Finite Group Databasebeta

← #29734445

Subgroups — #29734445

Order 15616.

968 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840373C211yesmaximal, characteristic
2C2×#16840171C211yesmaximal, characteristic, complemented
3C2×#16840112C211yesmaximal, characteristic, complemented
4#21451643C211yesmaximal, characteristic
5#21447021C212yesmaximal
6#21447014C212yesmaximal
7#21446783C211yesmaximal, characteristic, complemented
8#21446089C212yesmaximal, complemented
9#21446075C212yesmaximal, complemented
10#21445588C211yesmaximal, characteristic
11C61×#1960C211yesmaximal, characteristic, complemented
12#16840373—24no—
13#16840373—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519772—21no—
16C2×#14519772—21no—
17C2×#14519769C2211yescharacteristic
18C2×#14519767—21no—
19C2×#14519767—21no—
20C2×#14519753C2211yescharacteristic
21C2×#14519753C2211yescharacteristic
22C2×Q8×Dic61C2211yescharacteristic
23C2×Q8×Dic61C2211yescharacteristic
24C2×#14519734C2211yescharacteristic
25#16840171—24no—
26#16840112—24no—
27C2×#14519711C2212yes—
28C2×#14519709C2212yes—
29C2×#14519701C2212yes—
30C2×#14519701C2212yes—
31C2×#14519689C2212yes—
32C2×C4×#92418C2211yescharacteristic
33#16839702—21no—
34#16839699—21no—
35#16839683—21no—
36#16839675—21no—
37#16839661C2212yes—
38#16839619C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41#16839516C2212yes—
42C2×C4⋊Q8×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C2×#103×C61C2212yes—
45C2×#97×C61C2211yescharacteristic
46C61×#382C2211yescharacteristic
47C22×Dic122—21no—
48C22×C4×D61C2311yescharacteristic
49C22×#92418C2312yes—
50#14519777—22no—
51#14519777—22no—
52#14519772—28no—
53#14519772—28no—
54#14519769—22no—
55#14519769D414yes—
56#14519767—28no—
57#14519767—28no—
58#14519753—22no—
59#14519753—22no—
60#14519753—22no—
61#14519753—22no—
62Q8×Dic61—22no—
63Q8×Dic61—22no—
64Q8×Dic61—22no—
65Q8×Dic61—22no—
66#14519734—22no—
67#14519734—22no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196D411yescharacteristic
80C2×#11758196D411yescharacteristic
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—22no—
84C2×#11758198D411yescharacteristic
85C2×#11758198D411yescharacteristic
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61C2311yescharacteristic
89#14519711—24no—
90#14519709—24no—
91#14519701—24no—
92#14519701—24no—
93#14519689—24no—
94C4×#92418—22no—
95#14519684—22no—
96#14519684—22no—
97C22×Q8×C61C2311yescharacteristic
98C22×Q8×C61C2311yescharacteristic
99C22×C8×C61C2312yes—
100C4⋊Q8×C61—22no—
101C4⋊Q8×C61D414yes—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61C2312yes—
105C2×C42×C61C2311yescharacteristic
106#103×C61—24no—
107#103×C61—24no—
108#97×C61—22no—
109#95×C61—21no—
110#95×C61—21no—
111C23×D61C2×D411yescharacteristic
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61—22no—
118C2×Q8×C61C2×D412yes—
119C2×Q8×C61C2×D412yes—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2411yescharacteristic
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—22no—
129C22×Dic61C2×D411yescharacteristic
130C22×Dic61C2×D411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×C4×D61—22no—
135C2×C4×D61—22no—
136C2×Dic122—42no—
137C2×Dic122—82no—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C2×C8×C61#4114yes—
141C4⋊C4×C61—24no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C42×C61C2×D412yes—
146C42×C61C2×D412yes—
147#11758198—24no—
148#11758198—24no—
149#11758198—22no—
150#11758198—22no—
