Finite Group Databasebeta

← #30639095

Subgroups — #30639095

Order 16256.

310 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 79 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#22550674C211yesmaximal, characteristic
2#22550642C211yesmaximal, characteristic, complemented
3#22549893C211yesmaximal, characteristic, complemented
4#22549873C211yesmaximal, characteristic, complemented
5#22549647C211yesmaximal, characteristic, complemented
6#22549641C211yesmaximal, characteristic, complemented
7#408×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240C412yescomplemented
10#16861239—21no—
11#16861235C412yescomplemented
12#16861234—21no—
13#16861229—22no—
14#16861217—22no—
15#16861215—22no—
16C2×#14523232C2211yescharacteristic
17C2×#14523232C2211yescharacteristic
18M16×D127—22no—
19#16861168C2211yescharacteristic
20#16861136—22no—
21#16861134—21no—
22#16861131—21no—
23C2×#41×C127C2211yescharacteristic
24C2×M16×C127C2211yescharacteristic
25C22×C8×C127C2211yescharacteristic
26C22⋊C8×C127—22no—
27#14523248—21no—
28#14523248—21no—
29#14523248C2×C412yes—
30#14523248D412yes—
31C2×Dic254D411yescharacteristic
32C2×D508D411yescharacteristic
33C2×C22⋊D127—21no—
34C2×C4×D127—21no—
35#14523238—22no—
36#14523236—22no—
37#14523235—22no—
38#14523233—22no—
39#14523232—21no—
40#14523232—21no—
41#14523232C2×C412yes—
42#14523232D412yes—
43#14523231—24no—
44C2×C127⋊C8—21no—
45C2×C127⋊C8—21no—
46C8×D127—24no—
47C4⋊Dic127—21no—
48C22⋊Dic127—21no—
49C4×Dic127—21no—
50#41×C127—21no—
51#41×C127—21no—
52#41×C127—21no—
53#41×C127—21no—
54C2×Q8×C127C2×C411yescharacteristic
55C2×D4×C127—21no—
56C2×D4×C127C2×C411yescharacteristic
57C22×C4×C127—21no—
58C22×C4×C127C2311yescharacteristic
59M16×C127—21no—
60M16×C127—21no—
61C2×C8×C127—21no—
62C2×C8×C127—21no—
63C2×C8×C127—21no—
64C2×C8×C127D412yes—
65C22×D127—21no—
66Dic254—21no—
67Dic254C22⋊C412yes—
68D508—21no—
69D508C22⋊C412yes—
70C22⋊D127—22no—
71C22⋊D127—22no—
72C2×Dic127—21no—
73C2×Dic127—41no—
74C4×D127—22no—
75C4×D127—22no—
76C127⋊C8—22no—
77C127⋊C8—22no—
78C23×C127—21no—
79C23×C127#4111yescharacteristic
80Q8×C127—21no—
81Q8×C127—21no—
82D4×C127—21no—
83D4×C127—21no—
84D4×C127—41no—
85D4×C127—41no—
86C2×C4×C127—21no—
87C2×C4×C127—41no—
88C2×C4×C127—41no—
89C2×C4×C127C22×C411yescharacteristic
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127#4111yescharacteristic
95C8×C127—22no—
96C8×C127—41no—
97C22×C127—41no—
98C22×C127—42no—
99C22×C127—41no—
100C22×C127C4×D411yescharacteristic
101C22×C127#12111yescharacteristic
102C22×C127#12311yescharacteristic
103C2×D127—22no—
104C2×D127—81no—
105C4×C127—41no—
106C4×C127C2×C22⋊C411yescharacteristic
107C4×C127C4×D411yescharacteristic
108C4×C127#12011yescharacteristic
109C4×C127#12111yescharacteristic
110Dic127—22no—
111Dic127—81no—
112C2×C127—21no—
113C2×C127—21no—
114C2×C127—21no—
115C2×C127—81no—
116C2×C127#38611yescharacteristic
117D127—42no—
