Finite Group Databasebeta

← #29665176

Subgroups — #29665176

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456757C211yesmaximal, characteristic, complemented
2#21455612C211yesmaximal, characteristic, complemented
3C2×#16839958C211yesmaximal, characteristic
4#21454842C211yesmaximal, characteristic, complemented
5#21445231C214yesmaximal, complemented
6#21445163C214yesmaximal, complemented
7#21444589C211yesmaximal, characteristic, complemented
8#21444486C211yesmaximal, characteristic, complemented
9C61×#1542C211yesmaximal, characteristic, complemented
10C22×D488—21no—
11C2×#14519774C2211yescharacteristic
12C2×C4×D244C2211yescharacteristic
13#16840136C2212yescomplemented
14#16840110C2212yescomplemented
15#16840059—22no—
16#16840049C2212yescomplemented
17#16840039C2212yescomplemented
18#16839962—22no—
19#16839958—22no—
20#16839958C2214yes—
21#16839912—22no—
22C8×D244—22no—
23#16839827C2214yescomplemented
24#16839815C2211yescharacteristic
25#16839805—21no—
26#16839786C2211yescharacteristic
27#16839766—21no—
28C2×#14519699—21no—
29C22⋊C8×D61—22no—
30#16839512C2214yescomplemented
31#16839505C2214yescomplemented
32C61×#533C2211yescharacteristic
33C61×#431C2211yescharacteristic
34C2×C4⋊C8×C61C2211yescharacteristic
35C61×#333C2212yescomplemented
36C61×#332C2212yescomplemented
37C22×D244—21no—
38C22×D244—21no—
39C22×D244—21no—
40C22×D244D411yescharacteristic
41C22×C4×D61—21no—
42C2×#11758201—24no—
43C2×D488—24no—
44C2×D488—44no—
45C2×#11758194—21no—
46C2×C8×D61—24no—
47#14519774—21no—
48#14519774—21no—
49#14519774—21no—
50#14519774C2312yes—
51C4×D244—21no—
52C4×D244—21no—
53C4×D244D412yes—
54C4×D244C2312yes—
55C2×C4⋊Dic61D411yescharacteristic
56C2×#11758198—21no—
57#14519710—22no—
58#14519708—22no—
59C4⋊#92418—21no—
60C4⋊#92418D412yes—
61C4⋊#92418C2312yes—
62#14519699—24no—
63#14519697—22no—
64#14519697—22no—
65#14519697—24no—
66C22⋊#92418D412yes—
67#14519689—24no—
68C4×#92418C2312yes—
69C22×C8×C61—21no—
70C4⋊Q8×C61C2311yescharacteristic
71C4⋊Q8×C61C2311yescharacteristic
72#125×C61—21no—
73C22⋊Q8×C61—21no—
74C22⋊Q8×C61—21no—
75C4×Q8×C61—21no—
76C4×Q8×C61—21no—
77#117×C61—21no—
78C2×C42×C61C2311yescharacteristic
79C4⋊C8×C61—21no—
80C4⋊C8×C61C2312yes—
81C4⋊C8×C61C2312yes—
82C22⋊C8×C61—21no—
83C4×C8×C61—21no—
84C23×D61—21no—
85C23×D61—41no—
86C2×Q8×C61—21no—
87C2×Q8×C61—21no—
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61#4111yescharacteristic
91C22×Dic61—21no—
92C2×D244—21no—
93C2×D244—22no—
94C2×D244—22no—
95C2×D244—22no—
96C2×D244—22no—
97C2×D244—42no—
98C2×D244—42no—
99C2×D244—41no—
100C2×D244—41no—
101C2×D244—41no—
102C2×D244—82no—
103C2×D244—82no—
104C2×D244SD1614yescomplemented
105C2×D244C2×D412yes—
106C2×C4×D61—22no—
107C2×C4×D61—22no—
108C2×C4×D61—82no—
109C2×C8×C61—22no—
110C2×C8×C61—22no—
111C2×C8×C61—42no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61—21no—
115C4⋊C4×C61—21no—
116C4⋊C4×C61—41no—
117C4⋊C4×C61—41no—
118C22⋊C4×C61—41no—
119C42×C61—41no—
120C42×C61C2×D411yescharacteristic
121C42×C61C2×D411yescharacteristic
122C42×C61#4111yescharacteristic
123C42×C61C2411yescharacteristic
124#11758201—84no—
125#11758198—22no—
126#11758198—22no—
127C4⋊Dic61—21no—
