Finite Group Databasebeta

← #27150107

Subgroups — #27150107

Order 12416.

524 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 153 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×#15690801C211yesmaximal, characteristic, complemented
2C2×#15690797C211yesmaximal, characteristic
3#20260641C211yesmaximal, characteristic, complemented
4#20260640C211yesmaximal, characteristic, complemented
5#20260638C211yesmaximal, characteristic
6#20260628C214yesmaximal, complemented
7#20260623C214yesmaximal, complemented
8C42×Dic97C211yesmaximal, characteristic, complemented
9#390×C97C211yesmaximal, characteristic
10C22×Dic194C2211yescharacteristic
11#15690801—24no—
12#15690797—24no—
13C2×C4⋊Dic97C2212yes—
14C2×C4⋊Dic97C2211yescharacteristic
15C2×#13095490C2212yes—
16C2×C22⋊Dic97C2212yes—
17C2×C22⋊Dic97C2214yes—
18C2×C22⋊Dic97C2212yes—
19C2×C4×Dic97C2212yes—
20C2×C4×Dic97C2212yes—
21C2×C4×Dic97C2211yescharacteristic
22C2×C4×Dic97C2211yescharacteristic
23#15690717C2214yes—
24#15690717C2214yes—
25C22×D4×C97C2211yescharacteristic
26C2×C4⋊C4×C97C2211yescharacteristic
27C2×C22⋊C4×C97C2214yes—
28C2×C42×C97C2211yescharacteristic
29C2×Dic194—22no—
30C2×Dic194—22no—
31C2×Dic194—22no—
32C22×Dic97C2312yes—
33C22×Dic97C2312yes—
34C22×Dic97C2311yescharacteristic
35C22×Dic97C2311yescharacteristic
36C22×Dic97C2312yes—
37C4⋊Dic97—22no—
38C4⋊Dic97—24no—
39#13095490—24no—
40C22⋊Dic97—24no—
41C22⋊Dic97—24no—
42C22⋊Dic97—28no—
43C4×Dic97—24no—
44C4×Dic97—22no—
45C4×Dic97—24no—
46C4×Dic97D414yes—
47C24×C97C2312yes—
48C2×D4×C97—22no—
49C2×D4×C97—22no—
50C2×D4×C97—22no—
51C22×C4×C97C2312yes—
52C22×C4×C97C2312yes—
53C22×C4×C97C2311yescharacteristic
54C4⋊C4×C97—22no—
55C22⋊C4×C97—28no—
56C42×C97—22no—
57C23×C97—22no—
58C23×C97—24no—
59C23×C97—22no—
60C23×C97—22no—
61C23×C97—24no—
62C23×C97C2411yescharacteristic
63D4×C97—44no—
64C2×C4×C97—22no—
65C2×C4×C97—22no—
66C2×C4×C97—22no—
67C2×C4×C97—24no—
68C2×C4×C97C2×D412yes—
69C2×C4×C97#4114yes—
70C2×C4×C97#4112yes—
71C2×C4×C97#4112yes—
72C2×Dic97—22no—
73C2×Dic97—22no—
74C2×Dic97—22no—
75C2×Dic97—22no—
76C2×Dic97—22no—
77C2×Dic97—22no—
78C2×Dic97—24no—
79C2×Dic97—22no—
80C2×Dic97—22no—
81C2×Dic97C2×D412yes—
82C2×Dic97C2×D412yes—
83C2×Dic97#4114yes—
84Dic194—44no—
85C22×C97—42no—
86C22×C97—44no—
87C22×C97—44no—
88C22×C97—42no—
89C22×C97—42no—
90C22×C97C22×D411yescharacteristic
91C22×C97C2×#4112yes—
92C22×C97C2×#4111yescharacteristic
93C22×C97C2×#4112yes—
94C22×C97C2×#4111yescharacteristic
95C4×C97—24no—
96C4×C97—42no—
97C4×C97#12414yes—
98Dic97—24no—
99Dic97—22no—
100Dic97—22no—
101Dic97—42no—
102Dic97—42no—
103C2×C97—82no—
104C2×C97C2×#12411yescharacteristic
105C2×C97#52912yes—
106C2×C97#52912yes—
