Finite Group Databasebeta

← #27141949

Subgroups — #27141949

Order 12416.

272 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 83 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#20261212C211yesmaximal, characteristic
2#20260775C211yesmaximal, characteristic
3C2×#15690728C211yesmaximal, characteristic
4#20260640C211yesmaximal, characteristic
5#20260496C211yesmaximal, characteristic
6#20260494C211yesmaximal, characteristic
7#428×C97C211yesmaximal, characteristic
8C22×Dic194C2211yescharacteristic
9C2×#13095495C2211yescharacteristic
10C4×Dic194—21no—
11C4×Dic194—21no—
12#15690775—21no—
13#15690775—21no—
14#15690769C2211yescharacteristic
15#15690766—22no—
16C2×C4×Dic97C2211yescharacteristic
17#15690743—21no—
18#15690741—21no—
19#15690728C414yescomplemented
20#15690717—22no—
21C2×M16×C97C2211yescharacteristic
22#117×C97C2211yescharacteristic
23C2×C4⋊C4×C97C2211yescharacteristic
24#107×C97—21no—
25#106×C97—21no—
26C2×Dic194—21no—
27C2×Dic194—21no—
28C2×Dic194—21no—
29C2×Dic194—21no—
30C2×Dic194C2×C412yes—
31C2×Dic194Q812yes—
32C22×Dic97—21no—
33C22×Dic97D412yes—
34#13095495—21no—
35#13095495C2×C412yes—
36C2×#11715895—21no—
37C4⋊Dic97D411yescharacteristic
38C4⋊Dic97D411yescharacteristic
39#13095490—22no—
40#13095490—22no—
41C22⋊Dic97—21no—
42C4×Dic97—22no—
43C4×Dic97—22no—
44C4×Dic97—21no—
45C4×Dic97—21no—
46C4×Dic97—21no—
47C22×C4×C97—21no—
48C22×C4×C97C2311yescharacteristic
49M16×C97—21no—
50M16×C97C2×C412yes—
51C2×C8×C97—21no—
52C4⋊C4×C97—21no—
53C4⋊C4×C97—21no—
54C4⋊C4×C97D411yescharacteristic
55C4⋊C4×C97D411yescharacteristic
56C22⋊C4×C97—21no—
57C42×C97—21no—
58C23×C97C2×D411yescharacteristic
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97—42no—
62C2×C4×C97—41no—
63C2×C4×C97C22×C411yescharacteristic
64C2×C4×C97C2×D411yescharacteristic
65C2×C4×C97C2×D411yescharacteristic
66C2×C4×C97C2×Q811yescharacteristic
67C2×C4×C97#4111yescharacteristic
68C2×C4×C97#4111yescharacteristic
69C2×Dic97—22no—
70C2×Dic97—22no—
71C2×Dic97—22no—
72C2×Dic97—21no—
73C2×Dic97—21no—
74C2×Dic97—41no—
75C2×Dic97—41no—
76C2×Dic97C4⋊C414yes—
77Dic194—22no—
78Dic194—22no—
79Dic194—22no—
80Dic194—22no—
81C8×C97—22no—
82#11715895—22no—
83C22×C97—21no—
84C22×C97—21no—
85C22×C97C2×C4⋊C411yescharacteristic
86C22×C97#12011yescharacteristic
87C22×C97#12311yescharacteristic
88C4×C97—41no—
89C4×C97—41no—
90C4×C97—81no—
91C4×C97C4×D411yescharacteristic
92C4×C97C4×D411yescharacteristic
93C4×C97C22⋊Q811yescharacteristic
94C4×C97C22⋊Q811yescharacteristic
95Dic97—24no—
96Dic97—42no—
97Dic97—41no—
98Dic97—41no—
99C2×C97—21no—
100C2×C97—21no—
101C2×C97#38511yescharacteristic
102C2×C97#45411yescharacteristic
103C2×C97#45611yescharacteristic
104#1857—971nomaximal, Sylow
105C97#185711yescharacteristic, complemented, Sylow
106#519—971no—
107#428—971no—
108#428D9711yescharacteristic
109C2×#101—971no—
110#391—971no—
111#339—971no—
112#339—971no—
113C22×Q8—971no—
114C2×M16—971no—
115C2×M16C2×D9711yescharacteristic
116C4×Q8—1941no—
117C4×Q8—1941no—
118C4×Q8—1941no—
119C4×Q8—1941no—
120#117—971no—
121#117—1942no—
122#117C2×D9711yescharacteristic
123C2×C4⋊C4C2×D9711yescharacteristic
124C2×C42—971no—
125#107—21no—
126#107—1941no—
127#106—21no—
128#106—1941no—
129#101—974no—
130#95—1942no—
131C2×Q8—972no—
132C2×Q8—972no—
133C2×Q8—1941no—
134C2×Q8—1941no—
135C2×Q8—1941no—
136C2×Q8—1941no—
137C22×C4—21no—
138C22×C4—972no—
139C22×C4—1941no—
140C22×C4C22×D9711yescharacteristic
141M16—21no—
142M16—972no—
143M16—1941no—
144M16C4×D9712yes—
145C2×C8—21no—
146C2×C8—1941no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—971no—
150C4⋊C4—971no—
151C4⋊C4—1942no—
152C4⋊C4—1942no—
153C4⋊C4C22⋊D9711yescharacteristic
154C4⋊C4C22⋊D9711yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—1941no—
157C42—21no—
158C42—1942no—
159C42—1942no—
160C42—1941no—
161C42—1941no—
162C42—1941no—
163C23D4×D9711yescharacteristic
164Q8—1942no—
165Q8—1942no—
166Q8—1942no—
167Q8—1942no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—974no—
173C2×C4—3881no—
174C2×C4—3881no—
175C2×C4—1942no—
176C2×C4—1941no—
177C2×C4—1941no—
178C2×C4—1942no—
179C2×C4—1942no—
180C2×C4C2×C4×D9711yescharacteristic
181C2×C4D4×D9711yescharacteristic
182C2×C4C2×C22⋊D9711yescharacteristic
183C2×C4Q8×D9711yescharacteristic
184C2×C4#1309551111yescharacteristic
185C2×C4#1309551211yescharacteristic
186C8—22no—
187C8—1942no—
188C22—21no—
189C22—21no—
190C22C4⋊C4×D9711yescharacteristic
191C22#1569077811yescharacteristic
192C22#1569079511yescharacteristic
193C4—41no—
194C4—41no—
195C4—81no—
196C4—3882no—
197C4—3881no—
198C4—3881no—
199C4—1944no—
200C4C4×C22⋊D9711yescharacteristic
201C4#1569077211yescharacteristic
202C4#1569078611yescharacteristic
203C4#1569078811yescharacteristic
204C2—21no—
205C2—21no—
206C2#2026060611yescharacteristic
207C2#2026090611yescharacteristic
208C2#2026091611yescharacteristic
subgroupG/H
centerC22#15690778
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#428×C97C2
socleC22×C97#120
prime radicalC23×C97C2×D4
cosocleC22×C4×C97C23
hypercenter#428D97
hypocenterC97#1857
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
19. #15690728C4semidirect
105. C97#1857semidirect

Maximal subgroups

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

subgroupindex
#202612122
#202607752
C2×#156907282
#202606402
#202604962
#202604942
#428×C972
#185797