Finite Group Databasebeta

← #30600320

Subgroups — #30600320

Order 15616.

1062 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 122 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×D61⋊(C22⋊C8)C211yesmaximal, characteristic, complemented
2C2×#16840613C212yesmaximal, complemented
3#21460993C211yesmaximal, characteristic, complemented
4#21460692C212yesmaximal, complemented
5#21454098C211yesmaximal, characteristic
6C22×D61⋊C8—21no—
7C22×C22⋊#6923—21no—
8C2×#14519834C2212yes—
9D61⋊(C22⋊C8)—24no—
10C2×C22⋊#92420—21no—
11C2×(C2×#6923)⋊C4—21no—
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—
26C2×D61⋊C8—24no—
27C2×D61⋊C8—42no—
28C2×D61⋊C8—42no—
29C22×#92420—22no—
30C2×C22⋊#6923—24no—
31C2×C22⋊#6923—42no—
32C2×C22⋊#6923—42no—
33#14519834—22no—
34#14519834D414yescomplemented
35C22⋊#92420—24no—
36(C2×#6923)⋊C4—22no—
37(C2×#6923)⋊C4—22no—
38(C2×#6923)⋊C4—42no—
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—42no—
55#14519745—42no—
56C2×C4⋊Dic61C2×C411yescharacteristic
57C2×C4⋊Dic61C2×C411yescharacteristic
58C2×C22⋊Dic61—21no—
59C2×C22⋊Dic61—21no—
60C2×C22⋊Dic61—21no—
61C2×#11758198—21no—
62C2×#11758198—21no—
63#14519684—21no—
64#14519684—21no—
65#14519684—21no—
66C22×D4×C61—21no—
67C22×D4×C61C2×C411yescharacteristic
68C23×C4×C61C2×C411yescharacteristic
69#121×C61—42no—
70C2×C4⋊C4×C61C2×C411yescharacteristic
71C2×C22⋊C4×C61—21no—
72C24×C61—21no—
73C24×C61C22⋊C411yescharacteristic
74C23×D61—22no—
75C23×D61—22no—
76C23×D61—22no—
77C23×D61—22no—
78C23×D61—41no—
79C23×D61—41no—
80C23×D61C22⋊C411yescharacteristic
81C23×D61D812yes—
82C23×D61SD1612yes—
83C23×D61C2×D411yescharacteristic
84C22×#6923—24no—
85C22×#6923—42no—
86C22×#6923—42no—
87C22×#6923—42no—
88C22×#6923—42no—
89C22×#6923—42no—
90C22×#6923—42no—
91C2×D4×C61—22no—
92C2×D4×C61—41no—
93C2×D4×C61—41no—
94C2×D4×C61—42no—
95C2×D4×C61—42no—
96C2×D4×C61—42no—
97C22×C4×C61—22no—
98C22×C4×C61—22no—
99C22×C4×C61—21no—
100C22×C4×C61—41no—
101C22×C4×C61C22⋊C411yescharacteristic
102C22×C4×C61C22×C411yescharacteristic
103C2×C22⋊D61—42no—
104C2×C22⋊D61—42no—
105C2×C22⋊D61—42no—
106C2×C22⋊D61—42no—
107C2×C22⋊D61—41no—
108C2×C22⋊D61—41no—
109C2×C22⋊D61—42no—
110C22×Dic61—21no—
111C22×Dic61—21no—
112C22×Dic61—21no—
113C22×Dic61—21no—
114C22×Dic61—41no—
115C22×Dic61—41no—
116C22×Dic61C22⋊C411yescharacteristic
117C22×Dic61C2×D411yescharacteristic
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—21no—
121C2×C4×D61—21no—
122C2×C4×D61—22no—
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—48no—
127C2×C4×D61—48no—
128C2×C4×D61—41no—
129C2×C4×D61—41no—
130C2×C4×D61—41no—
131C2×C4×D61—41no—
132C2×C4×D61C2×D412yes—
133C2×C4×D61#4112yes—
134C22⋊#6923—48no—
135C2×#92420—24no—
136C2×#92420—42no—
137C2×#92420—42no—
138D61⋊C8—48no—
139C4⋊C4×C61—41no—
