Finite Group Databasebeta

← #30600326

Subgroups — #30600326

Order 15616.

952 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 108 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×#16840734C211yesmaximal, characteristic
2C2×#16840616C212yesmaximal, complemented
3#21460993C211yesmaximal, characteristic
4#21460692C212yesmaximal, complemented
5#21454103C211yesmaximal, characteristic
6C22×#11758217—21no—
7C22×C22⋊#6923—21no—
8#16840734—24no—
9C2×#14519834—21no—
10C2×#14519834C2212yes—
11C2×(C2×#6923)⋊C4—21no—
12C2×(C2×#6923)⋊C4C2212yes—
13#16840616—24no—
14C23×C4×D61C2211yescharacteristic
15C2×#14519753C411yescharacteristic, complemented
16C2×#14519753C2211yescharacteristic
17#16839634—21no—
18#16839633—21no—
19#16839633—21no—
20#16839622C411yescharacteristic, complemented
21#16839619C411yescharacteristic, complemented
22#16839619C2211yescharacteristic
23#16839617—21no—
24C2×C22⋊Q8×C61C411yescharacteristic, complemented
25C23×#6923—22no—
26C2×#11758217—24no—
27C2×#11758217—42no—
28C2×#11758217—42no—
29C22×#92420—22no—
30C2×C22⋊#6923—24no—
31C2×C22⋊#6923—42no—
32C2×C22⋊#6923—42no—
33#14519834—24no—
34#14519834—22no—
35#14519834D414yes—
36(C2×#6923)⋊C4—22no—
37(C2×#6923)⋊C4—22no—
38(C2×#6923)⋊C4—42no—
39C24×D61D411yescharacteristic
40C23×Dic61D411yescharacteristic
41C22×C4×D61—22no—
42C22×C4×D61—22no—
43C22×C4×D61—42no—
44C22×C4×D61—42no—
45C22×C4×D61C2×C411yescharacteristic
46C22×C4×D61D411yescharacteristic
47C22×C4×D61D411yescharacteristic
48C22×C4×D61C2311yescharacteristic
49#14519753—22no—
50#14519753—22no—
51#14519753—42no—
52C2×C4⋊Dic61C2×C411yescharacteristic
53C2×C4⋊Dic61C2×C411yescharacteristic
54C2×#11758196—21no—
55C2×#11758196—21no—
56C2×C22⋊Dic61—21no—
57C2×#11758198—21no—
58C2×#11758198—21no—
59C2×#11758198—21no—
60C2×#11758198—21no—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64C22×Q8×C61C2×C411yescharacteristic
65C23×C4×C61C2×C411yescharacteristic
66C22⋊Q8×C61—42no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61C2×C411yescharacteristic
69C2×C22⋊C4×C61—21no—
70C24×C61C22⋊C411yescharacteristic
71C23×D61—22no—
72C23×D61—22no—
73C23×D61—21no—
74C23×D61—21no—
75C23×D61—22no—
76C23×D61—22no—
77C23×D61—41no—
78C23×D61—41no—
79C23×D61C22⋊C411yescharacteristic
80C23×D61C2×D411yescharacteristic
81C22×#6923—24no—
82C22×#6923—42no—
83C22×#6923—42no—
84C22×#6923—42no—
85C22×#6923—42no—
86C22×#6923—42no—
87C22×#6923—42no—
88C2×Q8×C61—22no—
89C2×Q8×C61—41no—
90C2×Q8×C61—41no—
91C22×C4×C61—22no—
92C22×C4×C61—22no—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61—41no—
96C22×C4×C61C22⋊C411yescharacteristic
97C22×C4×C61C22×C411yescharacteristic
98C22×Dic61—22no—
99C22×Dic61—22no—
100C22×Dic61—21no—
101C22×Dic61—21no—
102C22×Dic61—21no—
103C22×Dic61—21no—
104C22×Dic61C22⋊C411yescharacteristic
