Finite Group Databasebeta

← #30616259

Subgroups — #30616259

Order 15616.

1598 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840743C212yesmaximal, complemented
2C2×#16840741C212yesmaximal, complemented
3#21462299C211yesmaximal, characteristic, complemented
4#21462257C211yesmaximal, characteristic, complemented
5#21461960C214yesmaximal, complemented
6#21461959C214yesmaximal, complemented
7C2×#16840342C211yesmaximal, characteristic
8C2×#14519856C2212yes—
9C2×#14519856C2212yes—
10#16840743—28no—
11#16840741—28no—
12C2×#14519836C2214yes—
13C2×#14519836C2218yes—
14C2×#14519835C2214yes—
15#16840713—24no—
16#16840704C2214yes—
17C4⋊C4×#6923—24no—
18C4×C22⋊#6923C2214yes—
19C22×#11758226C2211yescharacteristic
20C22×#11758222C2211yescharacteristic
21#16840342—24no—
22#16840342—24no—
23C2×#14519755C412yescomplemented
24C2×#14519753C411yescharacteristic, complemented
25C2×#14519752C411yescharacteristic, complemented
26C2×#14519752C2212yes—
27C2×#14519751C411yescharacteristic, complemented
28C2×#14519750C412yescomplemented
29C2×#14519750C2212yes—
30C2×C4⋊C4×D61C2211yescharacteristic
31C2×C22⋊Q8×C61C411yescharacteristic, complemented
32#14519856—24no—
33#14519856—24no—
34#14519856—24no—
35#14519856D418yescomplemented
36C2×C4⋊#6923—22no—
37C2×C4⋊#6923—22no—
38C2×C4⋊#6923—24no—
39C2×C22⋊#6923C2314yes—
40C2×C22⋊#6923C2314yes—
41C2×C4×#6923—22no—
42C2×C4×#6923—22no—
43C2×C4×#6923—24no—
44#14519836—28no—
45#14519836—216no—
46#14519835—28no—
47(C2×#6923)⋊C4—24no—
48(C2×#6923)⋊C4—24no—
49C2×#11758226—24no—
50C2×#11758226—22no—
51C2×#11758226—22no—
52C2×#11758226—24no—
53C2×#11758222—24no—
54C2×#11758222—22no—
55C2×#11758222—22no—
56C2×#11758222—24no—
57C22×Dic122C2×C411yescharacteristic
58C22×D244C2×C411yescharacteristic
59C22×D244C2×C412yes—
60C22×D244C2312yes—
61C22×D244C2311yescharacteristic
62C22×C22⋊D61C2×C411yescharacteristic
63C22×C22⋊D61C2×C412yes—
64C22×C22⋊D61C2311yescharacteristic
65C22×C4×D61C2×C412yes—
66C22×C4×D61C2×C411yescharacteristic
67C22×C4×D61C2312yes—
68C22×C4×D61C2311yescharacteristic
69#14519755—28no—
70#14519753—24no—
71#14519752—24no—
72#14519752—24no—
73#14519752—24no—
74#14519751—24no—
75#14519750—24no—
76#14519750—24no—
77#14519750—28no—
78C2×C4⋊Dic61C2×C411yescharacteristic
79C2×#11758196C2×C412yes—
80C2×#11758196C2×C412yes—
81C4⋊C4×D61D414yes—
82C4⋊C4×D61D414yes—
83C2×#11758198C2×C412yes—
84C2×#11758198C2×C412yes—
85C2×#11758198C2×C412yes—
86C22×Q8×C61C2×C411yescharacteristic
87C23×C4×C61C2×C411yescharacteristic
88C22⋊Q8×C61—24no—
89C2×C4⋊C4×C61C2×C412yes—
90C2×C4⋊C4×C61C2×C411yescharacteristic
91C2×C22⋊C4×C61C2×C412yes—
92C24×C61C22×C411yescharacteristic
93C23×D61C22×C412yes—
94C23×D61C22×C411yescharacteristic
95C23×D61C2411yescharacteristic
96C22×#6923—24no—
97C22×#6923—24no—
98C2×Q8×C61—22no—
99C2×Q8×C61—24no—
100C22×C4×C61—22no—
101C22×C4×C61—24no—
102C22×C4×C61C22×C412yes—
103C22×C4×C61C22×C411yescharacteristic
104C22×C4×C61C22×C412yes—
105C22×C4×C61C22×C411yescharacteristic
106C2×C22⋊D61—22no—
