Finite Group Databasebeta

← #29656104

Subgroups — #29656104

Order 15616.

1176 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456272C211yesmaximal, characteristic, complemented
2#21455299C212yesmaximal, complemented
3#21444891C214yesmaximal, complemented
4#21444878C212yesmaximal, complemented
5#21444873C212yesmaximal, complemented
6#21444554C211yesmaximal, characteristic, complemented
7#21444258C212yesmaximal, complemented
8C61×#3481C211yesmaximal, characteristic, complemented
9C2×#14519802—22no—
10C2×#14519802—22no—
11C2×#14519774C2211yescharacteristic, complemented
12#16840152—22no—
13#16840138C2212yes—
14#16840136C2212yescomplemented
15#16839962—22no—
16#16839962C2214yescomplemented
17#16839957—22no—
18#16839957C2214yes—
19#16839786C2212yescomplemented
20#16839766—22no—
21C4×#11758194C2211yescharacteristic, complemented
22#16839511C2212yescomplemented
23#16839506C2212yescomplemented
24Q8⋊#92418C2214yescomplemented
25D4⋊#92418C2214yescomplemented
26C61×#559C2211yescharacteristic, complemented
27C61×#554C2212yescomplemented
28C61×#553—22no—
29C61×#551—22no—
30C61×#546—22no—
31D42×C61C2211yescharacteristic, complemented
32C61×#532C2211yescharacteristic
33C61×#529—22no—
34C61×#525—22no—
35C4×#41×C61C2212yes—
36C2×C4×D4×C61—22no—
37C22×D244—22no—
38C22×D244—21no—
39#14519802—48no—
40#14519802—48no—
41C2×#11758201—24no—
42C2×#11758201—24no—
43C2×#11758199—24no—
44C2×#11758199—24no—
45C2×#11758194—21no—
46C2×#11758194—22no—
47#14519774—22no—
48#14519774—21no—
49#14519774C2312yes—
50C4⋊#92418—22no—
51C4⋊#92418C2314yes—
52#14519697—24no—
53#14519697—24no—
54#14519688—21no—
55C4×#92418C2312yes—
56C2×#41×C61D412yes—
57C2×#41×C61D412yes—
58C22×D4×C61—22no—
59C23×C4×C61—22no—
60C4⋊Q8×C61C2311yescharacteristic
61#127×C61C2311yescharacteristic
62#126×C61—24no—
63#125×C61—22no—
64#124×C61—22no—
65#124×C61D412yes—
66#123×C61—24no—
67#123×C61—22no—
68#123×C61—22no—
69C22⋊Q8×C61—22no—
70C22⋊Q8×C61—24no—
71#121×C61—24no—
72#121×C61—22no—
73#121×C61—22no—
74#120×C61—22no—
75#120×C61—22no—
76C4×Q8×C61D412yes—
77C4×Q8×C61C2312yes—
78C4×D4×C61—24no—
79C4×D4×C61—24no—
80C4×D4×C61—22no—
81C4×D4×C61—22no—
82C4×D4×C61—22no—
83C4×D4×C61—22no—
84C4×D4×C61—22no—
85C4×D4×C61D412yes—
86C4×D4×C61C2312yes—
87#117×C61—22no—
88#117×C61—22no—
89#117×C61D412yes—
90C2×C4⋊C4×C61—22no—
91C2×C22⋊C4×C61—24no—
92C2×C42×C61—22no—
93C2×C42×C61C2311yescharacteristic
94C24×C61—22no—
95C23×D61—41no—
96#41×C61—24no—
97#41×C61—42no—
98#41×C61—44no—
99C2×Q8×C61C2×D412yes—
100C2×Q8×C61C2×D412yes—
101C2×D4×C61—24no—
102C2×D4×C61—22no—
103C2×D4×C61—22no—
104C2×D4×C61—22no—
105C2×D4×C61—44no—
106C2×D4×C61—44no—
107C2×D4×C61C2×D412yes—
108C2×D4×C61C2×D412yes—
109C22×C4×C61—24no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—84no—
116C22×C4×C61C2×D412yes—
117C22×C4×C61C2×D411yescharacteristic
118C2×D244—24no—
119C2×D244—22no—
120C2×D244—44no—
121C2×D244—44no—
122C2×D244—44no—
123C2×D244—42no—
124C2×D244—41no—
125C4⋊C4×C61—24no—
126C4⋊C4×C61—24no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61C2×D412yes—
131C4⋊C4×C61C2×D412yes—
132C22⋊C4×C61—24no—
133C22⋊C4×C61—24no—
134C22⋊C4×C61—22no—
135C22⋊C4×C61—22no—
136C22⋊C4×C61—22no—
137C22⋊C4×C61—22no—
138C22⋊C4×C61—24no—
