Finite Group Databasebeta

← #30676423

Subgroups — #30676423

Order 16768.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16862279C211yesmaximal, characteristic, complemented
2C2×#16862276C211yesmaximal, characteristic, complemented
3C2×#16862272C211yesmaximal, characteristic, complemented
4#22553858C211yesmaximal, characteristic, complemented
5C2×#16862233C211yesmaximal, characteristic, complemented
6#22553703C212yesmaximal, complemented
7#22553700C212yesmaximal, complemented
8#22553695C212yesmaximal, complemented
9#22553693C212yesmaximal, complemented
10M16×Dic131C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C131C211yesmaximal, characteristic, complemented
12C2×#14523513C2211yescharacteristic
13C22×Dic262C2211yescharacteristic
14#16862279—24no—
15#16862276—24no—
16#16862272—24no—
17C2×Dic524C2212yescomplemented
18C2×#14523503C2212yescomplemented
19C2×#14523502C2212yescomplemented
20C2×#14523501C2212yes—
21C2×#14523501C2212yescomplemented
22C2×#14523498C2212yes—
23C2×#14523497C2211yescharacteristic
24#16862244C2211yescharacteristic, complemented
25#16862242C2211yescharacteristic, complemented
26#16862235C2211yescharacteristic, complemented
27#16862233—22no—
28#16862233C2212yes—
29#16862233C2212yes—
30#16862233C2211yescharacteristic, complemented
31#16862218—22no—
32C4×C22⋊D131—22no—
33C2×C22⋊Dic131—22no—
34C2×C4×Dic131C2211yescharacteristic
35#16862156C2212yescomplemented
36C8×Dic131C2212yescomplemented
37C2×#41×C131C2211yescharacteristic
38C22×D4×C131C2211yescharacteristic
39AGL(1,ℤ/8)×C131—24no—
40C2×SD16×C131C2212yes—
41C2×D8×C131C2212yes—
42C2×M16×C131C2211yescharacteristic
43#14523513—22no—
44#14523513—22no—
45C2×Dic262—21no—
46C2×Dic262—22no—
47C2×Dic262—22no—
48C2×Dic262C2312yes—
49C2×Dic262C2311yescharacteristic
50C2×D524C2311yescharacteristic
51C2×C22⋊D131—22no—
52C2×C22⋊D131—21no—
53C22×Dic131—21no—
54C22×Dic131D412yes—
55C2×C4×D131—21no—
56Dic524—24no—
57#14523503—24no—
58#14523502—24no—
59#14523501—24no—
60#14523501—24no—
61#14523498—24no—
62#14523497D414yescomplemented
63C2×C131⋊C8C2312yes—
64Dic131⋊C4—22no—
65C22⋊Dic131—22no—
66C22⋊Dic131—24no—
67C22⋊Dic131—24no—
68(C2×D131)⋊C4—22no—
69C4×Dic131D412yes—
70C4×Dic131C2311yescharacteristic
71C4×Dic131C2311yescharacteristic
72C24×C131—21no—
73#41×C131—22no—
74#41×C131—22no—
75C2×Q8×C131C2311yescharacteristic
76C2×D4×C131—21no—
77C2×D4×C131—22no—
78C2×D4×C131—22no—
79C2×D4×C131—21no—
80C2×D4×C131C2312yes—
81C2×D4×C131C2311yescharacteristic
82C22×C4×C131—21no—
83C22×C4×C131C2311yescharacteristic
84SD16×C131—24no—
85D8×C131—24no—
86M16×C131D414yescomplemented
87C2×C8×C131C2312yes—
88C22×D131—21no—
89Dic262—21no—
90Dic262—22no—
91Dic262—22no—
92Dic262—21no—
93Dic262—42no—
94D524—21no—
95D524—21no—
96C22⋊D131—44no—
97C22⋊D131—42no—
98C2×Dic131—22no—
99C2×Dic131—21no—
100C2×Dic131—42no—
101C2×Dic131—42no—
102C2×Dic131C2×D412yes—
103C2×Dic131C2×D412yes—
104C4×D131—42no—
105C131⋊C8C2×D414yes—
106C23×C131—21no—
107C23×C131—22no—
108C23×C131—44no—
109C23×C131—42no—
110C23×C131C2×D411yescharacteristic
111Q8×C131—21no—
112Q8×C131—21no—
113D4×C131—21no—
114D4×C131—22no—
115D4×C131—22no—
116D4×C131—21no—
117D4×C131—42no—
118D4×C131—42no—
119C2×C4×C131—21no—
120C2×C4×C131—42no—
121C2×C4×C131C2×D411yescharacteristic
122C2×C4×C131C2×D412yes—
123C2×C4×C131C2×D412yes—
124C2×C4×C131C2411yescharacteristic
125C8×C131C2×D414yes—
126C22×C131—22no—
127C22×C131—41no—
128C22×C131—42no—
129C22×C131—44no—
130C22×C131—42no—
131C22×C131—84no—
132C22×C131#12712yes—
133C22×C131C22×D411yescharacteristic
134C2×D131—41no—
135C2×D131—42no—
136C4×C131—41no—
137C4×C131#12712yes—
138C4×C131C22×D411yescharacteristic
139C4×C131C22×D411yescharacteristic
140Dic131—22no—
141Dic131—22no—
142Dic131—41no—
143Dic131—42no—
144C2×C131—22no—
145C2×C131—81no—
146C2×C131—82no—
147C2×C131C2×#12711yescharacteristic
148C2×C131#57512yes—
