Finite Group Databasebeta

← #32033533

Subgroups — #32033533

Order 18176.

1150 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×#17986556C211yesmaximal, characteristic, complemented
2C2×#17986289C211yesmaximal, characteristic, complemented
3C2×#17986279C211yesmaximal, characteristic, complemented
4#22589678C212yesmaximal, complemented
5#22589486C212yesmaximal, complemented
6#22589292C211yesmaximal, characteristic, complemented
7#22589288C211yesmaximal, characteristic, complemented
8#22588663C212yesmaximal
9#22588537C212yesmaximal, complemented
10#22588320C211yesmaximal, characteristic, complemented
11C4×#121×C71C211yesmaximal, characteristic
12#17986556—24no—
13#17986556—24no—
14C2×#14556128C2211yescharacteristic
15C2×#14556128C2211yescharacteristic
16C2×#14556120C2212yes—
17C2×#14556107—21no—
18C2×#14556107—21no—
19C2×C4×Dic142—21no—
20C2×C4×Dic142—21no—
21C2×#14556087C2212yes—
22#17986289—24no—
23#17986279—24no—
24C2×#14556061C2212yes—
25C2×#14556059C2212yes—
26C2×C4⋊(C71⋊C8)C2211yescharacteristic
27C2×C4⋊(C71⋊C8)C2211yescharacteristic
28C2×#14556049C2212yes—
29C2×#14556049C2212yes—
30C2×C22⋊(C71⋊C8)C2212yes—
31#17985761C2212yes—
32#17985761C2212yes—
33#17985761C2212yes—
34#17985761C2212yes—
35#17985754C2212yes—
36C2×#121×C71C2211yescharacteristic
37C2×C4×D4×C71—21no—
38C2×C4×D4×C71C2211yescharacteristic
39C22×C42×C71C2211yescharacteristic
40#390×C71C2211yescharacteristic
41#389×C71C2211yescharacteristic
42#388×C71—21no—
43#386×C71—21no—
44#380×C71C2211yescharacteristic
45C4×C4⋊C4×C71C2211yescharacteristic
46C4×C22⋊C4×C71—21no—
47C22×Dic142—21no—
48C22×Dic142—21no—
49C22×C71⋊C8C2312yes—
50C22×C71⋊C8C2312yes—
51#14556128—22no—
52#14556128—22no—
53#14556128D414yescomplemented
54#14556128D414yescomplemented
55#14556120—24no—
56#14556120—24no—
57#14556107—28no—
58#14556107—28no—
59C4×Dic142—28no—
60C4×Dic142—28no—
61#14556087—24no—
62#14556087—24no—
63C2×C4⋊Dic71—21no—
64C2×C4⋊Dic71—21no—
65C2×C4⋊Dic71C2312yes—
66C2×C4⋊Dic71C2312yes—
67C2×Dic71⋊C4—22no—
68C2×Dic71⋊C4—22no—
69C2×C22⋊Dic71—22no—
70C2×C4×Dic71—22no—
71#14556061—24no—
72#14556059—24no—
73C4⋊(C71⋊C8)—22no—
74C4⋊(C71⋊C8)—22no—
75#14556049—24no—
76#14556049—24no—
77C22⋊(C71⋊C8)—24no—
78C22×D4×C71—21no—
79C22×D4×C71C2311yescharacteristic
80C23×C4×C71—21no—
81C23×C4×C71D411yescharacteristic
82C23×C4×C71D411yescharacteristic
83C23×C4×C71C2311yescharacteristic
84#121×C71—22no—
85#121×C71Q814yescomplemented
86C4×D4×C71—22no—
87C4×D4×C71—22no—
88C4×D4×C71—44no—
89C2×C4⋊C4×C71—21no—
90C2×C4⋊C4×C71C2311yescharacteristic
91C2×C4⋊C4×C71C2311yescharacteristic
92C2×C4⋊C4×C71C2311yescharacteristic
93C2×C22⋊C4×C71—21no—
94C2×C22⋊C4×C71—21no—
95C2×C22⋊C4×C71—21no—
96C2×C22⋊C4×C71—21no—
97C2×C22⋊C4×C71—21no—
98C2×C42×C71—22no—
99C2×C42×C71—22no—
100C2×C42×C71—21no—
101C2×C42×C71D411yescharacteristic
102C2×C42×C71D411yescharacteristic
103C2×C42×C71C2311yescharacteristic
104C2×C42×C71C2311yescharacteristic
105#95×C71—21no—
106#95×C71—21no—
107#95×C71—21no—
108C2×Dic142—24no—
109C2×Dic142—24no—
110C2×Dic142—42no—
111C2×Dic142—42no—
112C2×Dic142—42no—
113C2×Dic142—42no—
114C22×Dic71—22no—
115C22×Dic71—22no—
116C2×C71⋊C8—22no—
117C2×C71⋊C8—22no—
118C2×C71⋊C8—22no—
119C2×C71⋊C8—22no—
120C2×C71⋊C8—22no—
