Finite Group Databasebeta

← #29600374

Subgroups — #29600374

Order 15616.

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

normal subgroups only

This group has 312 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#21456198C211yesmaximal, characteristic
2#356×D61C212yesmaximal
3#21443992C211yesmaximal, characteristic, complemented
4#21443991C212yesmaximal, complemented
5C61×#1413C211yesmaximal, characteristic, complemented
6#16840318C411yescharacteristic
7C4⋊Q8×D61C2211yescharacteristic
8#16840132C411yescharacteristic
9#16840131C411yescharacteristic
10#16840130C2211yescharacteristic
11#16840111—22no—
12#16839769—22no—
13#101×D61—22no—
14#16839714C411yescharacteristic
15#16839561C2212yes—
16#16839560—22no—
17C61×#491C2211yescharacteristic
18C61×#356C2212yes—
19C2×Q8×D61D412yes—
20C2×#11758189—21no—
21M16×D61—42no—
22#14519777C2×C411yescharacteristic
23#14519776—21no—
24#14519776—41no—
25#14519775D412yes—
26#14519769C2×C411yescharacteristic
27#14519755—41no—
28#14519742—41no—
29#14519735—21no—
30C4⋊C4×D61—41no—
31C42×D61C2×C411yescharacteristic
32#14519702—22no—
33#14519701—22no—
34#14519700—22no—
35#14519696—22no—
36#14519695—42no—
37#14519688C2×C411yescharacteristic
38C8×Dic61—21no—
39C4⋊Q8×C61C2311yescharacteristic
40#125×C61C2×C411yescharacteristic
41#103×C61—22no—
42#101×C61—22no—
43#97×C61C2×C411yescharacteristic
44C2×Q8×C61C2×D412yes—
45Q8×D61—42no—
46Q8×D61—42no—
47C2×C4×D61—22no—
48C2×C4×D61—21no—
49C2×C4×D61C22⋊C411yescharacteristic
50C2×Dic122—22no—
51C2×Dic122—41no—
52C2×Dic122C22⋊C412yes—
53M16×C61—42no—
54C2×C8×C61—21no—
55C4⋊C4×C61—21no—
56C4⋊C4×C61—41no—
57C4⋊C4×C61—41no—
58C42×C61C22×C411yescharacteristic
59#11758198—41no—
60C4⋊Dic61—21no—
61C4⋊Dic61—41no—
62#11758196—42no—
63#11758196—41no—
64#11758196—41no—
65C4×Dic61—21no—
66C4×Dic61—41no—
67C4×Dic61C22⋊C411yescharacteristic
68#11758194—42no—
69C2×#92418—21no—
70#11758189—42no—
71#11758189—82no—
72C8×D61—82no—
73C22×D61#9911yescharacteristic
74Q8×C61—42no—
75C2×C4×C61—22no—
76C2×C4×C61—21no—
77C2×C4×C61—41no—
78C2×C4×C61C2×C22⋊C411yescharacteristic
79C2×Dic61—21no—
80C2×Dic61—22no—
81C2×Dic61—41no—
82C2×Dic61#9911yescharacteristic
83C4×D61—22no—
84C4×D61—42no—
85C4×D61—42no—
86C4×D61—42no—
87Dic122—42no—
88Dic122—82no—
89Dic122—82no—
90C8×C61—41no—
91C8×C61—82no—
92#92418—41no—
93#92418—82no—
94#13599—611nomaximal, Sylow
95C22×C61C2×#9911yescharacteristic
96C2×D61—41no—
97C2×D61#35412yes—
98C4×C61—21no—
99C4×C61—41no—
100C4×C61—42no—
101C4×C61—81no—
102Dic61—22no—
103Dic61—21no—
104Dic61—42no—
105Dic61—81no—
106#3139—611no—
107C2×#356—612no—
108#1413—611no—
109#1413—612no—
110#1413D6111yescharacteristic, complemented
111C2×C61—21no—
112C2×C61C2×#35411yescharacteristic
113D61—22no—
114#533—611no—
115C2×C4⋊Q8—611no—
116#491—611no—
117#491—611no—
118#491—611no—
119#491C2×D6111yescharacteristic
120#487—1222no—
121#419—1222no—
122C2×#101—1222no—
123C4×M16—611no—
124#356—612no—
125#356—1222no—
126#356C2×D6112yes—
127C61#1359911yescharacteristic, complemented, Sylow
128C22×Q8—612no—
129C2×M16—1221no—
130C2×M16—2442no—
131C4⋊Q8—612no—
132C4⋊Q8—611no—
133C4⋊Q8—1221no—
134C4⋊Q8—2441no—
135C4⋊Q8C22×D6111yescharacteristic
136#125—611no—
137#125C4×D6111yescharacteristic
138C22⋊Q8—2441no—
139C4×Q8—2441no—
140#117—1221no—
141C2×C4⋊C4—2441no—
142C2×C42—611no—
143#103—22no—
144#103—1222no—
145#103—1222no—
146#103—1222no—
147#101—22no—
148#101—1222no—
149#101—2442no—
150#97—611no—
151#97C4×D6111yescharacteristic
152C4×C8—1221no—
153C2×Q8—612no—
154C2×Q8—1222no—
155C2×Q8—2441no—
156C2×Q8—2442no—
157C2×Q8—2442no—
158C2×Q8D4×D6112yes—
159C22×C4—611no—
160C22×C4—1221no—
161C22×C4—1222no—
162M16—42no—
163M16—4882no—
164M16—2442no—
165M16—2442no—
166C2×C8—21no—
167C2×C8—1221no—
168C2×C8—4882no—
169C4⋊C4—21no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—1221no—
173C4⋊C4—2442no—
174C4⋊C4—2441no—
175C4⋊C4—2441no—
176C4⋊C4—2441no—
177C22⋊C4—2441no—
178C42—611no—
179C42—1221no—
180C42—2441no—
181C42C2×C4×D6111yescharacteristic
182C23—611no—
183Q8—42no—
184Q8—4882no—
185Q8—4882no—
186Q8—2442no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—611no—
191C2×C4—1222no—
192C2×C4—1221no—
193C2×C4—1222no—
194C2×C4—2442no—
195C2×C4—2442no—
196C2×C4—2442no—
197C2×C4—2441no—
198C2×C4C22⋊C4×D6111yescharacteristic
199C8—41no—
200C8—82no—
201C8—4882no—
202C8—2441no—
203C22—612no—
204C22—2441no—
205C22#99×D6111yescharacteristic
206C4—21no—
207C4—41no—
208C4—42no—
209C4—81no—
210C4—1222no—
211C4—1221no—
212C4—4881no—
213C4—2442no—
214C2—21no—
215C2—1222no—
216C2#354×D6111yescharacteristic
subgroupG/H
centerC2#354×D61
commutatorC42×C61C22×C4
FrattiniC4⋊Q8C22×D61
FittingC61×#1413C2
socleC2×C61C2×#354
prime radicalC22×D61#99
cosocleC4⋊Q8×C61C23
hypercenter#1413D61
hypocenterC61#13599
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #1413

Complements

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

NH ≅ G/Ntype
3. #21443992C2semidirect
4. #21443991C2semidirect
5. C61×#1413C2semidirect
110. #1413D61semidirect
127. C61#13599semidirect

Maximal subgroups

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

subgroupindex
#214561982
#356×D612
#214439922
#214439912
C61×#14132
#1359961