Finite Group Databasebeta

← #27172267

Subgroups — #27172267

Order 12928.

326 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 87 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×#15692254C211yesmaximal, characteristic, complemented
2C2×#15692177C211yesmaximal, characteristic
3C2×#15692175C211yesmaximal, characteristic
4#20265078C211yesmaximal, characteristic
5#20264939C211yesmaximal, characteristic, complemented
6#20264931C211yesmaximal, characteristic, complemented
7#340×C101C211yesmaximal, characteristic, complemented
8C22×Dic202—21no—
9C22×D404C2211yescharacteristic
10C22×#11717579C2211yescharacteristic
11#15692254—22no—
12#15692254—22no—
13C2×C4⋊Dic101C2211yescharacteristic
14C2×#13108053—21no—
15C2×#13108051—21no—
16C2×#13108051—21no—
17C2×C4×Dic101C2211yescharacteristic
18#15692177—22no—
19#15692175—22no—
20C22×C8×C101C2211yescharacteristic
21C2×C4⋊C4×C101C2211yescharacteristic
22C2×C4⋊C4×C101C2211yescharacteristic
23C23×D101—21no—
24C2×Dic202—24no—
25C2×Dic202—42no—
26C2×Dic202—42no—
27C2×D404—21no—
28C2×D404—21no—
29C2×D404—21no—
30C2×D404—21no—
31C2×D404—21no—
32C2×D404Q812yescomplemented
33C22×Dic101—21no—
34C22×Dic101D411yescharacteristic
35C22×Dic101D411yescharacteristic
36C2×#11717579—21no—
37C2×#11717579—21no—
38C2×#11717579—21no—
39C4⋊Dic101—21no—
40C4⋊Dic101D412yes—
41#13108053—24no—
42#13108051—42no—
43#13108051—42no—
44C4×Dic101—21no—
45C4×Dic101—21no—
46C22×C4×C101—21no—
47C22×C4×C101—21no—
48C22×C4×C101C2311yescharacteristic
49C2×C8×C101—21no—
50C2×C8×C101—21no—
51C2×C8×C101—21no—
52C4⋊C4×C101—21no—
53C4⋊C4×C101—21no—
54C4⋊C4×C101—21no—
55C4⋊C4×C101D412yes—
56C23×C101C2×D411yescharacteristic
57C22×D101—22no—
58C22×D101—41no—
59C22×D101—41no—
60C22×D101—41no—
61C22×D101—41no—
62C22×D101—41no—
63C22×D101—41no—
64C2×C4×C101—22no—
65C2×C4×C101—41no—
66C2×C4×C101—41no—
67C2×C4×C101—41no—
68C2×C4×C101—41no—
69C2×C4×C101—41no—
70C2×C4×C101C2×D411yescharacteristic
71C2×C4×C101C2×D411yescharacteristic
72C2×C4×C101C2×Q811yescharacteristic
73C2×C4×C101#4111yescharacteristic
74C2×C4×C101#4111yescharacteristic
75C2×C4×C101#4111yescharacteristic
76C2×Dic101—21no—
77C2×Dic101—21no—
78C2×Dic101—21no—
79C2×Dic101—21no—
80C2×Dic101—22no—
81C2×Dic101—41no—
82C2×Dic101—41no—
83C2×Dic101D812yes—
84C2×Dic101SD1612yes—
85D404—22no—
86D404—22no—
87D404—41no—
88D404—41no—
89Dic202—48no—
90C8×C101—41no—
91#11717579—41no—
92C22×C101#12011yescharacteristic
93C22×C101C22⋊Q811yescharacteristic
94C22×C101#12411yescharacteristic
95C22×C101C2×D811yescharacteristic
96C22×C101C2×SD1611yescharacteristic
97C22×C101AGL(1,ℤ/8)11yescharacteristic
98C22×C101#13711yescharacteristic
99C2×D101—42no—
100C2×D101—42no—
101C2×D101—42no—
102C2×D101—81no—
103C2×D101—81no—
104C2×D101—81no—
105C2×D101—81no—
106C4×C101—42no—
107C4×C101—81no—
108C4×C101C22⋊Q811yescharacteristic
109C4×C101C22⋊Q811yescharacteristic
110C4×C101#12411yescharacteristic
111C4×C101#12411yescharacteristic
112Dic101—22no—
113Dic101—22no—
114Dic101—42no—
115C2×C101#39311yescharacteristic
116C2×C101#44711yescharacteristic
117C2×C101#44811yescharacteristic
118C2×C101#47411yescharacteristic
