Finite Group Databasebeta

← #29689467

Subgroups — #29689467

Order 15616.

990 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840391C211yesmaximal, characteristic
2#21448421C211yesmaximal, characteristic
3#21448419C211yesmaximal, characteristic
4#21446805C211yesmaximal, characteristic
5#21446804C211yesmaximal, characteristic
6#21446803C211yesmaximal, characteristic
7#21446802C214yesmaximal
8#21445895C214yesmaximal
9C61×#1996C211yesmaximal, characteristic
10#16840391—22no—
11#16840391—22no—
12#16840391—22no—
13#16840391—22no—
14C2×#14519776C412yescomplemented
15C2×#14519775C412yescomplemented
16C2×#14519767C412yescomplemented
17C2×#14519767C412yescomplemented
18C2×#14519756C2211yescharacteristic
19C2×#14519754C2212yes—
20C2×#14519733C2212yes—
21C22×#11758196C2211yescharacteristic
22#16839649—22no—
23#16839646—22no—
24#16839644—22no—
25#16839643—22no—
26#16839623C2212yes—
27#16839623C2212yes—
28#16839622C2211yescharacteristic
29#16839620C2212yes—
30C22⋊C4×Dic61C2212yes—
31C4×C22⋊Dic61C2211yescharacteristic
32#16839529C2214yes—
33#16839528C2214yes—
34#16839528C2214yes—
35C2×C22⋊Q8×C61C2211yescharacteristic
36C61×#380C2211yescharacteristic
37C4×C22⋊C4×C61C2211yescharacteristic
38C61×#342C2214yes—
39C22×Dic122C2×C412yes—
40C22×Dic122C2×C411yescharacteristic
41C23×Dic61C2311yescharacteristic
42C23×Dic61C2311yescharacteristic
43#14519776—24no—
44#14519776—24no—
45#14519775—24no—
46#14519775—24no—
47#14519767—24no—
48#14519767—24no—
49#14519767—24no—
50#14519767—24no—
51#14519756—21no—
52#14519756—21no—
53#14519756—21no—
54#14519756—21no—
55#14519754—24no—
56#14519754—24no—
57#14519733—24no—
58#14519733—24no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61C2×C412yes—
61C2×C4⋊Dic61C2×C411yescharacteristic
62C2×#11758196—22no—
63C2×#11758196C2×C412yes—
64C2×#11758196C2×C412yes—
65C2×#11758196C2×C412yes—
66C2×#11758196C2×C412yes—
67C2×#11758196C2×C412yes—
68C2×#11758196C2×C412yes—
69C2×#11758196C2×C412yes—
70C2×#11758196D412yes—
71C2×#11758196D412yes—
72C2×#11758196D412yes—
73C2×#11758196D412yes—
74C2×C22⋊Dic61C2312yes—
75C2×C22⋊Dic61C2312yes—
76C2×C22⋊Dic61C2312yes—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—22no—
79C2×C4×Dic61C2×C412yes—
80C2×C4×Dic61C2×C412yes—
81#14519684—21no—
82#14519684—21no—
83#14519684—22no—
84#14519684—22no—
85#14519684—24no—
86#14519684—24no—
87C22×Q8×C61C2×C411yescharacteristic
88C23×C4×C61C2311yescharacteristic
89C22⋊Q8×C61—21no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61—21no—
92C22⋊Q8×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61C2×C411yescharacteristic
95C2×C4⋊C4×C61C2×C412yes—
96C2×C22⋊C4×C61C2312yes—
97C2×C22⋊C4×C61C2312yes—
98C2×C22⋊C4×C61C2312yes—
99C2×C42×C61—21no—
100#95×C61—21no—
101#95×C61—21no—
102C24×C61C2411yescharacteristic
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C2×Q8×C61—81no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—81no—
109C22×C4×C61C22×C412yes—
110C22×C4×C61C22×C412yes—
111C22×C4×C61C22×C411yescharacteristic
112C22×C4×C61C22×C411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×Dic61—22no—
118C22×Dic61—22no—
119C22×Dic61—22no—
120C22×Dic61—22no—
121C22×Dic61—81no—
122C22×Dic61C22×C412yes—
123C22×Dic61C22×C412yes—
124C22×Dic61C22×C412yes—
125C22×Dic61C22×C412yes—
126C22×Dic61C2×D412yes—
127C22×Dic61C2×D412yes—
128C22×Dic61C2×D412yes—
129C22×Dic61C2×D412yes—
130C2×Dic122—21no—
131C2×Dic122—21no—
132C2×Dic122—24no—
133C2×Dic122—82no—