140C22⋊C4×C61—42no—
141C22⋊Dic61—42no—
142C22⋊Dic61—42no—
143C22⋊Dic61—41no—
144C22⋊Dic61—41no—
145#11758198—42no—
146#11758198—41no—
147#11758198—41no—
148C4⋊Dic61—22no—
149C4⋊Dic61—41no—
150C23×C61—21no—
151C23×C61—21no—
152C23×C61—21no—
153C23×C61—21no—
154C23×C61—21no—
155C23×C61—21no—
156C23×C61—41no—
157C23×C61—41no—
158C23×C61—41no—
159C23×C61—41no—
160C23×C61—41no—
161C23×C61—41no—
162C23×C61—41no—
163C23×C61#10212yes—
164C23×C61C2×C22⋊C411yescharacteristic
165C22×D61—21no—
166C22×D61—21no—
167C22×D61—21no—
168C22×D61—21no—
169C22×D61—21no—
170C22×D61—21no—
171C22×D61—21no—
172C22×D61—21no—
173C22×D61—24no—
174C22×D61—24no—
175C22×D61—24no—
176C22×D61—24no—
177C22×D61—42no—
178C22×D61—42no—
179C22×D61—44no—
180C22×D61—44no—
181C22×D61—44no—
182C22×D61—44no—
183C22×D61#9912yes—
184C22×D61#10212yes—
185C22×D61#10412yes—
186C22×D61#12112yes—
187C22×D61C2×D811yescharacteristic
188C22×D61C2×SD1611yescharacteristic
189C22×D61#13512yes—
190C2×#6923—44no—
191C2×#6923—44no—
192C2×#6923—44no—
193C2×#6923—82no—
194C2×#6923—82no—
195C2×#6923—82no—
196C2×#6923—82no—
197D4×C61—82no—
198D4×C61—84no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—44no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61#9912yes—
208C2×C4×C61C4×D412yes—
209C22⋊D61—84no—
210C22⋊D61—82no—
211C22⋊D61—82no—
212C2×Dic61—21no—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61—41no—
223C2×Dic61—41no—
224C2×Dic61—41no—
225C2×Dic61#12011yescharacteristic
226C2×Dic61#12311yescharacteristic
227C4×D61—24no—
228C4×D61—24no—
229C4×D61—42no—
230C4×D61—42no—
231C4×D61—44no—
232C4×D61—44no—
233#92420—44no—
234#9113—611nomaximal, Sylow
235C22×C61—22no—
236C22×C61—22no—
237C22×C61—22no—
238C22×C61—41no—
239C22×C61—41no—
240C22×C61—41no—
241C22×C61—41no—
242C22×C61—81no—
243C22×C61—81no—
244C22×C61—81no—
245C22×C61—81no—
246C22×C61—81no—
247C22×C61—81no—
248C22×C61—81no—
249C22×C61#38611yescharacteristic
250C22×C61C2×#9911yescharacteristic
251C22×C61C2×#9911yescharacteristic
252C22×C61C2×#10211yescharacteristic
253C22×C61#41611yescharacteristic
254C22×C61C2×#10411yescharacteristic
255C22×C61#42111yescharacteristic
256C2×D61—22no—
257C2×D61—22no—
258C2×D61—22no—
259C2×D61—22no—
260C2×D61—22no—
261C2×D61—21no—
262C2×D61—21no—
263C2×D61—22no—
264C2×D61—22no—
265C2×D61—22no—
266C2×D61—22no—
267C2×D61—21no—
268C2×D61—22no—
269C2×D61—21no—
270C2×D61—48no—
271C2×D61—48no—
272C2×D61#32714yescomplemented
273C2×D61#46512yes—
274C2×D61#46612yes—
275#6923—84no—
276C4×C61—22no—
277C4×C61—22no—
278C4×C61—81no—
279Dic61—21no—
280Dic61—21no—
281Dic61—41no—
282Dic61—81no—
283Dic61—81no—
284C22×C22⋊C8—611no—
285#2847—611no—
286C2×#327—612no—
287C22⋊#95—611no—
288#1294—612no—
289C2×C61—42no—
290C2×C61—161no—
291C2×C61C2×#32711yescharacteristic
