Finite Group Databasebeta

← #30600422

Subgroups — #30600422

Order 15616.

946 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 946 conjugacy classes of subgroups — too many to draw. Only its 116 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×#16840704C211yesmaximal, characteristic
2#21461642C211yesmaximal, characteristic
3C2×#16840615C212yesmaximal, complemented
4#21460692C212yesmaximal, complemented
5#21454107C211yesmaximal, characteristic
6C22×C4⋊#6923—21no—
7C22×C22⋊#6923—21no—
8C2×#14519834—21no—
9C2×#14519834C2212yes—
10C2×C22⋊#92420—21no—
11#16840704—24no—
12C2×(C2×#6923)⋊C4C2212yes—
13#16840615—24no—
14C23×C4×D61C2211yescharacteristic
15C2×#14519757C411yescharacteristic, complemented
16C2×#14519757C2211yescharacteristic
17C2×#14519756C411yescharacteristic, complemented
18#16839637—21no—
19#16839636—21no—
20#16839636—21no—
21#16839619C411yescharacteristic, complemented
22#16839619C2211yescharacteristic
23#16839617—21no—
24C61×#380C411yescharacteristic, complemented
25C23×#6923—22no—
26C22×#92420—22no—
27C2×C4⋊#6923—24no—
28C2×C4⋊#6923—42no—
29C2×C4⋊#6923—42no—
30C2×C22⋊#6923—24no—
31C2×C22⋊#6923—42no—
32C2×C22⋊#6923—42no—
33#14519834—22no—
34#14519834—22no—
35#14519834—42no—
36C22⋊#92420—42no—
37(C2×#6923)⋊C4—22no—
38(C2×#6923)⋊C4D414yes—
39C24×D61D411yescharacteristic
40C22×Dic122C2×C411yescharacteristic
41C23×Dic61D411yescharacteristic
42C22×C4×D61—22no—
43C22×C4×D61—22no—
44C22×C4×D61—42no—
45C22×C4×D61—42no—
46C22×C4×D61C2×C411yescharacteristic
47C22×C4×D61D411yescharacteristic
48C22×C4×D61D411yescharacteristic
49C22×C4×D61C2311yescharacteristic
50#14519757—22no—
51#14519757—22no—
52#14519757—42no—
53#14519756—42no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61C2×C411yescharacteristic
57C2×#11758196—21no—
58C2×C22⋊Dic61—21no—
59C2×#11758198—21no—
60C2×#11758198—21no—
61C2×#11758198—21no—
62C2×#11758198—21no—
63#14519684—21no—
64#14519684—21no—
65C23×C4×C61C2×C411yescharacteristic
66C2×C4⋊C4×C61C2×C411yescharacteristic
67C2×C4⋊C4×C61C2×C411yescharacteristic
68C2×C22⋊C4×C61—21no—
69#95×C61—21no—
70C24×C61C22⋊C411yescharacteristic
71C23×D61—21no—
72C23×D61—21no—
73C23×D61—21no—
74C23×D61—21no—
75C23×D61—41no—
76C23×D61—41no—
77C23×D61—41no—
78C23×D61—41no—
79C23×D61C22⋊C411yescharacteristic
80C23×D61SD1612yes—
81C23×D61Q1612yes—
82C23×D61C2×D411yescharacteristic
83C22×#6923—24no—
84C22×#6923—42no—
85C22×#6923—42no—
86C22×#6923—42no—
87C22×#6923—42no—
88C22×#6923—42no—
89C22×#6923—42no—
90C22×C4×C61—22no—
91C22×C4×C61—22no—
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—41no—
95C22×C4×C61C22⋊C411yescharacteristic
96C22×C4×C61C22×C411yescharacteristic
97C22×Dic61—22no—
98C22×Dic61—22no—
99C22×Dic61—21no—
100C22×Dic61—21no—
101C22×Dic61—41no—
102C22×Dic61C22⋊C411yescharacteristic
103C22×Dic61C2×D411yescharacteristic
104C2×C4×D61—21no—
105C2×C4×D61—21no—
106C2×C4×D61—22no—
107C2×C4×D61—22no—
108C2×C4×D61—22no—
109C2×C4×D61—22no—
110C2×C4×D61—22no—
111C2×C4×D61—22no—
112C2×C4×D61—48no—