134C2×Dic122—81no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—81no—
140C22⋊C4×C61—22no—
141C22⋊C4×C61—22no—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—24no—
144C22⋊C4×C61—22no—
145C42×C61—81no—
146C22⋊Dic61—22no—
147C22⋊Dic61—22no—
148C22⋊Dic61—22no—
149C22⋊Dic61—22no—
150C22⋊Dic61—24no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—24no—
154C4⋊Dic61—81no—
155#11758196—22no—
156#11758196—22no—
157#11758196—24no—
158#11758196—24no—
159#11758196—24no—
160#11758196—24no—
161#11758196—82no—
162#11758196C22⋊C414yes—
163#11758196C22⋊C414yes—
164#11758196C22⋊C414yes—
165#11758196C22⋊C414yes—
166C4×Dic61—24no—
167C4×Dic61—24no—
168C4×Dic61—82no—
169C4×Dic61—81no—
170C23×C61—41no—
171C23×C61—41no—
172C23×C61C23×C411yescharacteristic
173C23×C61C22×D411yescharacteristic
174C23×C61C22×D411yescharacteristic
175C23×C612−1+412yes—
176C23×C612−1+412yes—
177Q8×C61—81no—
178Q8×C61—81no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61C2×C22⋊C411yescharacteristic
196C2×C4×C61C2×C22⋊C411yescharacteristic
197C2×C4×C61C2×C22⋊C411yescharacteristic
198C2×C4×C61C2×C22⋊C411yescharacteristic
199C2×Dic61—24no—
200C2×Dic61—22no—
201C2×Dic61—24no—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61C2×C22⋊C414yes—
217C2×Dic61C2×C22⋊C414yes—
218Dic122—84no—
219Dic122—81no—
220Dic122—81no—
221#21828—611nomaximal, Sylow
222C22×C61—21no—
223C22×C61—21no—
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61C22×C22⋊C411yescharacteristic
231C22×C61#51912yes—
232C22×C61#53611yescharacteristic
233C22×C61#53612yes—
234C22×C61#53611yescharacteristic
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—42no—
240C4×C61—42no—
241C4×C61—82no—
242C4×C61—82no—
243Dic61—24no—
244Dic61—24no—
245Dic61—42no—
246Dic61—42no—
247Dic61—82no—
248Dic61—82no—
249C2×#544—611no—
250#2313—611no—
251#2313—611no—
252#1996—611no—
253#1996—611no—
254#1996—614no—
255#1996—611no—
256#1996D6111yescharacteristic
257#1820—614no—
258C2×C61—21no—
259C2×C61—22no—
260C2×C61—21no—
261C2×C61—22no—
262C2×C61#231011yescharacteristic
263C2×C61#288311yescharacteristic
264C2×C61#288411yescharacteristic
265#544—1222no—
266#544—1222no—
267#544—1222no—
268#544—1222no—
269C2×C4⋊Q8—612no—
270C2×C4⋊Q8—612no—
271C2×#125—612no—
272C2×#125—612no—
273C2×C22⋊Q8—611no—
274C2×C22⋊Q8—612no—
275C2×C22⋊Q8C2×D6111yescharacteristic
276C2×#117—612no—
277C22×C4⋊C4—611no—
278#387—1222no—
279#387—1222no—
280#387—1222no—
281#387—1222no—
282#381—612no—
283#381—612no—
284#380—611no—
285#380—612no—
286#380C2×D6111yescharacteristic
287C4×C22⋊C4—611no—
288C4×C22⋊C4—612no—
289C4×C22⋊C4C2×D6111yescharacteristic
290#342—614no—
291#342—614no—
292#342—614no—
293#342C2×D6114yes—
294C61#2182811yescharacteristic, complemented, Sylow
295C22×Q8—612no—
296C22×Q8—611no—
297C22×Q8C4×D6111yescharacteristic
298C23×C4—611no—
299C23×C4—611no—
300C23×C4C22×D6111yescharacteristic
301C4⋊Q8—1224no—
302C4⋊Q8—1224no—
303C4⋊Q8—1224no—
304C4⋊Q8—1224no—
305#125—1224no—
306#125—1224no—
307#125—1224no—
308#125—1224no—
309C22⋊Q8—21no—
310C22⋊Q8—21no—
311C22⋊Q8—21no—
312C22⋊Q8—21no—
313C22⋊Q8—1224no—
314C22⋊Q8—1224no—
315C22⋊Q8—1221no—
316C22⋊Q8—1221no—
317C22⋊Q8—1221no—
318C22⋊Q8—1221no—
319#117—1224no—
320#117—1224no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—612no—
323C2×C4⋊C4—612no—
324C2×C4⋊C4—612no—
325C2×C4⋊C4—612no—
326C2×C4⋊C4—612no—
327C2×C4⋊C4—612no—
328C2×C4⋊C4—612no—
329C2×C4⋊C4—612no—
330C2×C4⋊C4—612no—
331C2×C4⋊C4—611no—
332C2×C4⋊C4—612no—
333C2×C4⋊C4—612no—
334C2×C4⋊C4—612no—
