Finite Group Databasebeta

← #32003115

Subgroups — #32003115

Order 18176.

1520 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
1C22×C4×Dic142C211yesmaximal, characteristic, complemented
2C2×#17986137C211yesmaximal, characteristic
3C2×#17986099C212yesmaximal, complemented
4#22588698C212yesmaximal, complemented
5#22588658C211yesmaximal, characteristic
6#22588593C212yesmaximal, complemented
7#22588527C211yesmaximal, characteristic, complemented
8#22588274C211yesmaximal, characteristic, complemented
9#22588173C212yesmaximal, complemented
10#22588102C211yesmaximal, characteristic, complemented
11C4×#121×C71C211yesmaximal, characteristic
12C23×Dic142C2211yescharacteristic
13C22×#11765853C2212yes—
14C2×C4×Dic142—28no—
15C2×C4×Dic142—21no—
16C2×C4×Dic142—21no—
17C2×C4×Dic142C2212yes—
18C2×C4×Dic142C2212yes—
19C22×C4⋊Dic71C2211yescharacteristic
20C22×Dic71⋊C4—21no—
21C22×C4×Dic71—21no—
22#17986137C418yescomplemented
23#17986099—28no—
24C2×#14556061C2212yes—
25C2×#14556051C2212yes—
26C2×#14556051C2212yes—
27C2×#14556051C2212yes—
28C2×#14556049C2212yes—
29C2×C22⋊(C71⋊C8)C2211yescharacteristic
30C2×C22⋊(C71⋊C8)C2211yescharacteristic
31C2×#14556039C2212yes—
32#17985761C2212yes—
33#17985761C2212yes—
34#17985754C2212yes—
35C2×#121×C71C2211yescharacteristic
36C2×C4×D4×C71—21no—
37C2×C4×D4×C71C2211yescharacteristic
38C22×C42×C71C2211yescharacteristic
39#390×C71C2211yescharacteristic
40#389×C71C2211yescharacteristic
41#388×C71—21no—
42#386×C71—21no—
43#380×C71C2211yescharacteristic
44C4×C4⋊C4×C71C2211yescharacteristic
45C4×C22⋊C4×C71—21no—
46C22×Dic142—24no—
47C22×Dic142—24no—
48C22×Dic142—21no—
49C22×Dic142—22no—
50C22×Dic142C2×C414yescomplemented
51C22×Dic142C2312yes—
52C23×Dic71—21no—
53C23×Dic71—21no—
54C2×#11765853—24no—
55C2×#11765853—24no—
56C2×#11765853C2×C418yescomplemented
57C22×C71⋊C8C2312yes—
58C22×C71⋊C8C2312yes—
59C4×Dic142—28no—
60C4×Dic142—28no—
61C4×Dic142—24no—
62C4×Dic142—24no—
63C4×Dic142—24no—
64C4×Dic142D418yescomplemented
65C2×C4⋊Dic71—24no—
66C2×C4⋊Dic71—21no—
67C2×C4⋊Dic71C2312yes—
68C2×Dic71⋊C4—28no—
69C2×Dic71⋊C4—22no—
70C2×Dic71⋊C4—22no—
71C2×C4×Dic71—28no—
72C2×C4×Dic71—22no—
73C2×C4×Dic71—22no—
74#14556061—24no—
75#14556051—24no—
76#14556051—24no—
77#14556051C2×C418yescomplemented
78#14556049—24no—
79C22⋊(C71⋊C8)—22no—
80C22⋊(C71⋊C8)C2×C414yescomplemented
81#14556039—24no—
82C22×D4×C71—21no—
83C22×D4×C71C2311yescharacteristic
84C23×C4×C71—21no—
85C23×C4×C71D411yescharacteristic
86C23×C4×C71D411yescharacteristic
87C23×C4×C71C2311yescharacteristic
88#121×C71—22no—
89#121×C71C2×C414yes—
90C4×D4×C71—24no—
