Finite Group Databasebeta

← #30651414

Subgroups — #30651414

Order 16256.

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

normal subgroups only

This group has 336 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#22550601C211yesmaximal, characteristic, complemented
2#22550596C211yesmaximal, characteristic
3#22550229C212yesmaximal
4#22550228C211yesmaximal, characteristic, complemented
5#22550182C212yesmaximal, complemented
6#22550097C212yesmaximal
7#22550087C211yesmaximal, characteristic, complemented
8#22550000C211yesmaximal, characteristic, complemented
9#22549970C212yesmaximal, complemented
10C4×#14523236C211yesmaximal, characteristic
11#474×C127C211yesmaximal, characteristic
12C2×Dic508C2212yes—
13C2×#14523236C2211yescharacteristic
14#16861209C2211yescharacteristic
15#16861207—21no—
16#16861202—22no—
17#16861200C2212yes—
18#16861199—22no—
19#16861196C2212yes—
20#16861188—21no—
21Q8×Dic127—21no—
22C4×Dic254C2211yescharacteristic
23C4×Dic254C2211yescharacteristic
24#16861174C2212yes—
25#16861174C2212yes—
26#16861168—21no—
27#16861140C2211yescharacteristic
28Dic127⋊C8C2212yes—
29#16861134C2212yes—
30#16861132C2212yes—
31#16861132C2211yescharacteristic
32#16861132C2212yes—
33#16861130C2211yescharacteristic
34C4×C127⋊C8C2211yescharacteristic
35C8×Dic127C2212yes—
36C4⋊Q8×C127C2211yescharacteristic
37C4×D4×C127C2211yescharacteristic
38#107×C127C2212yes—
39C4⋊C8×C127C2211yescharacteristic
40#102×C127C2212yes—
41C2×Dic254D411yescharacteristic
42C2×Dic254C2312yes—
43C2×Dic254C2311yescharacteristic
44Dic508—24no—
45#14523236—22no—
46C2×C127⋊C8C2311yescharacteristic
47C2×C127⋊C8C2311yescharacteristic
48C4⋊Dic127—22no—
49C4⋊Dic127—21no—
50C4⋊Dic127—21no—
51C4⋊Dic127D411yescharacteristic
52C4⋊Dic127C2311yescharacteristic
53Dic127⋊C4—22no—
54Dic127⋊C4—22no—
55Dic127⋊C4—22no—
56Dic127⋊C4—21no—
57Dic127⋊C4D412yes—
58C22⋊Dic127—22no—
59C22⋊Dic127—22no—
60C4×Dic127—22no—
61C4×Dic127—22no—
62C4×Dic127—21no—
63C4×Dic127C2312yes—
64C2×Q8×C127—21no—
65C2×D4×C127C2311yescharacteristic
66C22×C4×C127—21no—
67C2×C8×C127C2312yes—
68C4⋊C4×C127—21no—
69C4⋊C4×C127C2312yes—
70C4⋊C4×C127C2311yescharacteristic
71C22⋊C4×C127—21no—
72C42×C127C2311yescharacteristic
73Dic254—22no—
74Dic254—21no—
75Dic254—22no—
76Dic254—22no—
77Dic254#4112yes—
78C2×Dic127—21no—
79C2×Dic127—22no—
80C2×Dic127—21no—
81C2×Dic127C2×D412yes—
82C2×Dic127C2×D412yes—
83C127⋊C8—21no—
84C127⋊C8—21no—
85C23×C127—21no—
86Q8×C127—42no—
87D4×C127—21no—
88D4×C127—21no—
89C2×C4×C127—21no—
90C2×C4×C127—22no—
91C2×C4×C127—42no—
92C2×C4×C127C2×D411yescharacteristic
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2411yescharacteristic
95C8×C127—22no—
96C22×C127—41no—
97C22×C127—41no—
98C22×C127—41no—
99C22×C127C22×D411yescharacteristic
100C4×C127—21no—
101C4×C127—41no—
102C4×C127—42no—
103C4×C127#13512yes—
104C4×C127C2×#4111yescharacteristic
105C4×C1272−1+411yescharacteristic
106Dic127—22no—
107Dic127—22no—
108Dic127—22no—
109Dic127—42no—
