Finite Group Databasebeta

← #29779567

Subgroups — #29779567

Order 15616.

862 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454127C211yesmaximal, characteristic
2C2×#16839669C211yesmaximal, characteristic
3#21447240C212yesmaximal, complemented
4#21446968C212yesmaximal, complemented
5#21446511C212yesmaximal, complemented
6#21446506C212yesmaximal, complemented
7#21445910C211yesmaximal, characteristic
8#21445907C211yesmaximal, characteristic
9#21445754C211yesmaximal, characteristic
10#21445583C211yesmaximal, characteristic
11C61×#1787C211yesmaximal, characteristic
12C2×#14519768C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14C22×C4×Dic61C2211yescharacteristic
15C2×#14519701C2212yes—
16C2×#14519701C2212yes—
17C2×C22⋊#92418C2211yescharacteristic
18C2×#14519687C2212yes—
19#16839669—21no—
20#16839669—21no—
21#16839669C2212yes—
22#16839669C2212yes—
23#16839661C2211yescharacteristic
24#16839658C2211yescharacteristic
25#16839649—21no—
26#16839645—21no—
27#16839637—21no—
28#16839634—21no—
29#16839627C2211yescharacteristic
30#16839622C2211yescharacteristic
31C2×#14519684—21no—
32C2×#14519684—21no—
33#16839525C2212yes—
34#16839525C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839516C2212yes—
38C23×Q8×C61C2211yescharacteristic
39C2×#103×C61C2212yes—
40C2×#103×C61C2212yes—
41C2×C22⋊C8×C61C2211yescharacteristic
42C61×#380C2211yescharacteristic
43C23×Dic61—21no—
44C23×Dic61D411yescharacteristic
45C23×Dic61D411yescharacteristic
46C22×#92418C2312yes—
47#14519768—42no—
48C2×C4⋊Dic61—22no—
49C2×C4⋊Dic61—22no—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2312yes—
54C2×C4⋊Dic61C2311yescharacteristic
55C2×C4⋊Dic61C2311yescharacteristic
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×C22⋊Dic61—21no—
59C2×C4×Dic61—42no—
60C2×C4×Dic61D411yescharacteristic
61C2×C4×Dic61D411yescharacteristic
62C2×C4×Dic61C2311yescharacteristic
63C2×C4×Dic61C2311yescharacteristic
64#14519701—24no—
65#14519701—42no—
66C22⋊#92418—22no—
67#14519687—42no—
68#14519684—21no—
69#14519684—22no—
70#14519684—22no—
71#14519684—22no—
72#14519684—22no—
73#14519684—21no—
74#14519684—21no—
75#14519684—21no—
76#14519684—21no—
77C22×Q8×C61—24no—
78C22×Q8×C61—21no—
79C22×Q8×C61—22no—
80C22×Q8×C61—22no—
81C22×Q8×C61—42no—
82C22×Q8×C61C2312yes—
83C23×C4×C61—21no—
84C23×C4×C61C2311yescharacteristic
85C22×C8×C61C2312yes—
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C22⋊C4×C61—21no—
89#103×C61—24no—
90#103×C61—42no—
91C22⋊C8×C61—22no—
92#95×C61—21no—
93C24×C61C2×D411yescharacteristic
94C2×Q8×C61—24no—
95C2×Q8×C61—22no—
96C2×Q8×C61—22no—
97C2×Q8×C61—22no—
98C2×Q8×C61—22no—
99C2×Q8×C61—22no—
100C2×Q8×C61—48no—
101C2×Q8×C61—48no—
102C2×Q8×C61—42no—
103C2×Q8×C61—42no—
104C2×Q8×C61#4114yes—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—24no—
108C22×C4×C61—22no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—42no—
112C22×C4×C61—42no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61#4112yes—
115C22×C4×C61#4112yes—
116C22×C4×C61C2411yescharacteristic
117C22×Dic61—22no—
118C22×Dic61—22no—
119C22×Dic61—22no—
120C22×Dic61—21no—
121C22×Dic61—21no—
122C22×Dic61—41no—
123C22×Dic61—41no—
124C22×Dic61—41no—
125C22×Dic61—41no—
126C22×Dic61—42no—
127C22×Dic61—42no—
128C22×Dic61—42no—
129C22×Dic61C2×D411yescharacteristic
130C22×Dic61C2×D411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C2×C8×C61—22no—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C22⋊C4×C61—42no—
139C22⋊Dic61—42no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C4⋊Dic61—22no—
