Finite Group Databasebeta

← #30669368

Subgroups — #30669368

Order 16768.

610 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16862229C211yesmaximal, characteristic
2C2×#16862229C211yesmaximal, characteristic
3C2×#16862205C211yesmaximal, characteristic, complemented
4#22553245C212yesmaximal, complemented
5#22553234C212yesmaximal, complemented
6#22553207C212yesmaximal, complemented
7#22553202C212yesmaximal, complemented
8#22553191C211yesmaximal, characteristic
9#22553185C211yesmaximal, characteristic, complemented
10#22553175C211yesmaximal, characteristic, complemented
11C22×C4⋊C4×C131C211yesmaximal, characteristic, complemented
12C22×C22⋊D131C2211yescharacteristic
13#16862229—24no—
14#16862229—24no—
15#16862205—24no—
16C2×C4⋊Dic131C2212yes—
17C2×Dic131⋊C4C2212yes—
18C2×Dic131⋊C4C2211yescharacteristic
19C2×Dic131⋊C4C2212yes—
20C2×Dic131⋊C4C2211yescharacteristic
21C2×C22⋊Dic131C2211yescharacteristic
22C2×C22⋊Dic131C2211yescharacteristic
23C2×C22⋊Dic131C2211yescharacteristic
24C2×(C2×D131)⋊C4C2212yes—
25C2×(C2×D131)⋊C4C2211yescharacteristic
26C2×(C2×D131)⋊C4C2212yes—
27C2×(C2×D131)⋊C4C2211yescharacteristic
28C2×C4×Dic131C2212yes—
29C131⋊#95C2212yes—
30C131⋊#95C2212yes—
31C131⋊#95C2212yes—
32C131⋊#95C2212yes—
33C23×C4×C131C2211yescharacteristic
34C23×C4×C131C2211yescharacteristic
35C23×C4×C131C2211yescharacteristic
36C2×C4⋊C4×C131—24no—
37C2×C4⋊C4×C131C2212yes—
38C2×C4⋊C4×C131C2212yes—
39C23×D131C2311yescharacteristic
40C2×C22⋊D131—22no—
41C2×C22⋊D131—22no—
42C2×C22⋊D131—22no—
43C22×Dic131C2311yescharacteristic
44C22×Dic131C2311yescharacteristic
45C22×Dic131C2312yes—
46C22×Dic131C2311yescharacteristic
47C22×Dic131C2312yes—
48C4⋊Dic131—24no—
49Dic131⋊C4—24no—
50Dic131⋊C4—22no—
51Dic131⋊C4—22no—
52Dic131⋊C4—24no—
53C22⋊Dic131—22no—
54C22⋊Dic131—22no—
55C22⋊Dic131D414yes—
56(C2×D131)⋊C4—24no—
57(C2×D131)⋊C4—22no—
58(C2×D131)⋊C4—22no—
59(C2×D131)⋊C4—24no—
60C4×Dic131—24no—
61C24×C131C2311yescharacteristic
62C22×C4×C131—22no—
63C22×C4×C131—22no—
64C22×C4×C131—22no—
65C22×C4×C131—24no—
66C22×C4×C131—22no—
67C22×C4×C131—22no—
68C22×C4×C131—22no—
69C22×C4×C131D414yes—
70C22×C4×C131C2312yes—
71C22×C4×C131C2311yescharacteristic
72C22×C4×C131C2311yescharacteristic
73C22×C4×C131C2312yes—
74C4⋊C4×C131—24no—
75C4⋊C4×C131—24no—
76C22×D131—21no—
77C22×D131—21no—
78C22×D131—22no—
79C22×D131—21no—
80C22×D131—22no—
81C22⋊D131—44no—
82C2×Dic131—22no—
83C2×Dic131—22no—
84C2×Dic131—22no—
85C2×Dic131—21no—
86C2×Dic131—21no—
87C2×Dic131—21no—
88C2×Dic131—21no—
89C2×Dic131—22no—
90C2×Dic131—22no—
91C2×Dic131—21no—
92C2×Dic131—22no—
93C2×Dic131—22no—
94C2×Dic131C2×D414yes—
95C23×C131—22no—
96C23×C131—22no—
97C23×C131C2×D412yes—
98C23×C131C2×D412yes—
99C23×C131#4112yes—
100C23×C131C2411yescharacteristic
101C2×C4×C131—28no—
102C2×C4×C131—28no—
103C2×C4×C131—22no—
104C2×C4×C131—21no—
105C2×C4×C131—21no—
106C2×C4×C131—21no—
107C2×C4×C131—21no—
108C2×C4×C131—22no—
109C2×C4×C131—22no—
110C2×C4×C131—22no—
111C2×C4×C131—22no—
112C2×C4×C131—44no—
113C2×C4×C131C2×D412yes—
114C2×C4×C131C2×D412yes—
115C2×C4×C131#4114yes—
116C22×C131—22no—
117C22×C131—22no—
118C22×C131—24no—
119C22×C131—24no—
120C22×C131#12114yes—
121C22×C131C22×D411yescharacteristic
122C22×C131C22×D411yescharacteristic
123C22×C131C2×#4111yescharacteristic
124C22×C131C2×#4112yes—
125C22×C1312−1+412yes—
126C2×D131—41no—
127C2×D131—41no—
128C2×D131—41no—
129C2×D131—42no—
130C2×D131—42no—
131C4×C131—22no—
132C4×C131—22no—
133C4×C131—24no—
134C4×C131—42no—
135Dic131—24no—
136Dic131—41no—
137Dic131—41no—
138Dic131—41no—
139Dic131—42no—
