Finite Group Databasebeta

← #27171579

Subgroups — #27171579

Order 12928.

358 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 83 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#20265689C211yesmaximal, characteristic
2C2×#15692212C211yesmaximal, characteristic, complemented
3C2×#15692183C211yesmaximal, characteristic, complemented
4C2×#15692181C211yesmaximal, characteristic, complemented
5#20265140C211yesmaximal, characteristic, complemented
6#20264936C211yesmaximal, characteristic, complemented
7#339×C101C211yesmaximal, characteristic, complemented
8C2×#13108074C2211yescharacteristic
9C22×C4×D101C2211yescharacteristic
10C2×#13108058C2211yescharacteristic
11#15692239—21no—
12#15692235C412yescomplemented
13#15692230C412yescomplemented
14#15692225—21no—
15C4×D404—22no—
16#15692212—22no—
17#15692212—22no—
18C2×#13108053—21no—
19C2×#13108051—21no—
20C2×C4×Dic101C2211yescharacteristic
21#15692183—22no—
22#15692181—22no—
23#15692167—22no—
24C2×M16×C101C2211yescharacteristic
25#117×C101C2211yescharacteristic
26C2×C42×C101C2211yescharacteristic
27C23×D101D411yescharacteristic
28#13108074—21no—
29#13108074—21no—
30#13108074—21no—
31#13108074—21no—
32C2×Dic202C2×C411yescharacteristic
33C2×D404—22no—
34C2×D404C2×C411yescharacteristic
35C2×C22⋊D101—21no—
36C22×Dic101—21no—
37C22×Dic101D411yescharacteristic
38C2×C4×D101—21no—
39C2×C4×D101—21no—
40C2×C4×D101—21no—
41C2×C4×D101—21no—
42C2×C4×D101—21no—
43C2×C4×D101C2×C412yes—
44C2×C4×D101D412yes—
45#13108058—21no—
46#13108058—21no—
47C2×#11717579—21no—
48C4⋊Dic101—22no—
49#13108053—24no—
50#13108051—24no—
51#13108051—22no—
52#13108051—22no—
53C4×Dic101—21no—
54C4×Dic101D412yescomplemented
55C22×C4×C101—21no—
56C22×C4×C101C2311yescharacteristic
57M16×C101—21no—
58M16×C101—21no—
59C2×C8×C101—21no—
60C4⋊C4×C101C2×C412yes—
61C22⋊C4×C101—21no—
62C42×C101—21no—
63C42×C101—21no—
64C42×C101D412yescomplemented
65C23×C101C2×D411yescharacteristic
66C22×D101—21no—
67C22×D101—21no—
68C22×D101—21no—
69C22×D101—42no—
70C22×D101C22⋊C412yes—
71C2×C4×C101—22no—
72C2×C4×C101—22no—
73C2×C4×C101—42no—
74C2×C4×C101C22×C411yescharacteristic
75C2×C4×C101C2×D411yescharacteristic
76C2×C4×C101C2×D411yescharacteristic
77C2×C4×C101C2×D411yescharacteristic
78C2×C4×C101#4111yescharacteristic
79C2×C4×C101#4111yescharacteristic
80C22⋊D101—41no—
81C22⋊D101—41no—
82C2×Dic101—21no—
83C2×Dic101—21no—
84C2×Dic101—21no—
85C2×Dic101—22no—
86C2×Dic101—42no—
87C2×Dic101C22⋊C412yes—
88D404—21no—
89D404—21no—
90D404—44no—
91C4×D101—22no—
92C4×D101—22no—
93C4×D101—22no—
94C4×D101—22no—
95C4×D101—41no—
96C4×D101—41no—
97Dic202—21no—
98Dic202—21no—
99C8×C101—41no—
100#11717579—41no—
101C22×C101—22no—
102C22×C101C2×C22⋊C411yescharacteristic
103C22×C101#12011yescharacteristic
104C22×C101#12311yescharacteristic
105C2×D101—22no—
106C2×D101—22no—
107C2×D101—22no—
108C2×D101—42no—
109C2×D101—42no—
110C2×D101—41no—
111C2×D101—42no—
112C4×C101—42no—
113C4×C101—42no—
114C4×C101C4×D411yescharacteristic
115C4×C101C4×D411yescharacteristic
116C4×C101#12111yescharacteristic
117C4×C101#12111yescharacteristic
118Dic101—22no—
119Dic101—42no—
120Dic101—41no—
121C2×C101—21no—
122C2×C101—21no—
123C2×C101#38611yescharacteristic
124C2×C101#42112yes—
125D101—42no—
126D101—81no—
127#1897—1011nomaximal, Sylow
