Finite Group Databasebeta

← #29576705

Subgroups — #29576705

Order 15616.

490 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456757C211yesmaximal, characteristic
2#21446643C211yesmaximal, characteristic
3#21444488C211yesmaximal, characteristic
4#21444383C211yesmaximal, characteristic, complemented
5#21443679C211yesmaximal, characteristic, complemented
6#21443650C211yesmaximal, characteristic, complemented
7C61×#2961C211yesmaximal, characteristic, complemented
8C2×#14519774C2211yescharacteristic
9#16840136C412yescomplemented
10#16840110C412yescomplemented
11#16839962—22no—
12#16839912—22no—
13#16839815C2211yescharacteristic
14#16839815C2211yescharacteristic
15C2×#14519699—21no—
16C2×#14519699—21no—
17#16839628C2211yescharacteristic
18#16839505—22no—
19#16839499—22no—
20C61×#533C2211yescharacteristic
21C61×#446—22no—
22C61×#446—22no—
23C61×#424—21no—
24C61×#423C2211yescharacteristic
25C61×#405—21no—
26C2×#97×C61C2211yescharacteristic
27C22×D244—21no—
28C22×D244—21no—
29C22×D244—21no—
30C2×#11758201—24no—
31#14519774—21no—
32#14519774—21no—
33#14519774—21no—
34#14519774C2×C412yes—
35C2×C4⋊Dic61—21no—
36C4⋊#92418—21no—
37C4⋊#92418—21no—
38#14519699—42no—
39#14519699—42no—
40#14519697—24no—
41C22⋊#92418—21no—
42C22⋊#92418—21no—
43C4×#92418—21no—
44C4×#92418C2×C412yes—
45#14519684—21no—
46#14519684—21no—
47C2×M16×C61—21no—
48C22×C8×C61D411yescharacteristic
49C22×C8×C61D411yescharacteristic
50C4⋊Q8×C61C2×C411yescharacteristic
51C4⋊Q8×C61C2×C411yescharacteristic
52#125×C61—21no—
53C22⋊Q8×C61—21no—
54C22⋊Q8×C61D412yes—
55C4×Q8×C61—21no—
56C4×Q8×C61D412yes—
57#117×C61—21no—
58C2×C42×C61C2311yescharacteristic
59C4⋊C8×C61—24no—
60C4⋊C8×C61—21no—
61C4⋊C8×C61D412yes—
62#97×C61—22no—
63#97×C61—21no—
64#97×C61—21no—
65C4×C8×C61—22no—
66C23×D61—41no—
67C2×Q8×C61—21no—
68C2×Q8×C61C22⋊C412yes—
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61#4111yescharacteristic
72C22×Dic61—41no—
73C2×D244—22no—
74C2×D244—22no—
75C2×D244—22no—
76C2×D244—41no—
77C2×D244—41no—
78C2×D244—41no—
79C2×D244—41no—
80C2×D244—41no—
81C2×D244—41no—
82C2×D244—41no—
83C2×D244—82no—
84C2×D244—82no—
85M16×C61—41no—
86M16×C61—41no—
87C2×C8×C61—21no—
88C2×C8×C61—21no—
89C2×C8×C61—22no—
90C2×C8×C61—21no—
91C2×C8×C61—21no—
92C2×C8×C61—21no—
93C2×C8×C61SD1612yescomplemented
94C4⋊C4×C61—21no—
95C4⋊C4×C61—21no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—22no—
98C4⋊C4×C61—22no—
99C4⋊C4×C61C22⋊C412yes—
100C22⋊C4×C61—22no—
101C42×C61—22no—
102C42×C61C22×C411yescharacteristic
103C42×C61C2×D411yescharacteristic
104C42×C61C2×D411yescharacteristic
105C42×C61#4111yescharacteristic
106#11758201—84no—
107C4⋊Dic61—41no—
108C4⋊Dic61—41no—
109C2×#92418—22no—
110C2×#92418—41no—
111C23×C61#12111yescharacteristic
112C22×D61—42no—
113C22×D61—82no—
114C22×D61—82no—
115C22×D61—81no—
116C22×D61—81no—
117Q8×C61—42no—
118Q8×C61—81no—
119C2×C4×C61—22no—
120C2×C4×C61—22no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61C2×C22⋊C411yescharacteristic
124C2×C4×C61C4×D411yescharacteristic
125C2×C4×C61C4×D411yescharacteristic
126C2×C4×C61#12011yescharacteristic
127C2×C4×C61#12111yescharacteristic
128C2×C4×C61#12311yescharacteristic
129C2×C4×C61C2×SD1611yescharacteristic
130C2×C4×C61#13511yescharacteristic
131C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C61#13711yescharacteristic
133C2×Dic61—81no—
134C2×Dic61—81no—
135C2×Dic61—81no—
136D244—44no—
137D244—42no—
138D244—42no—
139D244—82no—
140D244—82no—
141D244—82no—
142D244—81no—
143D244—81no—
144C8×C61—22no—
145C8×C61—41no—
146C8×C61—42no—
147#92418—42no—
148#92418—81no—
149#9294—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—21no—
152C22×C61#38611yescharacteristic
153C22×C61#46011yescharacteristic
154C22×C61#46411yescharacteristic
155C2×D61—82no—
156C2×D61—82no—
157C2×D61—82no—
158C2×D61—162no—
159C2×D61—162no—
160C4×C61—21no—
161C4×C61—21no—
162C4×C61—21no—
163C4×C61—21no—
164C4×C61—42no—
165C4×C61—42no—
166C4×C61C4×SD1611yescharacteristic
167C4×C61#44011yescharacteristic
168C4×C61#44811yescharacteristic
169C4×C61#44911yescharacteristic
170Dic61—161no—
171#3208—611no—
172#2961D6111yescharacteristic, complemented
173#1961—611no—
174#1545—611no—
175#1518—611no—
176#1286—611no—
177#1275—611no—
178C2×C61—21no—
