Finite Group Databasebeta

← #29597581

Subgroups — #29597581

Order 15616.

650 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454298C211yesmaximal, characteristic
2#21447514C211yesmaximal, characteristic, complemented
3#21447473C211yesmaximal, characteristic, complemented
4#21445789C211yesmaximal, characteristic
5#21443948C211yesmaximal, characteristic, complemented
6#21443942C211yesmaximal, characteristic, complemented
7C61×#3009C211yesmaximal, characteristic
8#16840529—21no—
9#16839779—21no—
10#16839767—21no—
11#16839746—22no—
12#16839746C2211yescharacteristic
13#16839746C2211yescharacteristic
14#16839746C2211yescharacteristic
15C2×#14519694—22no—
16C2×#14519694—21no—
17#16839736C2211yescharacteristic
18#16839561—22no—
19#16839552—22no—
20C2×H(5,𝔽2)×C61C2211yescharacteristic
21C2×#137×C61C411yescharacteristic, complemented
22C22×SD16×C61C411yescharacteristic, complemented
23C61×#536C2211yescharacteristic
24C61×#452—21no—
25C61×#452—21no—
26C61×#450—21no—
27C61×#450—21no—
28C61×#418C411yescharacteristic, complemented
29C61×#416C411yescharacteristic, complemented
30C61×#408C2211yescharacteristic
31#14519784—42no—
32#14519784—42no—
33C2×#11758194—24no—
34C2×#11758194—21no—
35C2×#11758194—21no—
36C2×#11758194—21no—
37#14519736—21no—
38#14519704—42no—
39#14519698—42no—
40#14519696—21no—
41#14519696—21no—
42#14519696D412yes—
43#14519694—24no—
44#14519694—21no—
45#14519694—21no—
46#14519694D412yes—
47#14519692—21no—
48#14519692—21no—
49H(5,𝔽2)×C61—42no—
50H(5,𝔽2)×C61—42no—
51C2×#41×C61—21no—
52C2×#41×C61C2×C411yescharacteristic
53C2×#41×C61D411yescharacteristic
54C2×#41×C61D411yescharacteristic
55C2×#41×C61C2311yescharacteristic
56C22×Q8×C61C2×C411yescharacteristic
57C22×D4×C61—21no—
58C22×D4×C61—21no—
59C22×D4×C61—21no—
60C22×D4×C61C2×C411yescharacteristic
61C22×D4×C61D411yescharacteristic
62C22×D4×C61D411yescharacteristic
63#137×C61—42no—
64C2×Q16×C61—21no—
65C2×SD16×C61—21no—
66C2×SD16×C61—21no—
67C2×SD16×C61—21no—
68C2×SD16×C61—42no—
69C2×M16×C61C2×C411yescharacteristic
70C22×C8×C61C2×C411yescharacteristic
71C4⋊Q8×C61—21no—
72#124×C61—21no—
73#123×C61—21no—
74#123×C61—21no—
75C22⋊Q8×C61—21no—
76C22⋊Q8×C61—21no—
77#117×C61C2×C411yescharacteristic
78#103×C61—21no—
79#102×C61—21no—
80#102×C61—21no—
81#102×C61—21no—
82C22⋊C8×C61—21no—
83C22⋊C8×C61—21no—
84C24×C61—21no—
85C24×C61—21no—
86C24×C61—21no—
87#41×C61—22no—
88#41×C61—22no—
89#41×C61—22no—
90#41×C61—22no—
91#41×C61—41no—
92#41×C61—41no—
93#41×C61—41no—
94#41×C61—41no—
95#41×C61—44no—
96C2×Q8×C61—21no—
97C2×Q8×C61—21no—
98C2×Q8×C61—41no—
99C2×Q8×C61—41no—
100C2×Q8×C61C22⋊C411yescharacteristic
101C2×Q8×C61C2×D411yescharacteristic
102C2×D4×C61—21no—
103C2×D4×C61—22no—
104C2×D4×C61—22no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—41no—
111C2×D4×C61—41no—
112C2×D4×C61—41no—
113C2×D4×C61—41no—
114C2×D4×C61—44no—
115C2×D4×C61—44no—
116C2×D4×C61—44no—
117C2×D4×C61C22⋊C411yescharacteristic
118C2×D4×C61C22⋊C411yescharacteristic
119C2×D4×C61C22⋊C411yescharacteristic
120C2×D4×C61C2×D411yescharacteristic
121C2×D4×C61#4111yescharacteristic
122C2×D4×C61#4111yescharacteristic
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61C22×C411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130Q16×C61—42no—
131SD16×C61—42no—
132SD16×C61—42no—
133SD16×C61—42no—
134M16×C61—41no—
135C2×C8×C61—21no—
136C2×C8×C61—21no—
137C2×C8×C61—41no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—41no—
140C4⋊C4×C61—41no—
141C22⋊C4×C61—21no—
142C22⋊C4×C61—41no—
143C22⋊C4×C61—41no—
