Finite Group Databasebeta

← #31997047

Subgroups — #31997047

Order 18176.

1438 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556134C211yesmaximal, characteristic
2C2×#17986036C211yesmaximal, characteristic, complemented
3C2×#107×D71C211yesmaximal, characteristic, complemented
4#22590929C212yesmaximal
5#22588372C212yesmaximal, complemented
6#22588371C212yesmaximal, complemented
7#22588369C211yesmaximal, characteristic, complemented
8#22588366C211yesmaximal, characteristic, complemented
9#22588366C211yesmaximal, characteristic, complemented
10#22588058C212yesmaximal, complemented
11C71×#1808C211yesmaximal, characteristic, complemented
12C22×#11765865C2211yescharacteristic
13C2×#14556134—24no—
14C2×#14556134—24no—
15C2×#14556134C418yescomplemented
16C22×#11765849C2211yescharacteristic
17C22×#11765848C2211yescharacteristic
18C22×#11765848C2211yescharacteristic
19C22×C8×D71C2211yescharacteristic
20C22×C8×D71C2211yescharacteristic
21C2×#14556086C2212yes—
22C2×C4⋊C4×D71C2212yes—
23#17986036—24no—
24C2×#14556062C2211yescharacteristic
25C2×#14556062C2211yescharacteristic
26C2×#14556061C2212yes—
27#107×D71—24no—
28C2×#14556059C2212yes—
29#17985903—22no—
30#17985899—22no—
31#17985877—22no—
32#17985875—22no—
33#17985859C2212yes—
34#17985856C2212yes—
35#17985856C2212yes—
36#17985762C2212yes—
37#17985761C2212yes—
38#17985761C2212yes—
39C23×C8×C71C2211yescharacteristic
40C2×#107×C71C2211yescharacteristic
41C2×#107×C71C2211yescharacteristic
42#380×C71C2212yes—
43#340×C71C2212yes—
44C2×#11765865—22no—
45C2×#11765865—22no—
46C2×#11765865—22no—
47C2×#11765865—22no—
48C2×#11765865C2×C414yes—
49C2×#11765865Q814yes—
50C22×Dic142D411yescharacteristic
51C22×D284D411yescharacteristic
52C22×C22⋊D71D411yescharacteristic
53C22×C22⋊D71D411yescharacteristic
54C22×C4×D71—22no—
55C22×C4×D71C2311yescharacteristic
56C22×C4×D71C2311yescharacteristic
57#14556134—232no—
58C2×#11765849—22no—
59C2×#11765849—22no—
60C2×#11765849C2×C414yescomplemented
61C2×#11765848—22no—
62C2×#11765848—22no—
63C2×#11765848—22no—
64C2×#11765848—22no—
65C2×#11765848C2×C414yes—
66C2×#11765848C2×C414yescomplemented
67C22×C71⋊C8C2311yescharacteristic
68C22×C71⋊C8C2311yescharacteristic
69C2×C8×D71—22no—
70C2×C8×D71—22no—
71C2×C8×D71C2×C414yes—
72C2×C8×D71C2×C414yescomplemented
73C2×C8×D71D414yes—
74C2×C8×D71D414yes—
75#14556086—24no—
76#14556086—24no—
77C2×C4⋊Dic71C2312yes—
78C2×C4⋊Dic71C2312yes—
79C2×Dic71⋊C4—22no—
80C4⋊C4×D71—24no—
81C4⋊C4×D71—24no—
82C2×C22⋊Dic71—22no—
83C2×(C2×D71)⋊C4—22no—
84C2×(C2×D71)⋊C4—22no—
85C2×(C2×D71)⋊C4—22no—
86C2×C4×Dic71—22no—
87#14556062—22no—
88#14556062—22no—
89#14556061—24no—
90#14556059—24no—
91C71⋊#95—22no—
92C71⋊#95—22no—
93C71⋊#95—22no—
94C23×C4×C71C2311yescharacteristic
95C22×C8×C71—22no—
96C22×C8×C71—22no—
97C22×C8×C71C2×C414yescomplemented
