Finite Group Databasebeta

← #30669291

Subgroups — #30669291

Order 16768.

864 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C22×#14523512C211yesmaximal, characteristic
2C2×#16862228C212yesmaximal, complemented
3C2×C4⋊C4×D131C211yesmaximal, characteristic, complemented
4#22553233C211yesmaximal, characteristic, complemented
5#22553232C211yesmaximal, characteristic, complemented
6#22553226C211yesmaximal, characteristic, complemented
7#22553188C212yesmaximal, complemented
8#22553187C212yesmaximal, complemented
9#22553184C212yesmaximal, complemented
10#22553174C211yesmaximal, characteristic, complemented
11#394×C131C211yesmaximal, characteristic, complemented
12C2×#14523512—28no—
13C2×#14523512—24no—
14C22×Dic262C2211yescharacteristic
15C22×C22⋊D131C2212yes—
16C23×Dic131C2212yes—
17C22×C4×D131C2211yescharacteristic
18C22×C4×D131C2211yescharacteristic
19C22×C4×D131C2211yescharacteristic
20#16862228—28no—
21C2×C4⋊Dic131C2211yescharacteristic
22C2×C4⋊Dic131C2212yes—
23C2×Dic131⋊C4C2211yescharacteristic
24C2×Dic131⋊C4C2212yes—
25C2×Dic131⋊C4C2211yescharacteristic
26C4⋊C4×D131—24no—
27C2×C22⋊Dic131C2212yes—
28C2×C22⋊Dic131C2212yes—
29C2×(C2×D131)⋊C4C2212yes—
30C131⋊#95C2212yes—
31C131⋊#95C2211yescharacteristic
32C131⋊#95C2212yes—
33C131⋊#95C2212yes—
34C22×D4×C131C2211yescharacteristic
35C2×C4⋊C4×C131C2211yescharacteristic
36C2×C22⋊C4×C131C2212yes—
37C2×C22⋊C4×C131C2212yes—
38#95×C131C2211yescharacteristic
39C23×D131C2311yescharacteristic
40#14523512—232no—
41C2×Dic262—24no—
42C2×Dic262—22no—
43C2×C22⋊D131—24no—
44C2×C22⋊D131—24no—
45C2×C22⋊D131D418yes—
46C22×Dic131—24no—
47C22×Dic131—24no—
48C22×Dic131—24no—
49C22×Dic131C2311yescharacteristic
50C22×Dic131C2312yes—
51C22×Dic131C2311yescharacteristic
52C22×Dic131C2312yes—
53C22×Dic131C2311yescharacteristic
54C2×C4×D131—24no—
55C2×C4×D131—22no—
56C2×C4×D131—22no—
57C2×C4×D131—24no—
58C2×C4×D131—22no—
59C2×C4×D131—22no—
60C2×C4×D131D414yes—
61C4⋊Dic131—24no—
62C4⋊Dic131—22no—
63Dic131⋊C4—24no—
64Dic131⋊C4—22no—
65Dic131⋊C4—22no—
66C22⋊Dic131—24no—
67C22⋊Dic131—24no—
68(C2×D131)⋊C4—24no—
69C24×C131C2312yes—
70C2×D4×C131—24no—
71C2×D4×C131—22no—
72C22×C4×C131C2311yescharacteristic
73C22×C4×C131C2312yes—
74C22×C4×C131C2311yescharacteristic
75C22×C4×C131C2311yescharacteristic
76C4⋊C4×C131—22no—
77C22⋊C4×C131—24no—
78C22⋊C4×C131—24no—
79C22×D131—21no—
80C22×D131—21no—
81C22×D131—22no—
82C22×D131C2×D414yes—
83C22×D131C2×D412yes—
84Dic262—28no—
85C22⋊D131—216no—
86C2×Dic131—216no—
87C2×Dic131—21no—
88C2×Dic131—22no—
89C2×Dic131—22no—
90C2×Dic131—22no—
91C2×Dic131—21no—
92C2×Dic131—22no—
93C2×Dic131—22no—
94C2×Dic131—21no—
95C2×Dic131—22no—
96C2×Dic131—22no—
97C2×Dic131C2×D412yes—
98C2×Dic131C2×D414yes—
99C2×Dic131#4112yescomplemented
100C4×D131—28no—
101C4×D131—28no—
102C4×D131—44no—
103C23×C131—22no—
104C23×C131—22no—
105C23×C131—22no—
106C23×C131—22no—
107C23×C131—22no—
108C23×C131—22no—
109C23×C131C2×D414yes—
110C23×C131C2411yescharacteristic
111D4×C131—28no—
112C2×C4×C131—21no—
113C2×C4×C131—22no—
114C2×C4×C131—21no—
115C2×C4×C131—22no—
116C2×C4×C131—22no—
117C2×C4×C131—21no—
118C2×C4×C131—22no—
119C2×C4×C131—22no—
120C2×C4×C131C2×D412yes—
121C2×C4×C131#4112yescomplemented
122C22×C131—24no—
123C22×C131—24no—
124C22×C131—24no—
125C22×C131—42no—
126C22×C131—42no—
127C22×C131—42no—
128C22×C131—42no—
129C22×C131C22×D411yescharacteristic
130C22×C131C22×D412yes—
131C22×C131C2×#4111yescharacteristic
132C22×C131C2×#4111yescharacteristic
133C22×C1312−1+412yes—
134C2×D131—22no—
135C2×D131—22no—
136C2×D131—24no—
137C2×D131—24no—
138C2×D131#12014yes—
139C4×C131—22no—
140C4×C131—22no—
141C4×C131—41no—
142C4×C131—42no—
143Dic131—22no—
144Dic131—24no—
145Dic131—22no—
146Dic131—41no—
147Dic131—42no—
148C2×C131—44no—
149C2×C131C2×#12011yescharacteristic
