Finite Group Databasebeta

← #29649312

Subgroups — #29649312

Order 15616.

612 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454949C212yesmaximal
2C8×#11758222C211yesmaximal, characteristic
3#21444610C212yesmaximal, complemented
4#21444606C214yesmaximal, complemented
5#21444573C212yesmaximal, complemented
6#21444567C211yesmaximal, characteristic, complemented
7#21444564C212yesmaximal, complemented
8C61×#1566C211yesmaximal, characteristic, complemented
9C4×#11758222C2211yescharacteristic
10C8×Dic122C411yescharacteristic
11C8×Dic122C411yescharacteristic
12#16839886—22no—
13#16839839—22no—
14#16839838—24no—
15C8×D244C411yescharacteristic
16C8×D244C411yescharacteristic
17C8×C22⋊D61C412yes—
18C8×C22⋊D61C412yes—
19#16839815C2212yes—
20#16839815C2211yescharacteristic
21#16839814C2214yes—
22#16839813C2212yes—
23#16839784C2214yes—
24C4×C8×D61C2212yes—
25#16839515C2212yes—
26#16839514C2214yes—
27#16839513C2211yescharacteristic
28#16839513C2211yescharacteristic
29#16839512C2214yes—
30C61×#423C2212yes—
31C2×C4×C8×C61C2211yescharacteristic
32C61×#334C2212yes—
33C61×#334C2212yes—
34C2×#11758222Q811yescharacteristic
35C2×#11758189—24no—
36C2×C8×D61C2×C412yes—
37C2×C8×D61C2×C412yes—
38C4×Dic122C2×C411yescharacteristic
39C4×Dic122D411yescharacteristic
40C4×D244C2×C411yescharacteristic
41C4×D244D411yescharacteristic
42C4×C22⋊D61C2×C412yes—
43C4×C22⋊D61D412yes—
44#14519736Q811yescharacteristic
45#14519734Q812yes—
46C42×D61C2312yes—
47C4⋊#92418C2×C411yescharacteristic
48C4⋊#92418C2×C411yescharacteristic
49C4⋊#92418C2312yes—
50C4⋊#92418C2312yes—
51#14519706—24no—
52#14519706C2×C412yes—
53#14519706C2×C412yes—
54C22⋊#92418—22no—
55C22⋊#92418C2×C412yes—
56#14519689—24no—
57#14519689C2×C412yes—
58#14519689C2×C412yes—
59C4×#92418—22no—
60C4×#92418C2312yes—
61C8×Dic61—24no—
62C8×Dic61C2×C412yes—
63C8×Dic61C2×C412yes—
64C2×M16×C61—22no—
65C22×C8×C61C2×C411yescharacteristic
66C22×C8×C61C2×C411yescharacteristic
67C2×C42×C61C2311yescharacteristic
68C4⋊C8×C61—22no—
69C4⋊C8×C61C2314yes—
70C4×C8×C61C2×C411yescharacteristic
71C4×C8×C61C2×C411yescharacteristic
72C4×C8×C61C2311yescharacteristic
73C4×C8×C61C2311yescharacteristic
74C22×C4×C61C22×C411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×Q811yescharacteristic
77C2×C22⋊D61C4⋊C412yes—
78#11758222—24no—
79C2×D244C4⋊C411yescharacteristic
80C2×C4×D61C22×C412yes—
81C2×C4×D61C2×D412yes—
82C2×C4×D61C2×Q812yes—
83C2×Dic122C4⋊C411yescharacteristic
84M16×C61—82no—
85C2×C8×C61—22no—
86C2×C8×C61—24no—
87C2×C8×C61—41no—
88C2×C8×C61C22×C411yescharacteristic
89C2×C8×C61C22×C411yescharacteristic
90C2×C8×C61C22×C411yescharacteristic
91C2×C8×C61C22×C411yescharacteristic
92C42×C61C22×C411yescharacteristic
93C42×C61C2×D411yescharacteristic
94C42×C61C2×Q811yescharacteristic
95C42×C61C2411yescharacteristic
96C22⋊Dic61C4⋊C412yes—
