Finite Group Databasebeta

← #32014802

Subgroups — #32014802

Order 18176.

894 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985993C211yesmaximal, characteristic
2#22596786C211yesmaximal, characteristic
3#22589050C212yesmaximal, complemented
4#22588988C212yesmaximal, complemented
5#22588614C212yesmaximal, complemented
6#22588439C211yesmaximal, characteristic, complemented
7#22588376C211yesmaximal, characteristic, complemented
8#22588283C211yesmaximal, characteristic
9#22588186C212yesmaximal, complemented
10#22588119C211yesmaximal, characteristic, complemented
11C71×#2924C211yesmaximal, characteristic, complemented
12C22×#11765865C2211yescharacteristic
13C2×#14556134C2212yes—
14C23×C71⋊C8C2211yescharacteristic
15C2×#14556063C2212yescomplemented
16C2×#14556054C2212yes—
17C2×#14556054C2212yes—
18#17985993C411yescharacteristic, complemented
19#17985993C411yescharacteristic, complemented
20#17985993C411yescharacteristic, complemented
21#17985993C411yescharacteristic, complemented
22#17985993C2211yescharacteristic
23#17985993C2211yescharacteristic
24#17985993C2211yescharacteristic
25#17985993C2211yescharacteristic
26C2×#14556051C411yescharacteristic, complemented
27C2×#14556051C411yescharacteristic, complemented
28C2×#14556048C411yescharacteristic, complemented
29C2×#14556048C411yescharacteristic, complemented
30#17985965C2212yes—
31#17985964—22no—
32#17985963C2211yescharacteristic
33#17985961—21no—
34C22⋊C8×D71—21no—
35C2×#14556038C2212yescomplemented
36C4×C22⋊Dic71C2211yescharacteristic
37#17985763C2212yescomplemented
38#17985758C2212yescomplemented
39#17985758C2212yescomplemented
40#17985753C2212yescomplemented
41C2×#117×C71C2211yescharacteristic
42#433×C71—22no—
43#432×C71—21no—
44#431×C71—21no—
45#424×C71C2212yescomplemented
46#407×C71C2211yescharacteristic
47C2×C22⋊C8×C71C2211yescharacteristic
48#405×C71C2212yescomplemented
49C2×#11765865—24no—
50C2×#11765865—21no—
51C2×#11765865—21no—
52C2×#11765865—42no—
53C2×#11765865C2×C411yescharacteristic
54C2×#11765865C2×C411yescharacteristic
55C2×#11765865C2311yescharacteristic
56C2×#11765865C2311yescharacteristic
57C22×Dic142C2×C411yescharacteristic
58C22×D284C2×C411yescharacteristic
59C22×C22⋊D71—21no—
60C22×C4×D71—21no—
61#14556134—24no—
62#14556134—42no—
63C2×#11765849C2312yes—
64C2×#11765848—22no—
65C22×C71⋊C8—42no—
66C22×C71⋊C8C2×C411yescharacteristic
67C22×C71⋊C8C2×C411yescharacteristic
68C22×C71⋊C8C2×C411yescharacteristic
69C22×C71⋊C8C2×C411yescharacteristic
70C22×C71⋊C8C2311yescharacteristic
71C22×C71⋊C8C2311yescharacteristic
72C2×C8×D71—22no—
73C2×C22⋊Dic71—21no—
74C2×C4×Dic71C2312yes—
75#14556063—42no—
76#14556054—24no—
77#14556054—42no—
78#14556051C2×C411yescharacteristic
79#14556051C2×C411yescharacteristic
80#14556051C2×C411yescharacteristic
81#14556051C2×C411yescharacteristic
82#14556051C2×C411yescharacteristic
83#14556051C2×C411yescharacteristic
84#14556051C2×C411yescharacteristic
