Finite Group Databasebeta

← #30649098

Subgroups — #30649098

Order 16256.

372 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550507C211yesmaximal, characteristic, complemented
2#22550447C211yesmaximal, characteristic
3#22550243C212yesmaximal, complemented
4#22550234C212yesmaximal, complemented
5#22550229C212yesmaximal, complemented
6#22550225C212yesmaximal, complemented
7#22549886C211yesmaximal, characteristic, complemented
8#22549872C211yesmaximal, characteristic
9C2×#16861132C211yesmaximal, characteristic, complemented
10#22549834C211yesmaximal, characteristic, complemented
11#417×C127C211yesmaximal, characteristic
12C22×Dic254C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14#16861208—21no—
15#16861208C2212yes—
16#16861207C2212yes—
17#16861200—21no—
18#16861200C2212yes—
19#16861196C2212yes—
20#16861190—22no—
21#16861188—21no—
22#16861188—21no—
23#16861188—22no—
24#16861168C2211yescharacteristic
25#16861168C2211yescharacteristic
26#16861134C2212yes—
27#16861134C2212yes—
28#16861133C2211yescharacteristic
29#16861133C2211yescharacteristic
30#16861132C2212yes—
31#16861132C2212yes—
32#16861130C2211yescharacteristic
33#16861130C2211yescharacteristic
34#16861119C2212yes—
35C8×Dic127C2212yes—
36C2×#41×C127C2211yescharacteristic
37C2×M16×C127C2211yescharacteristic
38C2×C4⋊C4×C127C2211yescharacteristic
39#103×C127C2212yes—
40#102×C127C2212yes—
41C2×Dic254—22no—
42C2×Dic254—21no—
43C2×Dic254—22no—
44C2×Dic254—22no—
45C2×Dic254C2312yes—
46C22×Dic127—21no—
47C2×C127⋊C8—22no—
48C2×C127⋊C8C2311yescharacteristic
49C2×C127⋊C8C2311yescharacteristic
50C4⋊Dic127—22no—
51C4⋊Dic127D411yescharacteristic
52C4⋊Dic127D411yescharacteristic
53C4⋊Dic127C2311yescharacteristic
54C4⋊Dic127C2311yescharacteristic
55Dic127⋊C4—22no—
56Dic127⋊C4—22no—
57Dic127⋊C4—22no—
58Dic127⋊C4—22no—
59C22⋊Dic127—22no—
60C22⋊Dic127—22no—
61C22⋊Dic127—21no—
62C22⋊Dic127D412yes—
63C4×Dic127—21no—
64C4×Dic127C2312yes—
65#41×C127—22no—
66#41×C127—22no—
67C2×Q8×C127C2311yescharacteristic
68C2×D4×C127—21no—
69C2×D4×C127C2311yescharacteristic
70C22×C4×C127—21no—
71C22×C4×C127—21no—
72C22×C4×C127C2311yescharacteristic
73M16×C127—22no—
74C2×C8×C127C2312yes—
75C4⋊C4×C127—21no—
76C4⋊C4×C127C2312yes—
77Dic254—22no—
78Dic254—22no—
79Dic254—42no—
80Dic254—82no—
81C2×Dic127—21no—
82C2×Dic127—21no—
83C2×Dic127—22no—
84C2×Dic127—42no—
85C2×Dic127C2×D412yes—
86C2×Dic127C2×D412yes—
87C127⋊C8—21no—
88C127⋊C8—21no—
89C23×C127—21no—
90C23×C127C2×D411yescharacteristic
91Q8×C127—21no—
92Q8×C127—21no—
93D4×C127—21no—
94D4×C127—21no—
95D4×C127—42no—
96C2×C4×C127—22no—
97C2×C4×C127—21no—
98C2×C4×C127—42no—
99C2×C4×C127—81no—
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127#4112yes—
102C2×C4×C127#4112yes—
103C2×C4×C127C2411yescharacteristic
104C8×C127—22no—
105C22×C127—21no—
106C22×C127—21no—
107C22×C127—41no—
108C22×C127—41no—
109C22×C127—41no—
110C22×C127#12412yes—
111C22×C127C22×D411yescharacteristic
112C4×C127—41no—
113C4×C127—42no—
114C4×C127#12412yes—
115C4×C127C2×#4111yescharacteristic
116C4×C127C2×#4111yescharacteristic
117Dic127—22no—
118Dic127—22no—
119Dic127—42no—
120Dic127—41no—
121Dic127—41no—
122C2×C127—22no—
123C2×C127—81no—
