Finite Group Databasebeta

← #27150216

Subgroups — #27150216

Order 12416.

704 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 209 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×#15690858C211yesmaximal, characteristic
2#20261501C212yesmaximal, complemented
3#20260718C214yesmaximal, complemented
4#20260717C212yesmaximal, complemented
5C2×#15690729C211yesmaximal, characteristic, complemented
6#101×D97C214yesmaximal, complemented
7C2×#101×C97C211yesmaximal, characteristic, complemented
8#15690858—24no—
9#15690858—24no—
10C2×#13095512C412yes—
11C2×#13095512C2212yes—
12C2×#13095511C414yes—
13C2×#13095511C2211yescharacteristic
14C2×#13095511C2211yescharacteristic
15C2×Q8×D97C412yes—
16C2×Q8×D97C2212yes—
17#15690818—24no—
18#15690816—24no—
19C2×#13095495C2212yes—
20M16×D97C2218yes—
21#15690729C2214yes—
22#15690728C2214yes—
23#15690726C2214yes—
24C22×Q8×C97C2211yescharacteristic
25C2×M16×C97C2212yes—
26#101×C97C2214yes—
27#13095512—28no—
28#13095512—28no—
29#13095511—22no—
30#13095511—22no—
31#13095511—28no—
32#13095511—28no—
33#13095511D414yes—
34#13095511D414yes—
35C2×Dic194C2×C414yes—
36C2×Dic194C2×C411yescharacteristic
37C2×Dic194C2311yescharacteristic
38Q8×D97—28no—
39Q8×D97—28no—
40C2×D388C2×C414yes—
41C2×D388C2×C411yescharacteristic
42C2×D388C2311yescharacteristic
43C2×C22⋊D97C2×C412yes—
44C2×C22⋊D97C2×C412yes—
45C2×C4×D97C2×C414yes—
46C2×C4×D97C2×C412yes—
47C2×C4×D97C2×C414yes—
48C2×C4×D97C2312yes—
49#13095495—22no—
50#13095495C2314yes—
51#13095494—28no—
52C2×#11715895—22no—
53C8×D97—28no—
54C2×Q8×C97—22no—
55C2×Q8×C97—22no—
56C2×Q8×C97C2×C412yes—
57C2×Q8×C97C2312yes—
58C22×C4×C97C2×C412yes—
59C22×C4×C97C2311yescharacteristic
60M16×C97—22no—
61M16×C97C2314yes—
62C2×C8×C97—22no—
63C23×C97C22×C411yescharacteristic
64C22×D97C22×C412yes—
65C22×D97C22×C412yes—
66Q8×C97—24no—
67Q8×C97—24no—
68C2×C4×C97—22no—
69C2×C4×C97—22no—
70C2×C4×C97C22×C414yes—
71C2×C4×C97C22×C411yescharacteristic
72C2×C4×C97C2×D412yes—
73C2×C4×C97C2×D412yes—
74C2×C4×C97C2411yescharacteristic
75C22⋊D97—22no—
76C22⋊D97—22no—
77C22⋊D97C22⋊C414yes—
78C22⋊D97C22⋊C414yes—
79C2×Dic97C22×C412yes—
80C2×Dic97C22×C412yes—
81D388—28no—
82D388—22no—
83D388C22⋊C414yes—
84C4×D97—28no—
85C4×D97—24no—
86C4×D97—28no—
87C4×D97C2×D418yes—
88Dic194—22no—
89Dic194—28no—
90Dic194C22⋊C414yes—
91C8×C97—24no—
92#11715895—24no—
93C22×C97—41no—
94C22×C97C2×C22⋊C411yescharacteristic
95C22×C97C2×C22⋊C411yescharacteristic
96C22×C97C23×C411yescharacteristic
97C2×D97—22no—
98C2×D97—42no—
99C2×D97—42no—
100C2×D97C2×C22⋊C414yes—
101C4×C97—24no—
102C4×C97C2×C22⋊C412yes—
103C4×C97C22×D412yes—
104Dic97—22no—
105Dic97C2×C22⋊C414yes—
106C2×C97—21no—
107C2×C97—21no—
108C2×C97—21no—
109C2×C97C22×C22⋊C411yescharacteristic
