Finite Group Databasebeta

← #29838101

Subgroups — #29838101

Order 15616.

504 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458805C211yesmaximal, characteristic, complemented
2#21458804C211yesmaximal, characteristic, complemented
3#21448091C211yesmaximal, characteristic, complemented
4#21448089C211yesmaximal, characteristic, complemented
5#21448077C211yesmaximal, characteristic
6#21447880C211yesmaximal, characteristic, complemented
7#21447879C211yesmaximal, characteristic, complemented
8#21447872C211yesmaximal, characteristic
9#21447870C211yesmaximal, characteristic, complemented
10#21447868C211yesmaximal, characteristic, complemented
11#21447726C211yesmaximal, characteristic, complemented
12#21447683C211yesmaximal, characteristic, complemented
13#21447680C211yesmaximal, characteristic, complemented
14#21447675C211yesmaximal, characteristic, complemented
15C61×#2210C211yesmaximal, characteristic
16#16840482—21no—
17#16840481—21no—
18C2×Dic488C2211yescharacteristic
19C2×Dic488C2211yescharacteristic
20C2×#14519723C2211yescharacteristic
21C2×#14519723C2211yescharacteristic
22C2×#14519721C2211yescharacteristic
23C2×#14519721C2211yescharacteristic
24C2×#14519721C2211yescharacteristic
25#16840145C2211yescharacteristic
26#16840145C2211yescharacteristic
27#16840145C2211yescharacteristic
28#16840142C2211yescharacteristic
29#16840140C2211yescharacteristic
30#16839983—21no—
31#16839983—21no—
32#16839982—21no—
33#16839981—21no—
34#16839967—21no—
35#16839965—21no—
36C8×Dic122C2211yescharacteristic
37C4×Dic244C2211yescharacteristic
38Q16×Dic61C2211yescharacteristic
39#16839856C2211yescharacteristic
40#16839856C2211yescharacteristic
41D8×Dic61C2211yescharacteristic
42#16839587C2211yescharacteristic
43#16839580C2211yescharacteristic
44#16839580C2211yescharacteristic
45#16839567C2211yescharacteristic
46#16839567C2211yescharacteristic
47#16839565C2211yescharacteristic
48#16839563C2211yescharacteristic
49#16839538C2211yescharacteristic
50#16839537C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C2×SD32×C61C2211yescharacteristic
53C2×D16×C61C2211yescharacteristic
54C61×#495C2211yescharacteristic
55C4×D8×C61C2211yescharacteristic
56C61×C4⋊C16C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C2×Dic244—21no—
60C2×Dic244C2311yescharacteristic
61C2×Dic244C2311yescharacteristic
62C2×Dic244C2311yescharacteristic
63C2×#11758202—21no—
64C2×#11758200—21no—
65C2×#11758200—21no—
66C2×#11758200—21no—
67#14519777—21no—
68#14519775—21no—
69#14519770—41no—
70#14519770—41no—
71#14519770—41no—
72#14519770—41no—
73#14519764—21no—
74#14519764—21no—
75#14519764—21no—
76#14519756—41no—
77#14519754—41no—
78Q8×Dic61—21no—
79C4×Dic122—21no—
80C4×Dic122D411yescharacteristic
81#14519742—21no—
82#14519742—21no—
83D4×Dic61—21no—
84#14519736—41no—
85#14519733—41no—
86Dic488—41no—
87Dic488—41no—
88#14519723—22no—
89#14519723—41no—
