Finite Group Databasebeta

← #30651447

Subgroups — #30651447

Order 16256.

372 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 115 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#22550596C211yesmaximal, characteristic, complemented
2#22550541C211yesmaximal, characteristic, complemented
3#22550234C212yesmaximal, complemented
4#22550228C211yesmaximal, characteristic, complemented
5#22550170C212yesmaximal, complemented
6#22550090C212yesmaximal
7#22550082C211yesmaximal, characteristic, complemented
8#22550003C211yesmaximal, characteristic, complemented
9#22549975C212yesmaximal, complemented
10C4×#14523233C211yesmaximal, characteristic
11#474×C127C211yesmaximal, characteristic, complemented
12C2×Q8×D127—21no—
13C2×#14523237C2212yes—
14C2×#14523233C2211yescharacteristic
15#16861209C2211yescharacteristic
16#16861208C2212yes—
17#16861202—22no—
18#16861196C2212yes—
19#16861188—21no—
20#16861187—22no—
21#16861185—21no—
22C4×Dic254C2211yescharacteristic
23#16861174C2212yes—
24C4×D508C2211yescharacteristic
25#16861171C2212yes—
26#16861168—21no—
27#16861142C2211yescharacteristic
28Dic127⋊C8C2212yes—
29#16861134C2212yes—
30#16861132C2212yes—
31#16861132C2212yes—
32#16861130C2211yescharacteristic
33#16861129C2211yescharacteristic
34#16861119C2212yes—
35C4×C127⋊C8C2211yescharacteristic
36C4⋊Q8×C127C2211yescharacteristic
37C4×D4×C127C2211yescharacteristic
38#107×C127C2212yes—
39C4⋊C8×C127C2211yescharacteristic
40#102×C127C2212yes—
41C2×Dic254—22no—
42C2×Dic254D411yescharacteristic
43C2×Dic254C2312yes—
44Q8×D127—44no—
45C2×D508C2311yescharacteristic
46C2×C4×D127—22no—
47C2×C4×D127—21no—
48#14523237—24no—
49#14523233—22no—
50C2×C127⋊C8C2311yescharacteristic
51C2×C127⋊C8C2311yescharacteristic
52C4⋊Dic127—21no—
53C4⋊Dic127—21no—
54C4⋊Dic127D411yescharacteristic
55C4⋊Dic127C2311yescharacteristic
56Dic127⋊C4—22no—
57Dic127⋊C4—22no—
58Dic127⋊C4D412yes—
59C22⋊Dic127—22no—
60C22⋊Dic127—22no—
61(C2×D127)⋊C4—22no—
62(C2×D127)⋊C4—21no—
63C4×Dic127—22no—
64C4×Dic127C2312yes—
65C2×Q8×C127—21no—
66C2×D4×C127C2311yescharacteristic
67C22×C4×C127—21no—
68C2×C8×C127C2312yes—
69C4⋊C4×C127—21no—
70C4⋊C4×C127C2312yes—
71C4⋊C4×C127C2311yescharacteristic
72C22⋊C4×C127—21no—
73C42×C127C2311yescharacteristic
74C22×D127—21no—
75Dic254—22no—
76Dic254—22no—
77Dic254—22no—
78Dic254—44no—
79D508—21no—
80D508#4112yes—
81C2×Dic127—22no—
82C2×Dic127—21no—
83C2×Dic127C2×D412yes—
84C2×Dic127C2×D412yes—
85C4×D127—44no—
86C4×D127—42no—
87C127⋊C8—21no—
88C127⋊C8—21no—
89C23×C127—21no—
90Q8×C127—42no—
91D4×C127—21no—
92D4×C127—21no—
93C2×C4×C127—21no—
94C2×C4×C127—22no—
95C2×C4×C127—42no—
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2411yescharacteristic
99C8×C127—22no—
100C22×C127—41no—
101C22×C127—41no—
102C22×C127—41no—
103C22×C127C22×D411yescharacteristic
104C2×D127—22no—
105C2×D127—41no—
106C2×D127—41no—
107C4×C127—21no—
108C4×C127—41no—
109C4×C127—42no—
110C4×C127AGL(1,ℤ/8)12yes—
111C4×C127C2×#4111yescharacteristic
112C4×C1272−1+411yescharacteristic
113Dic127—22no—
114Dic127—22no—
115Dic127—42no—
116Dic127—41no—
117C2×C127—81no—
118C2×C127#54811yescharacteristic
