Finite Group Databasebeta

← #29574285

Subgroups — #29574285

Order 15616.

390 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21445990C211yesmaximal, characteristic, complemented
2#21444607C211yesmaximal, characteristic, complemented
3#21444258C211yesmaximal, characteristic
4#21444201C211yesmaximal, characteristic
5#21443708C211yesmaximal, characteristic, complemented
6#21443655C211yesmaximal, characteristic, complemented
7C61×#2990C211yesmaximal, characteristic
8#16839815C2211yescharacteristic
9C2×#14519712—22no—
10#16839786C2211yescharacteristic
11C4×#11758194C2211yescharacteristic
12C2×C4×#92418C2211yescharacteristic
13#16839514—21no—
14#16839512—21no—
15Q8⋊#92418—21no—
16Q8⋊#92418C411yescharacteristic
17Q8⋊#92418C411yescharacteristic
18D4⋊#92418—21no—
19D4⋊#92418C411yescharacteristic
20D4⋊#92418C411yescharacteristic
21C4×#41×C61C2211yescharacteristic
22C61×#446—21no—
23C61×#435—22no—
24C8×D4×C61—21no—
25C61×#432—21no—
26C61×#431C2211yescharacteristic
27C61×#423C2211yescharacteristic
28C2×C4⋊C8×C61—21no—
29C2×#11758194—21no—
30C2×#11758194—21no—
31C22×#92418—21no—
32#14519712—44no—
33C4⋊#92418—21no—
34C4⋊#92418—21no—
35C4⋊#92418C2×C411yescharacteristic
36C4⋊#92418C2×C411yescharacteristic
37C22⋊#92418—21no—
38#14519688—21no—
39C4×#92418—21no—
40C4×#92418—21no—
41C4×#92418—21no—
42C4×#92418C2×C411yescharacteristic
43C4×#92418C2×C411yescharacteristic
44C2×#41×C61Q811yescharacteristic
45C2×M16×C61D412yes—
46C22×C8×C61—22no—
47C4×Q8×C61C2×C411yescharacteristic
48C4×Q8×C61D411yescharacteristic
49C4×D4×C61—21no—
50C4×D4×C61—21no—
51C4×D4×C61C2×C411yescharacteristic
52C4×D4×C61D411yescharacteristic
53#117×C61—21no—
54#117×C61Q811yescharacteristic
55C2×C42×C61—21no—
56C2×C42×C61C2311yescharacteristic
57C4⋊C8×C61—22no—
58C4⋊C8×C61—21no—
59C4⋊C8×C61—21no—
60C4⋊C8×C61D411yescharacteristic
61C4⋊C8×C61D411yescharacteristic
62C22⋊C8×C61—22no—
63C22⋊C8×C61—21no—
64#97×C61—22no—
65C4×C8×C61—21no—
66#41×C61—22no—
67#41×C61—41no—
68C2×Q8×C61C4⋊C411yescharacteristic
69C2×D4×C61—21no—
70C2×D4×C61C4⋊C411yescharacteristic
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61C2×Q811yescharacteristic
76C22×C4×C61#4111yescharacteristic
77M16×C61—42no—
78C2×C8×C61—21no—
79C2×C8×C61—21no—
80C2×C8×C61—22no—
81C2×C8×C61—82no—
82C4⋊C4×C61—21no—
83C4⋊C4×C61—21no—
84C4⋊C4×C61C4⋊C411yescharacteristic
85C4⋊C4×C61C4⋊C411yescharacteristic
86C22⋊C4×C61—21no—
87C22⋊C4×C61—21no—
88C22⋊C4×C61—21no—
89C42×C61—21no—
90C42×C61—21no—
91C42×C61—21no—
92C42×C61C22×C411yescharacteristic
93C42×C61C2×D411yescharacteristic
94C42×C61C2×D411yescharacteristic
95C42×C61#4111yescharacteristic
96#11758194—22no—
97#11758194—41no—
98#11758194—42no—
99C2×#92418—22no—
100C2×#92418—22no—
101C2×#92418—21no—
102C2×#92418—21no—
103C2×#92418—21no—
104C2×#92418—21no—
105C2×#92418—41no—
106C2×#92418—41no—
107C23×C61—21no—
108C23×C61C22⋊Q811yescharacteristic
109Q8×C61—21no—
110Q8×C61—21no—
111D4×C61—21no—
112D4×C61—42no—
113D4×C61#10812yescomplemented
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—42no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61C2×C4⋊C411yescharacteristic
129C2×C4×C61C4×D411yescharacteristic
130C2×C4×C61C4×D411yescharacteristic
131C2×C4×C61#12011yescharacteristic
132C2×C4×C61C22⋊Q811yescharacteristic
133C2×C4×C61#12311yescharacteristic
134C8×C61—41no—
135C8×C61—42no—
136C8×C61—41no—
137#92418—22no—
138#92418—22no—
139#92418—42no—
140#92418—41no—
141#92418—41no—
142#92418—81no—
143#8898—611nomaximal, Sylow
144C22×C61—21no—
145C22×C61—22no—
146C22×C61—21no—
147C22×C61—21no—
148C22×C61—41no—
149C22×C61—41no—
150C22×C61#38511yescharacteristic
151C4×C61—21no—
152C4×C61—21no—
153C4×C61—21no—
154C4×C61—41no—
155C4×C61—41no—
156C4×C61—41no—
157C4×C61C2×#10811yescharacteristic
158C4×C61#43011yescharacteristic
159C4×C61#44311yescharacteristic
160C4×C61#44411yescharacteristic
161C4×C61#45311yescharacteristic
162C4×C61#45611yescharacteristic
163#2990D6111yescharacteristic
164#1848—611no—
165#1574—611no—