179C2×C61—21no—
180C2×C61#188011yescharacteristic
181C2×C61#190311yescharacteristic
182C2×C61#191011yescharacteristic
183D61—162no—
184#533C2×D6111yescharacteristic
185C2×#127—611no—
186#492—612no—
187#486—612no—
188#460—1222no—
189#450—1222no—
190#446—22no—
191#446—22no—
192#431—611no—
193#431—611no—
194#424—21no—
195#423C2×D6111yescharacteristic
196C2×#102—1221no—
197C2×#102—1221no—
198#405—21no—
199C2×#97C2×D6111yescharacteristic
200#382—611no—
201#331—1222no—
202#328—1222no—
203C61#929411yescharacteristic, complemented, Sylow
204C22×D4—1221no—
205C22×D4—1221no—
206C22×D4—1221no—
207C2×SD16—1224no—
208C2×M16—21no—
209C22×C8C22⋊D6111yescharacteristic
210C22×C8C22⋊D6111yescharacteristic
211C4⋊Q8C4×D6111yescharacteristic
212C4⋊Q8C4×D6111yescharacteristic
213#127—612no—
214#127—1221no—
215#127—1221no—
216#127—1221no—
217#125—21no—
218C22⋊Q8—21no—
219C22⋊Q8D24412yes—
220C4×Q8—21no—
221C4×Q8C22⋊D6112yes—
222#117—21no—
223C2×C4⋊C4—1221no—
224C2×C42C22×D6111yescharacteristic
225C4⋊C8—24no—
226C4⋊C8—21no—
227C4⋊C8—1221no—
228C4⋊C8—1221no—
229C4⋊C8C22⋊D6112yes—
230#102—1224no—
231#102—2442no—
232#102—2442no—
233C22⋊C8—1221no—
234C22⋊C8—1221no—
235#97—22no—
236#97—21no—
237#97—21no—
238C4×C8—22no—
239C4×C8—612no—
240C4×C8—1221no—
241#95—1221no—
242#95—1221no—
243C24—2441no—
244C2×Q8—21no—
245C2×Q8#1175819812yes—
246C2×D4—1222no—
247C2×D4—1222no—
248C2×D4—1222no—
249C2×D4—4882no—
250C2×D4—4882no—
251C2×D4—2441no—
252C2×D4—2441no—
253C2×D4—2441no—
254C2×D4—2441no—
255C2×D4—2441no—
256C2×D4—2441no—
257C2×D4—2441no—
258C22×C4—2441no—
259C22×C4C2×D24411yescharacteristic
260C22×C4#1175822211yescharacteristic
261C22×C4C2×C22⋊D6111yescharacteristic
262SD16—4884no—
263M16—41no—
264M16—41no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—22no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—1222no—
272C2×C8—2441no—
273C2×C8#1175820112yescomplemented
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—2441no—
280C4⋊C4—2441no—
281C4⋊C4#1175819812yes—
282C22⋊C4—22no—
283C42—22no—
284C42C2×C4×D6111yescharacteristic
285C42#1175822211yescharacteristic
286C42C2×C22⋊D6111yescharacteristic
287C42C2×C22⋊D6111yescharacteristic
288C23—4882no—
289C23—4882no—
290C23—4881no—
291C23—4881no—
292C23—2442no—
293C23#1451975111yescharacteristic
294Q8—42no—
295Q8—81no—
296D4—4882no—
297D4—4882no—
298D4—4882no—
299D4—4881no—
300D4—4881no—
301D4—2444no—
302D4—2442no—
303D4—2442no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—4881no—
309C2×C4—4881no—
310C2×C4—4881no—
311C2×C4C2×#1175819811yescharacteristic
312C2×C4C4×C22⋊D6111yescharacteristic
313C2×C4C4×C22⋊D6111yescharacteristic
314C2×C4#1451974611yescharacteristic
315C2×C4#1451975111yescharacteristic
316C2×C4#1451976011yescharacteristic
317C2×C4C2×#1175820111yescharacteristic
318C2×C4#1451979711yescharacteristic
319C2×C4#1451980211yescharacteristic
320C2×C4#1451980811yescharacteristic
321C8—22no—
322C8—41no—
323C8—42no—
324C8—4881no—
325C8—2442no—
326C22—21no—
327C22—21no—
328C22—9762no—
329C22—9762no—
330C22—4882no—
331C22—4882no—
332C22—4882no—
333C22#1683964111yescharacteristic
334C22#1683996211yescharacteristic
335C22#1683998411yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—42no—
342C4—9761no—
343C4C4×#1175820111yescharacteristic
344C4#1683986011yescharacteristic
345C4#1683990111yescharacteristic
346C4#1683990711yescharacteristic
347C2—21no—
348C2—21no—
349C2—9762no—
350C2#2144617411yescharacteristic
351C2#2144629911yescharacteristic
352C2#2144634611yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2961C2
socleC22×C61#386
prime radicalC2×#14519774C22
cosocleC2×C42×C61C23
hypercenter#2961D61
hypocenterC61#9294
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2961

Complements

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

NH ≅ G/Ntype
4. #21444383C2semidirect
5. #21443679C2semidirect
6. #21443650C2semidirect
7. C61×#2961C2semidirect
9. #16840136C4semidirect
10. #16840110C4semidirect
93. C2×C8×C61SD16semidirect
172. #2961D61semidirect
203. C61#9294semidirect
273. C2×C8#11758201semidirect

Maximal subgroups

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

subgroupindex
#214567572
#214466432
#214444882
#214443832
#214436792
#214436502
C61×#29612
#929461