Finite Group Databasebeta

← #32031122

Subgroups — #32031122

Order 18176.

1138 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986327C211yesmaximal, characteristic, complemented
2C22×#14556059C211yesmaximal, characteristic, complemented
3#22592648C211yesmaximal, characteristic
4#22589670C212yesmaximal, complemented
5#22589533C212yesmaximal, complemented
6#22588764C211yesmaximal, characteristic, complemented
7#22588633C211yesmaximal, characteristic
8#22588608C212yesmaximal, complemented
9#22588533C212yesmaximal, complemented
10#22588430C211yesmaximal, characteristic, complemented
11C2×#554×C71C211yesmaximal, characteristic
12C23×C71⋊C8C2211yescharacteristic
13C2×#14556109—21no—
14C22×C4⋊Dic71C2211yescharacteristic
15#17986327—24no—
16C2×#14556059—24no—
17C2×#14556059C2212yes—
18C2×#14556059C2212yes—
19C2×#14556059C2212yes—
20C2×#14556052C2212yes—
21C2×#14556052C2212yes—
22C2×C22⋊(C71⋊C8)C2211yescharacteristic
23C2×C22⋊(C71⋊C8)C2212yes—
24#17985929—21no—
25#17985924—21no—
26#17985897C2212yes—
27#17985860—21no—
28#17985859C2211yescharacteristic
29#17985859C2212yes—
30#17985761C2212yes—
31#17985761C2212yes—
32#17985761C2212yes—
33#17985754C2212yes—
34#554×C71—24no—
35#554×C71—24no—
36C2×C4⋊Q8×C71C2211yescharacteristic
37C2×#125×C71—21no—
38C2×C22⋊Q8×C71—21no—
39C2×C22⋊Q8×C71—21no—
40C2×C22⋊Q8×C71C2211yescharacteristic
41C2×C22⋊Q8×C71C2211yescharacteristic
42C2×C4×Q8×C71C2211yescharacteristic
43C2×C4×D4×C71—21no—
44C2×C4×D4×C71C2211yescharacteristic
45C22×C4⋊C4×C71C2211yescharacteristic
46C22×C4⋊C4×C71C2211yescharacteristic
47C23×Dic71—21no—
48C22×C71⋊C8—24no—
49C22×C71⋊C8—22no—
50C22×C71⋊C8C2312yes—
51C22×C71⋊C8C2312yes—
52#14556109—44no—
53C2×C4⋊Dic71—22no—
54C2×C4⋊Dic71—22no—
55C2×C4⋊Dic71—21no—
56C2×C4⋊Dic71C2312yes—
57C2×C4⋊Dic71C2312yes—
58C2×Dic71⋊C4—22no—
59C2×C22⋊Dic71—21no—
60C2×C22⋊Dic71—22no—
61#14556059—24no—
62#14556059—24no—
63#14556059—24no—
64#14556052—24no—
65#14556052—24no—
66C22⋊(C71⋊C8)—24no—
67C22⋊(C71⋊C8)—22no—
68C71⋊#95—22no—
69C71⋊#95—21no—
70C71⋊#95—22no—
71C22×Q8×C71—21no—
72C22×Q8×C71C2311yescharacteristic
73C22×D4×C71D411yescharacteristic
74C23×C4×C71—21no—
75C23×C4×C71—21no—
76C23×C4×C71C2311yescharacteristic
77C23×C4×C71C2311yescharacteristic
78C4⋊Q8×C71—22no—
79C4⋊Q8×C71—22no—
80#125×C71—28no—
81C22⋊Q8×C71—28no—
82C22⋊Q8×C71—28no—
83C22⋊Q8×C71—22no—
84C22⋊Q8×C71—22no—
85C22⋊Q8×C71—22no—
86C22⋊Q8×C71D414yes—
87C4×Q8×C71—22no—
88C4×Q8×C71—22no—
89C4×D4×C71—28no—
90C4×D4×C71—22no—
91C4×D4×C71—22no—
92C2×C4⋊C4×C71—21no—
93C2×C4⋊C4×C71—21no—
94C2×C4⋊C4×C71—22no—
95C2×C4⋊C4×C71—21no—
96C2×C4⋊C4×C71—21no—
97C2×C4⋊C4×C71—21no—
98C2×C4⋊C4×C71—21no—
99C2×C4⋊C4×C71—22no—
100C2×C4⋊C4×C71—22no—
101C2×C4⋊C4×C71D411yescharacteristic
102C2×C4⋊C4×C71D414yes—
103C2×C4⋊C4×C71C2311yescharacteristic
104C2×C4⋊C4×C71C2311yescharacteristic
105C2×C4⋊C4×C71C2311yescharacteristic
106C2×C22⋊C4×C71—21no—
107C2×C22⋊C4×C71—21no—
108C2×C22⋊C4×C71D411yescharacteristic
