Finite Group Databasebeta

← #30639060

Subgroups — #30639060

Order 16256.

324 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16861229C211yesmaximal, characteristic
2C2×#16861215C211yesmaximal, characteristic, complemented
3C2×#16861136C211yesmaximal, characteristic, complemented
4C2×#16861135C211yesmaximal, characteristic, complemented
5#22549640C211yesmaximal, characteristic, complemented
6#22549630C211yesmaximal, characteristic, complemented
7#408×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861229—22no—
10#16861229—22no—
11C2×#14523238C411yescharacteristic, complemented
12C2×#14523236C411yescharacteristic, complemented
13C2×#14523235C411yescharacteristic, complemented
14C2×#14523233C411yescharacteristic, complemented
15#16861215—22no—
16#16861215—22no—
17C2×#14523232C2211yescharacteristic
18C2×#14523231—21no—
19C22×C127⋊C8C2211yescharacteristic
20C2×C8×D127—21no—
21C2×C4×Dic127C2211yescharacteristic
22#16861136—22no—
23#16861135—22no—
24C2×#41×C127C2211yescharacteristic
25C2×M16×C127C2211yescharacteristic
26C22×C8×C127C2211yescharacteristic
27C22⋊C8×C127—21no—
28C22⋊C8×C127—21no—
29#14523248—21no—
30#14523248—21no—
31#14523248D412yes—
32#14523248D412yes—
33C2×Dic254C2×C411yescharacteristic
34C2×D508C2×C411yescharacteristic
35C2×C22⋊D127—21no—
36C22×Dic127—21no—
37C2×C4×D127—21no—
38#14523238—22no—
39#14523236—22no—
40#14523235—22no—
41#14523233—22no—
42#14523232—21no—
43#14523232D412yes—
44#14523231—24no—
45C2×C127⋊C8—21no—
46C2×C127⋊C8—21no—
47C2×C127⋊C8—21no—
48C2×C127⋊C8C2×C411yescharacteristic
49C2×C127⋊C8C2×C411yescharacteristic
50C8×D127—24no—
51C4×Dic127—21no—
52C4×Dic127—21no—
53#41×C127—21no—
54#41×C127—21no—
55#41×C127—21no—
56#41×C127—21no—
57C2×Q8×C127C2×C411yescharacteristic
58C2×D4×C127—21no—
59C2×D4×C127C2×C411yescharacteristic
60C22×C4×C127—21no—
61C22×C4×C127C2311yescharacteristic
62M16×C127—21no—
63M16×C127—21no—
64C2×C8×C127—21no—
65C2×C8×C127—21no—
66C2×C8×C127—21no—
67C2×C8×C127D411yescharacteristic
68C2×C8×C127D411yescharacteristic
69C22×D127—21no—
70Dic254—21no—
71Dic254C22⋊C412yes—
72D508—21no—
73D508C22⋊C412yes—
74C22⋊D127—22no—
75C22⋊D127—22no—
76C2×Dic127—21no—
77C2×Dic127—22no—
78C2×Dic127—41no—
79C2×Dic127—41no—
80C4×D127—22no—
81C4×D127—22no—
82C127⋊C8—21no—
83C127⋊C8—21no—
84C127⋊C8—22no—
85C23×C127—21no—
86C23×C127#4111yescharacteristic
87Q8×C127—21no—
88Q8×C127—21no—
89D4×C127—21no—
90D4×C127—21no—
91D4×C127—41no—
92D4×C127—41no—
93C2×C4×C127—21no—
94C2×C4×C127—41no—
95C2×C4×C127—41no—
96C2×C4×C127C22×C411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127#4111yescharacteristic
102C8×C127—21no—
103C8×C127—21no—
104C8×C127—41no—
105C22×C127—21no—
106C22×C127—21no—
107C22×C127—41no—
108C22×C127—41no—
109C22×C127—41no—
110C22×C127C4×D411yescharacteristic
111C22×C127#12111yescharacteristic
112C22×C127#12311yescharacteristic
113C2×D127—22no—
114C2×D127—41no—
115C2×D127—41no—
116C4×C127—41no—
117C4×C127C2×C22⋊C411yescharacteristic
118C4×C127C4×D411yescharacteristic
119C4×C127#12011yescharacteristic
120C4×C127#12111yescharacteristic
121Dic127—22no—
122Dic127—42no—
123C2×C127—21no—
124C2×C127—21no—
125C2×C127—81no—
126C2×C127#38611yescharacteristic
127C2×C127#44311yescharacteristic
128C2×C127#44411yescharacteristic
129D127—42no—