139C22⋊C4×C61—22no—
140C42×C61—22no—
141C42×C61—22no—
142C42×C61—22no—
143C42×C61C2×D412yes—
144C42×C61C2×D411yescharacteristic
145C42×C61C2411yescharacteristic
146#11758201—48no—
147#11758201—48no—
148#11758199—48no—
149#11758199—48no—
150#11758194—24no—
151#11758194—44no—
152C2×#92418—22no—
153C2×#92418—24no—
154C23×C61—22no—
155C23×C61—22no—
156C23×C61—48no—
157C23×C61—82no—
158C23×C61#12011yescharacteristic
159C22×D61—42no—
160C22×D61—84no—
161C22×D61—82no—
162Q8×C61—22no—
163Q8×C61—22no—
164Q8×C61—42no—
165D4×C61—22no—
166D4×C61—42no—
167D4×C61—44no—
168D4×C61—48no—
169D4×C61—42no—
170D4×C61—42no—
171D4×C61—82no—
172D4×C61AGL(1,ℤ/8)14yescomplemented
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—22no—
180C2×C4×C61—44no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—44no—
184C2×C4×C61—44no—
185C2×C4×C61—82no—
186C2×C4×C61—84no—
187C2×C4×C61#12012yes—
188C2×C4×C61#12011yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C22×D412yes—
191D244—44no—
192D244—44no—
193D244—44no—
194D244—84no—
195D244—82no—
196#92418—24no—
197#92418—44no—
198#18527—611nomaximal, Sylow
199C22×C61—22no—
200C22×C61—24no—
201C22×C61—21no—
202C22×C61—42no—
203C22×C61—42no—
204C22×C61—44no—
205C22×C61—42no—
206C22×C61—88no—
207C22×C61C2×#12011yescharacteristic
208C2×D61—84no—
209C2×D61—82no—
210C2×D61—164no—
211C4×C61—22no—
212C4×C61—21no—
213C4×C61—42no—
214C4×C61—42no—
215C4×C61—42no—
216C4×C61—42no—
217C4×C61#45312yes—
218C4×C61C2×AGL(1,ℤ/8)12yes—
219C4×C61#57512yes—
220#3481D6111yescharacteristic, complemented
221#3151—611no—
222#3038—612no—
223#1622—614no—
224#1620—612no—
225#1619—612no—
226#1560—611no—
227#1489—612no—
228C2×C61—41no—
229C2×C61—44no—
230C2×C61—82no—
231C2×C61#300612yes—
232C2×C61C2×#45311yescharacteristic
233D61—162no—
234C2×AGL(1,ℤ/8)—1222no—
235C2×AGL(1,ℤ/8)—1222no—
236#559C2×D6111yescharacteristic, complemented
237#554C2×D6112yescomplemented
238#553—22no—
239#551—22no—
240#546—22no—
241D42C2×D6111yescharacteristic, complemented
242#532C2×D6111yescharacteristic
243C2×#127—611no—
244#529—22no—
245#525—22no—
246C4×#41C2×D6112yes—
247C2×C4×D4—22no—
248#496—1222no—
249#493—612no—
250#492—612no—
251#460—614no—
252#460—1222no—
253#459—614no—
254#459—1222no—
255#423—612no—
256#418—1222no—
257C4×M16—611no—
258#333—612no—
259#331—612no—
260Q8⋊C8—614no—
261D4⋊C8—614no—
262C61#1852711yescharacteristic, complemented, Sylow
263C2×#41C22⋊D6112yes—
264C2×#41C22⋊D6112yes—
265C22×D4—22no—
266C22×D4—1221no—
267C22×D4—1222no—
268C23×C4—22no—
269AGL(1,ℤ/8)—2448no—
270AGL(1,ℤ/8)—2448no—
271C2×SD16—1224no—
272C2×SD16—1224no—
273C2×D8—1224no—
274C2×D8—1224no—
275C2×M16—1221no—
276C2×M16—1222no—
277C4⋊Q8C22×D6111yescharacteristic
278#127—612no—
279#127—1222no—
280#127—1221no—
281#127C22×D6111yescharacteristic
282#126—24no—
283#125—22no—
284#124—22no—
285#124C22⋊D6112yes—
286#123—22no—
287#123—24no—
288#123—22no—
289C22⋊Q8—24no—
290C22⋊Q8—22no—
291#121—24no—
292#121—22no—
293#121—22no—
294#120—22no—
295#120—22no—
296C4×Q8C22⋊D6112yes—
297C4×Q8C22×D6112yes—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4C22⋊D6112yes—
306C4×D4C22×D6112yes—
307#117—22no—
308#117—22no—
309#117C22⋊D6112yes—
310C2×C4⋊C4—22no—
311C2×C22⋊C4—24no—
