Finite Group Databasebeta

← #30617897

Subgroups — #30617897

Order 15616.

1302 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#431C211yesmaximal, characteristic
2#21462268C211yesmaximal, characteristic
3#21462000C214yesmaximal, complemented
4#21461998C214yesmaximal, complemented
5#21461722C214yesmaximal
6C2×C4⋊Q8×D61C211yesmaximal, characteristic
7C2×#14519856—21no—
8#16840781—21no—
9#16840780—21no—
10#16840779—21no—
11#16840778—21no—
12D61⋊(C4⋊C8)C2212yes—
13D61⋊(C22⋊C8)C2212yes—
14C4×D61⋊C8C2212yes—
15#16840708C2212yes—
16#16840706—21no—
17#16840704—21no—
18#16840704C2212yes—
19C4×C4⋊#6923—21no—
20C4×C4⋊#6923C2212yes—
21#16840635C22116yes—
22C22×Q8×D61—21no—
23C2×#14519777C411yescharacteristic, complemented
24C2×#14519776—21no—
25C2×#14519776C412yescomplemented
26C2×#14519775—21no—
27C4⋊Q8×D61—21no—
28C4⋊Q8×D61—21no—
29C4⋊Q8×D61—21no—
30C4⋊Q8×D61—21no—
31C4⋊Q8×D61C412yescomplemented
32C4⋊Q8×D61C412yescomplemented
33C4⋊Q8×D61C2212yes—
34C4⋊Q8×D61C2212yes—
35C2×C4⋊C4×D61—21no—
36C2×C4⋊C4×D61C2212yes—
37C2×C42×D61C2211yescharacteristic
38C2×C4⋊Q8×C61C411yescharacteristic, complemented
39C2×D61⋊C8C2314yes—
40#14519856—44no—
41C2×C4⋊#6923—22no—
42C2×C4⋊#6923—22no—
43C2×C4⋊#6923C2314yes—
44C2×C22⋊#6923—22no—
45C2×C4×#6923—22no—
46#14519844—22no—
47C4⋊#92420—22no—
48#14519834—22no—
49C22⋊#92420—22no—
50C4×#92420—22no—
51C4×#92420—22no—
52(C2×#6923)⋊C4—22no—
53(C2×#6923)⋊C4—22no—
54C22×Dic122—21no—
55C22×Dic122—22no—
56C2×Q8×D61—22no—
57C2×Q8×D61—22no—
58C2×Q8×D61—22no—
59C2×Q8×D61—22no—
60C2×Q8×D61—44no—
61C2×Q8×D61—44no—
62C22×C4×D61—22no—
63C22×C4×D61D411yescharacteristic
64C22×C4×D61D411yescharacteristic
65C22×C4×D61C2311yescharacteristic
66#14519777—21no—
67#14519777—21no—
68#14519777C2×C412yes—
69#14519777C2×C412yes—
70#14519776—22no—
71#14519776—22no—
72#14519776—22no—
73#14519776—22no—
74#14519776—22no—
75#14519776—22no—
76#14519776C2×C414yes—
77#14519776C2×C414yes—
78#14519775—22no—
79#14519775—22no—
80#14519775—22no—
81#14519775—22no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61C2×C412yes—
84C2×#11758196—22no—
85C2×#11758196—22no—
86C4⋊C4×D61—22no—
87C4⋊C4×D61—22no—
88C4⋊C4×D61—22no—
89C4⋊C4×D61—22no—
90C4⋊C4×D61—22no—
91C4⋊C4×D61—22no—
92C4⋊C4×D61C2×C414yes—
93C4⋊C4×D61C2314yes—
94C2×C4×Dic61C2×C411yescharacteristic
95C2×C4×Dic61D411yescharacteristic
96C2×C4×Dic61D411yescharacteristic
97C42×D61C2×C411yescharacteristic
98C42×D61C2×C411yescharacteristic
99C42×D61D411yescharacteristic
100C42×D61D411yescharacteristic
101C42×D61D411yescharacteristic
102C42×D61D411yescharacteristic
103C42×D61C2311yescharacteristic
104C42×D61C2311yescharacteristic
105C22×Q8×C61—21no—
106C4⋊Q8×C61—21no—
107C4⋊Q8×C61—21no—
108C4⋊Q8×C61C2×C412yes—
109C4⋊Q8×C61C2×C412yes—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61C2×C412yes—
112C2×C42×C61C2×C411yescharacteristic
113C23×D61C2×D411yescharacteristic
114C22×#6923—24no—
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61—44no—
118C22×C4×C61—22no—
119C22×C4×C61C22⋊C411yescharacteristic
120C22×C4×C61C22⋊C411yescharacteristic
121C22×C4×C61C22×C411yescharacteristic
122C22×Dic61—22no—
123C22×Dic61C22⋊C411yescharacteristic
