Finite Group Databasebeta

← #30598215

Subgroups — #30598215

Order 15616.

1276 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461642C211yesmaximal, characteristic
2C2×#16840616C212yesmaximal, complemented
3C2×#16840615C212yesmaximal, complemented
4#21460993C211yesmaximal, characteristic
5C2×C22⋊Q8×D61C211yesmaximal, characteristic
6C22×C22⋊#6923—21no—
7C2×#14519834—21no—
8C2×#14519834C2212yes—
9C2×C22⋊#92420—21no—
10C2×(C2×#6923)⋊C4—21no—
11C2×(C2×#6923)⋊C4C2212yes—
12#16840616—24no—
13#16840615—24no—
14C22×Q8×D61C411yescharacteristic, complemented
15C23×C4×D61C2211yescharacteristic
16C2×#14519757C2211yescharacteristic
17C2×#14519756C411yescharacteristic, complemented
18C2×#14519755—21no—
19C2×#14519754—21no—
20C2×#14519753C2211yescharacteristic
21C22⋊Q8×D61—24no—
22C22⋊Q8×D61—24no—
23C2×C4⋊C4×D61—21no—
24C2×C4⋊C4×D61C411yescharacteristic, complemented
25C2×C22⋊C4×D61—21no—
26C2×C22⋊Q8×C61C411yescharacteristic, complemented
27C23×#6923—22no—
28C22×#92420—22no—
29C2×C22⋊#6923—24no—
30C2×C22⋊#6923—42no—
31C2×C22⋊#6923—42no—
32#14519834—22no—
33#14519834—22no—
34#14519834—42no—
35C22⋊#92420—42no—
36(C2×#6923)⋊C4—22no—
37(C2×#6923)⋊C4—22no—
38(C2×#6923)⋊C4—42no—
39C24×D61D411yescharacteristic
40C22×Dic122—21no—
41C22×Dic122C2×C411yescharacteristic
42C2×Q8×D61—22no—
43C2×Q8×D61—22no—
44C2×Q8×D61—42no—
45C2×Q8×D61—42no—
46C23×Dic61D411yescharacteristic
47C22×C4×D61—21no—
48C22×C4×D61—21no—
49C22×C4×D61—22no—
50C22×C4×D61—22no—
51C22×C4×D61—42no—
52C22×C4×D61—42no—
53C22×C4×D61C2×C411yescharacteristic
54C22×C4×D61D411yescharacteristic
55C22×C4×D61D411yescharacteristic
56C22×C4×D61C2311yescharacteristic
57#14519757—22no—
58#14519757D414yes—
59#14519756—22no—
60#14519756—22no—
61#14519755—28no—
62#14519754—28no—
63#14519753—22no—
64#14519753D414yes—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61C2×C411yescharacteristic
67C2×#11758196—21no—
68C2×#11758196—21no—
69C2×#11758196—21no—
70C4⋊C4×D61—28no—
71C4⋊C4×D61—22no—
72C4⋊C4×D61—22no—
73C2×C22⋊Dic61—21no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C22⋊C4×D61—28no—
77C22×Q8×C61C2×C411yescharacteristic
78C23×C4×C61C2×C411yescharacteristic
79C22⋊Q8×C61—22no—
80C22⋊Q8×C61—22no—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61C2×C411yescharacteristic
83C2×C22⋊C4×C61—21no—
84C24×C61C22⋊C411yescharacteristic
85C23×D61—22no—
86C23×D61—22no—
87C23×D61—22no—
88C23×D61—22no—
89C23×D61—41no—
90C23×D61—41no—
91C23×D61C22⋊C411yescharacteristic
92C23×D61SD1612yes—
93C23×D61Q1612yes—
94C23×D61C2×D411yescharacteristic
95C22×#6923—24no—
96C22×#6923—42no—
97C22×#6923—42no—
98C22×#6923—42no—
99C22×#6923—42no—
100C22×#6923—42no—
101C22×#6923—42no—
102C2×Q8×C61—21no—
103C2×Q8×C61—41no—
104C2×Q8×C61—41no—
105C2×Q8×C61C22⋊C412yescomplemented
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—41no—
111C22×C4×C61—41no—
112C22×C4×C61C22⋊C411yescharacteristic
113C22×C4×C61C22×C411yescharacteristic
114C22×Dic61—21no—
115C22×Dic61—21no—
116C22×Dic61—22no—
117C22×Dic61—22no—
118C22×Dic61—41no—
119C22×Dic61C22⋊C411yescharacteristic