130#1876—1271nomaximal, Sylow
131C127#187611yescharacteristic, complemented, Sylow
132C2×#135—1271no—
133C2×#131—1271no—
134C2×#104—1271no—
135C2×#104—1271no—
136#408D12711yescharacteristic, complemented
137#341—1271no—
138#336—1271no—
139C2×#41—1271no—
140C2×#41C2×D12711yescharacteristic
141#135—2542no—
142#135—2542no—
143C2×Q16—1271no—
144C2×SD16—1271no—
145C2×SD16—1271no—
146C2×D8—1271no—
147#131—2542no—
148#131—2542no—
149C2×M16—1271no—
150C2×M16—2541no—
151C2×M16C2×D12711yescharacteristic
152C22×C8—1271no—
153C22×C8—2541no—
154C22×C8C2×D12711yescharacteristic
155C2×C42—1271no—
156#104—2542no—
157#104—2542no—
158C22⋊C8—21no—
159C22⋊C8—21no—
160#41—21no—
161#41—21no—
162#41—21no—
163#41—21no—
164#41—1272no—
165#41—1272no—
166#41—2541no—
167#41—2541no—
168C2×Q8—1271no—
169C2×Q8C4×D12711yescharacteristic
170C2×D4—21no—
171C2×D4—1271no—
172C2×D4—2541no—
173C2×D4C4×D12711yescharacteristic
174C22×C4—21no—
175C22×C4—2541no—
176C22×C4—2541no—
177C22×C4C22×D12711yescharacteristic
178Q16—2542no—
179SD16—2542no—
180SD16—2542no—
181D8—2542no—
182M16—21no—
183M16—21no—
184M16—1272no—
185M16—2544no—
186M16—2541no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—1271no—
191C2×C8—1271no—
192C2×C8—2544no—
193C2×C8—2541no—
194C2×C8—2541no—
195C2×C8—2541no—
196C2×C8C22⋊D12711yescharacteristic
197C2×C8C22⋊D12711yescharacteristic
198C42—2541no—
199C42—2541no—
200C23—21no—
201C23—2541no—
202C23#1452324811yescharacteristic
203Q8—21no—
204Q8—21no—
205Q8—1272no—
206Q8—2541no—
207D4—21no—
208D4—21no—
209D4—41no—
210D4—41no—
211D4—1272no—
212D4—2542no—
213D4—2542no—
214D4—2541no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—5081no—
219C2×C4—5081no—
220C2×C4—2542no—
221C2×C4—2541no—
222C2×C4—2542no—
223C2×C4—2542no—
224C2×C4C2×C4×D12711yescharacteristic
225C2×C4D4×D12711yescharacteristic
226C2×C4D4×D12711yescharacteristic
227C2×C4D4×D12711yescharacteristic
228C2×C4C2×C22⋊D12711yescharacteristic
229C2×C4#1452324711yescharacteristic
230C8—21no—
231C8—21no—
232C8—41no—
233C8—2541no—
234C8—2541no—
235C8—2542no—
236C22—21no—
237C22—21no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—5081no—
242C22—5081no—
243C22—2542no—
244C22C4×C22⋊D12711yescharacteristic
245C22#1686118211yescharacteristic
246C22#1686119111yescharacteristic
247C4—41no—
248C4—5082no—
249C4—2542no—
250C4C22⋊C4×D12711yescharacteristic
251C4(C22⋊D127)⋊C411yescharacteristic
252C4#1686117711yescharacteristic
253C4#1686118011yescharacteristic
254C2—21no—
255C2—21no—
256C2—81no—
257C2—5082no—
258C2#2254975211yescharacteristic
259C2#2254999111yescharacteristic
260C2#2254999611yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#408×C127C2
socleC22×C127C4×D4
prime radicalC2×#16861229C2
cosocleC22×C4×C127C23
hypercenter#408D127
hypocenterC127#1876
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C2×#16861215C2semidirect
3. C2×#16861136C2semidirect
4. C2×#16861135C2semidirect
5. #22549640C2semidirect
6. #22549630C2semidirect
7. #408×C127C2semidirect
11. C2×#14523238C4semidirect
12. C2×#14523236C4semidirect
13. C2×#14523235C4semidirect
14. C2×#14523233C4semidirect
131. C127#1876semidirect
136. #408D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612292
C2×#168612152
C2×#168611362
C2×#168611352
#225496402
#225496302
#408×C1272
#1876127