124C22×Dic61C22⋊C411yescharacteristic
125C22×Dic61C2×D411yescharacteristic
126Q8×D61—48no—
127Q8×D61—48no—
128Q8×D61—44no—
129Q8×D61—44no—
130Q8×D61—44no—
131Q8×D61—44no—
132C2×C4×D61—24no—
133C2×C4×D61—24no—
134C2×C4×D61—24no—
135C2×C4×D61—24no—
136C2×C4×D61—44no—
137C2×C4×D61—44no—
138C2×C4×D61C22⋊C411yescharacteristic
139C2×C4×D61C22⋊C411yescharacteristic
140C2×C4×D61C22⋊C411yescharacteristic
141C2×C4×D61C22⋊C411yescharacteristic
142C2×C4×D61D814yes—
143C2×C4×D61SD1614yes—
144C2×C4×D61C22×C411yescharacteristic
145C2×C4×D61C2×D411yescharacteristic
146C2×C4×D61C2×D411yescharacteristic
147C2×C4×D61C2×D411yescharacteristic
148C2×C4×D61C2×D411yescharacteristic
149C2×C4×D61C2×D411yescharacteristic
150C2×C4×D61C2×D411yescharacteristic
151C2×C4×D61C2411yescharacteristic
152C2×Dic122—22no—
153C2×Dic122—22no—
154C2×Dic122—24no—
155C2×Dic122—24no—
156C2×Dic122—44no—
157C2×Dic122—48no—
158C22⋊#6923—44no—
159C2×#92420—24no—
160C4⋊#6923—28no—
161C4⋊#6923—44no—
162C4⋊#6923—44no—
163C4×#6923—44no—
164D61⋊C8—28no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61C22×C414yes—
169C42×C61C22⋊C411yescharacteristic
170C42×C61C22⋊C411yescharacteristic
171C42×C61C22×C411yescharacteristic
172C42×C61C22×C411yescharacteristic
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61C22×C414yes—
177#11758196—24no—
178#11758196—24no—
179#11758196—24no—
180#11758196—24no—
181C4×Dic61C22⋊C411yescharacteristic
182C4×Dic61C22⋊C411yescharacteristic
183C4×Dic61C22⋊C411yescharacteristic
184C4×Dic61C22⋊C411yescharacteristic
185C4×Dic61C22⋊C411yescharacteristic
186C4×Dic61C22⋊C411yescharacteristic
187C4×Dic61C22×C411yescharacteristic
188C4×Dic61C22×C411yescharacteristic
189C4×Dic61C2×D411yescharacteristic
190C4×Dic61C2×D411yescharacteristic
191C4×Dic61C2×D411yescharacteristic
192C4×Dic61C2×D411yescharacteristic
193C23×C61C2×C22⋊C411yescharacteristic
194C22×D61—22no—
195C22×D61—22no—
196C22×D61C2×C22⋊C411yescharacteristic
197C22×D61C2×D812yes—
198C22×D61C2×SD1612yes—
199C22×D61C22×D411yescharacteristic
200C2×#6923—44no—
201C2×#6923—44no—
202C2×#6923—44no—
203Q8×C61—44no—
204Q8×C61—44no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—24no—
208C2×C4×C61—44no—
209C2×C4×C61#10214yes—
210C2×C4×C61C2×C22⋊C411yescharacteristic
211C2×C4×C61C2×C22⋊C411yescharacteristic
212C2×C4×C61C2×C22⋊C411yescharacteristic
213C2×C4×C61C2×C22⋊C411yescharacteristic
214C2×C4×C61C2×C22⋊C411yescharacteristic
215C2×C4×C61C23×C411yescharacteristic
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—24no—
219C2×Dic61—44no—
220C2×Dic61#10214yes—
221C2×Dic61C2×C22⋊C411yescharacteristic
222C2×Dic61C2×C22⋊C411yescharacteristic
223C2×Dic61C2×C22⋊C411yescharacteristic
224C2×Dic61C2×C22⋊C411yescharacteristic
225C2×Dic61C2×C22⋊C411yescharacteristic
226C2×Dic61C22×D411yescharacteristic
227C4×D61—22no—
228C4×D61—22no—
229C4×D61—22no—
230C4×D61—22no—
231C4×D61—22no—
232C4×D61—22no—
233C4×D61—44no—
234C4×D61—44no—
235C4×D61—48no—
236C4×D61#10214yes—
237C4×D61#10214yes—
238C4×D61C2×D812yes—
239C4×D61C2×D812yes—
240C4×D61C2×SD1612yes—
241C4×D61C2×SD1612yes—
242C4×D61#13714yes—
243C4×D612−1+414yes—
244Dic122—44no—