98C22×C8×C71C2311yescharacteristic
99C22×C8×C71C2311yescharacteristic
100C2×C4⋊C4×C71C2312yes—
101C2×C4⋊C4×C71C2312yes—
102C2×C22⋊C4×C71—22no—
103#107×C71—22no—
104#107×C71—22no—
105#95×C71—22no—
106C23×D71C2×D411yescharacteristic
107C23×D71C2×D411yescharacteristic
108#11765865—28no—
109#11765865—28no—
110#11765865—28no—
111#11765865—28no—
112C2×Dic142—22no—
113C2×Dic142—22no—
114C2×Dic142C4⋊C414yes—
115C2×D284—22no—
116C2×D284—22no—
117C2×D284C4⋊C414yes—
118C2×C22⋊D71—22no—
119C2×C22⋊D71—22no—
120C2×C22⋊D71—22no—
121C2×C22⋊D71—22no—
122C2×C22⋊D71C4⋊C414yes—
123C2×C22⋊D71C4⋊C414yes—
124C22×Dic71—22no—
125C22×Dic71—22no—
126C22×Dic71C2×D411yescharacteristic
127C22×Dic71C2×D411yescharacteristic
128C2×C4×D71—21no—
129C2×C4×D71—21no—
130C2×C4×D71—21no—
131C2×C4×D71—21no—
132C2×C4×D71—44no—
133C2×C4×D71—82no—
134C2×C4×D71—82no—
135C2×C4×D71C22×C412yes—
136C2×C4×D71C22×C412yes—
137C2×C4×D71C2×D412yes—
138C2×C4×D71C2×D412yes—
139C2×C4×D71C2×Q812yes—
140C2×C4×D71C2×Q812yes—
141C2×C4×D71#4112yes—
142C2×C4×D71#4112yes—
143#11765849—28no—
144#11765848—28no—
145#11765848—28no—
146C2×C71⋊C8—21no—
147C2×C71⋊C8—21no—
148C2×C71⋊C8C22×C412yes—
149C2×C71⋊C8C22×C412yes—
150C2×C71⋊C8C2×D412yes—
151C2×C71⋊C8C2×D412yes—
152C8×D71—28no—
153C8×D71C22⋊C4116yescomplemented
154C4⋊Dic71—22no—
155C4⋊Dic71—22no—
156C4⋊Dic71—22no—
157C4⋊Dic71—22no—
158Dic71⋊C4—44no—
159C22⋊Dic71—82no—
160(C2×D71)⋊C4—44no—
161(C2×D71)⋊C4—82no—
162(C2×D71)⋊C4—82no—
163C4×Dic71—44no—
164C24×C71C2×D411yescharacteristic
165C22×C4×C71—21no—
166C22×C4×C71—21no—
167C22×C4×C71—21no—
168C22×C4×C71—21no—
169C22×C4×C71—22no—
170C22×C4×C71—22no—
171C22×C4×C71C22×C412yes—
172C22×C4×C71C2×D411yescharacteristic
173C22×C4×C71C2×Q812yes—
174C22×C4×C71C2411yescharacteristic
175C2×C8×C71—21no—
176C2×C8×C71—21no—
177C2×C8×C71—28no—
178C2×C8×C71C22×C412yes—
179C2×C8×C71C22×C412yes—
180C2×C8×C71C2×D412yes—
181C2×C8×C71C2×D412yes—
182C4⋊C4×C71—22no—
183C4⋊C4×C71—22no—
184C4⋊C4×C71—22no—
185C4⋊C4×C71—22no—
186C22⋊C4×C71—82no—
187C23×C71—21no—
188C23×C71—21no—
189C23×C71—21no—
190C23×C71—21no—
191C23×C71—21no—
192C23×C71—21no—
193C23×C71C2×C4⋊C412yes—
194C23×C71C22×D411yescharacteristic
195C22×D71—21no—
196C22×D71—21no—
197C22×D71—21no—
198C22×D71—21no—
199C22×D71—21no—
200C22×D71—21no—
201C22×D71—21no—
202C22×D71—21no—
203C22×D71—21no—
204C22×D71—21no—
205C22×D71C2×C4⋊C412yes—
206C22×D71C2×C4⋊C412yes—
207C22×D71#12112yes—
208C22×D71#12112yes—
209C2×C4×C71—21no—
210C2×C4×C71—21no—
211C2×C4×C71—22no—
212C2×C4×C71—22no—
213C2×C4×C71—22no—
214C2×C4×C71—22no—
215C2×C4×C71—42no—
216C2×C4×C71—42no—
217C2×C4×C71—42no—
218C2×C4×C71—42no—
219C2×C4×C71—42no—
220C2×C4×C71—42no—
221C2×C4×C71C2×C4⋊C412yes—
222C2×C4×C71C23×C411yescharacteristic
223C2×C4×C71C22×D411yescharacteristic