119C2×C101#47611yescharacteristic
120C2×C101#48611yescharacteristic
121C2×C101#48911yescharacteristic
122D101—82no—
123#2023—1011nomaximal, Sylow
124C101#202311yescharacteristic, complemented, Sylow
125C2×C4⋊Q8—1011no—
126C2×#102—1011no—
127C2×#102—1011no—
128#390—1011no—
129#340—1011no—
130#340D10111yescharacteristic, complemented
131#336—1011no—
132C22×Q8—2021no—
133C22×D4—1011no—
134C22×C8—1011no—
135C22×C8C2×D10111yescharacteristic
136C4⋊Q8—2022no—
137C4⋊Q8—2022no—
138C2×C4⋊C4—1011no—
139C2×C4⋊C4—2021no—
140C2×C4⋊C4C2×D10111yescharacteristic
141C2×C4⋊C4C2×D10111yescharacteristic
142C2×C22⋊C4—2021no—
143C2×C22⋊C4—2021no—
144C2×C42—1011no—
145#102—2022no—
146#102—2022no—
147C24—2021no—
148C2×Q8—4042no—
149C2×Q8—4042no—
150C2×Q8—2024no—
151C2×D4—1012no—
152C2×D4—2021no—
153C2×D4—2021no—
154C2×D4—2021no—
155C2×D4—2021no—
156C2×D4—2021no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—1011no—
160C22×C4—1011no—
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—1012no—
173C4⋊C4—2024no—
174C4⋊C4—2021no—
175C4⋊C4D40412yes—
176C22⋊C4—4042no—
177C22⋊C4—4042no—
178C42—2021no—
179C42—2021no—
180C23—4041no—
181C23—4041no—
182C23—4041no—
183C23—4041no—
184C23—4041no—
185C23—4041no—
186C23—2022no—
187C23D4×D10111yescharacteristic
188Q8—4048no—
189D4—4041no—
190D4—4041no—
191D4—2022no—
192D4—2022no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—1012no—
200C2×C4—1012no—
201C2×C4—4041no—
202C2×C4—4041no—
203C2×C4—2022no—
204C2×C4—2021no—
205C2×C4—2021no—
206C2×C4—2021no—
207C2×C4—2021no—
208C2×C4D4×D10111yescharacteristic
209C2×C4C2×D40411yescharacteristic
210C2×C4Q8×D10111yescharacteristic
211C2×C4#1310807311yescharacteristic
212C2×C4#1310807411yescharacteristic
213C2×C4#1310807411yescharacteristic
214C8—41no—
215C8—4041no—
216C22—8081no—
217C22—8081no—
218C22—8081no—
219C22—8081no—
220C22—4042no—
221C22—4042no—
222C22—4042no—
223C22#1569222511yescharacteristic
224C22#1569223311yescharacteristic
225C22#1569224111yescharacteristic
226C22D8×D10111yescharacteristic
227C22SD16×D10111yescharacteristic
228C22#1569228211yescharacteristic
229C22#1569228711yescharacteristic
230C4—42no—
231C4—81no—
232C4—4042no—
233C4—2022no—
234C4—2022no—
235C4#1569223311yescharacteristic
236C4#1569223511yescharacteristic
237C4#1569224111yescharacteristic
238C4#1569224311yescharacteristic
239C2—8082no—
240C2#2026509011yescharacteristic
241C2#2026531211yescharacteristic
242C2#2026531711yescharacteristic
243C2#2026545611yescharacteristic
244C2#2026546611yescharacteristic
245C2#2026552211yescharacteristic
246C2#2026553811yescharacteristic
subgroupG/H
centerC23D4×D101
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#340×C101C2
socleC23×C101C2×D4
prime radicalC22×D404C22
hypercenter#340D101
hypocenterC101#2023
LCSG ▷ C22×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
1. C2×#15692254C2semidirect
5. #20264939C2semidirect
6. #20264931C2semidirect
7. #340×C101C2semidirect
32. C2×D404Q8semidirect
124. C101#2023semidirect
130. #340D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922542
C2×#156921772
C2×#156921752
#202650782
#202649392
#202649312
#340×C1012
#2023101