Finite Group Databasebeta

← #29697128

Subgroups — #29697128

Order 15616.

994 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839758C211yesmaximal, characteristic
2#21448710C211yesmaximal, characteristic, complemented
3#21446512C212yesmaximal, complemented
4#21446450C212yesmaximal, complemented
5#21446080C212yesmaximal, complemented
6#21445911C211yesmaximal, characteristic, complemented
7#21445842C211yesmaximal, characteristic, complemented
8#21445759C211yesmaximal, characteristic
9#21445644C212yesmaximal, complemented
10#21445597C211yesmaximal, characteristic, complemented
11C61×#2339C211yesmaximal, characteristic
12C23×#92418C2211yescharacteristic
13C2×#14519736C2211yescharacteristic
14C2×#14519708C2212yes—
15#16839758C418yescomplemented
16C2×#14519701C2212yes—
17C2×#14519701C2211yescharacteristic
18C2×#14519701C2211yescharacteristic
19C2×#14519698C2212yes—
20C2×#14519698C2211yescharacteristic
21C2×#14519698C2211yescharacteristic
22C2×C22⋊#92418C2211yescharacteristic
23C2×#14519688C2212yes—
24#16839660C2212yes—
25#16839656—22no—
26#16839629—21no—
27#16839622C2211yescharacteristic
28C4×C4⋊Dic61C2212yes—
29#16839613—21no—
30#16839524C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34C22×#41×C61C2211yescharacteristic
35C2×C4×D4×C61C2212yes—
36C61×#391C2212yes—
37C61×#386—22no—
38C61×#381—21no—
39C61×#380C2211yescharacteristic
40C4×C22⋊C4×C61C2211yescharacteristic
41C2×#95×C61—21no—
42C22×#92418—42no—
43C22×#92418C2×C414yescomplemented
44C22×#92418C2312yes—
45C22×#92418C2311yescharacteristic
46C22×#92418C2311yescharacteristic
47#14519736—22no—
48#14519736C2×C414yescomplemented
49C2×C4⋊Dic61C2312yes—
50C2×C4⋊Dic61C2311yescharacteristic
51C2×C4⋊Dic61C2311yescharacteristic
52C2×#11758196—22no—
53C2×C22⋊Dic61—21no—
54C2×C22⋊Dic61—21no—
55C2×C4×Dic61—21no—
56C2×C4×Dic61—22no—
57#14519708—42no—
58#14519701—42no—
59#14519701C2×C414yescomplemented
60#14519701C2×C414yescomplemented
61#14519698—24no—
62#14519698C2×C414yes—
63#14519698C2×C414yes—
64C22⋊#92418—22no—
65#14519688—42no—
66#14519684—21no—
67#14519684—22no—
68#14519684—22no—
69C2×#41×C61—22no—
70C2×#41×C61—24no—
71C2×#41×C61—42no—
72C2×#41×C61C2×C414yes—
73C22×Q8×C61C2311yescharacteristic
74C22×D4×C61—21no—
75C22×D4×C61C2311yescharacteristic
76C23×C4×C61—21no—
77C23×C4×C61—22no—
78C23×C4×C61C2311yescharacteristic
79C4×D4×C61—24no—
80C4×D4×C61—42no—
81C2×C4⋊C4×C61C2312yes—
82C2×C22⋊C4×C61—22no—
83C2×C22⋊C4×C61—21no—
84C2×C22⋊C4×C61D412yes—
85C2×C42×C61C2312yes—
86#95×C61—21no—
87#95×C61—22no—
88#95×C61—22no—
89#95×C61D411yescharacteristic
90#95×C61D411yescharacteristic
91C24×C61—21no—
92C24×C61C2×D411yescharacteristic
93#41×C61—28no—
94#41×C61—44no—
95#41×C61—48no—
96C2×Q8×C61—21no—
97C2×Q8×C61—22no—
98C2×Q8×C61—22no—
99C2×Q8×C61C22×C412yes—
100C2×D4×C61—24no—
101C2×D4×C61—21no—
102C2×D4×C61—22no—
103C2×D4×C61—44no—
104C2×D4×C61C22×C412yes—
105C2×D4×C61#4114yes—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—21no—
109C22×C4×C61—24no—
110C22×C4×C61—22no—
111C22×C4×C61—44no—
112C22×C4×C61—44no—
113C22×C4×C61—44no—
114C22×C4×C61—82no—
115C22×C4×C61C22×C412yes—
116C22×C4×C61C2×D412yes—
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61#4112yes—
120C22×C4×C61C2411yescharacteristic
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—22no—
126C22⋊C4×C61—42no—
127C22⋊C4×C61—44no—
128C22⋊C4×C61—81no—
129C42×C61—22no—
130C42×C61—22no—
131C22⋊Dic61—24no—
132C22⋊Dic61—81no—
133C4⋊Dic61—21no—
134C4⋊Dic61—21no—