90#14519721—22no—
91#14519721—22no—
92#14519721—41no—
93C2×#11758185C2311yescharacteristic
94C2×#11758185C2311yescharacteristic
95#14519711C2311yescharacteristic
96C4⋊#92418D411yescharacteristic
97#14519706D411yescharacteristic
98#14519706D411yescharacteristic
99#14519702—21no—
100#14519701—21no—
101#14519700—21no—
102#14519700—21no—
103#14519698—21no—
104C8×Dic61C2311yescharacteristic
105C8×Dic61C2311yescharacteristic
106C2×Q16×C61C2311yescharacteristic
107C2×SD16×C61—21no—
108C2×D8×C61C2311yescharacteristic
109C2×D8×C61C2311yescharacteristic
110#124×C61—21no—
111C4×D4×C61—21no—
112SD32×C61—41no—
113D16×C61—41no—
114C2×C16×C61C2311yescharacteristic
115C2×C16×C61C2311yescharacteristic
116#107×C61C2311yescharacteristic
117#102×C61—21no—
118C4×C8×C61C2311yescharacteristic
119C2×Q8×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C22×C4×C61—41no—
123C22×Dic61—41no—
124C2×Dic122—21no—
125C2×Dic122—21no—
126C2×Dic122—21no—
127C2×Dic122D811yescharacteristic
128Q16×C61—21no—
129Q16×C61—21no—
130SD16×C61—81no—
131D8×C61—21no—
132D8×C61—21no—
133D8×C61—21no—
134D8×C61—21no—
135C2×C8×C61C2×D411yescharacteristic
136C2×C8×C61C2411yescharacteristic
137C4⋊C4×C61—21no—
138C22⋊C4×C61—41no—
139C22⋊C4×C61—41no—
140C22⋊C4×C61—41no—
141C42×C61C2×D411yescharacteristic
142C22⋊Dic61—41no—
143C22⋊Dic61—41no—
144C22⋊Dic61—41no—
145C22⋊Dic61—41no—
146C22⋊Dic61—41no—
147C22⋊Dic61—41no—
148C22⋊Dic61—41no—
149#11758202—81no—
150#11758200—81no—
151#11758200—81no—
152#11758200—81no—
153C4⋊Dic61—21no—
154C4⋊Dic61—41no—
155C4⋊Dic61—41no—
156C4⋊Dic61—41no—
157C4⋊Dic61D811yescharacteristic
158#11758196—41no—
159#11758196—41no—
160#11758196—41no—
161#11758196—41no—
162#11758196—41no—
163#11758196D811yescharacteristic
164#11758196D811yescharacteristic
165C4×Dic61—41no—
166C4×Dic61—41no—
167C4×Dic61—41no—
168C4×Dic61—41no—
169C4×Dic61C2×D411yescharacteristic
170C4×Dic61C2×D411yescharacteristic
171C2×#92418C2×D411yescharacteristic
172C2×#92418C2×D411yescharacteristic
173Dic244—21no—
174Dic244—21no—
175Dic244—21no—
176Dic244—21no—
177Dic244—21no—
178Dic244—21no—
179Dic244—81no—
180C16×C61—21no—
181C16×C61—21no—
182#11758185—21no—
183#11758185#4112yes—
184C23×C61—41no—
185C23×C61—41no—
186Q8×C61—41no—
187Q8×C61—41no—
188D4×C61—41no—
189D4×C61—41no—
190D4×C61—41no—
191D4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—82no—
195C2×C4×C61C2×D811yescharacteristic
196C2×C4×C61C2×D811yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—82no—
203C2×Dic61C2×D811yescharacteristic
204C2×Dic61C2×D811yescharacteristic
205C2×Dic61C2×D811yescharacteristic
206C2×Dic61C2×D811yescharacteristic
207Dic122—41no—
208Dic122—41no—
209Dic122—41no—
210Dic122—41no—
211Dic122—41no—
212Dic122—41no—
213Dic122—41no—
214C8×C61—21no—
215C8×C61C2×#4111yescharacteristic
216C8×C61H(5,𝔽2)11yescharacteristic
217#92418—21no—
218#92418—21no—
219#33969—611nomaximal, Sylow
220C22×C61—81no—