140C2×C131—24no—
141C2×C131C2×#12111yescharacteristic
142C2×C131#52511yescharacteristic
143C2×C131#53611yescharacteristic
144C2×C131#54712yes—
145C2×C131#55212yes—
146D131—81no—
147C131#241311yescharacteristic, complemented, Sylow
148#2413—1311nomaximal, Sylow
149C2×#123—1311no—
150C2×#123—1311no—
151C2×#117—1311no—
152C22×C4⋊C4D13111yescharacteristic, complemented
153#398—1312no—
154#396—1312no—
155#389—1312no—
156#388—1312no—
157#386—1311no—
158#385—1311no—
159#381—1311no—
160C22×D4—1311no—
161C23×C4C2×D13111yescharacteristic
162C23×C4C2×D13111yescharacteristic
163C23×C4C2×D13111yescharacteristic
164#123—2624no—
165#123—2624no—
166#117—2624no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—1312no—
169C2×C4⋊C4—1311no—
170C2×C4⋊C4—1312no—
171C2×C4⋊C4—1311no—
172C2×C4⋊C4—1312no—
173C2×C4⋊C4C2×D13112yes—
174C2×C4⋊C4C2×D13112yes—
175C2×C22⋊C4—1311no—
176C2×C22⋊C4—1311no—
177C2×C22⋊C4—1311no—
178C2×C22⋊C4—1312no—
179C2×C22⋊C4—1311no—
180C2×C22⋊C4—1312no—
181C2×C22⋊C4—1311no—
182C2×C42—1312no—
183#95—1312no—
184#95—1312no—
185#95—1312no—
186#95—1312no—
187C24—1311no—
188C24C22×D13111yescharacteristic
189C2×D4—2622no—
190C2×D4—2622no—
191C2×D4—2622no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—1311no—
200C22×C4—1312no—
201C22×C4—1311no—
202C22×C4—1311no—
203C22×C4—1312no—
204C22×C4C22⋊D13114yes—
205C22×C4C22×D13112yes—
206C22×C4C22×D13111yescharacteristic
207C22×C4C22×D13111yescharacteristic
208C22×C4C22×D13112yes—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—2624no—
212C4⋊C4—2624no—
213C4⋊C4—2622no—
214C4⋊C4—2622no—
215C4⋊C4—2624no—
216C22⋊C4—2624no—
217C22⋊C4—2622no—
218C22⋊C4—2622no—
219C22⋊C4—2622no—
220C22⋊C4—2622no—
221C22⋊C4—2624no—
222C22⋊C4—1314no—
223C42—2624no—
224C23—22no—
225C23—22no—
226C23—2621no—
227C23—2621no—
228C23—2621no—
229C23—2622no—
230C23—2622no—
231C23D4×D13112yes—
232C23C2×C22⋊D13112yes—
233C23#1452351212yes—
234C23C23×D13111yescharacteristic
235D4—5244no—
236C2×C4—28no—
237C2×C4—28no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—44no—
248C2×C4—2622no—
249C2×C4—2622no—
250C2×C4—2622no—
251C2×C4—2621no—
252C2×C4—2621no—
253C2×C4—2621no—
254C2×C4—2621no—
255C2×C4—2621no—
256C2×C4—2622no—
257C2×C4—2622no—
258C2×C4—2622no—
259C2×C4—2622no—
260C2×C4—1314no—
261C2×C4C2×C22⋊D13112yes—
262C2×C4C2×C22⋊D13112yes—
263C2×C4#1452351314yes—
264C22—22no—
265C22—24no—
266C22—24no—
267C22—22no—
268C22—5241no—
269C22—5242no—
270C22—5242no—
271C22—5241no—
272C22—5241no—
273C22#1686221814yes—
274C22C2×D4×D13111yescharacteristic
275C22C22×C22⋊D13111yescharacteristic
276C22C2×#1452351211yescharacteristic
277C22C2×#1452351312yes—
278C22#1686229412yes—
279C4—22no—
280C4—22no—
281C4—24no—
282C4—42no—
283C4—2624no—
284C4—5241no—
285C4—5242no—
286C4—5241no—
287C4—5241no—
288C2—24no—
289C2—10481no—
290C2C2×#1686221811yescharacteristic
291C2#2255382311yescharacteristic
292C2#2255388611yescharacteristic
293C2#2255396312yes—
294C2#2255399712yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC22×C4⋊C4×C131C2
prime radicalC22×C22⋊D131C22
hypercenterC22×C4⋊C4D131
hypocenterC131#2413
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#16862205C2semidirect
4. #22553245C2semidirect
5. #22553234C2semidirect
6. #22553207C2semidirect
7. #22553202C2semidirect
9. #22553185C2semidirect
10. #22553175C2semidirect
11. C22×C4⋊C4×C131C2semidirect
147. C131#2413semidirect
152. C22×C4⋊C4D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622292
C2×#168622052
#225532452
#225532342
#225532072
#225532022
#225531912
#225531852
#225531752
C22×C4⋊C4×C1312
#2413131