143C4⋊Dic61—22no—
144C4⋊Dic61—42no—
145C4⋊Dic61—42no—
146C4⋊Dic61—42no—
147#11758196—42no—
148#11758196—42no—
149C4×Dic61—22no—
150C4×Dic61—22no—
151C4×Dic61—41no—
152C4×Dic61—41no—
153C2×#92418—22no—
154C2×#92418—22no—
155C2×#92418—22no—
156C23×C61—21no—
157C23×C61—21no—
158C23×C61—21no—
159C23×C61—21no—
160C23×C61#12412yes—
161C23×C61#13712yes—
162C23×C61#13712yes—
163C23×C61C22×D411yescharacteristic
164Q8×C61—24no—
165Q8×C61—24no—
166Q8×C61—48no—
167Q8×C61—42no—
168Q8×C61—42no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—24no—
172C2×C4×C61—24no—
173C2×C4×C61—24no—
174C2×C4×C61—48no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61#12412yes—
184C2×C4×C61C2×#4112yes—
185C2×C4×C61C2×#4111yescharacteristic
186C2×C4×C61C2×#4111yescharacteristic
187C2×C4×C61H(5,𝔽2)12yes—
188C2×Dic61—21no—
189C2×Dic61—21no—
190C2×Dic61—21no—
191C2×Dic61—21no—
192C2×Dic61—21no—
193C2×Dic61—21no—
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—21no—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—44no—
201C2×Dic61—44no—
202C2×Dic61—41no—
203C2×Dic61—42no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—84no—
212C8×C61—42no—
213#92418—42no—
214#29112—611nomaximal, Sylow
215C22×C61—22no—
216C22×C61—22no—
217C22×C61—22no—
218C22×C61—22no—
219C22×C61—22no—
220C22×C61—22no—
221C22×C61#49114yes—
222C22×C61#52511yescharacteristic
223C22×C61C2×#12411yescharacteristic
224C22×C61#53411yescharacteristic
225C22×C61C2×#13711yescharacteristic
226C22×C61C2×#13711yescharacteristic
227C22×C61#57511yescharacteristic
228C22×C61#57511yescharacteristic
229C4×C61—22no—
230C4×C61—44no—
231C4×C61—81no—
232C4×C61—81no—
233C4×C61#55112yes—
234C4×C61#55112yes—
235Dic61—41no—
236Dic61—41no—
237Dic61—41no—
238Dic61—41no—
239Dic61—81no—
240Dic61—82no—
241Dic61—81no—
242#2851—611no—
243C2×#391—611no—
244#2074—612no—
245#2026—612no—
246#1935—612no—
247#1935—612no—
248#1824—611no—
249#1824—611no—
250#1787—611no—
251#1787D6111yescharacteristic
252#1749—611no—
253C2×C61—24no—
254C2×C61#256111yescharacteristic
255C2×C61#311711yescharacteristic
256C2×C61#311711yescharacteristic
257C2×C61C2×#49111yescharacteristic
258C2×C61#314011yescharacteristic
259C2×C61#319611yescharacteristic
260C2×C61#319611yescharacteristic
261C23×Q8C2×D6111yescharacteristic
262C2×#125—611no—
263C22×C4⋊C4—611no—
264C22×C42—611no—
265C2×#103—612no—
266C2×#103—612no—
267C2×#103C2×D6112yes—
268C2×#103C2×D6112yes—
269C2×C22⋊C8—611no—
270C2×C22⋊C8C2×D6111yescharacteristic
271C2×#97—612no—
272#391—612no—
273#391—612no—
274#391—1221no—
275#391—1221no—
276#389—611no—
277#389—611no—
278#387—1221no—
279#387—1221no—
280#385—1221no—
281#385—1221no—
282#382—611no—
283#380—611no—
284#380C2×D6111yescharacteristic
285C2×#95—1221no—
286C2×#95—1221no—
287#340—612no—
288#340—612no—
289#340—612no—
290#340—612no—
291#336—612no—
292C61#2911211yescharacteristic, complemented, Sylow
293C22×Q8—24no—
294C22×Q8—21no—
295C22×Q8—22no—
296C22×Q8—22no—
297C22×Q8—42no—
298C22×Q8C22×D6112yes—
299C23×C4—21no—
300C23×C4—611no—
301C23×C4—611no—
302C23×C4—1221no—
303C23×C4C22×D6111yescharacteristic
304C22×C8—612no—
305C22×C8C22×D6112yes—
306#125—2442no—
307C2×C4⋊C4—612no—
308C2×C4⋊C4—611no—
309C2×C4⋊C4—611no—
310C2×C4⋊C4—1222no—
311C2×C4⋊C4—1222no—
312C2×C4⋊C4—1221no—
313C2×C4⋊C4—1221no—
314C2×C4⋊C4—1221no—
315C2×C4⋊C4—1221no—
316C2×C4⋊C4—1221no—
317C2×C4⋊C4C22×D6111yescharacteristic
318C2×C4⋊C4C22×D6111yescharacteristic
319C2×C22⋊C4—21no—
320C2×C22⋊C4—1221no—
321C2×C42—611no—
