Finite Group Databasebeta

← #27142225

Subgroups — #27142225

Order 12416.

346 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 93 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#20261210C211yesmaximal, characteristic, complemented
2#20260752C211yesmaximal, characteristic, complemented
3#20260740C211yesmaximal, characteristic, complemented
4#20260709C211yesmaximal, characteristic, complemented
5#20260506C211yesmaximal, characteristic, complemented
6#20260492C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C97C211yesmaximal, characteristic
8C2×#13095495C2211yescharacteristic
9D4×Dic97—22no—
10D4×Dic97—22no—
11#15690769C2211yescharacteristic
12#15690769C2211yescharacteristic
13#15690769C2211yescharacteristic
14C2×C22⋊Dic97—22no—
15#15690737—22no—
16#15690731—22no—
17#15690725—22no—
18C2×#41×C97C2211yescharacteristic
19C22×D4×C97C2211yescharacteristic
20AGL(1,ℤ/8)×C97—22no—
21AGL(1,ℤ/8)×C97—22no—
22C2×SD16×C97C412yescomplemented
23C2×D8×C97C412yescomplemented
24C2×M16×C97C2211yescharacteristic
25C22×Dic97—24no—
26#13095495—21no—
27#13095495—21no—
28C2×#11715895—21no—
29C4⋊Dic97—21no—
30C4⋊Dic97—21no—
31C4⋊Dic97—21no—
32C22⋊Dic97—24no—
33C22⋊Dic97—21no—
34C22⋊Dic97—21no—
35C22⋊Dic97D412yes—
36C4×Dic97—21no—
37C4×Dic97—21no—
38C4×Dic97D412yes—
39C24×C97—21no—
40#41×C97—21no—
41#41×C97—21no—
42#41×C97—21no—
43#41×C97—21no—
44C2×Q8×C97C2×C411yescharacteristic, complemented
45C2×D4×C97—21no—
46C2×D4×C97—21no—
47C2×D4×C97—21no—
48C2×D4×C97—21no—
49C2×D4×C97C2×C412yes—
50C2×D4×C97C2×C411yescharacteristic, complemented
51C2×D4×C97D412yes—
52C2×D4×C97D412yes—
53C22×C4×C97—21no—
54C22×C4×C97C2311yescharacteristic
55SD16×C97—24no—
56D8×C97—24no—
57M16×C97—21no—
58M16×C97—21no—
59C2×C8×C97C2×C412yescomplemented
60C23×C97—22no—
61C23×C97—22no—
62C23×C97—22no—
63C23×C97—21no—
64C23×C97—82no—
65C23×C97C2×D411yescharacteristic
66Q8×C97—21no—
67Q8×C97—21no—
68D4×C97—21no—
69D4×C97—22no—
70D4×C97—21no—
71D4×C97—22no—
72D4×C97—22no—
73D4×C97—41no—
74D4×C97—41no—
75D4×C97C22⋊C414yes—
76C2×C4×C97—21no—
77C2×C4×C97—41no—
78C2×C4×C97—41no—
79C2×C4×C97C22×C411yescharacteristic
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97#4111yescharacteristic
84C2×C4×C97#4111yescharacteristic
85C2×Dic97—22no—
86C2×Dic97—22no—
87C2×Dic97—41no—
88C2×Dic97—84no—
89C8×C97—22no—
90#11715895—41no—
91C22×C97—24no—
92C22×C97—41no—
93C22×C97—44no—
94C22×C97—41no—
95C22×C97—82no—
96C22×C97—82no—
97C22×C97—81no—
98C22×C97C4×D411yescharacteristic
99C22×C97#12011yescharacteristic
100C22×C97#12111yescharacteristic
101C4×C97—41no—
102C4×C97C2×C22⋊C411yescharacteristic
103C4×C97C4×D411yescharacteristic
104C4×C97#12111yescharacteristic
105C4×C97#12311yescharacteristic
106Dic97—42no—
107Dic97—42no—
108Dic97—81no—
109C2×C97—21no—
110C2×C97—21no—
111C2×C97—21no—
112C2×C97—44no—
113C2×C97—81no—
114C2×C97#38611yescharacteristic
115#1904—971nomaximal, Sylow
116C97#190411yescharacteristic, complemented, Sylow
117C2×AGL(1,ℤ/8)D9711yescharacteristic
118#518—971no—
119#421—971no—
120#418—971no—
121#410—971no—