118#1881—1271nomaximal, Sylow
119C127#188111yescharacteristic, complemented, Sylow
120#575—1271no—
121#568—1271no—
122#421—1271no—
123#416—1271no—
124#408D12711yescharacteristic, complemented
125#344—1271no—
126#341—1271no—
127C2×#41—1271no—
128C2×#41C2×D12711yescharacteristic
129#137—1272no—
130#137—2541no—
131AGL(1,ℤ/8)—1272no—
132AGL(1,ℤ/8)—2541no—
133#135—2542no—
134#131—2542no—
135#131—2542no—
136C2×M16—1271no—
137C2×M16—1271no—
138C2×M16—2542no—
139C2×M16C2×D12711yescharacteristic
140C22×C8C2×D12711yescharacteristic
141#117—1271no—
142#104—2542no—
143#103—2541no—
144#102—2541no—
145C22⋊C8—22no—
146#41—21no—
147#41—21no—
148#41—21no—
149#41—21no—
150#41—1272no—
151#41—1272no—
152#41—2541no—
153#41—2541no—
154C2×Q8—1271no—
155C2×Q8C4×D12711yescharacteristic
156C2×D4—21no—
157C2×D4—1271no—
158C2×D4—2541no—
159C2×D4C4×D12711yescharacteristic
160C22×C4—21no—
161C22×C4—2541no—
162C22×C4C22×D12711yescharacteristic
163Q16—2542no—
164SD16—2542no—
165SD16—2542no—
166D8—2542no—
167M16—21no—
168M16—21no—
169M16—1272no—
170M16—1272no—
171M16—2544no—
172M16—2541no—
173M16—2541no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—2544no—
178C2×C8—2541no—
179C2×C8—2541no—
180C2×C8C22⋊D12712yes—
181C4⋊C4—2541no—
182C22⋊C4—2541no—
183C42—2541no—
184C23—21no—
185C23—2541no—
186C23#1452324811yescharacteristic
187Q8—21no—
188Q8—21no—
189Q8—1272no—
190Q8—2541no—
191D4—21no—
192D4—21no—
193D4—41no—
194D4—41no—
195D4—1272no—
196D4—2542no—
197D4—2542no—
198D4—2541no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—5081no—
203C2×C4—2541no—
204C2×C4—2542no—
205C2×C4—2542no—
206C2×C4C2×C4×D12711yescharacteristic
207C2×C4D4×D12711yescharacteristic
208C2×C4D4×D12711yescharacteristic
209C2×C4D4×D12711yescharacteristic
210C2×C4C2×C22⋊D12711yescharacteristic
211C2×C4#1452324711yescharacteristic
212C8—22no—
213C8—41no—
214C8—2542no—
215C8—2542no—
216C22—41no—
217C22—42no—
218C22—41no—
219C22—10161no—
220C22—2542no—
221C22C4×C22⋊D12711yescharacteristic
222C22#1686118211yescharacteristic
223C22#1686119111yescharacteristic
224C4—41no—
225C4—10161no—
226C4—2542no—
227C4C22⋊C4×D12711yescharacteristic
228C4(C22⋊D127)⋊C411yescharacteristic
229C4#1686117711yescharacteristic
230C4#1686118011yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—5082no—
236C2#2254975211yescharacteristic
subgroupG/H
centerC4#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#408×C127C2
socleC2×C127#386
prime radical#22550674C2
cosocleC22×C4×C127C23
hypercenter#408D127
hypocenterC127#1881
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #22550642C2semidirect
3. #22549893C2semidirect
4. #22549873C2semidirect
5. #22549647C2semidirect
6. #22549641C2semidirect
7. #408×C127C2semidirect
9. #16861240C4semidirect
11. #16861235C4semidirect
119. C127#1881semidirect
124. #408D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
#225506422
#225498932
#225498732
#225496472
#225496412
#408×C1272
#1881127