Finite Group Databasebeta

← #30609417

Subgroups — #30609417

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461006C212yesmaximal
2#21461005C212yesmaximal
3#21461004C212yesmaximal
4#21460999C212yesmaximal
5#21460998C212yesmaximal
6#21460995C211yesmaximal, characteristic
7#21460995C211yesmaximal, characteristic
8#21460994C212yesmaximal
9#382×D61C211yesmaximal, characteristic
10C22×C4⋊#6923C2212yes—
11C22×C4×#6923C2212yes—
12C2×(C2×#6923)⋊C4C2212yes—
13C2×(C2×#6923)⋊C4C2212yes—
14C2×(C2×#6923)⋊C4C2212yes—
15C2×(C2×#6923)⋊C4C2212yes—
16C2×(C2×#6923)⋊C4C2212yes—
17C2×(C2×#6923)⋊C4C2212yes—
18C2×(C2×#6923)⋊C4C2211yescharacteristic
19C2×(C2×#6923)⋊C4C2212yes—
20C2×(C2×#6923)⋊C4C2211yescharacteristic
21C2×(C2×#6923)⋊C4C2211yescharacteristic
22C2×(C2×#6923)⋊C4C2211yescharacteristic
23C2×(C2×#6923)⋊C4C2212yes—
24C2×(C2×#6923)⋊C4C2212yes—
25C2×(C2×#6923)⋊C4C2212yes—
26C2×C4⋊C4×D61C2211yescharacteristic
27C2×C4⋊C4×D61C2211yescharacteristic
28C2×C42×D61C2211yescharacteristic
29#16839628C411yescharacteristic, complemented
30#16839627C411yescharacteristic, complemented
31#16839627C411yescharacteristic, complemented
32#16839626C412yescomplemented
33#16839626C412yescomplemented
34#95×D61C2212yes—
35#95×D61C2212yes—
36C61×#382C411yescharacteristic, complemented
37C23×#6923C2312yes—
38C23×#6923C2312yes—
39C23×#6923C2312yes—
40C23×#6923C2312yes—
41C2×C4⋊#6923—24no—
42C2×C4⋊#6923—42no—
43C2×C4⋊#6923—42no—
44C2×C4×#6923—24no—
45C2×C4×#6923—42no—
46C2×C4×#6923—42no—
47(C2×#6923)⋊C4—24no—
48(C2×#6923)⋊C4—24no—
49(C2×#6923)⋊C4—24no—
50(C2×#6923)⋊C4—24no—
51(C2×#6923)⋊C4—24no—
52(C2×#6923)⋊C4—24no—
53(C2×#6923)⋊C4—41no—
54(C2×#6923)⋊C4—41no—
55(C2×#6923)⋊C4—41no—
56(C2×#6923)⋊C4—41no—
57(C2×#6923)⋊C4—42no—
58(C2×#6923)⋊C4—42no—
59(C2×#6923)⋊C4—42no—
60(C2×#6923)⋊C4—42no—
61C22×C4×D61C2311yescharacteristic
62C22×C4×D61C2311yescharacteristic
63C22×C4×D61C2312yes—
64C22×C4×D61C2312yes—
65C22×C4×D61C2311yescharacteristic
66C2×C4⋊Dic61C2×C411yescharacteristic
67C2×C4⋊Dic61C2×C411yescharacteristic
68C2×#11758196C2×C412yes—
69C2×#11758196C2×C412yes—
70C4⋊C4×D61—42no—
71C4⋊C4×D61—42no—
72C2×C4×Dic61C2×C411yescharacteristic
73C2×C4×Dic61C2×C411yescharacteristic
74C2×C4×Dic61C2×C411yescharacteristic
75C42×D61—24no—
76#14519684C2×C412yes—
77#14519684C2×C412yes—
78#14519684C2×C412yes—
79#14519684C2×C412yes—
80#14519684C2×C412yes—
81#14519684C2×C412yes—
82C2×C4⋊C4×C61C2×C411yescharacteristic
83C2×C4⋊C4×C61C2×C411yescharacteristic
84C2×C42×C61C2×C411yescharacteristic
85#95×C61C2×C412yes—
86#95×C61C2×C412yes—
87C23×D61C2411yescharacteristic
88C22×#6923—22no—
89C22×#6923—22no—
90C22×#6923—22no—
91C22×#6923—22no—
92C22×#6923—22no—
93C22×#6923—22no—
94C22×#6923—22no—
95C22×#6923—22no—
96C22×#6923—22no—
97C22×#6923—22no—
98C22×#6923—22no—
99C22×#6923—22no—
