Finite Group Databasebeta

← #29607006

Subgroups — #29607006

Order 15616.

1318 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839682C211yesmaximal, characteristic
2C2×#16839668C211yesmaximal, characteristic
3#21444089C211yesmaximal, characteristic
4#21444085C212yesmaximal
5#21444078C212yesmaximal
6#21444075C211yesmaximal, characteristic
7#21444072C211yesmaximal, characteristic
8#21444052C212yesmaximal
9#21444046C212yesmaximal
10#21444046C211yesmaximal, characteristic
11C2×C4×C4⋊C4×C61C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14C22×#11758196C2212yes—
15#16839682C418yescomplemented
16#16839668—24no—
17C2×#14519684C2212yes—
18C2×#14519684C2212yes—
19C2×#14519684C2212yes—
20C2×#14519684C2212yes—
21C2×#14519684C2211yescharacteristic
22C2×#14519684C2212yes—
23C2×#14519684C2212yes—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C2×#14519684C2212yes—
28C2×#14519684C2211yescharacteristic
29C2×#14519684C2212yes—
30C22×C4⋊C4×C61C2211yescharacteristic
31C22×C4⋊C4×C61C2211yescharacteristic
32C22×C42×C61C2211yescharacteristic
33C22×C42×C61C2211yescharacteristic
34C22×C42×C61C2211yescharacteristic
35C4×C4⋊C4×C61—24no—
36C2×#95×C61C2211yescharacteristic
37C2×#95×C61C2211yescharacteristic
38C22×Dic122—22no—
39C22×Dic122—22no—
40C22×Dic122—42no—
41C22×Dic122—42no—
42C22×Dic122C2×C414yes—
43C23×Dic61C2312yes—
44C23×Dic61C2312yes—
45C23×Dic61C2312yes—
46C23×Dic61C2312yes—
47C2×C4⋊Dic61—22no—
48C2×C4⋊Dic61—42no—
49C2×#11758196—42no—
50C2×#11758196—42no—
51C2×#11758196C2×C418yes—
52#14519684—24no—
53#14519684—24no—
54#14519684—24no—
55#14519684—24no—
56#14519684—22no—
57#14519684—24no—
58#14519684—42no—
59#14519684—42no—
60#14519684—42no—
61#14519684—42no—
62#14519684—42no—
63#14519684C2×C418yes—
64#14519684C2×C414yes—
65C23×C4×C61C2311yescharacteristic
66C23×C4×C61C2311yescharacteristic
67C23×C4×C61C2311yescharacteristic
68C23×C4×C61C2311yescharacteristic
69C23×C4×C61C2311yescharacteristic
70C23×C4×C61C2311yescharacteristic
71C23×C4×C61C2311yescharacteristic
72C2×C4⋊C4×C61—22no—
73C2×C4⋊C4×C61—22no—
74C2×C4⋊C4×C61—22no—
75C2×C4⋊C4×C61—22no—
76C2×C4⋊C4×C61—22no—
77C2×C4⋊C4×C61C2×C414yes—
78C2×C42×C61—22no—
79C2×C42×C61—22no—
80C2×C42×C61—22no—
81C2×C42×C61—42no—
82C2×C42×C61—42no—
83C2×C42×C61D414yes—
84C2×C42×C61Q814yescomplemented
85#95×C61—22no—
86#95×C61—22no—
87C24×C61C2411yescharacteristic
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—22no—
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61—22no—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61C22×C412yes—
113C22×C4×C61C22×C412yes—
114C22×C4×C61C22×C412yes—
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2×D412yes—
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×Q812yes—
119C22×C4×C61C2×Q812yes—
120C22×C4×C61C2×Q812yes—
121C22×C4×C61#4112yes—
122C22×C4×C61#4112yes—
123C22×C4×C61#4112yes—
124C22×C4×C61#4112yes—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—22no—
151C22×Dic61C22×C414yes—
152C22×Dic61C22×C414yes—
153C2×Dic122—28no—
154C2×Dic122—48no—
155C2×Dic122—44no—
156C2×Dic122—44no—
157C2×Dic122—48no—
158C4⋊C4×C61—28no—
159C4⋊C4×C61—28no—
160C42×C61—28no—
161C42×C61—44no—
162C42×C61—44no—
163C4⋊Dic61—44no—
164#11758196—48no—
165C23×C61C23×C411yescharacteristic
166C23×C61C22×D411yescharacteristic
167C23×C61C22×Q811yescharacteristic
168C23×C61C2×#4111yescharacteristic
169C23×C61C2×#4111yescharacteristic
170C23×C61C2×#4111yescharacteristic
171C23×C61C2×#4111yescharacteristic
172C23×C61H(5,𝔽2)12yes—