105C22×Dic61C2×D411yescharacteristic
106C2×C4×D61—22no—
107C2×C4×D61—22no—
108C2×C4×D61—22no—
109C2×C4×D61—22no—
110C2×C4×D61—21no—
111C2×C4×D61—21no—
112C2×C4×D61—41no—
113C2×C4×D61—41no—
114C2×C4×D61—41no—
115C2×C4×D61—41no—
116C2×C4×D61—41no—
117C2×C4×D61—41no—
118C2×C4×D61—48no—
119C2×C4×D61—48no—
120C2×C4×D61C2×D412yes—
121C2×C4×D61#4112yes—
122C22⋊#6923—48no—
123#11758217—48no—
124C2×#92420—24no—
125C2×#92420—42no—
126C2×#92420—42no—
127C4⋊C4×C61—42no—
128C4⋊C4×C61—41no—
129C22⋊C4×C61—42no—
130C22⋊Dic61—42no—
131#11758198—42no—
132#11758198—42no—
133#11758198—42no—
134#11758198—41no—
135#11758198—41no—
136C4⋊Dic61—22no—
137C4⋊Dic61—41no—
138#11758196—42no—
139#11758196—41no—
140#11758196—41no—
141C23×C61—21no—
142C23×C61—21no—
143C23×C61—21no—
144C23×C61—21no—
145C23×C61—21no—
146C23×C61—21no—
147C23×C61#9912yes—
148C23×C61C2×C22⋊C411yescharacteristic
149C22×D61—21no—
150C22×D61—21no—
151C22×D61—21no—
152C22×D61—21no—
153C22×D61—24no—
154C22×D61—24no—
155C22×D61—21no—
156C22×D61—21no—
157C22×D61—21no—
158C22×D61—21no—
159C22×D61—21no—
160C22×D61—21no—
161C22×D61—44no—
162C22×D61—44no—
163C22×D61—44no—
164C22×D61—44no—
165C22×D61—42no—
166C22×D61—42no—
167C22×D61—42no—
168C22×D61—42no—
169C22×D61#10412yes—
170C22×D61#12112yes—
171C22×D61AGL(1,ℤ/8)11yescharacteristic
172C22×D61#13712yes—
173C22×D61#13711yescharacteristic
174C2×#6923—44no—
175C2×#6923—44no—
176C2×#6923—44no—
177C2×#6923—82no—
178C2×#6923—82no—
179C2×#6923—82no—
180C2×#6923—82no—
181Q8×C61—82no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—44no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61C4×D412yes—
195C2×Dic61—24no—
196C2×Dic61—24no—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61#9912yes—
208C2×Dic61#12011yescharacteristic
209C2×Dic61#12311yescharacteristic
210C4×D61—42no—
211C4×D61—42no—
212C4×D61—44no—
213C4×D61—44no—
214C4×D61—42no—
215C4×D61—42no—
216#92420—44no—
217#9115—611nomaximal, Sylow
218C22×C61—22no—
219C22×C61—22no—
220C22×C61—22no—
221C22×C61—41no—
222C22×C61—41no—
223C22×C61—41no—
224C22×C61—41no—
225C22×C61#38611yescharacteristic
226C22×C61C2×#9911yescharacteristic
227C22×C61C2×#9911yescharacteristic
228C22×C61#41711yescharacteristic
229C22×C61#41911yescharacteristic
230C22×C61C2×#10411yescharacteristic
231C22×C61#42111yescharacteristic
232C2×D61—22no—
233C2×D61—22no—
234C2×D61—21no—
235C2×D61—21no—
236C2×D61—22no—
237C2×D61—22no—
238C2×D61—21no—
239C2×D61—21no—
240C2×D61—22no—
241C2×D61—22no—
242C2×D61—22no—
243C2×D61—48no—
244C2×D61—48no—
245C2×D61—41no—
246C2×D61—41no—
247C2×D61—41no—
248C2×D61—41no—
249C2×D61#46812yes—
250C2×D61#47012yes—
251#6923—84no—
252C4×C61—22no—
253C4×C61—41no—
254C4×C61—81no—
255C4×C61—81no—
256Dic61—21no—
257Dic61—22no—
258Dic61—21no—
259Dic61—81no—
260Dic61—81no—
261C2×#407—611no—
