Finite Group Databasebeta

← #32003117

Subgroups — #32003117

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
1C2×#17986138C211yesmaximal, characteristic
2C2×#17986097C212yesmaximal, complemented
3C4×#14556102C211yesmaximal, characteristic
4#22588695C212yesmaximal, complemented
5#22588657C211yesmaximal, characteristic
6#22588594C212yesmaximal, complemented
7#22588528C211yesmaximal, characteristic, complemented
8#22588275C211yesmaximal, characteristic, complemented
9#22588170C212yesmaximal, complemented
10#22588102C211yesmaximal, characteristic, complemented
11C22×C4×Q8×C71C211yesmaximal, characteristic, complemented
12C22×#11765852C2212yes—
13C2×#14556102C2211yescharacteristic
14C2×C4×D284C2211yescharacteristic
15C2×C4×C22⋊D71—21no—
16#17986138C418yescomplemented
17#17986097—28no—
18C2×#14556061C2212yes—
19C2×#14556052C2212yes—
20C2×#14556052C2212yes—
21C2×#14556052C2212yes—
22C2×#14556048C2212yes—
23C2×C22⋊(C71⋊C8)C2211yescharacteristic
24C2×C22⋊(C71⋊C8)C2211yescharacteristic
25C2×#14556039C2212yes—
26#17985901C2211yescharacteristic
27#17985897C2211yescharacteristic
28#17985890—21no—
29#17985878—21no—
30#17985859C2211yescharacteristic
31C4×C4⋊Dic71C2211yescharacteristic
32C4×(C2×D71)⋊C4—21no—
33#17985761C2212yes—
34#17985761C2212yes—
35#17985754C2212yes—
36C23×Q8×C71C2211yescharacteristic
37C2×C4×Q8×C71—28no—
38C2×C4×Q8×C71—21no—
39C2×C4×Q8×C71—21no—
40C2×C4×Q8×C71C2212yes—
41C2×C4×Q8×C71C2212yes—
42C22×C4⋊C4×C71—21no—
43C22×C4⋊C4×C71C2211yescharacteristic
44C22×C42×C71—21no—
45C22×C42×C71C2211yescharacteristic
46C22×D284C2311yescharacteristic
47C22×C22⋊D71—21no—
48C22×C4×D71—21no—
49C2×#11765852—24no—
50C2×#11765852—24no—
51C2×#11765852C2×C418yescomplemented
52C22×C71⋊C8C2312yes—
53C22×C71⋊C8C2312yes—
54#14556102—22no—
55#14556102C2×C414yes—
56C4×D284—24no—
57C4×C22⋊D71—44no—
58C2×C4⋊Dic71C2311yescharacteristic
59C2×C4⋊Dic71C2311yescharacteristic
60C2×C4⋊Dic71C2311yescharacteristic
61C2×Dic71⋊C4—21no—
62C2×C22⋊Dic71—21no—
63C2×(C2×D71)⋊C4—21no—
64C2×(C2×D71)⋊C4—21no—
65C2×(C2×D71)⋊C4—21no—
66C2×(C2×D71)⋊C4—21no—
67C2×C4×Dic71—21no—
68#14556061—24no—
69#14556052—24no—
70#14556052—24no—
71#14556052C2×C418yescomplemented
72#14556048—24no—
73C22⋊(C71⋊C8)—22no—
74C22⋊(C71⋊C8)C2×C414yescomplemented
75#14556039—24no—
76C71⋊#95—21no—
77C71⋊#95—21no—
78C71⋊#95—21no—
79C22×Q8×C71—24no—
80C22×Q8×C71—24no—
81C22×Q8×C71—21no—
82C22×Q8×C71—22no—
83C22×Q8×C71C2×C414yescomplemented
84C22×Q8×C71C2312yes—
85C23×C4×C71—21no—
86C23×C4×C71—21no—
87C23×C4×C71D411yescharacteristic
88C23×C4×C71D411yescharacteristic
89C23×C4×C71C2311yescharacteristic
90C4×Q8×C71—28no—
91C4×Q8×C71—28no—
92C4×Q8×C71—24no—
93C4×Q8×C71—24no—
94C4×Q8×C71—24no—
95C4×Q8×C71D418yescomplemented
96C2×C4⋊C4×C71—28no—
97C2×C4⋊C4×C71—22no—
98C2×C4⋊C4×C71—22no—
99C2×C4⋊C4×C71—21no—
100C2×C4⋊C4×C71—24no—
101C2×C4⋊C4×C71C2312yes—
102C2×C42×C71—28no—
103C2×C42×C71—22no—
104C2×C42×C71—22no—
105C2×C42×C71—24no—
106C2×C42×C71D411yescharacteristic
107C2×C42×C71D411yescharacteristic
108C2×C42×C71C2311yescharacteristic