173C23×C61H(5,𝔽2)12yes—
174C23×C612−1+412yes—
175C23×C612−1+412yes—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—24no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—24no—
188C2×C4×C61—24no—
189C2×C4×C61—24no—
190C2×C4×C61—22no—
191C2×C4×C61—24no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—24no—
198C2×C4×C61—24no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—41no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61C4×D414yes—
211C2×C4×C61C4×Q814yes—
212C2×C4×C61C22⋊Q814yes—
213C2×Dic61—24no—
214C2×Dic61—24no—
215C2×Dic61—24no—
216C2×Dic61—24no—
217C2×Dic61—24no—
218C2×Dic61—24no—
219C2×Dic61—42no—
220C2×Dic61—42no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241Dic122—416no—
242#14320—611nomaximal, Sylow
243C22×C61C2×C4×D411yescharacteristic
244C22×C61C2×C4×Q811yescharacteristic
245C22×C61C4×#4111yescharacteristic
246C22×C61#51812yes—
247C22×C61#51912yes—
248C22×C61C2×C22⋊Q811yescharacteristic
249C22×C61#52511yescharacteristic
250C22×C61#52911yescharacteristic
251C22×C61#53311yescharacteristic
252C22×C61#53411yescharacteristic
253C22×C61#53511yescharacteristic
254C22×C61#53611yescharacteristic
255C22×C61#53611yescharacteristic
256C22×C61#53812yes—
257C22×C61#53912yes—
258C22×C61#53912yes—
259C22×C61#54112yes—
260C22×C61#54212yes—
261C22×C61#54212yes—
262C22×C61#54212yes—
263C22×C61#54212yes—
264C22×C61#54412yes—
265C22×C61#54412yes—
266C4×C61—24no—
267C4×C61—24no—
268C4×C61—24no—
269C4×C61—42no—
270C4×C61—42no—
271C4×C61—42no—
272C4×C61—42no—
273Dic61—44no—
274Dic61—44no—
275Dic61—82no—
276Dic61—82no—
277C2×#395—611no—
278C2×#391—611no—
279C2×C4×C4⋊C4D6111yescharacteristic
280#1446—611no—
281#1445—612no—
282#1442—611no—
283#1442—612no—
284#1441—611no—
285#1436—612no—
286#1434—611no—
287#1434—612no—
288C2×C61C4×C22⋊Q811yescharacteristic
289C2×C61#232111yescharacteristic
290C2×C61#232711yescharacteristic
291C2×C61#233112yes—
292C2×C61#233212yes—
293C2×C61#262711yescharacteristic
294C2×C61#262811yescharacteristic
295C2×C61#263211yescharacteristic
296C2×C61#263811yescharacteristic
297C2×C61#265012yes—
298C2×C61#266612yes—
299C23×Q8—611no—
300C22×C4⋊C4—612no—
301C22×C4⋊C4—611no—
302C22×C4⋊C4C2×D6111yescharacteristic
303C22×C4⋊C4C2×D6111yescharacteristic
304C22×C42C2×D6111yescharacteristic
305C22×C42C2×D6111yescharacteristic
306C22×C42C2×D6111yescharacteristic
307#395—618no—
308#391—1224no—
309C4×C4⋊C4—24no—
310C2×#95—612no—
311C2×#95—611no—
312C2×#95—612no—
313C2×#95—612no—
314C2×#95—612no—
315C2×#95—611no—
316C2×#95—612no—
317C2×#95—612no—
318C2×#95—612no—
319C2×#95—612no—
320C2×#95—612no—
321C2×#95—612no—
322C2×#95—612no—
323C2×#95C2×D6111yescharacteristic
324C2×#95C2×D6111yescharacteristic
325C61#1432011yescharacteristic, complemented, Sylow
326C22×Q8—614no—
327C22×Q8—1222no—
328C22×Q8—1222no—
329C22×Q8—2442no—
330C22×Q8—2442no—
331C23×C4—612no—
332C23×C4—612no—
333C23×C4—612no—
334C23×C4—612no—
335C23×C4C22×D6111yescharacteristic
336C23×C4C22×D6111yescharacteristic
337C23×C4C22×D6111yescharacteristic
338C23×C4C22×D6111yescharacteristic
339C23×C4C22×D6111yescharacteristic
340C23×C4C22×D6111yescharacteristic
341C23×C4C22×D6111yescharacteristic
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—618no—
348C2×C4⋊C4—1222no—
349C2×C4⋊C4—2442no—
350C2×C4⋊C4—2442no—
351C2×C4⋊C4—2442no—
352C2×C4⋊C4C4×D6114yes—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—42no—
357C2×C42—42no—
358C2×C42Dic12214yescomplemented
359C2×C42C22⋊D6114yes—
