Finite Group Databasebeta

← #30602497

Subgroups — #30602497

Order 15616.

1298 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840748C211yesmaximal, characteristic
2C2×#16840741C211yesmaximal, characteristic
3C2×#16840633C212yesmaximal, complemented
4C2×#16840629C212yesmaximal, complemented
5C2×C4×#11758222C211yesmaximal, characteristic
6C2×D61⋊M16C2212yes—
7C2×#14519856C2212yes—
8#16840748—22no—
9#16840748C414yescomplemented
10C2×#14519838—21no—
11C2×#14519837—21no—
12#16840741—22no—
13#16840741C414yescomplemented
14C2×#14519836—21no—
15C2×#14519835—21no—
16#16840633—24no—
17#16840629—24no—
18C22×#11758222C2211yescharacteristic
19C4×#11758222—24no—
20C4×#11758222—24no—
21C2×C4×Dic122—21no—
22C2×C4×D244—21no—
23C2×C4×C22⋊D61—21no—
24C2×C4×C22⋊D61—21no—
25C2×#14519736C411yescharacteristic, complemented
26C2×#14519734C411yescharacteristic, complemented
27C2×#14519734C2211yescharacteristic
28C2×C42×D61C411yescharacteristic, complemented
29C2×C42×D61C2211yescharacteristic
30C22×C42×C61C411yescharacteristic, complemented
31C2×#11758213—22no—
32D61⋊M16—22no—
33D61⋊M16—22no—
34D61⋊M16—22no—
35D61⋊M16C2×C414yescomplemented
36C2×#11758217—22no—
37C2×D61⋊C8—22no—
38#14519856—22no—
39#14519856—22no—
40#14519856—22no—
41#14519856C2×C414yescomplemented
42C2×C4⋊#6923—22no—
43C2×C22⋊#6923—22no—
44C2×C4×#6923—22no—
45#14519838—24no—
46#14519837—24no—
47#14519836—24no—
48#14519835—24no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222—22no—
52C2×#11758222—22no—
53C2×#11758222C2×C412yes—
54C2×#11758222C2×C412yes—
55C2×#11758222D412yes—
56C2×#11758222Q812yes—
57C22×Dic122D411yescharacteristic
58C22×D244D411yescharacteristic
59C22×C22⋊D61D411yescharacteristic
60C22×C22⋊D61D411yescharacteristic
61C22×C4×D61—21no—
62C22×C4×D61—21no—
63C22×C4×D61C2×C411yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C4×Dic122—28no—
66C4×D244—28no—
67C4×C22⋊D61—28no—
68C4×C22⋊D61—28no—
69#14519736—22no—
70#14519736—22no—
71#14519734—22no—
72#14519734—22no—
73#14519734—22no—
74#14519734D414yes—
75C2×C4⋊Dic61—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×C22⋊Dic61—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61C2×C411yescharacteristic
84C2×C4×Dic61C2×C411yescharacteristic
85C42×D61—22no—
86C42×D61—22no—
87C42×D61—22no—
88C42×D61D414yes—
89C23×C4×C61—21no—
90C23×C4×C61C2×C411yescharacteristic
91C2×C42×C61—21no—
92C2×C42×C61—22no—
93C2×C42×C61—22no—
94C2×C42×C61C2×C411yescharacteristic
95C2×C42×C61C2×C411yescharacteristic
96C24×C61C22⋊C411yescharacteristic
97C23×D61C22⋊C411yescharacteristic
98C23×D61C2×D411yescharacteristic
99C22×#6923—42no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—24no—
106C22×C4×C61—21no—
107C22×C4×C61—44no—
108C22×C4×C61C4212yes—
109C22×C4×C61C22⋊C411yescharacteristic
110C22×C4×C61C4⋊C412yes—
111C22×C4×C61C22×C411yescharacteristic
112C2×C22⋊D61—22no—
113C2×C22⋊D61—22no—
114C2×C22⋊D61—22no—
115C2×C22⋊D61—22no—
116C2×C22⋊D61C22⋊C412yes—
117C2×C22⋊D61C22⋊C412yes—
118C2×C22⋊D61C4⋊C412yes—
119C2×C22⋊D61C4⋊C412yes—
120C22×Dic61—21no—
121C22×Dic61—21no—
122C22×Dic61C22⋊C411yescharacteristic
123C22×Dic61C2×D411yescharacteristic
