Finite Group Databasebeta

← #30648699

Subgroups — #30648699

Order 16256.

386 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 133 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#22550447C211yesmaximal, characteristic
2#22550217C212yesmaximal, complemented
3#22550217C212yesmaximal, complemented
4#22550211C212yesmaximal, complemented
5#22550206C212yesmaximal, complemented
6#22549916C211yesmaximal, characteristic, complemented
7#22549912C211yesmaximal, characteristic, complemented
8#22549884C211yesmaximal, characteristic, complemented
9C2×#16861132C211yesmaximal, characteristic, complemented
10#22549843C211yesmaximal, characteristic
11#520×C127C211yesmaximal, characteristic
12C22×Dic254C2211yescharacteristic
13C2×#14523232C2211yescharacteristic
14C22×C127⋊C8C2211yescharacteristic
15#16861207—21no—
16#16861196—21no—
17#16861190—22no—
18#16861188C2212yes—
19#16861188C2212yes—
20#16861168C2211yescharacteristic
21#16861142C2211yescharacteristic
22#16861142C2211yescharacteristic
23#16861142C2212yes—
24#16861140C2211yescharacteristic
25#16861140C2211yescharacteristic
26#16861140C2212yes—
27#16861132C2212yes—
28#16861132C2212yes—
29#16861132C2212yes—
30#16861132C2212yes—
31C22⋊(C127⋊C8)C2212yes—
32C22⋊(C127⋊C8)C2212yes—
33C2×#41×C127C2211yescharacteristic
34C4×Q8×C127—21no—
35C4×D4×C127—22no—
36C4×D4×C127—21no—
37#117×C127C2211yescharacteristic
38#117×C127C2212yes—
39C2×C4⋊C4×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2212yes—
41C2×Dic254—21no—
42C2×Dic254—22no—
43C2×Dic254—22no—
44C2×Dic254C2312yes—
45C22×Dic127—21no—
46#14523232—22no—
47C2×C127⋊C8—22no—
48C2×C127⋊C8C2311yescharacteristic
49C2×C127⋊C8C2311yescharacteristic
50C2×C127⋊C8C2312yes—
51C4⋊Dic127C2311yescharacteristic
52C4⋊Dic127C2311yescharacteristic
53Dic127⋊C4—22no—
54Dic127⋊C4—22no—
55C22⋊Dic127—22no—
56C22⋊Dic127—22no—
57C22⋊Dic127—21no—
58C4×Dic127—21no—
59#41×C127—22no—
60#41×C127—22no—
61C2×Q8×C127D411yescharacteristic
62C2×D4×C127D412yes—
63C2×D4×C127D411yescharacteristic
64C22×C4×C127—22no—
65C22×C4×C127—21no—
66C22×C4×C127C2312yes—
67C22×C4×C127C2311yescharacteristic
68C4⋊C4×C127—22no—
69C4⋊C4×C127—21no—
70C4⋊C4×C127C2312yes—
71C4⋊C4×C127C2311yescharacteristic
72C4⋊C4×C127C2311yescharacteristic
73C22⋊C4×C127—22no—
74C22⋊C4×C127—21no—
75C42×C127—22no—
76C42×C127—21no—
77Dic254—22no—
78Dic254—22no—
79Dic254—42no—
80C2×Dic127—21no—
81C2×Dic127—21no—
82C2×Dic127—22no—
83C2×Dic127—42no—
84C127⋊C8—21no—
85C127⋊C8—21no—
86C127⋊C8—22no—
87C23×C127C2×D412yes—
88C23×C127C2×D411yescharacteristic
89Q8×C127—22no—
90D4×C127—24no—
91D4×C127—22no—
92C2×C4×C127—22no—
93C2×C4×C127—22no—
94C2×C4×C127—21no—
95C2×C4×C127—22no—
96C2×C4×C127—21no—
97C2×C4×C127—81no—
98C2×C4×C127—82no—
99C2×C4×C127C2×D412yes—
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127#4112yes—
102C2×C4×C127#4112yes—
103C2×C4×C127C2411yescharacteristic
104C22×C127—21no—
105C22×C127—21no—
106C22×C127—22no—
107C22×C127—22no—
108C22×C127—22no—
109C22×C127#12312yes—
110C22×C127C22×D411yescharacteristic
111C4×C127—22no—
112C4×C127—42no—
113C4×C127—41no—
114C4×C127—41no—
115C4×C127#12312yes—
116C4×C127C2×#4111yescharacteristic
117C4×C127C2×#4111yescharacteristic
118Dic127—42no—
119Dic127—41no—
120Dic127—41no—
121C2×C127—22no—
122C2×C127—42no—