128C4⋊Dic61C2×D412yes—
129#11758194—42no—
130C2×#92418—22no—
131C2×#92418C2×D414yes—
132D488—48no—
133C8×D61—84no—
134C23×C61#12111yescharacteristic
135C22×D61—22no—
136C22×D61—24no—
137C22×D61—42no—
138C22×D61—81no—
139C22×D61—81no—
140C22×D61—82no—
141C22×D61—82no—
142C22×D61—82no—
143Q8×C61—42no—
144Q8×C61—81no—
145C2×C4×C61—22no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61#12111yescharacteristic
150C2×C4×C61#12111yescharacteristic
151C2×C4×C61#12111yescharacteristic
152C2×C4×C61C2×SD1612yes—
153C2×C4×C61#13512yes—
154C2×C4×C61C22×D411yescharacteristic
155C2×C4×C61C22×D411yescharacteristic
156C2×C4×C61C2×#4111yescharacteristic
157C2×Dic61—22no—
158C2×Dic61—81no—
159D244—22no—
160D244—24no—
161D244—42no—
162D244—42no—
163D244—44no—
164D244—82no—
165D244—82no—
166D244—82no—
167D244—82no—
168D244C2×SD1614yes—
169C4×D61—44no—
170C4×D61—82no—
171C8×C61—42no—
172C8×C61—42no—
173#92418—24no—
174#92418—42no—
175#19384—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61#46512yes—
179C22×C61C2×#12111yescharacteristic
180C2×D61—24no—
181C2×D61—42no—
182C2×D61—42no—
183C2×D61—84no—
184C2×D61—82no—
185C2×D61—82no—
186C2×D61—82no—
187C2×D61—164no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—81no—
192C4×C61—81no—
193C4×C61#46512yes—
194C4×C61C22×SD1611yescharacteristic
195C4×C61C2×#13511yescharacteristic
196C4×C61#57612yes—
197Dic61—42no—
198#3208—611no—
199#3077—611no—
200C2×#459—611no—
201#2972—611no—
202#1677—614no—
203#1665—614no—
204#1570—611no—
205#1545—611no—
206#1542D6111yescharacteristic, complemented
207C2×C61—22no—
208C2×C61#304111yescharacteristic
209C2×C61C2×#46511yescharacteristic
210C2×C61#306811yescharacteristic
211D61—44no—
212D61—162no—
213C22×D8—1221no—
214#533C2×D6111yescharacteristic
215C2×#127—611no—
216C2×C4×D4—611no—
217#492—612no—
218#486—612no—
219#478—1222no—
220#476—612no—
221#474—612no—
222#460—1222no—
223#459—614no—
224#459—1222no—
225#450—1222no—
226C8×D4—614no—
227C8×D4—1222no—
228#431—611no—
229#431C2×D6111yescharacteristic
230#428—1221no—
231#423—611no—
232C2×C4⋊C8C2×D6111yescharacteristic
233#418—1221no—
234C2×#102—1221no—
235C2×C22⋊C8—1222no—
236#334—614no—
237#333C2×D6112yescomplemented
238#332C2×D6112yescomplemented
239#331—614no—
240C61#1938411yescharacteristic, complemented, Sylow
241C22×D4—611no—
242C22×D4—1221no—
243C22×D4—1221no—
244C22×D4—1221no—
245C23×C4—1221no—
246C2×SD16—1224no—
247C2×D8—1224no—
248C2×D8—2444no—
249C2×M16—1221no—
250C22×C8—21no—
251C22×C8—1224no—
252C4⋊Q8C22×D6111yescharacteristic
253C4⋊Q8C22×D6111yescharacteristic
254#127—612no—
255#127—1221no—
256#127—1221no—
257#127—1221no—
258#125—21no—
259C22⋊Q8—21no—
260C22⋊Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×D4—612no—
264C4×D4—612no—
265C4×D4—1221no—
266C4×D4—1221no—
267#117—21no—
268C2×C4⋊C4—611no—
269C2×C22⋊C4—1221no—
270C2×C42C22×D6111yescharacteristic
271#107—1222no—
272#106—1222no—
273C4⋊C8—21no—
274C4⋊C8—612no—
275C4⋊C8—612no—
276C4⋊C8—1221no—
277C4⋊C8C22×D6112yes—
278C4⋊C8C22×D6112yes—
279#102—1224no—
280#102—1222no—
281#102—1222no—
282#102—1224no—
283C22⋊C8—21no—
284C22⋊C8—612no—
285C22⋊C8—1224no—
286C4×C8—21no—
287C4×C8—612no—
288C24—1221no—
