Finite Group Databasebeta

← #30615519

Subgroups — #30615519

Order 15616.

1382 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840830C211yesmaximal, characteristic
2C2×#16840772C211yesmaximal, characteristic
3C2×#16840755C211yesmaximal, characteristic
4#21462299C211yesmaximal, characteristic
5#21461785C214yesmaximal, complemented
6#21461783C214yesmaximal, complemented
7#21461506C211yesmaximal, characteristic
8#21461484C211yesmaximal, characteristic
9C2×#103×D61C211yesmaximal, characteristic
10C2×D61⋊M16C2211yescharacteristic
11#16840830—24no—
12#16840830—24no—
13C2×Q8×#6923—22no—
14C2×Q8×#6923C2214yes—
15C2×#14519856—22no—
16C2×#14519856C2211yescharacteristic
17C2×#14519856C2211yescharacteristic
18C2×#14519856C2211yescharacteristic
19#16840772—24no—
20C2×#14519846C2212yes—
21C2×#14519845C2212yes—
22#16840755—24no—
23C2×#14519840C2214yes—
24#16840713—22no—
25#16840713C2214yes—
26#16840704—22no—
27#16840635C2214yes—
28(C2×#6923)⋊C8C2212yes—
29(C2×#6923)⋊C8C2212yes—
30C22×Q8×D61C2211yescharacteristic
31C22×C8×D61C2211yescharacteristic
32C2×C4⋊C4×D61C2211yescharacteristic
33C2×#14519702C411yescharacteristic, complemented
34C2×#14519701C411yescharacteristic, complemented
35C2×#14519700C411yescharacteristic, complemented
36#103×D61C412yescomplemented
37#103×D61C412yescomplemented
38#103×D61C2212yes—
39#103×D61C2212yes—
40C2×#103×C61C411yescharacteristic, complemented
41C2×#11758213—21no—
42D61⋊M16—22no—
43D61⋊M16—22no—
44C2×#11758217—21no—
45C2×D61⋊C8C2312yes—
46Q8×#6923—216no—
47Q8×#6923—28no—
48Q8×#6923—28no—
49#14519856—216no—
50#14519856—22no—
51#14519856—22no—
52#14519856—22no—
53#14519856—22no—
54#14519856—22no—
55#14519856D414yes—
56C2×C4⋊#6923—24no—
57C2×C4⋊#6923—24no—
58C2×C4⋊#6923D412yes—
59C2×C4⋊#6923C2312yes—
60C2×C4⋊#6923C2312yes—
61C2×C22⋊#6923—21no—
62C2×C22⋊#6923—21no—
63C2×C22⋊#6923D412yes—
64C2×C4×#6923—24no—
65C2×C4×#6923—21no—
66C2×C4×#6923—21no—
67C2×C4×#6923C2312yes—
68#14519846—24no—
69#14519845—24no—
70#14519840—28no—
71(C2×#6923)⋊C4—24no—
72C22×Dic122—21no—
73C22×Dic122C2×C411yescharacteristic
74C2×Q8×D61—21no—
75C2×Q8×D61—22no—
76C2×Q8×D61—22no—
77C2×Q8×D61—21no—
78C2×Q8×D61C2×C412yes—
79C2×Q8×D61D414yes—
80C2×Q8×D61Q814yes—
81C2×Q8×D61C2312yes—
82C22×C4×D61—21no—
83C22×C4×D61—21no—
84C22×C4×D61C2311yescharacteristic
85C22×#92418C2×C411yescharacteristic
86C2×C8×D61—22no—
87C2×C8×D61—22no—
88C2×C8×D61C2×C411yescharacteristic
89C2×C8×D61C2×C411yescharacteristic
90C2×C8×D61C2311yescharacteristic
91C2×C8×D61C2311yescharacteristic
92C2×C4⋊Dic61C2×C411yescharacteristic
93C2×#11758196—21no—
94C4⋊C4×D61—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61C2×C412yes—
97C4⋊C4×D61C2312yes—
98#14519702C2×C412yes—
99#14519702C2×C412yes—
100#14519701C2×C412yes—
101#14519701C2×C412yes—
102#14519700C2×C412yes—
103#14519700C2×C412yes—
104C22×Q8×C61C2×C411yescharacteristic
105C22×C8×C61C2×C411yescharacteristic
106C2×C4⋊C4×C61C2×C411yescharacteristic
107#103×C61C2×C412yes—
108#103×C61C2×C412yes—
109C23×D61C2×D411yescharacteristic
110C22×#6923—22no—
111C22×#6923—22no—
112C22×#6923C2×D414yes—
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61C4⋊C414yescomplemented
116C2×Q8×C61C22×C412yes—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61C22×C411yescharacteristic