123C2×C127C2×#12311yescharacteristic
124C2×C127#57711yescharacteristic
125C2×C127#57811yescharacteristic
126#3097—1271nomaximal, Sylow
127C127#309711yescharacteristic, complemented, Sylow
128#536—1271no—
129#520D12711yescharacteristic
130#485—1272no—
131#485—1272no—
132#484—1272no—
133#483—1272no—
134#428—1271no—
135#427—1271no—
136#419—1271no—
137C2×#103—1271no—
138#408—1271no—
139C2×#41C2×D12711yescharacteristic
140C22×Q8—1271no—
141C2×M16—1271no—
142C22×C8—1271no—
143C4⋊Q8—2541no—
144#124—2541no—
145#123—2542no—
146C22⋊Q8—1272no—
147C22⋊Q8—1272no—
148C4×Q8—21no—
149C4×D4—22no—
150C4×D4—21no—
151#117—1271no—
152#117C2×D12711yescharacteristic
153#117C2×D12712yes—
154C2×C4⋊C4C2×D12711yescharacteristic
155C2×C4⋊C4C2×D12712yes—
156#107—1271no—
157#107—1271no—
158#107—1272no—
159#106—1271no—
160#106—1271no—
161#106—1272no—
162#103—1272no—
163#103—1272no—
164#103—1272no—
165#103—1272no—
166C22⋊C8—1272no—
167C22⋊C8—1272no—
168#41—22no—
169#41—22no—
170C2×Q8—1272no—
171C2×Q8—2542no—
172C2×Q8—2542no—
173C2×Q8—2541no—
174C2×Q8C22⋊D12711yescharacteristic
175C2×D4C22⋊D12712yes—
176C2×D4C22⋊D12711yescharacteristic
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—2541no—
180C22×C4C22×D12712yes—
181C22×C4C22×D12711yescharacteristic
182M16—2542no—
183C2×C8—1271no—
184C2×C8—1272no—
185C2×C8—1271no—
186C2×C8—2542no—
187C4⋊C4—22no—
188C4⋊C4—21no—
189C4⋊C4—1271no—
190C4⋊C4—1271no—
191C4⋊C4—2542no—
192C4⋊C4—2542no—
193C4⋊C4C22×D12712yes—
194C4⋊C4C22×D12711yescharacteristic
195C4⋊C4C22×D12711yescharacteristic
196C22⋊C4—22no—
197C22⋊C4—21no—
198C22⋊C4—2541no—
199C22⋊C4—2542no—
200C22⋊C4—2542no—
201C42—22no—
202C42—21no—
203C42—2541no—
204C23C2×C22⋊D12712yes—
205C23C2×C22⋊D12711yescharacteristic
206Q8—22no—
207Q8—5082no—
208Q8—2542no—
209Q8—2542no—
210D4—24no—
211D4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—81no—
218C2×C4—82no—
219C2×C4—5082no—
220C2×C4—2541no—
221C2×C4—2541no—
222C2×C4—2542no—
223C2×C4C2×C22⋊D12712yes—
224C2×C4C2×C22⋊D12711yescharacteristic
225C2×C4#1452324812yes—
226C2×C4#1452324812yes—
227C2×C4C23×D12711yescharacteristic
228C8—2541no—
229C8—2541no—
230C8—2542no—
231C22—21no—
232C22—21no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22#1686119212yes—
237C22C22×C22⋊D12711yescharacteristic
238C4—22no—
239C4—42no—
240C4—41no—
241C4—41no—
242C4—5082no—
243C4—5081no—
244C4—5081no—
245C4#1686119212yes—
246C4C2×#1452324811yescharacteristic
247C4C2×#1452324811yescharacteristic
248C2—22no—
249C2—42no—
250C2C2×#1686119211yescharacteristic
251C2#2255068711yescharacteristic
252C2#2255069111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#520×C127C2
socleC22×C127C22×D4
prime radicalC2×#41×C127C22
hypercenter#520D127
hypocenterC127#3097
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ #520

Complements

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

NH ≅ G/Ntype
2. #22550217C2semidirect
3. #22550217C2semidirect
4. #22550211C2semidirect
5. #22550206C2semidirect
6. #22549916C2semidirect
7. #22549912C2semidirect
8. #22549884C2semidirect
9. C2×#16861132C2semidirect
127. C127#3097semidirect

Maximal subgroups

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

subgroupindex
#225504472
#225502172
#225502112
#225502062
#225499162
#225499122
#225498842
C2×#168611322
#225498432
#520×C1272
#3097127