Finite Group Databasebeta

← #30604051

Subgroups — #30604051

Order 15616.

360 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462181C211yesmaximal, characteristic, complemented
2#21462177C211yesmaximal, characteristic, complemented
3#21460742C212yesmaximal, complemented
4#21460741C212yesmaximal, complemented
5#21454529C211yesmaximal, characteristic
6#16840683—21no—
7#16840683—21no—
8C22⋊#11758187C2212yes—
9C22⋊#11758187C2212yes—
10#16840651—21no—
11#16840648—21no—
12C2×#14519783C411yescharacteristic
13#16840308C411yescharacteristic
14#16839838—21no—
15#16839838—21no—
16C8×C22⋊D61—21no—
17C8×C22⋊D61—21no—
18#16839785C411yescharacteristic
19#16839730C2211yescharacteristic
20#16839729C2211yescharacteristic
21C2×C8×Dic61C2211yescharacteristic
22C2×#131×C61C411yescharacteristic
23C2×#11758187—22no—
24C2×#11758187—22no—
25C2×#11758222C2×C411yescharacteristic
26#14519783—42no—
27C2×#11758194C2×C411yescharacteristic
28C2×#11758189—21no—
29C22×#92418C2311yescharacteristic
30C2×C8×D61—21no—
31#14519775C811yescharacteristic
32#14519773C811yescharacteristic
33#14519766C811yescharacteristic
34#14519764C811yescharacteristic
35#14519749—21no—
36#14519749—21no—
37C4×C22⋊D61—21no—
38C4×C22⋊D61—21no—
39C2×C4×Dic61C2×C411yescharacteristic
40#14519706—21no—
41#14519706—21no—
42C22⋊#92418—21no—
43C22⋊#92418—21no—
44#14519689—21no—
45#14519689—21no—
46C8×Dic61D411yescharacteristic
47C8×Dic61D411yescharacteristic
48C8×Dic61D411yescharacteristic
49C8×Dic61D411yescharacteristic
50C2×#41×C61C2×C411yescharacteristic
51#131×C61—42no—
52C2×M16×C61—21no—
53C2×M16×C61C2×C411yescharacteristic
54C22×C8×C61—21no—
55C22×C8×C61C2×C411yescharacteristic
56#41×C61—41no—
57#41×C61—41no—
58C2×Q8×C61C2×C811yescharacteristic
59C2×D4×C61—21no—
60C2×D4×C61C2×C811yescharacteristic
61C22×C4×C61—21no—
62C22×C4×C61C22×C411yescharacteristic
63C2×C22⋊D61—21no—
64C2×C22⋊D61—21no—
65C2×C22⋊D61—21no—
66C22×Dic61M1611yescharacteristic
67C22×Dic61M1611yescharacteristic
68#11758222—41no—
69#11758222—41no—
70C2×D244C2×C811yescharacteristic
71C2×C4×D61—21no—
72C2×Dic122C2×C811yescharacteristic
73M16×C61—42no—
74M16×C61—41no—
75C2×C8×C61—22no—
76C2×C8×C61—21no—
77C2×C8×C61—42no—
78C2×C8×C61C22⋊C411yescharacteristic
79C2×C8×C61C22⋊C411yescharacteristic
80C22⋊Dic61—21no—
81C22⋊Dic61—21no—
82#11758198—21no—
83#11758198—21no—
84#11758196—21no—
85#11758196—21no—
86C4×Dic61C22⋊C411yescharacteristic
87C4×Dic61C22⋊C411yescharacteristic
88C4×Dic61C2×C811yescharacteristic
89C4×Dic61C2×C811yescharacteristic
90#11758194—41no—
91C2×#92418—21no—
92C2×#92418—41no—
93C2×#92418C2×D411yescharacteristic
94C2×#92418C2×D411yescharacteristic
95#11758189—42no—
96C8×D61—42no—
97#11758187—42no—
98#11758187—42no—
99C23×C61—21no—
100C23×C61C2×M1611yescharacteristic
101C22×D61—21no—
102Q8×C61—41no—
103D4×C61—41no—
104D4×C61—42no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61C2×C22⋊C411yescharacteristic
111C2×C4×C61C22×C811yescharacteristic
112C22⋊D61—42no—
113C22⋊D61—42no—
114C22⋊D61—42no—
115C2×Dic61—22no—
116C2×Dic61—21no—
117C2×Dic61—41no—
118C2×Dic61C22⋊C811yescharacteristic
119C2×Dic61C22⋊C811yescharacteristic
120C2×Dic61C22⋊C811yescharacteristic
121C2×Dic61C22⋊C811yescharacteristic
122D244—41no—
123C4×D61—41no—
124C4×D61—41no—
125Dic122—41no—
126C8×C61—21no—
127C8×C61—41no—
128C8×C61#10012yes—
129#92418—21no—
130#92418—21no—
131#92418—41no—
132#11492—611nomaximal, Sylow
133C22×C61—21no—
134C22×C61—21no—
135C22×C61—21no—
136C22×C61—41no—
137C22×C61—41no—
138C22×C61—41no—
139C22×C61C2×C22⋊C811yescharacteristic
140C2×D61—41no—
141C2×D61—41no—
142C2×D61—41no—
143C4×C61—21no—