119C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C127#57811yescharacteristic
121D127—42no—
122#3367—1271nomaximal, Sylow
123C127#336711yescharacteristic, complemented, Sylow
124#556—1271no—
125D4×Q8—1271no—
126#489—1272no—
127#487—1271no—
128#477—1272no—
129#474D12711yescharacteristic, complemented
130#463—1272no—
131#461—1271no—
132#446—1271no—
133#440—1272no—
134C4×D8—1271no—
135C22×Q8—2541no—
136C2×SD16—1272no—
137C2×D8—1271no—
138C4⋊Q8—1272no—
139C4⋊Q8—1271no—
140C4⋊Q8C2×D12711yescharacteristic
141#126—2542no—
142#124—1272no—
143C22⋊Q8—2542no—
144C22⋊Q8—2541no—
145C22⋊Q8—2541no—
146C4×Q8—1271no—
147C4×Q8—1272no—
148C4×D4—1272no—
149C4×D4—1271no—
150C4×D4C2×D12711yescharacteristic
151#117—2541no—
152#107—1271no—
153#107C2×D12712yes—
154C4⋊C8—1272no—
155C4⋊C8C2×D12711yescharacteristic
156#103—1272no—
157#103—1272no—
158#103—1272no—
159#102—1271no—
160#102—1271no—
161#102C2×D12712yes—
162#97—1272no—
163C4×C8—1271no—
164C2×Q8—21no—
165C2×Q8—1271no—
166C2×Q8—1272no—
167C2×Q8—5084no—
168C2×Q8—2542no—
169C2×D4—1271no—
170C2×D4C22×D12711yescharacteristic
171C22×C4—21no—
172C22×C4—2542no—
173C22×C4—2541no—
174SD16—2544no—
175D8—2542no—
176C2×C8—1271no—
177C2×C8—1271no—
178C2×C8C22×D12712yes—
179C4⋊C4—21no—
180C4⋊C4—1272no—
181C4⋊C4—1271no—
182C4⋊C4—1271no—
183C4⋊C4—2542no—
184C4⋊C4—2542no—
185C4⋊C4—2541no—
186C4⋊C4—2541no—
187C4⋊C4C22×D12711yescharacteristic
188C4⋊C4C22×D12712yes—
189C22⋊C4—21no—
190C22⋊C4—2542no—
191C22⋊C4—2542no—
192C22⋊C4—2542no—
193C22⋊C4—2541no—
194C42—1272no—
195C42—2542no—
196C42C22×D12711yescharacteristic
197C23—21no—
198C23—2541no—
199Q8—42no—
200Q8—5084no—
201Q8—2542no—
202Q8—2542no—
203Q8—2542no—
204D4—21no—
205D4—21no—
206D4—1272no—
207D4—2541no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—1272no—
212C2×C4—1272no—
213C2×C4—5084no—
214C2×C4—5082no—
215C2×C4—2542no—
216C2×C4—2541no—
217C2×C4D4×D12711yescharacteristic
218C2×C4D4×D12711yescharacteristic
219C2×C4C23×D12711yescharacteristic
220C8—22no—
221C8—2541no—
222C8—2541no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—5081no—
227C22—5081no—
228C22—2542no—
229C22C2×D4×D12711yescharacteristic
230C4—21no—
231C4—41no—
232C4—42no—
233C4—5082no—
234C4—5081no—
235C4—2542no—
236C4—2542no—
237C4#1686123312yes—
238C4#41×D12711yescharacteristic
239C4#1686125711yescharacteristic
240C2—81no—
241C2—5082no—
242C2#2255053611yescharacteristic
243C2C2×#1686123311yescharacteristic
244C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#474×C127C2
socleC22×C127C22×D4
prime radicalC2×#14523233C22
hypercenter#474D127
hypocenterC127#3367
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #22550596C2semidirect
2. #22550541C2semidirect
3. #22550234C2semidirect
4. #22550228C2semidirect
5. #22550170C2semidirect
7. #22550082C2semidirect
8. #22550003C2semidirect
9. #22549975C2semidirect
11. #474×C127C2semidirect
123. C127#3367semidirect
129. #474D127semidirect

Maximal subgroups

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

subgroupindex
#225505962
#225505412
#225502342
#225502282
#225501702
#225500902
#225500822
#225500032
#225499752
C4×#145232332
#474×C1272
#3367127