Finite Group Databasebeta

← #29607011

Subgroups — #29607011

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×#16839669C211yesmaximal, characteristic
2C2×C4⋊C4×Dic61C211yesmaximal, characteristic
3#21444090C211yesmaximal, characteristic
4#21444087C212yesmaximal
5#21444076C212yesmaximal
6#21444076C211yesmaximal, characteristic
7#21444072C211yesmaximal, characteristic
8#21444051C212yesmaximal
9#21444044C212yesmaximal
10#21444044C211yesmaximal, characteristic
11C2×C61×#395C211yesmaximal, characteristic
12C22×C4⋊Dic61C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14C22×C4×Dic61C2211yescharacteristic
15C22×C4×Dic61C2211yescharacteristic
16C22×C4×Dic61C2211yescharacteristic
17#16839669—24no—
18C4⋊C4×Dic61—24no—
19C2×#14519684C2212yes—
20C2×#14519684C2212yes—
21C2×#14519684C2212yes—
22C2×#14519684C2212yes—
23C2×#14519684C2211yescharacteristic
24C2×#14519684C2212yes—
25C2×#14519684C2211yescharacteristic
26C2×#14519684C2211yescharacteristic
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2212yes—
31C2×#14519684C2212yes—
32C23×Q8×C61C2211yescharacteristic
33C22×C4⋊C4×C61C2211yescharacteristic
34C22×C4⋊C4×C61C2212yes—
35C61×#395C418yescomplemented
36C2×#95×C61C2212yes—
37C2×#95×C61C2211yescharacteristic
38C23×Dic61C2312yes—
39C23×Dic61C2312yes—
40C23×Dic61C2311yescharacteristic
41C23×Dic61C2311yescharacteristic
42C23×Dic61C2311yescharacteristic
43C23×Dic61C2311yescharacteristic
44C2×C4⋊Dic61—22no—
45C2×C4⋊Dic61—22no—
46C2×C4⋊Dic61—22no—
47C2×C4⋊Dic61—22no—
48C2×C4⋊Dic61—42no—
49C2×C4×Dic61—22no—
50C2×C4×Dic61—22no—
51C2×C4×Dic61—22no—
52C2×C4×Dic61—42no—
53C2×C4×Dic61—42no—
54C2×C4×Dic61D414yes—
55C2×C4×Dic61Q814yescomplemented
56#14519684—22no—
57#14519684—22no—
58#14519684—24no—
59#14519684—22no—
60#14519684—24no—
61#14519684—24no—
62#14519684—24no—
63#14519684—24no—
64#14519684—42no—
65#14519684—42no—
66#14519684—42no—
67#14519684—42no—
68#14519684—42no—
69C22×Q8×C61—22no—
70C22×Q8×C61—22no—
71C22×Q8×C61—42no—
72C22×Q8×C61—42no—
73C22×Q8×C61C2×C414yes—
74C23×C4×C61C2311yescharacteristic
75C23×C4×C61C2312yes—
76C23×C4×C61C2312yes—
77C23×C4×C61C2311yescharacteristic
78C23×C4×C61C2311yescharacteristic
79C2×C4⋊C4×C61—22no—
80C2×C4⋊C4×C61—22no—
81C2×C4⋊C4×C61—42no—
82C2×C4⋊C4×C61—42no—
83C2×C4⋊C4×C61C2×C414yes—
84C2×C4⋊C4×C61C2×C418yes—
85#95×C61C2×C414yes—
86#95×C61C2×C418yes—
87C24×C61C2411yescharacteristic
88C2×Q8×C61—28no—
89C2×Q8×C61—48no—
90C2×Q8×C61—44no—
91C2×Q8×C61—44no—
92C2×Q8×C61—48no—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—21no—
96C22×C4×C61—21no—
97C22×C4×C61—22no—
98C22×C4×C61—22no—
99C22×C4×C61—21no—
100C22×C4×C61—22no—
101C22×C4×C61—21no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
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×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61C22×C414yes—
116C22×C4×C61C22×C414yes—
117C22×C4×C61C22×C412yes—
118C22×C4×C61C22×C412yes—
119C22×C4×C61C22×C412yes—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×Q812yes—
122C22×C4×C61#4112yes—
123C22×C4×C61#4112yes—
124C22×Dic61—22no—
125C22×Dic61—22no—
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