113C2×C4×D61—48no—
114C2×C4×D61—41no—
115C2×C4×D61—41no—
116C2×C4×D61—41no—
117C2×C4×D61—41no—
118C2×C4×D61C2×D412yes—
119C2×C4×D61#4112yes—
120C2×Dic122—22no—
121C2×Dic122—41no—
122C2×Dic122—41no—
123C22⋊#6923—48no—
124C2×#92420—42no—
125C2×#92420—42no—
126C2×#92420—42no—
127C4⋊#6923—48no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—41no—
130C22⋊C4×C61—42no—
131C22⋊Dic61—42no—
132#11758198—42no—
133#11758198—42no—
134#11758198—41no—
135#11758198—41no—
136#11758198—41no—
137#11758198—41no—
138C4⋊Dic61—42no—
139C4⋊Dic61—42no—
140C4⋊Dic61—41no—
141#11758196—42no—
142C23×C61—21no—
143C23×C61—21no—
144C23×C61—21no—
145C23×C61—21no—
146C23×C61—21no—
147C23×C61—21no—
148C23×C61#10312yes—
149C23×C61C2×C22⋊C411yescharacteristic
150C22×D61—21no—
151C22×D61—21no—
152C22×D61—21no—
153C22×D61—21no—
154C22×D61—21no—
155C22×D61—21no—
156C22×D61—21no—
157C22×D61—21no—
158C22×D61—21no—
159C22×D61—21no—
160C22×D61—22no—
161C22×D61—22no—
162C22×D61—22no—
163C22×D61—22no—
164C22×D61—42no—
165C22×D61—42no—
166C22×D61—42no—
167C22×D61—42no—
168C22×D61—42no—
169C22×D61—42no—
170C22×D61—42no—
171C22×D61—42no—
172C22×D61—42no—
173C22×D61—42no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61#10312yes—
177C22×D61#12011yescharacteristic
178C22×D61#12311yescharacteristic
179C22×D61C2×SD1611yescharacteristic
180C22×D61C2×Q1611yescharacteristic
181C22×D61AGL(1,ℤ/8)11yescharacteristic
182C22×D61#13711yescharacteristic
183C2×#6923—44no—
184C2×#6923—44no—
185C2×#6923—44no—
186C2×#6923—82no—
187C2×#6923—82no—
188C2×#6923—82no—
189C2×#6923—82no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—44no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61#9912yes—
202C2×C4×C61C4×D412yes—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—44no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61#9912yes—
215C2×Dic61#12112yes—
216C4×D61—24no—
217C4×D61—24no—
218C4×D61—42no—
219C4×D61—42no—
220C4×D61—44no—
221C4×D61—44no—
222Dic122—82no—
223#92420—82no—
224#9151—611nomaximal, Sylow
225C22×C61—22no—
226C22×C61—22no—
227C22×C61—22no—
228C22×C61—41no—
229C22×C61—41no—
230C22×C61—41no—
231C22×C61—41no—
232C22×C61#38611yescharacteristic
233C22×C61C2×#9911yescharacteristic
234C22×C61C2×#9911yescharacteristic
235C22×C61C2×#10311yescharacteristic
236C22×C61#41711yescharacteristic
237C22×C61#42111yescharacteristic
238C22×C61#42111yescharacteristic
239C2×D61—21no—
240C2×D61—21no—
241C2×D61—21no—
242C2×D61—21no—
243C2×D61—21no—
244C2×D61—21no—
245C2×D61—21no—
246C2×D61—21no—
247C2×D61—21no—
248C2×D61—21no—
249C2×D61—21no—
250C2×D61—21no—
251C2×D61—22no—
252C2×D61—22no—
253C2×D61—22no—
254C2×D61—44no—
255C2×D61—44no—
256C2×D61—44no—
257C2×D61—44no—
258C2×D61—41no—
259C2×D61—41no—
260C2×D61—41no—
261C2×D61—41no—
262C2×D61#45011yescharacteristic
263C2×D61#45111yescharacteristic
264C2×D61#48311yescharacteristic
265C2×D61#48411yescharacteristic
266#6923—84no—
267C4×C61—22no—
268C4×C61—22no—
269C4×C61—81no—
270C4×C61—81no—
271Dic61—22no—
272Dic61—22no—
