Finite Group Databasebeta

← #27142270

Subgroups — #27142270

Order 12416.

314 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 79 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#20261306C211yesmaximal, characteristic
2#20261212C211yesmaximal, characteristic, complemented
3#20260752C211yesmaximal, characteristic
4C2×#15690728C211yesmaximal, characteristic, complemented
5#20260492C211yesmaximal, characteristic, complemented
6#20260491C211yesmaximal, characteristic, complemented
7#421×C97C211yesmaximal, characteristic
8C22×Dic194C2211yescharacteristic
9C2×#13095495C2211yescharacteristic
10#15690808C412yescomplemented
11#15690797C412yescomplemented
12#15690791—22no—
13#15690789—22no—
14C4×Dic194—22no—
15#15690775—22no—
16#15690769C2211yescharacteristic
17#15690769C2211yescharacteristic
18#15690766—22no—
19#15690737—22no—
20#15690728—22no—
21C2×#41×C97C2211yescharacteristic
22C2×M16×C97C2211yescharacteristic
23#117×C97C2211yescharacteristic
24#104×C97—22no—
25C2×Dic194—21no—
26C2×Dic194—21no—
27C2×Dic194—21no—
28C2×Dic194—21no—
29C2×Dic194C2×C412yes—
30C2×Dic194D412yes—
31C22×Dic97D412yes—
32#13095495—21no—
33#13095495—21no—
34C2×#11715895—21no—
35C4⋊Dic97—21no—
36C4⋊Dic97—21no—
37#13095490—24no—
38#13095490—24no—
39C22⋊Dic97—21no—
40C22⋊Dic97—21no—
41C22⋊Dic97—24no—
42C4×Dic97—24no—
43C4×Dic97C2×C412yes—
44C4×Dic97D412yes—
45#41×C97—21no—
46#41×C97—21no—
47#41×C97—21no—
48#41×C97—21no—
49C2×Q8×C97C2×C411yescharacteristic
50C2×D4×C97—21no—
51C2×D4×C97C2×C411yescharacteristic
52C22×C4×C97—21no—
53C22×C4×C97C2311yescharacteristic
54M16×C97—21no—
55M16×C97—21no—
56C2×C8×C97—21no—
57C4⋊C4×C97—21no—
58C22⋊C4×C97—21no—
59C42×C97D412yes—
60C23×C97—21no—
61C23×C97C2×D411yescharacteristic
62Q8×C97—21no—
63Q8×C97—21no—
64D4×C97—21no—
65D4×C97—21no—
66D4×C97—41no—
67D4×C97—41no—
68C2×C4×C97—21no—
69C2×C4×C97—22no—
70C2×C4×C97—41no—
71C2×C4×C97—41no—
72C2×C4×C97C22×C411yescharacteristic
73C2×C4×C97C2×D411yescharacteristic
74C2×C4×C97C2×D411yescharacteristic
75C2×C4×C97C2×D411yescharacteristic
76C2×C4×C97#4111yescharacteristic
77C2×C4×C97#4111yescharacteristic
78C2×Dic97—22no—
79C2×Dic97—22no—
80C2×Dic97—22no—
81C2×Dic97—22no—
82C2×Dic97C22⋊C414yes—
83Dic194—22no—
84Dic194—22no—
85Dic194—22no—
86Dic194—22no—
87C8×C97—41no—
88#11715895—41no—
89C22×C97—41no—
90C22×C97—42no—
91C22×C97—41no—
92C22×C97C2×C22⋊C411yescharacteristic
93C22×C97#12011yescharacteristic
94C22×C97#12311yescharacteristic
95C4×C97—42no—
96C4×C97—41no—
97C4×C97C4×D411yescharacteristic
98C4×C97C4×D411yescharacteristic
99C4×C97#12111yescharacteristic
100C4×C97#12111yescharacteristic
101Dic97—24no—
102Dic97—42no—
103Dic97—42no—
104C2×C97—21no—
105C2×C97—21no—
106C2×C97—21no—
107C2×C97—81no—
108C2×C97#38611yescharacteristic
109#1912—971nomaximal, Sylow
110C97#191211yescharacteristic, complemented, Sylow
111#536—971no—
112#519—971no—
113#421—971no—
114#421D9711yescharacteristic
115C2×#101—971no—
116H(3,ℤ/4)—971no—
117H(3,ℤ/4)—971no—
118C2×#41C2×D9711yescharacteristic
119C22×Q8—971no—
120C2×M16—971no—
121C2×M16C2×D9711yescharacteristic
122C4⋊Q8—972no—
123#124—972no—
124#123—1942no—
125C22⋊Q8—1942no—
126C4×Q8—1942no—
127C4×Q8—1942no—
128#117—971no—
129#117—971no—
130#117—1942no—
131#117C2×D9711yescharacteristic
132#104—22no—
133#104—1942no—
134#101—1942no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—972no—
140C2×Q8—972no—
141C2×Q8—1941no—
142C2×Q8—1941no—
143C2×Q8—1941no—
144C2×Q8—1941no—
145C2×Q8C4×D9711yescharacteristic
146C2×D4—21no—
147C2×D4C4×D9711yescharacteristic
148C22×C4—21no—
149C22×C4—972no—
150C22×C4C22×D9711yescharacteristic
151M16—21no—
152M16—21no—
153M16—1941no—
154M16—1941no—
155C2×C8—21no—
156C2×C8—1941no—
157C4⋊C4—21no—
158C4⋊C4—1944no—
159C4⋊C4—1944no—
160C4⋊C4—1941no—
161C4⋊C4—1941no—
162C22⋊C4—21no—
163C22⋊C4—1944no—
164C22⋊C4—1941no—
165C22⋊C4—1941no—
166C42—972no—
167C42—972no—
168C42—1944no—
169C42C22⋊D9712yes—
170C23—21no—
171C23D4×D9711yescharacteristic
172Q8—21no—
173Q8—21no—
174Q8—1942no—
175Q8—1942no—
176Q8—1942no—
177Q8—1942no—
178D4—21no—
179D4—21no—
180D4—41no—
181D4—41no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—974no—
187C2×C4—1942no—
188C2×C4—1942no—
189C2×C4—1942no—
190C2×C4—1942no—
191C2×C4C2×C4×D9711yescharacteristic
192C2×C4D4×D9711yescharacteristic
193C2×C4D4×D9711yescharacteristic
194C2×C4C2×C22⋊D9711yescharacteristic
195C2×C4#1309551011yescharacteristic
196C2×C4#1309551111yescharacteristic
197C8—41no—
198C8—3881no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C22⋊C4×D9711yescharacteristic
203C22#1569077811yescharacteristic
204C22#1569079211yescharacteristic
205C4—42no—
206C4—41no—
207C4—3882no—
208C4—3882no—
209C4—1944no—
210C4#1569077011yescharacteristic
211C4C4×C22⋊D9711yescharacteristic
212C4#1569078111yescharacteristic
213C4#1569078311yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—81no—
218C2#2026061111yescharacteristic
subgroupG/H
centerC2#20260611
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#421×C97C2
socleC2×C97#386
prime radicalC2×#41×C97C22
cosocleC22×C4×C97C23
hypercenter#421D97
hypocenterC97#1912
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #20261212C2semidirect
4. C2×#15690728C2semidirect
5. #20260492C2semidirect
6. #20260491C2semidirect
10. #15690808C4semidirect
11. #15690797C4semidirect
110. C97#1912semidirect

Maximal subgroups

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

subgroupindex
#202613062
#202612122
#202607522
C2×#156907282
#202604922
#202604912
#421×C972
#191297