109C2×C42×C71C2311yescharacteristic
110C23×D71—21no—
111C2×D284—22no—
112C2×D284—22no—
113C2×D284—21no—
114C2×D284C22×C412yes—
115C2×C22⋊D71—24no—
116C2×C22⋊D71—42no—
117C2×C22⋊D71—42no—
118C22×Dic71—21no—
119C22×Dic71—21no—
120C22×Dic71—21no—
121C2×C4×D71—42no—
122C2×C4×D71—42no—
123C2×C4×D71—42no—
124#11765852—216no—
125C2×C71⋊C8—22no—
126C2×C71⋊C8—22no—
127C2×C71⋊C8—22no—
128C2×C71⋊C8—22no—
129C2×C71⋊C8—22no—
130C2×C71⋊C8C22×C414yes—
131C4⋊Dic71—21no—
132C4⋊Dic71—22no—
133C4⋊Dic71—22no—
134C4⋊Dic71C22×C412yes—
135Dic71⋊C4—42no—
136C22⋊Dic71—42no—
137(C2×D71)⋊C4—24no—
138(C2×D71)⋊C4—42no—
139(C2×D71)⋊C4—42no—
140(C2×D71)⋊C4—42no—
141C4×Dic71—42no—
142C24×C71C2×D411yescharacteristic
143C2×Q8×C71—24no—
144C2×Q8×C71—216no—
145C2×Q8×C71—24no—
146C2×Q8×C71—216no—
147C2×Q8×C71—24no—
148C2×Q8×C71—22no—
149C2×Q8×C71—22no—
150C2×Q8×C71—22no—
151C2×Q8×C71C22×C414yes—
152C2×Q8×C71C2×D414yes—
153C2×Q8×C71#4114yes—
154C22×C4×C71—24no—
155C22×C4×C71—24no—
156C22×C4×C71—24no—
157C22×C4×C71—22no—
158C22×C4×C71—22no—
159C22×C4×C71—22no—
160C22×C4×C71—22no—
161C22×C4×C71—22no—
162C22×C4×C71—22no—
163C22×C4×C71—24no—
164C22×C4×C71—22no—
165C22×C4×C71—22no—
166C22×C4×C71—22no—
167C22×C4×C71—41no—
168C22×C4×C71—42no—
169C22×C4×C71—42no—
170C22×C4×C71C22×C412yes—
171C22×C4×C71C2×D411yescharacteristic
172C22×C4×C71C2×D411yescharacteristic
173C22×C4×C71C2×D411yescharacteristic
174C22×C4×C71C2×D411yescharacteristic
175C22×C4×C71C2×D411yescharacteristic
176C22×C4×C71#4112yes—
177C22×C4×C71C2411yescharacteristic
178C4⋊C4×C71—28no—
179C4⋊C4×C71—24no—
180C4⋊C4×C71—28no—
181C4⋊C4×C71—22no—
182C4⋊C4×C71C2×D414yes—
183C42×C71—28no—
184C42×C71—28no—
185C42×C71—22no—
186C42×C71—22no—
187C42×C71—22no—
188C42×C71C2×D414yes—
189C23×C71—21no—
190C23×C71—21no—
191C23×C71—21no—
192C23×C71—21no—
193C23×C71C4×D412yes—
194C23×C71AGL(1,ℤ/8)12yes—
195C23×C71#13712yes—
196C23×C71C22×D411yescharacteristic
197C22×D71—22no—
198C22×D71—41no—
199C22×D71—41no—
200C22×D71—41no—
201C22×D71—41no—
202C22×D71—41no—
203C22×D71—41no—
204Q8×C71—216no—
205Q8×C71—24no—
206Q8×C71—24no—
207Q8×C71—24no—
208Q8×C71C4×D418yescomplemented
209C2×C4×C71—28no—
210C2×C4×C71—216no—
211C2×C4×C71—24no—
212C2×C4×C71—21no—
213C2×C4×C71—24no—
214C2×C4×C71—21no—
215C2×C4×C71—24no—
216C2×C4×C71—21no—
217C2×C4×C71—21no—
218C2×C4×C71—21no—
219C2×C4×C71—21no—
220C2×C4×C71—21no—
221C2×C4×C71—21no—
222C2×C4×C71—24no—
223C2×C4×C71—24no—
224C2×C4×C71—21no—
225C2×C4×C71—24no—
226C2×C4×C71—21no—
227C2×C4×C71—21no—
228C2×C4×C71—44no—
229C2×C4×C71—48no—
230C2×C4×C71—42no—
231C2×C4×C71—42no—
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
240C22⋊D71—48no—
241C2×Dic71—22no—
242C2×Dic71—41no—
243C2×Dic71—41no—
244C2×Dic71—41no—
245C2×Dic71—41no—
246C2×Dic71—41no—
247C2×Dic71—41no—
248C2×Dic71—41no—
249C2×Dic71—41no—
250D284—24no—
251D284—42no—
252C4×D71—84no—
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#44014yescomplemented
262C22×C71C2×C4×D411yescharacteristic
263C22×C71C2×#12011yescharacteristic
264C22×C71#52511yescharacteristic