273Dic61—81no—
274Dic61—81no—
275#2848—611no—
276C2×#380—611no—
277#1779—611no—
278C2×#328—612no—
279#1294—612no—
280C2×C61—42no—
281C2×C61#151311yescharacteristic
282C2×C61#151311yescharacteristic
283C2×C61#152011yescharacteristic
284C2×C61#152011yescharacteristic
285C2×C61#187211yescharacteristic
286C2×C61#189711yescharacteristic
287C2×C61#189811yescharacteristic
288D61—21no—
289D61—21no—
290D61—21no—
291D61—21no—
292D61—42no—
293C24×C4—611no—
294C2×C22⋊Q8—611no—
295C2×C22⋊Q8—611no—
296C2×C22⋊Q8—611no—
297C22×C4⋊C4—1221no—
298C22×C22⋊C4—1221no—
299C2×C22⋊C8—612no—
300C2×C22⋊C8—1221no—
301C2×C22⋊C8—1221no—
302#385—1221no—
303#385—1221no—
304#385—1221no—
305#380—611no—
306#380—611no—
307#380—1224no—
308#380#692311yescharacteristic, complemented
309#379—1221no—
310C2×#95—612no—
311#328—1224no—
312C61#915111yescharacteristic, complemented, Sylow
313C25—611no—
314C22×Q8—611no—
315C23×C4—611no—
316C23×C4—611no—
317C23×C4—611no—
318C23×C4—611no—
319C23×C4—611no—
320C23×C4—1222no—
321C23×C4—1222no—
322C23×C4—1222no—
323C23×C4—2442no—
324C23×C4—2442no—
325C23×C4C2×#692311yescharacteristic
326C22×C8—1222no—
327C22⋊Q8—1222no—
328C22⋊Q8—1222no—
329C22⋊Q8—2442no—
330C22⋊Q8—2442no—
331C2×C4⋊C4—611no—
332C2×C4⋊C4—1221no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4—1224no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—2442no—
337C2×C4⋊C4—2442no—
338C2×C4⋊C4C2×#692311yescharacteristic
339C2×C4⋊C4C2×#692311yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—1221no—
342C2×C22⋊C4—1221no—
343C2×C22⋊C4—1221no—
344C2×C22⋊C4—1221no—
345C2×C22⋊C4—1224no—
346C2×C22⋊C4—1221no—
347C2×C22⋊C4—2442no—
348C2×C22⋊C4—2442no—
349C22⋊C8—1222no—
350C22⋊C8—1222no—
351C22⋊C8—2442no—
352C22⋊C8—2442no—
353#95—21no—
354#95—614no—
355#95—1221no—
356#95—1221no—
357#95—1222no—
358C24—611no—
359C24—611no—
360C24—612no—
361C24—612no—
362C24—1221no—
363C24—1221no—
364C24—1221no—
365C24—1221no—
366C24—2441no—
367C24—2441no—
368C24—2441no—
369C24—2441no—
370C24C22⋊#692311yescharacteristic
371C2×Q8—1222no—
372C2×Q8—2441no—
373C2×Q8—2441no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—41no—
379C22×C4—611no—
380C22×C4—611no—
381C22×C4—612no—
382C22×C4—612no—
383C22×C4—1222no—
384C22×C4—1222no—
385C22×C4—1222no—
386C22×C4—1221no—
387C22×C4—1221no—
388C22×C4—1222no—
389C22×C4—1222no—
390C22×C4—1222no—
391C22×C4—1222no—
392C22×C4—1222no—
393C22×C4—1224no—
394C22×C4—1221no—
395C22×C4—1221no—
396C22×C4—2441no—
397C22×C4—2441no—
398C22×C4—2441no—
399C22×C4—2441no—
400C22×C4—2448no—
401C22×C4—2448no—
402C22×C4—2441no—
403C22×C4—2442no—
404C22×C4—2442no—
405C22×C4—2442no—
406C22×C4—2442no—
407C22×C4—2442no—
408C22×C4—2442no—
409C22×C4C22⋊#692311yescharacteristic
410C22×C4C22×#692311yescharacteristic
411C2×C8—2442no—
412C2×C8—2442no—
413C2×C8—2442no—
414C4⋊C4—22no—
415C4⋊C4—41no—
416C4⋊C4—2441no—
417C4⋊C4—2442no—
418C4⋊C4—2442no—
419C4⋊C4—2448no—
420C4⋊C4—2442no—
421C22⋊C4—42no—
422C22⋊C4—2441no—
423C22⋊C4—2441no—
424C22⋊C4—2441no—