135C4⋊Dic61—22no—
136C4⋊Dic61—22no—
137C4⋊Dic61C22×C412yes—
138C4⋊Dic61C22×C412yes—
139#11758196—44no—
140C4×Dic61—24no—
141C4×Dic61—82no—
142C2×#92418—22no—
143C2×#92418—22no—
144C2×#92418—22no—
145C2×#92418—22no—
146C2×#92418—22no—
147C2×#92418—44no—
148C2×#92418C22×C412yes—
149C2×#92418C22×C412yes—
150C23×C61—22no—
151C23×C61—24no—
152C23×C61—22no—
153C23×C61—22no—
154C23×C61—21no—
155C23×C61—21no—
156C23×C61—42no—
157C23×C61—41no—
158C23×C61—41no—
159C23×C61C4×D412yes—
160C23×C61C22×D411yescharacteristic
161Q8×C61—22no—
162Q8×C61—22no—
163Q8×C61—42no—
164D4×C61—22no—
165D4×C61—24no—
166D4×C61—48no—
167D4×C61#11714yes—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—24no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—48no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—88no—
185C2×C4×C61C4×D412yes—
186C2×C4×C61C23×C411yescharacteristic
187C2×C4×C61C2×#4112yes—
188C2×C4×C61C2×#4111yescharacteristic
189C2×C4×C61H(5,𝔽2)12yes—
190C2×Dic61—22no—
191C2×Dic61—22no—
192C2×Dic61—42no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198#92418—22no—
199#92418—22no—
200#92418—42no—
201#22568—611nomaximal, Sylow
202C22×C61—22no—
203C22×C61—24no—
204C22×C61—22no—
205C22×C61—22no—
206C22×C61—42no—
207C22×C61—42no—
208C22×C61—42no—
209C22×C61—44no—
210C22×C61—44no—
211C22×C61—42no—
212C22×C61C2×C4×D411yescharacteristic
213C22×C61#54612yes—
214C22×C61#57512yes—
215C22×C61#57711yescharacteristic
216C22×C61#57711yescharacteristic
217C4×C61—22no—
218C4×C61—42no—
219C4×C61—42no—
220C4×C61—42no—
221C4×C61—82no—
222C4×C61C2×#11711yescharacteristic
223C4×C61#51811yescharacteristic
224C4×C61#55112yes—
225Dic61—42no—
226Dic61—42no—
227Dic61—82no—
228C2×#416—611no—
229#2347—611no—
230#2339D6111yescharacteristic
231#1935—612no—
232#1927—612no—
233#1864—612no—
234#1825—611no—
235#1807—611no—
236#1788—611no—
237#1762—612no—
238#1752—611no—
239C2×C61—42no—
240C2×C61—44no—
241C2×C61#234211yescharacteristic
242C2×C61#294311yescharacteristic
243C2×C61#295211yescharacteristic
244C2×C61#330712yes—
245C2×C61#331112yes—
246C22×#41C2×D6111yescharacteristic
247C23×C8—611no—
248C2×C4×D4C2×D6112yes—
249C2×#117—611no—
250C2×#106—612no—
251#416—618no—
252C2×#103—611no—
253C2×#103—611no—
254C2×#103—612no—
255C2×#102—611no—
256C2×#102—611no—
257C2×#102—612no—
258C2×C22⋊C8—611no—
259C2×#97—612no—
260#391C2×D6112yes—
261#389—612no—
262#388—1222no—
263#386—22no—
264#383—1221no—
265#381—21no—
266#380—611no—
267#380C2×D6111yescharacteristic
268C4×C4⋊C4—612no—
269C4×C4⋊C4—1221no—
270C4×C22⋊C4C2×D6111yescharacteristic
271C2×#95—21no—
272#340—612no—
273#340—612no—
274#340—612no—
275#336—612no—
276C61#2256811yescharacteristic, complemented, Sylow
277C2×#41—22no—
278C2×#41—24no—
279C2×#41—42no—
280C2×#41C4×D6114yes—
281C22×Q8C22×D6111yescharacteristic
282C22×D4—21no—
283C22×D4C22×D6111yescharacteristic
284C23×C4—21no—
285C23×C4—22no—
286C23×C4C22×D6111yescharacteristic
287C22×C8—611no—
288C22×C8—611no—
289C22×C8—614no—
290C22×C8—612no—
291C22×C8—2442no—
292C4×D4—24no—
293C4×D4—42no—
294#117—614no—
295#117—1222no—
296C2×C4⋊C4—611no—
297C2×C4⋊C4—611no—
298C2×C4⋊C4—612no—
299C2×C4⋊C4—1222no—
300C2×C4⋊C4C22×D6112yes—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—1221no—
304C2×C22⋊C4—1221no—
305C2×C22⋊C4C22⋊D6112yes—
306C2×C42—1221no—
307C2×C42—1222no—
308C2×C42C22×D6112yes—
309#106—2442no—
310#103—614no—
311#103—614no—
312#103—2442no—
313#102—614no—
314#102—614no—
315#102—1224no—
316C22⋊C8—1222no—