289C24—2441no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×D4—612no—
293C2×D4—614no—
294C2×D4—1222no—
295C2×D4—1222no—
296C2×D4—1222no—
297C2×D4—1221no—
298C2×D4—1222no—
299C2×D4—4882no—
300C2×D4—4882no—
301C2×D4—2441no—
302C2×D4—2441no—
303C2×D4—2441no—
304C2×D4—2442no—
305C2×D4—2442no—
306C22×C4—1221no—
307C22×C4—1222no—
308C22×C4—1222no—
309C22×C4—4882no—
310C22×C4#1175822211yescharacteristic
311C22×C4D4×D6111yescharacteristic
312C22×C4D4×D6111yescharacteristic
313SD16—4884no—
314D8—2448no—
315M16—2442no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—42no—
319C2×C8—614no—
320C2×C8—1222no—
321C2×C8—4884no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—612no—
329C4⋊C4—1221no—
330C22⋊C4—41no—
331C22⋊C4—1222no—
332C22⋊C4—1222no—
333C42—41no—
334C42#1175822211yescharacteristic
335C42D4×D6111yescharacteristic
336C42D4×D6111yescharacteristic
337C42C23×D6111yescharacteristic
338C23—1224no—
339C23—1222no—
340C23—4882no—
341C23—4882no—
342C23—4881no—
343C23—4881no—
344C23—4882no—
345C23—2442no—
346C23#1451975011yescharacteristic
347Q8—42no—
348Q8—81no—
349D4—614no—
350D4—1224no—
351D4—1222no—
352D4—4882no—
353D4—4882no—
354D4—4882no—
355D4—4882no—
356D4—2442no—
357D4—2442no—
358D4—2444no—
359C2×C4—22no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—1222no—
364C2×C4—4882no—
365C2×C4—4881no—
366C2×C4—2444no—
367C2×C4#1451975011yescharacteristic
368C2×C4#1451975011yescharacteristic
369C2×C4#1451975011yescharacteristic
370C2×C4SD16×D6112yes—
371C2×C4#1451980012yes—
372C2×C4C2×D4×D6111yescharacteristic
373C2×C4C2×D4×D6111yescharacteristic
374C2×C4C2×#1175822211yescharacteristic
375C8—42no—
376C8—42no—
377C8—1224no—
378C8—2442no—
379C22—21no—
380C22—21no—
381C22—1224no—
382C22—9764no—
383C22—4884no—
384C22—4882no—
385C22—4882no—
386C22—4882no—
387C22—2442no—
388C22—2442no—
389C22#1683999412yes—
390C22C2×#1451975011yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—81no—
395C4—81no—
396C4—2442no—
397C4#1683999412yes—
398C4C2×SD16×D6111yescharacteristic
399C4C2×#1451980011yescharacteristic
400C4#1684056412yes—
401C2—22no—
402C2—9762no—
403C2—2444no—
404C2#2145532511yescharacteristic
405C2C2×#1683999411yescharacteristic
406C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1542C2
socleC22×C61C2×#121
prime radicalC2×#16839958C2
hypercenter#1542D61
hypocenterC61#19384
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
1. #21456757C2semidirect
2. #21455612C2semidirect
4. #21454842C2semidirect
5. #21445231C2semidirect
6. #21445163C2semidirect
7. #21444589C2semidirect
8. #21444486C2semidirect
9. C61×#1542C2semidirect
13. #16840136C22semidirect
14. #16840110C22semidirect
16. #16840049C22semidirect
17. #16840039C22semidirect
23. #16839827C22semidirect
30. #16839512C22semidirect
31. #16839505C22semidirect
35. C61×#333C22semidirect
36. C61×#332C22semidirect
104. C2×D244SD16semidirect
206. #1542D61semidirect
237. #333C2×D61semidirect
238. #332C2×D61semidirect
240. C61#19384semidirect

Maximal subgroups

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

subgroupindex
#214567572
#214556122
C2×#168399582
#214548422
#214452312
#214451632
#214445892
#214444862
C61×#15422
#1938461