97#11758198C4⋊C412yes—
98#11758198C4⋊C412yes—
99C4⋊Dic61C4⋊C411yescharacteristic
100C4⋊Dic61C4⋊C411yescharacteristic
101#11758196C4⋊C412yes—
102#11758196C4⋊C412yes—
103C4×Dic61C22×C412yes—
104C4×Dic61C2×D412yes—
105C4×Dic61C2×Q812yes—
106C2×#92418—22no—
107C2×#92418—22no—
108C2×#92418C22×C412yes—
109C2×#92418C22×C412yes—
110#11758189—84no—
111C8×D61—24no—
112C8×D61—42no—
113C23×C61C2×C4⋊C411yescharacteristic
114C22×D61C2×C4⋊C412yes—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—22no—
118C2×C4×C61—22no—
119C2×C4×C61C2×C4⋊C411yescharacteristic
120C2×C4×C61C2×C4⋊C411yescharacteristic
121C2×C4×C61C2×C4⋊C411yescharacteristic
122C2×C4×C61C23×C411yescharacteristic
123C2×C4×C61C22×D411yescharacteristic
124C2×C4×C61C22×Q811yescharacteristic
125C22⋊D61—24no—
126C2×Dic61C2×C4⋊C412yes—
127C2×Dic61C2×C4⋊C412yes—
128C2×Dic61C2×C4⋊C412yes—
129D244—22no—
130C4×D61—22no—
131C4×D61—22no—
132C4×D61—24no—
133C4×D61—24no—
134Dic122#10814yes—
135C8×C61—21no—
136C8×C61—21no—
137C8×C61—21no—
138C8×C61—44no—
139C8×C61#13112yes—
140#92418—22no—
141#92418—22no—
142#92418—42no—
143#92418—42no—
144#17835—611nomaximal, Sylow
145C22×C61—21no—
146C22×C61—21no—
147C22×C61—21no—
148C22×C61C22×C4⋊C411yescharacteristic
149C2×D61—22no—
150C2×D61—22no—
151C2×D61—22no—
152C4×C61—21no—
153C4×C61—21no—
154C4×C61—21no—
155C4×C61C2×#10812yes—
156C4×C61C2×#13111yescharacteristic
157C4×C61#56811yescharacteristic
158C4×C61#57712yes—
159Dic61—22no—
160Dic61—22no—
161Dic61C2×#10814yes—
162#2991—612no—
163C8×#41—611no—
164#1574—612no—
165#1574—614no—
166#1566—612no—
167#1566D6111yescharacteristic, complemented
168#1564—611no—
169#1564—612no—
170C2×C61—41no—
171C2×C61#290611yescharacteristic
172C2×C61#291211yescharacteristic
173C2×C61C22×#10811yescharacteristic
174D61—42no—
175C4×#41—611no—
176C8×Q8—611no—
177C8×Q8—611no—
178C8×Q8—1222no—
179#436—1222no—
180#436—1224no—
181C8×D4—612no—
182C8×D4—612no—
183C8×D4—611no—
184C8×D4—611no—
185#431—611no—
186#431—612no—
187#431—612no—
188#431—614no—
189#423—614no—
190#423C2×D6112yes—
191C2×C4×C8—612no—
192C2×C4×C8C2×D6111yescharacteristic
193#335—614no—
194#335—612no—
195#334—614no—
196#334—611no—
197#334—611no—
198#334C2×D6112yes—
199#334C2×D6112yes—
200C61#1783511yescharacteristic, complemented, Sylow
201C2×#41—611no—
202C2×M16—22no—
203C2×M16—1224no—
204C22×C8—612no—
205C22×C8—612no—
206C22×C8C4×D6111yescharacteristic
207C22×C8C4×D6111yescharacteristic
208C4×Q8—611no—
209C4×Q8—611no—
210C4×D4—612no—
211C4×D4—612no—
212C4×D4—611no—
213C4×D4—611no—
214#117—611no—
215#117—612no—
216C2×C42—612no—
217C2×C42C22×D6111yescharacteristic
218C4⋊C8—22no—
219C4⋊C8—612no—
220C4⋊C8—612no—
221C4⋊C8—611no—
222C4⋊C8—611no—
223C4⋊C8—612no—
224C4⋊C8—612no—
225C4⋊C8—1224no—
226C4⋊C8C22×D6114yes—
227C22⋊C8—612no—
228C22⋊C8—612no—
229C22⋊C8—612no—
230C22⋊C8—1224no—
