Finite Group Databasebeta

← #30679107

Subgroups — #30679107

Order 16768.

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

normal subgroups only

This group has 360 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#22554126C211yesmaximal, characteristic, complemented
2#22554113C211yesmaximal, characteristic, complemented
3#22553755C211yesmaximal, characteristic, complemented
4#22553751C212yesmaximal, complemented
5#22553746C212yesmaximal, complemented
6#22553742C211yesmaximal, characteristic, complemented
7#22553741C212yesmaximal, complemented
8C2×#16862190C211yesmaximal, characteristic, complemented
9SD32×D131C212yesmaximal, complemented
10#22553730C211yesmaximal, characteristic, complemented
11#509×C131C211yesmaximal, characteristic, complemented
12#16862293—21no—
13#16862291—21no—
14#16862276—21no—
15#16862275—22no—
16#16862272—21no—
17#16862270—22no—
18#16862268C2211yescharacteristic
19#16862265C2212yes—
20#16862265C2212yescomplemented
21C2×Dic524C2211yescharacteristic, complemented
22Q16×D131C2212yescomplemented
23D8×D131C2212yes—
24#16862252C2211yescharacteristic
25Dic1048C2212yescomplemented
26#16862192C2211yescharacteristic, complemented
27#16862191C2212yescomplemented
28#16862190C2212yescomplemented
29#16862190C2212yescomplemented
30#16862190C2211yescharacteristic, complemented
31#16862189C2211yescharacteristic, complemented
32#16862187C2211yescharacteristic, complemented
33#16862185C2212yescomplemented
34C2×C131⋊C16C2211yescharacteristic, complemented
35C16×D131C2212yescomplemented
36#135×C131C2211yescharacteristic, complemented
37C2×D8×C131C2211yescharacteristic
38SD32×C131C2212yescomplemented
39D16×C131C2212yescomplemented
40M32×C131C2211yescharacteristic, complemented
41#14523513—21no—
42#14523513—41no—
43#14523513D411yescharacteristic
44#14523512—22no—
45#14523512—22no—
46#14523512—41no—
47#14523512—42no—
48C2×Dic262—21no—
49C2×Dic262—41no—
50Q8×D131—22no—
51C2×C22⋊D131—42no—
52D4×D131—22no—
53D4×D131—41no—
54Dic524—21no—
55Dic524C2312yes—
56Dic524C2311yescharacteristic
57#14523503—22no—
58#14523502—21no—
59#14523501—22no—
60#14523501—22no—
61D1048C2311yescharacteristic
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×C131C2312yes—
74D8×C131C2311yescharacteristic
75C2×C8×C131C2311yescharacteristic
76C16×C131C2312yes—
77C22×D131—42no—
78Dic262—21no—
79Dic262—22no—
80Dic262—41no—
81Dic262—42no—
82Dic262D811yescharacteristic
83D524—21no—
84D524D811yescharacteristic
85C22⋊D131—41no—
86C22⋊D131—42no—
87C22⋊D131—42no—
88C22⋊D131—42no—
89C22⋊D131D812yes—
90C2×Dic131—41no—
91C2×Dic131—42no—
92C2×Dic131—42no—
93C4×D131—42no—
94C4×D131—41no—
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—162no—
113C2×D131C2×D812yes—
114C4×C131—41no—
115C4×C131C2×D811yescharacteristic
116C4×C131C22×D411yescharacteristic
117Dic131—41no—
118Dic131—42no—
119Dic131C2×D812yes—
120C2×C131—21no—
121C2×C131—82no—
122C2×C131—81no—
123C2×C131C22×D811yescharacteristic
124D131—22no—
125D131—81no—
126C131#341611yescharacteristic, complemented, Sylow
127#3416—1311nomaximal, Sylow
128#578—1311no—
129#576—1311no—
130#510—1312no—
131#510—1311no—
132#509—1312no—
133#509D13111yescharacteristic, complemented
134#508—1311no—
135#508—1312no—
136C2×SD32—1311no—
137C2×SD32—1312no—
138#504—1311no—
1392−1+4—2621no—
140H(5,𝔽2)—2621no—
141#137—2622no—
