Finite Group Databasebeta

← #29662759

Subgroups — #29662759

Order 15616.

642 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456398C211yesmaximal, characteristic, complemented
2#21455879C212yesmaximal, complemented
3#21445359C212yesmaximal, complemented
4#21445352C212yesmaximal, complemented
5#21445347C211yesmaximal, characteristic, complemented
6#21445342C211yesmaximal, characteristic, complemented
7#21445303C211yesmaximal, characteristic, complemented
8#21445298C211yesmaximal, characteristic, complemented
9#21444567C211yesmaximal, characteristic, complemented
10#21444479C212yesmaximal, complemented
11C61×#2965C211yesmaximal, characteristic, complemented
12#16840312C2212yes—
13#16840167C2211yescharacteristic
14#16840139C2211yescharacteristic
15#16840109C2212yes—
16#16840086—22no—
17#16840057C2212yes—
18#16840052C2212yes—
19#16839998—21no—
20#16839998—21no—
21#16839962C2212yes—
22#16839957C2212yes—
23#16839907—22no—
24C4×#11758201—22no—
25C4×D488—21no—
26C4×D488—21no—
27C4×#11758199—22no—
28#16839815C2212yes—
29#16839515C2212yes—
30#16839513C2211yescharacteristic
31#16839513C2211yescharacteristic
32#16839510C2212yes—
33#16839510C2212yes—
34#16839505C2212yes—
35#16839505C2212yes—
36C4×#41×C61C2211yescharacteristic
37C61×#446C2211yescharacteristic
38C61×#446C2211yescharacteristic
39C61×#436C2211yescharacteristic
40C61×#436C2211yescharacteristic
41C61×#435—21no—
42C61×#435—21no—
43C61×#431—21no—
44C61×#431C2211yescharacteristic
45C4×M16×C61—21no—
46C2×C4×C8×C61C2211yescharacteristic
47C2×#11758222—22no—
48C2×#11758201—22no—
49C2×D488—21no—
50C2×D488—21no—
51C2×#11758199—22no—
52#14519777C2312yes—
53#14519774C2312yes—
54#14519765—22no—
55#14519751—22no—
56#14519751—22no—
57C4×D244—22no—
58C4×D244—22no—
59#14519711—21no—
60#14519711—21no—
61#14519710—22no—
62#14519708—22no—
63C4⋊#92418C2312yes—
64C4⋊#92418C2312yes—
65#14519701—22no—
66#14519699—22no—
67#14519699—22no—
68#14519698—22no—
69#14519697—22no—
70#14519697—22no—
71C22⋊#92418—22no—
72C4×#92418—22no—
73C2×#41×C61D411yescharacteristic
74C2×M16×C61—21no—
75C2×M16×C61—21no—
76C22×C8×C61D411yescharacteristic
77C22×C8×C61D411yescharacteristic
78C4×Q8×C61D411yescharacteristic
79C4×Q8×C61C2311yescharacteristic
80C4×D4×C61—21no—
81C4×D4×C61—21no—
82C4×D4×C61D411yescharacteristic
83C4×D4×C61C2311yescharacteristic
84#117×C61—21no—
85#117×C61D411yescharacteristic
86C2×C42×C61—21no—
87C2×C42×C61C2311yescharacteristic
88C4⋊C8×C61—21no—
89C4⋊C8×C61—21no—
90C4⋊C8×C61C2311yescharacteristic
91C4⋊C8×C61C2311yescharacteristic
92C22⋊C8×C61—21no—
93C22⋊C8×C61—21no—
94C22⋊C8×C61—21no—
95#97×C61—21no—
96#97×C61—21no—
97C4×C8×C61—21no—
98C4×C8×C61D411yescharacteristic
99C4×C8×C61D411yescharacteristic
100C4×C8×C61C2311yescharacteristic
101C4×C8×C61C2311yescharacteristic
102#41×C61—42no—
103C2×Q8×C61C2×D411yescharacteristic
104C2×D4×C61—21no—
105C2×D4×C61C2×D411yescharacteristic
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C2×C22⋊D61—42no—
113#11758222—84no—
114C2×D244—22no—
115C2×D244—22no—
116C2×D244—22no—
117C2×C4×D61—42no—
118C2×Dic122—22no—
119M16×C61—42no—
120M16×C61—42no—
121C2×C8×C61—22no—
122C2×C8×C61—21no—
123C2×C8×C61—21no—
124C2×C8×C61—41no—
125C2×C8×C61C2×D411yescharacteristic
126C2×C8×C61C2×D411yescharacteristic
127C2×C8×C61C2×D411yescharacteristic
128C2×C8×C61C2×D411yescharacteristic
129C4⋊C4×C61—21no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61C2×D411yescharacteristic
132C4⋊C4×C61C2×D411yescharacteristic
