Finite Group Databasebeta

← #30670441

Subgroups — #30670441

Order 16768.

512 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#16862224C211yesmaximal, characteristic
2C2×C4⋊C4×D131C211yesmaximal, characteristic
3#22553252C211yesmaximal, characteristic, complemented
4#22553251C211yesmaximal, characteristic, complemented
5#22553250C211yesmaximal, characteristic, complemented
6#22553216C212yesmaximal, complemented
7#22553176C212yesmaximal, complemented
8#22553174C212yesmaximal
9#22553174C212yesmaximal
10#22553159C211yesmaximal, characteristic
11#400×C131C211yesmaximal, characteristic, complemented
12C22×Dic262C2211yescharacteristic
13C22×C4×D131C2212yes—
14C22×C4×D131C2211yescharacteristic
15#16862224—24no—
16C2×C4⋊Dic131C2211yescharacteristic
17C2×Dic131⋊C4C2211yescharacteristic
18C2×Dic131⋊C4C2212yes—
19C2×Dic131⋊C4C2211yescharacteristic
20C4⋊C4×D131—24no—
21C2×(C2×D131)⋊C4C2212yes—
22C2×(C2×D131)⋊C4C2211yescharacteristic
23C2×(C2×D131)⋊C4C2211yescharacteristic
24C2×C4×Dic131C2212yes—
25C131⋊#95C2211yescharacteristic
26C131⋊#95C2212yes—
27C131⋊#95C2212yes—
28C131⋊#95C2212yes—
29C131⋊#95C2212yes—
30C131⋊#95C2212yes—
31C131⋊#95C2212yes—
32C2×C4⋊C4×C131C2211yescharacteristic
33C2×C4⋊C4×C131C2211yescharacteristic
34#95×C131C2211yescharacteristic
35#95×C131C2212yes—
36#95×C131C2212yes—
37C23×D131C2311yescharacteristic
38C2×Dic262—22no—
39C2×Dic262—24no—
40C22×Dic131C2312yes—
41C22×Dic131C2311yescharacteristic
42C22×Dic131C2312yes—
43C22×Dic131C2311yescharacteristic
44C22×Dic131C2311yescharacteristic
45C2×C4×D131—24no—
46C2×C4×D131—24no—
47C2×C4×D131—28no—
48C2×C4×D131D414yes—
49C4⋊Dic131—22no—
50Dic131⋊C4—22no—
51Dic131⋊C4—24no—
52Dic131⋊C4—22no—
53(C2×D131)⋊C4—22no—
54(C2×D131)⋊C4—22no—
55(C2×D131)⋊C4—24no—
56C4×Dic131—24no—
57C22×C4×C131C2312yes—
58C22×C4×C131C2311yescharacteristic
59C22×C4×C131C2312yes—
60C22×C4×C131C2311yescharacteristic
61C22×C4×C131C2311yescharacteristic
62C4⋊C4×C131—22no—
63C4⋊C4×C131—22no—
64C22×D131—22no—
65C22×D131—21no—
66C22×D131—21no—
67C22×D131C2×D412yes—
68C22×D131#4114yes—
69Dic262—44no—
70C2×Dic131—21no—
71C2×Dic131—21no—
72C2×Dic131—22no—
73C2×Dic131—22no—
74C2×Dic131—22no—
75C2×Dic131—24no—
76C2×Dic131—22no—
77C2×Dic131—21no—
78C2×Dic131—21no—
79C2×Dic131—22no—
80C2×Dic131C2×D412yes—
81C2×Dic131#4114yes—
82C4×D131—28no—
83C4×D131—48no—
84C23×C131C2411yescharacteristic
85C2×C4×C131—22no—
86C2×C4×C131—21no—
87C2×C4×C131—21no—
88C2×C4×C131—22no—
89C2×C4×C131—22no—
90C2×C4×C131—22no—
91C2×C4×C131—24no—
92C2×C4×C131—22no—
93C2×C4×C131—21no—
94C2×C4×C131—22no—
95C2×C4×C131—21no—
96C2×C4×C131C2×D412yes—
97C22×C131C22×D411yescharacteristic
98C22×C131C2×#4112yes—
99C22×C131C2×#4112yes—
100C22×C1312−1+411yescharacteristic
101C22×C1312−1+411yescharacteristic
102C2×D131—22no—
103C2×D131—22no—
104C2×D131—24no—
105C2×D131—24no—
106C2×D131#12414yes—
107C4×C131—22no—
108C4×C131—41no—
109C4×C131—42no—
110C4×C131—41no—
111C4×C131—42no—
112Dic131—41no—
113Dic131—42no—
114Dic131—41no—
115Dic131—42no—
116Dic131#12314yes—