245Dic122—44no—
246Dic122—48no—
247Dic122—48no—
248#92420—44no—
249#21833—611nomaximal, Sylow
250C22×C61—21no—
251C22×C61—21no—
252C22×C61C2×#10212yes—
253C22×C61C22×C22⋊C411yescharacteristic
254C2×D61—21no—
255C2×D61—21no—
256C2×D61—24no—
257C2×D61—24no—
258C2×D61C2×#10212yes—
259C2×D61#53612yes—
260C2×D61C22×D811yescharacteristic
261C2×D61C22×SD1611yescharacteristic
262C2×D61C2×#13712yes—
263#6923—84no—
264C4×C61—21no—
265C4×C61—21no—
266C4×C61—44no—
267C4×C61C2×#10212yes—
268C4×C61C2×#10212yes—
269C4×C61#41912yes—
270C4×C61#51912yes—
271Dic61—21no—
272Dic61—21no—
273Dic61—44no—
274Dic61C2×#10212yes—
275Dic61C2×#10212yes—
276Dic61#41912yes—
277Dic61#53612yes—
278C22×C4⋊Q8—611no—
279C2×#431—611no—
280#2285—611no—
281#1990—614no—
282#1988—614no—
283#1823—614no—
284C2×C61—22no—
285C2×C61#231011yescharacteristic
286C2×C61C22×#10211yescharacteristic
287C2×C61C2×#41911yescharacteristic
288D61—22no—
289D61#322612yes—
290D61#322912yes—
291C23×Q8—1221no—
292C2×C4⋊Q8—612no—
293C2×C4⋊Q8—612no—
294C2×C4⋊Q8—612no—
295C2×C4⋊Q8—612no—
296C2×C4⋊Q8—612no—
297C2×C4⋊Q8—611no—
298C2×C4⋊Q8—1221no—
299C2×C4⋊Q8—1221no—
300C2×C4⋊Q8—1221no—
301C2×C4⋊Q8—1221no—
302C2×C4⋊Q8—1221no—
303C2×C4⋊Q8—1221no—
304C2×C4⋊Q8#692311yescharacteristic, complemented
305C2×#117—1221no—
306C22×C4⋊C4—612no—
307C22×C4⋊C4—1221no—
308C22×C42—611no—
309#431—1221no—
310#431—1221no—
311#431—1221no—
312#431—1221no—
313C2×C4⋊C8—612no—
314C2×C22⋊C8—612no—
315C2×C4×C8—612no—
316#384—612no—
317#382—1221no—
318#380—612no—
319#380—1221no—
320C4×C4⋊C4—612no—
321C4×C4⋊C4—1221no—
322#340—6116no—
323C61#2183311yescharacteristic, complemented, Sylow
324C22×Q8—21no—
325C22×Q8—1222no—
326C22×Q8—1222no—
327C22×Q8—1222no—
328C22×Q8—1222no—
329C22×Q8—1222no—
330C22×Q8—1221no—
331C22×Q8—2444no—
332C22×Q8—2444no—
333C23×C4—611no—
334C23×C4—611no—
335C23×C4—611no—
336C23×C4—1222no—
337C22×C8—614no—
338C4⋊Q8—21no—
339C4⋊Q8—21no—
340C4⋊Q8—614no—
341C4⋊Q8—614no—
342C4⋊Q8—612no—
343C4⋊Q8—612no—
344C4⋊Q8—1222no—
345C4⋊Q8—1222no—
346C4⋊Q8—1222no—
347C4⋊Q8—1222no—
348C4⋊Q8—1222no—
349C4⋊Q8—1222no—
350C4⋊Q8—1222no—
351C4⋊Q8—1222no—
352C4⋊Q8—1222no—
353C4⋊Q8—1222no—
354C4⋊Q8—1221no—
355C4⋊Q8—1221no—
356C4⋊Q8C2×#692312yes—
357C4⋊Q8C2×#692312yes—
358#117—2444no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—612no—
361C2×C4⋊C4—614no—
362C2×C4⋊C4—614no—
363C2×C4⋊C4—614no—
364C2×C4⋊C4—1222no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4C2×#692312yes—
376C2×C22⋊C4—1222no—
377C2×C42—611no—
378C2×C42—611no—
379C2×C42—611no—
380C2×C42—611no—
381C2×C42—611no—
382C2×C42—611no—
383C2×C42—611no—
384C2×C42—611no—
385C2×C42—611no—
386C2×C42—611no—
387C2×C42—611no—
388C2×C42—1222no—
389C2×C42C2×#692311yescharacteristic
390C4⋊C8—1222no—
391C4⋊C8—1222no—
392C22⋊C8—1222no—
393C22⋊C8—1222no—
394C4×C8—1222no—
395C4×C8—1222no—
396#95—1222no—
397#95—1222no—
398C24—611no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—44no—
402C2×Q8—1224no—
403C2×Q8—1224no—
404C2×Q8—1222no—
405C2×Q8—1222no—
406C2×Q8—2448no—
407C2×Q8—2448no—
408C2×Q8—2448no—
409C2×Q8—2444no—
410C2×Q8—2444no—
411C2×Q8—2444no—
412C2×Q8—2444no—
413C2×Q8—2444no—
414C22×C4—22no—
