Finite Group Databasebeta

← #30596658

Subgroups — #30596658

Order 15616.

1026 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#407C211yesmaximal, characteristic
2C2×#16840704C211yesmaximal, characteristic
3#21460681C214yesmaximal, complemented
4#21448200C211yesmaximal, characteristic
5C2×D61⋊M16C412yescomplemented
6C22×C4⋊#6923—21no—
7C22×C22⋊#6923—21no—
8#16840734—21no—
9#16840733—21no—
10#16840733—21no—
11#16840732—21no—
12D61⋊(C22⋊C8)C412yescomplemented
13D61⋊(C22⋊C8)C2212yes—
14#16840704—21no—
15#16840704—21no—
16#16840704C412yescomplemented
17#16840704C2212yes—
18C2×(C2×#6923)⋊C4C412yescomplemented
19C23×C4×D61C2211yescharacteristic
20C2×#14519734—21no—
21#16839624C412yescomplemented
22#16839621C2212yes—
23#16839620—21no—
24#16839617—21no—
25C4×#11758198—21no—
26C4×C22⋊Dic61C411yescharacteristic, complemented
27C4×C22⋊C4×C61C411yescharacteristic, complemented
28C23×#6923—22no—
29C2×#11758213—22no—
30D61⋊M16—44no—
31C2×#11758217—22no—
32C2×D61⋊C8C2×C414yescomplemented
33C2×C4⋊#6923—22no—
34C2×C4⋊#6923—22no—
35C2×C4⋊#6923—44no—
36C2×C22⋊#6923—22no—
37C2×C22⋊#6923—22no—
38C2×C22⋊#6923—44no—
39#14519834—22no—
40#14519834—22no—
41C22⋊#92420—22no—
42C22⋊#92420—22no—
43(C2×#6923)⋊C4—22no—
44(C2×#6923)⋊C4C2×C414yes—
45C24×D61D411yescharacteristic
46C23×Dic61D411yescharacteristic
47C22×C4×D61—44no—
48C22×C4×D61C2×C411yescharacteristic
49C22×C4×D61C2×C411yescharacteristic
50C22×C4×D61C2×C411yescharacteristic
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C22×C4×D61D411yescharacteristic
55C22×C4×D61D411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61Q811yescharacteristic
58C22×C4×D61Q811yescharacteristic
59C22×C4×D61C2311yescharacteristic
60#14519734—44no—
61C2×#11758196—22no—
62C2×C22⋊Dic61—21no—
63C2×#11758198—22no—
64C2×#11758198—22no—
65C2×C4×Dic61—21no—
66#14519684—22no—
67#14519684C2×C412yescomplemented
68C23×C4×C61C2×C411yescharacteristic
69C2×C22⋊C4×C61—21no—
70C2×C42×C61—21no—
71#95×C61C2×C412yescomplemented
72C24×C61C22⋊C411yescharacteristic
73C23×D61—22no—
74C23×D61—22no—
75C23×D61—42no—
76C23×D61—42no—
77C23×D61C22⋊C411yescharacteristic
78C23×D61C22⋊C411yescharacteristic
79C23×D61C22⋊C411yescharacteristic
80C23×D61C4⋊C411yescharacteristic
81C23×D61C4⋊C411yescharacteristic
82C23×D61C2×D411yescharacteristic
83C22×#6923—24no—
84C22×#6923—44no—
85C22×#6923—44no—
86C22×#6923—44no—
87C22×C4×C61—22no—
88C22×C4×C61—42no—
89C22×C4×C61C4211yescharacteristic
90C22×C4×C61C4211yescharacteristic
91C22×C4×C61C22⋊C411yescharacteristic
92C22×C4×C61C4⋊C411yescharacteristic
93C22×C4×C61C4⋊C411yescharacteristic
94C22×C4×C61C22×C411yescharacteristic
95C22×Dic61—22no—
96C22×Dic61—42no—
97C22×Dic61C22⋊C411yescharacteristic
98C22×Dic61C22⋊C411yescharacteristic
99C22×Dic61C22⋊C411yescharacteristic
100C22×Dic61C4⋊C411yescharacteristic
101C22×Dic61C4⋊C411yescharacteristic
102C22×Dic61C2×D411yescharacteristic
103C2×C4×D61—24no—
104C2×C4×D61—24no—
105C2×C4×D61—24no—
106C2×C4×D61—42no—
107C2×C4×D61—42no—
108C2×C4×D61—42no—
109C2×C4×D61—42no—
110C2×C4×D61—42no—
111C2×C4×D61—42no—