322C2×C42—611no—
323C2×C42—611no—
324C2×C42—611no—
325C2×C42—2442no—
326#103—24no—
327#103—42no—
328#103—1224no—
329#103—2442no—
330C22⋊C8—22no—
331C22⋊C8—1222no—
332#97—2442no—
333#95—21no—
334#95—1222no—
335#95—1222no—
336#95—1222no—
337#95—1222no—
338#95—1221no—
339#95—1221no—
340#95—1221no—
341#95—1221no—
342#95—1221no—
343C24D4×D6111yescharacteristic
344C2×Q8—24no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8#1175822414yes—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—611no—
364C22×C4—611no—
365C22×C4—611no—
366C22×C4—611no—
367C22×C4—1222no—
368C22×C4—1222no—
369C22×C4—1221no—
370C22×C4—1222no—
371C22×C4—1221no—
372C22×C4—2441no—
373C22×C4—2441no—
374C22×C4—2441no—
375C22×C4—2441no—
376C22×C4—2442no—
377C22×C4—2442no—
378C22×C4—2442no—
379C22×C4#1175822212yes—
380C22×C4D4×D6111yescharacteristic
381C22×C4#1175822412yes—
382C22×C4C23×D6111yescharacteristic
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—1222no—
387C2×C8—1222no—
388C2×C8—1222no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—1222no—
392C4⋊C4—1222no—
393C4⋊C4—1222no—
394C4⋊C4—1222no—
395C4⋊C4—2442no—
396C4⋊C4—2442no—
397C4⋊C4—2442no—
398C4⋊C4—2442no—
399C4⋊C4—2442no—
400C22⋊C4—42no—
401C22⋊C4—2442no—
402C42—1222no—
403C42—1222no—
404C42—2441no—
405C42—2441no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23#1451976312yes—
411C23#1451980612yes—
412C23#1451980912yes—
413C23C2×D4×D6111yescharacteristic
414Q8—24no—
415Q8—24no—
416Q8—48no—
417Q8—42no—
418Q8—42no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—48no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—1221no—
434C2×C4—1221no—
435C2×C4—1221no—
436C2×C4—1221no—
437C2×C4—1221no—
438C2×C4—1221no—
439C2×C4—1221no—
440C2×C4—1221no—
441C2×C4—1221no—
442C2×C4—1221no—
443C2×C4—1221no—
444C2×C4—1221no—
445C2×C4—4884no—
446C2×C4—2441no—
447C2×C4—2442no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4—2441no—
451C2×C4—2441no—
452C2×C4—2442no—
453C2×C4—2442no—
454C2×C4—2441no—
455C2×C4—2444no—
456C2×C4—2444no—
457C2×C4#1451976312yes—
458C2×C4C2×#1175822412yes—
459C2×C4C2×#1175822411yescharacteristic
460C2×C4C2×#1175822211yescharacteristic
461C2×C4#1451982212yes—
462C8—42no—
463C8—2442no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22#1684012914yes—
471C22#1684026811yescharacteristic
472C22C2×#1451976311yescharacteristic
473C22#1684032211yescharacteristic
474C22C2×#1451980611yescharacteristic
475C22C2×#1451980911yescharacteristic
476C22#1684056111yescharacteristic
477C22#1684056111yescharacteristic
478C4—22no—
479C4—44no—
480C4—81no—
481C4—81no—
482C4—4881no—
483C4—4882no—
484C4—4881no—
485C4—2441no—
486C4—2441no—
487C4—2441no—
488C4—2441no—
489C4#1684043812yes—
490C4#1684044312yes—
491C2—24no—
492C2#2145097411yescharacteristic
493C2#2145598511yescharacteristic
494C2#2145599111yescharacteristic
495C2C2×#1684012911yescharacteristic
496C2#2145620711yescharacteristic
497C2#2145664611yescharacteristic
498C2#2145665211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1787C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1787D61
hypocenterC61#29112
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
3. #21447240C2semidirect
4. #21446968C2semidirect
5. #21446511C2semidirect
6. #21446506C2semidirect
292. C61#29112semidirect

Maximal subgroups

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

subgroupindex
#214541272
C2×#168396692
#214472402
#214469682
#214465112
#214465062
#214459102
#214459072
#214457542
#214455832
C61×#17872
#2911261