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×Dic61—22no—
152C22×Dic61C2×D412yes—
153C22×Dic61C2×D412yes—
154C22×Dic61C2×Q812yes—
155C22×Dic61C2×Q812yes—
156C22×Dic61#4112yes—
157C22×Dic61#4112yes—
158C4⋊C4×C61—28no—
159C4⋊C4×C61—48no—
160C4⋊Dic61—28no—
161C4⋊Dic61—44no—
162C4×Dic61—28no—
163C4×Dic61—44no—
164C4×Dic61—44no—
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—
176Q8×C61—416no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—24no—
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—24no—
187C2×C4×C61—22no—
188C2×C4×C61—24no—
189C2×C4×C61—24no—
190C2×C4×C61—24no—
191C2×C4×C61—24no—
192C2×C4×C61—24no—
193C2×C4×C61—24no—
194C2×C4×C61—24no—
195C2×C4×C61—41no—
196C2×C4×C61—42no—
197C2×C4×C61—41no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61C4×D414yes—
207C2×C4×C61C4×Q814yes—
208C2×Dic61—22no—
209C2×Dic61—22no—
210C2×Dic61—24no—
211C2×Dic61—24no—
212C2×Dic61—22no—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—22no—
217C2×Dic61—24no—
218C2×Dic61—24no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
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—
241C2×Dic61C22⋊Q814yes—
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—42no—
269C4×C61—44no—
270C4×C61—44no—
271Dic61—24no—
272Dic61—42no—
273Dic61—42no—
274Dic61—42no—
275Dic61—82no—
276Dic61—82no—
277C2×#395D6111yescharacteristic
278C2×#391—611no—
279C2×C4×C4⋊C4—611no—
280#1446—611no—
281#1445—612no—
282#1442—612no—
283#1442—611no—
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×Q8C2×D6111yescharacteristic
300C22×C4⋊C4—611no—
301C22×C4⋊C4—611no—
302C22×C4⋊C4C2×D6111yescharacteristic
303C22×C4⋊C4C2×D6112yes—
304C22×C42—611no—
305C22×C42—611no—
306C22×C42—611no—
307#395Dic6118yescomplemented
308#391—1224no—
309C4×C4⋊C4—1224no—
310C2×#95—612no—
311C2×#95—612no—
312C2×#95—612no—
313C2×#95—611no—
314C2×#95—612no—
315C2×#95—612no—
316C2×#95—611no—
317C2×#95—611no—
318C2×#95—612no—
319C2×#95—612no—
320C2×#95—612no—
321C2×#95—612no—
322C2×#95—612no—
323C2×#95C2×D6112yes—
324C2×#95C2×D6111yescharacteristic
325C61#1432011yescharacteristic, complemented, Sylow
326C22×Q8—22no—
327C22×Q8—22no—
328C22×Q8—42no—
329C22×Q8—42no—
330C22×Q8C2×Dic6114yes—
331C23×C4—611no—
332C23×C4—611no—
333C23×C4—611no—
334C23×C4—611no—
335C23×C4—612no—
336C23×C4—612no—
337C23×C4C22×D6111yescharacteristic
338C23×C4C22×D6112yes—
339C23×C4C22×D6112yes—
340C23×C4C22×D6111yescharacteristic
341C23×C4C22×D6111yescharacteristic
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—42no—
345C2×C4⋊C4—42no—
346C2×C4⋊C4—1222no—
347C2×C4⋊C4—1222no—
348C2×C4⋊C4—1222no—
349C2×C4⋊C4—1222no—
350C2×C4⋊C4—2442no—
351C2×C4⋊C4C2×Dic6114yes—
352C2×C4⋊C4C2×Dic6118yes—
353C2×C42—614no—
354C2×C42—614no—
355C2×C42—1222no—
356C2×C42—1222no—
357C2×C42—1222no—
358C2×C42—2442no—
359C2×C42—2442no—
360#95—1222no—
361#95—1224no—
362#95—1224no—
363#95—1224no—
364#95—1224no—
365#95—1222no—
366#95—1222no—
367#95—1224no—
368#95—2442no—
369#95—2442no—
370#95—2442no—
371#95—2442no—
372#95—2442no—
373#95C2×Dic6114yes—
374#95C2×Dic6118yes—
375C24C23×D6111yescharacteristic
376C2×Q8—28no—
377C2×Q8—48no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—48no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—612no—