124#11758222—24no—
125#11758222—24no—
126#11758222—24no—
127#11758222—24no—
128#11758222—24no—
129#11758222—24no—
130#11758222—24no—
131#11758222—24no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244C22⋊C412yes—
135C2×D244C4⋊C412yes—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—24no—
141C2×C4×D61—24no—
142C2×C4×D61—44no—
143C2×C4×D61—44no—
144C2×C4×D61C4212yes—
145C2×C4×D61C4⋊C412yes—
146C2×C4×D61C22×C411yescharacteristic
147C2×C4×D61C22×C411yescharacteristic
148C2×C4×D61C2×D412yes—
149C2×C4×D61C2×D412yes—
150C2×C4×D61C2×D411yescharacteristic
151C2×C4×D61C2×Q811yescharacteristic
152C2×C4×D61#4112yes—
153C2×C4×D61#4112yes—
154C2×Dic122—22no—
155C2×Dic122—22no—
156C2×Dic122C22⋊C412yes—
157C2×Dic122C4⋊C412yes—
158C22⋊#6923—24no—
159C22⋊#6923—42no—
160#11758217—24no—
161#11758217—42no—
162C2×#92420—42no—
163C4⋊#6923—24no—
164C4⋊#6923—42no—
165C4×#6923—24no—
166C4×#6923—42no—
167#11758213—24no—
168#11758213—42no—
169D61⋊C8—24no—
170D61⋊C8—42no—
171C42×C61—24no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61C22⋊C412yes—
175C42×C61C22⋊C412yes—
176C22⋊Dic61—24no—
177#11758198—24no—
178#11758198—24no—
179C4⋊Dic61—24no—
180#11758196—24no—
181#11758196—24no—
182C4×Dic61—24no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185C4×Dic61—24no—
186C4×Dic61C22⋊C412yes—
187C4×Dic61C22⋊C412yes—
188C23×C61—21no—
189C23×C61—21no—
190C23×C61—42no—
191C23×C61#9512yes—
192C23×C61C2×C22⋊C411yescharacteristic
193C22×D61—21no—
194C22×D61—21no—
195C22×D61—42no—
196C22×D61—42no—
197C22×D61#9512yes—
198C22×D61C2×C22⋊C411yescharacteristic
199C22×D61C2×C4⋊C411yescharacteristic
200C22×D61#12012yes—
201C22×D61#12312yes—
202C2×#6923—44no—
203C2×#6923—84no—
204C2×C4×C61—24no—
205C2×C4×C61—22no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—48no—
214C2×C4×C61—42no—
215C2×C4×C61#9512yes—
216C2×C4×C61C2×C4211yescharacteristic
217C2×C4×C61C2×C22⋊C411yescharacteristic
218C2×C4×C61C2×C22⋊C411yescharacteristic
219C2×C4×C61C2×C4⋊C411yescharacteristic
220C2×C4×C61#11711yescharacteristic
221C2×C4×C61#11711yescharacteristic
222C22⋊D61—24no—
223C22⋊D61—24no—
224C22⋊D61—24no—
225C22⋊D61—24no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61#9512yes—
233C2×Dic61C2×C22⋊C411yescharacteristic
234C2×Dic61C2×C4⋊C411yescharacteristic
235C2×Dic61#12012yes—
236C2×Dic61#12312yes—
237D244—24no—
238D244—24no—
239C4×D61—22no—
240C4×D61—22no—
241C4×D61—22no—
242C4×D61—22no—
243C4×D61—48no—
244C4×D61—48no—
245C4×D61C4×D412yes—
246C4×D61C4×D412yes—
247C4×D61C4×D412yes—
248C4×D61C4×D412yes—
249C4×D61#12112yes—
250C4×D61#12112yes—
251C4×D61C22⋊Q812yes—
252C4×D61C22⋊Q812yes—
253Dic122—24no—
254Dic122—24no—
255#92420—44no—
256#10758—611nomaximal, Sylow
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61C2×#9511yescharacteristic
263C22×C61#37911yescharacteristic
264C22×C61#38111yescharacteristic
265C2×D61—24no—
266C2×D61—22no—
267C2×D61—42no—
268C2×D61—41no—
269C2×D61—41no—
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61#38512yes—
274C2×D61#38612yes—
275#6923—84no—
276C4×C61—24no—
277C4×C61—22no—
278C4×C61—42no—
279C4×C61C4×C22⋊C411yescharacteristic