124C2×C127C2×#12411yescharacteristic
125C2×C127#57711yescharacteristic
126C2×C127#57811yescharacteristic
127#3137—1271nomaximal, Sylow
128C127#313711yescharacteristic, complemented, Sylow
129#544—1271no—
130#536—1271no—
131#491—1272no—
132#489—1272no—
133#488—1272no—
134#487—1272no—
135#419—1271no—
136#417D12711yescharacteristic
137#416—1271no—
138C2×#103—1271no—
139#405—1271no—
140C2×#41C2×D12711yescharacteristic
141C22×Q8—1271no—
142C2×M16C2×D12711yescharacteristic
143C22×C8—1271no—
144C4⋊Q8—1272no—
145C4⋊Q8—1272no—
146C4⋊Q8—2541no—
147#125—1272no—
148#125—2541no—
149#124—1272no—
150#123—2542no—
151C22⋊Q8—2542no—
152C22⋊Q8—2541no—
153C22⋊Q8—2541no—
154#117—1271no—
155#117—1271no—
156C2×C4⋊C4C2×D12711yescharacteristic
157#103—1271no—
158#103—1271no—
159#103—1272no—
160#103—1272no—
161#103—1272no—
162#103—1272no—
163#103C2×D12712yes—
164#102—1271no—
165#102—1271no—
166#102C2×D12712yes—
167#97—1272no—
168C4×C8—1272no—
169#41—22no—
170#41—22no—
171C2×Q8—1272no—
172C2×Q8—2542no—
173C2×Q8—2542no—
174C2×Q8—2542no—
175C2×Q8—2541no—
176C2×Q8C22×D12711yescharacteristic
177C2×D4—21no—
178C2×D4C22×D12711yescharacteristic
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—2541no—
182C22×C4C22×D12711yescharacteristic
183M16—22no—
184C2×C8—1271no—
185C2×C8—1271no—
186C2×C8—2542no—
187C2×C8C22×D12712yes—
188C4⋊C4—21no—
189C4⋊C4—1271no—
190C4⋊C4—1271no—
191C4⋊C4—1271no—
192C4⋊C4—1271no—
193C4⋊C4—2542no—
194C4⋊C4—2542no—
195C4⋊C4—2542no—
196C4⋊C4—2542no—
197C4⋊C4—2542no—
198C4⋊C4C22×D12712yes—
199C22⋊C4—1272no—
200C22⋊C4—2542no—
201C22⋊C4—2542no—
202C22⋊C4—2541no—
203C42—1272no—
204C42—2541no—
205C23—21no—
206C23D4×D12711yescharacteristic
207Q8—21no—
208Q8—21no—
209Q8—10162no—
210Q8—5082no—
211Q8—2542no—
212Q8—2542no—
213D4—21no—
214D4—21no—
215D4—42no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—81no—
220C2×C4—1272no—
221C2×C4—1272no—
222C2×C4—5082no—
223C2×C4—2541no—
224C2×C4—2541no—
225C2×C4—2542no—
226C2×C4D4×D12711yescharacteristic
227C2×C4#1452324712yes—
228C2×C4#1452324812yes—
229C2×C4C23×D12711yescharacteristic
230C8—22no—
231C8—2541no—
232C8—2541no—
233C22—21no—
234C22—21no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22#1686119512yes—
239C22C2×D4×D12711yescharacteristic
240C4—41no—
241C4—42no—
242C4—5082no—
243C4—5081no—
244C4—5081no—
245C4—2542no—
246C4—2542no—
247C4#1686119512yes—
248C4C2×#1452324711yescharacteristic
249C4C2×#1452324811yescharacteristic
250C2—22no—
251C2—81no—
252C2C2×#1686119511yescharacteristic
253C2#2255068611yescharacteristic
254C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#417×C127C2
socleC22×C127C22×D4
prime radicalC2×#41×C127C22
hypercenter#417D127
hypocenterC127#3137
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #22550507C2semidirect
3. #22550243C2semidirect
4. #22550234C2semidirect
5. #22550229C2semidirect
6. #22550225C2semidirect
7. #22549886C2semidirect
9. C2×#16861132C2semidirect
10. #22549834C2semidirect
128. C127#3137semidirect

Maximal subgroups

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

subgroupindex
#225505072
#225504472
#225502432
#225502342
#225502292
#225502252
#225498862
#225498722
C2×#168611322
#225498342
#417×C1272
#3137127