85#14556051C2×C411yescharacteristic
86#14556048C2×C411yescharacteristic
87#14556048C2×C411yescharacteristic
88#14556048C2×C411yescharacteristic
89#14556048C2×C411yescharacteristic
90#14556048C2×C411yescharacteristic
91#14556048C2×C411yescharacteristic
92#14556048C2×C411yescharacteristic
93#14556048C2×C411yescharacteristic
94C22⋊(C71⋊C8)—21no—
95C22⋊(C71⋊C8)—21no—
96(C2×D71)⋊C8—22no—
97(C2×D71)⋊C8—22no—
98#14556038—42no—
99C71⋊#95—21no—
100C23×C4×C71C2311yescharacteristic
101C2×M16×C71C2312yes—
102C22×C8×C71C2312yes—
103#117×C71—21no—
104#117×C71—21no—
105#117×C71C2×C411yescharacteristic
106#117×C71C2×C411yescharacteristic
107#117×C71C2311yescharacteristic
108#117×C71C2311yescharacteristic
109C2×C4⋊C4×C71C2×C411yescharacteristic
110C2×C4⋊C4×C71C2×C411yescharacteristic
111C2×C22⋊C4×C71—21no—
112C2×C42×C71—21no—
113C4⋊C8×C71—22no—
114C4⋊C8×C71—22no—
115C22⋊C8×C71—21no—
116C22⋊C8×C71—21no—
117C22⋊C8×C71D411yescharacteristic
118C22⋊C8×C71D411yescharacteristic
119C22⋊C8×C71D412yes—
120#97×C71—22no—
121C4×C8×C71—22no—
122C23×D71—21no—
123#11765865—22no—
124#11765865—22no—
125#11765865—22no—
126#11765865—22no—
127#11765865—22no—
128#11765865—41no—
129#11765865—41no—
130#11765865—44no—
131#11765865—44no—
132#11765865#4114yes—
133C2×Dic142—22no—
134C2×Dic142—21no—
135C2×Dic142—41no—
136C2×Dic142C22×C411yescharacteristic
137C2×Dic142C22×C411yescharacteristic
138C2×D284—22no—
139C2×D284—21no—
140C2×D284—41no—
141C2×D284C22×C411yescharacteristic
142C2×D284C22×C411yescharacteristic
143C2×C22⋊D71—21no—
144C2×C22⋊D71—21no—
145C2×C22⋊D71—21no—
146C2×C22⋊D71—21no—
147C2×C22⋊D71—44no—
148C22×Dic71—21no—
149C22×Dic71—22no—
150C2×C4×D71—21no—
151C2×C4×D71—21no—
152C2×C4×D71—21no—
153C2×C4×D71—21no—
154C2×C4×D71—44no—
155#11765849—22no—
156#11765849—22no—
157#11765848—44no—
158C2×C71⋊C8—22no—
159C2×C71⋊C8—22no—
160C2×C71⋊C8—22no—
161C2×C71⋊C8—41no—
162C2×C71⋊C8—41no—
163C2×C71⋊C8—41no—
164C2×C71⋊C8—41no—
165C2×C71⋊C8C22×C411yescharacteristic
166C2×C71⋊C8C22×C411yescharacteristic
167C2×C71⋊C8C22×C411yescharacteristic
168C2×C71⋊C8C22×C411yescharacteristic
169C8×D71—44no—
170C22⋊Dic71—42no—
171C4×Dic71—22no—
172C4×Dic71—22no—
173C24×C71#4111yescharacteristic
174C22×C4×C71—22no—
175C22×C4×C71—22no—
176C22×C4×C71—21no—
177C22×C4×C71—21no—
178C22×C4×C71C22×C411yescharacteristic
179C22×C4×C71C22×C411yescharacteristic
180C22×C4×C71C2×D411yescharacteristic
181C22×C4×C71C2×D411yescharacteristic
182C22×C4×C71#4111yescharacteristic
183C22×C4×C71C2411yescharacteristic
184M16×C71—22no—
185M16×C71—22no—
186C2×C8×C71—22no—
187C2×C8×C71—22no—
188C2×C8×C71—22no—
189C2×C8×C71C2×D412yes—
190C2×C8×C71C2×D412yes—
191C4⋊C4×C71—21no—
192C4⋊C4×C71—21no—
193C4⋊C4×C71C22×C411yescharacteristic
194C4⋊C4×C71C22×C411yescharacteristic
195C4⋊C4×C71C22×C411yescharacteristic
196C4⋊C4×C71C22×C411yescharacteristic
197C22⋊C4×C71—21no—
198C22⋊C4×C71—21no—
199C22⋊C4×C71—21no—
200C22⋊C4×C71—21no—