224C2×C4×C71C22×D411yescharacteristic
225C2×C4×C71C22×Q811yescharacteristic
226C2×C4×C71C2×#4111yescharacteristic
227C2×C4×C71C2×#4111yescharacteristic
228C22⋊D71—28no—
229C22⋊D71—28no—
230C2×Dic71—21no—
231C2×Dic71—21no—
232C2×Dic71—42no—
233C2×Dic71—42no—
234C2×Dic71—42no—
235C2×Dic71—42no—
236C2×Dic71—42no—
237C2×Dic71—42no—
238C2×Dic71C2×C4⋊C412yes—
239C2×Dic71C2×C4⋊C412yes—
240C2×Dic71#12112yes—
241C2×Dic71#12112yes—
242D284—28no—
243C4×D71—22no—
244C4×D71—22no—
245C4×D71—22no—
246C4×D71—22no—
247C4×D71—88no—
248C4×D71C2×C22⋊C414yes—
249C4×D71#11714yes—
250C4×D71C22⋊Q814yes—
251C4×D71C22⋊Q814yes—
252Dic142—28no—
253C8×C71—22no—
254C8×C71C2×C22⋊C414yes—
255C71⋊C8—22no—
256C71⋊C8C2×C22⋊C414yes—
257C22×C71—22no—
258C22×C71—22no—
259C22×C71—22no—
260C22×C71—41no—
261C22×C71—41no—
262C22×C71—41no—
263C22×C71—41no—
264C22×C71C22×C4⋊C411yescharacteristic
265C22×C71C2×#12111yescharacteristic
266C22×C71C2×#12111yescharacteristic
267C22×C71#57611yescharacteristic
268C22×C71#57611yescharacteristic
269C22×C71#57811yescharacteristic
270C22×C71#57811yescharacteristic
271C2×D71—22no—
272C2×D71—22no—
273C2×D71—22no—
274C2×D71—22no—
275C2×D71—22no—
276C2×D71—22no—
277C2×D71—22no—
278C2×D71—22no—
279C2×D71—22no—
280C2×D71—41no—
281C2×D71—41no—
282C2×D71—41no—
283C2×D71—41no—
284C2×D71#38014yes—
285C4×C71—22no—
286C4×C71—82no—
287C4×C71—82no—
288C4×C71C22×C22⋊C411yescharacteristic
289C4×C71C2×#11711yescharacteristic
290C4×C71C2×C22⋊Q811yescharacteristic
291C4×C71C2×C22⋊Q811yescharacteristic
292Dic71—22no—
293Dic71—82no—
294Dic71—82no—
295Dic71#38014yes—
296#20663—711nomaximal, Sylow
297C2×C71—42no—
298C2×C71C2×#38011yescharacteristic
299C2×C71#291311yescharacteristic
300C2×C71#291311yescharacteristic
301C2×C71#305811yescharacteristic
302C2×C71#305911yescharacteristic
303C2×C71#306111yescharacteristic
304C2×C71#306111yescharacteristic
305D71—24no—
306D71—42no—
307C22×#131—711no—
308C2×#427—711no—
309C22×#107—711no—
310#2311—712no—
311#1809—712no—
312#1809—712no—
313#1808—711no—
314#1808—711no—
315#1808—711no—
316#1808D7111yescharacteristic, complemented
317#1736—712no—
318C71#2066311yescharacteristic, complemented, Sylow
319C22×#41—711no—
320C2×#131—718no—
321C2×#131—1424no—
322C2×#131—1424no—
323C22×M16—711no—
324C22×M16—711no—
325C22×M16—711no—
326C23×C8—711no—
327C23×C8—711no—
328C23×C8C2×D7111yescharacteristic
329C2×#117—712no—
330C22×C4⋊C4—712no—
331#427—1424no—
332C2×#107—711no—
333C2×#107—711no—
334C2×#107—712no—
335C2×#107—1424no—
336C2×#107C2×D7111yescharacteristic
337C2×#107C2×D7111yescharacteristic
338C2×#106—712no—
339#391—1422no—
340#390—1422no—
341#386—1422no—
342#386—1422no—
343#380—712no—
344#380—712no—
345#380—712no—
346#380C2×D7112yes—
347#340—712no—
348#340—712no—
349#340—712no—
350#340C2×D7112yes—
351C2×#41—714no—
