Finite Group Databasebeta

← #30600420

Subgroups — #30600420

Order 15616.

1088 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊#6923C211yesmaximal, characteristic, complemented
2#21461642C211yesmaximal, characteristic, complemented
3C2×#16840613C212yesmaximal, complemented
4#21460692C212yesmaximal, complemented
5#21454098C211yesmaximal, characteristic
6C22×C22⋊#6923—21no—
7C22×C4×#6923—21no—
8C2×#14519834—21no—
9C2×#14519834C2212yes—
10C2×C22⋊#92420—21no—
11C4×C22⋊#6923—24no—
12C2×(C2×#6923)⋊C4C2212yes—
13#16840613—24no—
14C23×C4×D61C2211yescharacteristic
15C2×#14519748C411yescharacteristic, complemented
16C2×#14519748C2211yescharacteristic
17C2×#14519745—21no—
18#16839642—21no—
19#16839638—21no—
20#16839638—21no—
21#16839622C411yescharacteristic, complemented
22#16839619C411yescharacteristic, complemented
23#16839619C2211yescharacteristic
24C2×#121×C61C411yescharacteristic, complemented
25C23×#6923—22no—
26C22×#92420—22no—
27C2×C22⋊#6923—24no—
28C2×C22⋊#6923—24no—
29C2×C22⋊#6923—24no—
30C2×C4×#6923—24no—
31C2×C4×#6923—42no—
32C2×C4×#6923—42no—
33#14519834—22no—
34#14519834—22no—
35#14519834—42no—
36C22⋊#92420—42no—
37(C2×#6923)⋊C4—22no—
38(C2×#6923)⋊C4D414yescomplemented
39C24×D61D411yescharacteristic
40C22×C22⋊D61—21no—
41C22×C22⋊D61—21no—
42C23×Dic61D411yescharacteristic
43C22×C4×D61—22no—
44C22×C4×D61—22no—
45C22×C4×D61—42no—
46C22×C4×D61—42no—
47C22×C4×D61C2×C411yescharacteristic
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51#14519748—22no—
52#14519748—22no—
53#14519748—42no—
54#14519745—44no—
55C2×C4⋊Dic61C2×C411yescharacteristic
56C2×C4⋊Dic61C2×C411yescharacteristic
57C2×C22⋊Dic61—21no—
58C2×C22⋊Dic61—21no—
59C2×C22⋊Dic61—21no—
60C2×#11758198—21no—
61C2×#11758198—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684—21no—
65C22×D4×C61—21no—
66C22×D4×C61C2×C411yescharacteristic
67C23×C4×C61C2×C411yescharacteristic
68#121×C61—42no—
69C2×C4⋊C4×C61C2×C411yescharacteristic
70C2×C22⋊C4×C61—21no—
71C24×C61—21no—
72C24×C61C22⋊C411yescharacteristic
73C23×D61—22no—
74C23×D61—22no—
75C23×D61—21no—
76C23×D61—21no—
77C23×D61—22no—
78C23×D61—22no—
79C23×D61—41no—
80C23×D61—41no—
81C23×D61C22⋊C411yescharacteristic
82C23×D61C2×D411yescharacteristic
83C22×#6923—24no—
84C22×#6923—24no—
85C22×#6923—24no—
86C22×#6923—42no—
87C22×#6923—42no—
88C22×#6923—42no—
89C22×#6923—42no—
90C2×D4×C61—22no—
91C2×D4×C61—41no—
92C2×D4×C61—41no—
93C2×D4×C61—42no—
94C2×D4×C61—42no—
95C2×D4×C61—42no—
96C22×C4×C61—22no—
97C22×C4×C61—22no—
98C22×C4×C61—21no—
99C22×C4×C61—41no—
100C22×C4×C61C22⋊C411yescharacteristic
101C22×C4×C61C22×C411yescharacteristic
102C2×C22⋊D61—42no—
103C2×C22⋊D61—42no—
104C2×C22⋊D61—42no—
105C2×C22⋊D61—41no—
106C2×C22⋊D61—41no—
107C2×C22⋊D61—42no—
108C2×C22⋊D61—42no—
109C22×Dic61—22no—
110C22×Dic61—22no—
111C22×Dic61—21no—
112C22×Dic61—21no—
113C22×Dic61—41no—
114C22×Dic61C22⋊C411yescharacteristic
