Finite Group Databasebeta

← #30670959

Subgroups — #30670959

Order 16768.

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

normal subgroups only

This group has 676 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×#16862242C211yesmaximal, characteristic
2C2×#16862232C212yesmaximal, complemented
3C2×#16862211C211yesmaximal, characteristic, complemented
4#22553230C211yesmaximal, characteristic, complemented
5#22553229C211yesmaximal, characteristic, complemented
6#22553226C211yesmaximal, characteristic, complemented
7#22553212C212yesmaximal, complemented
8#22553212C212yesmaximal, complemented
9#22553212C212yesmaximal, complemented
10#22553163C211yesmaximal, characteristic, complemented
11#394×C131C211yesmaximal, characteristic, complemented
12C22×Dic262C2211yescharacteristic
13C22×D524C2211yescharacteristic
14C22×C22⋊D131C2212yes—
15C22×C22⋊D131C2212yes—
16#16862242—24no—
17#16862232—28no—
18#16862211—24no—
19C2×Dic131⋊C4C2212yes—
20C2×Dic131⋊C4C2211yescharacteristic
21C2×Dic131⋊C4C2211yescharacteristic
22C2×C22⋊Dic131C2212yes—
23C2×C22⋊Dic131C2212yes—
24C2×(C2×D131)⋊C4C2212yes—
25C2×(C2×D131)⋊C4C2212yes—
26C2×(C2×D131)⋊C4C2212yes—
27C2×(C2×D131)⋊C4C2212yes—
28C2×C4×Dic131C2212yes—
29C2×C4×Dic131C2211yescharacteristic
30C2×C4×Dic131C2211yescharacteristic
31C2×C4×Dic131C2211yescharacteristic
32C131⋊#95C2211yescharacteristic
33C22×D4×C131C2211yescharacteristic
34C2×C4⋊C4×C131C2211yescharacteristic
35C2×C22⋊C4×C131C2212yes—
36C2×C22⋊C4×C131C2212yes—
37#95×C131C2211yescharacteristic
38C23×D131C2312yes—
39C2×Dic262—22no—
40C2×Dic262—22no—
41C2×Dic262D414yescomplemented
42C2×D524—22no—
43C2×D524—42no—
44C2×C22⋊D131—24no—
45C2×C22⋊D131—24no—
46C2×C22⋊D131—24no—
47C2×C22⋊D131—24no—
48C2×C22⋊D131—24no—
49C2×C22⋊D131—24no—
50C22×Dic131C2312yes—
51C22×Dic131C2311yescharacteristic
52C22×Dic131C2311yescharacteristic
53C22×Dic131C2311yescharacteristic
54C22×Dic131C2311yescharacteristic
55Dic131⋊C4—22no—
56Dic131⋊C4—24no—
57Dic131⋊C4—22no—
58C22⋊Dic131—24no—
59C22⋊Dic131—42no—
60(C2×D131)⋊C4—24no—
61(C2×D131)⋊C4—24no—
62(C2×D131)⋊C4—42no—
63(C2×D131)⋊C4—42no—
64C4×Dic131—22no—
65C4×Dic131—22no—
66C4×Dic131—24no—
67C4×Dic131D414yes—
68C24×C131C2312yes—
69C2×D4×C131—22no—
70C2×D4×C131—42no—
71C22×C4×C131C2312yes—
72C22×C4×C131C2311yescharacteristic
73C22×C4×C131C2311yescharacteristic
74C22×C4×C131C2311yescharacteristic
75C4⋊C4×C131—22no—
76C22⋊C4×C131—24no—
77C22⋊C4×C131—42no—
78C22×D131—22no—
79C22×D131—22no—
80C22×D131—22no—
81C22×D131—22no—
82C22×D131—22no—
83C22×D131—22no—
84C22×D131—22no—
85Dic262—28no—
86D524—44no—
87C22⋊D131—48no—
88C22⋊D131—48no—
89C2×Dic131—21no—
90C2×Dic131—21no—
91C2×Dic131—22no—
92C2×Dic131—21no—
93C2×Dic131—22no—
94C2×Dic131—22no—
95C2×Dic131—22no—
96C2×Dic131C2×D412yes—
97C2×Dic131C2×D412yes—
98C2×Dic131C2×D412yes—
99C2×Dic131C2×D412yes—
100C2×Dic131#4112yes—
101C2×Dic131#4114yes—
102C23×C131—22no—
103C23×C131—22no—
104C23×C131—22no—
105C23×C131—22no—
106C23×C131—22no—
107C23×C131—22no—
108C23×C131—22no—
109C23×C131C2411yescharacteristic
110D4×C131—44no—
111C2×C4×C131—21no—
112C2×C4×C131—21no—
113C2×C4×C131—22no—
114C2×C4×C131—22no—
115C2×C4×C131—21no—
116C2×C4×C131—22no—
117C2×C4×C131—22no—
118C2×C4×C131—22no—
119C2×C4×C131C2×D412yes—
120C2×C4×C131C2×D412yes—
121C22×C131—42no—
122C22×C131—42no—
123C22×C131—42no—
124C22×C131—42no—
125C22×C131—42no—
126C22×C131—42no—
127C22×C131—42no—
128C22×C131C22×D411yescharacteristic
129C22×C131C22×D411yescharacteristic
130C22×C131C2×#4111yescharacteristic
131C22×C131C2×#4112yes—
132C22×C131H(5,𝔽2)12yes—
133C2×D131—42no—
134C2×D131—42no—
135C2×D131—42no—
136C2×D131—42no—
137C2×D131—42no—
138C2×D131—42no—
139C2×D131—42no—
140C4×C131—22no—
141C4×C131—22no—
142C4×C131—41no—
143C4×C131—42no—
