Finite Group Databasebeta

← #29685964

Subgroups — #29685964

Order 15616.

1322 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839945C211yesmaximal, characteristic
2C4×#14519807C212yesmaximal, complemented
3#21448535C211yesmaximal, characteristic, complemented
4#21446212C211yesmaximal, characteristic, complemented
5#21446188C212yesmaximal, complemented
6#21446052C212yesmaximal, complemented
7#21446034C211yesmaximal, characteristic, complemented
8#21445778C211yesmaximal, characteristic
9#21445629C212yesmaximal, complemented
10C4×#14519696C211yesmaximal, characteristic, complemented
11C4×2−1+4×C61C211yesmaximal, characteristic
12C2×#14519807C2212yes—
13C2×#14519775C2211yescharacteristic
14C2×C4×Dic122C2211yescharacteristic
15#16839945C418yescomplemented
16#16839867—22no—
17#16839862—22no—
18C4×#11758202—22no—
19C4×#11758200—22no—
20#16839805C2212yes—
21C2×#14519704C2212yes—
22C2×#14519704C2212yes—
23#16839770C2212yes—
24#16839763C2212yes—
25C2×#14519696C2211yescharacteristic
26C2×#14519696C2211yescharacteristic
27C4×#11758194C2212yes—
28#16839669C2211yescharacteristic
29#16839667C2211yescharacteristic
30#16839646—22no—
31#16839643—22no—
32#16839627C2211yescharacteristic
33#16839608C2211yescharacteristic
34#16839531C2212yes—
35#16839521C2212yes—
36#16839521C2212yes—
37C2×2−1+4×C61C2211yescharacteristic
38C61×#520—22no—
39C61×#520—22no—
40C61×#520C2212yes—
41C61×#519—22no—
42C61×#519C2211yescharacteristic
43C4×#41×C61—22no—
44C4×#41×C61—22no—
45C4×#41×C61C2212yes—
46C2×C4×Q8×C61—22no—
47C2×C4×Q8×C61C2211yescharacteristic
48C22×Dic122C2311yescharacteristic
49#14519807—24no—
50#14519807C2×C418yescomplemented
51C2×#11758202—22no—
52C2×#11758200—22no—
53C2×#11758194C2312yes—
54C2×#11758194C2312yes—
55#14519775—22no—
56#14519775C2×C414yescomplemented
57C4×Dic122—22no—
58C4×Dic122—22no—
59C2×C4⋊Dic61C2311yescharacteristic
60C2×#11758196—22no—
61C2×#11758196—21no—
62C2×C4×Dic61—21no—
63C2×C4×Dic61C2311yescharacteristic
64C2×C4×Dic61C2311yescharacteristic
65#14519710—22no—
66#14519708—22no—
67#14519704—24no—
68#14519704C2×C418yescomplemented
69#14519701—22no—
70#14519700—22no—
71#14519700—22no—
72#14519698—22no—
73#14519696—22no—
74#14519696C2×C414yescomplemented
75#14519688—22no—
76C4×#92418—22no—
77#14519684—21no—
78#14519684—21no—
79#14519684—22no—
80#14519684—22no—
81#14519684—22no—
822−1+4×C61—24no—
832−1+4×C61—22no—
842−1+4×C61C2×C414yes—
85C2×#41×C61—22no—
86C2×#41×C61—22no—
87C2×#41×C61C2312yes—
88C22×Q8×C61—22no—
89C22×Q8×C61C2311yescharacteristic
90C4×Q8×C61—22no—
91C4×Q8×C61—24no—
92C4×Q8×C61—22no—
93C4×Q8×C61—22no—
94C4×Q8×C61—22no—
95C4×Q8×C61—22no—
96C4×Q8×C61—22no—
97C4×Q8×C61—22no—
98C4×Q8×C61D412yes—
99C4×Q8×C61D412yes—
100C4×D4×C61—22no—
101C4×D4×C61—24no—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104C4×D4×C61—22no—
105C4×D4×C61—24no—
106C4×D4×C61—22no—
107C4×D4×C61D412yes—