201C42×C71—21no—
202C42×C71—21no—
203C42×C71—21no—
204C42×C71—21no—
205C23×C71—22no—
206C23×C71—22no—
207C23×C71—41no—
208C23×C71C4×D411yescharacteristic
209C23×C71C4×D411yescharacteristic
210C23×C71C2×#4111yescharacteristic
211C22×D71—21no—
212C22×D71—21no—
213C22×D71—42no—
214C22×D71—42no—
215C22×D71—81no—
216C2×C4×C71—22no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—22no—
222C2×C4×C71—22no—
223C2×C4×C71—21no—
224C2×C4×C71—21no—
225C2×C4×C71—21no—
226C2×C4×C71—21no—
227C2×C4×C71—42no—
228C2×C4×C71—81no—
229C2×C4×C71C4×D411yescharacteristic
230C2×C4×C71C4×D411yescharacteristic
231C2×C4×C71C23×C411yescharacteristic
232C2×C4×C71C22×D411yescharacteristic
233C2×C4×C71C2×#4112yes—
234C2×C4×C71H(5,𝔽2)12yes—
235C22⋊D71—24no—
236C22⋊D71—42no—
237C22⋊D71—42no—
238C22⋊D71—42no—
239C2×Dic71—21no—
240C2×Dic71—21no—
241C2×Dic71—42no—
242C2×Dic71—42no—
243C2×Dic71—42no—
244C2×Dic71—42no—
245D284—21no—
246D284—21no—
247D284—21no—
248D284—42no—
249D284#11712yes—
250C4×D71—24no—
251C4×D71—42no—
252C4×D71—42no—
253C4×D71—42no—
254Dic142—21no—
255Dic142—21no—
256Dic142—21no—
257Dic142—42no—
258Dic142#11712yes—
259C8×C71—22no—
260C8×C71—22no—
261C8×C71—42no—
262C71⋊C8—21no—
263C71⋊C8—21no—
264C71⋊C8—21no—
265C71⋊C8—21no—
266C71⋊C8—42no—
267C22×C71—21no—
268C22×C71—21no—
269C22×C71—21no—
270C22×C71—21no—
271C22×C71—21no—
272C22×C71—21no—
273C22×C71—21no—
274C22×C71—44no—
275C22×C71#44412yes—
276C22×C71C2×C4×D411yescharacteristic
277C22×C71#55112yes—
278C2×D71—22no—
279C2×D71—41no—
280C2×D71—41no—
281C2×D71—41no—
282C2×D71—41no—
283C2×D71—41no—
284C2×D71—84no—
285C4×C71—21no—
286C4×C71—21no—
287C4×C71—41no—
288C4×C71—41no—
289C4×C71—41no—
290C4×C71—41no—
291C4×C71C2×#11711yescharacteristic
292C4×C71#51811yescharacteristic
293C4×C71#54612yes—
294Dic71—22no—
295Dic71—41no—
296Dic71—82no—
297#22586—711nomaximal, Sylow
298C2×C71—22no—
299C2×C71—22no—
300C2×C71—41no—
301C2×C71#234211yescharacteristic
302C2×C71C2×#44411yescharacteristic
303C2×C71#295711yescharacteristic
304D71—42no—
305D71—81no—
306#2924D7111yescharacteristic, complemented
307C2×#416—711no—
308#2880—711no—
309#1938—712no—
310#1929—712no—
311#1866—712no—
312#1826—711no—
313#1810—711no—
314#1789—711no—
315#1765—712no—
316#1752—711no—
317C71#2258611yescharacteristic, complemented, Sylow
318C22×#41—711no—
319C2×#131—712no—
320C23×C8—711no—
321C2×#117C2×D7111yescharacteristic
322#433—22no—
323#432—21no—
324#431—21no—
325C2×#108—712no—
326#424C2×D7112yescomplemented
327C2×#104—712no—
328C2×#104—712no—
329#416—711no—
330#416—711no—
331#416—711no—
332#416—711no—
333#416—711no—
334#416—711no—
335#416—711no—
336#416—711no—
337C2×#103—711no—
338C2×#103—711no—
339C2×#102—711no—
340C2×#102—711no—
341#408—712no—
342#408—1422no—
343#407—711no—
344#407—1421no—
345#407C2×D7111yescharacteristic
346C2×C22⋊C8—1421no—
347C2×C22⋊C8C2×D7111yescharacteristic
348#405C2×D7112yescomplemented
349C2×#97—712no—