109C2×C22⋊C4×C71D411yescharacteristic
110C2×C42×C71—21no—
111C2×C42×C71C2311yescharacteristic
112C22×Dic71—22no—
113C22×Dic71—22no—
114C22×Dic71—42no—
115C22×Dic71—42no—
116C22×Dic71—42no—
117C2×C71⋊C8—28no—
118C2×C71⋊C8—22no—
119C2×C71⋊C8—22no—
120C2×C71⋊C8—22no—
121C2×C71⋊C8—22no—
122C2×C71⋊C8—22no—
123C2×C71⋊C8#4114yes—
124C4⋊Dic71—22no—
125C4⋊Dic71—22no—
126C4⋊Dic71—22no—
127C4⋊Dic71—22no—
128C4⋊Dic71—42no—
129Dic71⋊C4—44no—
130C22⋊Dic71—42no—
131C22⋊Dic71—44no—
132C24×C71C2×D411yescharacteristic
133C24×C71C2×D411yescharacteristic
134C2×Q8×C71—24no—
135C2×Q8×C71—21no—
136C2×Q8×C71—22no—
137C2×Q8×C71—22no—
138C2×Q8×C71—82no—
139C2×Q8×C71C2×D412yes—
140C2×D4×C71—22no—
141C2×D4×C71—22no—
142C2×D4×C71SD1614yes—
143C22×C4×C71—22no—
144C22×C4×C71—22no—
145C22×C4×C71—21no—
146C22×C4×C71—21no—
147C22×C4×C71—21no—
148C22×C4×C71—21no—
149C22×C4×C71—24no—
150C22×C4×C71—24no—
151C22×C4×C71—21no—
152C22×C4×C71—21no—
153C22×C4×C71—22no—
154C22×C4×C71—22no—
155C22×C4×C71—82no—
156C22×C4×C71—82no—
157C22×C4×C71C2×D411yescharacteristic
158C22×C4×C71C2×D411yescharacteristic
159C22×C4×C71C2×D411yescharacteristic
160C22×C4×C71C2×D412yes—
161C22×C4×C71C2×D412yes—
162C22×C4×C71C2×D411yescharacteristic
163C22×C4×C71C2411yescharacteristic
164C4⋊C4×C71—24no—
165C4⋊C4×C71—24no—
166C4⋊C4×C71—24no—
167C4⋊C4×C71—24no—
168C4⋊C4×C71—24no—
169C4⋊C4×C71—24no—
170C4⋊C4×C71—22no—
171C4⋊C4×C71—21no—
172C4⋊C4×C71—21no—
173C4⋊C4×C71—21no—
174C4⋊C4×C71C2×D412yes—
175C4⋊C4×C71C2×D412yes—
176C4⋊C4×C71C2×D412yes—
177C22⋊C4×C71—24no—
178C22⋊C4×C71—24no—
179C22⋊C4×C71—22no—
180C22⋊C4×C71—22no—
181C42×C71—24no—
182C42×C71—21no—
183C42×C71—21no—
184C23×C71—22no—
185C23×C71—22no—
186C23×C71—21no—
187C23×C71—22no—
188C23×C71—21no—
189C23×C71—22no—
190C23×C71#12012yes—
191C23×C71C2×SD1612yes—
192C23×C71C2×SD1612yes—
193C23×C71#13712yes—
194C23×C71C22×D411yescharacteristic
195Q8×C71—24no—
196Q8×C71—41no—
197Q8×C71—41no—
198Q8×C71—84no—
199D4×C71—28no—
200C2×C4×C71—21no—
201C2×C4×C71—22no—
202C2×C4×C71—21no—
203C2×C4×C71—22no—
204C2×C4×C71—22no—
205C2×C4×C71—24no—
206C2×C4×C71—22no—
207C2×C4×C71—21no—
208C2×C4×C71—22no—
209C2×C4×C71—21no—
210C2×C4×C71—22no—
211C2×C4×C71—22no—
212C2×C4×C71—22no—
213C2×C4×C71—22no—
214C2×C4×C71—42no—
215C2×C4×C71—42no—
216C2×C4×C71—42no—
217C2×C4×C71—42no—
218C2×C4×C71—41no—
219C2×C4×C71—41no—
220C2×C4×C71—42no—
221C2×C4×C71—84no—
222C2×C4×C71—84no—
223C2×C4×C71#12012yes—
224C2×C4×C71C2×SD1612yes—
225C2×C4×C71C22×D411yescharacteristic
226C2×C4×C71C22×D411yescharacteristic
227C2×C4×C71C2×#4112yes—
228C2×C4×C712−1+412yes—
229C2×Dic71—42no—
230C2×Dic71—42no—
231C2×Dic71—42no—
232C2×Dic71—42no—
233C2×Dic71—42no—
234C2×Dic71—42no—
235C2×Dic71—84no—
236C71⋊C8—24no—
237C71⋊C8—42no—
238C22×C71—24no—
239C22×C71—24no—
240C22×C71—22no—
241C22×C71—22no—
242C22×C71—24no—
243C22×C71—22no—
244C22×C71—42no—
245C22×C71—41no—
246C22×C71—41no—
247C22×C71#44814yes—
248C22×C71C2×#12011yescharacteristic
249C22×C71#54712yes—
