Finite Group Databasebeta

← #30670271

Subgroups — #30670271

Order 16768.

584 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 151 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×(C22⋊D131)⋊C4C211yesmaximal, characteristic
2C2×C4⋊C4×D131C211yesmaximal, characteristic, complemented
3#22553252C211yesmaximal, characteristic, complemented
4#22553250C211yesmaximal, characteristic, complemented
5#22553248C211yesmaximal, characteristic, complemented
6#22553207C212yesmaximal, complemented
7#22553188C212yesmaximal
8#22553183C212yesmaximal, complemented
9#22553171C211yesmaximal, characteristic, complemented
10#22553170C212yesmaximal, complemented
11#399×C131C211yesmaximal, characteristic, complemented
12C22×C22⋊D131C2211yescharacteristic
13C23×Dic131C2211yescharacteristic
14C22×C4×D131C2211yescharacteristic
15C22×C4×D131C2212yes—
16(C22⋊D131)⋊C4—24no—
17C2×C4⋊Dic131C2211yescharacteristic
18C2×Dic131⋊C4C2212yes—
19C2×Dic131⋊C4C2212yes—
20C2×Dic131⋊C4C2211yescharacteristic
21C2×Dic131⋊C4C2212yes—
22C4⋊C4×D131—24no—
23C2×C22⋊Dic131C2212yes—
24C2×(C2×D131)⋊C4C2212yes—
25C2×(C2×D131)⋊C4C2211yescharacteristic
26C2×C4×Dic131C2211yescharacteristic
27C131⋊#95C2212yes—
28C131⋊#95C2212yes—
29C131⋊#95C2211yescharacteristic
30C131⋊#95C2212yes—
31C131⋊#95C2212yes—
32C2×C4⋊C4×C131C2211yescharacteristic
33C2×C4⋊C4×C131C2212yes—
34C2×C22⋊C4×C131C2211yescharacteristic
35C2×C22⋊C4×C131C2212yes—
36#95×C131C2211yescharacteristic
37C23×D131C2311yescharacteristic
38C2×C22⋊D131—22no—
39C2×C22⋊D131—22no—
40C2×C22⋊D131Q814yes—
41C22×Dic131—22no—
42C22×Dic131—24no—
43C22×Dic131C2312yes—
44C22×Dic131C2311yescharacteristic
45C22×Dic131C2312yes—
46C22×Dic131C2311yescharacteristic
47C22×Dic131C2311yescharacteristic
48C2×C4×D131—24no—
49C2×C4×D131—24no—
50C2×C4×D131—42no—
51C2×C4×D131—42no—
52C2×C4×D131D414yes—
53C4⋊Dic131—22no—
54Dic131⋊C4—24no—
55Dic131⋊C4—24no—
56Dic131⋊C4—22no—
57Dic131⋊C4—42no—
58C22⋊Dic131—42no—
59(C2×D131)⋊C4—22no—
60(C2×D131)⋊C4—24no—
61C4×Dic131—22no—
62C24×C131C2311yescharacteristic
63C22×C4×C131C2312yes—
64C22×C4×C131C2312yes—
65C22×C4×C131C2311yescharacteristic
66C22×C4×C131C2311yescharacteristic
67C4⋊C4×C131—22no—
68C4⋊C4×C131—42no—
69C22⋊C4×C131—22no—
70C22⋊C4×C131—42no—
71C22×D131—22no—
72C22×D131—22no—
73C22×D131C2×D412yes—
74C22×D131C2×Q812yes—
75C22×D131#4112yes—
76C22⋊D131—28no—
77C2×Dic131—21no—
78C2×Dic131—21no—
79C2×Dic131—21no—
80C2×Dic131—22no—
81C2×Dic131—22no—
82C2×Dic131—22no—
83C2×Dic131—22no—
84C2×Dic131—22no—
85C2×Dic131—22no—
86C2×Dic131—22no—
87C2×Dic131—44no—
88C2×Dic131C2×D412yes—
89C2×Dic131C2×Q812yes—
90C2×Dic131#4114yes—
91C4×D131—28no—
92C4×D131—48no—
93C23×C131—21no—
94C23×C131—22no—
95C23×C131—21no—
96C23×C131—22no—
97C23×C131C2×Q812yes—
98C23×C131C2411yescharacteristic
99C2×C4×C131—22no—
100C2×C4×C131—21no—
101C2×C4×C131—22no—
102C2×C4×C131—22no—
103C2×C4×C131—22no—
104C2×C4×C131—22no—
105C2×C4×C131—22no—
106C2×C4×C131—22no—
107C2×C4×C131—22no—
108C2×C4×C131C2×D412yes—
109C22×C131—22no—
110C22×C131—24no—
111C22×C131—41no—
112C22×C131—41no—
113C22×C131—42no—
114C22×C131C22×D411yescharacteristic
115C22×C131C22×Q811yescharacteristic
116C22×C131C2×#4111yescharacteristic
117C22×C131C2×#4112yes—
118C22×C131H(5,𝔽2)12yes—
119C2×D131—22no—
120C2×D131—22no—
121C2×D131—24no—
122C2×D131—24no—
123C2×D131C22⋊Q814yes—
124C4×C131—22no—
125C4×C131—41no—
126C4×C131—42no—
127C4×C131—42no—
128Dic131—22no—
129Dic131—41no—
130Dic131—42no—
131Dic131—42no—