350C4×C22⋊C4—711no—
351#341—712no—
352#339—712no—
353#339—712no—
354#336—712no—
355C2×#41—2842no—
356C2×#41—711no—
357C2×#41—711no—
358C2×#41—711no—
359C2×#41—711no—
360C2×#41—1424no—
361C2×#41—1421no—
362C2×#41—1421no—
363C22×Q8—711no—
364C22×D4—711no—
365C22×D4—1421no—
366C23×C4—1421no—
367C23×C4C22×D7111yescharacteristic
368#131—2842no—
369#131—1424no—
370C2×M16—712no—
371C2×M16—1422no—
372C2×M16C22×D7112yes—
373C22×C8—2842no—
374C22×C8—711no—
375C22×C8—711no—
376C22×C8—711no—
377C22×C8—711no—
378C22×C8—711no—
379C22×C8—711no—
380C22×C8—1422no—
381C22×C8C22×D7112yes—
382#117—21no—
383#117—21no—
384#117C4×D7111yescharacteristic
385#117C4×D7111yescharacteristic
386#117C22×D7111yescharacteristic
387#117C22×D7111yescharacteristic
388C2×C4⋊C4C4×D7111yescharacteristic
389C2×C4⋊C4C4×D7111yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—1421no—
392C2×C42—21no—
393C2×C42—712no—
394#108—2842no—
395C4⋊C8—22no—
396C4⋊C8—22no—
397#104—2842no—
398#104—1424no—
399#103—711no—
400#103—711no—
401#103—711no—
402#103—711no—
403#103—711no—
404#103—711no—
405#103—711no—
406#103—711no—
407#102—711no—
408#102—711no—
409#102—711no—
410#102—711no—
411#102—711no—
412#102—711no—
413#102—711no—
414#102—711no—
415C22⋊C8—21no—
416C22⋊C8—21no—
417C22⋊C8—1421no—
418C22⋊C8—1421no—
419C22⋊C8—1422no—
420C22⋊C8—1422no—
421C22⋊C8C22⋊D7111yescharacteristic
422C22⋊C8C22⋊D7111yescharacteristic
423C22⋊C8C22⋊D7112yes—
424#97—22no—
425#97—2842no—
426C4×C8—22no—
427#95—1421no—
428C24—1421no—
429C24#1176586511yescharacteristic
430#41—2844no—
431#41—2841no—
432#41—2841no—
433#41—2844no—
434#41—714no—
435#41—1422no—
436#41—1422no—
437#41—1422no—
438#41—1422no—
439#41—1422no—
440C2×Q8—2841no—
441C2×Q8—711no—
442C2×Q8—711no—
443C2×Q8—1422no—
444C2×Q8—1421no—
445C2×D4—2841no—
446C2×D4—2844no—
447C2×D4—711no—
448C2×D4—711no—
449C2×D4—1422no—
450C2×D4—1421no—
451C2×D4—1421no—
452C2×D4—1421no—
453C2×D4—1421no—
454C2×D4—1421no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—2844no—
460C22×C4—1422no—
461C22×C4—1421no—
462C22×C4—1421no—
463C22×C4—1421no—
464C22×C4—1421no—
465C22×C4—1421no—
466C22×C4C2×C4×D7111yescharacteristic
467C22×C4C2×C4×D7111yescharacteristic
468C22×C4C2×C22⋊D7111yescharacteristic
469C22×C4C2×C22⋊D7111yescharacteristic
470C22×C4#1176586511yescharacteristic
471C22×C4C23×D7111yescharacteristic
472M16—22no—
473M16—22no—
474M16—2844no—
475M16—1422no—
476M16—1422no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—2844no—
481C2×C8—2841no—
482C2×C8—2841no—
483C2×C8—2841no—
484C2×C8—2841no—
485C2×C8—711no—
486C2×C8—711no—
487C2×C8—711no—
488C2×C8—711no—
489C2×C8—1422no—
490C2×C8—1422no—
491C2×C8—1422no—
492C2×C8C2×C22⋊D7112yes—
493C2×C8C2×C22⋊D7112yes—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4C2×C4×D7111yescharacteristic
497C4⋊C4C2×C4×D7111yescharacteristic
498C4⋊C4C2×C4×D7111yescharacteristic
