Finite Group Databasebeta

← #30639850

Subgroups — #30639850

Order 16256.

342 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C2×#16861242C211yesmaximal, characteristic
2C2×#16861240C211yesmaximal, characteristic
3C22×#14523238C211yesmaximal, characteristic, complemented
4C2×#16861127C211yesmaximal, characteristic, complemented
5#22549844C211yesmaximal, characteristic
6#22549640C211yesmaximal, characteristic, complemented
7C2×#101×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9C22×Dic254C2211yescharacteristic
10#16861242—22no—
11#16861242—22no—
12#16861240—22no—
13#16861240—22no—
14C2×Dic508—21no—
15C2×#14523238—22no—
16C2×#14523238—21no—
17C2×#14523238—22no—
18C2×#14523238—22no—
19C2×#14523237—21no—
20C2×#14523235—21no—
21C2×#14523232C2211yescharacteristic
22C22×C127⋊C8C2211yescharacteristic
23#16861127—22no—
24(C2×D127)⋊C8—22no—
25C22×Q8×C127C2211yescharacteristic
26C2×M16×C127C2211yescharacteristic
27C2×M16×C127C2211yescharacteristic
28#101×C127—22no—
29#14523248—21no—
30#14523248—21no—
31#14523248—21no—
32#14523248—21no—
33C2×Dic254—21no—
34C2×Dic254—21no—
35C2×Dic254—21no—
36C2×Dic254—21no—
37C2×Dic254—21no—
38C2×Dic254D411yescharacteristic
39C2×Dic254D412yes—
40C2×D508D411yescharacteristic
41C2×C22⋊D127—21no—
42C22×Dic127—21no—
43C2×C4×D127—21no—
44Dic508—24no—
45#14523238—28no—
46#14523238—24no—
47#14523237—24no—
48#14523235—24no—
49#14523232—21no—
50#14523232D412yes—
51C2×C127⋊C8—21no—
52C2×C127⋊C8—21no—
53C2×C127⋊C8—21no—
54C2×C127⋊C8D412yes—
55C2×Q8×C127—21no—
56C2×Q8×C127—21no—
57C2×Q8×C127—21no—
58C2×Q8×C127—21no—
59C2×Q8×C127—21no—
60C2×Q8×C127D412yes—
61C22×C4×C127—21no—
62C22×C4×C127C2311yescharacteristic
63M16×C127—21no—
64M16×C127—21no—
65M16×C127—21no—
66M16×C127D412yes—
67C2×C8×C127—21no—
68C2×C8×C127—21no—
69C22×D127—21no—
70Dic254—22no—
71Dic254—22no—
72Dic254—21no—
73Dic254—21no—
74Dic254—41no—
75Dic254—41no—
76D508—21no—
77D508—21no—
78C22⋊D127—41no—
79C22⋊D127—41no—
80C2×Dic127—21no—
81C2×Dic127—22no—
82C2×Dic127—41no—
83C2×Dic127—41no—
84C4×D127—41no—
85C4×D127—41no—
86C127⋊C8—22no—
87C127⋊C8—22no—
88C23×C127#4111yescharacteristic
89Q8×C127—22no—
90Q8×C127—22no—
91Q8×C127—41no—
92Q8×C127—41no—
93C2×C4×C127—22no—
94C2×C4×C127—41no—
95C2×C4×C127—41no—
96C2×C4×C127C2×D411yescharacteristic
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2×D411yescharacteristic
100C2×C4×C127C2×D411yescharacteristic
101C2×C4×C127C2×D411yescharacteristic
102C8×C127—22no—
103C8×C127—41no—
104C22×C127—22no—
105C22×C127#12111yescharacteristic
106C22×C127#12111yescharacteristic
107C22×C127#12111yescharacteristic
108C2×D127—41no—
109C2×D127—42no—
110C4×C127—42no—
111C4×C127#12011yescharacteristic
112C4×C127#12011yescharacteristic
113C4×C127#12011yescharacteristic
114C4×C127#12711yescharacteristic
115Dic127—41no—
116Dic127—42no—
117C2×C127—21no—
118C2×C127—21no—
119C2×C127#39211yescharacteristic
120C2×C127#47312yes—
121D127—81no—
