Finite Group Databasebeta

← #27172049

Subgroups — #27172049

Order 12928.

336 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 85 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#20265723C211yesmaximal, characteristic
2C2×#15692183C211yesmaximal, characteristic, complemented
3C2×#15692182C211yesmaximal, characteristic, complemented
4C2×#15692181C211yesmaximal, characteristic, complemented
5M16×Dic101C211yesmaximal, characteristic, complemented
6C4×#13108053C211yesmaximal, characteristic, complemented
7C2×#104×C101C211yesmaximal, characteristic, complemented
8C2×#13108074C2211yescharacteristic
9C2×#13108058C2211yescharacteristic
10#15692253C411yescharacteristic, complemented
11#15692251C411yescharacteristic, complemented
12#15692244C411yescharacteristic, complemented
13#15692242C411yescharacteristic, complemented
14#15692227—21no—
15#15692227—21no—
16C4×C22⋊D101—21no—
17C4×C22⋊D101—21no—
18C2×#13108053—21no—
19C2×C4×Dic101C2211yescharacteristic
20C2×C4×Dic101C2211yescharacteristic
21#15692183—22no—
22#15692182—22no—
23#15692181—22no—
24#15692165—21no—
25C8×Dic101—21no—
26#15692162—21no—
27C2×#41×C101C2211yescharacteristic
28C2×M16×C101C2211yescharacteristic
29C2×C42×C101C2211yescharacteristic
30#104×C101—22no—
31#13108074—21no—
32#13108074—21no—
33#13108074—21no—
34#13108074—21no—
35C2×Dic202C2×C411yescharacteristic
36C2×D404C2×C411yescharacteristic
37C2×C22⋊D101—21no—
38C2×C22⋊D101—21no—
39C2×C22⋊D101—21no—
40C22×Dic101—21no—
41C22×Dic101D411yescharacteristic
42C22×Dic101D411yescharacteristic
43C2×C4×D101—21no—
44#13108058—21no—
45#13108058D412yescomplemented
46C2×#11717579—21no—
47#13108053—24no—
48#13108053—21no—
49#13108053—21no—
50C22⋊Dic101—21no—
51C22⋊Dic101—21no—
52#13108051—21no—
53#13108051—21no—
54C4×Dic101—21no—
55C4×Dic101—21no—
56C4×Dic101C2×C411yescharacteristic
57C4×Dic101C2×C411yescharacteristic
58C4×Dic101D411yescharacteristic
59C4×Dic101D411yescharacteristic
60#41×C101—21no—
61#41×C101—21no—
62#41×C101—21no—
63#41×C101—21no—
64C2×Q8×C101C2×C411yescharacteristic
65C2×D4×C101—21no—
66C2×D4×C101C2×C411yescharacteristic
67C22×C4×C101—21no—
68C22×C4×C101—21no—
69C22×C4×C101C2311yescharacteristic
70M16×C101—21no—
71M16×C101D412yescomplemented
72C2×C8×C101—21no—
73C42×C101—21no—
74C42×C101—21no—
75C23×C101—21no—
76C23×C101C2×D411yescharacteristic
77C22×D101—21no—
78Q8×C101—21no—
79Q8×C101—21no—
80D4×C101—21no—
81D4×C101—21no—
82D4×C101—41no—
83D4×C101—41no—
84C2×C4×C101—22no—
85C2×C4×C101—21no—
86C2×C4×C101—41no—
87C2×C4×C101—41no—
88C2×C4×C101—41no—
89C2×C4×C101—41no—
90C2×C4×C101C22×C411yescharacteristic
91C2×C4×C101C2×D411yescharacteristic
92C2×C4×C101C2×D411yescharacteristic
93C2×C4×C101C2×D411yescharacteristic
94C2×C4×C101#4111yescharacteristic
95C2×C4×C101#4111yescharacteristic
96C22⋊D101—42no—
97C22⋊D101—41no—
98C22⋊D101—41no—
99C22⋊D101—42no—
100C2×Dic101—22no—
101C2×Dic101—22no—
102C2×Dic101—21no—
103C2×Dic101—41no—
104C2×Dic101—41no—
105C2×Dic101—41no—
106C2×Dic101C22⋊C411yescharacteristic
107C2×Dic101C22⋊C411yescharacteristic
108C2×Dic101C22⋊C411yescharacteristic
109C2×Dic101C22⋊C411yescharacteristic
110D404—21no—
111D404—21no—
112C4×D101—41no—
113C4×D101—41no—
114Dic202—21no—
115Dic202—21no—
116C8×C101—22no—
117#11717579—22no—
118C22×C101—21no—
119C22×C101—21no—
120C22×C101—41no—
121C22×C101—41no—
122C22×C101—41no—
123C22×C101C2×C22⋊C411yescharacteristic
124C22×C101#12411yescharacteristic
125C22×C101#12711yescharacteristic
126C2×D101—41no—
127C2×D101—41no—
128C2×D101—41no—
129C4×C101—42no—
130C4×C101—41no—
131C4×C101C4×D411yescharacteristic
132C4×C101C4×D411yescharacteristic
133C4×C101#12111yescharacteristic
134C4×C101#12111yescharacteristic
135Dic101—21no—
136Dic101—21no—
137Dic101—21no—
138Dic101—42no—
139Dic101—41no—
140Dic101#10412yescomplemented
141C2×C101—21no—
142C2×C101—21no—