499C4⋊C4C2×C4×D7111yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—2842no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—1422no—
510C42—1422no—
511C23—22no—
512C23—22no—
513C23—41no—
514C23—2842no—
515C23—2842no—
516C23—5681no—
517C23—1421no—
518C23—1421no—
519C23C4×C22⋊D7111yescharacteristic
520C23C4×C22⋊D7111yescharacteristic
521C23C2×#1176586511yescharacteristic
522Q8—2842no—
523Q8—712no—
524Q8—1421no—
525Q8—1421no—
526Q8—1421no—
527D4—2842no—
528D4—2842no—
529D4—2842no—
530D4—2842no—
531D4—712no—
532D4—1424no—
533D4—1421no—
534D4—1421no—
535D4—1421no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—42no—
548C2×C4—81no—
549C2×C4—2842no—
550C2×C4—2842no—
551C2×C4—2842no—
552C2×C4—2842no—
553C2×C4—2842no—
554C2×C4—2842no—
555C2×C4—2842no—
556C2×C4—1421no—
557C2×C4—1421no—
558C2×C4—1424no—
559C2×C4C4×C22⋊D7111yescharacteristic
560C2×C4C4×C22⋊D7111yescharacteristic
561C2×C4C22×C4×D7111yescharacteristic
562C2×C4C22×C22⋊D7111yescharacteristic
563C2×C4#41×D7112yes—
564C2×C4#1455617412yes—
565C8—22no—
566C8—22no—
567C8—42no—
568C8—2842no—
569C8—1421no—
570C8—1421no—
571C8—1421no—
572C8—1421no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—44no—
581C22—2841no—
582C22—2841no—
583C22—2841no—
584C22—2841no—
585C22—2841no—
586C22—5684no—
587C22—1422no—
588C22#1798611912yes—
589C22C2×C4×C22⋊D7111yescharacteristic
590C22#1798667312yes—
591C4—21no—
592C4—21no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—2841no—
598C4—5682no—
599C4—1422no—
600C4#117×D7111yescharacteristic
601C4#1798647311yescharacteristic
602C4#1798663712yes—
603C2—22no—
604C2—22no—
605C2—41no—
606C2—2842no—
607C2—5681no—
608C2#2259119011yescharacteristic
609C2C2×#1798611911yescharacteristic
610C2#2259725511yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2924C2
socleC22×C71C2×C4×D4
prime radicalC22×#11765865C22
cosocleC22×C4×C71C24
hypercenter#2924D71
hypocenterC71#22586
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ #2924

Complements

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

NH ≅ G/Ntype
3. #22589050C2semidirect
4. #22588988C2semidirect
5. #22588614C2semidirect
6. #22588439C2semidirect
7. #22588376C2semidirect
9. #22588186C2semidirect
10. #22588119C2semidirect
11. C71×#2924C2semidirect
15. C2×#14556063C22semidirect
18. #17985993C4semidirect
19. #17985993C4semidirect
20. #17985993C4semidirect
21. #17985993C4semidirect
26. C2×#14556051C4semidirect
27. C2×#14556051C4semidirect
28. C2×#14556048C4semidirect
29. C2×#14556048C4semidirect
35. C2×#14556038C22semidirect
37. #17985763C22semidirect
38. #17985758C22semidirect
39. #17985758C22semidirect
40. #17985753C22semidirect
45. #424×C71C22semidirect
48. #405×C71C22semidirect
306. #2924D71semidirect
317. C71#22586semidirect
326. #424C2×D71semidirect
348. #405C2×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179859932
#225967862
#225890502
#225889882
#225886142
#225884392
#225883762
#225882832
#225881862
#225881192
C71×#29242
#2258671