122#2020—1271nomaximal, Sylow
123C127#202011yescharacteristic, complemented, Sylow
124C2×#137—1271no—
125C2×#137—1271no—
126C22×Q16—1271no—
127C2×#101—1271no—
128C2×#101D12711yescharacteristic, complemented
129#408—1271no—
130#341—1271no—
131C2×#41—1271no—
132C22×Q8—1271no—
133C22×Q8C2×D12711yescharacteristic
134#137—2542no—
135#137—2542no—
136#137—2542no—
137#137—2542no—
138C2×Q16—2541no—
139C2×Q16—2542no—
140C2×Q16—2542no—
141C2×Q16—2541no—
142C2×Q16—2542no—
143C2×SD16—2541no—
144C2×SD16—2541no—
145C2×M16—1271no—
146C2×M16C2×D12711yescharacteristic
147C2×M16C2×D12711yescharacteristic
148C22×C8—1271no—
149#101—22no—
150#101—2542no—
151C22⋊C8—2542no—
152#41—2541no—
153#41—2541no—
154#41—2541no—
155#41—2541no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—1271no—
162C2×Q8—1272no—
163C2×Q8—2541no—
164C2×Q8—2541no—
165C2×Q8—2541no—
166C2×Q8—2541no—
167C2×Q8—2541no—
168C2×Q8D50812yes—
169C2×D4—1271no—
170C2×D4—2541no—
171C22×C4—21no—
172C22×C4—2541no—
173C22×C4—2541no—
174C22×C4C22×D12711yescharacteristic
175Q16—2544no—
176Q16—2548no—
177Q16—2544no—
178SD16—2544no—
179SD16—2544no—
180M16—21no—
181M16—21no—
182M16—21no—
183M16—1272no—
184M16—2541no—
185M16C22⋊D12712yes—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—1272no—
189C2×C8—2541no—
190C2×C8—2541no—
191C2×C8—2541no—
192C23—2541no—
193C23#1452324811yescharacteristic
194Q8—22no—
195Q8—22no—
196Q8—41no—
197Q8—41no—
198Q8—5081no—
199Q8—5081no—
200Q8—2542no—
201Q8—2541no—
202Q8—2542no—
203Q8—2541no—
204D4—5081no—
205D4—5081no—
206D4—2541no—
207D4—2541no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—5081no—
212C2×C4—5081no—
213C2×C4—5081no—
214C2×C4—5081no—
215C2×C4—2542no—
216C2×C4—2541no—
217C2×C4D4×D12711yescharacteristic
218C2×C4D4×D12711yescharacteristic
219C2×C4D4×D12711yescharacteristic
220C2×C4D4×D12711yescharacteristic
221C2×C4C2×C22⋊D12711yescharacteristic
222C2×C4C2×D50811yescharacteristic
223C8—22no—
224C8—41no—
225C8—2542no—
226C8—2542no—
227C22—22no—
228C22—5081no—
229C22—5082no—
230C22#1686118211yescharacteristic
231C22#1686118211yescharacteristic
232C22#1686118311yescharacteristic
233C4—42no—
234C4—5081no—
235C4—5082no—
236C4#1686117711yescharacteristic
237C4#1686117911yescharacteristic
238C4#1686117911yescharacteristic
239C4#1686120511yescharacteristic
240C2—21no—
241C2—21no—
242C2—10161no—
243C2#2254978511yescharacteristic
244C2#2255015012yes—
subgroupG/H
centerC22#16861182
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#101×C127C2
socleC22×C127#121
prime radicalC2×#14523248C22
hypercenterC2×#101D127
hypocenterC127#2020
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
3. C22×#14523238C2semidirect
4. C2×#16861127C2semidirect
6. #22549640C2semidirect
7. C2×#101×C127C2semidirect
123. C127#2020semidirect
128. C2×#101D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612422
C2×#168612402
C22×#145232382
C2×#168611272
#225498442
#225496402
C2×#101×C1272
#2020127