100C22×#6923—22no—
101C22×#6923—22no—
102C22×#6923—22no—
103C22×#6923—22no—
104C22×#6923—22no—
105C22×#6923—22no—
106C22×#6923—22no—
107C22×#6923—22no—
108C22×#6923—22no—
109C22×#6923—22no—
110C22×#6923—22no—
111C22×#6923—22no—
112C22×#6923—22no—
113C22×#6923—22no—
114C22×#6923#4114yes—
115C22×#6923#4114yes—
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C22×C411yescharacteristic
118C22×C4×C61C22×C412yes—
119C22×C4×C61C22×C412yes—
120C22×C4×C61C22×C411yescharacteristic
121C22×Dic61C22×C411yescharacteristic
122C22×Dic61C22×C411yescharacteristic
123C22×Dic61C22×C412yes—
124C22×Dic61C22×C412yes—
125C22×Dic61C22×C411yescharacteristic
126C2×C4×D61—22no—
127C2×C4×D61—21no—
128C2×C4×D61—21no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61#4114yes—
150C4⋊#6923—48no—
151C4×#6923—48no—
152C4⋊C4×C61—41no—
153C4⋊C4×C61—41no—
154C42×C61—22no—
155C4⋊Dic61—41no—
156C4⋊Dic61—41no—
157#11758196—42no—
158#11758196—42no—
159C4×Dic61—22no—
160C4×Dic61—22no—
161C4×Dic61—22no—
162C23×C61C23×C411yescharacteristic
163C22×D61C2×#4112yes—
164C22×D61C2×#4112yes—
165C22×D61C2×#4112yes—
166C22×D61H(5,𝔽2)12yes—
167C22×D61H(5,𝔽2)11yescharacteristic
168C22×D61H(5,𝔽2)11yescharacteristic
169C22×D612−1+411yescharacteristic
170C22×D612−1+411yescharacteristic
171C22×D612−1+411yescharacteristic
172C22×D612−1+411yescharacteristic
173C2×#6923—24no—
174C2×#6923—24no—
175C2×#6923—24no—
176C2×#6923—24no—
177C2×#6923—24no—
178C2×#6923—24no—
179C2×#6923—42no—
180C2×#6923—42no—
181C2×#6923—42no—
182C2×#6923—42no—
183C2×#6923—42no—
184C2×#6923—42no—
185C2×#6923—42no—
186C2×#6923—42no—
187C2×#6923—42no—
188C2×#6923—42no—
189C2×#6923—42no—
190C2×#6923—42no—
191C2×#6923—42no—
192C2×#6923—42no—
193C2×#6923—42no—
194C2×#6923—42no—
195C2×#6923—42no—
196C2×#6923—42no—
197C2×#6923—42no—
198C2×#6923—42no—
199C2×#6923—42no—
200C2×#6923—42no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61#11712yes—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61—21no—
220C2×Dic61—22no—
221C2×Dic61—22no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61#11712yes—
231C4×D61—24no—
232C4×D61—24no—
233C4×D61—42no—
234C4×D61—42no—
235C4×D61—42no—
236C4×D61—42no—
237C4×D61—42no—
238C4×D61—42no—
239C4×D61—42no—
240C4×D61—42no—
241C4×D61—42no—
242C4×D61—42no—
243C4×D61—42no—
244C4×D61—42no—
245#14477—611nomaximal, Sylow
246C22×C61C2×#11711yescharacteristic
247C22×C61C4×#4111yescharacteristic
248C22×C61C4×#4111yescharacteristic
249C22×C61#51811yescharacteristic
250C22×C61#51911yescharacteristic
251C22×C61#52011yescharacteristic
252C22×C61#52011yescharacteristic
253C2×D61#53812yes—
254C2×D61#53812yes—
255C2×D61#53912yes—
256C2×D61#53912yes—
257C2×D61#54212yes—
258C2×D61#54212yes—
259C2×D61#55212yes—
260C2×D61#55212yes—
261C2×D61#55212yes—
262C2×D61#55312yes—
263C2×D61#55512yes—
264C2×D61#55612yes—
