Finite Group Databasebeta

← #30602522

Subgroups — #30602522

Order 15616.

1254 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840831C211yesmaximal, characteristic
2D61⋊#413C211yesmaximal, characteristic
3#21461508C212yesmaximal, complemented
4#21461500C212yesmaximal, complemented
5#410×D61C211yesmaximal, characteristic
6C2×D61⋊M16C2212yes—
7#16840831—24no—
8#16840831—24no—
9C2×Q8×#6923—21no—
10C2×D4×#6923—22no—
11C2×D4×#6923—21no—
12C2×#14519856C412yescomplemented
13C2×#14519856C2211yescharacteristic
14C22×C4⋊#6923C412yescomplemented
15C22×C4⋊#6923C2211yescharacteristic
16#16840749—22no—
17#16840748C414yescomplemented
18D61⋊#101—21no—
19D61⋊#100—21no—
20#16840635—24no—
21#16840633—24no—
22C2×#41×D61C2211yescharacteristic
23#117×D61C2212yes—
24#16839736C411yescharacteristic, complemented
25#16839735C412yescomplemented
26#16839734—22no—
27#99×D61—22no—
28C61×#410C411yescharacteristic, complemented
29C23×#6923—22no—
30C23×#6923—21no—
31C2×#11758213—22no—
32D61⋊M16—22no—
33D61⋊M16—22no—
34D61⋊M16—22no—
35D61⋊M16C2×C414yescomplemented
36C2×#11758217—22no—
37C2×D61⋊C8—22no—
38Q8×#6923—28no—
39D4×#6923—216no—
40D4×#6923—28no—
41#14519856—22no—
42#14519856—24no—
43#14519856—24no—
44#14519856D414yes—
45C2×C4⋊#6923—22no—
46C2×C4⋊#6923—21no—
47C2×C4⋊#6923—22no—
48C2×C4⋊#6923—24no—
49C2×C4⋊#6923—24no—
50C2×C4⋊#6923C2×C411yescharacteristic
51C2×C4⋊#6923C2×C412yes—
52C2×C4⋊#6923C2×C411yescharacteristic
53C2×C4⋊#6923D414yes—
54C2×C22⋊#6923—22no—
55C2×C22⋊#6923—21no—
56C2×C4×#6923—22no—
57C2×C4×#6923—21no—
58#14519838—24no—
59#14519837—24no—
60C2×#11758226Q811yescharacteristic
61C2×#11758222C2×C412yes—
62C2×#11758222C2×C411yescharacteristic
63C2×#11758224D412yes—
64C2×#11758224Q811yescharacteristic
65#41×D61—22no—
66#41×D61—22no—
67#41×D61—22no—
68#41×D61—22no—
69C2×Q8×D61D411yescharacteristic
70C2×D4×D61D412yes—
71C2×D4×D61D411yescharacteristic
72C22×C4×D61C2×C412yes—
73C22×C4×D61C2311yescharacteristic
74#14519736C2×C412yescomplemented
75#14519735—22no—
76#14519734—22no—
77#14519733—22no—
78C4⋊C4×D61—22no—
79C22⋊C4×D61—22no—
80C42×D61—22no—
81#14519692—22no—
82#14519691—24no—
83C2×#41×C61C2×C411yescharacteristic
84#117×C61C2×C412yescomplemented
85#99×C61—22no—
86C23×D61C22⋊C412yes—
87C23×D61C2×D411yescharacteristic
88C22×#6923—21no—
89C22×#6923—22no—
90C22×#6923—21no—
91C22×#6923—28no—
92C22×#6923—24no—
93C22×#6923—82no—
94C22×#6923—84no—
95#41×C61—22no—
96#41×C61—22no—
97C2×Q8×C61C4⋊C411yescharacteristic
98C2×D4×C61C22⋊C412yes—
99C2×D4×C61C4⋊C411yescharacteristic
100C22×C4×C61C4212yes—
101C22×C4×C61C22×C411yescharacteristic
102C2×C22⋊D61C22⋊C412yes—
103C2×C22⋊D61C22⋊C412yes—
104C2×C22⋊D61C4⋊C412yes—
105C22×Dic61C4⋊C412yes—
106C22×Dic61C2×D411yescharacteristic
107#11758226—22no—
108#11758226—22no—
109Q8×D61—22no—
110Q8×D61—22no—
111#11758224—24no—
112#11758224—24no—
113#11758224—22no—
114#11758224—22no—
115D4×D61—24no—
116D4×D61—24no—
117D4×D61—22no—
118D4×D61—22no—
119#11758222—22no—
120#11758222—22no—
121#11758222—24no—
122#11758222—24no—
