Finite Group Databasebeta

← #30603017

Subgroups — #30603017

Order 15616.

1118 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519842C212yesmaximal
2C2×C4×#11758213C211yesmaximal, characteristic, complemented
3C2×#16840632C212yesmaximal, complemented
4C4×(C2×#6923)⋊C4C211yesmaximal, characteristic, complemented
5#21448308C211yesmaximal, characteristic
6C22×#11758213C2212yes—
7C22×C4×#6923C2212yes—
8C2×#14519842—24no—
9C2×#14519842—24no—
10C2×#14519842C418yescomplemented
11C4×#11758213—24no—
12C2×#14519832—21no—
13C2×C4×#92420—21no—
14C2×(C2×#6923)⋊C4—21no—
15C2×(C2×#6923)⋊C4—21no—
16#16840632—24no—
17C22×#11758226C2211yescharacteristic
18C2×#14519735C2211yescharacteristic
19C2×C42×D61C2211yescharacteristic
20#16839669C411yescharacteristic, complemented
21#16839664C411yescharacteristic, complemented
22#16839663C411yescharacteristic, complemented
23#16839640—21no—
24#16839640—21no—
25#16839609—21no—
26#16839609—21no—
27C61×#391C411yescharacteristic, complemented
28C23×#6923—22no—
29C2×#11758213—22no—
30C2×#11758213—22no—
31C2×#11758213—22no—
32C2×#11758213C2×C414yes—
33C2×#11758213D414yes—
34C2×#11758213D414yes—
35C22×#92420—22no—
36C2×C4×#6923—22no—
37C2×C4×#6923—22no—
38C2×C4×#6923—22no—
39C2×C4×#6923—22no—
40C2×C4×#6923—22no—
41C2×C4×#6923C2×C414yes—
42#14519842—216no—
43#14519832—24no—
44C4×#92420—24no—
45(C2×#6923)⋊C4—24no—
46(C2×#6923)⋊C4—24no—
47C2×#11758226—22no—
48C2×#11758226—22no—
49C2×#11758226—22no—
50C2×#11758226—22no—
51C2×#11758226—22no—
52C2×#11758226C2×C414yes—
53C22×D244—21no—
54C22×D244C2×C411yescharacteristic
55C22×C4×D61—21no—
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61C2311yescharacteristic
59#14519735—22no—
60#14519735—22no—
61C2×C4⋊Dic61C2×C411yescharacteristic
62C2×#11758198—21no—
63C2×#11758198—21no—
64C2×#11758198—21no—
65C2×#11758198—21no—
66C2×#11758198—21no—
67C2×C4×Dic61—21no—
68C2×C4×Dic61C2×C411yescharacteristic
69C2×C4×Dic61D411yescharacteristic
70C2×C4×Dic61D411yescharacteristic
71C42×D61—22no—
72C42×D61—22no—
73#14519684—21no—
74#14519684—21no—
75C22×Q8×C61C2×C411yescharacteristic
76C2×C4⋊C4×C61C2×C411yescharacteristic
77C2×C42×C61C2×C411yescharacteristic
78#95×C61—21no—
79#95×C61—21no—
80C23×D61—21no—
81C23×D61C2×D411yescharacteristic
82C22×#6923—24no—
83C22×#6923—24no—
84C22×#6923—24no—
85C22×#6923—42no—
86C22×#6923—42no—
87C22×#6923—42no—
88C22×#6923—42no—
89C2×Q8×C61—21no—
90C2×Q8×C61—22no—
91C2×Q8×C61—22no—
92C2×Q8×C61C4212yes—
93C22×C4×C61—21no—
94C22×C4×C61—21no—
95C22×C4×C61C22⋊C411yescharacteristic
96C22×C4×C61C22⋊C411yescharacteristic
97C22×C4×C61C22×C411yescharacteristic
98C22×Dic61—21no—
99C22×Dic61C22⋊C411yescharacteristic
100C22×Dic61C22⋊C411yescharacteristic
101C22×Dic61C2×D411yescharacteristic
102#11758226—24no—
103#11758226—24no—
104#11758226—24no—
105#11758226—24no—
106#11758226—42no—
107#11758226—42no—
108#11758226—42no—
109#11758226—42no—
110C2×D244—22no—
111C2×D244—22no—
112C2×D244—24no—
113C2×D244—21no—
114C2×D244—42no—
115C2×D244—41no—