262#2848—611no—
263C2×#328—612no—
264C22⋊#95—611no—
265#1294—612no—
266C2×C61—42no—
267C2×C61#150411yescharacteristic
268C2×C61#150811yescharacteristic
269C2×C61#151111yescharacteristic
270C2×C61#151311yescharacteristic
271C2×C61#187111yescharacteristic
272C2×C61#189411yescharacteristic
273C2×C61#189811yescharacteristic
274D61—22no—
275D61—22no—
276D61—42no—
277C24×C4—611no—
278C22×M16—1221no—
279C2×C22⋊Q8—611no—
280C2×C22⋊Q8—611no—
281C2×C22⋊Q8#692311yescharacteristic, complemented
282C22×C22⋊C4—1221no—
283#407—1224no—
284C2×C22⋊C8—612no—
285C2×C22⋊C8—1221no—
286#385—1221no—
287#385—1221no—
288#385—1221no—
289#380—611no—
290#380—611no—
291#380—611no—
292#379—1221no—
293C2×#95—612no—
294C2×#95—1221no—
295#328—1224no—
296C61#911511yescharacteristic, complemented, Sylow
297C25—611no—
298C22×Q8C2×#692311yescharacteristic
299C23×C4—611no—
300C23×C4—611no—
301C23×C4—611no—
302C23×C4—611no—
303C23×C4—611no—
304C23×C4—1222no—
305C23×C4—1222no—
306C23×C4—1222no—
307C23×C4—2442no—
308C23×C4—2442no—
309C23×C4C2×#692311yescharacteristic
310C2×M16—1224no—
311C2×M16—2442no—
312C2×M16—2442no—
313C22×C8—1222no—
314C22⋊Q8—42no—
315C22⋊Q8—1222no—
316C22⋊Q8—1222no—
317C22⋊Q8—2442no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—611no—
320C2×C4⋊C4—611no—
321C2×C4⋊C4—1221no—
322C2×C4⋊C4—1221no—
323C2×C4⋊C4C2×#692311yescharacteristic
324C2×C22⋊C4—21no—
325C2×C22⋊C4—1221no—
326C2×C22⋊C4—1221no—
327C2×C22⋊C4—1221no—
328C2×C22⋊C4—1221no—
329C2×C22⋊C4—1224no—
330C2×C22⋊C4—1221no—
331C2×C22⋊C4—2442no—
332C2×C22⋊C4—2442no—
333C22⋊C8—614no—
334C22⋊C8—1222no—
335C22⋊C8—1224no—
336#95—1221no—
337#95—1221no—
338#95—1222no—
339#95—1222no—
340#95—1221no—
341#95—2442no—
342C24—611no—
343C24—611no—
344C24—1222no—
345C24—1221no—
346C24—1221no—
347C24—1222no—
348C24—1222no—
349C24—1222no—
350C24—2441no—
351C24—2441no—
352C24C22⋊#692311yescharacteristic
353C2×Q8—22no—
354C2×Q8—41no—
355C2×Q8—41no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—41no—
361C22×C4—611no—
362C22×C4—611no—
363C22×C4—612no—
364C22×C4—612no—
365C22×C4—1222no—
366C22×C4—1221no—
367C22×C4—1222no—
368C22×C4—1221no—
369C22×C4—1222no—
370C22×C4—1221no—
371C22×C4—1222no—
372C22×C4—1221no—
373C22×C4—1222no—
374C22×C4—1222no—
375C22×C4—1224no—
376C22×C4—1221no—
377C22×C4—1221no—
378C22×C4—2441no—
379C22×C4—2441no—
380C22×C4—2441no—
381C22×C4—2441no—
382C22×C4—2441no—
383C22×C4—2441no—
384C22×C4—2448no—
385C22×C4—2448no—
386C22×C4—2442no—
387C22×C4—2442no—
388C22×C4—2442no—
389C22×C4—2442no—
390C22×C4—2442no—
391C22×C4—2442no—
392C22×C4C22⋊#692311yescharacteristic
393C22×C4C22×#692311yescharacteristic
394M16—2448no—
395C2×C8—1224no—
396C2×C8—2442no—
397C2×C8—2442no—
398C4⋊C4—41no—
399C4⋊C4—42no—
400C4⋊C4—1222no—
401C4⋊C4—2441no—
402C4⋊C4—2441no—
403C4⋊C4—2441no—
404C4⋊C4—2442no—