120C22×Dic61—21no—
121C22×Dic61—21no—
122C22×Dic61#4111yescharacteristic
123Q8×D61—24no—
124Q8×D61—24no—
125Q8×D61—24no—
126Q8×D61—24no—
127Q8×D61—24no—
128Q8×D61—22no—
129Q8×D61—22no—
130Q8×D61C4⋊C418yescomplemented
131Q8×D61C2×D414yes—
132Q8×D61C2×Q814yes—
133C2×C4×D61—22no—
134C2×C4×D61—24no—
135C2×C4×D61—24no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—82no—
140C2×C4×D61C22×C411yescharacteristic
141C2×C4×D61C2×D412yes—
142C2×C4×D61C2×D412yes—
143C2×C4×D61C2×D411yescharacteristic
144C2×C4×D61C2×Q812yes—
145C2×C4×D61#4112yes—
146C2×C4×D61#4111yescharacteristic
147C2×C4×D61C2411yescharacteristic
148C2×Dic122—22no—
149C2×Dic122—21no—
150C2×Dic122—22no—
151C2×Dic122—24no—
152C2×Dic122—24no—
153C2×Dic122—22no—
154C2×Dic122C4⋊C414yescomplemented
155C2×Dic122C22×C412yes—
156C22⋊#6923—24no—
157C22⋊#6923—24no—
158C22⋊#6923—42no—
159#11758217—42no—
160C2×#92420—22no—
161C4⋊#6923—216no—
162C4⋊#6923—24no—
163C4⋊#6923—22no—
164C4⋊#6923—22no—
165C4⋊#6923—22no—
166C4⋊#6923—84no—
167C4⋊#6923C2×D414yes—
168C4×#6923—216no—
169C4×#6923—24no—
170C4×#6923—22no—
171C4×#6923—22no—
172C4×#6923—42no—
173#11758213—42no—
174D61⋊C8—22no—
175D61⋊C8—22no—
176C2×C8×C61—22no—
177C2×C8×C61C22×C411yescharacteristic
178C2×C8×C61C22×C411yescharacteristic
179C4⋊C4×C61—21no—
180C4⋊C4×C61C22×C412yes—
181C4⋊Dic61—21no—
182C4⋊Dic61C22×C412yes—
183#11758196—22no—
184#11758196—22no—
185C2×#92418—22no—
186C2×#92418C22×C411yescharacteristic
187C2×#92418C22×C411yescharacteristic
188C8×D61—22no—
189C8×D61—22no—
190C8×D61—21no—
191C8×D61—21no—
192C8×D61—21no—
193C8×D61—21no—
194C23×C61C4×D411yescharacteristic
195C22×D61—22no—
196C22×D61—22no—
197C22×D61C4×D411yescharacteristic
198C22×D61#12012yes—
199C22×D61C22⋊Q812yes—
200C22×D61C22×D411yescharacteristic
201C2×#6923—24no—
202C2×#6923—24no—
203C2×#6923—24no—
204C2×#6923—24no—
205C2×#6923—42no—
206C2×#6923—42no—
207C2×#6923—42no—
208C2×#6923—42no—
209C2×#6923—42no—
210Q8×C61—22no—
211Q8×C61—24no—
212Q8×C61C2×C4⋊C414yes—
213C2×C4×C61—22no—
214C2×C4×C61—24no—
215C2×C4×C61—22no—
216C2×C4×C61—81no—
217C2×C4×C61C2×C4⋊C412yes—
218C2×C4×C61C4×D412yes—
219C2×C4×C61C4×D411yescharacteristic
220C2×C4×C61C23×C411yescharacteristic
221C2×Dic61—22no—
222C2×Dic61—24no—
223C2×Dic61—22no—
224C2×Dic61—81no—
225C2×Dic61C4×D411yescharacteristic
226C2×Dic61C22⋊Q812yes—
227C2×Dic61#12312yes—
228C2×Dic61C2×#4111yescharacteristic
229C4×D61—24no—
230C4×D61—24no—
231C4×D61—28no—
232C4×D61—44no—
233C4×D61—82no—
234C4×D61C2×C4⋊C412yes—
235C4×D61C4×D412yes—
236C4×D61#12012yes—
237C4×D61C22⋊Q812yes—
238C4×D61C22⋊Q812yes—
239C4×D61#12312yes—
240C4×D61C22×D411yescharacteristic
241C4×D61C22×D411yescharacteristic
242C4×D61C22×Q811yescharacteristic
243C4×D61C2×#4111yescharacteristic
244Dic122—22no—
245Dic122—24no—
246Dic122—24no—
247Dic122—24no—
248Dic122—28no—
249Dic122C2×C4⋊C414yes—
250#92420—42no—
251C8×C61—21no—
252C8×C61—21no—
253#92418—21no—
254#92418—21no—
255#20880—611nomaximal, Sylow
256C22×C61—21no—
257C22×C61—21no—
258C22×C61#38512yes—
259C22×C61C2×C4×D411yescharacteristic
260C2×D61—21no—
261C2×D61—21no—
262C2×D61—24no—
263C2×D61—24no—
264C2×D61#38512yes—
265C2×D61C2×#12011yescharacteristic