115C22×Dic61C2×D411yescharacteristic
116C2×C4×D61—21no—
117C2×C4×D61—21no—
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—22no—
123C2×C4×D61—22no—
124C2×C4×D61—48no—
125C2×C4×D61—48no—
126C2×C4×D61—41no—
127C2×C4×D61—41no—
128C2×C4×D61—41no—
129C2×C4×D61—41no—
130C2×C4×D61C2×D412yes—
131C2×C4×D61#4112yes—
132C22⋊#6923—216no—
133C2×#92420—42no—
134C2×#92420—42no—
135C2×#92420—42no—
136C4×#6923—48no—
137C4⋊C4×C61—41no—
138C22⋊C4×C61—42no—
139C22⋊Dic61—42no—
140C22⋊Dic61—42no—
141C22⋊Dic61—42no—
142#11758198—42no—
143#11758198—42no—
144C4⋊Dic61—22no—
145C4⋊Dic61—41no—
146C23×C61—21no—
147C23×C61—21no—
148C23×C61—21no—
149C23×C61—21no—
150C23×C61—21no—
151C23×C61—21no—
152C23×C61—41no—
153C23×C61—41no—
154C23×C61—41no—
155C23×C61—41no—
156C23×C61—41no—
157C23×C61—41no—
158C23×C61—41no—
159C23×C61#10412yes—
160C23×C61C2×C22⋊C411yescharacteristic
161C22×D61—21no—
162C22×D61—21no—
163C22×D61—21no—
164C22×D61—21no—
165C22×D61—24no—
166C22×D61—24no—
167C22×D61—21no—
168C22×D61—21no—
169C22×D61—21no—
170C22×D61—21no—
171C22×D61—21no—
172C22×D61—21no—
173C22×D61—44no—
174C22×D61—44no—
175C22×D61—44no—
176C22×D61—44no—
177C22×D61—42no—
178C22×D61—42no—
179C22×D61—42no—
180C22×D61—42no—
181C22×D61#10412yes—
182C22×D61#12011yescharacteristic
183C22×D61#12311yescharacteristic
184C22×D61#13512yes—
185C22×D61AGL(1,ℤ/8)12yes—
186C2×#6923—28no—
187C2×#6923—44no—
188C2×#6923—44no—
189C2×#6923—44no—
190C2×#6923—44no—
191C2×#6923—44no—
192C2×#6923—44no—
193D4×C61—82no—
194D4×C61—84no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—44no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61#9912yes—
204C2×C4×C61C4×D412yes—
205C22⋊D61—84no—
206C22⋊D61—82no—
207C22⋊D61—82no—
208C2×Dic61—21no—
209C2×Dic61—21no—
210C2×Dic61—21no—
211C2×Dic61—21no—
212C2×Dic61—44no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61#9912yes—
220C2×Dic61#12112yes—
221C4×D61—24no—
222C4×D61—24no—
223C4×D61—42no—
224C4×D61—42no—
225C4×D61—44no—
226C4×D61—44no—
227#92420—82no—
228#9150—611nomaximal, Sylow
229C22×C61—22no—
230C22×C61—22no—
231C22×C61—22no—
232C22×C61—41no—
233C22×C61—41no—
234C22×C61—41no—
235C22×C61—41no—
236C22×C61—81no—
237C22×C61—81no—
238C22×C61—81no—
239C22×C61—81no—
240C22×C61—81no—
241C22×C61—81no—
242C22×C61—81no—
243C22×C61#38611yescharacteristic
244C22×C61C2×#9911yescharacteristic
245C22×C61C2×#9911yescharacteristic
246C22×C61#41611yescharacteristic
247C22×C61#41811yescharacteristic
248C22×C61C2×#10411yescharacteristic
249C22×C61C2×#10411yescharacteristic
250C2×D61—21no—
251C2×D61—21no—
252C2×D61—21no—
253C2×D61—21no—
254C2×D61—22no—
255C2×D61—22no—
256C2×D61—22no—
257C2×D61—22no—
258C2×D61—22no—
259C2×D61—22no—
260C2×D61—22no—
261C2×D61—48no—
262C2×D61—48no—
263C2×D61—41no—
264C2×D61—41no—
265C2×D61—41no—
266C2×D61—41no—
267C2×D61#44912yes—
268C2×D61#48212yes—
269#6923—48no—
270C4×C61—22no—
271C4×C61—22no—
272C4×C61—81no—
273Dic61—22no—
274Dic61—22no—