107C2×C22⋊D61—21no—
108C2×C22⋊D61—21no—
109C2×C22⋊D61—22no—
110C2×C22⋊D61—22no—
111C2×C22⋊D61—24no—
112C2×C22⋊D61—24no—
113C2×C22⋊D61—82no—
114C22×Dic61C22×C411yescharacteristic
115C22×Dic61C22×C412yes—
116C22×Dic61C22×C411yescharacteristic
117#11758226—44no—
118#11758226—44no—
119#11758226—44no—
120#11758226—44no—
121#11758222—44no—
122#11758222—44no—
123#11758222—44no—
124#11758222—44no—
125C2×D244—22no—
126C2×D244—21no—
127C2×D244—21no—
128C2×D244—24no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—24no—
133C2×D244—22no—
134C2×D244—24no—
135C2×D244—24no—
136C2×D244—82no—
137C2×C4×D61—22no—
138C2×C4×D61—24no—
139C2×C4×D61—24no—
140C2×C4×D61—28no—
141C2×C4×D61—82no—
142C2×C4×D61C2×D414yes—
143C2×C4×D61C2×D414yes—
144C2×C4×D61C2×D412yes—
145C2×C4×D61C2×D414yes—
146C2×C4×D61C2×D412yes—
147C2×C4×D61#4114yes—
148C2×Dic122—22no—
149C2×Dic122—24no—
150C22⋊#6923—24no—
151C22⋊#6923—24no—
152C22⋊#6923—24no—
153C22⋊#6923C2×D418yes—
154C4⋊#6923—28no—
155C4⋊#6923—44no—
156C4⋊#6923—84no—
157C4×#6923—28no—
158C4×#6923—44no—
159C4×#6923—84no—
160C4⋊C4×C61—24no—
161C4⋊C4×C61C22⋊C414yes—
162C22⋊C4×C61—24no—
163#11758198—24no—
164#11758198—24no—
165#11758198—24no—
166C4⋊Dic61C22⋊C414yes—
167#11758196—24no—
168#11758196C22⋊C418yes—
169C23×C61—21no—
170C23×C61—22no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61C23×C411yescharacteristic
174C22×D61—21no—
175C22×D61—22no—
176C22×D61—24no—
177C22×D61—22no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—42no—
182C22×D61—42no—
183C22×D61—41no—
184C22×D61C22×D411yescharacteristic
185C22×D61C22×D411yescharacteristic
186C22×D61C22×D412yes—
187C22×D61C2×#4112yes—
188C2×#6923—28no—
189C2×#6923—44no—
190C2×#6923—44no—
191C2×#6923—44no—
192C2×#6923—44no—
193C2×#6923—44no—
194Q8×C61—44no—
195C2×C4×C61—22no—
196C2×C4×C61—21no—
197C2×C4×C61—22no—
198C2×C4×C61—24no—
199C2×C4×C61—42no—
200C2×C4×C61—44no—
201C2×C4×C61C2×C22⋊C414yes—
202C2×C4×C61C2×C22⋊C412yes—
203C2×C4×C61C4×D414yes—
204C22⋊D61—42no—
205C22⋊D61—41no—
206C22⋊D61—41no—
207C22⋊D61—44no—
208C22⋊D61—84no—
209C2×Dic61—21no—
210C2×Dic61—22no—
211C2×Dic61—42no—
212C2×Dic61C2×C22⋊C414yes—
213C2×Dic61C2×C22⋊C412yes—
214C2×Dic61C4×D414yes—
215D244—42no—
216D244—44no—
217D244—41no—
218D244—42no—
219D244—41no—
220D244—42no—
221D244—44no—
222D244—84no—
223C4×D61—44no—
224C4×D61—44no—
225C4×D61—44no—
226C4×D61—48no—
227C4×D61#12114yes—
228C4×D61#12114yes—
229C4×D61#12414yes—
230C4×D61#12714yes—
231Dic122—44no—
232#21214—611nomaximal, Sylow
233C22×C61—21no—
234C22×C61—21no—
235C22×C61—41no—
236C22×C61—41no—
237C22×C61—41no—
238C22×C61—82no—
239C22×C61C22×C22⋊C411yescharacteristic
240C22×C61C2×C4×D412yes—
241C2×D61—21no—
242C2×D61—21no—
243C2×D61—21no—
244C2×D61—21no—
245C2×D61—42no—
246C2×D61—42no—
247C2×D61—41no—
248C2×D61—42no—
249C2×D61—41no—
250C2×D61—42no—
251C2×D61—42no—
252C2×D61—41no—
253C2×D61—84no—
254C2×D61—82no—
255C2×D61C2×#12112yes—
256C2×D61C2×#12411yescharacteristic
257C2×D61C2×#12711yescharacteristic