149D131—81no—
150C131#315311yescharacteristic, complemented, Sylow
151#3153—1311nomaximal, Sylow
152C2×#137—1311no—
153C2×#137—1311no—
154C2×AGL(1,ℤ/8)—1311no—
155C2×AGL(1,ℤ/8)D13111yescharacteristic, complemented
156#532—1311no—
157C2×#124—1311no—
158#497—1312no—
159#496—1312no—
160#495—1312no—
161#495—1312no—
162C4×M16—1311no—
163C2×#41—1311no—
164C2×#41C2×D13111yescharacteristic
165C22×Q8—1311no—
166C22×D4C2×D13111yescharacteristic
167#137—2624no—
168#137—2624no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—2624no—
171C2×Q16—1312no—
172C2×Q16—1312no—
173C2×SD16—1312no—
174C2×SD16—1312no—
175C2×SD16—1312no—
176C2×SD16C2×D13112yes—
177C2×D8—1312no—
178C2×D8C2×D13112yes—
179C2×M16—1311no—
180C2×M16C2×D13111yescharacteristic
181C4⋊Q8—1311no—
182#127—1311no—
183#124—2622no—
184#124—1311no—
185#124—1312no—
186#124—1312no—
187#124—1311no—
188#121—2622no—
189C4×D4—2622no—
190C2×C22⋊C4—2622no—
191C2×C42—1311no—
192#97—1312no—
193C4×C8—1312no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—2622no—
198#41—2622no—
199C2×Q8—2621no—
200C2×Q8—2622no—
201C2×Q8—2622no—
202C2×Q8—1312no—
203C2×Q8—1311no—
204C2×Q8C22×D13111yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—2621no—
210C2×D4—2622no—
211C2×D4—1311no—
212C2×D4C22×D13112yes—
213C2×D4C22×D13111yescharacteristic
214C22×C4—21no—
215C22×C4—2621no—
216C22×C4—2621no—
217C22×C4—1312no—
218C22×C4C22×D13111yescharacteristic
219Q16—2624no—
220Q16—2624no—
221SD16—24no—
222SD16—2624no—
223SD16—2624no—
224SD16—2624no—
225D8—24no—
226D8—2624no—
227M16—1314no—
228M16C22⋊D13114yescomplemented
229C2×C8—1312no—
230C2×C8C22×D13112yes—
231C4⋊C4—2622no—
232C22⋊C4—2622no—
233C22⋊C4—2624no—
234C22⋊C4—2624no—
235C22⋊C4—2622no—
236C42—1312no—
237C42—1311no—
238C42—1311no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—2621no—
244C23D4×D13111yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—2621no—
248Q8—2622no—
249Q8—2622no—
250Q8—2621no—
251Q8—5242no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—2621no—
259D4—2621no—
260D4—5244no—
261D4—5242no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—2622no—
265C2×C4—2621no—
266C2×C4—5242no—
267C2×C4—5242no—
268C2×C4—5242no—
269C2×C4—1312no—
270C2×C4—1312no—
271C2×C4D4×D13111yescharacteristic
272C2×C4D4×D13112yes—
273C2×C4C2×C22⋊D13112yes—
274C2×C4C23×D13111yescharacteristic
275C8—1314no—
276C8C2×C22⋊D13114yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—5241no—
284C22—5242no—
285C22#1686224212yes—
286C22C2×D4×D13111yescharacteristic
287C4—41no—
288C4—2622no—
289C4—2622no—
290C4—5241no—
291C4—5242no—
292C4#1686224212yes—
293C4C2×D4×D13111yescharacteristic
294C4C22×C22⋊D13111yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—10481no—
299C2C2×#1686224211yescharacteristic
300C2#2255410812yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC2×AGL(1,ℤ/8)×C131C2
socleC22×C131C22×D4
hypercenterC2×AGL(1,ℤ/8)D131
hypocenterC131#3153
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16862279C2semidirect
2. C2×#16862276C2semidirect
3. C2×#16862272C2semidirect
4. #22553858C2semidirect
5. C2×#16862233C2semidirect
6. #22553703C2semidirect
7. #22553700C2semidirect
8. #22553695C2semidirect
9. #22553693C2semidirect
10. M16×Dic131C2semidirect
11. C2×AGL(1,ℤ/8)×C131C2semidirect
17. C2×Dic524C22semidirect
18. C2×#14523503C22semidirect
19. C2×#14523502C22semidirect
21. C2×#14523501C22semidirect
24. #16862244C22semidirect
25. #16862242C22semidirect
26. #16862235C22semidirect
30. #16862233C22semidirect
35. #16862156C22semidirect
36. C8×Dic131C22semidirect
62. #14523497D4semidirect
86. M16×C131D4semidirect
150. C131#3153semidirect
155. C2×AGL(1,ℤ/8)D131semidirect
228. M16C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622792
C2×#168622762
C2×#168622722
#225538582
C2×#168622332
#225537032
#225537002
#225536952
#225536932
M16×Dic1312
C2×AGL(1,ℤ/8)×C1312
#3153131