C8—2842no—
409C22⋊C8—1422no—
410#95—1422no—
411#95—1421no—
412#95—1422no—
413C24C2×C22⋊D7111yescharacteristic
414C24C2×C22⋊D7111yescharacteristic
415C2×Q8—24no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—28no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—44no—
429C2×Q8—48no—
430C2×Q8—48no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8C2×C22⋊D7112yes—
434C2×Q8C2×C22⋊D7112yes—
435C2×Q8C2×C22⋊D7112yes—
436C2×Q8#1176586414yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4#1176585514yes—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—2842no—
455C22×C4—5682no—
456C22×C4—1422no—
457C22×C4—1422no—
458C22×C4C2×C22⋊D7111yescharacteristic
459C22×C4C2×C22⋊D7111yescharacteristic
460C22×C4C2×C22⋊D7112yes—
461C22×C4C2×C22⋊D7112yes—
462C22×C4C2×C22⋊D7111yescharacteristic
463C22×C4C2×C22⋊D7111yescharacteristic
464C22×C4C23×D7111yescharacteristic
465C2×C8—2844no—
466C2×C8—1422no—
467C2×C8—1422no—
468C2×C8—1422no—
469C2×C8—1422no—
470C2×C8—1422no—
471C2×C8—1422no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—2842no—
480C4⋊C4—5682no—
481C4⋊C4—1422no—
482C4⋊C4—1422no—
483C4⋊C4—1422no—
484C4⋊C4—1422no—
485C4⋊C4C2×C22⋊D7112yes—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—2842no—
491C22⋊C4—5682no—
492C42—24no—
493C42—21no—
494C42—21no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23#1455609712yes—
502C23C2×#1176585512yes—
503C23C2×#1176585512yes—
504C23#1455615912yes—
505C23C22×C22⋊D7111yescharacteristic
506Q8—24no—
507Q8—22no—
508Q8—24no—
509Q8—22no—
510Q8—22no—
511Q8—48no—
512Q8—48no—
513Q8—42no—
514Q8—41no—
515Q8—41no—
516Q8#1455610914yes—
517D4—28no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—48no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—2842no—
542C2×C4—2842no—
543C2×C4—2842no—
544C2×C4—2842no—
545C2×C4—2842no—
546C2×C4—2842no—
547C2×C4—5684no—
548C2×C4#1455609712yes—
549C2×C4C2×#1176585512yes—
550C2×C4C22×C22⋊D7111yescharacteristic
551C2×C4C22×C22⋊D7111yescharacteristic
552C2×C4C2×#1176586412yes—
553C2×C4#1455617312yes—
554C8—2842no—
555C8—2842no—
556C22—24no—
557C22—24no—
558C22—22no—
559C22—22no—
560C22—24no—
561C22—22no—
562C22—42no—
563C22—41no—
564C22—41no—
565C22#1798615514yes—
566C22C2×#1455609711yescharacteristic
567C22#1798664312yes—
568C22C22×#1176585511yescharacteristic
569C22C2×#1455615911yescharacteristic
570C22#1798680712yes—
571C4—22no—
572C4—22no—
573C4—24no—
574C4—41no—
575C4—42no—
576C4—42no—
577C4—41no—
578C4—42no—
579C4—5682no—
580C4—5682no—
581C4C2×#1455610911yescharacteristic
582C4#1798655811yescharacteristic
583C4#1798664312yes—
584C2—24no—
585C2—42no—
586C2#2259166311yescharacteristic
587C2C2×#1798615511yescharacteristic
588C2#2259762611yescharacteristic
589C2#2260021612yes—
590C2#2260051112yes—
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#24042
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×#17986332C2semidirect
2. C22×#14556052C2semidirect
4. #22589583C2semidirect
5. #22589540C2semidirect
6. #22588764C2semidirect
8. #22588594C2semidirect
9. #22588275C2semidirect
10. #22588275C2semidirect
307. C71#24042semidirect

Maximal subgroups

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

subgroupindex
C2×#179863322
C22×#145560522
#225926572
#225895832
#225895402
#225887642
#225886332
#225885942
#225882752
C2×D4×Q8×C712
#2404271