144C4×C61—41no—
145C4×C61#41011yescharacteristic
146C4×C61C2×#10011yescharacteristic
147Dic61—21no—
148Dic61—21no—
149Dic61—21no—
150Dic61—41no—
151Dic61#35012yes—
152#2931—611no—
153#2163—611no—
154#2162—611no—
155#1315—612no—
156#1315—612no—
157C2×C61—41no—
158C2×C61—81no—
159C2×C61#146011yescharacteristic
160C2×C61C2×#35011yescharacteristic
161C2×C61#211711yescharacteristic
162D61—81no—
163C2×#131—611no—
164C2×#131#692311yescharacteristic
165#532—611no—
166#436—1221no—
167#436—1221no—
168C8×D4—1221no—
169C8×D4—1221no—
170#423—611no—
171#408—611no—
172#408—611no—
173C2×C4×C8—611no—
174C4⋊C16—1221no—
175C4⋊C16—1221no—
176C22⋊C16—612no—
177C22⋊C16—612no—
178#346—1221no—
179C4×C16—1221no—
180C61#1149211yescharacteristic, complemented, Sylow
181C2×#41—611no—
182C2×#41C2×#692311yescharacteristic
183#131—42no—
184#131—2442no—
185C2×M16—21no—
186C2×M16—611no—
187C2×M16—1221no—
188C2×M16C2×#692311yescharacteristic
189C22×C8—21no—
190C22×C8—611no—
191C22×C8—1221no—
192C22×C8C2×#692311yescharacteristic
193C4⋊Q8—611no—
194#127—611no—
195#124—611no—
196#124—611no—
197#121—1221no—
198#121—1221no—
199C4×D4—1221no—
200C4×D4—1221no—
201C2×C42—611no—
202C2×C16—1222no—
203C2×C16—1222no—
204C4⋊C8—1221no—
205C4⋊C8—1221no—
206C22⋊C8—1221no—
207C22⋊C8—1221no—
208C22⋊C8—1221no—
209C22⋊C8—1221no—
210C4×C8—611no—
211C4×C8—611no—
212C4×C8—611no—
213C4×C8—611no—
214#41—41no—
215#41—41no—
216#41—2441no—
217#41—2441no—
218C2×Q8—611no—
219C2×Q8D61⋊C811yescharacteristic
220C2×D4—21no—
221C2×D4—611no—
222C2×D4—1221no—
223C2×D4—1221no—
224C2×D4—1221no—
225C2×D4D61⋊C811yescharacteristic
226C22×C4—21no—
227C22×C4—611no—
228C22×C4—611no—
229C22×C4—1221no—
230C22×C4C22×#692311yescharacteristic
231M16—42no—
232M16—41no—
233M16—2442no—
234M16—2441no—
235C2×C8—22no—
236C2×C8—21no—
237C2×C8—42no—
238C2×C8—611no—
239C2×C8—611no—
240C2×C8—1221no—
241C2×C8—2442no—
242C2×C8—2441no—
243C2×C8C22⋊#692311yescharacteristic
244C2×C8C22⋊#692311yescharacteristic
245C4⋊C4—1221no—
246C4⋊C4—1221no—
247C22⋊C4—1221no—
248C22⋊C4—1221no—
249C22⋊C4—1221no—
250C22⋊C4—1221no—
251C42—611no—
252C42—611no—
253C42—611no—
254C42—611no—
255C16—2442no—
256C16—2442no—
257C23—21no—
258C23—1221no—
259C23D61⋊M1611yescharacteristic
260Q8—41no—
261Q8—2441no—
262D4—41no—
263D4—42no—
264D4—2441no—
265D4—2442no—
266D4—2442no—
267D4—2442no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—611no—
274C2×C4—611no—
275C2×C4—611no—
276C2×C4—611no—
277C2×C4—1222no—
278C2×C4—1221no—
279C2×C4—2441no—
280C2×C4—2441no—
281C2×C4—2441no—
282C2×C4C2×C22⋊#692311yescharacteristic
283C2×C4C2×D61⋊C811yescharacteristic
284C8—21no—
285C8—41no—
286C8—1221no—
287C8—1221no—
288C8—2441no—
289C8#1451983712yes—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—2441no—
297C22—2441no—
298C22—2441no—
299C22D61⋊(C22⋊C8)11yescharacteristic
300C4—21no—
301C4—41no—
302C4—612no—
303C4—1221no—
304C4—1221no—
305C4—1221no—
306C4—2441no—
307C4#1684074111yescharacteristic
308C4C2×#1451983711yescharacteristic
309C2—41no—
310C2—81no—
311C2—4881no—
312C2#2146105811yescharacteristic
313C2D61⋊#35011yescharacteristic
314C2#2146206211yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC2×#131×C61C4
socleC22×C61C2×C22⋊C8
prime radical#16840308C4
cosocleC22×#92418C23
hypercenterC2×#131#6923
hypocenterC61#11492
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. #21462181C2semidirect
2. #21462177C2semidirect
3. #21460742C2semidirect
4. #21460741C2semidirect
180. C61#11492semidirect

Maximal subgroups

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

subgroupindex
#214621812
#214621772
#214607422
#214607412
#214545292
#1149261