128C101#189711yescharacteristic, complemented, Sylow
129#525—1011no—
130C2×#117—1011no—
131C2×#104—1011no—
132C2×#104—1011no—
133#407—1011no—
134#339—1011no—
135#339D10111yescharacteristic, complemented
136C2×#41—1011no—
137C23×C4—1011no—
138C2×M16—1011no—
139C2×M16C2×D10111yescharacteristic
140#123—2021no—
141C22⋊Q8—1012no—
142#121—1012no—
143#120—2021no—
144C4×D4—2022no—
145#117—2022no—
146#117—2022no—
147#117C2×D10111yescharacteristic
148C2×C4⋊C4—2021no—
149C2×C22⋊C4—2021no—
150C2×C42—1011no—
151C2×C42C2×D10111yescharacteristic
152#104—2022no—
153#104—2022no—
154C22⋊C8—2022no—
155C24—1011no—
156#41—2021no—
157#41—2021no—
158#41—2021no—
159#41—2021no—
160C2×Q8—1011no—
161C2×D4—1011no—
162C2×D4—2022no—
163C2×D4—2021no—
164C22×C4—21no—
165C22×C4—1011no—
166C22×C4—1012no—
167C22×C4—1012no—
168C22×C4—2021no—
169C22×C4—2021no—
170C22×C4—2021no—
171C22×C4—2021no—
172C22×C4—2021no—
173C22×C4—2021no—
174C22×C4C22×D10111yescharacteristic
175M16—21no—
176M16—21no—
177M16—2021no—
178M16—2021no—
179C2×C8—21no—
180C2×C8—2021no—
181C4⋊C4—2022no—
182C4⋊C4—2024no—
183C4⋊C4C4×D10112yes—
184C22⋊C4—21no—
185C22⋊C4—2022no—
186C22⋊C4—2022no—
187C22⋊C4—2024no—
188C42—21no—
189C42—21no—
190C42—1012no—
191C42—2021no—
192C42D40412yescomplemented
193C23—1012no—
194C23—4042no—
195C23—2021no—
196C23—2021no—
197C23—2021no—
198C23D4×D10111yescharacteristic
199Q8—2021no—
200Q8—2021no—
201D4—4044no—
202D4—4041no—
203D4—4041no—
204D4—2021no—
205D4—2021no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—1012no—
210C2×C4—4042no—
211C2×C4—4041no—
212C2×C4—4041no—
213C2×C4—2021no—
214C2×C4—2022no—
215C2×C4—2021no—
216C2×C4—2021no—
217C2×C4—2022no—
218C2×C4—2022no—
219C2×C4—2022no—
220C2×C4—2022no—
221C2×C4C2×C4×D10111yescharacteristic
222C2×C4D4×D10111yescharacteristic
223C2×C4D4×D10111yescharacteristic
224C2×C4C2×D40411yescharacteristic
225C2×C4#1310807411yescharacteristic
226C2×C4#1310807511yescharacteristic
227C8—41no—
228C8—4041no—
229C22—22no—
230C22—4042no—
231C22—4042no—
232C22—4041no—
233C22—4042no—
234C22—2022no—
235C22—2022no—
236C22—2022no—
237C22C22⋊C4×D10111yescharacteristic
238C22#1569222511yescharacteristic
239C22#1569224011yescharacteristic
240C4—42no—
241C4—42no—
242C4—4042no—
243C4—4041no—
244C4—2022no—
245C4#1569221711yescharacteristic
246C4C4×D40411yescharacteristic
247C4#1569222811yescharacteristic
248C4#1569223011yescharacteristic
249C2—21no—
250C2—21no—
251C2—8081no—
252C2—4042no—
253C2#2026505511yescharacteristic
254C2#2026519212yes—
subgroupG/H
centerC2×C4D4×D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#339×C101C2
socleC22×C101C2×C22⋊C4
prime radical#20265689C2
cosocleC22×C4×C101C23
hypercenter#339D101
hypocenterC101#1897
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. C2×#15692212C2semidirect
3. C2×#15692183C2semidirect
4. C2×#15692181C2semidirect
5. #20265140C2semidirect
6. #20264936C2semidirect
7. #339×C101C2semidirect
12. #15692235C4semidirect
13. #15692230C4semidirect
54. C4×Dic101D4semidirect
64. C42×C101D4semidirect
128. C101#1897semidirect
135. #339D101semidirect
192. C42D404semidirect

Maximal subgroups

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

subgroupindex
#202656892
C2×#156922122
C2×#156921832
C2×#156921812
#202651402
#202649362
#339×C1012
#1897101