Finite Group Databasebeta

← #30648701

Subgroups — #30648701

Order 16256.

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

normal subgroups only

This group has 386 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#22550452C211yesmaximal, characteristic
2#22550357C211yesmaximal, characteristic
3#22550220C212yesmaximal, complemented
4#22550219C212yesmaximal, complemented
5#22550213C212yesmaximal, complemented
6#22550208C212yesmaximal, complemented
7#22549912C211yesmaximal, characteristic, complemented
8#22549886C211yesmaximal, characteristic, complemented
9C2×#16861132C211yesmaximal, characteristic, complemented
10#22549844C211yesmaximal, characteristic
11#428×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×Dic254C2211yescharacteristic
14C22×C127⋊C8C2211yescharacteristic
15#16861209—21no—
16#16861197—21no—
17#16861193—22no—
18#16861189C2212yes—
19#16861189C2212yes—
20#16861174—21no—
21#16861171—21no—
22C4×C22⋊D127—22no—
23#16861168C2211yescharacteristic
24#16861167C2212yes—
25C4⋊C4×D127C2212yes—
26#16861142C2211yescharacteristic
27#16861142C2211yescharacteristic
28#16861140C2211yescharacteristic
29#16861140C2211yescharacteristic
30#16861134C2212yes—
31#16861134C2212yes—
32#16861132C2212yes—
33#16861132C2212yes—
34(C2×D127)⋊C8C2212yes—
35(C2×D127)⋊C8C2212yes—
36C2×M16×C127C2211yescharacteristic
37#117×C127C2211yescharacteristic
38C2×C4⋊C4×C127C2211yescharacteristic
39#107×C127C2212yes—
40#106×C127C2212yes—
41#14523248—22no—
42#14523248—22no—
43C2×Dic254—21no—
44C2×Dic254—22no—
45C2×Dic254—22no—
46C2×Dic254D411yescharacteristic
47C2×Dic254C2312yes—
48C2×D508D411yescharacteristic
49C2×C22⋊D127D412yes—
50C22×Dic127—21no—
51C2×C4×D127—22no—
52C2×C4×D127C2312yes—
53C2×C127⋊C8—22no—
54C2×C127⋊C8C2311yescharacteristic
55C2×C127⋊C8C2311yescharacteristic
56C4⋊Dic127—22no—
57C4⋊Dic127—22no—
58C4⋊Dic127C2311yescharacteristic
59C4⋊Dic127C2311yescharacteristic
60Dic127⋊C4—22no—
61C22⋊Dic127—21no—
62(C2×D127)⋊C4—22no—
63(C2×D127)⋊C4—22no—
64(C2×D127)⋊C4—22no—
65C4×Dic127—22no—
66C4×Dic127—21no—
67C22×C4×C127—21no—
68C22×C4×C127C2311yescharacteristic
69M16×C127—22no—
70C2×C8×C127C2312yes—
71C4⋊C4×C127—21no—
72C4⋊C4×C127C2311yescharacteristic
73C4⋊C4×C127C2311yescharacteristic
74C4⋊C4×C127C2312yes—
75C22⋊C4×C127—21no—
76C42×C127—21no—
77C22×D127C2×D412yes—
78Dic254—22no—
79Dic254—22no—
80Dic254—22no—
81Dic254—42no—
82D508—22no—
83C22⋊D127—24no—
84C2×Dic127—21no—
85C2×Dic127—21no—
86C2×Dic127—22no—
87C2×Dic127—42no—
88C2×Dic127C2×D412yes—
89C4×D127—22no—
90C4×D127—22no—
91C4×D127—82no—
92C127⋊C8—21no—
93C127⋊C8—21no—
94C23×C127C2×D411yescharacteristic
95C2×C4×C127—21no—
96C2×C4×C127—21no—
97C2×C4×C127—22no—
98C2×C4×C127—81no—
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127#4112yes—
101C2×C4×C127#4112yes—
102C2×C4×C127C2411yescharacteristic
103C8×C127—22no—
104C22×C127—21no—
105C22×C127—21no—
106C22×C127#12312yes—
107C22×C127C22×D411yescharacteristic
108C2×D127—22no—
109C2×D127—22no—
110C2×D127—22no—
111C4×C127—41no—
112C4×C127—42no—
113C4×C127—41no—
114C4×C127#12312yes—
115C4×C127C2×#4111yescharacteristic
116C4×C127C2×#4111yescharacteristic
117Dic127—22no—
118Dic127—42no—
119Dic127—41no—
120Dic127—41no—
121C2×C127—22no—
122C2×C127C2×#12311yescharacteristic