121C2×C71⋊C8—22no—
122C4⋊Dic71—24no—
123C4⋊Dic71—24no—
124C4⋊Dic71—22no—
125C4⋊Dic71—22no—
126C4⋊Dic71C2×D414yes—
127C4⋊Dic71C2×D414yes—
128Dic71⋊C4—28no—
129Dic71⋊C4—28no—
130C22⋊Dic71—28no—
131C4×Dic71—28no—
132C24×C71—21no—
133C24×C71C2×D411yescharacteristic
134C2×D4×C71—24no—
135C2×D4×C71—21no—
136C2×D4×C71—21no—
137C2×D4×C71—21no—
138C2×D4×C71—21no—
139C2×D4×C71—21no—
140C2×D4×C71—42no—
141C2×D4×C71—42no—
142C2×D4×C71C2×Q812yes—
143C22×C4×C71—21no—
144C22×C4×C71—21no—
145C22×C4×C71—21no—
146C22×C4×C71—21no—
147C22×C4×C71—21no—
148C22×C4×C71—22no—
149C22×C4×C71—22no—
150C22×C4×C71—22no—
151C22×C4×C71—22no—
152C22×C4×C71—21no—
153C22×C4×C71—21no—
154C22×C4×C71—41no—
155C22×C4×C71—41no—
156C22×C4×C71—42no—
157C22×C4×C71—42no—
158C22×C4×C71—42no—
159C22×C4×C71C2×D411yescharacteristic
160C22×C4×C71C2×D411yescharacteristic
161C22×C4×C71C2×D411yescharacteristic
162C22×C4×C71C2×D411yescharacteristic
163C22×C4×C71C2×D411yescharacteristic
164C22×C4×C71C2×Q812yes—
165C22×C4×C71#4112yes—
166C22×C4×C71C2411yescharacteristic
167C4⋊C4×C71—21no—
168C4⋊C4×C71—21no—
169C4⋊C4×C71—21no—
170C4⋊C4×C71—21no—
171C4⋊C4×C71—21no—
172C4⋊C4×C71—42no—
173C4⋊C4×C71C2×Q812yes—
174C22⋊C4×C71—24no—
175C22⋊C4×C71—42no—
176C22⋊C4×C71—42no—
177C22⋊C4×C71—42no—
178C22⋊C4×C71—42no—
179C42×C71—22no—
180C42×C71—22no—
181C42×C71—21no—
182C42×C71—21no—
183C42×C71—42no—
184C42×C71C2×D412yes—
185C42×C71C2×D412yes—
186C23×C71—21no—
187C23×C71—21no—
188C23×C71—21no—
189C23×C71—21no—
190C23×C71—21no—
191C23×C71—21no—
192C23×C71—22no—
193C23×C71—41no—
194C23×C71—41no—
195C23×C71—41no—
196C23×C71—41no—
197C23×C71—41no—
198C23×C71—41no—
199C23×C71C22⋊Q812yes—
200C23×C71C22×D411yescharacteristic
201D4×C71—22no—
202D4×C71—22no—
203D4×C71—48no—
204D4×C71—41no—
205D4×C71—41no—
206C2×C4×C71—21no—
207C2×C4×C71—21no—
208C2×C4×C71—21no—
209C2×C4×C71—21no—
210C2×C4×C71—22no—
211C2×C4×C71—22no—
212C2×C4×C71—22no—
213C2×C4×C71—22no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—22no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—44no—
222C2×C4×C71—41no—
223C2×C4×C71—41no—
224C2×C4×C71—44no—
225C2×C4×C71—41no—
226C2×C4×C71—41no—
227C2×C4×C71—41no—
228C2×C4×C71—41no—
229C2×C4×C71—41no—
230C2×C4×C71—41no—
231C2×C4×C71—84no—
232C2×C4×C71#12012yes—
233C2×C4×C71#12012yes—
234C2×C4×C71C22⋊Q812yes—
235C2×C4×C71#13512yes—
236C2×C4×C71AGL(1,ℤ/8)12yes—
237C2×C4×C71C22×D411yescharacteristic
238C2×C4×C71C22×D411yescharacteristic
239C2×C4×C71C22×Q811yescharacteristic
240C2×C4×C71C2×#4111yescharacteristic
241C2×C4×C712−1+411yescharacteristic
242C2×C4×C712−1+411yescharacteristic
243C2×Dic71—24no—
244C2×Dic71—24no—
245C2×Dic71—42no—
246C2×Dic71—42no—
247C2×Dic71—42no—
248C2×Dic71—42no—
249Dic142—48no—
250Dic142—48no—
251C71⋊C8—42no—
252C71⋊C8—42no—
253C22×C71—22no—
254C22×C71—22no—
255C22×C71—22no—
256C22×C71—41no—
257C22×C71—41no—
258C22×C71—41no—
259C22×C71—41no—
260C22×C71—42no—
261C22×C71—42no—
262C22×C71—42no—
263C22×C71—81no—
264C22×C71—81no—
265C22×C71—81no—
266C22×C71—81no—
267C22×C71C2×#12011yescharacteristic
268C22×C71C2×C22⋊Q811yescharacteristic
269C22×C71#53611yescharacteristic
270C22×C71C2×#13511yescharacteristic
271C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C71#57511yescharacteristic
