Finite Group Databasebeta

← #27171141

Subgroups — #27171141

Order 12928.

356 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 101 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×#15692217C211yesmaximal, characteristic
2C2×#15692184C211yesmaximal, characteristic, complemented
3C2×#15692177C211yesmaximal, characteristic
4C2×#15692175C211yesmaximal, characteristic
5#20265042C211yesmaximal, characteristic, complemented
6#20264939C211yesmaximal, characteristic, complemented
7#340×C101C211yesmaximal, characteristic, complemented
8C22×D404C2211yescharacteristic
9C22×C4×D101—21no—
10C22×#11717579C2211yescharacteristic
11#15692217—22no—
12#15692217C414yescomplemented
13C2×C4⋊Dic101C2211yescharacteristic
14C2×#13108051—21no—
15C2×C4×Dic101C2211yescharacteristic
16#15692184—22no—
17#15692177—22no—
18#15692175—22no—
19#15692162—21no—
20#15692162—21no—
21C22×C8×C101C2211yescharacteristic
22C2×C4⋊C4×C101C2211yescharacteristic
23C2×C4⋊C4×C101C2211yescharacteristic
24C23×D101—21no—
25C2×D404—21no—
26C2×D404—21no—
27C2×D404—21no—
28C2×D404C2×C412yes—
29C2×D404D412yes—
30C2×D404Q812yes—
31C22×Dic101—21no—
32C22×Dic101D411yescharacteristic
33C22×Dic101D411yescharacteristic
34C2×C4×D101—24no—
35C2×C4×D101—42no—
36C2×C4×D101—42no—
37C2×#11717579—21no—
38C2×#11717579—21no—
39C2×#11717579—21no—
40C4⋊Dic101—21no—
41C4⋊Dic101—21no—
42#13108051—24no—
43C4×Dic101C2×C412yes—
44C4×Dic101D412yes—
45C22×C4×C101—21no—
46C22×C4×C101—21no—
47C22×C4×C101C2311yescharacteristic
48C2×C8×C101—21no—
49C2×C8×C101—21no—
50C2×C8×C101—21no—
51C4⋊C4×C101—21no—
52C4⋊C4×C101—21no—
53C4⋊C4×C101—21no—
54C4⋊C4×C101C2×C412yes—
55C23×C101C2×D411yescharacteristic
56C22×D101—22no—
57C22×D101—22no—
58C22×D101—22no—
59C22×D101—41no—
60C22×D101—41no—
61C22×D101—41no—
62C22×D101—41no—
63C2×C4×C101—22no—
64C2×C4×C101—41no—
65C2×C4×C101—41no—
66C2×C4×C101—41no—
67C2×C4×C101—41no—
68C2×C4×C101—41no—
69C2×C4×C101C22×C411yescharacteristic
70C2×C4×C101C2×D411yescharacteristic
71C2×C4×C101C2×D411yescharacteristic
72C2×C4×C101C2×Q811yescharacteristic
73C2×C4×C101#4111yescharacteristic
74C2×C4×C101#4111yescharacteristic
75C2×Dic101—21no—
76C2×Dic101—21no—
77C2×Dic101—21no—
78C2×Dic101—21no—
79C2×Dic101—41no—
80C2×Dic101—41no—
81C2×Dic101—41no—
82C2×Dic101C22⋊C412yes—
83C2×Dic101C22⋊C412yes—
84D404—22no—
85D404—22no—
86D404—22no—
87D404C4⋊C414yes—
88C4×D101—48no—
89C8×C101—41no—
90#11717579—41no—
91C22×C101C2×C22⋊C411yescharacteristic
92C22×C101#12111yescharacteristic
93C22×C101C22⋊Q811yescharacteristic
94C22×C101#13511yescharacteristic
95C22×C101#13511yescharacteristic
96C22×C101AGL(1,ℤ/8)11yescharacteristic
97C22×C101AGL(1,ℤ/8)11yescharacteristic
98C2×D101—24no—
99C2×D101—42no—
100C2×D101—42no—
101C2×D101—42no—
102C2×D101—42no—
103C2×D101—42no—
104C2×D101—42no—
105C4×C101—42no—
106C4×C101—81no—
107C4×C101C2×C4⋊C411yescharacteristic
108C4×C101#11711yescharacteristic
109C4×C101#12111yescharacteristic
110C4×C101C22⋊Q811yescharacteristic
111Dic101—22no—
112Dic101—22no—
113Dic101—81no—
114C2×C101#38011yescharacteristic
115C2×C101#41611yescharacteristic
116C2×C101#41811yescharacteristic
117C2×C101#46311yescharacteristic
118C2×C101#46311yescharacteristic
119C2×C101#47811yescharacteristic
