Finite Group Databasebeta

← #29734442

Subgroups — #29734442

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×#16840374C211yesmaximal, characteristic
2C2×#16840171C211yesmaximal, characteristic, complemented
3C2×#16840113C211yesmaximal, characteristic, complemented
4#21451640C211yesmaximal, characteristic
5#21447023C212yesmaximal
6#21447016C212yesmaximal
7#21446784C211yesmaximal, characteristic, complemented
8#21446639C211yesmaximal, characteristic, complemented
9#21446090C212yesmaximal, complemented
10#21446076C212yesmaximal, complemented
11C61×#1750C211yesmaximal, characteristic
12#16840374—24no—
13#16840374—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16C2×#14519770—21no—
17C2×#14519770—21no—
18C2×#14519769C2211yescharacteristic
19C2×#14519767—21no—
20C2×#14519767—21no—
21C2×#14519756C2211yescharacteristic
22C2×#14519756C2211yescharacteristic
23C2×C4×Dic122C2211yescharacteristic
24C2×C4×Dic122C2211yescharacteristic
25#16840171—24no—
26#16840113—24no—
27C2×#14519711C2212yes—
28C2×#14519709C2212yes—
29C2×#14519702C2212yes—
30C2×#14519702C2212yes—
31C2×#14519702C2212yes—
32C2×#14519702C2212yes—
33#16839701—21no—
34#16839698—21no—
35#16839682—21no—
36#16839674—21no—
37#16839660C2212yes—
38#16839628C2211yescharacteristic
39#16839622C2212yes—
40#16839525C2212yes—
41#16839525C2212yes—
42C2×#117×C61C2211yescharacteristic
43C2×C22⋊C8×C61C2212yes—
44C2×#97×C61C2211yescharacteristic
45C2×C4×C8×C61C2211yescharacteristic
46C61×#336C2212yes—
47C22×Dic122—21no—
48C22×Dic122C2311yescharacteristic
49C22×Dic122C2311yescharacteristic
50#14519777—22no—
51#14519777—22no—
52#14519777—22no—
53#14519777D414yes—
54#14519770—28no—
55#14519770—28no—
56#14519769—22no—
57#14519769D414yes—
58#14519767—28no—
59#14519767—28no—
60#14519756—22no—
61#14519756—22no—
62#14519756—22no—
63#14519756—22no—
64C4×Dic122—22no—
65C4×Dic122—22no—
66C4×Dic122—22no—
67C4×Dic122—22no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2312yes—
72C2×C4⋊Dic61C2312yes—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—22no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61—22no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87#14519711—24no—
88#14519709—24no—
89#14519702—24no—
90#14519702—24no—
91#14519702—24no—
92#14519702—24no—
93#14519684—22no—
94#14519684—21no—
95#14519684—21no—
96#14519684—22no—
97C23×C4×C61C2311yescharacteristic
98C22×C8×C61C2312yes—
99C22×C8×C61C2312yes—
100#117×C61—22no—
101#117×C61—22no—
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C22⋊C4×C61D411yescharacteristic
105C2×C22⋊C4×C61D411yescharacteristic
106C2×C42×C61C2311yescharacteristic
107C2×C42×C61C2311yescharacteristic
108C22⋊C8×C61—24no—
109#97×C61—22no—
110C4×C8×C61—22no—
111C24×C61C2×D411yescharacteristic
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—21no—
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C2×Dic122—22no—
129C2×Dic122—22no—
130C2×Dic122—22no—
131C2×Dic122—22no—
132C2×Dic122—21no—
133C2×Dic122—21no—
134C2×Dic122—42no—
135C2×Dic122—82no—
136C2×Dic122C2×D412yes—
137C2×Dic122C2×D412yes—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C2×C8×C61#4114yes—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C22⋊C4×C61—22no—
147C22⋊C4×C61—22no—
148C42×C61—21no—
149C42×C61—21no—
150C42×C61C2×D412yes—
151C42×C61C2×D412yes—