151#11758198—82no—
152C4⋊Dic61—24no—
153C4⋊Dic61—24no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—22no—
157C4⋊Dic61—22no—
158C4⋊Dic61—42no—
159C4⋊Dic61—82no—
160C4⋊Dic61C2×D412yes—
161C4⋊Dic61C2×D412yes—
162#11758196—24no—
163#11758196—24no—
164#11758196—24no—
165#11758196—24no—
166#11758196—22no—
167#11758196—22no—
168C4×Dic61—24no—
169C4×Dic61—24no—
170C4×Dic61—21no—
171C4×Dic61—21no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—22no—
175C23×C61C22×D411yescharacteristic
176C22×D61—21no—
177C22×D61—21no—
178C22×D61—21no—
179C22×D61—22no—
180C22×D61—22no—
181Q8×C61—24no—
182Q8×C61—24no—
183Q8×C61—42no—
184Q8×C61—42no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61#12012yes—
197C2×C4×C61#12012yes—
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C2×#4112yes—
201C2×C4×C612−1+412yes—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—21no—
207C2×Dic61—21no—
208C2×Dic61—21no—
209C2×Dic61—22no—
210C2×Dic61—22no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C4×D61—42no—
217C4×D61—42no—
218Dic122—84no—
219C8×C61—24no—
220#92418—42no—
221#25686—611nomaximal, Sylow
222C22×C61C2×#12011yescharacteristic
223C22×C61#53211yescharacteristic
224C22×C61#53611yescharacteristic
225C22×C61#57712yes—
226C22×C61#57812yes—
227C2×D61—42no—
228C2×D61—41no—
229C2×D61—42no—
230C2×D61—41no—
231C2×D61—41no—
232C4×C61—22no—
233C4×C61—22no—
234C4×C61—42no—
235C4×C61—42no—
236C4×C61—82no—
237C4×C61#54712yes—
238C4×C61#54812yes—
239Dic61—41no—
240Dic61—41no—
241Dic61—42no—
242Dic61—41no—
243Dic61—42no—
244Dic61—82no—
245C2×#541—611no—
246C2×#501—611no—
247C2×#487—611no—
248#2628—611no—
249#2036—612no—
250#2035—612no—
251#1991—611no—
252#1960D6111yescharacteristic, complemented
253#1865—612no—
254#1863—612no—
255#1750—611no—
256C2×C61#243211yescharacteristic
257C2×C61#301011yescharacteristic
258C2×C61#301111yescharacteristic
259C2×C61#317812yes—
260C2×C61#324212yes—
261D61—81no—
262#541—1224no—
263#541—1224no—
264C2×C4⋊Q8—611no—
265C2×C4⋊Q8C2×D6111yescharacteristic
266C2×#126—1221no—
267C2×#126—1221no—
268C2×#125—611no—
269C2×#125—1221no—
270C2×#125—1221no—
271C2×C22⋊Q8—611no—
272C2×C22⋊Q8—611no—
273C2×C4×Q8—611no—
274C2×C4×Q8—611no—
275C2×#117—611no—
276#501—1224no—
277#487—1224no—
278C2×#107—612no—
279C2×#106—612no—
280C2×#103—612no—
281C2×#103—612no—
282C2×#103C2×D6112yes—
283C2×#103C2×D6112yes—
284C2×C22⋊C8—612no—
285C2×#97C2×D6111yescharacteristic
286C2×C4×C8—611no—
287#400—1221no—
288#399—1221no—
289#395—1221no—
290#393—1221no—
291#389—612no—
292#382C2×D6111yescharacteristic
293#380—612no—
294#340—612no—
295#340—612no—
296#336—612no—
297C61#2568611yescharacteristic, complemented, Sylow
298C22×Q8—1221no—
299C22×Q8C22×D6111yescharacteristic
300C22×Q8C22×D6111yescharacteristic
301C23×C4—611no—
302C22×C8—612no—
303C22×C8C22×D6112yes—
304C4⋊Q8—22no—
305C4⋊Q8—1222no—
306C4⋊Q8—1222no—
307C4⋊Q8D24414yes—
308#126—1228no—
309#126—1228no—
310#125—614no—
311#125—1228no—
312#125—1228no—
313#125—1222no—
314C22⋊Q8—1222no—
315C22⋊Q8—1222no—
316C22⋊Q8—1222no—
317C22⋊Q8—1222no—
318C4×Q8—1222no—
319C4×Q8—1222no—
320C4×Q8—1222no—
321C4×Q8—1222no—
322#117—1222no—
323#117—1222no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—611no—
327C2×C4⋊C4—611no—
328C2×C4⋊C4—611no—
329C2×C4⋊C4—611no—
330C2×C4⋊C4—612no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4—1221no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—1221no—
