Finite Group Databasebeta

← #27142218

Subgroups — #27142218

Order 12416.

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

normal subgroups only

This group has 324 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#20261502C211yesmaximal, characteristic, complemented
2#20260752C211yesmaximal, characteristic, complemented
3#20260740C211yesmaximal, characteristic, complemented
4C2×#15690727C211yesmaximal, characteristic, complemented
5#20260506C211yesmaximal, characteristic, complemented
6#20260493C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C97C211yesmaximal, characteristic
8#15690817—22no—
9#15690817—22no—
10C2×#13095495—22no—
11C2×#13095495C2211yescharacteristic
12C2×#13095495C2211yescharacteristic
13C2×#13095495C2211yescharacteristic
14#15690769C2211yescharacteristic
15#15690737—22no—
16#15690731—22no—
17#15690727—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
25#13095495—24no—
26#13095495—21no—
27#13095495—21no—
28#13095495—21no—
29#13095495—21no—
30#13095495—21no—
31#13095495D412yes—
32C2×#11715895—24no—
33C2×#11715895—21no—
34C2×#11715895—21no—
35C2×#11715895D412yes—
36C4⋊Dic97—21no—
37C22⋊Dic97—21no—
38C4×Dic97—21no—
39C24×C97—21no—
40#41×C97—21no—
41#41×C97—21no—
42#41×C97D412yes—
43#41×C97D412yes—
44C2×Q8×C97C2×C411yescharacteristic
45C2×D4×C97—21no—
46C2×D4×C97—21no—
47C2×D4×C97—21no—
48C2×D4×C97—21no—
49C2×D4×C97—21no—
50C2×D4×C97—21no—
51C2×D4×C97C2×C412yes—
52C2×D4×C97C2×C411yescharacteristic
53C22×C4×C97—21no—
54C22×C4×C97C2311yescharacteristic
55SD16×C97—24no—
56D8×C97—24no—
57M16×C97—21no—
58M16×C97—21no—
59C2×C8×C97C2×C412yes—
60C23×C97—21no—
61C23×C97—22no—
62C23×C97—41no—
63C23×C97—41no—
64C23×C97—82no—
65C23×C97#4111yescharacteristic
66Q8×C97—21no—
67Q8×C97C22⋊C412yes—
68D4×C97—22no—
69D4×C97—22no—
70D4×C97—22no—
71D4×C97—21no—
72D4×C97—22no—
73D4×C97—41no—
74D4×C97—41no—
75D4×C97C22⋊C412yes—
76C2×C4×C97—21no—
77C2×C4×C97—22no—
78C2×C4×C97—22no—
79C2×C4×C97C22×C411yescharacteristic
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97C2×D411yescharacteristic
84C2×C4×C97#4111yescharacteristic
85C2×Dic97—41no—
86C8×C97—22no—
87#11715895—22no—
88#11715895—22no—
89#11715895—41no—
90C22×C97—22no—
91C22×C97—41no—
92C22×C97—42no—
93C22×C97—81no—
94C22×C97—82no—
95C22×C97—82no—
96C22×C97—82no—
97C22×C97C4×D411yescharacteristic
98C22×C97#12111yescharacteristic
99C22×C97#12311yescharacteristic
100C4×C97—22no—
101C4×C97C2×C22⋊C411yescharacteristic
102C4×C97C4×D411yescharacteristic
103C4×C97#12011yescharacteristic
104C4×C97#12111yescharacteristic
105Dic97—81no—
106C2×C97—21no—
107C2×C97—21no—
108C2×C97—21no—
109C2×C97—42no—
110C2×C97—82no—
111C2×C97#38611yescharacteristic
112#1903—971nomaximal, Sylow
113C97#190311yescharacteristic, complemented, Sylow
114C2×AGL(1,ℤ/8)D9711yescharacteristic
115#568—971no—
116#421—971no—
117#418—971no—
118C2×#100—971no—
119#344—971no—
120#338—971no—
121C2×#41C2×D9711yescharacteristic
122C22×D4C2×D9711yescharacteristic
123AGL(1,ℤ/8)—22no—
124AGL(1,ℤ/8)—22no—
125C2×SD16Dic9712yescomplemented
126C2×D8Dic9712yescomplemented
127#131—1942no—
128#131—1942no—
129C2×M16—971no—
130C2×M16—971no—
131C2×M16—971no—
132C2×M16—1942no—
133C2×M16C2×D9711yescharacteristic
134#117—971no—
135#104—1942no—
136#102—1942no—
137#100—1942no—
138C24—21no—
139#41—21no—
140#41—21no—
141#41C22⋊D9712yes—
142#41C22⋊D9712yes—
143C2×Q8C2×Dic9711yescharacteristic
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4C2×Dic9712yes—
151C2×D4C2×Dic9711yescharacteristic
152C22×C4—21no—
153C22×C4C22×D9711yescharacteristic
154SD16—24no—
155D8—24no—
156M16—21no—
157M16—21no—
158M16—972no—
159M16—1941no—
160M16—1941no—
161M16—1944no—
162M16—1941no—
163M16—1941no—
164M16—1941no—
165C2×C8—972no—
166C2×C8—1941no—
167C2×C8—1944no—
168C2×C8—1941no—
169C2×C8C2×Dic9712yes—
170C4⋊C4—1941no—
171C22⋊C4—1941no—
172C42—1941no—
173C23—21no—
174C23—22no—
175C23—41no—
176C23—41no—
177C23—82no—
178C23#1309551011yescharacteristic
179Q8—21no—
180Q8C22⋊Dic9712yes—
181D4—22no—
182D4—22no—
183D4—22no—
184D4—21no—
185D4—22no—
186D4—41no—
187D4—41no—
188D4C22⋊Dic9712yes—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—3881no—
193C2×C4C22×Dic9711yescharacteristic
194C2×C4D4×D9711yescharacteristic
195C2×C4D4×D9711yescharacteristic
196C2×C4C2×C22⋊D9711yescharacteristic
197C2×C4C2×C22⋊D9711yescharacteristic
198C2×C4#1309551011yescharacteristic
199C8—22no—
200C8—3881no—
201C8—1942no—
202C8—1942no—
203C22—22no—
204C22—41no—
205C22—42no—
206C22—81no—
207C22—82no—
208C22—82no—
209C22—82no—
210C22D4×Dic9711yescharacteristic
211C22#1569078211yescharacteristic
212C22#1569079111yescharacteristic
213C4—22no—
214C4—7761no—
215C4C2×C22⋊Dic9711yescharacteristic
216C4D4×Dic9711yescharacteristic
217C4#1569077811yescharacteristic
218C4#1569078211yescharacteristic
219C2—21no—
220C2—21no—
221C2—21no—
222C2—42no—
223C2—82no—
224C2#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#1903
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. #20261502C2semidirect
2. #20260752C2semidirect
3. #20260740C2semidirect
4. C2×#15690727C2semidirect
5. #20260506C2semidirect
6. #20260493C2semidirect
22. C2×SD16×C97C4semidirect
23. C2×D8×C97C4semidirect
113. C97#1903semidirect
125. C2×SD16Dic97semidirect
126. C2×D8Dic97semidirect

Maximal subgroups

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

subgroupindex
#202615022
#202607522
#202607402
C2×#156907272
#202605062
#202604932
C2×AGL(1,ℤ/8)×C972
#190397