123C2×D244C4⋊C412yes—
124C2×D244C4⋊C411yescharacteristic
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—42no—
130C2×C4×D61C4212yes—
131C2×C4×D61C22⋊C412yes—
132C2×C4×D61C4⋊C412yes—
133C2×C4×D61C22×C411yescharacteristic
134C2×C4×D61C22×C411yescharacteristic
135C2×C4×D61C2×D412yes—
136C2×C4×D61C2×D412yes—
137C2×C4×D61C2×D411yescharacteristic
138C2×C4×D61C2×Q811yescharacteristic
139C2×C4×D61#4112yes—
140C2×C4×D61#4112yes—
141C2×Dic122C22⋊C412yes—
142C2×Dic122C4⋊C411yescharacteristic
143C22⋊#6923—28no—
144C22⋊#6923—24no—
145#11758217—24no—
146#11758217—42no—
147C2×#92420—42no—
148C4⋊#6923—28no—
149C4⋊#6923—22no—
150C4⋊#6923—21no—
151C4⋊#6923—21no—
152C4⋊#6923—24no—
153C4⋊#6923C22⋊C412yes—
154C4⋊#6923C22⋊C414yes—
155C4⋊#6923C22⋊C412yes—
156C4×#6923—28no—
157C4×#6923—24no—
158#11758213—24no—
159#11758213—42no—
160D61⋊C8—24no—
161D61⋊C8—42no—
162C4⋊C4×C61—22no—
163C22⋊C4×C61—22no—
164C42×C61—22no—
165C22⋊Dic61—22no—
166#11758198—42no—
167C4⋊Dic61—22no—
168#11758196—42no—
169C4×Dic61—22no—
170C4×Dic61—42no—
171C23×C61#9512yes—
172C23×C61C2×C22⋊C411yescharacteristic
173C22×D61—22no—
174C22×D61—22no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—42no—
178C22×D61—42no—
179C22×D61#9512yes—
180C22×D61C2×C22⋊C411yescharacteristic
181C22×D61C2×C4⋊C411yescharacteristic
182C22×D61#12012yes—
183C22×D61#12312yes—
184C2×#6923—22no—
185C2×#6923—24no—
186C2×#6923—22no—
187C2×#6923—42no—
188C2×#6923—44no—
189C2×#6923—42no—
190C2×#6923—84no—
191C2×#6923—88no—
192Q8×C61—22no—
193D4×C61—24no—
194D4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—42no—
198C2×C4×C61#9512yes—
199C2×C4×C61C2×C4211yescharacteristic
200C2×C4×C61C2×C4⋊C411yescharacteristic
201C2×C4×C61C4×D412yes—
202C2×C4×C61C4×D412yes—
203C22⋊D61—24no—
204C22⋊D61—24no—
205C22⋊D61—24no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—42no—
209C2×Dic61#9512yes—
210C2×Dic61C2×C22⋊C411yescharacteristic
211C2×Dic61C2×C4⋊C411yescharacteristic
212C2×Dic61#12112yes—
213C2×Dic61#12112yes—
214D244—24no—
215D244—22no—
216C4×D61—22no—
217C4×D61—22no—
218C4×D61—22no—
219C4×D61—22no—
220C4×D61—22no—
221C4×D61—22no—
222C4×D61—82no—
223C4×D61—82no—
224C4×D61C2×C22⋊C411yescharacteristic
225C4×D61C2×C22⋊C411yescharacteristic
226C4×D61#11711yescharacteristic
227C4×D61#11711yescharacteristic
228C4×D61C4×D412yes—
229C4×D61C4×D412yes—
230C4×D61#12012yes—
231C4×D61C22⋊Q812yes—
232C4×D61C22⋊Q812yes—
233C4×D61#12312yes—
234Dic122—22no—
235Dic122—24no—
236#92420—44no—
237#10765—611nomaximal, Sylow
238C22×C61—22no—
239C22×C61—41no—
240C22×C61—42no—
241C22×C61C2×#9511yescharacteristic
242C22×C61#38612yes—
243C2×D61—22no—
244C2×D61—21no—
245C2×D61—22no—
246C2×D61—21no—
247C2×D61—22no—
248C2×D61—41no—
249C2×D61—42no—
250C2×D61—44no—
251C2×D61—42no—
252C2×D61—42no—
253C2×D61—42no—
254C2×D61#37911yescharacteristic
255C2×D61#38111yescharacteristic
256C2×D61#38512yes—
257#6923—44no—
258#6923—42no—
259#6923—42no—
260C4×C61—22no—
261C4×C61—82no—
262C4×C61C4×C22⋊C411yescharacteristic
263C4×C61C4×C22⋊C411yescharacteristic