143C2×C101—81no—
144C2×C101#39011yescharacteristic
145C2×C101C2×#10411yescharacteristic
146C2×C101#42111yescharacteristic
147D101—81no—
148#1980—1011nomaximal, Sylow
149C101#198011yescharacteristic, complemented, Sylow
150#532—1011no—
151C2×#104—1011no—
152C2×#104—1011no—
153C2×#104—1011no—
154C2×#104D10111yescharacteristic, complemented
155C4×M16—1011no—
156C4×C4⋊C4—1011no—
157C2×#41—1011no—
158C2×#41C2×D10111yescharacteristic
159C2×M16—1011no—
160C2×M16C2×D10111yescharacteristic
161C4⋊Q8—1011no—
162#127—1011no—
163#124—1011no—
164#124—1011no—
165#121—2021no—
166#121—2021no—
167C4×D4—2021no—
168C4×D4—2021no—
169C2×C4⋊C4—2021no—
170C2×C42—1011no—
171C2×C42—1011no—
172C2×C42C2×D10111yescharacteristic
173#104—22no—
174#104—2022no—
175#104—2022no—
176#104—2022no—
177#97—2021no—
178C4×C8—2021no—
179#95—2021no—
180#41—21no—
181#41—21no—
182#41—21no—
183#41—21no—
184#41—2021no—
185#41—2021no—
186#41—2021no—
187#41—2021no—
188C2×Q8—1011no—
189C2×Q8C4×D10111yescharacteristic
190C2×D4—21no—
191C2×D4—1011no—
192C2×D4—2021no—
193C2×D4—2021no—
194C2×D4—2021no—
195C2×D4C4×D10111yescharacteristic
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—1011no—
199C22×C4—1011no—
200C22×C4—2021no—
201C22×C4—2021no—
202C22×C4C22×D10111yescharacteristic
203M16—21no—
204M16—1012no—
205M16—2021no—
206M16C22⋊D10112yescomplemented
207C2×C8—21no—
208C2×C8—2021no—
209C4⋊C4—2021no—
210C4⋊C4—2021no—
211C4⋊C4—2024no—
212C22⋊C4—2021no—
213C22⋊C4—2021no—
214C22⋊C4—2021no—
215C22⋊C4—2021no—
216C42—21no—
217C42—21no—
218C42—1011no—
219C42—1011no—
220C42—1011no—
221C42—1011no—
222C42—2021no—
223C42—2021no—
224C23—21no—
225C23—2021no—
226C23D4×D10111yescharacteristic
227Q8—21no—
228Q8—21no—
229Q8—2021no—
230Q8—2021no—
231D4—21no—
232D4—21no—
233D4—41no—
234D4—41no—
235D4—4042no—
236D4—4041no—
237D4—4041no—
238D4—4042no—
239D4—2021no—
240D4—2021no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—1011no—
248C2×C4—1011no—
249C2×C4—1011no—
250C2×C4—1011no—
251C2×C4—4041no—
252C2×C4—4041no—
253C2×C4—4041no—
254C2×C4—4041no—
255C2×C4—4041no—
256C2×C4—2022no—
257C2×C4—2022no—
258C2×C4—2021no—
259C2×C4C2×C4×D10111yescharacteristic
260C2×C4D4×D10111yescharacteristic
261C2×C4D4×D10111yescharacteristic
262C2×C4C2×C22⋊D10111yescharacteristic
263C2×C4#1310807311yescharacteristic
264C2×C4#1310807411yescharacteristic
265C8—22no—
266C8—2022no—
267C22—21no—
268C22—21no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—4041no—
273C22—4041no—
274C22—4041no—
275C22C22⋊C4×D10111yescharacteristic
276C22#1569224111yescharacteristic
277C22#1569225111yescharacteristic
278C4—42no—
279C4—41no—
280C4—1012no—
281C4—4042no—
282C4—4041no—
283C4—2021no—
284C4—2021no—
285C4—2021no—
286C4#1569221511yescharacteristic
287C4C4×C22⋊D10111yescharacteristic
288C4#1569222711yescharacteristic
289C4#1569222811yescharacteristic
290C2—21no—
291C2—21no—
292C2—81no—
293C2—8081no—
294C2#2026507711yescharacteristic
295C2#104×D10111yescharacteristic
296C2#2026519211yescharacteristic
subgroupG/H
centerC2×C4D4×D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
FittingC2×#104×C101C2
socleC22×C101C2×C22⋊C4
prime radical#20265723C2
cosocleC22×C4×C101C23
hypercenterC2×#104D101
hypocenterC101#1980
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#15692183C2semidirect
3. C2×#15692182C2semidirect
4. C2×#15692181C2semidirect
5. M16×Dic101C2semidirect
6. C4×#13108053C2semidirect
7. C2×#104×C101C2semidirect
10. #15692253C4semidirect
11. #15692251C4semidirect
12. #15692244C4semidirect
13. #15692242C4semidirect
45. #13108058D4semidirect
71. M16×C101D4semidirect
140. Dic101#104semidirect
149. C101#1980semidirect
154. C2×#104D101semidirect
206. M16C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202657232
C2×#156921832
C2×#156921822
C2×#156921812
M16×Dic1012
C4×#131080532
C2×#104×C1012
#1980101