250C22×C71C22×SD1611yescharacteristic
251C22×C71C2×#13711yescharacteristic
252C22×C71#57712yes—
253C4×C71—22no—
254C4×C71—22no—
255C4×C71—41no—
256C4×C71—42no—
257C4×C71—41no—
258C4×C71—42no—
259C4×C71—42no—
260C4×C71—42no—
261C4×C71#52511yescharacteristic
262C4×C71#53611yescharacteristic
263C4×C71#54712yes—
264Dic71—82no—
265Dic71—82no—
266#24097—711nomaximal, Sylow
267C2×C71—24no—
268C2×C71—42no—
269C2×C71#240611yescharacteristic
270C2×C71C2×#44811yescharacteristic
271C2×C71#300911yescharacteristic
272C2×C71#333512yes—
273C2×C71#334412yes—
274C2×#554D7111yescharacteristic
275C2×#483—711no—
276C22×#106—711no—
277#2486—711no—
278#2058—712no—
279#2035—712no—
280#1892—711no—
281#1869—711no—
282#1865—712no—
283#1853—712no—
284#1824—711no—
285C71#2409711yescharacteristic, complemented, Sylow
286C23×C8—711no—
287#554—24no—
288#554—24no—
289C2×C4⋊Q8C2×D7111yescharacteristic
290C2×#125—21no—
291C2×#123—1421no—
292C2×C22⋊Q8—21no—
293C2×C22⋊Q8—21no—
294C2×C22⋊Q8C2×D7111yescharacteristic
295C2×C22⋊Q8C2×D7111yescharacteristic
296C2×C4×Q8C2×D7111yescharacteristic
297C2×C4×D4—21no—
298C2×C4×D4C2×D7111yescharacteristic
299C22×C4⋊C4—711no—
300C22×C4⋊C4C2×D7111yescharacteristic
301C22×C4⋊C4C2×D7111yescharacteristic
302#483—1424no—
303C2×#106—712no—
304C2×#106—712no—
305C2×#106—712no—
306C2×#106—1424no—
307C2×#103—712no—
308C2×#103—712no—
309C2×C22⋊C8—711no—
310C2×C22⋊C8—712no—
311#398—1421no—
312#397—1421no—
313#389—712no—
314#381—1421no—
315#380—711no—
316#380—712no—
317#340—712no—
318#340—712no—
319#340—712no—
320#336—712no—
321C22×Q8—21no—
322C22×Q8C22×D7111yescharacteristic
323C22×D4C22⋊D7111yescharacteristic
324C23×C4—21no—
325C23×C4—21no—
326C23×C4—1421no—
327C23×C4C22×D7111yescharacteristic
328C23×C4C22×D7111yescharacteristic
329C22×C8—712no—
330C22×C8—712no—
331C22×C8—1424no—
332C22×C8—1422no—
333C4⋊Q8—22no—
334C4⋊Q8—22no—
335#125—28no—
336#123—2844no—
337C22⋊Q8—28no—
338C22⋊Q8—28no—
339C22⋊Q8—22no—
340C22⋊Q8—22no—
341C22⋊Q8—22no—
342C22⋊Q8C22⋊D7114yes—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×D4—28no—
346C4×D4—22no—
347C4×D4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—712no—
358C2×C4⋊C4—712no—
359C2×C4⋊C4—1421no—
360C2×C4⋊C4—1422no—
361C2×C4⋊C4—1422no—
362C2×C4⋊C4—1422no—
363C2×C4⋊C4C22⋊D7111yescharacteristic
364C2×C4⋊C4C22⋊D7114yes—
365C2×C4⋊C4C22×D7111yescharacteristic
366C2×C4⋊C4C22×D7111yescharacteristic
367C2×C4⋊C4C22×D7111yescharacteristic
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—1421no—
371C2×C22⋊C4—1422no—
372C2×C22⋊C4C22⋊D7111yescharacteristic
373C2×C22⋊C4C22⋊D7111yescharacteristic
374C2×C42—21no—
375C2×C42C22×D7111yescharacteristic
376#106—1424no—
377#106—1424no—
378#106—1424no—
379#103—1424no—
380#103—1424no—
381C22⋊C8—1424no—
382C22⋊C8—1422no—
383#95—1422no—
384#95—1421no—
385#95—1422no—
386C24C2×C22⋊D7111yescharacteristic
387C24C2×C22⋊D7111yescharacteristic
388C2×Q8—24no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—82no—