265C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C71C2×#13711yescharacteristic
267C22×C71C2×#13711yescharacteristic
268C22×C71#57511yescharacteristic
269C2×D71—42no—
270C2×D71—42no—
271C2×D71—42no—
272C2×D71—81no—
273C2×D71—81no—
274C2×D71—81no—
275C2×D71—81no—
276C4×C71—22no—
277C4×C71—22no—
278C4×C71—22no—
279C4×C71—28no—
280C4×C71—22no—
281C4×C71—41no—
282C4×C71—44no—
283C4×C71C2×C4×D412yes—
284C4×C71#52512yes—
285Dic71—42no—
286Dic71—81no—
287Dic71—81no—
288#21454—711nomaximal, Sylow
289C2×C71—24no—
290C2×C71C4×#12011yescharacteristic
291C2×C71C2×#44011yescharacteristic
292C2×C71C4×#13711yescharacteristic
293C2×C71#300411yescharacteristic
294C2×C71#300611yescharacteristic
295C2×C71#311211yescharacteristic
296C2×C71#311711yescharacteristic
297D71—82no—
298C22×C4×Q8D7111yescharacteristic, complemented
299C2×#448—711no—
300C2×#440—712no—
301C4×#121—711no—
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×Q8C2×D7111yescharacteristic
311C22×SD16—712no—
312C2×#121—711no—
313C2×C4×Q8—28no—
314C2×C4×Q8—21no—
315C2×C4×Q8—21no—
316C2×C4×Q8C2×D7112yes—
317C2×C4×Q8C2×D7112yes—
318C2×C4×D4—711no—
319C2×C4×D4—1421no—
320C22×C4⋊C4—21no—
321C22×C4⋊C4C2×D7111yescharacteristic
322C22×C42—21no—
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#390—711no—
335#389—711no—
336#388—1421no—
337#386—1421no—
338#380—711no—
339C4×C4⋊C4—711no—
340C4×C22⋊C4—1421no—
341#340—712no—
342#340—712no—
343#336—712no—
344C22×Q8—24no—
345C22×Q8—24no—
346C22×Q8—21no—
347C22×Q8—22no—
348C22×Q8C4×D7114yescomplemented
349C22×Q8C22×D7112yes—
350C22×D4—711no—
351C22×D4—1421no—
352C23×C4—21no—
353C23×C4—21no—
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—714no—
364#121—1422no—
365C4×Q8—28no—
366C4×Q8—28no—
367C4×Q8—24no—
368C4×Q8—24no—
369C4×Q8—24no—
370C4×Q8C22⋊D7118yescomplemented
371C4×D4—2844no—
372C4×D4—1424no—
373C2×C4⋊C4—28no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4—711no—
379C2×C4⋊C4—711no—
380C2×C4⋊C4—711no—
381C2×C4⋊C4—1421no—
382C2×C4⋊C4C22×D7112yes—
383C2×C22⋊C4—1421no—
384C2×C22⋊C4—1421no—
385C2×C22⋊C4—1421no—
386C2×C22⋊C4—1421no—
387C2×C22⋊C4—1421no—
388C2×C42—28no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—24no—
392C2×C42—1421no—
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—1421no—
406#95—1421no—
407#95—1421no—
408C24—1421no—
409C24C2×C22⋊D7111yescharacteristic
410C2×Q8—24no—
411C2×Q8—216no—
412C2×Q8—24no—
413C2×Q8—216no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—24no—
418C2×Q8C2×C4×D7114yes—
419C2×Q8C2×C22⋊D7114yes—
420C2×Q8#1176586514yes—
421C2×D4—2842no—
422C2×D4—2842no—
423C2×D4—712no—
424C2×D4—1424no—
425C2×D4—1422no—
426C2×D4—1421no—
427C2×D4—1422no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—2842no—
445C22×C4—2842no—
446C22×C4—2842no—
447C22×C4—1421no—
448C22×C4—1421no—
449C22×C4—1421no—
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—28no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—28no—
469C4⋊C4—2842no—
470C4⋊C4—712no—
471C4⋊C4—1422no—
472C4⋊C4—1422no—
473C4⋊C4—1421no—
474C4⋊C4C2×C22⋊D7114yes—
475C22⋊C4—2842no—
476C22⋊C4—2842no—