133C22⋊C4×C61—21no—
134C22⋊C4×C61—21no—
135C22⋊C4×C61—21no—
136C42×C61—21no—
137C42×C61—21no—
138C42×C61—21no—
139C42×C61C2×D411yescharacteristic
140C42×C61C2×D411yescharacteristic
141C42×C61#4111yescharacteristic
142C42×C61C2411yescharacteristic
143#11758201—44no—
144#11758199—44no—
145#11758198—42no—
146#11758198—42no—
147C4⋊Dic61—22no—
148C4⋊Dic61—22no—
149C2×#92418—22no—
150C2×#92418—22no—
151D488—42no—
152D488—42no—
153C23×C61—21no—
154C23×C61#12111yescharacteristic
155C22×D61—42no—
156Q8×C61—22no—
157D4×C61—22no—
158D4×C61—42no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—22no—
162C2×C4×C61—22no—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—42no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—42no—
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C22×D411yescharacteristic
175C2×C4×C61C2×#4111yescharacteristic
176C22⋊D61—84no—
177C2×Dic61—42no—
178D244—42no—
179D244—42no—
180D244—42no—
181D244—42no—
182D244—82no—
183C4×D61—84no—
184Dic122—82no—
185C8×C61—21no—
186C8×C61—21no—
187C8×C61—21no—
188C8×C61—41no—
189C8×C61—41no—
190C8×C61#13512yes—
191#92418—42no—
192#92418—42no—
193#19160—611nomaximal, Sylow
194C22×C61—21no—
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—41no—
198C22×C61—41no—
199C22×C61—41no—
200C22×C61C2×#12111yescharacteristic
201C2×D61—82no—
202C2×D61—82no—
203C2×D61—82no—
204C4×C61—21no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—41no—
208C4×C61—41no—
209C4×C61—41no—
210C4×C61#47212yes—
211C4×C61C2×#13511yescharacteristic
212C4×C61#57511yescharacteristic
213C4×C61#57512yes—
214Dic61—82no—
215#3169—611no—
216#3106—612no—
217#2965D6111yescharacteristic, complemented
218#1698—612no—
219#1697—612no—
220#1696—611no—
221#1695—611no—
222#1688—611no—
223#1687—611no—
224#1564—611no—
225#1543—612no—
226C2×C61—41no—
227C2×C61—81no—
228C2×C61#303511yescharacteristic
229C2×C61#305411yescharacteristic
230C2×C61C2×#47211yescharacteristic
231D61—162no—
232#532—612no—
233C4×#41C2×D6111yescharacteristic
234#500—611no—
235#493—611no—
236#486—612no—
237#482—1222no—
238#477—612no—
239#476—612no—
240#466—1221no—
241#466—1221no—
242#460—612no—
243#459—612no—
244#449—1222no—
245#446C2×D6111yescharacteristic
246#446C2×D6111yescharacteristic
247C4×SD16—1222no—
248C4×D8—1222no—
249C4×D8—1221no—
250C4×D8—1221no—
251#436C2×D6111yescharacteristic
252#436C2×D6111yescharacteristic
253#435—21no—
254#435—21no—
255#431—21no—
256#431—612no—
257#431C2×D6111yescharacteristic
258C4×M16—21no—
259C2×C4×C8C2×D6111yescharacteristic
260#335—612no—
261#334—611no—
262#334—611no—
263#333—612no—
264#333—612no—
265#331—612no—
266#331—612no—
267C61#1916011yescharacteristic, complemented, Sylow
268C2×#41—1222no—
269C2×#41D24411yescharacteristic
270C2×SD16—1222no—
271C2×D8—1222no—
272C2×D8—1221no—
273C2×D8—1221no—
274C2×M16—21no—
275C2×M16—21no—
276C22×C8C22⋊D6111yescharacteristic
277C22×C8C22⋊D6111yescharacteristic
278C4⋊Q8—612no—
279#127—612no—
280#124—1222no—
281#121—1222no—
282#121—1222no—
283C4×Q8D24411yescharacteristic
284C4×Q8C22×D6111yescharacteristic
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—1222no—
288C4×D4—1222no—
289C4×D4D24411yescharacteristic
290C4×D4C22×D6111yescharacteristic
291#117—21no—
292#117D24411yescharacteristic
293C2×C42—21no—
294C2×C42C22×D6111yescharacteristic
295#107—1221no—
296#107—1221no—
297#107—1222no—
298#106—1222no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8—612no—
302C4⋊C8—612no—
303C4⋊C8C22×D6111yescharacteristic