312C2×C42—22no—
313C2×C42C22×D6111yescharacteristic
314C4⋊C8—614no—
315C4⋊C8—1222no—
316#102—1224no—
317#102—1224no—
318#97—1221no—
319C4×C8—612no—
320C24—22no—
321C24—2441no—
322#41—24no—
323#41—42no—
324#41—44no—
325C2×Q8C2×C22⋊D6112yes—
326C2×Q8C2×C22⋊D6112yes—
327C2×D4—24no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—1224no—
334C2×D4—1222no—
335C2×D4—2444no—
336C2×D4—2444no—
337C2×D4—2444no—
338C2×D4—2442no—
339C2×D4—2441no—
340C2×D4C2×C22⋊D6112yes—
341C2×D4C2×C22⋊D6112yes—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—84no—
349C22×C4C2×C22⋊D6112yes—
350C22×C4C2×C22⋊D6111yescharacteristic
351SD16—2448no—
352SD16—2448no—
353D8—2448no—
354D8—2448no—
355M16—1224no—
356M16—2444no—
357C2×C8—1222no—
358C2×C8—1224no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4C2×C22⋊D6112yes—
365C4⋊C4C2×C22⋊D6112yes—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—24no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42C2×C22⋊D6111yescharacteristic
378C42C2×C22⋊D6112yes—
379C42C23×D6111yescharacteristic
380C23—22no—
381C23—22no—
382C23—48no—
383C23—82no—
384C23—4882no—
385C23—4884no—
386C23—2442no—
387C23#1451974611yescharacteristic
388Q8—22no—
389Q8—22no—
390Q8—42no—
391D4—22no—
392D4—42no—
393D4—44no—
394D4—48no—
395D4—42no—
396D4—42no—
397D4—82no—
398D4—4884no—
399D4—4882no—
400D4—2444no—
401D4—2444no—
402D4—2444no—
403D4#1451980214yescomplemented
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—82no—
417C2×C4—84no—
418C2×C4#1451974612yes—
419C2×C4#1451974611yescharacteristic
420C2×C4C22×C22⋊D6111yescharacteristic
421C2×C4C22×C22⋊D6112yes—
422C8—1224no—
423C8—2444no—
424C22—21no—
425C22—22no—
426C22—24no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—44no—
431C22—88no—
432C22—9764no—
433C22—4882no—
434C22—4884no—
435C22C2×#1451974611yescharacteristic
436C4—21no—
437C4—22no—
438C4—42no—
439C4—42no—
440C4—42no—
441C4—42no—
442C4#1683993012yes—
443C4C2×#1451980212yes—
444C4#1684056012yes—
445C2—41no—
446C2—44no—
447C2—82no—
448C2—9762no—
449C2#2145507112yes—
450C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3481C2
socleC22×C61C2×#120
hypercenter#3481D61
hypocenterC61#18527
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3481

Complements

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

NH ≅ G/Ntype
1. #21456272C2semidirect
2. #21455299C2semidirect
3. #21444891C2semidirect
4. #21444878C2semidirect
5. #21444873C2semidirect
6. #21444554C2semidirect
7. #21444258C2semidirect
8. C61×#3481C2semidirect
11. C2×#14519774C22semidirect
14. #16840136C22semidirect
16. #16839962C22semidirect
19. #16839786C22semidirect
21. C4×#11758194C22semidirect
22. #16839511C22semidirect
23. #16839506C22semidirect
24. Q8⋊#92418C22semidirect
25. D4⋊#92418C22semidirect
26. C61×#559C22semidirect
27. C61×#554C22semidirect
31. D42×C61C22semidirect
172. D4×C61AGL(1,ℤ/8)semidirect
220. #3481D61semidirect
236. #559C2×D61semidirect
237. #554C2×D61semidirect
241. D42C2×D61semidirect
262. C61#18527semidirect
403. D4#14519802semidirect

Maximal subgroups

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

subgroupindex
#214562722
#214552992
#214448912
#214448782
#214448732
#214445542
#214442582
C61×#34812
#1852761