91C4×D4×C71—44no—
92C2×C4⋊C4×C71—21no—
93C2×C4⋊C4×C71C2311yescharacteristic
94C2×C4⋊C4×C71C2311yescharacteristic
95C2×C4⋊C4×C71C2311yescharacteristic
96C2×C22⋊C4×C71—21no—
97C2×C22⋊C4×C71—21no—
98C2×C22⋊C4×C71—21no—
99C2×C22⋊C4×C71—21no—
100C2×C22⋊C4×C71—21no—
101C2×C42×C71—24no—
102C2×C42×C71—21no—
103C2×C42×C71D411yescharacteristic
104C2×C42×C71D411yescharacteristic
105C2×C42×C71C2311yescharacteristic
106C2×C42×C71C2311yescharacteristic
107#95×C71—21no—
108#95×C71—21no—
109#95×C71—21no—
110C2×Dic142—24no—
111C2×Dic142—216no—
112C2×Dic142—24no—
113C2×Dic142—24no—
114C2×Dic142—22no—
115C2×Dic142—22no—
116C2×Dic142—22no—
117C2×Dic142—216no—
118C2×Dic142C22×C414yes—
119C2×Dic142C2×D414yes—
120C2×Dic142#4114yes—
121C22×Dic71—24no—
122C22×Dic71—24no—
123C22×Dic71—24no—
124C22×Dic71—22no—
125C22×Dic71—24no—
126C22×Dic71—22no—
127C22×Dic71—42no—
128C22×Dic71—42no—
129#11765853—216no—
130C2×C71⋊C8—22no—
131C2×C71⋊C8—22no—
132C2×C71⋊C8—22no—
133C2×C71⋊C8—22no—
134C2×C71⋊C8—22no—
135C2×C71⋊C8C22×C414yes—
136C4⋊Dic71—24no—
137C4⋊Dic71—22no—
138C4⋊Dic71C2×D414yes—
139Dic71⋊C4—28no—
140Dic71⋊C4—28no—
141C4×Dic71—28no—
142C4×Dic71—28no—
143C24×C71—21no—
144C24×C71C2×D411yescharacteristic
145C2×D4×C71—24no—
146C2×D4×C71—22no—
147C2×D4×C71—22no—
148C2×D4×C71—21no—
149C2×D4×C71—42no—
150C2×D4×C71—42no—
151C2×D4×C71C22×C412yes—
152C22×C4×C71—22no—
153C22×C4×C71—22no—
154C22×C4×C71—22no—
155C22×C4×C71—22no—
156C22×C4×C71—22no—
157C22×C4×C71—22no—
158C22×C4×C71—21no—
159C22×C4×C71—22no—
160C22×C4×C71—21no—
161C22×C4×C71—21no—
162C22×C4×C71—41no—
163C22×C4×C71—42no—
164C22×C4×C71—42no—
165C22×C4×C71—42no—
166C22×C4×C71C22×C412yes—
167C22×C4×C71C2×D411yescharacteristic
168C22×C4×C71C2×D411yescharacteristic
169C22×C4×C71C2×D411yescharacteristic
170C22×C4×C71C2×D411yescharacteristic
171C22×C4×C71C2×D411yescharacteristic
172C22×C4×C71#4112yes—
173C22×C4×C71C2411yescharacteristic
174C4⋊C4×C71—21no—
175C4⋊C4×C71—22no—
176C4⋊C4×C71—22no—
177C4⋊C4×C71—42no—
178C4⋊C4×C71C22×C412yes—
179C22⋊C4×C71—24no—
180C22⋊C4×C71—42no—
181C22⋊C4×C71—42no—
182C22⋊C4×C71—42no—
183C22⋊C4×C71—42no—
184C42×C71—22no—
185C42×C71—22no—
186C42×C71—22no—
187C42×C71—42no—
188C42×C71C2×D414yes—
189C23×C71—21no—
190C23×C71—21no—
191C23×C71—21no—
192C23×C71—21no—
193C23×C71—22no—
194C23×C71—41no—
195C23×C71—41no—
196C23×C71—41no—
197C23×C71—41no—
198C23×C71—41no—
199C23×C71—41no—
200C23×C71C4×D412yes—
201C23×C71AGL(1,ℤ/8)12yes—
202C23×C71#13712yes—
203C23×C71C22×D411yescharacteristic
204D4×C71—24no—
205D4×C71—48no—
206D4×C71—42no—
207C2×C4×C71—28no—
208C2×C4×C71—21no—
209C2×C4×C71—21no—