108C4×D4×C61D412yes—
109#117×C61—24no—
110#117×C61—24no—
111#117×C61—22no—
112#117×C61—22no—
113#117×C61—22no—
114#117×C61C2312yes—
115C2×C4⋊C4×C61—24no—
116C2×C4⋊C4×C61—22no—
117C2×C4⋊C4×C61D412yes—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C42×C61—24no—
120C2×C42×C61—22no—
121C2×C42×C61D412yes—
122C2×C42×C61C2311yescharacteristic
123#41×C61—28no—
124#41×C61—28no—
125#41×C61—28no—
126#41×C61—28no—
127#41×C61—22no—
128#41×C61—22no—
129#41×C61C22×C414yes—
130#41×C61#4114yes—
131C2×Q8×C61—22no—
132C2×Q8×C61—28no—
133C2×Q8×C61—28no—
134C2×Q8×C61—22no—
135C2×Q8×C61—22no—
136C2×Q8×C61C22×C412yes—
137C2×Q8×C61C2×D412yes—
138C2×Q8×C61#4112yes—
139C2×D4×C61—22no—
140C2×D4×C61—22no—
141C2×D4×C61C2×D412yes—
142C22×C4×C61—24no—
143C22×C4×C61—22no—
144C22×C4×C61—24no—
145C22×C4×C61—22no—
146C22×C4×C61—22no—
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C2×Dic122—22no—
157C2×Dic122—22no—
158C2×Dic122—21no—
159C2×Dic122C22×C412yes—
160C4⋊C4×C61—24no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—24no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61C2×D412yes—
168C4⋊C4×C61C2×D412yes—
169C22⋊C4×C61—24no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C42×C61—24no—
174C42×C61—24no—
175C42×C61—22no—
176C42×C61—22no—
177C42×C61—22no—
178C42×C61—22no—
179C42×C61—22no—
180C42×C61C2×D412yes—
181C42×C61C2×D412yes—
182#11758202—28no—
183#11758200—28no—
184C4⋊Dic61—22no—
185#11758196—28no—
186#11758196—41no—
187#11758196—41no—
188C4×Dic61—21no—
189C4×Dic61—22no—
190C4×Dic61—42no—
191C4×Dic61C22×C412yes—
192#11758194—22no—
193#11758194—22no—
194#11758194—22no—
195#11758194C22×C414yes—
196C2×#92418—22no—
197C2×#92418—22no—
198C23×C61—22no—
199C23×C61C22×D411yescharacteristic
200Q8×C61—28no—
201Q8×C61—24no—
202Q8×C61—22no—
203Q8×C61—24no—
204Q8×C61C4×D414yes—
205D4×C61—28no—
206D4×C61—24no—
207D4×C61—24no—
208D4×C61—22no—
209D4×C61C4×D414yes—
210C2×C4×C61—216no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—24no—
215C2×C4×C61—24no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—44no—
222C2×C4×C61—42no—
223C2×C4×C61—82no—
224C2×C4×C61—84no—
225C2×C4×C61C4×D414yes—
226C2×C4×C61#12012yes—
227C2×C4×C61#12011yescharacteristic
228C2×C4×C61#12011yescharacteristic
229C2×C4×C61C23×C411yescharacteristic
230C2×C4×C61C22×D412yes—
231C2×C4×C61C2×#4112yes—
232C2×C4×C61C2×#4111yescharacteristic
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—42no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244Dic122—24no—
245Dic122—42no—
246#92418—24no—
247#92418—42no—
248#21476—611nomaximal, Sylow
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—24no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61C2×C4×D411yescharacteristic
255C22×C61C2×#12011yescharacteristic
256C22×C61#52511yescharacteristic
257C4×C61—21no—
258C4×C61—21no—
259C4×C61—22no—
260C4×C61—24no—
261C4×C61—24no—
262C4×C61—42no—
263C4×C61—42no—