352C2×#41—714no—
353C2×#41—1422no—
354C2×#41—1422no—
355C2×#41—1422no—
356C2×#41—1422no—
357C22×Q8—711no—
358C22×D4—711no—
359C22×D4—711no—
360C22×D4—711no—
361C23×C4—711no—
362C23×C4—711no—
363C23×C4—1422no—
364C23×C4C22×D7111yescharacteristic
365#131—14232no—
366C2×M16—714no—
367C2×M16—714no—
368C2×M16—714no—
369C2×M16—1422no—
370C2×M16—1422no—
371C2×M16—1422no—
372C2×M16—1422no—
373C2×M16—1422no—
374C2×M16—1422no—
375C22×C8—22no—
376C22×C8—22no—
377C22×C8—711no—
378C22×C8—714no—
379C22×C8—714no—
380C22×C8—714no—
381C22×C8—711no—
382C22×C8—714no—
383C22×C8—1422no—
384C22×C8—1422no—
385C22×C8C4×D7114yescomplemented
386C22×C8C22×D7111yescharacteristic
387C22×C8C22×D7111yescharacteristic
388#117—1424no—
389#117—1424no—
390C2×C4⋊C4—712no—
391C2×C4⋊C4—712no—
392C2×C4⋊C4—1424no—
393C2×C4⋊C4—1424no—
394C2×C4⋊C4—1422no—
395C2×C4⋊C4C22×D7112yes—
396C2×C4⋊C4C22×D7112yes—
397C2×C22⋊C4—22no—
398C2×C22⋊C4—1422no—
399C2×C22⋊C4—1422no—
400C2×C22⋊C4—1422no—
401C2×C22⋊C4—1422no—
402C2×C42—1422no—
403#107—22no—
404#107—22no—
405#107—1424no—
406#107—1422no—
407#107—1422no—
408#106—1424no—
409#95—22no—
410#95—1422no—
411#95—1422no—
412#95—1422no—
413C24—711no—
414C24—711no—
415C24D4×D7111yescharacteristic
416#41—1428no—
417#41—1428no—
418#41—1428no—
419#41—1428no—
420C2×Q8—714no—
421C2×Q8—1422no—
422C2×Q8—1422no—
423C2×D4—714no—
424C2×D4—714no—
425C2×D4—714no—
426C2×D4—1422no—
427C2×D4—1422no—
428C2×D4—1422no—
429C2×D4—1422no—
430C2×D4—1422no—
431C2×D4—1422no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—2844no—
439C22×C4—5682no—
440C22×C4—5682no—
441C22×C4—711no—
442C22×C4—712no—
443C22×C4—712no—
444C22×C4—712no—
445C22×C4—712no—
446C22×C4—712no—
447C22×C4—712no—
448C22×C4—711no—
449C22×C4—712no—
450C22×C4—712no—
451C22×C4—1422no—
452C22×C4—1422no—
453C22×C4—1421no—
454C22×C4—1421no—
455C22×C4—1421no—
456C22×C4—1421no—
457C22×C4C2×C4×D7112yes—
458C22×C4D4×D7111yescharacteristic
459C22×C4Q8×D7112yes—
460C22×C4C23×D7111yescharacteristic
461M16—1428no—
462M16—1428no—
463M16—1428no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—28no—
467C2×C8—712no—
468C2×C8—712no—
469C2×C8—712no—
470C2×C8—7116no—
471C2×C8—712no—
472C2×C8—1421no—
473C2×C8—1421no—
474C2×C8—1428no—
475C2×C8C2×C4×D7112yes—
476C2×C8C2×C4×D7112yes—
477C2×C8D4×D7112yes—
478C2×C8D4×D7112yes—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—2844no—
484C4⋊C4—1422no—
485C4⋊C4—1422no—
486C4⋊C4—1422no—
487C4⋊C4—1422no—
488C22⋊C4—82no—
489C22⋊C4—2844no—
490C22⋊C4—5682no—
491C22⋊C4—5682no—
492C22⋊C4—5682no—
493C42—2844no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—712no—
501C23—712no—
502C23—712no—
503C23—712no—
504C23—1421no—
505C23—1421no—
506C23—1421no—
507C23—1421no—
508C23—1421no—
509C23—1421no—
510C23—1421no—
511C23—1421no—
512C23—1421no—
513C23—1421no—