425C22⋊C4—2441no—
426C22⋊C4—2442no—
427C22⋊C4—2442no—
428C22⋊C4—2442no—
429C22⋊C4—2448no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—611no—
437C23—611no—
438C23—611no—
439C23—611no—
440C23—611no—
441C23—611no—
442C23—612no—
443C23—1222no—
444C23—1222no—
445C23—1222no—
446C23—1222no—
447C23—1221no—
448C23—1221no—
449C23—1221no—
450C23—1221no—
451C23—1221no—
452C23—1221no—
453C23—1221no—
454C23—1221no—
455C23—1221no—
456C23—1221no—
457C23—2442no—
458C23—2442no—
459C23—2442no—
460C23—2442no—
461C23—2442no—
462C23—2442no—
463C23—2442no—
464C23—2442no—
465C23—2442no—
466C23—2442no—
467C23—2442no—
468C23—2442no—
469C23#1451984012yes—
470C23C2×C22⋊#692311yescharacteristic
471Q8—4882no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—44no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—612no—
484C2×C4—612no—
485C2×C4—1224no—
486C2×C4—1224no—
487C2×C4—1221no—
488C2×C4—1221no—
489C2×C4—1221no—
490C2×C4—1221no—
491C2×C4—4882no—
492C2×C4—4882no—
493C2×C4—4882no—
494C2×C4—4882no—
495C2×C4—2442no—
496C2×C4—2442no—
497C2×C4—2444no—
498C2×C4—2444no—
499C2×C4—2444no—
500C2×C4—2444no—
501C2×C4—2444no—
502C2×C4—2444no—
503C2×C4—2441no—
504C2×C4—2441no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4#1451983612yes—
510C2×C4D4×#692312yes—
511C8—4882no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—611no—
520C22—611no—
521C22—611no—
522C22—611no—
523C22—1221no—
524C22—1221no—
525C22—1221no—
526C22—1221no—
527C22—1221no—
528C22—1221no—
529C22—1221no—
530C22—1221no—
531C22—1221no—
532C22—1221no—
533C22—1221no—
534C22—1221no—
535C22—1222no—
536C22—1222no—
537C22—1222no—
538C22—2444no—
539C22—2444no—
540C22—2444no—
541C22—2444no—
542C22—2441no—
543C22—2441no—
544C22—2441no—
545C22—2441no—
546C22#1684071011yescharacteristic
547C22D61⋊#9911yescharacteristic
548C22C2×#1451983611yescharacteristic
549C22C2×#1451984011yescharacteristic
550C22#1684075311yescharacteristic
551C22#1684075911yescharacteristic
552C22#1684076011yescharacteristic
553C4—22no—
554C4—22no—
555C4—81no—
556C4—81no—
557C4—1222no—
558C4—1222no—
559C4—4884no—
560C4—4881no—
561C4—4881no—
562C2—42no—
563C2—1221no—
564C2—1221no—
565C2—1221no—
566C2—1221no—
567C2—2442no—
568C2#2146123511yescharacteristic
569C2#2146123611yescharacteristic
570C2#2146125711yescharacteristic
571C2#2146125811yescharacteristic
572C2#2146180411yescharacteristic
573C2#2146183911yescharacteristic
574C2#2146184111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#380C4
socleC23×C61C2×C22⋊C4
prime radicalC24×D61D4
cosocleC22×C4×D61C23
hypercenter#380#6923
hypocenterC61#9151
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. C2×#16840615C2semidirect
4. #21460692C2semidirect
15. C2×#14519757C4semidirect
17. C2×#14519756C4semidirect
21. #16839619C4semidirect
24. C61×#380C4semidirect
308. #380#6923semidirect
312. C61#9151semidirect

Maximal subgroups

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

subgroupindex
C2×#168407042
#214616422
C2×#168406152
#214606922
#214541072
#915161