266C2×D61C2×C22⋊Q811yescharacteristic
267C2×D61#57712yes—
268C2×D61#57812yes—
269#6923—44no—
270#6923—44no—
271#6923—82no—
272C4×C61—24no—
273C4×C61—42no—
274C4×C61#38512yes—
275C4×C61C22×C4⋊C411yescharacteristic
276C4×C61C2×C4×D411yescharacteristic
277Dic61—24no—
278Dic61—42no—
279Dic61#38512yes—
280Dic61C2×C22⋊Q811yescharacteristic
281Dic61C2×#12311yescharacteristic
282C2×#519—611no—
283C2×#427—611no—
284C2×#419—611no—
285C22×#103—611no—
286#2316—611no—
287#1865—614no—
288#1863—614no—
289#1736—611no—
290#1729—611no—
291C2×C61—22no—
292C2×C61C2×#38511yescharacteristic
293C2×C61#294911yescharacteristic
294C2×C61#295011yescharacteristic
295D61—22no—
296D61#301312yes—
297D61#308212yes—
298C23×Q8—611no—
299C22×M16—611no—
300C23×C8—611no—
301#519—1224no—
302#519—1224no—
303C2×C4×Q8—614no—
304C2×C4×Q8—1222no—
305C2×#117—611no—
306C2×#117—611no—
307C2×#117—611no—
308C2×#117—1222no—
309C22×C4⋊C4—611no—
310#427—1224no—
311C2×#107—612no—
312C2×#106—612no—
313#419—1224no—
314C2×#103—611no—
315C2×#103—611no—
316C2×#103—611no—
317C2×#103—612no—
318C2×#103—612no—
319C2×#103—612no—
320C2×#103—612no—
321C2×#103—614no—
322C2×#103#692311yescharacteristic, complemented
323#389—614no—
324#389—1222no—
325#380—1222no—
326#340—614no—
327#336—612no—
328#336—612no—
329C61#2088011yescharacteristic, complemented, Sylow
330C22×Q8—614no—
331C22×Q8—614no—
332C22×Q8—612no—
333C22×Q8—612no—
334C22×Q8—611no—
335C22×Q8—1221no—
336C22×Q8—1221no—
337C22×Q8—1222no—
338C22×Q8—1221no—
339C22×Q8—1222no—
340C22×Q8C2×#692311yescharacteristic
341C23×C4—611no—
342C23×C4—1221no—
343C23×C4—1221no—
344C2×M16—1221no—
345C2×M16—1222no—
346C2×M16—1222no—
347C2×M16—1221no—
348C22×C8—611no—
349C22×C8—611no—
350C22×C8—611no—
351C22×C8—611no—
352C22×C8—611no—
353C22×C8—612no—
354C22×C8—1222no—
355C22×C8—1222no—
356C22×C8C2×#692311yescharacteristic
357C4×Q8—1228no—
358C4×Q8—1228no—
359C4×Q8—12216no—
360#117—614no—
361#117—1222no—
362#117—1222no—
363#117—1222no—
364#117—12216no—
365#117—1222no—
366#117—1222no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—612no—
370C2×C4⋊C4—612no—
371C2×C4⋊C4—612no—
372C2×C4⋊C4—612no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1224no—
377C2×C4⋊C4—1224no—
378C2×C4⋊C4C2×#692311yescharacteristic
379C2×C22⋊C4—612no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1221no—
382C2×C42—612no—
383C2×C42—1221no—
384C2×C42—1224no—
385C2×C42—1221no—
386#107—1224no—
387#106—1224no—
388#103—612no—
389#103—612no—
390#103—612no—
391#103—612no—
392#103—612no—
393#103—612no—
394#103—1228no—
395#103C2×#692312yes—
396#103C2×#692312yes—
397#95—1224no—
398C24—611no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—614no—
402C2×Q8—618no—
403C2×Q8—614no—
404C2×Q8—614no—
405C2×Q8—612no—
406C2×Q8—1224no—
407C2×Q8—1224no—
408C2×Q8—1222no—
409C2×Q8—1222no—
410C2×Q8—1224no—
411C2×Q8—1222no—
412C2×Q8—1224no—
413C2×Q8—1224no—
414C2×Q8—1224no—
415C2×Q8—1224no—
416C2×Q8—1222no—
417C2×Q8—1222no—
418C2×Q8—1221no—
419C2×Q8C4⋊#692314yescomplemented
420C2×Q8C22×#692312yes—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—611no—
424C22×C4—612no—
425C22×C4—612no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—612no—
429C22×C4—612no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—614no—