110Dic127—41no—
111C2×C127—81no—
112C2×C127#54811yescharacteristic
113C2×C127C2×#13511yescharacteristic
114C2×C127#57811yescharacteristic
115#3364—1271nomaximal, Sylow
116C127#336411yescharacteristic, complemented, Sylow
117#557—1271no—
118#556—1271no—
119#488—1272no—
120#487—1271no—
121#479—1272no—
122#474D12711yescharacteristic
123#464—1272no—
124#462—1271no—
125C8×Q8—1271no—
126C4×Q16—1272no—
127C4×SD16—1271no—
128C2×Q16—1272no—
129C2×SD16—1271no—
130C4⋊Q8—1271no—
131C4⋊Q8—2541no—
132C4⋊Q8C2×D12711yescharacteristic
133#126—2542no—
134#125—1272no—
135#125—2542no—
136#124—1272no—
137C22⋊Q8—2541no—
138C4×Q8—1271no—
139C4×Q8—1272no—
140C4×Q8—1272no—
141C4×Q8—1271no—
142C4×Q8—2541no—
143C4×D4C2×D12711yescharacteristic
144#117—2541no—
145#107C2×D12712yes—
146#106—1271no—
147C4⋊C8—1272no—
148C4⋊C8C2×D12711yescharacteristic
149#103—1272no—
150#103—1271no—
151#103—1272no—
152#103—1272no—
153#102—1271no—
154#102C2×D12712yes—
155C4×C8—1272no—
156C4×C8—1271no—
157C2×Q8—21no—
158C2×Q8—1271no—
159C2×Q8—1271no—
160C2×Q8—1272no—
161C2×D4C22×D12711yescharacteristic
162C22×C4—21no—
163Q16—2544no—
164SD16—2542no—
165C2×C8—1271no—
166C2×C8—1271no—
167C2×C8C22×D12712yes—
168C4⋊C4—21no—
169C4⋊C4—1272no—
170C4⋊C4—1271no—
171C4⋊C4—1271no—
172C4⋊C4—2542no—
173C4⋊C4—2542no—
174C4⋊C4—2542no—
175C4⋊C4—2542no—
176C4⋊C4—2541no—
177C4⋊C4—2541no—
178C4⋊C4—2541no—
179C4⋊C4C22×D12712yes—
180C4⋊C4C22×D12711yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—2542no—
183C22⋊C4—2542no—
184C42—1272no—
185C42—2542no—
186C42—2542no—
187C42—2541no—
188C42C22×D12711yescharacteristic
189C23—21no—
190Q8—42no—
191Q8—1272no—
192Q8—2542no—
193Q8—2542no—
194Q8—2542no—
195Q8—2541no—
196D4—21no—
197D4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—1272no—
202C2×C4—1272no—
203C2×C4—2542no—
204C2×C4—2541no—
205C2×C4—2541no—
206C2×C4D4×D12711yescharacteristic
207C2×C4D4×D12711yescharacteristic
208C2×C4C23×D12711yescharacteristic
209C8—22no—
210C8—2541no—
211C8—2541no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22C2×D4×D12711yescharacteristic
216C4—21no—
217C4—41no—
218C4—42no—
219C4—5082no—
220C4—5081no—
221C4—2542no—
222C4—2542no—
223C4—2542no—
224C4#1686122812yes—
225C4#41×D12711yescharacteristic
226C4#1686125711yescharacteristic
227C2—81no—
228C2#2255053611yescharacteristic
229C2C2×#1686122811yescharacteristic
230C2#2255069211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#474×C127C2
socleC22×C127C22×D4
prime radicalC2×D4×C127C23
hypercenter#474D127
hypocenterC127#3364
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. #22550601C2semidirect
4. #22550228C2semidirect
5. #22550182C2semidirect
7. #22550087C2semidirect
8. #22550000C2semidirect
9. #22549970C2semidirect
116. C127#3364semidirect

Maximal subgroups

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

subgroupindex
#225506012
#225505962
#225502292
#225502282
#225501822
#225500972
#225500872
#225500002
#225499702
C4×#145232362
#474×C1272
#3364127