Finite Group Databasebeta

← #27186549

Subgroups — #27186549

Order 12928.

286 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 60 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#20266148C211yesmaximal, characteristic, complemented
2#20266119C211yesmaximal, characteristic, complemented
3#20266030C212yesmaximal, complemented
4#20266029C212yesmaximal, complemented
5#20265689C211yesmaximal, characteristic
6C2×#13108084—21no—
7C2×C4⋊#87938—21no—
8C2×C22⋊#87938—21no—
9#15692312C2212yes—
10C22⋊#11717581—21no—
11(C2×#87938)⋊C4C2212yes—
12C2×#13108074C411yescharacteristic, complemented
13C22×C4×D101C2211yescharacteristic
14#15692239—21no—
15#15692235C411yescharacteristic, complemented
16#15692235C2211yescharacteristic
17#15692230C411yescharacteristic, complemented
18#15692230C2211yescharacteristic
19#15692225—21no—
20C4×D404—21no—
21C4×D404—21no—
22#117×C101C411yescharacteristic, complemented
23C22×#87938—22no—
24#13108084—42no—
25C2×#11717581—22no—
26C4⋊#87938—42no—
27C22⋊#87938—42no—
28C23×D101D411yescharacteristic
29#13108074—42no—
30C2×Dic202C2×C411yescharacteristic
31C2×D404—21no—
32C2×D404—21no—
33C2×D404C2×C411yescharacteristic
34C2×C22⋊D101—21no—
35C22×Dic101D411yescharacteristic
36C2×C4×D101—21no—
37C2×C4×D101—42no—
38C2×C4×D101C2×C411yescharacteristic
39C2×C4×D101D411yescharacteristic
40C2×C4×D101D411yescharacteristic
41C2×C4×D101C2311yescharacteristic
42C4⋊Dic101—21no—
43C4⋊Dic101—21no—
44#13108051—21no—
45#13108051—21no—
46#13108051—21no—
47#13108051—21no—
48C22×C4×C101C2×C411yescharacteristic
49C4⋊C4×C101C2×C411yescharacteristic
50C4⋊C4×C101C2×C411yescharacteristic
51C22⋊C4×C101—21no—
52C42×C101—21no—
53C23×C101C22⋊C411yescharacteristic
54C22×D101—21no—
55C22×D101—21no—
56C22×D101—21no—
57C22×D101—41no—
58C22×D101—41no—
59C22×D101C22⋊C411yescharacteristic
60C22×D101C2×D411yescharacteristic
61C2×#87938—42no—
62C2×#87938—42no—
63C2×#87938—42no—
64C2×C4×C101—22no—
65C2×C4×C101—21no—
66C2×C4×C101—21no—
67C2×C4×C101C22⋊C411yescharacteristic
68C2×C4×C101C22×C411yescharacteristic
69C22⋊D101—42no—
70C2×Dic101—21no—
71C2×Dic101—41no—
72C2×Dic101C22⋊C411yescharacteristic
73C2×Dic101C2×D411yescharacteristic
74D404—41no—
75D404—42no—
76D404—41no—
77D404—41no—
78C4×D101—22no—
79C4×D101—22no—
80C4×D101—41no—
81C4×D101—41no—
82C4×D101—42no—
83Dic202—41no—
84#11717581—42no—
85C22×C101—21no—
86C22×C101—21no—
87C22×C101—21no—
88C22×C101C2×C22⋊C411yescharacteristic
89C2×D101—21no—
90C2×D101—21no—
91C2×D101—21no—
92C2×D101—21no—
93C2×D101—21no—
94C2×D101—42no—
95C2×D101—42no—
96C2×D101—41no—
97C2×D101—41no—
98C2×D101—41no—
99C2×D101AGL(1,ℤ/8)11yescharacteristic
100C2×D101#13711yescharacteristic
101#87938—82no—
102C4×C101—21no—
103C4×C101—41no—
104C4×C101—41no—
105C4×C101#9912yes—
106Dic101—21no—
107Dic101—21no—
108Dic101—41no—
109C2×C101—41no—
110C2×C101C2×#9911yescharacteristic
111C2×C101#41711yescharacteristic