280C4×C61C4×C22⋊C411yescharacteristic
281C4×C61#38011yescharacteristic
282C4×C61#38011yescharacteristic
283Dic61—22no—
284Dic61—42no—
285Dic61—42no—
286Dic61#38512yes—
287Dic61#38612yes—
288C2×C4×#41—611no—
289C2×#413—611no—
290C2×#410—611no—
291C2×#339—612no—
292C2×H(3,ℤ/4)—612no—
293C2×C61—21no—
294C2×C61—21no—
295C2×C61—42no—
296C2×C61C22⋊#9511yescharacteristic
297C2×C61#182712yes—
298D61—22no—
299D61—22no—
300D61—42no—
301C22×#41—611no—
302C22×M16—612no—
303C4×#41—1224no—
304C4×#41—1224no—
305C2×C4×Q8—1221no—
306C2×C4×D4—1221no—
307C2×C4×D4—1221no—
308C2×C4×D4—1221no—
309C2×#117—611no—
310C2×#117—611no—
311C2×#117—611no—
312C2×#117—612no—
313C22×C42—611no—
314C22×C42—611no—
315C22×C42#692311yescharacteristic, complemented
316#413—614no—
317#413—1222no—
318C2×#101—1221no—
319C2×#100—1221no—
320#410—614no—
321#410—1222no—
322C2×#99—1221no—
323C2×#99—1221no—
324#339—1224no—
325H(3,ℤ/4)—1224no—
326C61#1075811yescharacteristic, complemented, Sylow
327C2×#41—612no—
328C2×#41—612no—
329C2×#41—612no—
330C2×#41—612no—
331C2×#41—1222no—
332C2×#41—1222no—
333C2×#41—1222no—
334C2×#41—1222no—
335C22×Q8—611no—
336C22×D4—611no—
337C22×D4—611no—
338C22×D4—611no—
339C23×C4—21no—
340C23×C4—611no—
341C23×C4—611no—
342C23×C4—1221no—
343C23×C4—1221no—
344C23×C4C2×#692311yescharacteristic
345C2×M16—614no—
346C2×M16—1222no—
347C2×M16—1222no—
348C2×M16—1222no—
349C2×M16—1222no—
350C2×M16—1222no—
351C22×C8—1222no—
352C4×Q8—1228no—
353C4×D4—1228no—
354C4×D4—1228no—
355C4×D4—1228no—
356#117—614no—
357#117—614no—
358#117—1222no—
359#117—1222no—
360#117—1222no—
361#117—1222no—
362#117—1222no—
363#117—1222no—
364#117—1222no—
365#117—1222no—
366C2×C4⋊C4—1221no—
367C2×C4⋊C4—1221no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1222no—
370C2×C22⋊C4—1221no—
371C2×C22⋊C4—1221no—
372C2×C22⋊C4—1221no—
373C2×C22⋊C4—1222no—
374C2×C42—21no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—611no—
378C2×C42—611no—
379C2×C42—614no—
380C2×C42—1222no—
381C2×C42—1222no—
382C2×C42—1221no—
383C2×C42—1221no—
384C2×C42—1222no—
385C2×C42—1222no—
386C2×C42C2×#692311yescharacteristic
387C2×C42C2×#692311yescharacteristic
388#101—1224no—
389#100—1224no—
390#99—1224no—
391#99—1224no—
392C24—611no—
393C24—611no—
394C24C22⋊#692311yescharacteristic
395#41—1224no—
396#41—1224no—
397#41—1224no—
398#41—1224no—
399#41—1224no—
400#41—1224no—
401#41—1224no—
402#41—1224no—
403C2×Q8—612no—
404C2×Q8—612no—
405C2×Q8—1222no—
406C2×Q8—1222no—
407C2×D4—612no—
408C2×D4—612no—
409C2×D4—612no—
410C2×D4—612no—
411C2×D4—612no—
412C2×D4—612no—
413C2×D4—1222no—
414C2×D4—1222no—
415C2×D4—1222no—
416C2×D4—1222no—
417C2×D4—1222no—
418C2×D4—1222no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—44no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—612no—
431C22×C4—612no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—612no—
435C22×C4—612no—
436C22×C4—611no—
437C22×C4—612no—
438C22×C4—612no—
439C22×C4—1224no—
440C22×C4—1224no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—2444no—
448C22×C4—2444no—
449C22×C4—2442no—
450C22×C4C4×#692312yes—
451C22×C4C4⋊#692312yes—
452C22×C4C22⋊#692311yescharacteristic