210C2×C4×C71—21no—
211C2×C4×C71—21no—
212C2×C4×C71—21no—
213C2×C4×C71—21no—
214C2×C4×C71—21no—
215C2×C4×C71—21no—
216C2×C4×C71—24no—
217C2×C4×C71—24no—
218C2×C4×C71—22no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—44no—
223C2×C4×C71—41no—
224C2×C4×C71—41no—
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×C71C4×D412yes—
233C2×C4×C71#12012yes—
234C2×C4×C71#12012yes—
235C2×C4×C71#13712yes—
236C2×C4×C71C23×C411yescharacteristic
237C2×C4×C71C22×D412yes—
238C2×C4×C71C2×#4112yes—
239C2×C4×C71C2×#4111yescharacteristic
240C2×Dic71—216no—
241C2×Dic71—24no—
242C2×Dic71—24no—
243C2×Dic71—24no—
244C2×Dic71—24no—
245C2×Dic71—48no—
246C2×Dic71—42no—
247C2×Dic71—42no—
248Dic142—216no—
249Dic142—24no—
250Dic142—24no—
251Dic142—24no—
252Dic142C4×D418yescomplemented
253C71⋊C8—24no—
254C71⋊C8—42no—
255C22×C71—22no—
256C22×C71—22no—
257C22×C71—22no—
258C22×C71—22no—
259C22×C71—22no—
260C22×C71—22no—
261C22×C71—42no—
262C22×C71—42no—
263C22×C71—42no—
264C22×C71—81no—
265C22×C71—81no—
266C22×C71—81no—
267C22×C71—81no—
268C22×C71#44014yescomplemented
269C22×C71C2×C4×D411yescharacteristic
270C22×C71C2×#12011yescharacteristic
271C22×C71#52511yescharacteristic
272C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C71C2×#13711yescharacteristic
274C22×C71C2×#13711yescharacteristic
275C22×C71#57511yescharacteristic
276C4×C71—22no—
277C4×C71—22no—
278C4×C71—22no—
279C4×C71—22no—
280C4×C71—42no—
281C4×C71—41no—
282C4×C71—81no—
283C4×C71—81no—
284C4×C71C2×C4×D412yes—
285C4×C71#52512yes—
286Dic71—28no—
287Dic71—44no—
288#21454—711nomaximal, Sylow
289C2×C71—24no—
290C2×C71—82no—
291C2×C71C4×#12011yescharacteristic
292C2×C71C2×#44011yescharacteristic
293C2×C71C4×#13711yescharacteristic
294C2×C71#300411yescharacteristic
295C2×C71#300611yescharacteristic
296C2×C71#311211yescharacteristic
297C2×C71#311711yescharacteristic
298C22×C4×Q8—711no—
299C2×#448—711no—
300C2×#440—712no—
301C4×#121D7111yescharacteristic
302#1880—712no—
303#1874—711no—
304#1863—712no—
305#1852—711no—
306#1787—711no—
307#1763—712no—
308#1749—711no—
309C71#2145411yescharacteristic, complemented, Sylow
310C23×Q8—711no—
311C22×SD16—712no—
312C2×#121C2×D7111yescharacteristic
313C2×C4×Q8—712no—
314C2×C4×Q8—712no—
315C2×C4×Q8—1428no—
316C2×C4×Q8—1421no—
317C2×C4×Q8—1421no—
318C2×C4×D4—21no—
319C2×C4×D4C2×D7111yescharacteristic
320C22×C4⋊C4—711no—
321C22×C4⋊C4—1421no—
322C22×C42—1421no—
323C22×C42C2×D7111yescharacteristic
324#448—718no—
325#440—1428no—
326C2×#107—712no—
327C2×#103—712no—
328C2×#103—712no—
329C2×#103—712no—
330C2×#102—712no—
331C2×C22⋊C8—711no—
332C2×C22⋊C8—711no—