112C2×C4×D61—42no—
113C2×C4×D61—42no—
114C2×C4×D61—42no—
115C2×C4×D61C4211yescharacteristic
116C2×C4×D61C4211yescharacteristic
117C2×C4×D61C22⋊C411yescharacteristic
118C2×C4×D61C22⋊C411yescharacteristic
119C2×C4×D61C22⋊C411yescharacteristic
120C2×C4×D61C22⋊C411yescharacteristic
121C2×C4×D61C4⋊C411yescharacteristic
122C2×C4×D61C4⋊C411yescharacteristic
123C2×C4×D61C4⋊C411yescharacteristic
124C2×C4×D61C4⋊C411yescharacteristic
125C2×C4×D61C4⋊C411yescharacteristic
126C2×C4×D61C4⋊C411yescharacteristic
127C2×C4×D61C22×C411yescharacteristic
128C2×C4×D61C22×C411yescharacteristic
129C2×C4×D61C2×D411yescharacteristic
130C2×C4×D61C2×Q811yescharacteristic
131C22⋊#6923—44no—
132C22⋊#6923—44no—
133#11758217—44no—
134C2×#92420—24no—
135C4⋊#6923—44no—
136C4⋊#6923—44no—
137#11758213—44no—
138D61⋊C8—44no—
139C22⋊C4×C61—42no—
140C42×C61—42no—
141C22⋊Dic61—42no—
142#11758198—44no—
143#11758198—42no—
144#11758198—42no—
145#11758196—44no—
146C4×Dic61—42no—
147C23×C61—22no—
148C23×C61—22no—
149C23×C61—41no—
150C23×C61#9511yescharacteristic
151C23×C61#9511yescharacteristic
152C23×C61C2×C22⋊C411yescharacteristic
153C22×D61—22no—
154C22×D61—22no—
155C22×D61—21no—
156C22×D61—21no—
157C22×D61—22no—
158C22×D61—22no—
159C22×D61—22no—
160C22×D61—22no—
161C22×D61—21no—
162C22×D61—21no—
163C22×D61—22no—
164C22×D61—22no—
165C22×D61—41no—
166C22×D61—41no—
167C22×D61—41no—
168C22×D61—41no—
169C22×D61—41no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—48no—
173C22×D61—48no—
174C22×D61#9511yescharacteristic
175C22×D61#9511yescharacteristic
176C22×D61C2×C22⋊C411yescharacteristic
177C22×D61C2×C4⋊C411yescharacteristic
178C22×D61AGL(1,ℤ/8)12yes—
179C22×D61#13712yes—
180C2×#6923—44no—
181C2×#6923—44no—
182C2×#6923—44no—
183C2×#6923—88no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61#9511yescharacteristic
194C2×C4×C61#9511yescharacteristic
195C2×C4×C61C2×C4211yescharacteristic
196C2×C4×C61C2×C4⋊C411yescharacteristic
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206C2×Dic61#9511yescharacteristic
207C2×Dic61#9511yescharacteristic
208C2×Dic61C2×C22⋊C411yescharacteristic
209C2×Dic61C2×C4⋊C411yescharacteristic
210C4×D61—22no—
211C4×D61—22no—
212C4×D61—22no—
213C4×D61—22no—
214C4×D61—22no—
215C4×D61—22no—
216C4×D61—22no—
217C4×D61—22no—
218C4×D61—22no—
219C4×D61—41no—
220C4×D61—41no—
221C4×D61—41no—
222C4×D61—41no—
223C4×D61—41no—
224C4×D61—41no—
225C4×D61#9914yes—
226#92420—44no—
227#7983—611nomaximal, Sylow
228C22×C61—21no—
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—21no—
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—44no—
236C22×C61H(3,ℤ/4)12yes—
237C22×C61C2×#9511yescharacteristic
238C22×C61#41712yes—
239C2×D61—24no—
240C2×D61—24no—
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—42no—
251C2×D61—42no—
252C2×D61—44no—
253C2×D61—44no—
254C2×D61—44no—
255C2×D61—42no—
256C2×D61—42no—
257C2×D61#34212yes—
258C2×D61#41811yescharacteristic
259C2×D61#41911yescharacteristic
260C2×D61#42812yes—
261#6923—84no—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—82no—