337C2×C4⋊C4—1222no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4C22×D6112yes—
340C2×C22⋊C4—611no—
341C2×C22⋊C4—611no—
342C2×C22⋊C4—1221no—
343C2×C22⋊C4—1221no—
344C2×C22⋊C4—1222no—
345C2×C42—611no—
346C2×C42—1221no—
347C2×C42—1221no—
348C2×C42C22×D6111yescharacteristic
349#107—1224no—
350#106—1224no—
351#103—24no—
352#103—24no—
353#103—1224no—
354#103—1224no—
355C22⋊C8—1224no—
356#97—22no—
357C4×C8—1222no—
358#95—21no—
359#95—21no—
360#95—1222no—
361#95—1222no—
362C24—611no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—4882no—
370C2×Q8—2442no—
371C2×Q8C2×D24412yes—
372C2×Q8C2×D24412yes—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—611no—
377C22×C4—611no—
378C22×C4—611no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4—1221no—
383C22×C4—1222no—
384C22×C4—1222no—
385C22×C4—1222no—
386C22×C4D4×D6111yescharacteristic
387C22×C4D4×D6111yescharacteristic
388C22×C4C23×D6111yescharacteristic
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—1222no—
392C2×C8—1222no—
393C2×C8—1222no—
394C2×C8#1175822414yes—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—612no—
400C4⋊C4—612no—
401C4⋊C4—1224no—
402C4⋊C4—1224no—
403C4⋊C4—1224no—
404C4⋊C4—1224no—
405C4⋊C4—1221no—
406C4⋊C4—1221no—
407C4⋊C4—1222no—
408C4⋊C4—1222no—
409C4⋊C4—1224no—
410C4⋊C4—1224no—
411C4⋊C4—1222no—
412C4⋊C4—1222no—
413C4⋊C4—4882no—
414C4⋊C4—2442no—
415C22⋊C4—1224no—
416C22⋊C4—1224no—
417C22⋊C4—1222no—
418C22⋊C4—1222no—
419C22⋊C4—4882no—
420C42—1224no—
421C42—1224no—
422C42—1221no—
423C42—1221no—
424C42C2×D24412yes—
425C42D4×D6112yes—
426C23—1221no—
427C23—1221no—
428C23—1222no—
429C23—1222no—
430C23—1221no—
431C23C2×D4×D6111yescharacteristic
432Q8—24no—
433Q8—24no—
434Q8—42no—
435Q8—42no—
436Q8—4884no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—1221no—
449C2×C4—1221no—
450C2×C4—1222no—
451C2×C4—1222no—
452C2×C4—1221no—
453C2×C4—1221no—
454C2×C4—1222no—
455C2×C4—1222no—
456C2×C4—1221no—
457C2×C4—2442no—
458C2×C4—2442no—
459C2×C4—2442no—
460C2×C4—2442no—
461C2×C4—2442no—
462C2×C4—2442no—
463C2×C4—2442no—
464C2×C4#1451974712yes—
465C2×C4#1451974712yes—
466C2×C4C2×D4×D6111yescharacteristic
467C2×C4C22×D24411yescharacteristic
468C2×C4C2×#1175822412yes—
469C2×C4#1451982412yes—
470C8—24no—
471C8—2442no—
472C22—2441no—
473C22—2441no—
474C22—2442no—
475C22—2442no—
476C22—2441no—
477C22C2×#1451974711yescharacteristic
478C22#1684031011yescharacteristic
479C22#1684033211yescharacteristic
480C22#1684057112yes—
481C22#1684057812yes—
482C4—22no—
483C4—22no—
484C4—42no—
485C4—42no—
486C4—82no—
487C4—4882no—
488C4—2441no—
489C4—2442no—
490C4—2441no—
491C4—2442no—
492C4—2441no—
493C4#1684041612yes—
494C4#1684041812yes—
495C2—4881no—
496C2#2144944911yescharacteristic
497C2#2145510611yescharacteristic
498C2#2145511511yescharacteristic
499C2#2145648412yes—
500C2#2145708212yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1960C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1960D61
hypocenterC61#25686
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
2. C2×#16840171C2semidirect
3. C2×#16840112C2semidirect
7. #21446783C2semidirect
8. #21446089C2semidirect
9. #21446075C2semidirect
11. C61×#1960C2semidirect
252. #1960D61semidirect
297. C61#25686semidirect

Maximal subgroups

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

subgroupindex
C2×#168403732
C2×#168401712
C2×#168401122
#214516432
#214470212
#214470142
#214467832
#214460892
#214460752
#214455882
C61×#19602
#2568661