117C2×C131C2×#12311yescharacteristic
118C2×C131C2×#12411yescharacteristic
119C2×C131#53611yescharacteristic
120C2×C131#55212yes—
121C2×C131#55212yes—
122D131—24no—
123C131#250311yescharacteristic, complemented, Sylow
124#2503—1311nomaximal, Sylow
125C2×C22⋊Q8—1311no—
126C22×C4⋊C4—1311no—
127#400—1311no—
128#400—1311no—
129#400—1311no—
130#400D13111yescharacteristic, complemented
131#391—1312no—
132#382—1312no—
133#381—1312no—
134#381—1312no—
135C4×C22⋊C4—1311no—
136C22×Q8—1311no—
137C23×C4—1312no—
138C23×C4—1311no—
139C22⋊Q8—2624no—
140C2×C4⋊C4—2624no—
141C2×C4⋊C4—1312no—
142C2×C4⋊C4—1311no—
143C2×C4⋊C4—1311no—
144C2×C4⋊C4—1311no—
145C2×C4⋊C4C2×D13111yescharacteristic
146C2×C4⋊C4C2×D13111yescharacteristic
147C2×C22⋊C4—1311no—
148C2×C22⋊C4—1311no—
149C2×C22⋊C4—1312no—
150C2×C42—1312no—
151#95—1311no—
152#95—1312no—
153#95—1312no—
154#95—1312no—
155#95—1312no—
156#95—1312no—
157#95—1312no—
158#95C2×D13111yescharacteristic
159#95C2×D13112yes—
160#95C2×D13112yes—
161C24—1311no—
162C2×Q8—2622no—
163C2×Q8—2624no—
164C22×C4—2624no—
165C22×C4—2624no—
166C22×C4—2628no—
167C22×C4—1312no—
168C22×C4—1311no—
169C22×C4—1311no—
170C22×C4—1312no—
171C22×C4—1314no—
172C22×C4—1311no—
173C22×C4C22×D13112yes—
174C22×C4C22×D13111yescharacteristic
175C22×C4C22×D13111yescharacteristic
176C22×C4C22×D13112yes—
177C22×C4C22×D13111yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—2624no—
181C4⋊C4—2622no—
182C4⋊C4—2622no—
183C4⋊C4—2622no—
184C22⋊C4—2624no—
185C22⋊C4—2622no—
186C22⋊C4—2622no—
187C42—2624no—
188C23—2622no—
189C23—2621no—
190C23—2621no—
191C23—1312no—
192C23—1314no—
193C23C23×D13111yescharacteristic
194Q8—5244no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—2628no—
207C2×C4—2621no—
208C2×C4—2621no—
209C2×C4—2622no—
210C2×C4—2622no—
211C2×C4—2622no—
212C2×C4—2624no—
213C2×C4—2622no—
214C2×C4—2621no—
215C2×C4—2621no—
216C2×C4—2622no—
217C2×C4—5248no—
218C2×C4—1312no—
219C2×C4—1314no—
220C2×C4D4×D13112yes—
221C22—2622no—
222C22—2624no—
223C22—2624no—
224C22—2622no—
225C22—1314no—
226C22C2×D4×D13111yescharacteristic
227C22#41×D13112yes—
228C22#41×D13112yes—
229C22#1686229311yescharacteristic
230C22#1686229411yescharacteristic
231C4—22no—
232C4—41no—
233C4—42no—
234C4—41no—
235C4—42no—
236C4—5241no—
237C4—5242no—
238C4—5241no—
239C4—5242no—
240C4—1314no—
241C2—2624no—
242C2#123×D13111yescharacteristic
243C2#124×D13111yescharacteristic
244C2#2255388511yescharacteristic
245C2#2255399412yes—
246C2#2255399812yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#400×C131C2
prime radicalC23×D131C23
hypercenter#400D131
hypocenterC131#2503
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #22553252C2semidirect
4. #22553251C2semidirect
5. #22553250C2semidirect
6. #22553216C2semidirect
7. #22553176C2semidirect
11. #400×C131C2semidirect
123. C131#2503semidirect
130. #400D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622242
C2×C4⋊C4×D1312
#225532522
#225532512
#225532502
#225532162
#225531762
#225531742
#225531592
#400×C1312
#2503131