514C23C4⋊C4×D7112yes—
515C23C2×D4×D7111yescharacteristic
516Q8—1428no—
517D4—1428no—
518D4—1428no—
519D4—1428no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—2842no—
533C2×C4—2842no—
534C2×C4—2842no—
535C2×C4—2842no—
536C2×C4—2842no—
537C2×C4—2842no—
538C2×C4—5688no—
539C2×C4—712no—
540C2×C4—712no—
541C2×C4—712no—
542C2×C4—714no—
543C2×C4—714no—
544C2×C4—714no—
545C2×C4—714no—
546C2×C4—712no—
547C2×C4—1421no—
548C2×C4—1421no—
549C2×C4—1422no—
550C2×C4—1422no—
551C2×C4—1422no—
552C2×C4—1422no—
553C2×C4C4⋊C4×D7112yes—
554C2×C4C22×C4×D7111yescharacteristic
555C2×C4C2×D4×D7111yescharacteristic
556C2×C4C2×D4×D7111yescharacteristic
557C2×C4C2×Q8×D7111yescharacteristic
558C2×C4#41×D7111yescharacteristic
559C2×C4#41×D7111yescharacteristic
560C8—22no—
561C8—714no—
562C8—1422no—
563C8C22⋊C4×D7114yes—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—2841no—
572C22—2841no—
573C22—2841no—
574C22—2841no—
575C22—714no—
576C22—1422no—
577C22—1422no—
578C22—1422no—
579C22—1422no—
580C22—1422no—
581C22—1422no—
582C22—1422no—
583C22—1422no—
584C22—1422no—
585C22C2×C4⋊C4×D7111yescharacteristic
586C22#121×D7111yescharacteristic
587C22#121×D7111yescharacteristic
588C22#1798680011yescharacteristic
589C22#1798680111yescharacteristic
590C22#1798681211yescharacteristic
591C22#1798681311yescharacteristic
592C4—22no—
593C4—82no—
594C4—82no—
595C4—5682no—
596C4—5682no—
597C4—714no—
598C4—1422no—
599C4C2×C22⋊C4×D7111yescharacteristic
600C4#117×D7111yescharacteristic
601C4C22⋊Q8×D7111yescharacteristic
602C4C22⋊Q8×D7111yescharacteristic
603C2—42no—
604C2—2842no—
605C2—1424no—
606C2#380×D7111yescharacteristic
607C2#2259693611yescharacteristic
608C2#2259693711yescharacteristic
609C2#2259798311yescharacteristic
610C2#2259799011yescharacteristic
611C2#2259800511yescharacteristic
612C2#2259801111yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1808C2
socleC23×C71C22×D4
prime radicalC22×#11765865C22
cosocleC22×C4×C71C24
hypercenter#1808D71
hypocenterC71#20663
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1808

Complements

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

NH ≅ G/Ntype
2. C2×#17986036C2semidirect
3. C2×#107×D71C2semidirect
5. #22588372C2semidirect
6. #22588371C2semidirect
7. #22588369C2semidirect
8. #22588366C2semidirect
9. #22588366C2semidirect
10. #22588058C2semidirect
11. C71×#1808C2semidirect
15. C2×#14556134C4semidirect
60. C2×#11765849C2×C4semidirect
66. C2×#11765848C2×C4semidirect
72. C2×C8×D71C2×C4semidirect
97. C22×C8×C71C2×C4semidirect
153. C8×D71C22⋊C4semidirect
316. #1808D71semidirect
318. C71#20663semidirect
385. C22×C8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C22×#145561342
C2×#179860362
C2×#107×D712
#225909292
#225883722
#225883712
#225883692
#225883662
#225880582
C71×#18082
#2066371