123C2×C127#57711yescharacteristic
124C2×C127#57811yescharacteristic
125D127—42no—
126#3097—1271nomaximal, Sylow
127C127#309711yescharacteristic, complemented, Sylow
128#536—1271no—
129#520—1271no—
130#485—1272no—
131#485—1272no—
132#484—1272no—
133#483—1272no—
134#428D12711yescharacteristic, complemented
135#427—1271no—
136#419—1271no—
137C2×#103—1271no—
138#408—1271no—
139C2×#41—1271no—
140C22×Q8—1271no—
141C2×M16C2×D12711yescharacteristic
142C22×C8—1271no—
143C4⋊Q8—2541no—
144#124—2541no—
145#123—2542no—
146C22⋊Q8—1272no—
147C22⋊Q8—1272no—
148C4×Q8—2541no—
149C4×D4—2541no—
150C4×D4—2542no—
151#117—1271no—
152#117—1272no—
153#117C2×D12711yescharacteristic
154C2×C4⋊C4—1272no—
155C2×C4⋊C4C2×D12711yescharacteristic
156#107—1271no—
157#107—1271no—
158#107C2×D12712yes—
159#106—1271no—
160#106—1271no—
161#106C2×D12712yes—
162#103—1272no—
163#103—1272no—
164#103—1272no—
165#103—1272no—
166C22⋊C8—1272no—
167C22⋊C8—1272no—
168#41—2542no—
169#41—2542no—
170C2×Q8—1271no—
171C2×Q8—1272no—
172C2×Q8—2542no—
173C2×Q8—2542no—
174C2×Q8—2541no—
175C2×D4—1272no—
176C2×D4—1271no—
177C22×C4—21no—
178C22×C4—1272no—
179C22×C4—2542no—
180C22×C4—2541no—
181C22×C4C22×D12711yescharacteristic
182M16—22no—
183C2×C8—1271no—
184C2×C8—1271no—
185C2×C8—2542no—
186C2×C8C22×D12712yes—
187C4⋊C4—21no—
188C4⋊C4—1271no—
189C4⋊C4—1271no—
190C4⋊C4—2542no—
191C4⋊C4—2542no—
192C4⋊C4—2542no—
193C4⋊C4C22×D12712yes—
194C4⋊C4C22×D12711yescharacteristic
195C4⋊C4C22×D12711yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—2542no—
198C22⋊C4—2542no—
199C22⋊C4—2542no—
200C22⋊C4—2541no—
201C42—21no—
202C42—2542no—
203C42—2541no—
204C23—1272no—
205C23D4×D12711yescharacteristic
206Q8—5082no—
207Q8—2542no—
208Q8—2542no—
209Q8—2542no—
210D4—2544no—
211D4—2542no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—81no—
216C2×C4—1272no—
217C2×C4—10162no—
218C2×C4—5082no—
219C2×C4—2541no—
220C2×C4—2541no—
221C2×C4—2542no—
222C2×C4—2542no—
223C2×C4—2542no—
224C2×C4D4×D12711yescharacteristic
225C2×C4#1452324812yes—
226C2×C4#1452324912yes—
227C2×C4C23×D12711yescharacteristic
228C8—22no—
229C8—2541no—
230C8—2541no—
231C22—21no—
232C22—21no—
233C22—2542no—
234C22—2542no—
235C22—2542no—
236C22#1686119412yes—
237C22C2×D4×D12711yescharacteristic
238C4—42no—
239C4—41no—
240C4—41no—
241C4—5082no—
242C4—5081no—
243C4—5081no—
244C4—2542no—
245C4#1686119412yes—
246C4C2×#1452324811yescharacteristic
247C4C2×#1452324911yescharacteristic
248C2—22no—
249C2—5082no—
250C2C2×#1686119411yescharacteristic
251C2#2255068611yescharacteristic
252C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#428×C127C2
socleC22×C127C22×D4
prime radicalC2×#14523248C22
hypercenter#428D127
hypocenterC127#3097
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #22550220C2semidirect
4. #22550219C2semidirect
5. #22550213C2semidirect
6. #22550208C2semidirect
7. #22549912C2semidirect
8. #22549886C2semidirect
9. C2×#16861132C2semidirect
11. #428×C127C2semidirect
127. C127#3097semidirect
134. #428D127semidirect

Maximal subgroups

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

subgroupindex
#225504522
#225503572
#225502202
#225502192
#225502132
#225502082
#225499122
#225498862
C2×#168611322
#225498442
#428×C1272
#3097127