273C22×C71#57511yescharacteristic
274C4×C71—22no—
275C4×C71—22no—
276C4×C71—42no—
277C4×C71—41no—
278C4×C71—41no—
279C4×C71—81no—
280C4×C71—81no—
281C4×C71#44914yes—
282C4×C71#54811yescharacteristic
283C4×C71#54811yescharacteristic
284C4×C71D4×Q811yescharacteristic
285C4×C71D4×Q811yescharacteristic
286Dic71—44no—
287Dic71—44no—
288#24300—711nomaximal, Sylow
289C2×C71—42no—
290C2×C71—82no—
291C2×C71#240911yescharacteristic
292C2×C71C2×#44911yescharacteristic
293C2×C71#300511yescharacteristic
294C2×C71#307611yescharacteristic
295C2×C71#307711yescharacteristic
296C2×C71#323311yescharacteristic
297C2×C71#323511yescharacteristic
298C2×#533—711no—
299C2×#476—711no—
300C2×#474—711no—
301C4×#121D7111yescharacteristic
302#2059—712no—
303#2025—712no—
304#1988—711no—
305#1987—711no—
306#1875—712no—
307#1854—712no—
308#1797—711no—
309C71#2430011yescharacteristic, complemented, Sylow
310#533—1424no—
311#533—1424no—
312C2×C4⋊Q8—711no—
313C2×C4⋊Q8—711no—
314C2×#125—712no—
315C2×C22⋊Q8—1421no—
316C2×C22⋊Q8—1421no—
317C2×#121C2×D7111yescharacteristic
318C2×C4×Q8—1421no—
319C2×C4×Q8—1421no—
320C2×C4×D4—21no—
321C2×C4×D4C2×D7111yescharacteristic
322C2×#117—712no—
323C22×C42C2×D7111yescharacteristic
324#476—1424no—
325#474—1424no—
326C2×#107—712no—
327C2×#106—712no—
328C2×C4⋊C8—711no—
329C2×C4⋊C8—711no—
330C2×#102—712no—
331C2×#102—712no—
332C2×C22⋊C8—712no—
333#390C2×D7111yescharacteristic
334#389C2×D7111yescharacteristic
335#388—21no—
336#386—21no—
337#380C2×D7111yescharacteristic
338C4×C4⋊C4C2×D7111yescharacteristic
339C4×C22⋊C4—21no—
340#340—712no—
341#340—712no—
342#340—712no—
343#340—712no—
344#336—712no—
345C22×Q8—1421no—
346C22×Q8—1421no—
347C22×D4—21no—
348C22×D4C22×D7111yescharacteristic
349C23×C4—21no—
350C23×C4C22⋊D7111yescharacteristic
351C23×C4C22⋊D7111yescharacteristic
352C23×C4C22×D7111yescharacteristic
353C22×C8—712no—
354C22×C8—712no—
355C4⋊Q8—714no—
356C4⋊Q8—714no—
357C4⋊Q8—1422no—
358C4⋊Q8—1422no—
359#125—1424no—
360#125—1424no—
361C22⋊Q8—1428no—
362C22⋊Q8—1428no—
363#121—22no—
364#121Dic14214yescomplemented
365C4×Q8—1428no—
366C4×Q8—1428no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—44no—
370#117—1424no—
371#117—1424no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—712no—
374C2×C4⋊C4—712no—
375C2×C4⋊C4—1422no—
376C2×C4⋊C4—1422no—
377C2×C4⋊C4—1421no—
378C2×C4⋊C4—1421no—
379C2×C4⋊C4C22×D7111yescharacteristic
380C2×C4⋊C4C22×D7111yescharacteristic
381C2×C4⋊C4C22×D7111yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—1422no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—21no—
391C2×C42—1422no—
392C2×C42C22⋊D7111yescharacteristic
393C2×C42C22⋊D7111yescharacteristic
394C2×C42C22×D7111yescharacteristic
395C2×C42C22×D7111yescharacteristic
396#107—1424no—
397#106—1424no—
398C4⋊C8—1422no—
399C4⋊C8—1422no—
400#102—1424no—
401#102—1424no—
402C22⋊C8—1424no—
403#95—21no—
404#95—21no—
405#95—21no—
406C24—21no—
407C24C2×C22⋊D7111yescharacteristic
408C2×Q8—2842no—
409C2×Q8—2842no—
410C2×Q8—2842no—
411C2×Q8—2842no—
412C2×Q8—1424no—
413C2×Q8—1424no—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4C2×Dic14212yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—1422no—
440C22×C4—1422no—
441C22×C4C2×C22⋊D7111yescharacteristic