264C4×C61—42no—
265C4×C61C2×C4×D412yes—
266C4×C61#52512yes—
267Dic61—42no—
268Dic61—42no—
269Dic61—81no—
270Dic61—81no—
271C4×2−1+4D6111yescharacteristic
272C2×#456—611no—
273C4×#137—612no—
274#2327—611no—
275#1886—611no—
276#1882—612no—
277#1860—612no—
278#1857—611no—
279#1793—611no—
280#1759—612no—
281C4×#101—611no—
282C2×C61—21no—
283C2×C61—21no—
284C2×C61—44no—
285C2×C61C4×#12011yescharacteristic
286C2×C61#302011yescharacteristic
287C2×C61#302211yescharacteristic
288C2×2−1+4C2×D6111yescharacteristic
289C2×#137—612no—
290C2×C4⋊Q8—611no—
291#520—22no—
292#520—22no—
293#520C2×D6112yes—
294#519—22no—
295#519C2×D6111yescharacteristic
296C4×#41—22no—
297C4×#41—22no—
298C4×#41C2×D6112yes—
299C2×C4×Q8—22no—
300C2×C4×Q8—611no—
301C2×C4×Q8C2×D6111yescharacteristic
302#456—618no—
303#441—1222no—
304#440—1222no—
305C4×Q16—1222no—
306C4×SD16—1222no—
307#428—612no—
308C2×#104—612no—
309C2×#104—612no—
310#419—612no—
311#417—612no—
312C2×#101—611no—
313C2×#101—611no—
314C4×M16—612no—
315#391—611no—
316#391—611no—
317#387—1222no—
318#387—1222no—
319#382—611no—
320#376—611no—
321#343—612no—
322#339—612no—
323#339—612no—
324C61#2147611yescharacteristic, complemented, Sylow
3252−1+4—24no—
3262−1+4—22no—
3272−1+4C4×D6114yes—
328C2×#41—22no—
329C2×#41—22no—
330C2×#41C22×D6112yes—
331C22×Q8—22no—
332C22×Q8—611no—
333C22×Q8C22×D6111yescharacteristic
334#137—618no—
335#137—1224no—
336C2×Q16—1222no—
337C2×SD16—1222no—
338C2×M16—612no—
339C2×M16—612no—
340C4⋊Q8—614no—
341C4⋊Q8—1222no—
342C4×Q8—22no—
343C4×Q8—24no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×Q8—22no—
350C4×Q8—1222no—
351C4×Q8—1222no—
352C4×Q8C22⋊D6112yes—
353C4×Q8C22⋊D6112yes—
354C4×D4—22no—
355C4×D4—24no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—24no—
360C4×D4—22no—
361C4×D4C22⋊D6112yes—
362C4×D4C22⋊D6112yes—
363#117—24no—
364#117—24no—
365#117—22no—
366#117—22no—
367#117—22no—
368#117C22×D6112yes—
369C2×C4⋊C4—24no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4C22⋊D6112yes—
375C2×C4⋊C4C22×D6111yescharacteristic
376C2×C42—24no—
377C2×C42—22no—
378C2×C42—611no—
379C2×C42—611no—
380C2×C42—1221no—
381C2×C42C22⋊D6112yes—
382C2×C42C22×D6111yescharacteristic
383#107—1222no—
384#106—1222no—
385#104—618no—
386#104—1224no—
387#103—1222no—
388#103—1222no—
389#103—1222no—
390#102—1222no—
391#101—614no—
392#101—1222no—
393#97—1222no—
394C4×C8—1222no—
395#95—1221no—
396#95—1221no—
397#95—1222no—
398#95—1222no—
399#95—1222no—
400#41—28no—
401#41—28no—
402#41—28no—
403#41—22no—
404#41—22no—
405#41—28no—
406#41C2×C4×D6114yes—
407#41#1175822214yes—
408C2×Q8—22no—
409C2×Q8—28no—
410C2×Q8—28no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—612no—
414C2×Q8—1221no—
415C2×Q8—1222no—
416C2×Q8—1222no—
417C2×Q8C2×C4×D6112yes—
418C2×Q8#1175822212yes—
419C2×Q8C2×C22⋊D6112yes—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4C2×C22⋊D6112yes—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4C2×C22⋊D6112yes—