142#137—2621no—
143AGL(1,ℤ/8)—2622no—
144AGL(1,ℤ/8)—2621no—
145#135—1312no—
146#135—1312no—
147#135—1311no—
148#135C2×D13111yescharacteristic, complemented
149C2×Q16—1312no—
150C2×Q16—1311no—
151C2×D8—1312no—
152C2×D8C2×D13111yescharacteristic
153#131—1311no—
154Q32—1311no—
155Q32—1312no—
156SD32—1311no—
157SD32—1312no—
158SD32—1312no—
159SD32—1311no—
160SD32—1312no—
161SD32C2×D13112yescomplemented
162D16—1311no—
163D16C2×D13112yescomplemented
164M32—1312no—
165M32C2×D13111yescharacteristic, complemented
166C2×C16—1311no—
167C2×C16—1312no—
168#41—21no—
169#41—2622no—
170#41—2622no—
171#41—2621no—
172#41—5241no—
173#41—5241no—
174#41—5242no—
175#41—1311no—
176C2×Q8—2622no—
177C2×Q8—2621no—
178C2×Q8—5241no—
179C2×D4—21no—
180C2×D4—2622no—
181C2×D4—5241no—
182C2×D4—5242no—
183Q16—2622no—
184Q16—2621no—
185Q16—1312no—
186Q16—1311no—
187Q16C22×D13111yescharacteristic
188SD16—21no—
189SD16—2622no—
190SD16—2622no—
191SD16—2621no—
192D8—21no—
193D8—2622no—
194D8—1311no—
195D8C22×D13111yescharacteristic
196D8C22×D13112yes—
197M16—1311no—
198M16—1312no—
199C2×C8—1312no—
200C2×C8C22×D13111yescharacteristic
201C16—1311no—
202C16—1311no—
203C16C22×D13112yes—
204C23—41no—
205C23—5242no—
206Q8—21no—
207Q8—2621no—
208Q8—2622no—
209Q8—5242no—
210Q8—5241no—
211Q8—1311no—
212D4—22no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—2621no—
217D4—5242no—
218D4—5242no—
219D4—5241no—
220D4—5242no—
221D4—1312no—
222D4—1311no—
223C2×C4—41no—
224C2×C4—5242no—
225C2×C4—5241no—
226C2×C4—5242no—
227C2×C4—5242no—
228C2×C4—5241no—
229C2×C4—1312no—
230C2×C4D4×D13111yescharacteristic
231C8—1312no—
232C8D4×D13111yescharacteristic
233C8C23×D13111yescharacteristic
234C22—41no—
235C22—42no—
236C22—5241no—
237C22—161no—
238C22—20962no—
239C22—1312no—
240C22D8×D13111yescharacteristic
241C4—41no—
242C4—5241no—
243C4—5242no—
244C4—1312no—
245C4D8×D13111yescharacteristic
246C4C2×D4×D13111yescharacteristic
247C2—21no—
248C2—2622no—
249C2—82no—
250C2—81no—
251C2—10481no—
252C2C2×D8×D13111yescharacteristic
subgroupG/H
centerC2C2×D8×D131
commutator, cosocleC8×C131C24
FrattiniC8C23×D131
Fitting#509×C131C2
socleC2×C131C22×D8
hypercenter#509D131
hypocenterC131#3416
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. #22554126C2semidirect
2. #22554113C2semidirect
3. #22553755C2semidirect
4. #22553751C2semidirect
5. #22553746C2semidirect
6. #22553742C2semidirect
7. #22553741C2semidirect
8. C2×#16862190C2semidirect
9. SD32×D131C2semidirect
10. #22553730C2semidirect
11. #509×C131C2semidirect
20. #16862265C22semidirect
21. C2×Dic524C22semidirect
22. Q16×D131C22semidirect
25. Dic1048C22semidirect
26. #16862192C22semidirect
27. #16862191C22semidirect
28. #16862190C22semidirect
29. #16862190C22semidirect
30. #16862190C22semidirect
31. #16862189C22semidirect
32. #16862187C22semidirect
33. #16862185C22semidirect
34. C2×C131⋊C16C22semidirect
35. C16×D131C22semidirect
36. #135×C131C22semidirect
38. SD32×C131C22semidirect
39. D16×C131C22semidirect
40. M32×C131C22semidirect
126. C131#3416semidirect
133. #509D131semidirect
148. #135C2×D131semidirect
161. SD32C2×D131semidirect
163. D16C2×D131semidirect
165. M32C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225541262
#225541132
#225537552
#225537512
#225537462
#225537422
#225537412
C2×#168621902
SD32×D1312
#225537302
#509×C1312
#3416131