442C22×C4C2×C22⋊D7111yescharacteristic
443C22×C4C2×C22⋊D7111yescharacteristic
444C22×C4C2×C22⋊D7111yescharacteristic
445C22×C4C2×C22⋊D7111yescharacteristic
446C22×C4C2×Dic14212yes—
447C22×C4#1176586512yes—
448C22×C4C23×D7111yescharacteristic
449C2×C8—1422no—
450C2×C8—1422no—
451C2×C8—1422no—
452C2×C8—1422no—
453C2×C8—1422no—
454C2×C8—1422no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—42no—
461C4⋊C4—714no—
462C4⋊C4—714no—
463C4⋊C4—1428no—
464C4⋊C4—1428no—
465C4⋊C4—1422no—
466C4⋊C4—1422no—
467C4⋊C4—1424no—
468C4⋊C4—1424no—
469C4⋊C4C2×Dic14212yes—
470C22⋊C4—24no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—1428no—
476C42—21no—
477C42—21no—
478C42—22no—
479C42—22no—
480C42—42no—
481C42—1428no—
482C42D4×D7112yes—
483C42D4×D7112yes—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23#1455610712yes—
498C23C22×C22⋊D7111yescharacteristic
499Q8—2848no—
500Q8—2848no—
501D4—22no—
502D4—22no—
503D4—48no—
504D4—41no—
505D4—41no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—44no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—44no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—84no—
532C2×C4—2842no—
533C2×C4—2842no—
534C2×C4—2842no—
535C2×C4—2842no—
536C2×C4—1424no—
537C2×C4—1424no—
538C2×C4#1455609612yes—
539C2×C4#1455609612yes—
540C2×C4#1455610712yes—
541C2×C4#1455614812yes—
542C2×C4#1455615412yes—
543C2×C4C2×D4×D7111yescharacteristic
544C2×C4C2×D4×D7111yescharacteristic
545C2×C4C22×Dic14211yescharacteristic
546C2×C4C2×#1176586511yescharacteristic
547C2×C4#1455617511yescharacteristic
548C2×C4#1455617511yescharacteristic
549C8—2842no—
550C8—2842no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22C2×#1455609611yescharacteristic
566C22C2×#1455610711yescharacteristic
567C22#1798657011yescharacteristic
568C22C2×#1455614811yescharacteristic
569C22C2×#1455615411yescharacteristic
570C22#1798679711yescharacteristic
571C22#1798679711yescharacteristic
572C4—22no—
573C4—22no—
574C4—41no—
575C4—41no—
576C4—42no—
577C4—81no—
578C4—81no—
579C4—2844no—
580C4—2844no—
581C4#1798614214yes—
582C4#1798665811yescharacteristic
583C4#1798665811yescharacteristic
584C4D4×Dic14211yescharacteristic
585C4D4×Dic14211yescharacteristic
586C2—42no—
587C2—82no—
588C2#2259170511yescharacteristic
589C2C2×#1798614211yescharacteristic
590C2#2259758011yescharacteristic
591C2#2259812811yescharacteristic
592C2#2259813711yescharacteristic
593C2#2259951511yescharacteristic
594C2#2259953411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC4×#121×C71C2
socleC23×C71C22×D4
prime radicalC2×#121×C71C22
hypercenterC4×#121D71
hypocenterC71#24300
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#17986556C2semidirect
2. C2×#17986289C2semidirect
3. C2×#17986279C2semidirect
4. #22589678C2semidirect
5. #22589486C2semidirect
6. #22589292C2semidirect
7. #22589288C2semidirect
9. #22588537C2semidirect
10. #22588320C2semidirect
53. #14556128D4semidirect
54. #14556128D4semidirect
85. #121×C71Q8semidirect
309. C71#24300semidirect
364. #121Dic142semidirect

Maximal subgroups

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

subgroupindex
C2×#179865562
C2×#179862892
C2×#179862792
#225896782
#225894862
#225892922
#225892882
#225886632
#225885372
#225883202
C4×#121×C712
#2430071