150C2×C131#52511yescharacteristic
151C2×C131#53611yescharacteristic
152C2×C131#54712yes—
153C2×C131#54712yes—
154D131—24no—
155C131#240611yescharacteristic, complemented, Sylow
156#2406—1311nomaximal, Sylow
157C22×#41—1311no—
158C2×#123—1312no—
159C22×C4⋊C4—1311no—
160#395—1311no—
161#395—1311no—
162#394—1311no—
163#394D13111yescharacteristic, complemented
164#386—1312no—
165#385—1312no—
166#385—1312no—
167#381—1311no—
168C2×#41—2628no—
169C2×#41—2624no—
170C22×Q8—1311no—
171C22×D4—1312no—
172C22×D4C2×D13111yescharacteristic
173C23×C4—1312no—
174C23×C4—1311no—
175C23×C4—1311no—
176C23×C4—1311no—
177#123—2628no—
178C2×C4⋊C4—2624no—
179C2×C4⋊C4—1311no—
180C2×C4⋊C4—1311no—
181C2×C4⋊C4—1311no—
182C2×C4⋊C4—1312no—
183C2×C4⋊C4—1312no—
184C2×C4⋊C4C2×D13111yescharacteristic
185C2×C22⋊C4—1312no—
186C2×C22⋊C4—1312no—
187C2×C22⋊C4—1312no—
188C2×C22⋊C4C2×D13112yes—
189C2×C22⋊C4C2×D13112yes—
190#95—1312no—
191#95—1311no—
192#95—1312no—
193#95—1312no—
194#95C2×D13111yescharacteristic
195C24—1311no—
196C24C22×D13112yes—
197#41—26232no—
198C2×Q8—2624no—
199C2×Q8—2622no—
200C2×D4—24no—
201C2×D4—22no—
202C2×D4—2624no—
203C2×D4—2624no—
204C2×D4—1318no—
205C22×C4—2622no—
206C22×C4—2624no—
207C22×C4—2622no—
208C22×C4—2622no—
209C22×C4—2624no—
210C22×C4—2624no—
211C22×C4—2624no—
212C22×C4—2624no—
213C22×C4—2622no—
214C22×C4—1312no—
215C22×C4—1311no—
216C22×C4—1311no—
217C22×C4—1311no—
218C22×C4—1312no—
219C22×C4—1314no—
220C22×C4C22×D13112yes—
221C22×C4C22×D13111yescharacteristic
222C22×C4C22×D13111yescharacteristic
223C22×C4C22×D13111yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—2622no—
226C4⋊C4—2624no—
227C4⋊C4—2622no—
228C4⋊C4—2624no—
229C4⋊C4—2622no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—2624no—
233C22⋊C4—2624no—
234C22⋊C4—2624no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—2621no—
242C23—2621no—
243C23—2622no—
244C23—1314no—
245C23—1312no—
246C23D4×D13114yes—
247C23C23×D13111yescharacteristic
248Q8—2628no—
249D4—28no—
250D4—26216no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—2628no—
260C2×C4—2628no—
261C2×C4—26216no—
262C2×C4—2621no—
263C2×C4—2622no—
264C2×C4—2622no—
265C2×C4—2622no—
266C2×C4—2621no—
267C2×C4—2622no—
268C2×C4—2622no—
269C2×C4—2621no—
270C2×C4—2622no—
271C2×C4—2622no—
272C2×C4—5244no—
273C2×C4—1312no—
274C2×C4—1314no—
275C2×C4—1312no—
276C2×C4D4×D13112yes—
277C2×C4#1452351212yescomplemented
278C22—24no—
279C22—24no—
280C22—24no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—2622no—
286C22—2624no—
287C22—2624no—
288C22—2622no—
289C22—1314no—
290C22C2×D4×D13111yescharacteristic
291C22C2×D4×D13112yes—
292C22#41×D13111yescharacteristic
293C22C2×#1452351211yescharacteristic
294C22#1686229312yes—
295C4—22no—
296C4—22no—
297C4—42no—
298C4—41no—
299C4—2622no—
300C4—2624no—
301C4—2622no—
302C4—5242no—
303C4—5241no—
304C2—44no—
305C2—2624no—
306C2#120×D13111yescharacteristic
307C2#2255382211yescharacteristic
308C2#2255388211yescharacteristic
309C2#2255396012yes—
310C2#2255396212yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#394×C131C2
prime radicalC22×#14523512C2
hypercenter#394D131
hypocenterC131#2406
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#16862228C2semidirect
3. C2×C4⋊C4×D131C2semidirect
4. #22553233C2semidirect
5. #22553232C2semidirect
6. #22553226C2semidirect
7. #22553188C2semidirect
8. #22553187C2semidirect
9. #22553184C2semidirect
10. #22553174C2semidirect
11. #394×C131C2semidirect
99. C2×Dic131#41semidirect
121. C2×C4×C131#41semidirect
155. C131#2406semidirect
163. #394D131semidirect
277. C2×C4#14523512semidirect

Maximal subgroups

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

subgroupindex
C22×#145235122
C2×#168622282
C2×C4⋊C4×D1312
#225532332
#225532322
#225532262
#225531882
#225531872
#225531842
#225531742
#394×C1312
#2406131