264C4×C61#38512yes—
265Dic61—22no—
266Dic61—82no—
267Dic61#38011yescharacteristic
268Dic61#38011yescharacteristic
269Dic61#38612yes—
270C2×#520—611no—
271C2×#413—611no—
272C2×#410—611no—
273#1736—612no—
274#1733—612no—
275C2×C61—22no—
276C2×C61—21no—
277C2×C61—42no—
278C2×C61C22⋊#9511yescharacteristic
279D61—21no—
280D61—22no—
281D61—42no—
282D61#179511yescharacteristic
283D61#179611yescharacteristic
284C22×#41—611no—
285C22×M16—612no—
286#520—1224no—
287#520—1224no—
288C2×C4×Q8—1221no—
289C2×C4×D4—1221no—
290C2×C4×D4—1222no—
291C2×#117—611no—
292C2×#117—612no—
293C2×#117—612no—
294C22×C4⋊C4—611no—
295C22×C4⋊C4—612no—
296#413—614no—
297#413—1222no—
298C2×#101—1221no—
299C2×#100—1221no—
300#410—611no—
301#410—612no—
302#410—1222no—
303#410#692311yescharacteristic, complemented
304C2×#99—1222no—
305#340—1224no—
306#339—1224no—
307C61#1076511yescharacteristic, complemented, Sylow
308C2×#41—611no—
309C2×#41—611no—
310C2×#41—612no—
311C2×#41—612no—
312C2×#41—611no—
313C2×#41—1222no—
314C2×#41—1222no—
315C2×#41—1222no—
316C2×#41—1222no—
317C2×#41C2×#692311yescharacteristic
318C22×Q8—611no—
319C22×D4—611no—
320C22×D4—612no—
321C23×C4—611no—
322C23×C4—612no—
323C23×C4—1221no—
324C23×C4—1222no—
325C2×M16—614no—
326C2×M16—1222no—
327C2×M16—1222no—
328C2×M16—1222no—
329C2×M16—1222no—
330C2×M16—1222no—
331C22×C8—1222no—
332C4×Q8—1228no—
333C4×D4—1228no—
334C4×D4—12216no—
335#117—614no—
336#117—612no—
337#117—1222no—
338#117—1224no—
339#117—1224no—
340#117—1222no—
341#117—1222no—
342#117—1222no—
343#117C2×#692312yescomplemented
344C2×C4⋊C4—612no—
345C2×C4⋊C4—611no—
346C2×C4⋊C4—614no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—1222no—
349C2×C4⋊C4—1224no—
350C2×C4⋊C4—1224no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4—1222no—
354C2×C22⋊C4—1222no—
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1222no—
357C2×C42—1222no—
358C2×C42—1221no—
359C2×C42—1222no—
360#101—1224no—
361#100—1224no—
362#99—22no—
363#99—1224no—
364#99—1222no—
365C24—611no—
366C24—612no—
367#41—22no—
368#41—22no—
369#41—1224no—
370#41—1224no—
371#41—1224no—
372#41—1224no—
373#41—1222no—
374#41—1222no—
375#41—1222no—
376#41—1222no—
377#41—1222no—
378#41—1222no—
379C2×Q8—612no—
380C2×Q8—611no—
381C2×Q8—1222no—
382C2×Q8—1222no—
383C2×Q8C4⋊#692311yescharacteristic
384C2×D4—612no—
385C2×D4—612no—
386C2×D4—612no—
387C2×D4—612no—
388C2×D4—611no—
389C2×D4—1222no—
390C2×D4—1222no—
391C2×D4—1224no—
392C2×D4—1224no—
393C2×D4C4⋊#692311yescharacteristic
394C2×D4C22⋊#692312yes—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—611no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—612no—
401C22×C4—612no—
402C22×C4—612no—
403C22×C4—612no—
404C22×C4—612no—
405C22×C4—612no—
406C22×C4—612no—
407C22×C4—612no—
408C22×C4—1222no—
409C22×C4—1222no—
410C22×C4—1222no—
411C22×C4—1222no—
412C22×C4—1224no—
413C22×C4—1228no—
414C22×C4—1221no—
415C22×C4—1221no—
416C22×C4—1222no—
417C22×C4—4882no—
418C22×C4—4884no—
419C22×C4—2442no—
420C22×C4C4×#692312yes—
421C22×C4C22×#692311yescharacteristic