112C2×C101#42111yescharacteristic
113D101—21no—
114D101—21no—
115D101—41no—
116D101—81no—
117#1513—1011nomaximal, Sylow
118C101#151311yescharacteristic, complemented, Sylow
119#525—1011no—
120#407—1011no—
121#380—1011no—
122#328—1012no—
123#327—1012no—
124C2×#41—1011no—
125C23×C4—1011no—
126C2×M16—2021no—
127#123—2021no—
128C22⋊Q8—1011no—
129C22⋊Q8—1011no—
130#121—1011no—
131#121—1011no—
132#120—2021no—
133C4×D4—2021no—
134C4×D4—2021no—
135#117#8793811yescharacteristic, complemented
136C2×C4⋊C4—2021no—
137C2×C22⋊C4—2021no—
138C22⋊C8—1012no—
139C22⋊C8—2021no—
140#95—1012no—
141C24—1011no—
142#41—4042no—
143C2×Q8—1011no—
144C2×D4—1011no—
145C2×D4—2021no—
146C2×D4—2021no—
147C2×D4—2021no—
148C22×C4—1011no—
149C22×C4—1011no—
150C22×C4—1011no—
151C22×C4—1011no—
152C22×C4—1011no—
153C22×C4—4042no—
154C22×C4—2022no—
155C22×C4—2021no—
156C22×C4C2×#8793811yescharacteristic
157M16—4042no—
158C2×C8—2022no—
159C4⋊C4—4042no—
160C4⋊C4—2021no—
161C4⋊C4—2021no—
162C4⋊C4C2×#8793811yescharacteristic
163C4⋊C4C2×#8793811yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—4042no—
166C22⋊C4—2021no—
167C22⋊C4—2021no—
168C22⋊C4—2021no—
169C22⋊C4—2021no—
170C42—21no—
171C23—1011no—
172C23—1011no—
173C23—4041no—
174C23—4041no—
175C23—2021no—
176C23—2021no—
177C23—2021no—
178C23C22⋊#8793811yescharacteristic
179Q8—4041no—
180D4—4041no—
181D4—4041no—
182D4—4041no—
183D4—4042no—
184D4—4042no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—1011no—
189C2×C4—1011no—
190C2×C4—4041no—
191C2×C4—4041no—
192C2×C4—4042no—
193C2×C4—4042no—
194C2×C4—4042no—
195C2×C4—4042no—
196C2×C4—4041no—
197C2×C4—2021no—
198C2×C4—2022no—
199C2×C4—2022no—
200C2×C4C22⋊#8793811yescharacteristic
201C2×C4C22×#8793811yescharacteristic
202C8—4042no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—1011no—
207C22—1011no—
208C22—4042no—
209C22—4042no—
210C22—4041no—
211C22—4041no—
212C22—4041no—
213C22—2021no—
214C22—2021no—
215C22—2021no—
216C22—2021no—
217C22—2021no—
218C22C2×C22⋊#8793811yescharacteristic
219C4—21no—
220C4—41no—
221C4—41no—
222C4—8082no—
223C4—4041no—
224C4—2021no—
225C4—2021no—
226C4#1569231412yes—
227C2—41no—
228C2—8081no—
229C2—4041no—
230C2—2021no—
231C2—2021no—
232C2C2×#1569231411yescharacteristic
233C2#2026616811yescharacteristic
234C2#2026617411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#117×C101C4
socleC22×C101C2×C22⋊C4
prime radical#20265689C2
cosocleC2×C4×D101C23
hypercenter#117#87938
hypocenterC101#1513
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #20266148C2semidirect
2. #20266119C2semidirect
3. #20266030C2semidirect
4. #20266029C2semidirect
12. C2×#13108074C4semidirect
15. #15692235C4semidirect
17. #15692230C4semidirect
22. #117×C101C4semidirect
118. C101#1513semidirect
135. #117#87938semidirect

Maximal subgroups

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

subgroupindex
#202661482
#202661192
#202660302
#202660292
#202656892
#1513101