393C2×Q8C2×C22⋊D7112yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4#1176585214yes—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4—2842no—
412C22×C4—2842no—
413C22×C4—2842no—
414C22×C4—1422no—
415C22×C4—1422no—
416C22×C4C2×C22⋊D7111yescharacteristic
417C22×C4C2×C22⋊D7111yescharacteristic
418C22×C4C2×C22⋊D7112yes—
419C22×C4C2×C22⋊D7112yes—
420C22×C4C2×C22⋊D7111yescharacteristic
421C22×C4C2×C22⋊D7111yescharacteristic
422C22×C4C23×D7111yescharacteristic
423C2×C8—714no—
424C2×C8—1422no—
425C2×C8—1422no—
426C2×C8—1422no—
427C2×C8—1428no—
428C2×C8—1422no—
429C2×C8—1422no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—2842no—
441C4⋊C4—2844no—
442C4⋊C4—1422no—
443C4⋊C4—1422no—
444C4⋊C4—1422no—
445C4⋊C4—1422no—
446C4⋊C4C2×C22⋊D7112yes—
447C4⋊C4C2×C22⋊D7112yes—
448C4⋊C4C2×C22⋊D7112yes—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—2842no—
454C22⋊C4—2844no—
455C42—24no—
456C42—21no—
457C42—21no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—21no—
462C23—21no—
463C23—22no—
464C23#1455609712yes—
465C23C2×#1176585212yes—
466C23C2×#1176585212yes—
467C23#1455615912yes—
468C23C22×C22⋊D7111yescharacteristic
469Q8—24no—
470Q8—41no—
471Q8—41no—
472Q8—84no—
473D4—28no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4—2842no—
498C2×C4—2842no—
499C2×C4—2842no—
500C2×C4—2842no—
501C2×C4—2842no—
502C2×C4—2842no—
503C2×C4—5684no—
504C2×C4#1455609712yes—
505C2×C4C2×#1176585212yes—
506C2×C4C22×C22⋊D7111yescharacteristic
507C2×C4C22×C22⋊D7111yescharacteristic
508C2×C4#41×D7112yes—
509C2×C4#1455617512yes—
510C8—2842no—
511C8—1424no—
512C22—24no—
513C22—22no—
514C22—22no—
515C22—24no—
516C22—22no—
517C22—24no—
518C22—42no—
519C22—41no—
520C22—41no—
521C22#1798613814yes—
522C22C2×#1455609711yescharacteristic
523C22#1798664812yes—
524C22C22×#1176585211yescharacteristic
525C22C2×#1455615911yescharacteristic
526C22#1798681012yes—
527C4—22no—
528C4—22no—
529C4—41no—
530C4—42no—
531C4—41no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—5682no—
536C4—5682no—
537C4#1798650611yescharacteristic
538C4#1798657011yescharacteristic
539C4#1798664812yes—
540C2—24no—
541C2—42no—
542C2#2259167111yescharacteristic
543C2C2×#1798613811yescharacteristic
544C2#2259761711yescharacteristic
545C2#2260069112yes—
546C2#2260078912yes—
subgroupG/H
centerC23C22×C22⋊D71
commutator, cosocleC22×C4×C71C24
FrattiniC22×C4C23×D71
FittingC2×#554×C71C2
socleC23×C71C22×D4
prime radicalC22×D4×C71D4
hypercenterC2×#554D71
hypocenterC71#24097
LCSG ▷ C22×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×#17986327C2semidirect
2. C22×#14556059C2semidirect
4. #22589670C2semidirect
5. #22589533C2semidirect
6. #22588764C2semidirect
8. #22588608C2semidirect
9. #22588533C2semidirect
10. #22588430C2semidirect
285. C71#24097semidirect

Maximal subgroups

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

subgroupindex
C2×#179863272
C22×#145560592
#225926482
#225896702
#225895332
#225887642
#225886332
#225886082
#225885332
#225884302
C2×#554×C712
#2409771