Finite Group Databasebeta

← #30664446

Subgroups — #30664446

Order 16768.

238 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 63 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#22553848C211yesmaximal, characteristic
2#22553362C211yesmaximal, characteristic, complemented
3#22553157C211yesmaximal, characteristic, complemented
4#22553049C211yesmaximal, characteristic, complemented
5#22553047C211yesmaximal, characteristic
6#22553045C211yesmaximal, characteristic
7#328×C131C211yesmaximal, characteristic, complemented
8#16862241—21no—
9#16862237C411yescharacteristic, complemented
10#16862235C411yescharacteristic, complemented
11#16862230—21no—
12#16862220C2211yescharacteristic
13Q8×Dic131C411yescharacteristic, complemented
14#16862208C411yescharacteristic, complemented
15(C22⋊D131)⋊C4—21no—
16#16862204—21no—
17C2×C4×Dic131—21no—
18C2×C4×Dic131C2211yescharacteristic
19Dic131⋊C8—21no—
20C22⋊(C131⋊C8)C2211yescharacteristic
21C8×Dic131—21no—
22C131⋊#95—21no—
23C131⋊#95C2211yescharacteristic
24C22⋊Q8×C131C2211yescharacteristic
25C22⋊C8×C131C2211yescharacteristic
26#95×C131C2211yescharacteristic
27C2×D524C2×C411yescharacteristic
28C2×C22⋊D131—21no—
29C22×Dic131—21no—
30C22×Dic131D411yescharacteristic
31C22×Dic131D411yescharacteristic
32C2×C4×D131—21no—
33C2×C131⋊C8—21no—
34C4⋊Dic131—21no—
35C4⋊Dic131—21no—
36C4⋊Dic131C2×C411yescharacteristic
37Dic131⋊C4—21no—
38(C2×D131)⋊C4—21no—
39(C2×D131)⋊C4—21no—
40(C2×D131)⋊C4—21no—
41(C2×D131)⋊C4—21no—
42C4×Dic131—24no—
43C4×Dic131—21no—
44C4×Dic131C2×C411yescharacteristic
45C4×Dic131C2×C411yescharacteristic
46C4×Dic131D411yescharacteristic
47C4×Dic131D411yescharacteristic
48C2×Q8×C131C2×C411yescharacteristic
49C22×C4×C131—21no—
50C22×C4×C131C2311yescharacteristic
51C2×C8×C131—21no—
52C4⋊C4×C131—21no—
53C4⋊C4×C131C2×C411yescharacteristic
54C22⋊C4×C131—21no—
55C22×D131—21no—
56D524—41no—
57C22⋊D131—41no—
58C22⋊D131—41no—
59C2×Dic131—21no—
60C2×Dic131—21no—
61C2×Dic131—22no—
62C2×Dic131—22no—
63C2×Dic131—21no—
64C2×Dic131—41no—
65C2×Dic131—41no—
66C2×Dic131C22⋊C411yescharacteristic
67C2×Dic131C22⋊C411yescharacteristic
68C2×Dic131C22⋊C411yescharacteristic
69C2×Dic131C22⋊C411yescharacteristic
70C4×D131—42no—
71C131⋊C8—41no—
72C23×C131C2×D411yescharacteristic
73Q8×C131—41no—
74C2×C4×C131—22no—
75C2×C4×C131—21no—
76C2×C4×C131—21no—
77C2×C4×C131—41no—
78C2×C4×C131—41no—
79C2×C4×C131—41no—
80C2×C4×C131C22×C411yescharacteristic
81C2×C4×C131C2×D411yescharacteristic
82C8×C131—41no—
83C22×C131—21no—
84C22×C131—21no—
85C22×C131C2×C22⋊C411yescharacteristic
86C22×C131#13511yescharacteristic
87C22×C131#13511yescharacteristic
88C2×D131—41no—
89C2×D131—41no—
90C2×D131—41no—
91C4×C131—21no—
92C4×C131—21no—
93C4×C131—42no—
94C4×C131—41no—
95C4×C131—41no—
96Dic131—21no—
97Dic131—21no—
98Dic131—21no—
99Dic131—42no—
100Dic131—41no—
101Dic131#10412yes—
102C2×C131—21no—
103C2×C131—21no—