422M16—1224no—
423M16—1224no—
424M16—2442no—
425M16—2442no—
426C2×C8—1224no—
427C2×C8—2442no—
428C2×C8—2442no—
429C4⋊C4—22no—
430C4⋊C4—614no—
431C4⋊C4—612no—
432C4⋊C4—612no—
433C4⋊C4—1221no—
434C4⋊C4—1222no—
435C4⋊C4—1224no—
436C4⋊C4—1221no—
437C4⋊C4—1228no—
438C4⋊C4—1222no—
439C4⋊C4—2442no—
440C22⋊C4—22no—
441C22⋊C4—1224no—
442C22⋊C4—1228no—
443C22⋊C4—1222no—
444C22⋊C4—2442no—
445C42—22no—
446C42—1224no—
447C42—1228no—
448C42—1222no—
449C42—2442no—
450C23—612no—
451C23—611no—
452C23—612no—
453C23—611no—
454C23—612no—
455C23—1222no—
456C23—1222no—
457C23—2441no—
458C23—2441no—
459C23—2442no—
460C23—2442no—
461C23(C2×#6923)⋊C412yes—
462C23C2×C22⋊#692311yescharacteristic
463Q8—22no—
464Q8—1224no—
465Q8—1222no—
466D4—24no—
467D4—22no—
468D4—1224no—
469D4—1224no—
470D4—1224no—
471D4—1224no—
472D4—1222no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—42no—
476C2×C4—612no—
477C2×C4—612no—
478C2×C4—611no—
479C2×C4—611no—
480C2×C4—612no—
481C2×C4—612no—
482C2×C4—612no—
483C2×C4—612no—
484C2×C4—611no—
485C2×C4—611no—
486C2×C4—611no—
487C2×C4—611no—
488C2×C4—612no—
489C2×C4—612no—
490C2×C4—612no—
491C2×C4—1222no—
492C2×C4—1222no—
493C2×C4—1222no—
494C2×C4—1222no—
495C2×C4—1222no—
496C2×C4—1222no—
497C2×C4—1222no—
498C2×C4—1222no—
499C2×C4—1222no—
500C2×C4—1222no—
501C2×C4—1224no—
502C2×C4—4882no—
503C2×C4—4882no—
504C2×C4—4884no—
505C2×C4—4888no—
506C2×C4—2442no—
507C2×C4—2442no—
508C2×C4—2444no—
509C2×C4—2442no—
510C2×C4(C2×#6923)⋊C412yes—
511C2×C4C2×C4×#692311yescharacteristic
512C2×C4C2×C4⋊#692311yescharacteristic
513C2×C4D4×#692312yes—
514C2×C4D4×#692312yes—
515C8—2444no—
516C22—22no—
517C22—41no—
518C22—42no—
519C22—611no—
520C22—611no—
521C22—612no—
522C22—1222no—
523C22—1222no—
524C22—1221no—
525C22—1221no—
526C22—1222no—
527C22—2442no—
528C22—2441no—
529C22—2442no—
530C22—2442no—
531C22—2442no—
532C22—2444no—
533C22C2×(C2×#6923)⋊C411yescharacteristic
534C22#1684071012yes—
535C4—22no—
536C4—82no—
537C4—611no—
538C4—611no—
539C4—612no—
540C4—1222no—
541C4—4882no—
542C4—2444no—
543C4—2442no—
544C4—2442no—
545C4C22⋊C4×#692311yescharacteristic
546C4C22⋊C4×#692311yescharacteristic
547C4#1684070912yes—
548C2—22no—
549C2—21no—
550C2—42no—
551C2—611no—
552C2—611no—
553C2—1221no—
554C2—1222no—
555C2—2442no—
556C2#2146099211yescharacteristic
subgroupG/H
centerC2#21460992
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#410C4
socleC2×C61C22⋊#95
prime radicalC2×#41×D61C22
cosocleC22×C4×D61C23
hypercenter#410#6923
hypocenterC61#10765
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
3. #21461508C2semidirect
4. #21461500C2semidirect
12. C2×#14519856C4semidirect
14. C22×C4⋊#6923C4semidirect
17. #16840748C4semidirect
24. #16839736C4semidirect
25. #16839735C4semidirect
28. C61×#410C4semidirect
35. D61⋊M16C2×C4semidirect
74. #14519736C2×C4semidirect
84. #117×C61C2×C4semidirect
303. #410#6923semidirect
307. C61#10765semidirect
343. #117C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168408312
D61⋊#4132
#214615082
#214615002
#410×D612
#1076561