152C22⋊Dic61—24no—
153C22⋊Dic61—24no—
154C22⋊Dic61—82no—
155C4⋊Dic61—24no—
156C4⋊Dic61—24no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—22no—
160C4⋊Dic61—22no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61—42no—
164C4⋊Dic61C2×D412yes—
165C4⋊Dic61C2×D412yes—
166#11758196—24no—
167#11758196—24no—
168#11758196—24no—
169#11758196—24no—
170#11758196—24no—
171#11758196—24no—
172#11758196—82no—
173C4×Dic61—24no—
174C4×Dic61—24no—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61C22×D411yescharacteristic
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61#12012yes—
193C2×C4×C61#12012yes—
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C2×#4112yes—
197C2×C4×C612−1+412yes—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214Dic122—24no—
215Dic122—24no—
216Dic122—42no—
217Dic122—42no—
218Dic122—84no—
219C8×C61—24no—
220C8×C61—42no—
221#25686—611nomaximal, Sylow
222C22×C61—42no—
223C22×C61—41no—
224C22×C61—42no—
225C22×C61—41no—
226C22×C61—41no—
227C22×C61C2×#12011yescharacteristic
228C22×C61#53211yescharacteristic
229C22×C61#53611yescharacteristic
230C22×C61#57712yes—
231C22×C61#57812yes—
232C4×C61—22no—
233C4×C61—22no—
234C4×C61—41no—
235C4×C61—41no—
236C4×C61—41no—
237C4×C61#54712yes—
238C4×C61#54812yes—
239Dic61—42no—
240Dic61—42no—
241Dic61—42no—
242Dic61—42no—
243Dic61—82no—
244Dic61—82no—
245C2×#541—611no—
246C2×#501—611no—
247C2×#487—611no—
248#2628—611no—
249#2036—612no—
250#2035—612no—
251#1991—611no—
252#1960—611no—
253#1865—612no—
254#1863—612no—
255#1750D6111yescharacteristic
256C2×C61—81no—
257C2×C61#243211yescharacteristic
258C2×C61#301011yescharacteristic
259C2×C61#301111yescharacteristic
260C2×C61#317812yes—
261C2×C61#324212yes—
262#541—1224no—
263#541—1224no—
264C2×C4⋊Q8—611no—
265C2×C4⋊Q8—611no—
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×#117C2×D6111yescharacteristic
276#501—1224no—
277#487—1224no—
278C2×#107—612no—
279C2×#106—612no—
280C2×#103—612no—
281C2×#103—612no—
282C2×#103—612no—
283C2×#103—612no—
284C2×C22⋊C8C2×D6112yes—
285C2×#97C2×D6111yescharacteristic
286C2×C4×C8C2×D6111yescharacteristic
287#400—1221no—
288#399—1221no—
289#395—1221no—
290#393—1221no—
291#389—612no—
292#382—611no—
293#380—612no—
294#340—612no—
295#340—612no—
296#336C2×D6112yes—
297C61#2568611yescharacteristic, complemented, Sylow
298C22×Q8—611no—
299C22×Q8—611no—
300C22×Q8—1221no—
301C23×C4C22×D6111yescharacteristic
302C22×C8C22×D6112yes—
303C22×C8C22×D6112yes—
304C4⋊Q8—614no—
305C4⋊Q8—1222no—
306C4⋊Q8—1222no—
307C4⋊Q8—1222no—
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—22no—
323#117—22no—
324C2×C4⋊C4—611no—
325C2×C4⋊C4—611no—
326C2×C4⋊C4—612no—
327C2×C4⋊C4—612no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4—1221no—
330C2×C4⋊C4—1221no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4—1221no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—1222no—
337C2×C4⋊C4—1221no—
338C2×C4⋊C4D24411yescharacteristic
339C2×C4⋊C4D24411yescharacteristic
340C2×C22⋊C4—1221no—
341C2×C22⋊C4—1221no—
342C2×C22⋊C4—1222no—
343C2×C22⋊C4D24411yescharacteristic
344C2×C22⋊C4D24411yescharacteristic
345C2×C42—1221no—
346C2×C42—1221no—
347C2×C42C22×D6111yescharacteristic
348C2×C42C22×D6111yescharacteristic
349#107—1224no—
350#106—1224no—
351#103—1224no—
352#103—1224no—