477C22⋊C4—2842no—
478C22⋊C4—2842no—
479C22⋊C4—1424no—
480C42—28no—
481C42—22no—
482C42—28no—
483C42—22no—
484C42—22no—
485C42—2842no—
486C42C2×C22⋊D7114yes—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—2841no—
492C23—2841no—
493C23—2841no—
494C23—2841no—
495C23—2841no—
496C23—2841no—
497C23—1422no—
498C23C4×C22⋊D7112yes—
499C23#1455615312yes—
500C23#1455615912yes—
501C23C22×C22⋊D7111yescharacteristic
502Q8—216no—
503Q8—24no—
504Q8—24no—
505Q8—24no—
506Q8C4×C22⋊D7118yescomplemented
507D4—2848no—
508D4—2842no—
509D4—1424no—
510C2×C4—216no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—44no—
530C2×C4—48no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—2841no—
534C2×C4—2841no—
535C2×C4—2841no—
536C2×C4—2841no—
537C2×C4—2841no—
538C2×C4—2841no—
539C2×C4—2841no—
540C2×C4—2841no—
541C2×C4—5684no—
542C2×C4—1422no—
543C2×C4C4×C22⋊D7112yes—
544C2×C4#1455609712yes—
545C2×C4#1455609712yes—
546C2×C4#1455615912yes—
547C2×C4C22×C4×D7111yescharacteristic
548C2×C4C22×C22⋊D7112yes—
549C2×C4C2×#1176586512yes—
550C2×C4C2×#1176586511yescharacteristic
551C8—2842no—
552C8—1424no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—2842no—
560C22—2842no—
561C22—2842no—
562C22—5681no—
563C22—5681no—
564C22—5681no—
565C22—5681no—
566C22#1798609714yescomplemented
567C22C2×C4×C22⋊D7111yescharacteristic
568C22C2×#1455609711yescharacteristic
569C22#1798651011yescharacteristic
570C22C2×#1455615311yescharacteristic
571C22C2×#1455615911yescharacteristic
572C22C2×#1455615911yescharacteristic
573C22#1798679711yescharacteristic
574C4—28no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—22no—
579C4—41no—
580C4—44no—
581C4—2842no—
582C4—5681no—
583C4—5681no—
584C4C2×C4×C22⋊D7112yes—
585C4#1798651012yes—
586C2—24no—
587C2—5682no—
588C2C4×#1455609711yescharacteristic
589C2C2×#1798609711yescharacteristic
590C2C4×#1455615911yescharacteristic
591C2#2259757411yescharacteristic
592C2#2259759211yescharacteristic
593C2#2259846211yescharacteristic
594C2#2259851111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC22×C4×Q8×C71C2
socleC23×C71C22×D4
prime radicalC2×#14556102C22
cosocleC22×C4×C71C24
hypercenterC22×C4×Q8D71
hypocenterC71#21454
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#17986097C2semidirect
4. #22588695C2semidirect
6. #22588594C2semidirect
7. #22588528C2semidirect
8. #22588275C2semidirect
9. #22588170C2semidirect
10. #22588102C2semidirect
11. C22×C4×Q8×C71C2semidirect
16. #17986138C4semidirect
51. C2×#11765852C2×C4semidirect
71. #14556052C2×C4semidirect
74. C22⋊(C71⋊C8)C2×C4semidirect
83. C22×Q8×C71C2×C4semidirect
95. C4×Q8×C71D4semidirect
208. Q8×C71C4×D4semidirect
261. C22×C71#440semidirect
298. C22×C4×Q8D71semidirect
309. C71#21454semidirect
348. C22×Q8C4×D71semidirect
370. C4×Q8C22⋊D71semidirect
506. Q8C4×C22⋊D71semidirect
566. C22#17986097semidirect

Maximal subgroups

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

subgroupindex
C2×#179861382
C2×#179860972
C4×#145561022
#225886952
#225886572
#225885942
#225885282
#225882752
#225881702
#225881022
C22×C4×Q8×C712
#2145471