258#6923—48no—
259#6923—84no—
260C4×C61—24no—
261C4×C61—41no—
262C4×C61—42no—
263C4×C61#39014yes—
264Dic61—24no—
265Dic61—41no—
266Dic61#39014yes—
267C2×#537—611no—
268C2×#410—612no—
269C2×#410—612no—
270#2316—611no—
271#2274—611no—
272#1964—614no—
273#1964—614no—
274C2×C61—22no—
275C2×C61—81no—
276C2×C61C2×#39011yescharacteristic
277C2×C61#288311yescharacteristic
278C2×C61#288411yescharacteristic
279D61—22no—
280D61—21no—
281D61—21no—
282D61—81no—
283D61—82no—
284C22×#41—611no—
285C22×#41—611no—
286#537—1224no—
287#537—1224no—
288C2×C22⋊Q8—611no—
289C2×C22⋊Q8—612no—
290C2×C22⋊Q8#692311yescharacteristic, complemented
291C2×#121—611no—
292C2×#121—612no—
293C2×#121—611no—
294C2×#121—612no—
295C2×#121—612no—
296C2×#117—612no—
297C2×#117—612no—
298C22×C4⋊C4—611no—
299#410—1228no—
300#410—1228no—
301C2×#99—614no—
302C2×#99—614no—
303C2×#99—618no—
304#389—1224no—
305#380—614no—
306C4×C4⋊C4—1224no—
307C4×C22⋊C4—614no—
308C61#2121411yescharacteristic, complemented, Sylow
309C2×#41—1222no—
310C2×#41—1222no—
311C2×#41—1224no—
312C2×#41—1224no—
313C2×#41—1222no—
314C2×#41—1222no—
315C2×#41—1224no—
316C2×#41—1224no—
317C22×Q8—611no—
318C22×Q8C2×#692311yescharacteristic
319C22×D4—611no—
320C22×D4—611no—
321C22×D4—612no—
322C22×D4—611no—
323C22×D4—611no—
324C22×D4—612no—
325C22×D4—612no—
326C23×C4—611no—
327C23×C4—612no—
328C23×C4—612no—
329C23×C4—611no—
330C23×C4C2×#692311yescharacteristic
331C22⋊Q8—24no—
332C22⋊Q8—1228no—
333C22⋊Q8—1224no—
334#121—1228no—
335#121—1224no—
336#121—1224no—
337#121—1224no—
338#121—1224no—
339#121—1224no—
340#121—1224no—
341#117—618no—
342#117—1224no—
343#117—1224no—
344#117—1224no—
345C2×C4⋊C4—612no—
346C2×C4⋊C4—611no—
347C2×C4⋊C4—614no—
348C2×C4⋊C4—614no—
349C2×C4⋊C4—612no—
350C2×C4⋊C4—1222no—
351C2×C4⋊C4—1224no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4C2×#692312yes—
354C2×C4⋊C4C2×#692311yescharacteristic
355C2×C22⋊C4—614no—
356C2×C22⋊C4—612no—
357C2×C22⋊C4—612no—
358C2×C22⋊C4—612no—
359C2×C22⋊C4—614no—
360C2×C22⋊C4C2×#692312yes—
361C2×C42—1222no—
362C2×C42—1224no—
363C2×C42—1222no—
364#99—1228no—
365#99—1228no—
366#99—12216no—
367#95—1224no—
368#95—1224no—
369C24—611no—
370C24—611no—
371C24—612no—
372C24C22×#692311yescharacteristic
373#41—2444no—
374#41—2444no—
375#41—2444no—
376#41—2444no—
377#41—2444no—
378#41—2444no—
379#41—2444no—
380#41—2444no—
381C2×Q8—22no—
382C2×Q8—24no—
383C2×Q8—1222no—
384C2×Q8—1224no—
385C2×D4—1222no—
386C2×D4—1224no—
387C2×D4—1222no—
388C2×D4—1222no—
389C2×D4—1221no—
390C2×D4—1221no—
391C2×D4—1222no—
392C2×D4—1224no—
393C2×D4—1224no—
394C2×D4—1222no—
395C2×D4—1224no—
396C2×D4—1222no—
397C2×D4—1221no—
398C2×D4—1221no—
399C2×D4—1222no—
400C2×D4—1224no—
401C2×D4—1222no—
402C2×D4—1224no—
403C2×D4—4882no—
404C2×D4—4882no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—611no—
408C22×C4—612no—
409C22×C4—612no—
410C22×C4—614no—
411C22×C4—614no—
412C22×C4—612no—
413C22×C4—614no—
414C22×C4—614no—
415C22×C4—611no—
416C22×C4—1224no—
417C22×C4—1228no—
418C22×C4—1224no—