317#97—2442no—
318#95—21no—
319#95—22no—
320#95—22no—
321#95—1222no—
322#95—1221no—
323#95—1222no—
324#95C22⋊D6111yescharacteristic
325#95C22⋊D6111yescharacteristic
326C24—21no—
327C24C2×C22⋊D6111yescharacteristic
328#41—28no—
329#41—44no—
330#41—48no—
331C2×Q8—21no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8C2×C4×D6112yes—
335C2×D4—24no—
336C2×D4—21no—
337C2×D4—22no—
338C2×D4—44no—
339C2×D4C2×C4×D6112yes—
340C2×D4#1175822414yes—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—82no—
350C22×C4—1221no—
351C22×C4—1221no—
352C22×C4—1222no—
353C22×C4C2×C4×D6112yes—
354C22×C4#1175822212yes—
355C22×C4C2×C22⋊D6112yes—
356C22×C4C2×C22⋊D6112yes—
357C22×C4C2×C22⋊D6111yescharacteristic
358C22×C4C23×D6111yescharacteristic
359C2×C8—612no—
360C2×C8—612no—
361C2×C8—1222no—
362C2×C8—1222no—
363C2×C8—1222no—
364C2×C8—1222no—
365C2×C8—1222no—
366C2×C8—2444no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—612no—
370C4⋊C4—612no—
371C4⋊C4—1221no—
372C4⋊C4—1221no—
373C4⋊C4—1222no—
374C4⋊C4—1222no—
375C4⋊C4—2444no—
376C22⋊C4—42no—
377C22⋊C4—44no—
378C22⋊C4—81no—
379C22⋊C4—1224no—
380C22⋊C4—4881no—
381C42—22no—
382C42—22no—
383C42—1224no—
384C42—4882no—
385C23—22no—
386C23—22no—
387C23—21no—
388C23—21no—
389C23—24no—
390C23—22no—
391C23—42no—
392C23—41no—
393C23—41no—
394C23C4×C22⋊D6112yes—
395C23C22×C22⋊D6111yescharacteristic
396Q8—22no—
397Q8—22no—
398Q8—42no—
399D4—22no—
400D4—24no—
401D4—48no—
402D4#1451973314yes—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—48no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—88no—
420C2×C4—1222no—
421C2×C4—1222no—
422C2×C4—2442no—
423C2×C4—2442no—
424C2×C4—2442no—
425C2×C4—2442no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4C4×C22⋊D6112yes—
429C2×C4C22×C4×D6111yescharacteristic
430C2×C4C2×#1175822412yes—
431C2×C4C2×#1175822211yescharacteristic
432C2×C4#1451982212yes—
433C8—1222no—
434C8—1222no—
435C8—2442no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—24no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—44no—
444C22—44no—
445C22—42no—
446C22C2×C4×C22⋊D6111yescharacteristic
447C22#1684040612yes—
448C22#1684056012yes—
449C22#1684057011yescharacteristic
450C22#1684057311yescharacteristic
451C4—22no—
452C4—42no—
453C4—42no—
454C4—42no—
455C4—82no—
456C4—4882no—
457C4—2442no—
458C4—2442no—
459C4C2×#1451973311yescharacteristic
460C4#1684023511yescharacteristic
461C4#1684043812yes—
462C2—42no—
463C2—44no—
464C2#2144866411yescharacteristic
465C2#2145461711yescharacteristic
466C2#2145470011yescharacteristic
467C2#2145779512yes—
468C2#2145785512yes—
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2339C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenter#2339D61
hypocenterC61#22568
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2339

Complements

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

NH ≅ G/Ntype
2. #21448710C2semidirect
3. #21446512C2semidirect
4. #21446450C2semidirect
5. #21446080C2semidirect
6. #21445911C2semidirect
7. #21445842C2semidirect
9. #21445644C2semidirect
10. #21445597C2semidirect
15. #16839758C4semidirect
43. C22×#92418C2×C4semidirect
48. #14519736C2×C4semidirect
59. #14519701C2×C4semidirect
60. #14519701C2×C4semidirect
276. C61#22568semidirect

Maximal subgroups

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

subgroupindex
C2×#168397582
#214487102
#214465122
#214464502
#214460802
#214459112
#214458422
#214457592
#214456442
#214455972
C61×#23392
#2256861