144Dic131—22no—
145Dic131—41no—
146Dic131—42no—
147Dic131#12114yes—
148Dic131#12414yescomplemented
149C2×C131—82no—
150C2×C131C2×#12111yescharacteristic
151C2×C131C2×#12411yescharacteristic
152C2×C131#53511yescharacteristic
153C2×C131#55012yes—
154C2×C131#55912yes—
155D131—82no—
156C131#254811yescharacteristic, complemented, Sylow
157#2548—1311nomaximal, Sylow
158C2×#127—1311no—
159C2×#124—1312no—
160C2×C4×Q8—1311no—
161#394—1311no—
162#394—1311no—
163#394—1311no—
164#394D13111yescharacteristic, complemented
165#390—1312no—
166#390—1312no—
167#390—1312no—
168C4×C4⋊C4—1311no—
169C22×Q8—1311no—
170C22×D4—1312no—
171C22×D4—1312no—
172C22×D4—1311no—
173C22×D4C2×D13111yescharacteristic
174#127—2624no—
175#124—2628no—
176C4×Q8—2624no—
177C2×C4⋊C4—1311no—
178C2×C4⋊C4—1311no—
179C2×C4⋊C4—1312no—
180C2×C4⋊C4C2×D13111yescharacteristic
181C2×C22⋊C4—1312no—
182C2×C22⋊C4—1312no—
183C2×C22⋊C4—1312no—
184C2×C22⋊C4—1312no—
185C2×C22⋊C4—1312no—
186C2×C22⋊C4—1312no—
187C2×C22⋊C4C2×D13112yes—
188C2×C22⋊C4C2×D13112yes—
189C2×C42—1311no—
190C2×C42—1311no—
191C2×C42—1312no—
192C2×C42—1311no—
193#95—1311no—
194#95C2×D13111yescharacteristic
195C24—1312no—
196C24C22×D13112yes—
197C2×Q8—2622no—
198C2×Q8—2622no—
199C2×Q8—1314no—
200C2×D4—22no—
201C2×D4—42no—
202C2×D4—2624no—
203C2×D4—2624no—
204C2×D4—2624no—
205C2×D4—2624no—
206C2×D4—2624no—
207C2×D4—2624no—
208C2×D4—2622no—
209C2×D4—5242no—
210C22×C4—1312no—
211C22×C4—1311no—
212C22×C4—1311no—
213C22×C4—1311no—
214C22×C4—1311no—
215C22×C4C22×D13112yes—
216C22×C4C22×D13111yescharacteristic
217C22×C4C22×D13111yescharacteristic
218C22×C4C22×D13111yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—2624no—
221C4⋊C4—2622no—
222C4⋊C4—2622no—
223C22⋊C4—24no—
224C22⋊C4—42no—
225C22⋊C4—2624no—
226C22⋊C4—2624no—
227C22⋊C4—2624no—
228C22⋊C4—5242no—
229C22⋊C4—5242no—
230C22⋊C4—5242no—
231C42—2622no—
232C42—2622no—
233C42—2624no—
234C42—1314no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—2622no—
243C23—2622no—
244C23—2622no—
245C23—2622no—
246C23—2622no—
247C23—2622no—
248C23—2622no—
249C23C23×D13111yescharacteristic
250Q8—2628no—
251D4—44no—
252D4—5244no—
253D4—5248no—
254D4—5248no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—2621no—
264C2×C4—2621no—
265C2×C4—2622no—
266C2×C4—2621no—
267C2×C4—2622no—
268C2×C4—2622no—
269C2×C4—2622no—
270C2×C4—1312no—
271C2×C4—1312no—
272C2×C4—1312no—
273C2×C4—1312no—
274C2×C4—1314no—
275C2×C4—1312no—
276C2×C4D4×D13112yes—
277C2×C4D4×D13112yes—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—5242no—
286C22—5242no—
287C22—5242no—
288C22—5242no—
289C22—5242no—
290C22—5242no—
291C22—5242no—
292C22C2×D4×D13111yescharacteristic
293C22C2×D4×D13111yescharacteristic
294C22#41×D13112yes—
295C22#41×D13111yescharacteristic
296C22#1686229112yes—
297C4—22no—
298C4—22no—
299C4—41no—
300C4—42no—
301C4—2622no—
302C4—5241no—
303C4—5242no—
304C4—1314no—
305C4—1314no—
306C2—82no—
307C2—10482no—
308C2#121×D13111yescharacteristic
309C2#124×D13111yescharacteristic
310C2#2255387711yescharacteristic
311C2#2255398012yes—
312C2#2255404612yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#394×C131C2
prime radicalC2×#16862242C2
hypercenter#394D131
hypocenterC131#2548
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#16862232C2semidirect
3. C2×#16862211C2semidirect
4. #22553230C2semidirect
5. #22553229C2semidirect
6. #22553226C2semidirect
7. #22553212C2semidirect
8. #22553212C2semidirect
9. #22553212C2semidirect
10. #22553163C2semidirect
11. #394×C131C2semidirect
41. C2×Dic262D4semidirect
148. Dic131#124semidirect
156. C131#2548semidirect
164. #394D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622422
C2×#168622322
C2×#168622112
#225532302
#225532292
#225532262
#225532122
#225531632
#394×C1312
#2548131