419C22×C4—1222no—
420C22×C4—1224no—
421C22×C4—1224no—
422C22×C4—4882no—
423C22×C4C22×#692311yescharacteristic
424C22×C4C22×#692312yes—
425C22×C4C22×#692312yes—
426C22×C4C22×#692311yescharacteristic
427C4⋊C4—24no—
428C4⋊C4—618no—
429C4⋊C4—614no—
430C4⋊C4—1224no—
431C4⋊C4—1228no—
432C4⋊C4—4884no—
433C4⋊C4—2444no—
434C4⋊C4C22⋊#692314yes—
435C22⋊C4—24no—
436C22⋊C4—618no—
437C22⋊C4—1224no—
438C22⋊C4—1224no—
439C22⋊C4—1224no—
440C22⋊C4—1224no—
441C22⋊C4—1224no—
442C22⋊C4—1224no—
443C42—1228no—
444C42—4884no—
445C42—2444no—
446C23—21no—
447C23—22no—
448C23—41no—
449C23—41no—
450C23—611no—
451C23—611no—
452C23—612no—
453C23—612no—
454C23—1224no—
455C23—1222no—
456C23—1221no—
457C23—1222no—
458C23—2441no—
459C23—2441no—
460C23—2441no—
461C23—2442no—
462C23—2441no—
463C23—2442no—
464C23C23×#692311yescharacteristic
465Q8—44no—
466Q8—2444no—
467D4—4884no—
468D4—4884no—
469D4—2442no—
470D4—2441no—
471D4—2441no—
472D4—2444no—
473D4—2442no—
474D4—2442no—
475D4—2444no—
476D4—2442no—
477D4—2441no—
478D4—2441no—
479D4—2444no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—612no—
487C2×C4—614no—
488C2×C4—614no—
489C2×C4—614no—
490C2×C4—614no—
491C2×C4—614no—
492C2×C4—614no—
493C2×C4—1228no—
494C2×C4—1221no—
495C2×C4—1222no—
496C2×C4—2444no—
497C2×C4—2444no—
498C2×C4—2448no—
499C2×C4—2442no—
500C2×C4—2444no—
501C2×C4—2444no—
502C2×C4—2444no—
503C2×C4—2444no—
504C2×C4—2444no—
505C2×C4—2444no—
506C2×C4C2×C22⋊#692314yes—
507C2×C4C2×C22⋊#692312yes—
508C2×C4D4×#692314yes—
509C22—21no—
510C22—21no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—82no—
515C22—612no—
516C22—611no—
517C22—611no—
518C22—1221no—
519C22—1221no—
520C22—1221no—
521C22—1221no—
522C22—4884no—
523C22—4882no—
524C22—2442no—
525C22—2442no—
526C22—2442no—
527C22—2441no—
528C22—2441no—
529C22—2442no—
530C22—2441no—
531C22—2442no—
532C22C22×C22⋊#692311yescharacteristic
533C22C2×D4×#692312yes—
534C4—24no—
535C4—41no—
536C4—42no—
537C4—614no—
538C4—1224no—
539C4—4884no—
540C4—2448no—
541C4—2441no—
542C4#1684071414yes—
543C2—22no—
544C2—81no—
545C2—1221no—
546C2—1222no—
547C2—1221no—
548C2—4882no—
549C2—4881no—
550C2C2×#1684071411yescharacteristic
551C2#2146241911yescharacteristic
552C2#2146242011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840342C2
cosocleC23×D61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61#21214
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16840743C2semidirect
2. C2×#16840741C2semidirect
3. #21462299C2semidirect
4. #21462257C2semidirect
5. #21461960C2semidirect
6. #21461959C2semidirect
23. C2×#14519755C4semidirect
24. C2×#14519753C4semidirect
25. C2×#14519752C4semidirect
27. C2×#14519751C4semidirect
28. C2×#14519750C4semidirect
31. C2×C22⋊Q8×C61C4semidirect
35. #14519856D4semidirect
290. C2×C22⋊Q8#6923semidirect
308. C61#21214semidirect

Maximal subgroups

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

subgroupindex
C2×#168407432
C2×#168407412
#214622992
#214622572
#214619602
#214619592
C2×#168403422
#2121461