107C2×C97#53211yescharacteristic
108C2×C97#53211yescharacteristic
109#2861—971nomaximal, Sylow
110C97#286111yescharacteristic, complemented, Sylow
111C2×#125—971no—
112C2×#124—971no—
113#391—971no—
114#391—971no—
115#390—971no—
116#390D9711yescharacteristic
117#388—974no—
118#388—974no—
119C43—971no—
120C22×Q8—971no—
121C22×D4C2×D9711yescharacteristic
122#125—1944no—
123#124—1944no—
124C2×C4⋊C4—972no—
125C2×C4⋊C4—972no—
126C2×C4⋊C4—971no—
127C2×C4⋊C4C2×D9711yescharacteristic
128C2×C22⋊C4—972no—
129C2×C22⋊C4—972no—
130C2×C22⋊C4—974no—
131C2×C22⋊C4C2×D9714yes—
132C2×C42—971no—
133C2×C42—972no—
134C2×C42—972no—
135C2×C42—971no—
136C2×C42C2×D9711yescharacteristic
137#95—974no—
138#95—974no—
139C24C22×D9712yes—
140C2×Q8—1942no—
141C2×Q8—1942no—
142C2×Q8—1942no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—22no—
146C22×C4—972no—
147C22×C4—971no—
148C22×C4—971no—
149C22×C4—972no—
150C22×C4—972no—
151C22×C4C22×D9712yes—
152C22×C4C22×D9712yes—
153C22×C4C22×D9711yescharacteristic
154C4⋊C4—22no—
155C4⋊C4—1944no—
156C4⋊C4—1944no—
157C4⋊C4—1942no—
158C22⋊C4—28no—
159C22⋊C4—1948no—
160C22⋊C4—1944no—
161C22⋊C4—1944no—
162C42—22no—
163C42—974no—
164C42—1944no—
165C42—1944no—
166C42—1942no—
167C23—22no—
168C23—24no—
169C23—24no—
170C23—22no—
171C23—22no—
172C23C23×D9711yescharacteristic
173Q8—3884no—
174D4—44no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—972no—
180C2×C4—972no—
181C2×C4—974no—
182C2×C4—1942no—
183C2×C4—1942no—
184C2×C4—1944no—
185C2×C4—1942no—
186C2×C4—1942no—
187C2×C4—1942no—
188C2×C4—1942no—
189C2×C4—1942no—
190C2×C4—1942no—
191C2×C4D4×D9712yes—
192C2×C4#1309551014yes—
193C2×C4#1309551012yes—
194C2×C4#1309551112yes—
195C22—42no—
196C22—44no—
197C22—44no—
198C22—42no—
199C22—42no—
200C22C2×D4×D9711yescharacteristic
201C22#41×D9712yes—
202C22C2×#1309551011yescharacteristic
203C22C2×#1309551012yes—
204C22C2×#1309551111yescharacteristic
205C4—24no—
206C4—42no—
207C4—3882no—
208C4—3882no—
209C4—1944no—
210C4—1942no—
211C4—1942no—
212C4#1569079614yes—
213C2—82no—
214C2C2×#1569079611yescharacteristic
215C2#2026126612yes—
216C2#2026126712yes—
217C2#2026128311yescharacteristic
218C2#2026128411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#390×C97C2
prime radicalC22×D4×C97C22
hypercenter#390D97
hypocenterC97#2861
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#15690801C2semidirect
3. #20260641C2semidirect
4. #20260640C2semidirect
6. #20260628C2semidirect
7. #20260623C2semidirect
8. C42×Dic97C2semidirect
110. C97#2861semidirect

Maximal subgroups

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

subgroupindex
C2×#156908012
C2×#156907972
#202606412
#202606402
#202606382
#202606282
#202606232
C42×Dic972
#390×C972
#286197