335C2×C4⋊C4—1222no—
336C2×C4⋊C4—1221no—
337C2×C4⋊C4C4×D6111yescharacteristic
338C2×C4⋊C4C4×D6112yes—
339C2×C22⋊C4—612no—
340C2×C22⋊C4—612no—
341C2×C22⋊C4—612no—
342C2×C22⋊C4C22×D6112yes—
343C2×C22⋊C4C22×D6112yes—
344C2×C22⋊C4C22×D6112yes—
345C2×C42—21no—
346C2×C42—612no—
347C2×C42—612no—
348C2×C42—1222no—
349C2×C42—1221no—
350#95—21no—
351#95—21no—
352#95—1222no—
353#95—1222no—
354#95—1224no—
355#95—1224no—
356#95—1221no—
357#95—1221no—
358C24C23×D6111yescharacteristic
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—81no—
362C2×Q8—1224no—
363C2×Q8—1221no—
364C2×Q8—1221no—
365C2×Q8—4881no—
366C2×Q8—4882no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—81no—
370C22×C4—612no—
371C22×C4—612no—
372C22×C4—612no—
373C22×C4—612no—
374C22×C4—612no—
375C22×C4—612no—
376C22×C4—612no—
377C22×C4—612no—
378C22×C4—1222no—
379C22×C4—1222no—
380C22×C4—1222no—
381C22×C4—1222no—
382C22×C4—4881no—
383C22×C4C2×C4×D6112yes—
384C22×C4C2×C4×D6112yes—
385C22×C4C2×C4×D6111yescharacteristic
386C22×C4C2×C4×D6111yescharacteristic
387C22×C4D4×D6111yescharacteristic
388C22×C4D4×D6111yescharacteristic
389C22×C4D4×D6111yescharacteristic
390C22×C4D4×D6111yescharacteristic
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—81no—
396C4⋊C4—614no—
397C4⋊C4—614no—
398C4⋊C4—614no—
399C4⋊C4—614no—
400C4⋊C4—1224no—
401C4⋊C4—1224no—
402C4⋊C4—1224no—
403C4⋊C4—1224no—
404C4⋊C4—1224no—
405C4⋊C4—1222no—
406C4⋊C4—1222no—
407C4⋊C4—1221no—
408C4⋊C4—1221no—
409C4⋊C4—4882no—
410C4⋊C4—4881no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—24no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—1222no—
417C22⋊C4—1222no—
418C22⋊C4—1222no—
419C22⋊C4—1222no—
420C22⋊C4—1224no—
421C42—81no—
422C42—1224no—
423C42—1224no—
424C42—4881no—
425C42—4882no—
426C23—41no—
427C23—41no—
428C23C22×C4×D6111yescharacteristic
429C23C2×D4×D6111yescharacteristic
430C23C2×D4×D6111yescharacteristic
431C23#1451982412yes—
432C23#1451982512yes—
433Q8—81no—
434Q8—81no—
435Q8—4881no—
436Q8—4881no—
437Q8—4884no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—614no—
455C2×C4—614no—
456C2×C4—1224no—
457C2×C4—1222no—
458C2×C4—1224no—
459C2×C4—1222no—
460C2×C4—1222no—
461C2×C4—1222no—
462C2×C4—2442no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4C22⋊C4×D6111yescharacteristic
474C2×C4C22⋊C4×D6111yescharacteristic
475C2×C4C22⋊C4×D6111yescharacteristic
476C2×C4C22⋊C4×D6111yescharacteristic
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—42no—
484C22—42no—
485C22C2×C22⋊C4×D6111yescharacteristic
486C22#1684023812yes—
487C22#1684033211yescharacteristic
488C22#1684033512yes—
489C22#1684033711yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—42no—
495C4—42no—
496C4—82no—
497C4—82no—
498C4—1224no—
499C4—1224no—
500C4—4882no—
501C4—4882no—
502C4—2442no—
503C4—2442no—
504C2—21no—
505C2—22no—
506C2—21no—
507C2—22no—
508C2#2144840211yescharacteristic
509C2#2145427711yescharacteristic
510C2#2145428111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1996C2
socleC22×C61C22×C22⋊C4
prime radical, cosocleC24×C61C24
hypercenter#1996D61
hypocenterC61#21828
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
14. C2×#14519776C4semidirect
15. C2×#14519775C4semidirect
16. C2×#14519767C4semidirect
17. C2×#14519767C4semidirect
294. C61#21828semidirect

Maximal subgroups

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

subgroupindex
C2×#168403912
#214484212
#214484192
#214468052
#214468042
#214468032
#214468022
#214458952
C61×#19962
#2182861