292C2×C61#150111yescharacteristic
293C2×C61#151611yescharacteristic
294C2×C61#151911yescharacteristic
295C2×C61#187111yescharacteristic
296C2×C61#189311yescharacteristic
297C2×C61#189811yescharacteristic
298D61—22no—
299D61—24no—
300D61—22no—
301C24×C4—611no—
302C23×C8—1221no—
303C2×#121—611no—
304C2×#121—611no—
305C2×#121#692311yescharacteristic, complemented
306C2×#120—1221no—
307C22×C22⋊C4—1221no—
308C2×C22⋊C8—612no—
309C2×C22⋊C8—1224no—
310C2×C22⋊C8—1221no—
311#386—1221no—
312#386—1221no—
313#386—1221no—
314#380—611no—
315#380—611no—
316#380—611no—
317C2×#95—612no—
318C2×#95—1221no—
319#327—1224no—
320C61#911311yescharacteristic, complemented, Sylow
321C25—611no—
322C22×D4—21no—
323C22×D4—1221no—
324C22×D4—1221no—
325C22×D4C2×#692311yescharacteristic
326C23×C4—611no—
327C23×C4—611no—
328C23×C4—611no—
329C23×C4—611no—
330C23×C4—611no—
331C23×C4—1222no—
332C23×C4—1222no—
333C23×C4—1222no—
334C23×C4—2442no—
335C23×C4—2442no—
336C23×C4C2×#692311yescharacteristic
337C22×C8—1222no—
338C22×C8—1224no—
339C22×C8—2442no—
340C22×C8—2442no—
341#121—42no—
342#121—1222no—
343#121—1222no—
344#121—2442no—
345#120—2442no—
346#120—2442no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4C2×#692311yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1221no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1224no—
356C2×C22⋊C4—1221no—
357C2×C22⋊C4—2442no—
358C2×C22⋊C4—2442no—
359C22⋊C8—614no—
360C22⋊C8—1222no—
361C22⋊C8—1224no—
362#95—1221no—
363#95—1221no—
364#95—1222no—
365#95—1222no—
366#95—1221no—
367#95—2442no—
368C24—21no—
369C24—611no—
370C24—611no—
371C24—612no—
372C24—612no—
373C24—1222no—
374C24—1222no—
375C24—1222no—
376C24—1222no—
377C24—2441no—
378C24—2441no—
379C24C22⋊#692311yescharacteristic
380C2×D4—22no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—2442no—
387C2×D4—2441no—
388C2×D4—2441no—
389C2×D4—2442no—
390C2×D4—2442no—
391C2×D4—2442no—
392C2×D4—2442no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—41no—
397C22×C4—611no—
398C22×C4—612no—
399C22×C4—612no—
400C22×C4—611no—
401C22×C4—1221no—
402C22×C4—1221no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1221no—
407C22×C4—1221no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1224no—
412C22×C4—1221no—
413C22×C4—1221no—
414C22×C4—2441no—
415C22×C4—2441no—
416C22×C4—2441no—
417C22×C4—2441no—
418C22×C4—2448no—
419C22×C4—2448no—
420C22×C4—2441no—
421C22×C4—2441no—
422C22×C4—2442no—
423C22×C4—2442no—
424C22×C4—2442no—
425C22×C4—2442no—
426C22×C4—2442no—
427C22×C4—2442no—
428C22×C4C22⋊#692311yescharacteristic
429C22×C4C22×#692311yescharacteristic
430C2×C8—1224no—
431C2×C8—2448no—
432C2×C8—2442no—
433C2×C8—2442no—
434C4⋊C4—41no—
435C4⋊C4—1222no—
436C4⋊C4—2441no—
437C22⋊C4—42no—
438C22⋊C4—2442no—
439C22⋊C4—2441no—