405C22⋊C4—42no—
406C22⋊C4—2441no—
407C22⋊C4—2441no—
408C22⋊C4—2442no—
409C22⋊C4—2442no—
410C22⋊C4—2442no—
411C22⋊C4—2442no—
412C22⋊C4—2448no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—612no—
420C23—611no—
421C23—611no—
422C23—612no—
423C23—612no—
424C23—1224no—
425C23—1224no—
426C23—1221no—
427C23—1221no—
428C23—1221no—
429C23—1221no—
430C23—1221no—
431C23—1221no—
432C23—1221no—
433C23—1221no—
434C23—1221no—
435C23—1221no—
436C23—2444no—
437C23—2444no—
438C23—2442no—
439C23—2442no—
440C23—2444no—
441C23—2444no—
442C23—2442no—
443C23—2442no—
444C23#1451983512yes—
445C23C2×C22⋊#692311yescharacteristic
446Q8—82no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—44no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—611no—
460C2×C4—611no—
461C2×C4—612no—
462C2×C4—1224no—
463C2×C4—1224no—
464C2×C4—1221no—
465C2×C4—1221no—
466C2×C4—1221no—
467C2×C4—1221no—
468C2×C4—4882no—
469C2×C4—4882no—
470C2×C4—4882no—
471C2×C4—4882no—
472C2×C4—2442no—
473C2×C4—2442no—
474C2×C4—2444no—
475C2×C4—2444no—
476C2×C4—2442no—
477C2×C4—2442no—
478C2×C4—2444no—
479C2×C4—2444no—
480C2×C4—2444no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4—2441no—
484C2×C4—2441no—
485C2×C4—2441no—
486C2×C4—2441no—
487C2×C4D4×#692312yes—
488C8—2444no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—612no—
497C22—612no—
498C22—1222no—
499C22—1222no—
500C22—1221no—
501C22—1221no—
502C22—1222no—
503C22—1222no—
504C22—1221no—
505C22—1221no—
506C22—1222no—
507C22—1222no—
508C22—1222no—
509C22—2448no—
510C22—2448no—
511C22—2441no—
512C22—2441no—
513C22—2441no—
514C22—2441no—
515C22#1684071011yescharacteristic
516C22C2×#1451983511yescharacteristic
517C22D61⋊#9911yescharacteristic
518C22#1684075311yescharacteristic
519C22#1684075511yescharacteristic
520C22D61⋊#10411yescharacteristic
521C22#1684076111yescharacteristic
522C4—22no—
523C4—41no—
524C4—81no—
525C4—81no—
526C4—1221no—
527C4—1222no—
528C4—1221no—
529C4—4884no—
530C4—4881no—
531C4—4881no—
532C2—42no—
533C2—1222no—
534C2—1222no—
535C2—2442no—
536C2#2146120511yescharacteristic
537C2#2146121711yescharacteristic
538C2#2146122911yescharacteristic
539C2#2146123711yescharacteristic
540C2#2146180111yescharacteristic
541C2#2146183411yescharacteristic
542C2#2146184011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×D61D4
cosocleC22×C4×D61C23
hypercenterC2×C22⋊Q8#6923
hypocenterC61#9115
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16840616C2semidirect
4. #21460692C2semidirect
15. C2×#14519753C4semidirect
20. #16839622C4semidirect
21. #16839619C4semidirect
24. C2×C22⋊Q8×C61C4semidirect
281. C2×C22⋊Q8#6923semidirect
296. C61#9115semidirect

Maximal subgroups

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

subgroupindex
C2×#168407342
C2×#168406162
#214609932
#214606922
#214541032
#911561