144C42×C61—21no—
145C22⋊Dic61—41no—
146C4⋊Dic61—41no—
147C4×Dic61—41no—
148#11758194—22no—
149#11758194—22no—
150#11758194—44no—
151#11758194—41no—
152#11758194—41no—
153#11758194—41no—
154#11758194—41no—
155C2×#92418—44no—
156C2×#92418—41no—
157C2×#92418—41no—
158C2×#92418—41no—
159C23×C61—21no—
160C23×C61—21no—
161C23×C61—21no—
162C23×C61—42no—
163C23×C61—42no—
164C23×C61—42no—
165C23×C61—161no—
166C23×C61—161no—
167C23×C61—161no—
168C23×C61C4×D411yescharacteristic
169C23×C61#12111yescharacteristic
170C23×C61#12311yescharacteristic
171Q8×C61—21no—
172Q8×C61—41no—
173Q8×C61—41no—
174Q8×C61—41no—
175Q8×C61—81no—
176Q8×C61#9912yes—
177D4×C61—24no—
178D4×C61—21no—
179D4×C61—21no—
180D4×C61—21no—
181D4×C61—21no—
182D4×C61—21no—
183D4×C61—41no—
184D4×C61—41no—
185D4×C61—41no—
186D4×C61—42no—
187D4×C61—42no—
188D4×C61—42no—
189D4×C61—42no—
190D4×C61—42no—
191D4×C61#9912yes—
192C2×C4×C61—24no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—81no—
205C2×C4×C61C2×C22⋊C411yescharacteristic
206C2×C4×C61C4×D411yescharacteristic
207C2×C4×C61#12011yescharacteristic
208C2×C4×C61#12111yescharacteristic
209C2×Dic61—81no—
210C8×C61—41no—
211C8×C61—41no—
212#92418—42no—
213#92418—42no—
214#92418—81no—
215#13097—611nomaximal, Sylow
216C22×C61—21no—
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—22no—
220C22×C61—41no—
221C22×C61—41no—
222C22×C61—41no—
223C22×C61—41no—
224C22×C61—41no—
225C22×C61—81no—
226C22×C61—81no—
227C22×C61—81no—
228C22×C61—162no—
229C22×C61—162no—
230C22×C61—162no—
231C22×C61#38611yescharacteristic
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—22no—
235C4×C61—21no—
236C4×C61—81no—
237C4×C61—81no—
238C4×C61C2×#9911yescharacteristic
239C4×C61C2×#9911yescharacteristic
240Dic61—161no—
241#3009D6111yescharacteristic
242#2888—611no—
243#2135—611no—
244#2123—611no—
245#1795—611no—
246#1395—611no—
247#1392—611no—
248C2×C61—21no—
249C2×C61—41no—
250C2×C61—41no—
251C2×C61—41no—
252C2×C61—42no—
253C2×C61—81no—
254C2×C61—81no—
255C2×C61#189811yescharacteristic
256C2×H(5,𝔽2)C2×D6111yescharacteristic
257C2×#137Dic6111yescharacteristic, complemented
258C22×SD16Dic6111yescharacteristic, complemented
259#568—1221no—
260#536C2×D6111yescharacteristic
261#452—21no—
262#452—21no—
263#450—21no—
264#450—21no—
265#421—1221no—
266#418—1221no—
267#418Dic6111yescharacteristic, complemented
268#416Dic6111yescharacteristic, complemented
269#413—611no—
270#413—611no—
271#413—611no—
272#413—1222no—
273C2×#100—1222no—
274C2×#100—1221no—
275#410—611no—
276#408C2×D6111yescharacteristic
277#356—1222no—
278#352—1222no—
279C61#1309711yescharacteristic, complemented, Sylow
280H(5,𝔽2)—42no—
281H(5,𝔽2)—42no—
282C2×#41—21no—
283C2×#41C2×Dic6111yescharacteristic
284C2×#41C22⋊D6111yescharacteristic
285C2×#41C22⋊D6111yescharacteristic
286C2×#41C22×D6111yescharacteristic
287C22×Q8C2×Dic6111yescharacteristic
288C22×D4—21no—
289C22×D4—21no—
290C22×D4—21no—
291C22×D4C2×Dic6111yescharacteristic
292C22×D4C22⋊D6111yescharacteristic
293C22×D4C22⋊D6111yescharacteristic
294#137—42no—
295C2×Q16—21no—
296C2×SD16—21no—
297C2×SD16—21no—
298C2×SD16—21no—
299C2×SD16—42no—
300#131—2442no—
301#131—2442no—
302C2×M16—1224no—
303C2×M16—1221no—
304C2×M16—1221no—
305C2×M16—1221no—
306C2×M16C2×Dic6111yescharacteristic
307C22×C8C2×Dic6111yescharacteristic
308C4⋊Q8—21no—
309#124—21no—
310#123—21no—
311#123—21no—
312C22⋊Q8—21no—
313C22⋊Q8—21no—
314#117—1221no—
315#117C2×Dic6111yescharacteristic
316#104—2442no—
317#103—21no—
318#102—21no—
319#102—21no—
