Finite Group Databasebeta

← #29600353

Subgroups — #29600353

Order 15616.

398 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 59 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#21456277C211yesmaximal, characteristic
2#21447503C212yesmaximal
3#21443988C211yesmaximal, characteristic, complemented
4#21443987C212yesmaximal, complemented
5C61×#1411C211yesmaximal, characteristic, complemented
6C2×#14519809—22no—
7#16840313C2211yescharacteristic
8C4⋊Q8×D61C411yescharacteristic
9#16840155—22no—
10#16840141C2211yescharacteristic
11#16840136C411yescharacteristic
12#16840135C411yescharacteristic
13#16839745—22no—
14C4×#11758189C411yescharacteristic
15#16839561C2212yes—
16#16839558—22no—
17C61×#494C2211yescharacteristic
18C61×#356C2212yes—
19C2×#11758226D412yes—
20C2×Q8×D61—21no—
21#14519809—44no—
22#14519809—44no—
23C2×#11758202—22no—
24C2×#11758190—22no—
25C2×#11758201—22no—
26C2×#11758189—21no—
27M16×D61—42no—
28#14519777C2×C411yescharacteristic
29#14519776—41no—
30#14519775—21no—
31#14519774C2×C411yescharacteristic
32#14519766D412yes—
33#14519752—41no—
34#14519739—41no—
35C4⋊C4×D61—41no—
36C42×D61C2×C411yescharacteristic
37#14519696—22no—
38#14519693—42no—
39#14519687—21no—
40C4×#92418C2×C411yescharacteristic
41C2×Q16×C61—22no—
42C4⋊Q8×C61C2×C411yescharacteristic
43C4⋊Q8×C61C2311yescharacteristic
44#101×C61—22no—
45C4×C8×C61C2×C411yescharacteristic
46C2×Q8×C61—21no—
47C2×Q8×C61C2×D412yes—
48#11758226—42no—
49#11758226—42no—
50Q8×D61—42no—
51Q8×D61—42no—
52C2×D244—22no—
53C2×D244—41no—
54C2×D244C22⋊C412yes—
55C2×C4×D61—21no—
56C2×C4×D61—22no—
57C2×C4×D61—41no—
58C2×C4×D61C22⋊C411yescharacteristic
59C2×Dic122—21no—
60C2×Dic122—41no—
61Q16×C61—44no—
62M16×C61—42no—
63C2×C8×C61—21no—
64C4⋊C4×C61—41no—
65C4⋊C4×C61—41no—
66C42×C61C22×C411yescharacteristic
67#11758202—44no—
68#11758201—44no—
69#11758198—42no—
70C4⋊Dic61—41no—
71#11758196—42no—
72C4×Dic61—21no—
73C4×Dic61C22⋊C411yescharacteristic
74#11758194—42no—
75C2×#92418—21no—
76#11758190—44no—
77#11758189—24no—
78#11758189—82no—
79C8×D61—82no—
80C22×D61—22no—
81C22×D61#9911yescharacteristic
82Q8×C61—42no—
83Q8×C61—42no—
84C2×C4×C61—21no—
85C2×C4×C61—22no—
86C2×C4×C61—41no—
87C2×C4×C61C2×C22⋊C411yescharacteristic
88C2×Dic61—21no—
89C2×Dic61—41no—
90C2×Dic61#9911yescharacteristic
91D244—42no—
92D244—82no—
93D244—82no—
94C4×D61—22no—
95C4×D61—42no—
96C4×D61—42no—
97C4×D61—42no—
98C4×D61—81no—
99C4×D61—81no—
100Dic122—42no—
101Dic122—82no—
102C8×C61—22no—
103C8×C61—82no—
104#92418—22no—
105#92418—82no—
106#13592—611nomaximal, Sylow
107C22×C61C2×#9911yescharacteristic
108C2×D61—21no—
109C2×D61—41no—
110C2×D61—42no—
111C2×D61—82no—
112C4×C61—22no—
113C4×C61—21no—
114C4×C61—42no—
115C4×C61—81no—
116Dic61—41no—
117Dic61—81no—
118Dic61#35312yes—
119#3152—611no—
120#2132—612no—
121#1411—611no—