231C22⋊C8—1222no—
232C4×C8—612no—
233C4×C8—612no—
234C4×C8—612no—
235C4×C8—1224no—
236C4×C8—1222no—
237C4×C8C4×D6111yescharacteristic
238C4×C8C4×D6111yescharacteristic
239C4×C8C22×D6111yescharacteristic
240C4×C8C22×D6111yescharacteristic
241#41—1224no—
242C2×Q8—611no—
243C2×D4—612no—
244C2×D4—611no—
245C22×C4—612no—
246C22×C4—612no—
247C22×C4—612no—
248C22×C4C2×C4×D6111yescharacteristic
249C22×C4D4×D6111yescharacteristic
250C22×C4Q8×D6111yescharacteristic
251M16—82no—
252M16—4884no—
253C2×C8—22no—
254C2×C8—24no—
255C2×C8—41no—
256C2×C8—612no—
257C2×C8—612no—
258C2×C8—1224no—
259C2×C8—1222no—
260C2×C8—1222no—
261C2×C8—2442no—
262C2×C8C2×C4×D6111yescharacteristic
263C2×C8C2×C4×D6111yescharacteristic
264C2×C8C2×C4×D6111yescharacteristic
265C2×C8C2×C4×D6111yescharacteristic
266C4⋊C4—612no—
267C4⋊C4—612no—
268C4⋊C4—611no—
269C4⋊C4—611no—
270C22⋊C4—612no—
271C22⋊C4—612no—
272C22⋊C4—612no—
273C42—612no—
274C42—612no—
275C42—612no—
276C42C2×C4×D6111yescharacteristic
277C42D4×D6111yescharacteristic
278C42Q8×D6111yescharacteristic
279C42C23×D6111yescharacteristic
280C23—612no—
281C23C4⋊C4×D6111yescharacteristic
282Q8—614no—
283D4—1224no—
284D4—1222no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—612no—
290C2×C4—612no—
291C2×C4—612no—
292C2×C4—1224no—
293C2×C4—1222no—
294C2×C4—1222no—
295C2×C4—1224no—
296C2×C4C4⋊C4×D6111yescharacteristic
297C2×C4C4⋊C4×D6111yescharacteristic
298C2×C4C4⋊C4×D6111yescharacteristic
299C2×C4C22×C4×D6111yescharacteristic
300C2×C4C2×D4×D6111yescharacteristic
301C2×C4C2×Q8×D6111yescharacteristic
302C8—21no—
303C8—21no—
304C8—21no—
305C8—44no—
306C8—1222no—
307C8—1222no—
308C8—2442no—
309C8—2442no—
310C8#1451978512yes—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—1222no—
315C22—1222no—
316C22—1222no—
317C22C2×C4⋊C4×D6111yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—614no—
322C4—1222no—
323C4—1222no—
324C4#108×D6112yes—
325C4C2×#1451978511yescharacteristic
326C4#1684052811yescharacteristic
327C4#1684056812yes—
328C2—41no—
329C2—2442no—
330C2#2145438811yescharacteristic
331C2#2145441011yescharacteristic
332C2C2×#108×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1566C2
socleC22×C61C22×C4⋊C4
prime radicalC2×#11758222Q8
cosocleC42×C61C24
hypercenter#1566D61
hypocenterC61#17835
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1566

Complements

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

NH ≅ G/Ntype
3. #21444610C2semidirect
4. #21444606C2semidirect
5. #21444573C2semidirect
6. #21444567C2semidirect
7. #21444564C2semidirect
8. C61×#1566C2semidirect
167. #1566D61semidirect
200. C61#17835semidirect

Maximal subgroups

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

subgroupindex
#214549492
C8×#117582222
#214446102
#214446062
#214445732
#214445672
#214445642
C61×#15662
#1783561