104C2×C131#41011yescharacteristic
105C2×C131#41611yescharacteristic
106C2×C131C2×#10411yescharacteristic
107D131—81no—
108C131#150011yescharacteristic, complemented, Sylow
109#1500—1311nomaximal, Sylow
110#529—1311no—
111#431—1311no—
112#376—1311no—
113#328—1311no—
114#328D13111yescharacteristic, complemented
115#327—1311no—
116#327—1311no—
117#126—2621no—
118#125—1311no—
119#124—1311no—
120#123—2621no—
121C22⋊Q8C2×D13111yescharacteristic
122#121—1311no—
123C4×Q8—1311no—
124C4×D4—2621no—
125C4×D4—1311no—
126#117—2621no—
127C2×C42—2621no—
128C2×C42—1311no—
129C4⋊C8—2621no—
130C22⋊C8—1311no—
131C22⋊C8C2×D13111yescharacteristic
132C4×C8—2621no—
133#95—2621no—
134#95—1311no—
135#95C2×D13111yescharacteristic
136C2×Q8C4×D13111yescharacteristic
137C2×D4—2621no—
138C2×D4—1311no—
139C22×C4—21no—
140C22×C4—2621no—
141C22×C4—2621no—
142C22×C4—1311no—
143C22×C4—1311no—
144C22×C4C22×D13111yescharacteristic
145C2×C8—21no—
146C2×C8—2621no—
147C4⋊C4—21no—
148C4⋊C4—2621no—
149C4⋊C4—2621no—
150C4⋊C4—2621no—
151C4⋊C4—1311no—
152C4⋊C4C4×D13111yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—2621no—
155C22⋊C4—2621no—
156C22⋊C4—2621no—
157C22⋊C4—2621no—
158C42—2624no—
159C42—2621no—
160C42—1311no—
161C42—1311no—
162C42—1311no—
163C42—1311no—
164C23—2621no—
165C23D4×D13111yescharacteristic
166Q8—41no—
167D4—5241no—
168D4—5241no—
169D4—5241no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—2621no—
177C2×C4—2621no—
178C2×C4—2622no—
179C2×C4—2622no—
180C2×C4—2621no—
181C2×C4—5242no—
182C2×C4—5241no—
183C2×C4—5241no—
184C2×C4—1311no—
185C2×C4—1311no—
186C2×C4—1311no—
187C2×C4—1311no—
188C2×C4C2×C4×D13111yescharacteristic
189C2×C4D4×D13111yescharacteristic
190C8—41no—
191C8—5241no—
192C22—21no—
193C22—21no—
194C22—5241no—
195C22—5241no—
196C22—5241no—
197C22C22⋊C4×D13111yescharacteristic
198C22#1686226711yescharacteristic
199C22#1686226911yescharacteristic
200C4—21no—
201C4—21no—
202C4—42no—
203C4—41no—
204C4—41no—
205C4—2621no—
206C4—2621no—
207C4—2621no—
208C4—5242no—
209C4—5241no—
210C4—1312no—
211C2—21no—
212C2—21no—
213C2—10481no—
214C2#2255328411yescharacteristic
215C2#2255330711yescharacteristic
216C2#104×D13111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D131
commutatorC2×C4×C131C22×C4
FrattiniC22×C4C22×D131
Fitting#328×C131C2
socleC22×C131C2×C22⋊C4
prime radical#16862220C22
cosocleC22×C4×C131C23
hypercenter#328D131
hypocenterC131#1500
LCSG ▷ C2×C4×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #22553362C2semidirect
3. #22553157C2semidirect
4. #22553049C2semidirect
7. #328×C131C2semidirect
9. #16862237C4semidirect
10. #16862235C4semidirect
13. Q8×Dic131C4semidirect
14. #16862208C4semidirect
108. C131#1500semidirect
114. #328D131semidirect

Maximal subgroups

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

subgroupindex
#225538482
#225533622
#225531572
#225530492
#225530472
#225530452
#328×C1312
#1500131