116C2×D244—41no—
117C2×D244C4212yes—
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—24no—
123C2×C4×D61—41no—
124C2×C4×D61—41no—
125C2×C4×D61—41no—
126C2×C4×D61—42no—
127C2×C4×D61—41no—
128C2×C4×D61C22×C412yes—
129C2×C4×D61C2×D412yes—
130C2×C4×D61C2×D412yes—
131C2×C4×D61#4112yes—
132C2×C4×D61#4112yes—
133C2×C4×D61#4112yes—
134C2×#92420—24no—
135C2×#92420—24no—
136C2×#92420—24no—
137C4×#6923—24no—
138C4×#6923—24no—
139C4×#6923—42no—
140C4×#6923—42no—
141#11758213—24no—
142#11758213—24no—
143#11758213—24no—
144#11758213C22⋊C418yescomplemented
145C4⋊C4×C61—22no—
146C42×C61—22no—
147#11758198—42no—
148#11758198—42no—
149#11758198—41no—
150#11758198—41no—
151#11758198—42no—
152#11758198—42no—
153C4⋊Dic61—22no—
154C4×Dic61—22no—
155C4×Dic61—21no—
156C4×Dic61—21no—
157C4×Dic61—21no—
158C4×Dic61—21no—
159C4×Dic61—42no—
160C23×C61C2×C22⋊C411yescharacteristic
161C22×D61—21no—
162C22×D61—21no—
163C22×D61—21no—
164C22×D61—21no—
165C22×D61—22no—
166C22×D61—41no—
167C22×D61—41no—
168C22×D61—41no—
169C22×D61—41no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61C4×D412yes—
173C22×D61#12112yes—
174C22×D61#12411yescharacteristic
175C22×D61#12711yescharacteristic
176C2×#6923—28no—
177C2×#6923—44no—
178C2×#6923—44no—
179C2×#6923—44no—
180C2×#6923—44no—
181C2×#6923—44no—
182C2×#6923—44no—
183Q8×C61—24no—
184Q8×C61—42no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—22no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61#10412yescomplemented
197C2×C4×C61C2×C4211yescharacteristic
198C2×C4×C61#11711yescharacteristic
199C2×C4×C61C4×D412yes—
200C2×C4×C61C4×D412yes—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61#10412yescomplemented
210C2×Dic61C4×D412yes—
211C2×Dic61#12011yescharacteristic
212C2×Dic61#12112yes—
213C2×Dic61#12311yescharacteristic
214D244—24no—
215D244—44no—
216D244—44no—
217D244—42no—
218C4×D61—44no—
219C4×D61—44no—
220C4×D61—44no—
221C4×D61—44no—
222C4×D61C2×C22⋊C412yes—
223C4×D61#11712yes—
224C4×D61C4×D412yes—
225C4×D61C4×D412yes—
226C4×D61#12112yes—
227C4×D61#12312yes—
228C4×D61#12412yes—
229C4×D61#12612yes—
230#92420—28no—
231#10933—611nomaximal, Sylow
232C22×C61C4×C22⋊C411yescharacteristic
233C22×C61#38611yescharacteristic
234C22×C61#39011yescharacteristic
235C22×C61C2×#10411yescharacteristic
236C22×C61C2×#10411yescharacteristic
237C22×C61#42111yescharacteristic
238C22×C61#42111yescharacteristic
239C2×D61—22no—
240C2×D61—22no—
241C2×D61—21no—
242C2×D61—21no—
243C2×D61—21no—
244C2×D61—21no—
245C2×D61—22no—
246C2×D61—22no—
247C2×D61—42no—
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—81no—
251C2×D61—81no—
252C2×D61—81no—
253C2×D61—81no—
254C2×D61#38812yes—
255C2×D61#39012yes—
256#6923—48no—
257C4×C61—22no—
258C4×C61—41no—
259C4×C61—42no—
260C4×C61—81no—
261C4×C61C4×C22⋊C412yes—
262C4×C61#38812yes—
263Dic61—22no—
264Dic61—41no—
265Dic61—81no—
266Dic61#38612yes—
267Dic61#38812yes—
268C22×#104—612no—
269C2×C4×M16—611no—
270#2297—611no—
271C2×#339—612no—
272C4×#95—611no—