360#95—22no—
361#95—22no—
362#95—614no—
363#95—618no—
364#95—1224no—
365#95—1224no—
366#95—1222no—
367#95—1224no—
368#95—1224no—
369#95—1224no—
370#95—2442no—
371#95—2442no—
372#95—2442no—
373#95—2442no—
374#95—2442no—
375C24C23×D6111yescharacteristic
376C2×Q8—1228no—
377C2×Q8—2448no—
378C2×Q8—2444no—
379C2×Q8—2444no—
380C2×Q8—2448no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—614no—
406C22×C4—614no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1222no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—1222no—
415C22×C4—1222no—
416C22×C4—1222no—
417C22×C4—1222no—
418C22×C4—1222no—
419C22×C4—1222no—
420C22×C4—1222no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—1222no—
433C22×C4C2×Dic12212yes—
434C22×C4C2×Dic12212yes—
435C22×C4C2×Dic12212yes—
436C22×C4C2×C4×D6112yes—
437C22×C4C2×C4×D6112yes—
438C22×C4C2×C4×D6112yes—
439C22×C4#1175822212yes—
440C22×C4#1175822212yes—
441C22×C4#1175822212yes—
442C22×C4#1175822212yes—
443C22×C4C2×C22⋊D6112yes—
444C22×C4C2×C22⋊D6112yes—
445C22×C4C2×C22⋊D6112yes—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—2448no—
449C4⋊C4—2444no—
450C42—28no—
451C42—44no—
452C42—44no—
453C23C22×C4×D6111yescharacteristic
454C23C22×C22⋊D6111yescharacteristic
455C23C22×Dic12211yescharacteristic
456C23C2×#1175822211yescharacteristic
457C23C2×#1175822211yescharacteristic
458C23C2×#1175822211yescharacteristic
459C23C2×#1175822211yescharacteristic
460C23#1451982212yes—
461C23#1451982312yes—
462C23#1451982412yes—
463C23#1451982512yes—
464Q8—24416no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—1224no—
500C2×C4—1224no—
501C2×C4—1224no—
502C2×C4—1224no—
503C2×C4—1224no—
504C2×C4—1224no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4—2442no—
511C2×C4—2442no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4C4×C22⋊D6114yes—
528C2×C4C4×Dic12214yes—
529C2×C4#1451975614yes—
530C22C2×C4×C22⋊D6111yescharacteristic
531C22C2×C4×Dic12211yescharacteristic
532C22C4×#1175822211yescharacteristic
533C22#1684023412yes—
534C22#1684023812yes—
535C22C2×#1451975611yescharacteristic
536C22#1684027311yescharacteristic
537C22#1684029611yescharacteristic
538C22#1684031911yescharacteristic
539C22#1684032111yescharacteristic
540C22#1684032611yescharacteristic
541C22#1684033311yescharacteristic
542C22#1684033611yescharacteristic
543C22#1684034512yes—
544C22#1684035412yes—
545C22#1684035912yes—
546C22#1684037012yes—
547C22#1684037712yes—
548C22#1684038312yes—
549C22#1684038412yes—
550C22#1684038412yes—
551C22#1684039012yes—
552C22#1684039312yes—
553C4—24no—
554C4—24no—
555C4—24no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—4882no—
561C4—4882no—
562C4—2444no—
563C4—2444no—
564C2C4×#1451975611yescharacteristic
565C2#2144847311yescharacteristic
566C2#2144853511yescharacteristic
567C2#2144856312yes—
568C2#2144857012yes—
569C2#2145163211yescharacteristic
570C2#2145164011yescharacteristic
571C2#2145167911yescharacteristic
572C2#2145173611yescharacteristic
573C2#2145183112yes—
574C2#2145199812yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C4×C4⋊C4×C61C2
socle, prime radical, cosocleC24×C61C24
hypercenterC2×C4×C4⋊C4D61
hypocenterC61#14320
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
15. #16839682C4semidirect
84. C2×C42×C61Q8semidirect
325. C61#14320semidirect
358. C2×C42Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×#168396822
C2×#168396682
#214440892
#214440852
#214440782
#214440752
#214440722
#214440522
#214440462
C2×C4×C4⋊C4×C612
#1432061