265C2×D61#56211yescharacteristic
266C2×D61#56311yescharacteristic
267C2×D61#56311yescharacteristic
268C2×D61#56411yescharacteristic
269#6923—44no—
270#6923—44no—
271#6923—82no—
272#6923—82no—
273C4×C61—22no—
274C4×C61—42no—
275C4×C61—42no—
276C4×C61—41no—
277C4×C61—41no—
278Dic61—22no—
279Dic61—42no—
280Dic61—42no—
281Dic61—41no—
282Dic61—41no—
283C2×#382—611no—
284#1443—612no—
285#1442—612no—
286#1441—612no—
287#1436—612no—
288#1436—612no—
289#1434—612no—
290#1434—611no—
291#1434—611no—
292C2×C61#231311yescharacteristic
293C2×C61#233711yescharacteristic
294C2×C61#233711yescharacteristic
295C2×C61#234611yescharacteristic
296C2×C61#234711yescharacteristic
297C2×C61#237111yescharacteristic
298C2×C61#238211yescharacteristic
299D61#276512yes—
300D61#276712yes—
301D61#277512yes—
302D61#278112yes—
303C22×C4⋊C4—611no—
304C22×C4⋊C4—611no—
305C22×C4⋊C4—612no—
306C22×C42—611no—
307C22×C42—612no—
308#382—611no—
309#382—611no—
310#382—611no—
311#382—612no—
312#382—612no—
313#382#692311yescharacteristic, complemented
314C2×#95—612no—
315C2×#95—612no—
316C2×#95—612no—
317C2×#95—612no—
318C2×#95—612no—
319C2×#95—612no—
320C2×#95—612no—
321C2×#95—612no—
322C2×#95—611no—
323C2×#95—611no—
324C2×#95—612no—
325C2×#95—611no—
326C2×#95—611no—
327C2×#95—612no—
328C2×#95—612no—
329C2×#95—612no—
330C61#1447711yescharacteristic, complemented, Sylow
331C23×C4—611no—
332C23×C4—612no—
333C23×C4—612no—
334C23×C4—612no—
335C23×C4—612no—
336C23×C4—611no—
337C23×C4—611no—
338C23×C4—612no—
339C23×C4—612no—
340C2×C4⋊C4—612no—
341C2×C4⋊C4—612no—
342C2×C4⋊C4—611no—
343C2×C4⋊C4—611no—
344C2×C4⋊C4—1224no—
345C2×C4⋊C4—2442no—
346C2×C4⋊C4—2442no—
347C2×C4⋊C4—2442no—
348C2×C4⋊C4—2442no—
349C2×C4⋊C4C2×#692311yescharacteristic
350C2×C4⋊C4C2×#692311yescharacteristic
351C2×C42—611no—
352C2×C42—611no—
353C2×C42—611no—
354C2×C42—1224no—
355C2×C42—1224no—
356C2×C42—2442no—
357C2×C42—2442no—
358C2×C42C2×#692311yescharacteristic
359#95—612no—
360#95—612no—
361#95—612no—
362#95—612no—
363#95—612no—
364#95—612no—
365#95—1224no—
366#95—1224no—
367#95—1224no—
368#95—1224no—
369#95—1224no—
370#95—1224no—
371#95—2441no—
372#95—2441no—
373#95—2441no—
374#95—2441no—
375#95—2442no—
376#95—2442no—
377#95—2442no—
378#95—2442no—
379#95C2×#692312yes—
380#95C2×#692312yes—
381C24—611no—
382C22×C4—611no—
383C22×C4—614no—
384C22×C4—614no—
385C22×C4—614no—
386C22×C4—611no—
387C22×C4—611no—
388C22×C4—612no—
389C22×C4—612no—
390C22×C4—1222no—
391C22×C4—1221no—
392C22×C4—1221no—
393C22×C4—1222no—
394C22×C4—1222no—
395C22×C4—1222no—
396C22×C4—1222no—
397C22×C4—1221no—
398C22×C4—1221no—
399C22×C4—1222no—
400C22×C4—1222no—
401C22×C4—1222no—
402C22×C4—1222no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4—1222no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1221no—
412C22×C4—1221no—