273C2×C61#143711yescharacteristic
274C2×C61C4×#10411yescharacteristic
275C2×C61#175911yescharacteristic
276C2×C61#187111yescharacteristic
277C2×C61#187111yescharacteristic
278C2×C61#198011yescharacteristic
279C2×C61#198011yescharacteristic
280D61—22no—
281D61—22no—
282D61—82no—
283C22×#41—611no—
284C22×M16—612no—
285C2×#117—611no—
286C22×C42—611no—
287C22×C42—612no—
288C2×#104—618no—
289C2×#104—1224no—
290C2×#104—1224no—
291C4×M16—1224no—
292C2×#97—1221no—
293C2×C4×C8—1221no—
294#391—611no—
295#391#692311yescharacteristic, complemented
296#390—611no—
297#390—611no—
298#386—1221no—
299#386—1221no—
300C4×C22⋊C4—1221no—
301C4×C22⋊C4—1221no—
302C2×#95—1221no—
303C2×#95—1221no—
304#339—1224no—
305C61#1093311yescharacteristic, complemented, Sylow
306C2×#41—614no—
307C2×#41—1222no—
308C2×#41—1222no—
309C2×#41—1222no—
310C2×#41—1222no—
311C2×#41—1222no—
312C22×Q8C2×#692311yescharacteristic
313C22×D4—611no—
314C22×D4—1221no—
315C23×C4—611no—
316C23×C4—611no—
317C23×C4—611no—
318C23×C4—1221no—
319C23×C4—1222no—
320C2×M16—614no—
321C2×M16—614no—
322C2×M16—614no—
323C2×M16—1222no—
324C2×M16—1222no—
325C2×M16—1222no—
326C22×C8—1222no—
327#117—1222no—
328#117—1222no—
329C2×C4⋊C4—611no—
330C2×C4⋊C4C2×#692311yescharacteristic
331C2×C22⋊C4—1221no—
332C2×C22⋊C4—1221no—
333C2×C22⋊C4—1221no—
334C2×C22⋊C4—1221no—
335C2×C22⋊C4—1221no—
336C2×C42—611no—
337C2×C42—611no—
338C2×C42—611no—
339C2×C42—614no—
340C2×C42—1222no—
341C2×C42—1222no—
342C2×C42—1222no—
343C2×C42—1222no—
344C2×C42—1222no—
345C2×C42—1222no—
346C2×C42—1221no—
347C2×C42—1222no—
348C2×C42C2×#692311yescharacteristic
349#104—12216no—
350#97—1224no—
351C4×C8—1224no—
352#95—21no—
353#95—21no—
354#95—1221no—
355#95—1221no—
356#95—1224no—
357#95—1224no—
358C24—611no—
359C24—1221no—
360#41—1224no—
361#41—1224no—
362#41—1224no—
363#41—1224no—
364#41—2442no—
365#41—2442no—
366#41—2442no—
367#41—2442no—
368C2×Q8—21no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8C4×#692312yes—
372C2×D4—612no—
373C2×D4—1224no—
374C2×D4—1221no—
375C2×D4—1222no—
376C2×D4—1222no—
377C2×D4—2442no—
378C2×D4—2441no—
379C2×D4—2441no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—611no—
383C22×C4—612no—
384C22×C4—612no—
385C22×C4—612no—
386C22×C4—612no—
387C22×C4—612no—
388C22×C4—612no—
389C22×C4—611no—
390C22×C4—611no—
391C22×C4—1222no—
392C22×C4—1224no—
393C22×C4—1222no—
394C22×C4—1222no—
395C22×C4—1224no—
396C22×C4—1224no—
397C22×C4—1224no—
398C22×C4—1222no—
399C22×C4—1221no—
400C22×C4—2441no—
401C22×C4—2441no—
402C22×C4—2441no—
403C22×C4—2441no—
404C22×C4—2442no—
405C22×C4—2442no—
406C22×C4—2442no—
407C22×C4—2442no—
408C22×C4—2442no—
409C22×C4C22⋊#692311yescharacteristic
410C22×C4C22⋊#692311yescharacteristic
411C22×C4C22×#692311yescharacteristic
412M16—618no—
413M16—1224no—
414M16—1224no—
415M16—1224no—
416C2×C8—1224no—
417C2×C8—1224no—
418C2×C8—1224no—
419C4⋊C4—22no—
420C4⋊C4—1222no—
421C22⋊C4—2442no—
422C22⋊C4—2442no—
423C22⋊C4—2442no—
424C22⋊C4—2441no—
425C22⋊C4—2441no—
426C22⋊C4—2442no—