453C22×C4C22×#692311yescharacteristic
454M16—1224no—
455M16—1224no—
456M16—2442no—
457M16—2442no—
458C2×C8—1224no—
459C2×C8—2442no—
460C2×C8—2442no—
461C4⋊C4—1224no—
462C4⋊C4—1224no—
463C4⋊C4—1224no—
464C4⋊C4—1224no—
465C4⋊C4—2442no—
466C22⋊C4—1224no—
467C22⋊C4—1224no—
468C22⋊C4—1224no—
469C22⋊C4—1224no—
470C22⋊C4—2442no—
471C42—24no—
472C42—21no—
473C42—21no—
474C42—612no—
475C42—612no—
476C42—1224no—
477C42—1224no—
478C42—1224no—
479C42—1221no—
480C42—1221no—
481C42—2442no—
482C42C22⋊#692312yes—
483C42C22⋊#692312yes—
484C23—21no—
485C23—21no—
486C23—42no—
487C23—612no—
488C23—611no—
489C23—611no—
490C23—612no—
491C23—612no—
492C23—1221no—
493C23—1221no—
494C23—2442no—
495C23—2442no—
496C23(C2×#6923)⋊C412yes—
497C23C2×C22⋊#692311yescharacteristic
498Q8—1224no—
499Q8—1224no—
500D4—1224no—
501D4—1224no—
502D4—1224no—
503D4—1224no—
504D4—1224no—
505D4—1224no—
506C2×C4—24no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—48no—
516C2×C4—42no—
517C2×C4—612no—
518C2×C4—611no—
519C2×C4—611no—
520C2×C4—612no—
521C2×C4—612no—
522C2×C4—612no—
523C2×C4—612no—
524C2×C4—612no—
525C2×C4—612no—
526C2×C4—612no—
527C2×C4—612no—
528C2×C4—612no—
529C2×C4—612no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—4884no—
539C2×C4—2448no—
540C2×C4—2448no—
541C2×C4—2444no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4(C2×#6923)⋊C412yes—
545C2×C4C2×C4×#692311yescharacteristic
546C2×C4C2×C22⋊#692311yescharacteristic
547C2×C4C2×C22⋊#692311yescharacteristic
548C2×C4C2×C4⋊#692311yescharacteristic
549C2×C4#1451985611yescharacteristic
550C2×C4#1451985611yescharacteristic
551C8—2444no—
552C22—22no—
553C22—22no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—612no—
558C22—612no—
559C22—1224no—
560C22—1222no—
561C22—2442no—
562C22—2441no—
563C22—2441no—
564C22—2442no—
565C22—2442no—
566C22—2442no—
567C22C2×(C2×#6923)⋊C411yescharacteristic
568C22#1684070311yescharacteristic
569C22#1684070511yescharacteristic
570C4—24no—
571C4—22no—
572C4—42no—
573C4—612no—
574C4—612no—
575C4—1222no—
576C4—4884no—
577C4—2442no—
578C4—2442no—
579C4C4×C22⋊#692311yescharacteristic
580C4C4×C22⋊#692311yescharacteristic
581C4#1684070411yescharacteristic
582C4#1684070411yescharacteristic
583C2—21no—
584C2—21no—
585C2—42no—
586C2—1222no—
587C2—1222no—
588C2—2442no—
589C2#2146099311yescharacteristic
590C2#2146172812yes—
subgroupG/H
centerC2×C4#14519856
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC22×C42×C61C4
socleC22×C61#381
prime radicalC22×#11758222C22
cosocleC22×C4×D61C23
hypercenterC22×C42#6923
hypocenterC61#10758
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
3. C2×#16840633C2semidirect
4. C2×#16840629C2semidirect
9. #16840748C4semidirect
13. #16840741C4semidirect
25. C2×#14519736C4semidirect
26. C2×#14519734C4semidirect
28. C2×C42×D61C4semidirect
30. C22×C42×C61C4semidirect
35. D61⋊M16C2×C4semidirect
41. #14519856C2×C4semidirect
315. C22×C42#6923semidirect
326. C61#10758semidirect

Maximal subgroups

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

subgroupindex
C2×#168407482
C2×#168407412
C2×#168406332
C2×#168406292
C2×C4×#117582222
#1075861