122#344—971no—
123H(3,ℤ/4)—971no—
124C2×#41C2×D9711yescharacteristic
125C22×D4C2×D9711yescharacteristic
126AGL(1,ℤ/8)—22no—
127AGL(1,ℤ/8)—22no—
128C2×SD16Dic9712yescomplemented
129C2×D8Dic9712yescomplemented
130C2×M16—971no—
131C2×M16C2×D9711yescharacteristic
132C4×D4—1942no—
133C4×D4—1942no—
134#117—971no—
135#117—971no—
136#117—971no—
137C2×C22⋊C4—1942no—
138#104—1942no—
139#102—1942no—
140#99—1942no—
141C24—21no—
142#41—21no—
143#41—21no—
144#41—21no—
145#41—21no—
146C2×Q8C2×Dic9711yescharacteristic, complemented
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4C2×Dic9712yes—
152C2×D4C2×Dic9711yescharacteristic, complemented
153C2×D4C22⋊D9712yes—
154C2×D4C22⋊D9712yes—
155C22×C4—21no—
156C22×C4—1944no—
157C22×C4C22×D9711yescharacteristic
158SD16—24no—
159D8—24no—
160M16—21no—
161M16—21no—
162M16—1941no—
163M16—1941no—
164C2×C8—1941no—
165C2×C8C2×Dic9712yescomplemented
166C4⋊C4—1941no—
167C4⋊C4—1941no—
168C4⋊C4—1941no—
169C22⋊C4—972no—
170C22⋊C4—1941no—
171C22⋊C4—1944no—
172C22⋊C4—1941no—
173C42—972no—
174C42—1941no—
175C42—1941no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—21no—
180C23—82no—
181C23D4×D9711yescharacteristic
182Q8—21no—
183Q8—21no—
184D4—21no—
185D4—22no—
186D4—21no—
187D4—22no—
188D4—22no—
189D4—41no—
190D4—41no—
191D4C22⋊Dic9714yes—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—7764no—
196C2×C4—3881no—
197C2×C4—1942no—
198C2×C4—1942no—
199C2×C4C22×Dic9711yescharacteristic
200C2×C4D4×D9711yescharacteristic
201C2×C4C2×C22⋊D9711yescharacteristic
202C2×C4C2×C22⋊D9711yescharacteristic
203C2×C4#1309551011yescharacteristic
204C2×C4#1309551011yescharacteristic
205C8—22no—
206C8—3881no—
207C22—24no—
208C22—41no—
209C22—44no—
210C22—41no—
211C22—82no—
212C22—82no—
213C22—81no—
214C22D4×Dic9711yescharacteristic
215C22#1569077811yescharacteristic
216C22#1569078211yescharacteristic
217C4—41no—
218C4—7761no—
219C4—3882no—
220C4—3882no—
221C4C2×C22⋊Dic9711yescharacteristic
222C4D4×Dic9711yescharacteristic
223C4#1569078211yescharacteristic
224C4#1569079111yescharacteristic
225C2—21no—
226C2—21no—
227C2—21no—
228C2—44no—
229C2—81no—
230C2#2026061511yescharacteristic
subgroupG/H
centerC2#20260615
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting, prime radicalC2×AGL(1,ℤ/8)×C97C2
socleC2×C97#386
cosocleC22×C4×C97C23
hypercenterC2×AGL(1,ℤ/8)D97
hypocenterC97#1904
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #20261210C2semidirect
2. #20260752C2semidirect
3. #20260740C2semidirect
4. #20260709C2semidirect
5. #20260506C2semidirect
6. #20260492C2semidirect
22. C2×SD16×C97C4semidirect
23. C2×D8×C97C4semidirect
44. C2×Q8×C97C2×C4semidirect
50. C2×D4×C97C2×C4semidirect
59. C2×C8×C97C2×C4semidirect
116. C97#1904semidirect
128. C2×SD16Dic97semidirect
129. C2×D8Dic97semidirect
146. C2×Q8C2×Dic97semidirect
152. C2×D4C2×Dic97semidirect
165. C2×C8C2×Dic97semidirect

Maximal subgroups

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

subgroupindex
#202612102
#202607522
#202607402
#202607092
#202605062
#202604922
C2×AGL(1,ℤ/8)×C972
#190497