404C22×C4—612no—
405C22×C4—612no—
406C22×C4—612no—
407C22×C4—612no—
408C22×C4—612no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1221no—
412C22×C4—1221no—
413C22×C4—1221no—
414C22×C4—1221no—
415C22×C4—1221no—
416C22×C4—1221no—
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×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4D4×D6112yes—
438C22×C4#1175822412yes—
439C22×C4Q8×D6112yes—
440C22×C4#1175822612yes—
441C22×C4C22×Dic6114yes—
442C22×C4C22×Dic6114yes—
443C22×C4C22×Dic6112yes—
444C22×C4C22×Dic6112yes—
445C22×C4C22×Dic6112yes—
446C4⋊C4—28no—
447C4⋊C4—48no—
448C4⋊C4—1228no—
449C4⋊C4—2444no—
450C42—1228no—
451C42—2444no—
452C42—2444no—
453C23C23×Dic6111yescharacteristic
454C23C2×D4×D6111yescharacteristic
455C23C2×Q8×D6111yescharacteristic
456C23#41×D6111yescharacteristic
457C23#41×D6111yescharacteristic
458C23C2×#1175822411yescharacteristic
459C23C2×#1175822611yescharacteristic
460C23#1451982212yes—
461C23#1451982212yes—
462C23#1451982512yes—
463C23#1451982512yes—
464Q8—416no—
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—24no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—614no—
495C2×C4—1222no—
496C2×C4—1222no—
497C2×C4—1224no—
498C2×C4—1224no—
499C2×C4—1222no—
500C2×C4—1222no—
501C2×C4—1222no—
502C2×C4—1222no—
503C2×C4—1222no—
504C2×C4—1224no—
505C2×C4—1224no—
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—2441no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4D4×Dic6114yes—
529C2×C4Q8×Dic6114yes—
530C22C2×D4×Dic6111yescharacteristic
531C22C2×Q8×Dic6111yescharacteristic
532C22#41×Dic6111yescharacteristic
533C22#1684023712yes—
534C22#1684024012yes—
535C22C22⋊Q8×D6111yescharacteristic
536C22#1684027511yescharacteristic
537C22#1684029811yescharacteristic
538C22#1684031411yescharacteristic
539C22#1684032211yescharacteristic
540C22#1684032711yescharacteristic
541C22#1684033711yescharacteristic
542C22#1684033711yescharacteristic
543C22#1684034712yes—
544C22#1684035812yes—
545C22#1684035812yes—
546C22#1684037312yes—
547C22#1684038212yes—
548C22#1684038212yes—
549C22#1684038612yes—
550C22#1684038612yes—
551C22#1684038912yes—
552C22#1684038912yes—
553C4—24no—
554C4—24no—
555C4—42no—
556C4—44no—
557C4—44no—
558C4—1224no—
559C4—4882no—
560C4—4882no—
561C4—2442no—
562C4—2442no—
563C4—2442no—
564C2C22⋊Q8×Dic6111yescharacteristic
565C2#2144848011yescharacteristic
566C2#2144853911yescharacteristic
567C2#2144856812yes—
568C2#2144857712yes—
569C2#2145163111yescharacteristic
570C2#2145164211yescharacteristic
571C2#2145168511yescharacteristic
572C2#2145174511yescharacteristic
573C2#2145184012yes—
574C2#2145199712yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#395C2
socle, prime radical, cosocleC24×C61C24
hypercenterC2×#395D61
hypocenterC61#14320
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
35. C61×#395C4semidirect
55. C2×C4×Dic61Q8semidirect
307. #395Dic61semidirect
325. C61#14320semidirect

Maximal subgroups

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

subgroupindex
C2×#168396692
C2×C4⋊C4×Dic612
#214440902
#214440872
#214440762
#214440722
#214440512
#214440442
C2×C61×#3952
#1432061