320#102—21no—
321#102—2442no—
322#101—612no—
323#101—1221no—
324#101—1221no—
325#100—612no—
326#100—1224no—
327#100—1221no—
328#100—1221no—
329#99—1221no—
330#99—1221no—
331C22⋊C8—21no—
332C22⋊C8—21no—
333C24—21no—
334C24—21no—
335C24—21no—
336#41—22no—
337#41—22no—
338#41—22no—
339#41—22no—
340#41—41no—
341#41—41no—
342#41—41no—
343#41—41no—
344#41—44no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8C22⋊Dic6111yescharacteristic
350C2×Q8D4×D6111yescharacteristic
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4C22⋊Dic6111yescharacteristic
367C2×D4C22⋊Dic6111yescharacteristic
368C2×D4C22⋊Dic6111yescharacteristic
369C2×D4D4×D6111yescharacteristic
370C2×D4#1175822411yescharacteristic
371C2×D4#1175822411yescharacteristic
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4C22×Dic6111yescharacteristic
377C22×C4C2×C22⋊D6111yescharacteristic
378C22×C4C2×C22⋊D6111yescharacteristic
379Q16—42no—
380SD16—42no—
381SD16—42no—
382SD16—42no—
383M16—41no—
384M16—1222no—
385M16—1222no—
386M16—2441no—
387M16—2441no—
388M16—2444no—
389M16—2441no—
390M16—2441no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—41no—
394C2×C8—2441no—
395C2×C8—2444no—
396C2×C8—2441no—
397C2×C8—2441no—
398C4⋊C4—21no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—2441no—
402C22⋊C4—21no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—2441no—
406C42—21no—
407C42—2441no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—161no—
415C23—161no—
416C23—161no—
417C23D4×Dic6111yescharacteristic
418C23#1451974911yescharacteristic
419C23#1451975811yescharacteristic
420Q8—21no—
421Q8—41no—
422Q8—41no—
423Q8—41no—
424Q8—81no—
425Q8#1451969212yes—
426D4—24no—
427D4—21no—
428D4—21no—
429D4—21no—
430D4—21no—
431D4—21no—
432D4—41no—
433D4—41no—
434D4—41no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—42no—
439D4—42no—
440D4#1451969212yes—
441C2×C4—24no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—81no—
454C2×C4—4881no—
455C2×C4C2×C22⋊Dic6111yescharacteristic
456C2×C4D4×Dic6111yescharacteristic
457C2×C4#1451974511yescharacteristic
458C2×C4#1451974911yescharacteristic
459C8—41no—
460C8—41no—
461C8—4881no—
462C8—2442no—
463C8—2442no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—22no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—162no—
477C22—162no—
478C22—162no—
479C22#1683964211yescharacteristic
480C4—21no—
481C4—21no—
482C4—22no—
483C4—21no—
484C4—81no—
485C4—81no—
486C4—9761no—
487C4C2×#1451969211yescharacteristic
488C4C2×#1451969211yescharacteristic
489C2—21no—
490C2—41no—
491C2—41no—
492C2—41no—
493C2—42no—
494C2—81no—
495C2—81no—
496C2#2144627311yescharacteristic
subgroupG/H
centerC2#21446273
commutatorC22×C4×C61C22×C4
FrattiniC2×#41C22×D61
FittingC61×#3009C2
socleC2×C61#1898
prime radicalC2×H(5,𝔽2)×C61C22
cosocleC2×#41×C61C23
hypercenter#3009D61
hypocenterC61#13097
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ #3009

Complements

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

NH ≅ G/Ntype
2. #21447514C2semidirect
3. #21447473C2semidirect
5. #21443948C2semidirect
6. #21443942C2semidirect
21. C2×#137×C61C4semidirect
22. C22×SD16×C61C4semidirect
28. C61×#418C4semidirect
29. C61×#416C4semidirect
257. C2×#137Dic61semidirect
258. C22×SD16Dic61semidirect
267. #418Dic61semidirect
268. #416Dic61semidirect
279. C61#13097semidirect

Maximal subgroups

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

subgroupindex
#214542982
#214475142
#214474732
#214457892
#214439482
#214439422
C61×#30092
#1309761