353#103—1224no—
354#103—1224no—
355C22⋊C8—24no—
356#97—22no—
357C4×C8—22no—
358#95—1222no—
359#95—1221no—
360#95—1221no—
361#95—1222no—
362C24C2×D24411yescharacteristic
363C2×Q8—612no—
364C2×Q8—612no—
365C2×Q8—1221no—
366C2×Q8—1221no—
367C2×Q8—1222no—
368C2×Q8—1222no—
369C2×Q8—1222no—
370C2×Q8—1222no—
371C2×Q8—4882no—
372C2×Q8—2442no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—1221no—
378C22×C4—1221no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4—1222no—
382C22×C4—1222no—
383C22×C4C2×D24411yescharacteristic
384C22×C4C2×D24411yescharacteristic
385C22×C4C2×D24411yescharacteristic
386C22×C4C2×D24411yescharacteristic
387C22×C4C2×D24411yescharacteristic
388C22×C4C23×D6111yescharacteristic
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8#1175822214yes—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—612no—
398C4⋊C4—612no—
399C4⋊C4—1224no—
400C4⋊C4—1224no—
401C4⋊C4—1224no—
402C4⋊C4—1224no—
403C4⋊C4—1221no—
404C4⋊C4—1221no—
405C4⋊C4—1224no—
406C4⋊C4—1224no—
407C4⋊C4—1224no—
408C4⋊C4—1224no—
409C4⋊C4—1222no—
410C4⋊C4—1222no—
411C4⋊C4—1222no—
412C4⋊C4—1222no—
413C4⋊C4—4882no—
414C4⋊C4—2442no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—1224no—
418C22⋊C4—1224no—
419C22⋊C4—4882no—
420C42—21no—
421C42—21no—
422C42—1224no—
423C42—1224no—
424C42D4×D6112yes—
425C42D4×D6112yes—
426C23—21no—
427C23—21no—
428C23—22no—
429C23—22no—
430C23—21no—
431C23C22×D24411yescharacteristic
432Q8—1224no—
433Q8—1224no—
434Q8—4884no—
435Q8—2442no—
436Q8—2442no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—1222no—
449C2×C4—1222no—
450C2×C4—1222no—
451C2×C4—1222no—
452C2×C4—2442no—
453C2×C4—2442no—
454C2×C4—2442no—
455C2×C4—2442no—
456C2×C4—2442no—
457C2×C4—2442no—
458C2×C4—2441no—
459C2×C4—2441no—
460C2×C4—2442no—
461C2×C4—2442no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4#1451974712yes—
465C2×C4#1451974712yes—
466C2×C4C2×D4×D6111yescharacteristic
467C2×C4C2×D4×D6111yescharacteristic
468C2×C4C2×#1175822212yes—
469C2×C4#1451982412yes—
470C8—24no—
471C8—42no—
472C22—41no—
473C22—41no—
474C22—42no—
475C22—42no—
476C22—41no—
477C22C2×#1451974711yescharacteristic
478C22#1684031211yescharacteristic
479C22#1684033811yescharacteristic
480C22#1684057412yes—
481C22#1684057912yes—
482C4—22no—
483C4—22no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—4882no—
488C4—4882no—
489C4—2442no—
490C4—2442no—
491C4—2442no—
492C4—2442no—
493C4#1684041712yes—
494C4#1684042112yes—
495C2—81no—
496C2#2144944711yescharacteristic
497C2#2145510711yescharacteristic
498C2#2145511611yescharacteristic
499C2#2145648712yes—
500C2#2145708912yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1750C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1750D61
hypocenterC61#25686
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1750

Complements

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

NH ≅ G/Ntype
2. C2×#16840171C2semidirect
3. C2×#16840113C2semidirect
7. #21446784C2semidirect
8. #21446639C2semidirect
9. #21446090C2semidirect
10. #21446076C2semidirect
297. C61#25686semidirect

Maximal subgroups

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

subgroupindex
C2×#168403742
C2×#168401712
C2×#168401132
#214516402
#214470232
#214470162
#214467842
#214466392
#214460902
#214460762
C61×#17502
#2568661