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×C4—1222no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4C22×#692311yescharacteristic
440C22×C4C22×#692311yescharacteristic
441C22×C4C22×#692312yes—
442C22×C4C22×#692312yes—
443C22×C4C22×#692311yescharacteristic
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—2442no—
447C4⋊C4—2442no—
448C4⋊C4—2441no—
449C4⋊C4—2441no—
450C4⋊C4—2448no—
451C42—22no—
452C42—1222no—
453C42—1222no—
454C42—1222no—
455C42—2448no—
456C23—612no—
457C23—612no—
458C23—612no—
459C23—612no—
460C23—611no—
461C23—611no—
462C23—611no—
463C23—611no—
464C23—611no—
465C23—611no—
466C23C23×#692311yescharacteristic
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—612no—
482C2×C4—1224no—
483C2×C4—1224no—
484C2×C4—1221no—
485C2×C4—1221no—
486C2×C4—1224no—
487C2×C4—1224no—
488C2×C4—1224no—
489C2×C4—1224no—
490C2×C4—1224no—
491C2×C4—1224no—
492C2×C4—1221no—
493C2×C4—1221no—
494C2×C4—1222no—
495C2×C4—1222no—
496C2×C4—1221no—
497C2×C4—1221no—
498C2×C4—1222no—
499C2×C4—1222no—
500C2×C4—1222no—
501C2×C4—1222no—
502C2×C4—1221no—
503C2×C4—1221no—
504C2×C4—2442no—
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×C4—2442no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4—2442no—
537C2×C4—2442no—
538C2×C4#1451985612yes—
539C22—612no—
540C22—612no—
541C22—612no—
542C22—612no—
543C22—612no—
544C22—612no—
545C22—612no—
546C22—612no—
547C22—612no—
548C22—612no—
549C22—612no—
550C22—612no—
551C22—611no—
552C22—611no—
553C22—611no—
554C22—611no—
555C22C2×#1451985611yescharacteristic
556C22#41×#692311yescharacteristic
557C22#41×#692311yescharacteristic
558C22#1684082911yescharacteristic
559C22#1684083011yescharacteristic
560C22#1684083111yescharacteristic
561C22#1684083111yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—41no—
566C4—41no—
567C4—1222no—
568C4—4882no—
569C4—4882no—
570C4—2444no—
571C4—2444no—
572C4—2442no—
573C4—2442no—
574C4—2441no—
575C4—2441no—
576C2—612no—
577C2—612no—
578C2—612no—
579C2—612no—
580C2#2146229611yescharacteristic
581C2#2146232011yescharacteristic
582C2#2146232011yescharacteristic
583C2#2146232911yescharacteristic
584C2#2146233011yescharacteristic
585C2#2146235411yescharacteristic
586C2#2146236511yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC61×#382C4
prime radical, cosocleC23×D61C24
hypercenter#382#6923
hypocenterC61#14477
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
29. #16839628C4semidirect
30. #16839627C4semidirect
31. #16839627C4semidirect
32. #16839626C4semidirect
33. #16839626C4semidirect
36. C61×#382C4semidirect
313. #382#6923semidirect
330. C61#14477semidirect

Maximal subgroups

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

subgroupindex
#214610062
#214610052
#214610042
#214609992
#214609982
#214609952
#214609942
#382×D612
#1447761