304C4⋊C8C22×D6111yescharacteristic
305#103—1222no—
306#102—1222no—
307#102—1222no—
308#102—1222no—
309#102—1222no—
310#102—1222no—
311C22⋊C8—21no—
312C22⋊C8—21no—
313C22⋊C8—21no—
314C22⋊C8—1222no—
315#97—21no—
316#97—21no—
317C4×C8—21no—
318C4×C8—1222no—
319C4×C8C22⋊D6111yescharacteristic
320C4×C8C22⋊D6111yescharacteristic
321C4×C8C22×D6111yescharacteristic
322C4×C8C22×D6111yescharacteristic
323#41—42no—
324#41—4884no—
325C2×Q8—1222no—
326C2×Q8C2×D24411yescharacteristic
327C2×D4—21no—
328C2×D4—1222no—
329C2×D4—1222no—
330C2×D4—1222no—
331C2×D4—2442no—
332C2×D4C2×D24411yescharacteristic
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—2442no—
337C22×C4C2×D24411yescharacteristic
338C22×C4#1175822211yescharacteristic
339C22×C4C2×C22⋊D6111yescharacteristic
340SD16—2444no—
341D8—2442no—
342D8—2444no—
343D8—2442no—
344M16—42no—
345M16—42no—
346C2×C8—22no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—41no—
350C2×C8—1222no—
351C2×C8—1222no—
352C2×C8C2×C22⋊D6111yescharacteristic
353C2×C8C2×C22⋊D6111yescharacteristic
354C2×C8C2×C22⋊D6111yescharacteristic
355C2×C8C2×C22⋊D6111yescharacteristic
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—1222no—
359C4⋊C4—1222no—
360C4⋊C4C2×D24411yescharacteristic
361C4⋊C4C2×D24411yescharacteristic
362C22⋊C4—21no—
363C22⋊C4—21no—
364C22⋊C4—21no—
365C22⋊C4—2442no—
366C22⋊C4—2442no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42C2×D24411yescharacteristic
371C42#1175822211yescharacteristic
372C42C2×C22⋊D6111yescharacteristic
373C42C23×D6111yescharacteristic
374C23—21no—
375C23—2442no—
376C23#1451975111yescharacteristic
377Q8—22no—
378Q8—4882no—
379D4—22no—
380D4—42no—
381D4—4882no—
382D4—4884no—
383D4—2442no—
384D4—2442no—
385D4—2442no—
386D4—2442no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—4884no—
399C2×C4—2442no—
400C2×C4#1451975111yescharacteristic
401C2×C4#1451975111yescharacteristic
402C2×C4#1451975111yescharacteristic
403C2×C4C22×C22⋊D6111yescharacteristic
404C2×C4C22×D24411yescharacteristic
405C2×C4C2×#1175822211yescharacteristic
406C8—21no—
407C8—21no—
408C8—21no—
409C8—41no—
410C8—41no—
411C8—2442no—
412C8—2442no—
413C8#1451979712yes—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—4882no—
421C22—4882no—
422C22—4882no—
423C22C2×#1451975111yescharacteristic
424C4—21no—
425C4—21no—
426C4—21no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—4882no—
431C4#1684003212yes—
432C4C2×#1451979711yescharacteristic
433C4#1684056011yescharacteristic
434C4#1684056212yes—
435C2—41no—
436C2—81no—
437C2—9762no—
438C2#2145527411yescharacteristic
439C2#2145542811yescharacteristic
440C2C2×#1684003211yescharacteristic
subgroupG/H
centerC2×C4#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2965C2
socleC22×C61C2×#121
hypercenter#2965D61
hypocenterC61#19160
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2965

Complements

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

NH ≅ G/Ntype
1. #21456398C2semidirect
2. #21455879C2semidirect
3. #21445359C2semidirect
4. #21445352C2semidirect
5. #21445347C2semidirect
6. #21445342C2semidirect
7. #21445303C2semidirect
8. #21445298C2semidirect
9. #21444567C2semidirect
10. #21444479C2semidirect
11. C61×#2965C2semidirect
217. #2965D61semidirect
267. C61#19160semidirect

Maximal subgroups

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

subgroupindex
#214563982
#214558792
#214453592
#214453522
#214453472
#214453422
#214453032
#214452982
#214445672
#214444792
C61×#29652
#1916061