433C22×C4C2×C22⋊D6112yes—
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C23×D6111yescharacteristic
437Q16—1228no—
438SD16—1228no—
439M16—614no—
440M16—1222no—
441M16—1222no—
442M16—1222no—
443C2×C8—1222no—
444C2×C8—1222no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—1228no—
453C4⋊C4—1222no—
454C4⋊C4—2441no—
455C4⋊C4—2441no—
456C4⋊C4C2×C22⋊D6112yes—
457C4⋊C4C2×C22⋊D6112yes—
458C22⋊C4—24no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C42—24no—
463C42—24no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—612no—
470C42—1221no—
471C42—1222no—
472C42—2442no—
473C42C2×C22⋊D6112yes—
474C42C2×C22⋊D6112yes—
475C23—22no—
476C23C22×C22⋊D6111yescharacteristic
477Q8—28no—
478Q8—24no—
479Q8—24no—
480Q8—22no—
481Q8—1224no—
482Q8—2442no—
483Q8C4×C22⋊D6114yes—
484D4—28no—
485D4—24no—
486D4—24no—
487D4—22no—
488D4C4×C22⋊D6114yes—
489C2×C4—216no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—82no—
503C2×C4—84no—
504C2×C4—1222no—
505C2×C4—1222no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2442no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2441no—
515C2×C4C4×C22⋊D6114yes—
516C2×C4#1451974611yescharacteristic
517C2×C4#1451974611yescharacteristic
518C2×C4#1451974612yes—
519C2×C4C22×C4×D6111yescharacteristic
520C2×C4C22×C22⋊D6112yes—
521C2×C4C2×#1175822212yes—
522C2×C4C2×#1175822211yescharacteristic
523C8—1224no—
524C8—2442no—
525C22—21no—
526C22—21no—
527C22—24no—
528C22—42no—
529C22—42no—
530C22C2×C4×C22⋊D6111yescharacteristic
531C22C2×#1451974611yescharacteristic
532C22#1684027311yescharacteristic
533C4—21no—
534C4—21no—
535C4—22no—
536C4—24no—
537C4—24no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—4881no—
542C4—4881no—
543C4—2442no—
544C4—2442no—
545C4C2×C4×C22⋊D6112yes—
546C4#1684027312yes—
547C2—21no—
548C2—21no—
549C2—44no—
550C2C4×#1451974611yescharacteristic
551C2#2145518311yescharacteristic
552C2#2145519811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×2−1+4×C61C2
socleC22×C61C2×#120
prime radicalC2×2−1+4×C61C22
cosocleC22×C4×C61C24
hypercenterC4×2−1+4D61
hypocenterC61#21476
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×2−1+4

Complements

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

NH ≅ G/Ntype
2. C4×#14519807C2semidirect
3. #21448535C2semidirect
4. #21446212C2semidirect
5. #21446188C2semidirect
6. #21446052C2semidirect
7. #21446034C2semidirect
9. #21445629C2semidirect
10. C4×#14519696C2semidirect
15. #16839945C4semidirect
50. #14519807C2×C4semidirect
56. #14519775C2×C4semidirect
68. #14519704C2×C4semidirect
74. #14519696C2×C4semidirect
324. C61#21476semidirect

Maximal subgroups

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

subgroupindex
C2×#168399452
C4×#145198072
#214485352
#214462122
#214461882
#214460522
#214460342
#214457782
#214456292
C4×#145196962
C4×2−1+4×C612
#2147661