275Dic61—81no—
276Dic61—81no—
277#2847—611no—
278C2×C4×C22⋊C4—611no—
279#1779—611no—
280C2×#327—612no—
281#1294—612no—
282C2×C61—42no—
283C2×C61—161no—
284C2×C61#150111yescharacteristic
285C2×C61#150111yescharacteristic
286C2×C61#150711yescharacteristic
287C2×C61#150711yescharacteristic
288C2×C61#187511yescharacteristic
289C2×C61#189611yescharacteristic
290C2×C61#189811yescharacteristic
291D61—22no—
292D61—22no—
293D61—42no—
294C24×C4—611no—
295C2×#121—611no—
296C2×#121—611no—
297C2×#121#692311yescharacteristic, complemented
298C2×#120—1221no—
299C22×C22⋊C4—1221no—
300C22×C42—1221no—
301C2×C22⋊C8—612no—
302C2×C22⋊C8—1221no—
303C2×C22⋊C8—1221no—
304#386—1221no—
305#386—1221no—
306#386—1221no—
307#380—611no—
308#380—611no—
309#380—611no—
310C4×C22⋊C4—1224no—
311C2×#95—612no—
312#327—1224no—
313C61#915011yescharacteristic, complemented, Sylow
314C25—611no—
315C22×D4—21no—
316C22×D4—1221no—
317C22×D4—1221no—
318C22×D4C2×#692311yescharacteristic
319C23×C4—611no—
320C23×C4—611no—
321C23×C4—611no—
322C23×C4—611no—
323C23×C4—611no—
324C23×C4—1222no—
325C23×C4—1222no—
326C23×C4—1222no—
327C23×C4—2442no—
328C23×C4—2442no—
329C23×C4C2×#692311yescharacteristic
330C22×C8—1222no—
331#121—42no—
332#121—1222no—
333#121—1222no—
334#121—2442no—
335#120—2444no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4C2×#692311yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—1224no—
341C2×C22⋊C4—1224no—
342C2×C22⋊C4—1224no—
343C2×C22⋊C4—1221no—
344C2×C22⋊C4—1221no—
345C2×C22⋊C4—1221no—
346C2×C22⋊C4—1221no—
347C2×C22⋊C4—1221no—
348C2×C42—1224no—
349C2×C42—2442no—
350C2×C42—2442no—
351C22⋊C8—1222no—
352C22⋊C8—1222no—
353C22⋊C8—2442no—
354C22⋊C8—2442no—
355#95—614no—
356#95—1221no—
357#95—1221no—
358#95—1222no—
359#95—1221no—
360C24—21no—
361C24—611no—
362C24—611no—
363C24—1221no—
364C24—1222no—
365C24—1221no—
366C24—1222no—
367C24—1222no—
368C24—1222no—
369C24—2441no—
370C24—2441no—
371C24C22⋊#692311yescharacteristic
372C2×D4—22no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—2442no—
379C2×D4—2441no—
380C2×D4—2441no—
381C2×D4—2442no—
382C2×D4—2442no—
383C2×D4—2442no—
384C2×D4—2442no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—41no—
389C22×C4—611no—
390C22×C4—611no—
391C22×C4—612no—
392C22×C4—612no—
393C22×C4—1222no—
394C22×C4—1222no—
395C22×C4—1222no—
396C22×C4—1221no—
397C22×C4—1221no—
398C22×C4—1222no—
399C22×C4—1222no—
400C22×C4—1222no—
401C22×C4—1224no—
402C22×C4—1224no—
403C22×C4—1224no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1221no—
407C22×C4—1221no—
408C22×C4—2441no—
409C22×C4—2441no—
410C22×C4—2441no—
411C22×C4—2441no—
412C22×C4—2448no—
413C22×C4—2448no—
414C22×C4—2441no—
415C22×C4—2442no—
416C22×C4—2442no—
417C22×C4—2442no—
418C22×C4—2442no—
419C22×C4C22⋊#692311yescharacteristic
420C22×C4C22×#692311yescharacteristic
421C2×C8—2442no—
422C2×C8—2442no—
423C2×C8—2442no—
424C4⋊C4—41no—
425C4⋊C4—1222no—
426C4⋊C4—2441no—
