Finite Group Databasebeta

← #27187119

Subgroups — #27187119

Order 12928.

456 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C22×C4⋊#87938C211yesmaximal, characteristic, complemented
2C2×#15692317C212yesmaximal, complemented
3D101⋊(C22⋊C8)C211yesmaximal, characteristic, complemented
4#20266050C212yesmaximal, complemented
5#121×D101C211yesmaximal, characteristic
6C23×#87938—21no—
7C2×D101⋊C8C2212yes—
8C2×C4⋊#87938—24no—
9C2×C4⋊#87938—21no—
10C2×C4⋊#87938C2212yes—
11#15692317—24no—
12#15692312—21no—
13C22⋊#11717581—21no—
14C2×D4×D101—21no—
15C2×D4×D101C2211yescharacteristic
16C22×C4×D101C2211yescharacteristic
17#15692230C411yescharacteristic, complemented
18#15692229C411yescharacteristic, complemented
19#15692228—21no—
20#15692227—21no—
21#15692226C411yescharacteristic, complemented
22C4⋊C4×D101C2211yescharacteristic
23C22⋊C4×D101—21no—
24#121×C101C411yescharacteristic, complemented
25C22×#87938—24no—
26C22×#87938—22no—
27C22×#87938—42no—
28C22×#87938—42no—
29D101⋊C8—22no—
30D101⋊C8—22no—
31C2×#11717581—22no—
32C4⋊#87938—24no—
33C4⋊#87938—22no—
34C4⋊#87938D414yescomplemented
35C23×D101—21no—
36C23×D101D411yescharacteristic
37C2×D404—21no—
38C2×D404C2×C411yescharacteristic, complemented
39C2×C22⋊D101—21no—
40C2×C22⋊D101—21no—
41C2×C22⋊D101—21no—
42D4×D101—22no—
43D4×D101—22no—
44D4×D101—42no—
45D4×D101—42no—
46C22×Dic101D411yescharacteristic
47C2×C4×D101—22no—
48C2×C4×D101—22no—
49C2×C4×D101—21no—
50C2×C4×D101C2×C411yescharacteristic
51C2×C4×D101D411yescharacteristic
52C2×C4×D101D411yescharacteristic
53C2×C4×D101C2311yescharacteristic
54C4⋊Dic101C2×C411yescharacteristic, complemented
55#13108053—21no—
56C22⋊Dic101—21no—
57#13108051—21no—
58#13108051—21no—
59C2×D4×C101—21no—
60C2×D4×C101C2×C411yescharacteristic, complemented
61C22×C4×C101C2×C411yescharacteristic
62C4⋊C4×C101C2×C411yescharacteristic, complemented
63C22⋊C4×C101—21no—
64C23×C101—21no—
65C23×C101C22⋊C411yescharacteristic
66C22×D101—21no—
67C22×D101—21no—
68C22×D101—21no—
69C22×D101—21no—
70C22×D101—21no—
71C22×D101—41no—
72C22×D101—41no—
73C22×D101—41no—
74C22×D101—41no—
75C22×D101—41no—
76C22×D101—41no—
77C22×D101C22⋊C411yescharacteristic
78C22×D101D812yes—
79C22×D101SD1612yes—
80C22×D101C2×D411yescharacteristic
81C2×#87938—24no—
82C2×#87938—48no—
83C2×#87938—42no—
84C2×#87938—42no—
85D4×C101—22no—
86D4×C101—42no—
87C2×C4×C101—22no—
88C2×C4×C101—21no—
89C2×C4×C101C22⋊C411yescharacteristic
90C2×C4×C101C22×C411yescharacteristic
91C22⋊D101—42no—
92C22⋊D101—42no—
93C22⋊D101—42no—
94C2×Dic101—22no—
95C2×Dic101—21no—
96C2×Dic101C22⋊C411yescharacteristic
97C2×Dic101C2×D411yescharacteristic
98D404—22no—
99D404—42no—
100C4×D101—21no—
101C4×D101—21no—
102C4×D101—22no—
103C4×D101—22no—
104C4×D101—41no—
105C4×D101—41no—
106C4×D101C2×D412yes—
107C4×D101#4112yes—
108#11717581—42no—
109C22×C101—21no—
110C22×C101—21no—
111C22×C101—41no—
112C22×C101—41no—
113C22×C101—41no—
114C22×C101#10212yescomplemented
115C22×C101C2×C22⋊C411yescharacteristic
116C2×D101—21no—
117C2×D101—21no—
118C2×D101—22no—
119C2×D101—22no—
120C2×D101—22no—
121C2×D101—22no—
122C2×D101—41no—
123C2×D101—41no—
124C2×D101—41no—
125C2×D101—81no—
126C2×D101—81no—
127C2×D101—81no—
128C2×D101—81no—
129C2×D101#10212yescomplemented
130C2×D101#12011yescharacteristic
131C2×D101#12311yescharacteristic
132C2×D101C2×D811yescharacteristic
133C2×D101C2×SD1611yescharacteristic
134C2×D101AGL(1,ℤ/8)11yescharacteristic
135C2×D101#13711yescharacteristic
136#87938—44no—
137C4×C101—21no—
138C4×C101—41no—
139C4×C101C4×D412yes—
140Dic101—21no—
141Dic101—41no—
142Dic101#12112yes—
143C2×C101—22no—
144C2×C101—81no—
145C2×C101#38611yescharacteristic
146C2×C101C2×#10211yescharacteristic
147C2×C101#41711yescharacteristic
148D101—22no—
149D101—81no—
150D101#44711yescharacteristic
151D101#44811yescharacteristic