166#1489—611no—
167#1476—611no—
168#1299—611no—
169#1278—611no—
170C2×C61—41no—
171C2×C61—42no—
172C2×C61#185511yescharacteristic
173C2×C61#185911yescharacteristic
174C2×C61#186611yescharacteristic
175C4×#41C2×D6111yescharacteristic
176#446—21no—
177#435—22no—
178C8×D4—21no—
179#432—21no—
180#431—611no—
181#431C2×D6111yescharacteristic
182C2×#108—1222no—
183#423—611no—
184#423C2×D6111yescharacteristic
185C2×C4⋊C8—21no—
186C4×M16—611no—
187C2×C4×C8—611no—
188#335—1221no—
189#334—1221no—
190Q8⋊C8—611no—
191Q8⋊C8—611no—
192Q8⋊C8—1221no—
193D4⋊C8—611no—
194D4⋊C8—611no—
195D4⋊C8—1221no—
196C61#889811yescharacteristic, complemented, Sylow
197C2×#41Dic12211yescharacteristic
198C2×M16—1221no—
199C2×M16—1221no—
200C2×M16C22⋊D6112yes—
201C22×C8—22no—
202C22×C8—1221no—
203C4×Q8C4×D6111yescharacteristic
204C4×Q8C22⋊D6111yescharacteristic
205C4×D4—21no—
206C4×D4—21no—
207C4×D4C4×D6111yescharacteristic
208C4×D4C22⋊D6111yescharacteristic
209#117—21no—
210#117Dic12211yescharacteristic
211C2×C42—21no—
212C2×C42C22×D6111yescharacteristic
213#108—2444no—
214C4⋊C8—22no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—611no—
218C4⋊C8—611no—
219C4⋊C8—1221no—
220C4⋊C8—1221no—
221C4⋊C8C22⋊D6111yescharacteristic
222C4⋊C8C22⋊D6111yescharacteristic
223C22⋊C8—22no—
224C22⋊C8—21no—
225C22⋊C8—1221no—
226#97—22no—
227#97—1221no—
228C4×C8—21no—
229C4×C8—611no—
230C4×C8—611no—
231C4×C8—1221no—
232C4×C8—1221no—
233C4×C8—1221no—
234#41—22no—
235#41—41no—
236C2×Q8#1175819611yescharacteristic
237C2×D4—21no—
238C2×D4#1175819611yescharacteristic
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4C2×Dic12211yescharacteristic
243C22×C4#1175822211yescharacteristic
244C22×C4C2×C22⋊D6111yescharacteristic
245M16—42no—
246M16—1222no—
247M16—2442no—
248M16—2441no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—22no—
252C2×C8—82no—
253C2×C8—1222no—
254C2×C8—1222no—
255C2×C8—1221no—
256C2×C8—1221no—
257C2×C8—1221no—
258C2×C8—1221no—
259C2×C8—2441no—
260C2×C8—2441no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4#1175819611yescharacteristic
264C4⋊C4#1175819611yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42C2×C4×D6111yescharacteristic
272C42#1175822211yescharacteristic
273C42C2×C22⋊D6111yescharacteristic
274C42C2×C22⋊D6111yescharacteristic
275C23—21no—
276C23#1451975611yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—42no—
281D4#1451971212yescomplemented
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4C2×#1175819611yescharacteristic
297C2×C4C4×C22⋊D6111yescharacteristic
298C2×C4C4×C22⋊D6111yescharacteristic
299C2×C4#1451974611yescharacteristic
300C2×C4#1451975611yescharacteristic
301C2×C4#1451976011yescharacteristic
302C8—41no—
303C8—41no—
304C8—42no—
305C8—1222no—
306C8—1222no—
307C8—4881no—
308C8—2442no—
309C8—2441no—
310C8—2441no—
311C22—21no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—41no—
316C22—41no—
317C22#1683963511yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4C2×#1451971211yescharacteristic
325C4#1683981011yescharacteristic
326C4#1683987711yescharacteristic
327C4#1683988211yescharacteristic
328C4#1683993011yescharacteristic
329C4#1683994511yescharacteristic
330C2—41no—
331C2—42no—
332C2#2144602411yescharacteristic
333C2#2144604511yescharacteristic
334C2#2144609411yescharacteristic
subgroupG/H
centerC2×C4#14519746
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2990C2
socleC22×C61#385
prime radicalC2×#41×C61Q8
cosocleC2×C42×C61C23
hypercenter#2990D61
hypocenterC61#8898
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2990

Complements

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

NH ≅ G/Ntype
1. #21445990C2semidirect
2. #21444607C2semidirect
5. #21443708C2semidirect
6. #21443655C2semidirect
113. D4×C61#108semidirect
196. C61#8898semidirect
281. D4#14519712semidirect

Maximal subgroups

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

subgroupindex
#214459902
#214446072
#214442582
#214442012
#214437082
#214436552
C61×#29902
#889861