132Dic131#12314yes—
133C2×C131—42no—
134C2×C131C2×C22⋊Q811yescharacteristic
135C2×C131C2×#12311yescharacteristic
136C2×C131#53511yescharacteristic
137C2×C131#55312yes—
138C2×C131#55412yes—
139D131—24no—
140C131#248711yescharacteristic, complemented, Sylow
141#2487—1311nomaximal, Sylow
142C2×C4×D4—1311no—
143C22×C4⋊C4—1311no—
144#400—1311no—
145#400—1311no—
146#399—1311no—
147#399D13111yescharacteristic, complemented
148#389—1312no—
149#386—1312no—
150#385—1312no—
151#380—1312no—
152#380—1311no—
153C22×D4—1311no—
154C23×C4—1311no—
155C23×C4—1311no—
156C23×C4—1312no—
157C4×D4—2624no—
158C2×C4⋊C4—2624no—
159C2×C4⋊C4—1312no—
160C2×C4⋊C4—1311no—
161C2×C4⋊C4—1311no—
162C2×C4⋊C4—1312no—
163C2×C4⋊C4—1312no—
164C2×C4⋊C4C2×D13111yescharacteristic
165C2×C4⋊C4C2×D13112yes—
166C2×C22⋊C4—1311no—
167C2×C22⋊C4—1312no—
168C2×C22⋊C4—1312no—
169C2×C22⋊C4C2×D13111yescharacteristic
170C2×C22⋊C4C2×D13112yes—
171C2×C42—1311no—
172#95—1312no—
173#95—1312no—
174#95—1311no—
175#95—1312no—
176#95—1312no—
177#95C2×D13111yescharacteristic
178C24—1311no—
179C24C22×D13111yescharacteristic
180C2×D4—2622no—
181C2×D4—2622no—
182C2×D4—1314no—
183C22×C4—2624no—
184C22×C4—2624no—
185C22×C4—2622no—
186C22×C4—2624no—
187C22×C4—5242no—
188C22×C4—5242no—
189C22×C4—1311no—
190C22×C4—1312no—
191C22×C4—1312no—
192C22×C4—1311no—
193C22×C4—1314no—
194C22×C4—1311no—
195C22×C4C22×D13112yes—
196C22×C4C22×D13112yes—
197C22×C4C22×D13111yescharacteristic
198C22×C4C22×D13111yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—42no—
201C4⋊C4—2624no—
202C4⋊C4—2624no—
203C4⋊C4—2622no—
204C4⋊C4—2622no—
205C4⋊C4—5242no—
206C22⋊C4—22no—
207C22⋊C4—42no—
208C22⋊C4—2624no—
209C22⋊C4—2622no—
210C22⋊C4—5242no—
211C42—2622no—
212C23—21no—
213C23—22no—
214C23—21no—
215C23—22no—
216C23—2622no—
217C23—2622no—
218C23—1312no—
219C23—1312no—
220C23—1312no—
221C23Q8×D13112yes—
222C23C23×D13111yescharacteristic
223D4—2628no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—2628no—
234C2×C4—2621no—
235C2×C4—2621no—
236C2×C4—2621no—
237C2×C4—2622no—
238C2×C4—2622no—
239C2×C4—2622no—
240C2×C4—2622no—
241C2×C4—2622no—
242C2×C4—2622no—
243C2×C4—2622no—
244C2×C4—5248no—
245C2×C4—5244no—
246C2×C4—1312no—
247C2×C4—1312no—
248C2×C4—1314no—
249C2×C4D4×D13112yes—
250C22—24no—
251C22—22no—
252C22—41no—
253C22—42no—
254C22—41no—
255C22—2622no—
256C22—2624no—
257C22—2624no—
258C22—2622no—
259C22—1314no—
260C22C2×D4×D13111yescharacteristic
261C22C2×Q8×D13111yescharacteristic
262C22#41×D13111yescharacteristic
263C22#41×D13112yes—
264C22#1686229112yes—
265C4—22no—
266C4—42no—
267C4—41no—
268C4—42no—
269C4—2622no—
270C4—5242no—
271C4—5242no—
272C4—5241no—
273C4—1314no—
274C2—42no—
275C2—2624no—
276C2C22⋊Q8×D13111yescharacteristic
277C2#123×D13111yescharacteristic
278C2#2255387511yescharacteristic
279C2#2255400712yes—
280C2#2255401612yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#399×C131C2
prime radicalC22×C22⋊D131C22
hypercenter#399D131
hypocenterC131#2487
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D131C2semidirect
3. #22553252C2semidirect
4. #22553250C2semidirect
5. #22553248C2semidirect
6. #22553207C2semidirect
8. #22553183C2semidirect
9. #22553171C2semidirect
10. #22553170C2semidirect
11. #399×C131C2semidirect
140. C131#2487semidirect
147. #399D131semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D131)⋊C42
C2×C4⋊C4×D1312
#225532522
#225532502
#225532482
#225532072
#225531882
#225531832
#225531712
#225531702
#399×C1312
#2487131