120C22×Dic61C2×D411yescharacteristic
121Q8×D61—48no—
122Q8×D61—48no—
123C2×C4×D61—22no—
124C2×C4×D61—21no—
125C2×C4×D61—21no—
126C2×C4×D61—22no—
127C2×C4×D61—24no—
128C2×C4×D61—24no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—48no—
134C2×C4×D61—48no—
135C2×C4×D61—41no—
136C2×C4×D61—41no—
137C2×C4×D61—41no—
138C2×C4×D61—41no—
139C2×C4×D61—42no—
140C2×C4×D61—42no—
141C2×C4×D61—42no—
142C2×C4×D61—42no—
143C2×C4×D61C2×D412yes—
144C2×C4×D61C2×D412yes—
145C2×Dic122—24no—
146C2×Dic122—21no—
147C2×Dic122—42no—
148C2×Dic122—42no—
149C2×Dic122—41no—
150C2×Dic122—41no—
151C2×Dic122C22⋊C412yescomplemented
152C22⋊#6923—48no—
153C2×#92420—42no—
154C2×#92420—42no—
155C2×#92420—42no—
156C4⋊C4×C61—24no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61C22⋊C412yescomplemented
159C22⋊C4×C61—24no—
160C22⋊Dic61—24no—
161#11758198—24no—
162#11758198—24no—
163C4⋊Dic61—21no—
164C4⋊Dic61—24no—
165C4⋊Dic61C22⋊C412yescomplemented
166#11758196—24no—
167#11758196—24no—
168#11758196—24no—
169C23×C61—21no—
170C23×C61—21no—
171C23×C61—21no—
172C23×C61—21no—
173C23×C61—21no—
174C23×C61—21no—
175C23×C61#10312yes—
176C23×C61C2×C22⋊C411yescharacteristic
177C22×D61—21no—
178C22×D61—21no—
179C22×D61—21no—
180C22×D61—21no—
181C22×D61—21no—
182C22×D61—21no—
183C22×D61—21no—
184C22×D61—21no—
185C22×D61—24no—
186C22×D61—24no—
187C22×D61—24no—
188C22×D61—24no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—44no—
192C22×D61—44no—
193C22×D61—44no—
194C22×D61—44no—
195C22×D61#9912yes—
196C22×D61#10312yes—
197C22×D61#10412yes—
198C22×D61#12012yes—
199C22×D61C2×SD1611yescharacteristic
200C22×D61C2×Q1611yescharacteristic
201C22×D61#13712yes—
202C2×#6923—44no—
203C2×#6923—44no—
204C2×#6923—44no—
205C2×#6923—82no—
206C2×#6923—82no—
207C2×#6923—82no—
208C2×#6923—82no—
209Q8×C61—42no—
210Q8×C61—42no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61C2×C22⋊C411yescharacteristic
225C2×C4×C61C2×C22⋊C411yescharacteristic
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—44no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61#9912yes—
238C2×Dic61#12012yes—
239C4×D61—24no—
240C4×D61—24no—
241C4×D61—42no—
242C4×D61—42no—
243C4×D61—44no—
244C4×D61—44no—
245C4×D61—48no—
246C4×D61—48no—
247Dic122—48no—
248Dic122—42no—
249Dic122—42no—
250#92420—82no—
251#8329—611nomaximal, Sylow
252C22×C61—22no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61#37911yescharacteristic
260C22×C61C2×#9911yescharacteristic
261C22×C61C2×#9911yescharacteristic
262C22×C61C2×#10311yescharacteristic
263C22×C61#41911yescharacteristic
264C22×C61C2×#10411yescharacteristic
265C22×C61#42111yescharacteristic
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—22no—
269C2×D61—22no—
270C2×D61—22no—
271C2×D61—22no—
272C2×D61—21no—
273C2×D61—22no—
274C2×D61—21no—
275C2×D61—22no—
276C2×D61—22no—
277C2×D61—22no—
278C2×D61—22no—
279C2×D61—22no—
280C2×D61—48no—
281C2×D61—48no—
282C2×D61#32814yescomplemented
283C2×D61#44812yes—
284C2×D61#45112yes—
285#6923—84no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—42no—
289C4×C61—41no—
290C4×C61—42no—
291Dic61—22no—