440C22⋊C4—2441no—
441C22⋊C4—2441no—
442C22⋊C4—2441no—
443C22⋊C4—2442no—
444C22⋊C4—2448no—
445C22⋊C4—2442no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—612no—
460C23—612no—
461C23—612no—
462C23—611no—
463C23—611no—
464C23—612no—
465C23—612no—
466C23—1224no—
467C23—1224no—
468C23—1224no—
469C23—1224no—
470C23—1221no—
471C23—1221no—
472C23—1221no—
473C23—1221no—
474C23—1221no—
475C23—1221no—
476C23—1221no—
477C23—1221no—
478C23—2442no—
479C23—2442no—
480C23—2444no—
481C23—2444no—
482C23—2444no—
483C23—2444no—
484C23#1451983912yes—
485C23C2×C22⋊#692311yescharacteristic
486D4—82no—
487D4—84no—
488D4—4882no—
489D4—4882no—
490D4—4884no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—44no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—611no—
500C2×C4—611no—
501C2×C4—1224no—
502C2×C4—1224no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—4882no—
509C2×C4—4882no—
510C2×C4—4882no—
511C2×C4—4882no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4—2444no—
515C2×C4—2444no—
516C2×C4—2442no—
517C2×C4—2442no—
518C2×C4—2444no—
519C2×C4—2444no—
520C2×C4—2444no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4#1451983512yes—
528C2×C4D4×#692312yes—
529C8—2444no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—612no—
545C22—612no—
546C22—614no—
547C22—1221no—
548C22—1222no—
549C22—1221no—
550C22—1222no—
551C22—1222no—
552C22—1222no—
553C22—1222no—
554C22—1222no—
555C22—1221no—
556C22—1221no—
557C22—1222no—
558C22—1222no—
559C22—1222no—
560C22—1222no—
561C22—2448no—
562C22—2448no—
563C22#1684071011yescharacteristic
564C22C2×#1451983511yescharacteristic
565C22D61⋊#9911yescharacteristic
566C22C2×#1451983911yescharacteristic
567C22#1684075211yescharacteristic
568C22D61⋊#10411yescharacteristic
569C22#1684076011yescharacteristic
570C4—22no—
571C4—22no—
572C4—81no—
573C4—1221no—
574C4—1221no—
575C4—4884no—
576C4—4881no—
577C4—4881no—
578C4—2441no—
579C2—42no—
580C2—161no—
581C2—1222no—
582C2—1222no—
583C2—1224no—
584C2D61⋊#32711yescharacteristic
585C2#2146119411yescharacteristic
586C2#2146124411yescharacteristic
587C2#2146125511yescharacteristic
588C2#2146180311yescharacteristic
589C2#2146183311yescharacteristic
590C2#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#9113
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×D61⋊(C22⋊C8)C2semidirect
2. C2×#16840613C2semidirect
3. #21460993C2semidirect
4. #21460692C2semidirect
15. C2×#14519748C4semidirect
21. #16839622C4semidirect
22. #16839619C4semidirect
24. C2×#121×C61C4semidirect
34. #14519834D4semidirect
272. C2×D61#327semidirect
305. C2×#121#6923semidirect
320. C61#9113semidirect

Maximal subgroups

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

subgroupindex
C2×D61⋊(C22⋊C8)2
C2×#168406132
#214609932
#214606922
#214540982
#911361