333C2×#97—712no—
334#390C2×D7111yescharacteristic
335#389C2×D7111yescharacteristic
336#388—21no—
337#386—21no—
338#380C2×D7111yescharacteristic
339C4×C4⋊C4C2×D7111yescharacteristic
340C4×C22⋊C4—21no—
341#340—712no—
342#340—712no—
343#336—712no—
344C22×Q8—714no—
345C22×Q8—712no—
346C22×Q8—1424no—
347C22×Q8—1424no—
348C22×Q8—1421no—
349C22×Q8—1422no—
350C22×D4—21no—
351C22×D4C22×D7111yescharacteristic
352C23×C4—21no—
353C23×C4—1421no—
354C23×C4—1421no—
355C23×C4C22⋊D7111yescharacteristic
356C23×C4C22⋊D7111yescharacteristic
357C23×C4C22×D7111yescharacteristic
358C2×SD16—718no—
359C2×SD16—1424no—
360C2×SD16—1424no—
361C22×C8—712no—
362C22×C8—712no—
363#121—22no—
364#121C4×D7114yes—
365C4×Q8—718no—
366C4×Q8—1424no—
367C4×Q8—1424no—
368C4×Q8—1428no—
369C4×Q8—1428no—
370C4×Q8—1424no—
371C4×D4—24no—
372C4×D4—44no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—712no—
375C2×C4⋊C4—1424no—
376C2×C4⋊C4—1428no—
377C2×C4⋊C4—1421no—
378C2×C4⋊C4—1422no—
379C2×C4⋊C4—1422no—
380C2×C4⋊C4C22×D7111yescharacteristic
381C2×C4⋊C4C22×D7111yescharacteristic
382C2×C4⋊C4C22×D7111yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C42—24no—
389C2×C42—21no—
390C2×C42—1428no—
391C2×C42—1422no—
392C2×C42—1422no—
393C2×C42C22⋊D7111yescharacteristic
394C2×C42C22⋊D7111yescharacteristic
395C2×C42C22×D7111yescharacteristic
396C2×C42C22×D7111yescharacteristic
397#107—1424no—
398#103—718no—
399#103—1424no—
400#103—1424no—
401#102—1424no—
402C22⋊C8—714no—
403C22⋊C8—1422no—
404#97—1424no—
405#95—21no—
406#95—21no—
407#95—21no—
408C24—21no—
409C24C2×C22⋊D7111yescharacteristic
410C2×Q8—714no—
411C2×Q8—714no—
412C2×Q8—714no—
413C2×Q8—14216no—
414C2×Q8—1424no—
415C2×Q8—1424no—
416C2×Q8—14216no—
417C2×Q8—1422no—
418C2×Q8—1422no—
419C2×Q8—1422no—
420C2×Q8—1424no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4C2×C4×D7112yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—41no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—2842no—
443C22×C4—2842no—
444C22×C4—1424no—
445C22×C4—1422no—
446C22×C4—1424no—
447C22×C4—1424no—
448C22×C4—1422no—
449C22×C4—1424no—
450C22×C4C2×C4×D7112yes—
451C22×C4C2×C22⋊D7111yescharacteristic
452C22×C4C2×C22⋊D7111yescharacteristic
453C22×C4C2×C22⋊D7111yescharacteristic
454C22×C4C2×C22⋊D7111yescharacteristic
455C22×C4C2×C22⋊D7111yescharacteristic
456C22×C4#1176586512yes—
457C22×C4C23×D7111yescharacteristic
458SD16—14216no—
459C2×C8—714no—
460C2×C8—1422no—
461C2×C8—1422no—
462C2×C8—1422no—
463C2×C8—1422no—
464C2×C8—1422no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—42no—
469C4⋊C4—714no—
470C4⋊C4—1428no—
471C4⋊C4—1424no—