292Dic61—22no—
293Dic61—42no—
294Dic61—42no—
295C22×C22⋊Q8—611no—
296#1779—611no—
297C2×#328—612no—
298C2×#328—612no—
299C22⋊#95—611no—
300C2×C61—42no—
301C2×C61C2×#32811yescharacteristic
302C2×C61#150811yescharacteristic
303C2×C61#151111yescharacteristic
304C2×C61#152011yescharacteristic
305C2×C61#178111yescharacteristic
306C2×C61#178711yescharacteristic
307C2×C61#178911yescharacteristic
308D61—22no—
309D61—24no—
310D61—22no—
311C23×Q8—611no—
312C24×C4—611no—
313C2×C22⋊Q8—611no—
314C2×C22⋊Q8—611no—
315C2×C22⋊Q8—611no—
316C2×C22⋊Q8—1221no—
317C2×C22⋊Q8—1221no—
318C2×C22⋊Q8—1224no—
319C2×C22⋊Q8—1224no—
320C2×C22⋊Q8#692311yescharacteristic, complemented
321C22×C4⋊C4—611no—
322C22×C4⋊C4—1221no—
323C22×C22⋊C4—1221no—
324C22×C22⋊C4—1221no—
325C2×C22⋊C8—612no—
326C2×C22⋊C8—1221no—
327C2×C22⋊C8—1221no—
328C2×#95—612no—
329C2×#95—1221no—
330#328—1224no—
331#328—1224no—
332C61#832911yescharacteristic, complemented, Sylow
333C25—611no—
334C22×Q8—611no—
335C22×Q8—1221no—
336C22×Q8—1222no—
337C22×Q8—1222no—
338C22×Q8—2442no—
339C22×Q8—2442no—
340C22×Q8C2×#692311yescharacteristic
341C23×C4—611no—
342C23×C4—611no—
343C23×C4—611no—
344C23×C4—611no—
345C23×C4—611no—
346C23×C4—1222no—
347C23×C4—1222no—
348C23×C4—1221no—
349C23×C4—1221no—
350C23×C4—1222no—
351C23×C4—2442no—
352C23×C4—2442no—
353C23×C4C2×#692311yescharacteristic
354C22×C8—1222no—
355C22⋊Q8—22no—
356C22⋊Q8—22no—
357C22⋊Q8—614no—
358C22⋊Q8—614no—
359C22⋊Q8—1228no—
360C22⋊Q8—1222no—
361C22⋊Q8—1228no—
362C22⋊Q8—1222no—
363C22⋊Q8—1222no—
364C22⋊Q8—1222no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1228no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4C2×#692311yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—1228no—
377C2×C22⋊C4—1221no—
378C2×C22⋊C4—1221no—
379C2×C22⋊C4—1221no—
380C2×C22⋊C4—1224no—
381C2×C22⋊C4—2442no—
382C2×C22⋊C4—2442no—
383C22⋊C8—1222no—
384C22⋊C8—1222no—
385C22⋊C8—2442no—
386C22⋊C8—2442no—
387#95—1222no—
388#95—1222no—
389#95—2442no—
390C24—611no—
391C24—611no—
392C24—612no—
393C24—612no—
394C24—1222no—
395C24—1222no—
396C24—1222no—
397C24—1222no—
398C24—2441no—
399C24—2441no—
400C24C22⋊#692311yescharacteristic
401C2×Q8—21no—
402C2×Q8—41no—
403C2×Q8—41no—
404C2×Q8—612no—
405C2×Q8—1224no—
406C2×Q8—1221no—
407C2×Q8—2441no—
408C2×Q8—2441no—
409C2×Q8—2442no—
410C2×Q8—2442no—
411C2×Q8—2448no—
412C2×Q8—2448no—
413C2×Q8C22⋊#692312yescomplemented
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—612no—
423C22×C4—612no—
424C22×C4—1222no—
425C22×C4—1222no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1224no—
431C22×C4—1224no—
432C22×C4—1222no—
433C22×C4—1222no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—1224no—
439C22×C4—2441no—
440C22×C4—2441no—
441C22×C4—2441no—
442C22×C4—2441no—
443C22×C4—2448no—
444C22×C4—2448no—
445C22×C4—2442no—
446C22×C4—2442no—
447C22×C4—2442no—
448C22×C4—2442no—
449C22×C4—2441no—
450C22×C4—2442no—
451C22×C4—2442no—
452C22×C4—2442no—
453C22×C4—2442no—
454C22×C4—2442no—
455C22×C4—2442no—
456C22×C4C22⋊#692311yescharacteristic
457C22×C4C22×#692311yescharacteristic
458C2×C8—2442no—