415C22×C4—611no—
416C22×C4—614no—
417C22×C4—614no—
418C22×C4—611no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—1224no—
433C22×C4—1224no—
434C22×C4—1224no—
435C22×C4—1224no—
436C22×C4—1222no—
437C22×C4—1224no—
438C22×C4—2444no—
439C22×C4—2444no—
440C22×C4C22⋊#692311yescharacteristic
441C22×C4C22⋊#692311yescharacteristic
442C22×C4C22×#692311yescharacteristic
443C2×C8—1224no—
444C2×C8—1228no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—614no—
449C4⋊C4—1224no—
450C4⋊C4—1224no—
451C4⋊C4—1224no—
452C4⋊C4—1224no—
453C4⋊C4—1228no—
454C4⋊C4—1222no—
455C4⋊C4—1222no—
456C4⋊C4—1222no—
457C4⋊C4—2444no—
458C4⋊C4—2444no—
459C4⋊C4C22×#692314yes—
460C22⋊C4—2444no—
461C42—611no—
462C42—611no—
463C42—611no—
464C42—611no—
465C42—611no—
466C42—611no—
467C42—611no—
468C42—611no—
469C42—611no—
470C42—611no—
471C42—611no—
472C42—611no—
473C42—2444no—
474C42C22⋊#692311yescharacteristic
475C42C22⋊#692311yescharacteristic
476C42C22×#692311yescharacteristic
477C42C22×#692311yescharacteristic
478C23—611no—
479C23—612no—
480C23—612no—
481C23—611no—
482C23—1222no—
483C23—1222no—
484C23C2×C22⋊#692311yescharacteristic
485Q8—44no—
486Q8—44no—
487Q8—2448no—
488Q8—2448no—
489Q8—2444no—
490Q8—2444no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—44no—
495C2×C4—614no—
496C2×C4—611no—
497C2×C4—611no—
498C2×C4—611no—
499C2×C4—611no—
500C2×C4—614no—
501C2×C4—614no—
502C2×C4—612no—
503C2×C4—612no—
504C2×C4—612no—
505C2×C4—612no—
506C2×C4—614no—
507C2×C4—614no—
508C2×C4—611no—
509C2×C4—611no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1224no—
519C2×C4—2444no—
520C2×C4—2444no—
521C2×C4—2448no—
522C2×C4—2444no—
523C2×C4—2444no—
524C2×C4—2444no—
525C2×C4—2444no—
526C2×C4#1451983914yes—
527C2×C4C2×C22⋊#692311yescharacteristic
528C2×C4C2×C22⋊#692311yescharacteristic
529C2×C4C2×C22⋊#692311yescharacteristic
530C2×C4C2×C22⋊#692311yescharacteristic
531C2×C4C2×C22⋊#692311yescharacteristic
532C2×C4C23×#692311yescharacteristic
533C8—2444no—
534C22—21no—
535C22—21no—
536C22—612no—
537C22—611no—
538C22—611no—
539C22—612no—
540C22—612no—
541C22—1224no—
542C22—1224no—
543C22—1221no—
544C22—1221no—
545C22C2×#1451983912yes—
546C22C22×C22⋊#692311yescharacteristic
547C4—21no—
548C4—21no—
549C4—44no—
550C4—612no—
551C4—612no—
552C4—612no—
553C4—612no—
554C4—1221no—
555C4—1221no—
556C4—4884no—
557C4—2444no—
558C4C2×#1451983912yes—
559C4C2×#1451983912yes—
560C4#1684075512yes—
561C4#1684083012yes—
562C2—22no—
563C2—612no—
564C2—612no—
565C2—1222no—
566C2#2146229311yescharacteristic
567C2C22×#1451983911yescharacteristic
568C2C2×#1684075511yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×C4⋊Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenterC2×C4⋊Q8#6923
hypocenterC61#21833
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21462000C2semidirect
4. #21461998C2semidirect
23. C2×#14519777C4semidirect
25. C2×#14519776C4semidirect
31. C4⋊Q8×D61C4semidirect
32. C4⋊Q8×D61C4semidirect
38. C2×C4⋊Q8×C61C4semidirect
304. C2×C4⋊Q8#6923semidirect
323. C61#21833semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4312
#214622682
#214620002
#214619982
#214617222
C2×C4⋊Q8×D612
#2183361