427C22⋊C4—42no—
428C22⋊C4—12216no—
429C22⋊C4—2442no—
430C22⋊C4—2442no—
431C22⋊C4—2442no—
432C22⋊C4—2442no—
433C22⋊C4—2442no—
434C42—2448no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—611no—
449C23—611no—
450C23—612no—
451C23—612no—
452C23—612no—
453C23—1224no—
454C23—1224no—
455C23—1221no—
456C23—1221no—
457C23—1221no—
458C23—1221no—
459C23—1221no—
460C23—1221no—
461C23—1221no—
462C23—1221no—
463C23—1221no—
464C23—1221no—
465C23—2444no—
466C23—2444no—
467C23—2442no—
468C23—2442no—
469C23—2444no—
470C23—2444no—
471C23—2442no—
472C23—2442no—
473C23#1451984112yes—
474C23C2×C22⋊#692311yescharacteristic
475D4—82no—
476D4—84no—
477D4—4882no—
478D4—4882no—
479D4—4884no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—44no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—612no—
489C2×C4—612no—
490C2×C4—1224no—
491C2×C4—1224no—
492C2×C4—1221no—
493C2×C4—1221no—
494C2×C4—1221no—
495C2×C4—1221no—
496C2×C4—1228no—
497C2×C4—2442no—
498C2×C4—2442no—
499C2×C4—2444no—
500C2×C4—2444no—
501C2×C4—2444no—
502C2×C4—2441no—
503C2×C4—2441no—
504C2×C4—2441no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2444no—
509C2×C4—2444no—
510C2×C4—2444no—
511C2×C4—2444no—
512C2×C4—2444no—
513C2×C4—2444no—
514C2×C4#1451983512yes—
515C2×C4D4×#692312yes—
516C8—4882no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—612no—
532C22—612no—
533C22—1221no—
534C22—1221no—
535C22—1221no—
536C22—1221no—
537C22—1222no—
538C22—1222no—
539C22—1222no—
540C22—1222no—
541C22—1222no—
542C22—1222no—
543C22—1222no—
544C22—2448no—
545C22—2448no—
546C22—2441no—
547C22—2441no—
548C22—2441no—
549C22—2441no—
550C22#1684071011yescharacteristic
551C22C2×#1451983511yescharacteristic
552C22D61⋊#9911yescharacteristic
553C22#1684075211yescharacteristic
554C22#1684075411yescharacteristic
555C22C2×#1451984111yescharacteristic
556C22D61⋊#10411yescharacteristic
557C4—22no—
558C4—22no—
559C4—81no—
560C4—1222no—
561C4—1222no—
562C4—4881no—
563C4—4881no—
564C4—2448no—
565C2—42no—
566C2—161no—
567C2—1222no—
568C2—1222no—
569C2—2442no—
570C2#2146119411yescharacteristic
571C2#2146119511yescharacteristic
572C2#2146121411yescharacteristic
573C2#2146121511yescharacteristic
574C2#2146180711yescharacteristic
575C2#2146183711yescharacteristic
576C2#2146184011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#121×C61C4
socleC23×C61C2×C22⋊C4
prime radical#21454098C2
cosocleC22×C4×D61C23
hypercenterC2×#121#6923
hypocenterC61#9150
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C4×C22⋊#6923C2semidirect
2. #21461642C2semidirect
3. C2×#16840613C2semidirect
4. #21460692C2semidirect
15. C2×#14519748C4semidirect
21. #16839622C4semidirect
22. #16839619C4semidirect
24. C2×#121×C61C4semidirect
38. (C2×#6923)⋊C4D4semidirect
297. C2×#121#6923semidirect
313. C61#9150semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊#69232
#214616422
C2×#168406132
#214606922
#214540982
#915061