122#1411—612no—
123#1411D6111yescharacteristic, complemented
124C2×C61—21no—
125C2×C61C2×#35311yescharacteristic
126D61—41no—
127D61—82no—
128C2×#137—1222no—
129#532—611no—
130C2×C4⋊Q8—611no—
131#497—1222no—
132#494—611no—
133#494C2×D6111yescharacteristic
134#492—611no—
135#492—611no—
136#413—1222no—
137C4×M16—611no—
138#356—612no—
139#356C2×D6112yes—
140#355—1222no—
141C61#1359211yescharacteristic, complemented, Sylow
142C2×#41—612no—
143C22×Q8—1221no—
144#137—2444no—
145#137—2444no—
146C2×Q16—22no—
147C2×Q16—1222no—
148C2×SD16—1222no—
149C2×SD16—1222no—
150C2×M16—1221no—
151C2×M16—2442no—
152C4⋊Q8—611no—
153C4⋊Q8—1221no—
154C4⋊Q8—2441no—
155C4⋊Q8C4×D6111yescharacteristic
156C4⋊Q8C22×D6111yescharacteristic
157#127—611no—
158#124—612no—
159#121—2441no—
160C4×D4—2441no—
161C2×C4⋊C4—2441no—
162C2×C42—611no—
163#101—22no—
164#101—1222no—
165#100—2442no—
166#97—1221no—
167C4×C8—611no—
168C4×C8C4×D6111yescharacteristic
169#41—2442no—
170#41—2442no—
171C2×Q8—21no—
172C2×Q8—1221no—
173C2×Q8—2441no—
174C2×Q8—2442no—
175C2×Q8—2442no—
176C2×Q8D4×D6112yes—
177C2×D4—612no—
178C2×D4—1222no—
179C2×D4—2441no—
180C22×C4—611no—
181C22×C4—1222no—
182C22×C4—1221no—
183C22×C4—2441no—
184Q16—44no—
185Q16—2444no—
186SD16—2444no—
187SD16—2444no—
188M16—42no—
189M16—1224no—
190M16—4882no—
191M16—2442no—
192C2×C8—21no—
193C2×C8—1221no—
194C2×C8—4882no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—2441no—
198C4⋊C4—2442no—
199C22⋊C4—2442no—
200C42—611no—
201C42—1221no—
202C42C2×C4×D6111yescharacteristic
203C23—611no—
204C23—1222no—
205Q8—42no—
206Q8—42no—
207Q8—4882no—
208Q8—2442no—
209D4—4882no—
210D4—4882no—
211D4—2442no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—611no—
216C2×C4—1222no—
217C2×C4—1221no—
218C2×C4—4881no—
219C2×C4—4881no—
220C2×C4—2442no—
221C2×C4—2442no—
222C2×C4—2442no—
223C2×C4—2441no—
224C2×C4C22⋊C4×D6111yescharacteristic
225C8—22no—
226C8—82no—
227C8—1222no—
228C8—4882no—
229C22—1221no—
230C22—4882no—
231C22—2441no—
232C22—2442no—
233C22#99×D6111yescharacteristic
234C4—22no—
235C4—21no—
236C4—42no—
237C4—81no—
238C4—612no—
239C4—4881no—
240C4—2441no—
241C2—21no—
242C2—4882no—
243C2—2441no—
244C2#353×D6111yescharacteristic
subgroupG/H
centerC2#353×D61
commutatorC42×C61C22×C4
FrattiniC4⋊Q8C22×D61
FittingC61×#1411C2
socleC2×C61C2×#353
prime radical#16840313C22
cosocleC4⋊Q8×C61C23
hypercenter#1411D61
hypocenterC61#13592
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #1411

Complements

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

NH ≅ G/Ntype
3. #21443988C2semidirect
4. #21443987C2semidirect
5. C61×#1411C2semidirect
123. #1411D61semidirect
141. C61#13592semidirect

Maximal subgroups

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

subgroupindex
#214562772
#214475032
#214439882
#214439872
C61×#14112
#1359261