110D97—24no—
111D97—42no—
112#2889—971nomaximal, Sylow
113C97#288911yescharacteristic, complemented, Sylow
114C2×2−1+4—971no—
115#568—972no—
116#413—972no—
117#413—974no—
118C2×#101—971no—
119C2×#101—974no—
120C2×#101D9711yescharacteristic, complemented
1212−1+4—1944no—
1222−1+4—1944no—
123C2×#41—972no—
124C2×#41—972no—
125C2×#41—974no—
126C2×#41—971no—
127C2×#41—971no—
128C22×Q8—972no—
129C22×Q8—972no—
130C22×Q8C2×D9711yescharacteristic
131#131—1944no—
132#131—1944no—
133C2×M16—972no—
134C2×M16—978no—
135C2×M16C2×D9712yes—
136#101—974no—
137#101—974no—
138#101C2×D9714yes—
139#100—974no—
140#41—974no—
141#41—974no—
142#41—1948no—
143#41—1948no—
144#41—1948no—
145#41—1948no—
146#41—1942no—
147#41—1942no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—971no—
151C2×Q8—974no—
152C2×Q8—971no—
153C2×Q8—1948no—
154C2×Q8—1948no—
155C2×Q8C4×D9712yes—
156C2×Q8C22×D9712yes—
157C2×D4—972no—
158C2×D4—972no—
159C2×D4—971no—
160C2×D4—974no—
161C2×D4—971no—
162C22×C4—972no—
163C22×C4—974no—
164C22×C4—974no—
165C22×C4—972no—
166C22×C4C4×D9712yes—
167C22×C4C22×D9711yescharacteristic
168M16—22no—
169M16—974no—
170M16—1948no—
171M16—1942no—
172M16C22×D9714yes—
173C2×C8—22no—
174C2×C8—1948no—
175C2×C8—1942no—
176C23—972no—
177C23—972no—
178C23C2×C4×D9711yescharacteristic
179Q8—24no—
180Q8—24no—
181Q8—974no—
182Q8—1948no—
183Q8—1942no—
184D4—974no—
185D4—974no—
186D4—974no—
187D4—1948no—
188D4—1942no—
189D4—1942no—
190D4—1942no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—972no—
194C2×C4—978no—
195C2×C4—972no—
196C2×C4—1948no—
197C2×C4—1948no—
198C2×C4—1944no—
199C2×C4C2×C4×D9714yes—
200C2×C4C2×C4×D9711yescharacteristic
201C2×C4D4×D9712yes—
202C2×C4D4×D9712yes—
203C2×C4C23×D9711yescharacteristic
204C8—24no—
205C8—1944no—
206C22—41no—
207C22—974no—
208C22—3882no—
209C22—3882no—
210C22—1942no—
211C22C22⋊C4×D9711yescharacteristic
212C22C22⋊C4×D9711yescharacteristic
213C22C22×C4×D9711yescharacteristic
214C4—24no—
215C4—974no—
216C4—1942no—
217C4C22⋊C4×D9712yes—
218C4C2×D4×D9712yes—
219C2—21no—
220C2—21no—
221C2—21no—
222C2—3882no—
223C2—1944no—
224C2C2×C22⋊C4×D9711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D97
commutatorC22×C97C23×C4
FrattiniC2×C4C23×D97
FittingC2×#101×C97C2
socleC2×C97C22×C22⋊C4
prime radicalC2×#15690858C2
cosocleC2×C4×C97C24
hypercenterC2×#101D97
hypocenterC97#2889
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. #20261501C2semidirect
3. #20260718C2semidirect
4. #20260717C2semidirect
5. C2×#15690729C2semidirect
6. #101×D97C2semidirect
7. C2×#101×C97C2semidirect
113. C97#2889semidirect
120. C2×#101D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908582
#202615012
#202607182
#202607172
C2×#156907292
#101×D972
C2×#101×C972
#288997