427C42—22no—
428C42—1221no—
429C42—1221no—
430C42—1221no—
431C42—1221no—
432C42—1224no—
433C42—1224no—
434C42—1222no—
435C42—2442no—
436C42—2442no—
437C42—2442no—
438C23—612no—
439C23—611no—
440C23—611no—
441C23—612no—
442C23—1221no—
443C23—1221no—
444C23—1221no—
445C23—1221no—
446C23—1222no—
447C23—2441no—
448C23—2441no—
449C23—2441no—
450C23—2441no—
451C23—2441no—
452C23—2441no—
453C23C2×C22⋊#692311yescharacteristic
454Q8—24no—
455Q8—42no—
456D4—1224no—
457D4—2444no—
458D4—2444no—
459D4—2442no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—611no—
472C2×C4—611no—
473C2×C4—612no—
474C2×C4—612no—
475C2×C4—612no—
476C2×C4—612no—
477C2×C4—612no—
478C2×C4—612no—
479C2×C4—612no—
480C2×C4—612no—
481C2×C4—612no—
482C2×C4—612no—
483C2×C4—612no—
484C2×C4—1221no—
485C2×C4—1221no—
486C2×C4—1228no—
487C2×C4—1221no—
488C2×C4—1221no—
489C2×C4—1221no—
490C2×C4—2444no—
491C2×C4—2444no—
492C2×C4—2444no—
493C2×C4—2444no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4—2441no—
497C2×C4—2444no—
498C2×C4—2444no—
499C2×C4—2444no—
500C2×C4—2444no—
501C2×C4—2444no—
502C2×C4—2444no—
503C2×C4#1451984212yescomplemented
504C2×C4C2×C4×#692311yescharacteristic
505C2×C4#1451985611yescharacteristic
506C2×C4D4×#692312yes—
507C2×C4D4×#692312yes—
508C8—1228no—
509C22—612no—
510C22—612no—
511C22—1222no—
512C22—1222no—
513C22—1222no—
514C22—1222no—
515C22—1221no—
516C22—1221no—
517C22—1221no—
518C22—1221no—
519C22—4881no—
520C22—4881no—
521C22—4881no—
522C22—4881no—
523C22—2442no—
524C22—2442no—
525C22—2442no—
526C22C4×C22⋊#692311yescharacteristic
527C22#1684071011yescharacteristic
528C22#1684071411yescharacteristic
529C22D61⋊#10411yescharacteristic
530C22C2×#1451984211yescharacteristic
531C22#1684075911yescharacteristic
532C22#1684076111yescharacteristic
533C4—22no—
534C4—42no—
535C4—41no—
536C4—81no—
537C4—612no—
538C4—612no—
539C4—1222no—
540C4—4881no—
541C4—2448no—
542C4—2441no—
543C4C22⋊C4×#692312yes—
544C4#1684071212yes—
545C2—1222no—
546C2—1222no—
547C2—4882no—
548C2#2146100011yescharacteristic
549C2C4×#1451984211yescharacteristic
550C2#2146158311yescharacteristic
551C2#2146180011yescharacteristic
552C2#2146180111yescharacteristic
553C2#2146198211yescharacteristic
554C2#2146198411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC23×C61C2×C22⋊C4
prime radicalC22×#11758226C22
cosocleC22×C4×D61C23
hypercenter#391#6923
hypocenterC61#10933
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11758213C2semidirect
3. C2×#16840632C2semidirect
4. C4×(C2×#6923)⋊C4C2semidirect
10. C2×#14519842C4semidirect
20. #16839669C4semidirect
21. #16839664C4semidirect
22. #16839663C4semidirect
27. C61×#391C4semidirect
144. #11758213C22⋊C4semidirect
196. C2×C4×C61#104semidirect
209. C2×Dic61#104semidirect
295. #391#6923semidirect
305. C61#10933semidirect
503. C2×C4#14519842semidirect

Maximal subgroups

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

subgroupindex
C22×#145198422
C2×C4×#117582132
C2×#168406322
C4×(C2×#6923)⋊C42
#214483082
#1093361