266C4×C61#34212yes—
267Dic61—21no—
268Dic61—21no—
269Dic61—21no—
270Dic61—82no—
271Dic61C2×#9912yes—
272C2×#407—611no—
273#2280—611no—
274C2×#380—611no—
275#1289—614no—
276C2×C61—22no—
277C2×C61—22no—
278C2×C61—41no—
279C2×C61C2×H(3,ℤ/4)11yescharacteristic
280C2×C61#173611yescharacteristic
281C2×C61C2×#34211yescharacteristic
282D61—21no—
283D61—22no—
284D61—21no—
285D61—22no—
286D61—41no—
287C24×C4—611no—
288C22×M16—612no—
289C2×#117—1221no—
290C22×C4⋊C4—1221no—
291C22×C22⋊C4—1221no—
292#407—1221no—
293#407—1221no—
294#407—1221no—
295#407—1221no—
296C2×C22⋊C8—612no—
297C2×C22⋊C8—612no—
298#381—612no—
299#380—612no—
300#380—612no—
301#380—612no—
302#380—1221no—
303#380—1221no—
304#380—1221no—
305#379—1221no—
306C4×C22⋊C4—611no—
307C4×C22⋊C4—1221no—
308C4×C22⋊C4#692311yescharacteristic, complemented
309C2×#95—612no—
310C61#798311yescharacteristic, complemented, Sylow
311C25—611no—
312C23×C4—611no—
313C23×C4—611no—
314C23×C4—611no—
315C23×C4—611no—
316C23×C4—611no—
317C23×C4—611no—
318C23×C4—611no—
319C23×C4—611no—
320C23×C4—611no—
321C23×C4—611no—
322C23×C4—611no—
323C23×C4—611no—
324C23×C4—611no—
325C23×C4—1222no—
326C23×C4—2444no—
327C23×C4C2×#692311yescharacteristic
328C2×M16—1222no—
329C2×M16—1222no—
330C2×M16—2444no—
331C22×C8—614no—
332#117—2444no—
333C2×C4⋊C4—1222no—
334C2×C4⋊C4—1222no—
335C2×C4⋊C4—1222no—
336C2×C4⋊C4—2444no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—1222no—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4—1222no—
341C2×C22⋊C4—1222no—
342C2×C22⋊C4—1221no—
343C2×C22⋊C4—2444no—
344C2×C42—21no—
345C2×C42—1221no—
346C22⋊C8—1222no—
347C22⋊C8—1222no—
348C22⋊C8—1222no—
349C22⋊C8—1222no—
350#95—614no—
351#95—612no—
352#95—1222no—
353#95—1222no—
354#95C2×#692312yescomplemented
355C24—611no—
356C24—611no—
357C24—611no—
358C24—611no—
359C24—611no—
360C24—611no—
361C24—1222no—
362C24—1222no—
363C24—2442no—
364C24—2442no—
365C24C22⋊#692311yescharacteristic
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—611no—
369C22×C4—611no—
370C22×C4—611no—
371C22×C4—611no—
372C22×C4—611no—
373C22×C4—611no—
374C22×C4—611no—
375C22×C4—611no—
376C22×C4—611no—
377C22×C4—611no—
378C22×C4—611no—
379C22×C4—611no—
380C22×C4—611no—
381C22×C4—611no—
382C22×C4—611no—
383C22×C4—611no—
384C22×C4—611no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—611no—
388C22×C4—611no—
389C22×C4—611no—
390C22×C4—1224no—
391C22×C4—1224no—
392C22×C4—1224no—
393C22×C4—1224no—
394C22×C4—1222no—
395C22×C4—2442no—
396C22×C4—2442no—
397C22×C4—2442no—
398C22×C4—2442no—
399C22×C4—2442no—
400C22×C4—2442no—
401C22×C4—2442no—
402C22×C4—2442no—
403C22×C4—2442no—
404C22×C4—2442no—
405C22×C4—2444no—
406C22×C4—2444no—
407C22×C4—2444no—
408C22×C4C4×#692311yescharacteristic
409C22×C4C4×#692311yescharacteristic
410C22×C4C4⋊#692311yescharacteristic
411C22×C4C4⋊#692311yescharacteristic
412C22×C4C22⋊#692311yescharacteristic
413C22×C4C22×#692311yescharacteristic
414M16—2444no—
415M16—2444no—
416C2×C8—1224no—
417C2×C8—2444no—
418C4⋊C4—2444no—
419C4⋊C4—2444no—
420C4⋊C4—2444no—
421C22⋊C4—42no—
422C22⋊C4—2442no—