152D101#48011yescharacteristic
153D101#48111yescharacteristic
154#1874—1011nomaximal, Sylow
155C101#187411yescharacteristic, complemented, Sylow
156C2×#121—1011no—
157C22×C4⋊C4—1011no—
158C2×#102—1012no—
159C2×C22⋊C8—1011no—
160#340—1012no—
161C22×D4—1011no—
162C22×D4—2021no—
163C23×C4—1011no—
164C23×C4—2021no—
165C22×C8—1012no—
166#121—1011no—
167#121—1011no—
168#121—1011no—
169#121—2021no—
170#121—2021no—
171#121#8793811yescharacteristic, complemented
172C2×C4⋊C4—1012no—
173C2×C4⋊C4—1011no—
174C2×C4⋊C4—2024no—
175C2×C4⋊C4—2021no—
176C2×C22⋊C4—2021no—
177#102—2024no—
178C22⋊C8—2021no—
179C22⋊C8—2021no—
180C24—1011no—
181C24—2021no—
182C2×D4—21no—
183C2×D4—1011no—
184C2×D4—4042no—
185C2×D4—4042no—
186C2×D4—2021no—
187C2×D4—2022no—
188C2×D4—2022no—
189C2×D4—2021no—
190C2×D4—2021no—
191C2×D4—2021no—
192C2×D4C2×#8793811yescharacteristic, complemented
193C22×C4—1011no—
194C22×C4—1011no—
195C22×C4—1011no—
196C22×C4—1011no—
197C22×C4—1011no—
198C22×C4—4042no—
199C22×C4—4042no—
200C22×C4—2022no—
201C22×C4—2022no—
202C22×C4—2021no—
203C22×C4—2022no—
204C22×C4—2024no—
205C22×C4C2×#8793811yescharacteristic
206C2×C8—2022no—
207C2×C8—2022no—
208C2×C8—2022no—
209C4⋊C4—1014no—
210C4⋊C4—1011no—
211C4⋊C4—2021no—
212C4⋊C4—2022no—
213C4⋊C4—2024no—
214C4⋊C4C2×#8793811yescharacteristic, complemented
215C22⋊C4—21no—
216C22⋊C4—2021no—
217C22⋊C4—2021no—
218C22⋊C4—2021no—
219C23—21no—
220C23—1011no—
221C23—1012no—
222C23—1012no—
223C23—1011no—
224C23—4041no—
225C23—4041no—
226C23—4041no—
227C23—4041no—
228C23—4041no—
229C23—4041no—
230C23—2021no—
231C23—2021no—
232C23—2021no—
233C23—2021no—
234C23—2021no—
235C23C22⋊#8793811yescharacteristic
236D4—22no—
237D4—42no—
238D4—4042no—
239D4—4042no—
240D4—4042no—
241D4—4042no—
242D4—2022no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—1011no—
246C2×C4—1012no—
247C2×C4—1012no—
248C2×C4—1011no—
249C2×C4—4041no—
250C2×C4—4041no—
251C2×C4—4042no—
252C2×C4—4042no—
253C2×C4—4048no—
254C2×C4—2022no—
255C2×C4—2021no—
256C2×C4—2021no—
257C2×C4—2022no—
258C2×C4—2024no—
259C2×C4—2021no—
260C2×C4—2022no—
261C2×C4C22⋊#8793811yescharacteristic
262C2×C4C22×#8793811yescharacteristic
263C8—4042no—
264C22—21no—
265C22—21no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—8081no—
270C22—8081no—
271C22—8081no—
272C22—8081no—
273C22—1011no—
274C22—1011no—
275C22—1011no—
276C22—1011no—
277C22—1011no—
278C22—1011no—
279C22—1012no—
280C22—4041no—
281C22—4041no—
282C22—4041no—
283C22—2022no—
284C22—2022no—
285C22—2022no—
286C22—2022no—
287C22—2021no—
288C22—2021no—
289C22#1569231712yescomplemented
290C22C2×C22⋊#8793811yescharacteristic
291C4—21no—
292C4—41no—
293C4—1012no—
294C4—4044no—
295C4—4041no—
296C4—2021no—
297C4D4×#8793812yes—
298C2—22no—
299C2—81no—
300C2—8081no—
301C2—1011no—
302C2—1011no—
303C2—1011no—
304C2—1011no—
305C2—2022no—
306C2#2026612511yescharacteristic
307C2C2×#1569231711yescharacteristic
308C2#2026616811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#121×C101C4
socleC22×C101C2×C22⋊C4
prime radical#121×D101C2
cosocleC2×C4×D101C23
hypercenter#121#87938
hypocenterC101#1874
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊#87938C2semidirect
2. C2×#15692317C2semidirect
3. D101⋊(C22⋊C8)C2semidirect
4. #20266050C2semidirect
17. #15692230C4semidirect
18. #15692229C4semidirect
21. #15692226C4semidirect
24. #121×C101C4semidirect
34. C4⋊#87938D4semidirect
38. C2×D404C2×C4semidirect
54. C4⋊Dic101C2×C4semidirect
60. C2×D4×C101C2×C4semidirect
62. C4⋊C4×C101C2×C4semidirect
114. C22×C101#102semidirect
129. C2×D101#102semidirect
155. C101#1874semidirect
171. #121#87938semidirect
192. C2×D4C2×#87938semidirect
214. C4⋊C4C2×#87938semidirect
289. C22#15692317semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#879382
C2×#156923172
D101⋊(C22⋊C8)2
#202660502
#121×D1012
#1874101