Finite Group Databasebeta

← #30679101

Subgroups — #30679101

Order 16768.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#22554114C211yesmaximal, characteristic, complemented
2#22554113C211yesmaximal, characteristic, complemented
3#22553750C211yesmaximal, characteristic, complemented
4#22553749C212yesmaximal, complemented
5#22553746C212yesmaximal, complemented
6#22553743C212yesmaximal, complemented
7#22553742C211yesmaximal, characteristic, complemented
8C2×#16862187C211yesmaximal, characteristic, complemented
9D16×D131C212yesmaximal, complemented
10#22553730C211yesmaximal, characteristic, complemented
11#509×C131C211yesmaximal, characteristic, complemented
12#16862292—21no—
13#16862291—21no—
14#16862273—22no—
15#16862272—21no—
16#16862271—21no—
17#16862270—22no—
18#16862269C2212yescomplemented
19#16862268C2211yescharacteristic, complemented
20#16862265C2212yescomplemented
21C2×D1048C2211yescharacteristic, complemented
22D8×D131C2212yescomplemented
23D8×D131C2212yescomplemented
24#16862252C2211yescharacteristic
25#16862192C2211yescharacteristic, complemented
26#16862191C2212yescomplemented
27#16862190C2211yescharacteristic, complemented
28#16862189C2211yescharacteristic, complemented
29D2096C2212yescomplemented
30#16862187C2211yescharacteristic, complemented
31#16862187C2212yescomplemented
32#16862187C2212yescomplemented
33#16862185C2212yescomplemented
34C2×C131⋊C16C2211yescharacteristic, complemented
35C16×D131C2212yescomplemented
36#135×C131C2211yescharacteristic, complemented
37C2×D8×C131C2211yescharacteristic, complemented
38SD32×C131C2212yescomplemented
39D16×C131C2212yescomplemented
40M32×C131C2211yescharacteristic, complemented
41#14523514—22no—
42#14523513—21no—
43#14523513—41no—
44#14523513D411yescharacteristic
45#14523512—22no—
46#14523512—41no—
47C2×D524—21no—
48C2×D524—41no—
49C2×C22⋊D131—42no—
50D4×D131—22no—
51D4×D131—22no—
52D4×D131—41no—
53D4×D131—42no—
54Dic524C2311yescharacteristic
55#14523502—21no—
56#14523501—22no—
57#14523500—22no—
58D1048—21no—
59D1048C2311yescharacteristic
60D1048C2312yes—
61#14523498—22no—
62#14523498—22no—
63#14523497D411yescharacteristic
64#14523496D412yes—
65C8×D131C2312yes—
66C131⋊C16C2311yescharacteristic
67C131⋊C16C2311yescharacteristic
68#41×C131—21no—
69C2×D4×C131—21no—
70Q16×C131C2311yescharacteristic
71SD16×C131—21no—
72D8×C131—21no—
73D8×C131C2311yescharacteristic
74D8×C131C2312yes—
75C2×C8×C131C2311yescharacteristic
76C16×C131C2312yes—
77C22×D131—41no—
78C22×D131—42no—
79C22×D131—42no—
80Dic262—21no—
81Dic262D811yescharacteristic
82D524—21no—
83D524—22no—
84D524—42no—
85D524—41no—
86D524D811yescharacteristic
87C22⋊D131—41no—
88C22⋊D131—42no—
89C22⋊D131—42no—
90C22⋊D131—42no—
91C22⋊D131D812yes—
92C2×Dic131—42no—
93C4×D131—41no—
94C4×D131—42no—
95C4×D131C2×D412yes—
96C131⋊C8C2×D412yes—
97C23×C131—41no—
98Q8×C131—21no—
99D4×C131—22no—
100D4×C131—21no—
101D4×C131—41no—
102D4×C131—41no—
103C2×C4×C131—41no—
104C2×C4×C131C2×D411yescharacteristic
105C8×C131C2×D411yescharacteristic
106C8×C131C2411yescharacteristic
107C22×C131—41no—
108C22×C131—42no—
109C22×C131—161no—
110C22×C131C2×D811yescharacteristic
111C2×D131—41no—
112C2×D131—42no—
113C2×D131—162no—
114C2×D131—162no—
115C2×D131—161no—
116C2×D131C2×D812yes—
117C4×C131—41no—
118C4×C131C2×D811yescharacteristic
119C4×C131C22×D411yescharacteristic
120Dic131—41no—
121Dic131C2×D812yes—
122C2×C131—21no—
123C2×C131—81no—
124C2×C131—82no—
125C2×C131C22×D811yescharacteristic
126D131—22no—
127D131—81no—
128D131—82no—
129C131#341511yescharacteristic, complemented, Sylow
130#3415—1311nomaximal, Sylow
131#576—1311no—
132#576—1311no—
133#509—1312no—
134#509—1312no—
135#509—1311no—