120C2×C101#47811yescharacteristic
121D101—44no—
122#1804—1011nomaximal, Sylow
123C101#180411yescharacteristic, complemented, Sylow
124C2×C4×D4—1011no—
125C2×C4⋊C8—1011no—
126C2×#102—1011no—
127C2×#102—1011no—
128#382—1011no—
129#340—1011no—
130#340D10111yescharacteristic, complemented
131C22×D4—1011no—
132C23×C4—2021no—
133C22×C8—1011no—
134C22×C8C2×D10111yescharacteristic
135C4×D4—1014no—
136C4×D4—2022no—
137C2×C4⋊C4—1011no—
138C2×C4⋊C4C2×D10111yescharacteristic
139C2×C4⋊C4C2×D10111yescharacteristic
140C2×C22⋊C4—2021no—
141C2×C42—1011no—
142C4⋊C8—2022no—
143#102—2022no—
144#102—2022no—
145#95—2021no—
146#95—2021no—
147C24—2021no—
148C2×D4—1012no—
149C2×D4—1012no—
150C2×D4—1012no—
151C2×D4—2021no—
152C2×D4—2021no—
153C2×D4—2021no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—1011no—
157C22×C4—1011no—
158C22×C4—4042no—
159C22×C4—4042no—
160C22×C4—2024no—
161C22×C4—2021no—
162C22×C4C22×D10111yescharacteristic
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—2021no—
167C2×C8—2021no—
168C2×C8—2021no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—2021no—
173C4⋊C4—2021no—
174C4⋊C4C4×D10112yes—
175C22⋊C4—2024no—
176C42—1012no—
177C42—1012no—
178C23—4041no—
179C23—4041no—
180C23—4041no—
181C23—4041no—
182C23—2022no—
183C23—2022no—
184C23—2022no—
185C23D4×D10111yescharacteristic
186D4—1014no—
187D4—2022no—
188D4—2022no—
189D4—2022no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—1012no—
197C2×C4—1012no—
198C2×C4—4048no—
199C2×C4—4041no—
200C2×C4—4041no—
201C2×C4—4041no—
202C2×C4—2021no—
203C2×C4—2021no—
204C2×C4—2021no—
205C2×C4—2021no—
206C2×C4C2×C4×D10111yescharacteristic
207C2×C4D4×D10111yescharacteristic
208C2×C4D4×D10111yescharacteristic
209C2×C4Q8×D10111yescharacteristic
210C2×C4#1310807411yescharacteristic
211C2×C4#1310807411yescharacteristic
212C8—41no—
213C8—4041no—
214C22—4042no—
215C22—4042no—
216C22—4042no—
217C22—4042no—
218C22—4042no—
219C22—4042no—
220C22—2024no—
221C22C22⋊C4×D10111yescharacteristic
222C22#1569222811yescharacteristic
223C22#1569223311yescharacteristic
224C22#1569227611yescharacteristic
225C22#1569227811yescharacteristic
226C22#1569227911yescharacteristic
227C22#1569228211yescharacteristic
228C4—42no—
229C4—81no—
230C4—8081no—
231C4—2022no—
232C4—2022no—
233C4C4⋊C4×D10111yescharacteristic
234C4#1569221211yescharacteristic
235C4#1569222811yescharacteristic
236C4#1569223311yescharacteristic
237C2—4044no—
238C2#2026503611yescharacteristic
239C2#2026517211yescharacteristic
240C2#2026518011yescharacteristic
241C2#2026539111yescharacteristic
242C2#2026539511yescharacteristic
243C2#2026547611yescharacteristic
244C2#2026548011yescharacteristic
subgroupG/H
centerC23D4×D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#340×C101C2
socleC23×C101C2×D4
prime radicalC22×D404C22
cosocleC22×C4×C101C23
hypercenter#340D101
hypocenterC101#1804
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#15692184C2semidirect
5. #20265042C2semidirect
6. #20264939C2semidirect
7. #340×C101C2semidirect
12. #15692217C4semidirect
123. C101#1804semidirect
130. #340D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922172
C2×#156921842
C2×#156921772
C2×#156921752
#202650422
#202649392
#340×C1012
#1804101