459C2×C8—2442no—
460C2×C8—2442no—
461C4⋊C4—24no—
462C4⋊C4—21no—
463C4⋊C4—612no—
464C4⋊C4—1224no—
465C4⋊C4—1224no—
466C4⋊C4—1224no—
467C4⋊C4—1224no—
468C4⋊C4—1221no—
469C4⋊C4C22⋊#692312yescomplemented
470C22⋊C4—24no—
471C22⋊C4—1224no—
472C22⋊C4—1224no—
473C22⋊C4—1224no—
474C22⋊C4—2448no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—612no—
482C23—612no—
483C23—611no—
484C23—611no—
485C23—612no—
486C23—612no—
487C23—612no—
488C23—1224no—
489C23—1224no—
490C23—1224no—
491C23—1224no—
492C23—1221no—
493C23—1221no—
494C23—1221no—
495C23—1221no—
496C23—1221no—
497C23—1221no—
498C23—1221no—
499C23—1221no—
500C23—2442no—
501C23—2442no—
502C23—2444no—
503C23—2444no—
504C23—2444no—
505C23—2444no—
506C23#1451984012yes—
507C23C2×C22⋊#692311yescharacteristic
508Q8—42no—
509Q8—42no—
510Q8—2442no—
511Q8—2442no—
512Q8—2448no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—612no—
527C2×C4—612no—
528C2×C4—1224no—
529C2×C4—1224no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—4882no—
537C2×C4—4882no—
538C2×C4—4882no—
539C2×C4—4882no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2444no—
543C2×C4—2444no—
544C2×C4—2448no—
545C2×C4—2448no—
546C2×C4—2444no—
547C2×C4—2444no—
548C2×C4—2444no—
549C2×C4—2444no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4C2×C22⋊#692311yescharacteristic
555C2×C4C2×C22⋊#692311yescharacteristic
556C8—4882no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—612no—
565C22—612no—
566C22—614no—
567C22—1221no—
568C22—1222no—
569C22—1221no—
570C22—1222no—
571C22—1222no—
572C22—1222no—
573C22—1222no—
574C22—1222no—
575C22—1221no—
576C22—1221no—
577C22—1222no—
578C22—1222no—
579C22—1222no—
580C22—1222no—
581C22—2448no—
582C22—2448no—
583C22#1684070311yescharacteristic
584C22D61⋊#9911yescharacteristic
585C22D61⋊#9911yescharacteristic
586C22C2×#1451984011yescharacteristic
587C22#1684075511yescharacteristic
588C22D61⋊#10411yescharacteristic
589C22#1684075911yescharacteristic
590C4—21no—
591C4—21no—
592C4—41no—
593C4—42no—
594C4—42no—
595C4—1222no—
596C4—1222no—
597C4—4884no—
598C4—2442no—
599C4—2442no—
600C2—42no—
601C2—1222no—
602C2—1222no—
603C2—1224no—
604C2D61⋊#32811yescharacteristic
605C2#2146121711yescharacteristic
606C2#2146122811yescharacteristic
607C2#2146125711yescharacteristic
608C2#2146164611yescharacteristic
609C2#2146165811yescharacteristic
610C2#2146166111yescharacteristic
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#8329
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
3. C2×#16840615C2semidirect
14. C22×Q8×D61C4semidirect
17. C2×#14519756C4semidirect
24. C2×C4⋊C4×D61C4semidirect
26. C2×C22⋊Q8×C61C4semidirect
105. C2×Q8×C61C22⋊C4semidirect
151. C2×Dic122C22⋊C4semidirect
158. C4⋊C4×C61C22⋊C4semidirect
165. C4⋊Dic61C22⋊C4semidirect
282. C2×D61#328semidirect
320. C2×C22⋊Q8#6923semidirect
332. C61#8329semidirect
413. C2×Q8C22⋊#6923semidirect
469. C4⋊C4C22⋊#6923semidirect

Maximal subgroups

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

subgroupindex
#214616422
C2×#168406162
C2×#168406152
#214609932
C2×C22⋊Q8×D612
#832961