136#509D13111yescharacteristic, complemented
137#508—1311no—
138#508—1312no—
139C2×D16—1311no—
140C2×D16—1312no—
141#504—1311no—
142H(5,𝔽2)—2621no—
143H(5,𝔽2)—2621no—
144AGL(1,ℤ/8)—2622no—
145AGL(1,ℤ/8)—2622no—
146AGL(1,ℤ/8)—2621no—
147AGL(1,ℤ/8)—2621no—
148#135—1312no—
149#135—1312no—
150#135—1311no—
151#135C2×D13111yescharacteristic, complemented
152C2×D8—1312no—
153C2×D8—1312no—
154C2×D8—1311no—
155C2×D8C2×D13111yescharacteristic, complemented
156#131—1311no—
157Q32—1311no—
158SD32—1311no—
159SD32—1311no—
160SD32—1312no—
161SD32C2×D13112yescomplemented
162D16—1312no—
163D16—1311no—
164D16—1312no—
165D16—1312no—
166D16C2×D13112yescomplemented
167M32—1312no—
168M32C2×D13111yescharacteristic, complemented
169C2×C16—1311no—
170C2×C16—1312no—
171#41—21no—
172#41—2622no—
173#41—2622no—
174#41—2621no—
175#41—5241no—
176#41—5241no—
177#41—1311no—
178C2×D4—21no—
179C2×D4—2622no—
180C2×D4—2622no—
181C2×D4—2621no—
182C2×D4—5241no—
183C2×D4—5242no—
184C2×D4—5241no—
185C2×D4—5242no—
186Q16—1311no—
187Q16C22×D13111yescharacteristic
188SD16—21no—
189SD16—2622no—
190SD16—2622no—
191SD16—2621no—
192D8—21no—
193D8—2622no—
194D8—2622no—
195D8—2621no—
196D8—1311no—
197D8—1312no—
198D8C22×D13112yes—
199D8C22×D13111yescharacteristic
200M16—1311no—
201M16—1312no—
202C2×C8—1312no—
203C2×C8C22×D13111yescharacteristic
204C16—1311no—
205C16—1311no—
206C16C22×D13112yes—
207C23—41no—
208C23—5242no—
209C23—5242no—
210C23—5241no—
211Q8—21no—
212Q8—2621no—
213Q8—1311no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—2621no—
219D4—2622no—
220D4—5242no—
221D4—5242no—
222D4—5241no—
223D4—5242no—
224D4—5242no—
225D4—5241no—
226D4—1312no—
227D4—1311no—
228C2×C4—41no—
229C2×C4—5242no—
230C2×C4—5242no—
231C2×C4—5241no—
232C2×C4—1312no—
233C2×C4D4×D13111yescharacteristic
234C8—1312no—
235C8D4×D13111yescharacteristic
236C8C23×D13111yescharacteristic
237C22—41no—
238C22—42no—
239C22—5242no—
240C22—5241no—
241C22—161no—
242C22—20961no—
243C22—20962no—
244C22—20962no—
245C22—1312no—
246C22D8×D13111yescharacteristic
247C4—41no—
248C4—5241no—
249C4—1312no—
250C4D8×D13111yescharacteristic
251C4C2×D4×D13111yescharacteristic
252C2—21no—
253C2—2622no—
254C2—82no—
255C2—81no—
256C2—10482no—
257C2—10481no—
258C2C2×D8×D13111yescharacteristic
subgroupG/H
centerC2C2×D8×D131
commutator, cosocleC8×C131C24
FrattiniC8C23×D131
Fitting#509×C131C2
socleC2×C131C22×D8
hypercenter#509D131
hypocenterC131#3415
LCSG ▷ C8×C131 ▷ C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #22554114C2semidirect
2. #22554113C2semidirect
3. #22553750C2semidirect
4. #22553749C2semidirect
5. #22553746C2semidirect
6. #22553743C2semidirect
7. #22553742C2semidirect
8. C2×#16862187C2semidirect
9. D16×D131C2semidirect
10. #22553730C2semidirect
11. #509×C131C2semidirect
18. #16862269C22semidirect
19. #16862268C22semidirect
20. #16862265C22semidirect
21. C2×D1048C22semidirect
22. D8×D131C22semidirect
23. D8×D131C22semidirect
25. #16862192C22semidirect
26. #16862191C22semidirect
27. #16862190C22semidirect
28. #16862189C22semidirect
29. D2096C22semidirect
30. #16862187C22semidirect
31. #16862187C22semidirect
32. #16862187C22semidirect
33. #16862185C22semidirect
34. C2×C131⋊C16C22semidirect
35. C16×D131C22semidirect
36. #135×C131C22semidirect
37. C2×D8×C131C22semidirect
38. SD32×C131C22semidirect
39. D16×C131C22semidirect
40. M32×C131C22semidirect
129. C131#3415semidirect
136. #509D131semidirect
151. #135C2×D131semidirect
155. C2×D8C2×D131semidirect
161. SD32C2×D131semidirect
166. D16C2×D131semidirect
168. M32C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225541142
#225541132
#225537502
#225537492
#225537462
#225537432
#225537422
C2×#168621872
D16×D1312
#225537302
#509×C1312
#3415131