221C22×C61—81no—
222C22×C61—81no—
223C22×C61—81no—
224C22×C61—81no—
225C22×C61—81no—
226C22×C61C22×D811yescharacteristic
227C4×C61—21no—
228C4×C61—21no—
229C4×C61—81no—
230C4×C61—81no—
231C4×C61#55011yescharacteristic
232C4×C61#57611yescharacteristic
233Dic61—21no—
234Dic61—21no—
235Dic61—21no—
236Dic61—21no—
237Dic61—81no—
238Dic61—81no—
239Dic61—81no—
240Dic61—81no—
241#3393—611no—
242#3393—611no—
243#2251—611no—
244#2251—611no—
245#2249—611no—
246#2211—611no—
247#2211—611no—
248#2210—611no—
249#2210D6111yescharacteristic
250#2209—611no—
251#2209—611no—
252#2183—611no—
253#2175—611no—
254#2175—611no—
255C4×Q32—611no—
256C2×C61—161no—
257C2×C61—161no—
258C2×C61#339111yescharacteristic
259C2×C61#341411yescharacteristic
260C2×C61#341711yescharacteristic
261#556—1221no—
262#556—1221no—
263C2×Q32—611no—
264C2×Q32—611no—
265C2×Q32—611no—
266C2×Q32—611no—
267C2×SD32—611no—
268C2×SD32—611no—
269C2×SD32—611no—
270C2×SD32C2×D6111yescharacteristic
271C2×D16C2×D6111yescharacteristic
272#495—611no—
273#495—611no—
274#495—611no—
275#495C2×D6111yescharacteristic
276#494—611no—
277#494—611no—
278#464—1221no—
279#464—1221no—
280#464—1221no—
281#464—1221no—
282#461—1221no—
283#461—1221no—
284C8×Q8—611no—
285C4×Q16—611no—
286C4×Q16—611no—
287C4×Q16—611no—
288C4×Q16—611no—
289C4×D8—611no—
290C4×D8C2×D6111yescharacteristic
291#366—611no—
292C4⋊C16—611no—
293C4⋊C16—611no—
294C4⋊C16C2×D6111yescharacteristic
295#358—611no—
296#358—611no—
297#358—611no—
298#358C2×D6111yescharacteristic
299#357—611no—
300#357C2×D6111yescharacteristic
301#346—611no—
302#346—611no—
303C4×C16—611no—
304C61#3396911yescharacteristic, complemented, Sylow
305C2×Q16—611no—
306C2×Q16—611no—
307C2×Q16—611no—
308C2×Q16—1221no—
309C2×Q16—1221no—
310C2×Q16C22×D6111yescharacteristic
311C2×SD16—21no—
312C2×SD16—1221no—
313C2×SD16—1221no—
314C2×SD16—1221no—
315C2×D8C22×D6111yescharacteristic
316C2×D8C22×D6111yescharacteristic
317C4⋊Q8—1221no—
318C4⋊Q8—1221no—
319#126—2441no—
320#126—2441no—
321#126—2441no—
322#126—2441no—
323#124—21no—
324#124—1221no—
325#124—1221no—
326#124—1221no—
327C22⋊Q8—2441no—
328C22⋊Q8—2441no—
329C4×Q8—611no—
330C4×Q8—1221no—
331C4×Q8—1221no—
332C4×Q8—1221no—
333C4×Q8—1221no—
334C4×D4—21no—
335C4×D4—1221no—
336#117—2441no—
337#117—2441no—
338Q32—1222no—
339Q32—2441no—
340Q32—2441no—
341Q32—2441no—
342SD32—41no—
343SD32—1222no—
344SD32—1222no—
345SD32—2441no—
346D16—41no—
347C2×C16—611no—
348C2×C16—611no—
349C2×C16C22×D6111yescharacteristic
350C2×C16C22×D6111yescharacteristic
351#107—611no—
352#107C22×D6111yescharacteristic
353C4⋊C8—611no—
354C4⋊C8—611no—
355C4⋊C8—611no—
356#103—1221no—
357#103—1221no—
358#103—1221no—
359#103—1221no—
360#102—21no—
361#102—1221no—
362C4×C8—611no—
363C4×C8—611no—
364C4×C8C22×D6111yescharacteristic
365C2×Q8—21no—
366C2×Q8—611no—
367C2×Q8—1221no—
368C2×Q8—1221no—
369C2×Q8—1221no—
370C2×D4—21no—
371C2×D4—21no—