433C22×C4—1222no—
434C22×C4—1224no—
435C22×C4—1224no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—4882no—
444C22×C4C22×#692311yescharacteristic
445M16—2442no—
446M16—2442no—
447C2×C8—22no—
448C2×C8—611no—
449C2×C8—611no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1222no—
453C2×C8—1222no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1222no—
457C2×C8—1222no—
458C2×C8—1222no—
459C2×C8—1222no—
460C2×C8C22×#692311yescharacteristic
461C2×C8C22×#692311yescharacteristic
462C4⋊C4—21no—
463C4⋊C4—614no—
464C4⋊C4—612no—
465C4⋊C4—1222no—
466C4⋊C4—1222no—
467C4⋊C4—1224no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C4⋊C4—12216no—
472C4⋊C4—1221no—
473C4⋊C4—4884no—
474C4⋊C4C22×#692312yes—
475C22⋊C4—1224no—
476C22⋊C4—1224no—
477C22⋊C4—2442no—
478C42—1222no—
479C42—1222no—
480C42—1224no—
481C42—12216no—
482C42—2442no—
483C23—611no—
484C23—612no—
485C23—612no—
486C23—611no—
487C23—1222no—
488C23—1222no—
489C23D4×#692311yescharacteristic
490Q8—24no—
491Q8—22no—
492Q8—614no—
493Q8—1224no—
494Q8—1224no—
495Q8—1228no—
496Q8—1224no—
497Q8—1222no—
498Q8C2×C4⋊#692314yes—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—81no—
503C2×C4—612no—
504C2×C4—612no—
505C2×C4—611no—
506C2×C4—611no—
507C2×C4—612no—
508C2×C4—612no—
509C2×C4—612no—
510C2×C4—612no—
511C2×C4—612no—
512C2×C4—611no—
513C2×C4—611no—
514C2×C4—612no—
515C2×C4—611no—
516C2×C4—611no—
517C2×C4—1224no—
518C2×C4—1224no—
519C2×C4—1228no—
520C2×C4—1224no—
521C2×C4—1222no—
522C2×C4—1224no—
523C2×C4—1224no—
524C2×C4—1224no—
525C2×C4—1224no—
526C2×C4—1222no—
527C2×C4—4882no—
528C2×C4—4881no—
529C2×C4—2444no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4C2×C4⋊#692312yes—
536C2×C4D4×#692311yescharacteristic
537C2×C4D4×#692312yes—
538C2×C4C23×#692311yescharacteristic
539C8—21no—
540C8—21no—
541C8—1221no—
542C8—1221no—
543C8—2442no—
544C22—21no—
545C22—21no—
546C22—612no—
547C22—612no—
548C22—611no—
549C22—611no—
550C22—612no—
551C22—1224no—
552C22—1224no—
553C22—1221no—
554C22—1221no—
555C22#1684070912yes—
556C22C2×D4×#692311yescharacteristic
557C4—24no—
558C4—42no—
559C4—611no—
560C4—611no—
561C4—612no—
562C4—1224no—
563C4—4882no—
564C4—2444no—
565C4—2444no—
566C4—2442no—
567C4#1684070912yes—
568C4C22×C4⋊#692311yescharacteristic
569C4C2×D4×#692311yescharacteristic
570C2—22no—
571C2—612no—
572C2—612no—
573C2—1222no—
574C2C2×#1684070911yescharacteristic
575C2#2146257911yescharacteristic
576C2#2146258011yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×#103×C61C4
socleC22×C61C2×C4×D4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenterC2×#103#6923
hypocenterC61#20880
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
5. #21461785C2semidirect
6. #21461783C2semidirect
33. C2×#14519702C4semidirect
34. C2×#14519701C4semidirect
35. C2×#14519700C4semidirect
36. #103×D61C4semidirect
37. #103×D61C4semidirect
40. C2×#103×C61C4semidirect
115. C2×Q8×C61C4⋊C4semidirect
130. Q8×D61C4⋊C4semidirect
154. C2×Dic122C4⋊C4semidirect
322. C2×#103#6923semidirect
329. C61#20880semidirect
419. C2×Q8C4⋊#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168408302
C2×#168407722
C2×#168407552
#214622992
#214617852
#214617832
#214615062
#214614842
C2×#103×D612
#2088061