472C4⋊C4—1422no—
473C4⋊C4—1428no—
474C4⋊C4C2×C4×D7112yes—
475C22⋊C4—24no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—42no—
484C42—1428no—
485C42—1428no—
486C42D4×D7114yes—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23C4×C22⋊D7112yes—
499C23#1455615412yes—
500C23#1455615812yes—
501C23C22×C22⋊D7111yescharacteristic
502Q8—718no—
503Q8—14216no—
504Q8—1424no—
505Q8—1424no—
506Q8—1424no—
507D4—24no—
508D4—48no—
509D4—42no—
510C2×C4—28no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—84no—
535C2×C4—2848no—
536C2×C4—2842no—
537C2×C4—2842no—
538C2×C4—1424no—
539C2×C4—14216no—
540C2×C4—1424no—
541C2×C4—1424no—
542C2×C4—1424no—
543C2×C4C4×C22⋊D7112yes—
544C2×C4#1455609612yes—
545C2×C4#1455609612yes—
546C2×C4#1455615812yes—
547C2×C4C22×C4×D7111yescharacteristic
548C2×C4C2×D4×D7112yes—
549C2×C4#41×D7112yes—
550C2×C4C2×#1176586511yescharacteristic
551C8—2842no—
552C8—1424no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22#1798609914yescomplemented
567C22C2×C4×C22⋊D7111yescharacteristic
568C22C2×#1455609611yescharacteristic
569C22#1798650611yescharacteristic
570C22C2×#1455615411yescharacteristic
571C22C2×#1455615811yescharacteristic
572C22C2×#1455615811yescharacteristic
573C22#1798679711yescharacteristic
574C4—22no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—41no—
579C4—42no—
580C4—81no—
581C4—81no—
582C4—2844no—
583C4—1428no—
584C4C4×D4×D7112yes—
585C4#1798650812yes—
586C2—24no—
587C2—82no—
588C2C4×#1455609611yescharacteristic
589C2C2×#1798609911yescharacteristic
590C2C4×#1455615811yescharacteristic
591C2#2259757311yescharacteristic
592C2#2259759011yescharacteristic
593C2#2259845811yescharacteristic
594C2#2259850711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC4×#121×C71C2
socleC23×C71C22×D4
prime radicalC2×#121×C71C22
cosocleC22×C4×C71C24
hypercenterC4×#121D71
hypocenterC71#21454
LCSG ▷ C2×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. C22×C4×Dic142C2semidirect
3. C2×#17986099C2semidirect
4. #22588698C2semidirect
6. #22588593C2semidirect
7. #22588527C2semidirect
8. #22588274C2semidirect
9. #22588173C2semidirect
10. #22588102C2semidirect
22. #17986137C4semidirect
50. C22×Dic142C2×C4semidirect
56. C2×#11765853C2×C4semidirect
64. C4×Dic142D4semidirect
77. #14556051C2×C4semidirect
80. C22⋊(C71⋊C8)C2×C4semidirect
252. Dic142C4×D4semidirect
268. C22×C71#440semidirect
309. C71#21454semidirect
566. C22#17986099semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1422
C2×#179861372
C2×#179860992
#225886982
#225886582
#225885932
#225885272
#225882742
#225881732
#225881022
C4×#121×C712
#2145471