372C22×C4—41no—
373C22×C4—2441no—
374Q16—21no—
375Q16—21no—
376Q16—1221no—
377Q16—1221no—
378Q16—1221no—
379Q16—1221no—
380Q16—1221no—
381Q16—1221no—
382Q16—4881no—
383Q16—4881no—
384SD16—81no—
385SD16—4881no—
386SD16—4881no—
387SD16—4881no—
388D8—21no—
389D8—21no—
390D8—21no—
391D8—21no—
392C2×C8—611no—
393C2×C8—611no—
394C2×C8D4×D6111yescharacteristic
395C2×C8C23×D6111yescharacteristic
396C4⋊C4—21no—
397C4⋊C4—611no—
398C4⋊C4—611no—
399C4⋊C4—611no—
400C4⋊C4—1221no—
401C4⋊C4—2441no—
402C4⋊C4—2441no—
403C4⋊C4—2441no—
404C4⋊C4—2441no—
405C4⋊C4—2441no—
406C4⋊C4—2441no—
407C4⋊C4—2441no—
408C4⋊C4—2441no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—2441no—
413C22⋊C4—2441no—
414C22⋊C4—2441no—
415C22⋊C4—2441no—
416C22⋊C4—2441no—
417C22⋊C4—2441no—
418C22⋊C4—2441no—
419C42—611no—
420C42—611no—
421C42—2441no—
422C42—2441no—
423C42—2441no—
424C42—2441no—
425C42D4×D6111yescharacteristic
426C16—21no—
427C16—21no—
428C16—612no—
429C16—1221no—
430C23—41no—
431C23—41no—
432Q8—41no—
433Q8—41no—
434Q8—2441no—
435Q8—2441no—
436Q8—2441no—
437Q8—2441no—
438Q8—2441no—
439Q8—2441no—
440Q8—2441no—
441D4—41no—
442D4—41no—
443D4—41no—
444D4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—82no—
448C2×C4—611no—
449C2×C4—611no—
450C2×C4—611no—
451C2×C4—611no—
452C2×C4—4882no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4—2441no—
457C2×C4D8×D6111yescharacteristic
458C2×C4D8×D6111yescharacteristic
459C2×C4C2×D4×D6111yescharacteristic
460C8—21no—
461C8—1221no—
462C8—1221no—
463C8#41×D6111yescharacteristic
464C8#1451982211yescharacteristic
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22C2×D8×D6111yescharacteristic
472C4—21no—
473C4—21no—
474C4—81no—
475C4—81no—
476C4—1221no—
477C4—1221no—
478C4—1221no—
479C4—1221no—
480C4—4881no—
481C4—4881no—
482C4—4881no—
483C4—4881no—
484C4#1684043011yescharacteristic
485C4#1684056611yescharacteristic
486C2—161no—
487C2—161no—
488C2#2145879011yescharacteristic
489C2#2145895111yescharacteristic
490C2#2145896911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2210C2
socleC22×C61C22×D8
prime radicalC2×D16×C61C22
hypercenter#2210D61
hypocenterC61#33969
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2210

Complements

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

NH ≅ G/Ntype
1. #21458805C2semidirect
2. #21458804C2semidirect
3. #21448091C2semidirect
4. #21448089C2semidirect
6. #21447880C2semidirect
7. #21447879C2semidirect
9. #21447870C2semidirect
10. #21447868C2semidirect
11. #21447726C2semidirect
12. #21447683C2semidirect
13. #21447680C2semidirect
14. #21447675C2semidirect
304. C61#33969semidirect

Maximal subgroups

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

subgroupindex
#214588052
#214588042
#214480912
#214480892
#214480772
#214478802
#214478792
#214478722
#214478702
#214478682
#214477262
#214476832
#214476802
#214476752
C61×#22102
#3396961