423C22⋊C4—2442no—
424C22⋊C4—2444no—
425C22⋊C4—2442no—
426C22⋊C4—2444no—
427C22⋊C4—2444no—
428C42—42no—
429C42—2442no—
430C23—22no—
431C23—22no—
432C23—41no—
433C23—611no—
434C23—611no—
435C23—611no—
436C23—612no—
437C23—612no—
438C23—611no—
439C23—1222no—
440C23—1221no—
441C23—1221no—
442C23—1221no—
443C23—1222no—
444C23—1222no—
445C23—1222no—
446C23—1221no—
447C23—1222no—
448C23—1222no—
449C23—1222no—
450C23—1222no—
451C23—2441no—
452C23—2441no—
453C23—2441no—
454C23—2441no—
455C23—2441no—
456C23—2441no—
457C23—2441no—
458C23—2448no—
459C23—2448no—
460C23(C2×#6923)⋊C411yescharacteristic
461C23(C2×#6923)⋊C411yescharacteristic
462C23C2×C22⋊#692311yescharacteristic
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—611no—
473C2×C4—614no—
474C2×C4—611no—
475C2×C4—611no—
476C2×C4—611no—
477C2×C4—1222no—
478C2×C4—1222no—
479C2×C4—1222no—
480C2×C4—1222no—
481C2×C4—1222no—
482C2×C4—1222no—
483C2×C4—1222no—
484C2×C4—1222no—
485C2×C4—1222no—
486C2×C4—1222no—
487C2×C4—1222no—
488C2×C4—1222no—
489C2×C4—4888no—
490C2×C4—2441no—
491C2×C4—2441no—
492C2×C4—2441no—
493C2×C4—2441no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4—2441no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2444no—
502C2×C4—2444no—
503C2×C4—2444no—
504C2×C4—2442no—
505C2×C4(C2×#6923)⋊C411yescharacteristic
506C2×C4(C2×#6923)⋊C411yescharacteristic
507C2×C4C2×C4×#692311yescharacteristic
508C2×C4C2×C4⋊#692311yescharacteristic
509C8—2444no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—44no—
518C22—611no—
519C22—611no—
520C22—612no—
521C22—612no—
522C22—1224no—
523C22—1224no—
524C22—1221no—
525C22—1221no—
526C22—1221no—
527C22—1221no—
528C22—1221no—
529C22—1221no—
530C22—1221no—
531C22—1221no—
532C22—1221no—
533C22—2442no—
534C22—2442no—
535C22—2444no—
536C22—2444no—
537C22—2444no—
538C22—2442no—
539C22—2442no—
540C22#1684062912yes—
541C22C2×(C2×#6923)⋊C411yescharacteristic
542C22#1684075312yes—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—82no—
547C4—612no—
548C4—1221no—
549C4—1221no—
550C4—1221no—
551C4—4882no—
552C4—4884no—
553C4#1684063912yes—
554C2—22no—
555C2—22no—
556C2—41no—
557C2—1221no—
558C2—1221no—
559C2—1222no—
560C2—1222no—
561C2—2441no—
562C2C2×#1684062911yescharacteristic
563C2#2146150811yescharacteristic
564C2C2×#1684063911yescharacteristic
subgroupG/H
centerC22C2×(C2×#6923)⋊C4
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC4×C22⋊C4×C61C4
socleC22×C61C2×#95
prime radicalC24×D61D4
cosocleC22×C4×D61C23
hypercenterC4×C22⋊C4#6923
hypocenterC61#7983
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #21460681C2semidirect
5. C2×D61⋊M16C4semidirect
12. D61⋊(C22⋊C8)C4semidirect
16. #16840704C4semidirect
18. C2×(C2×#6923)⋊C4C4semidirect
21. #16839624C4semidirect
26. C4×C22⋊Dic61C4semidirect
27. C4×C22⋊C4×C61C4semidirect
32. C2×D61⋊C8C2×C4semidirect
67. #14519684C2×C4semidirect
71. #95×C61C2×C4semidirect
308. C4×C22⋊C4#6923semidirect
310. C61#7983semidirect
354. #95C2×#6923semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4072
C2×#168407042
#214606812
#214482002
#798361