Finite Group Databasebeta

← #398×D101 #27174361

Subgroups — #398×D101 #27174361

Order 12928.

692 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×D101C211yesmaximal, characteristic
2C2×C4⋊C4×D101C211yesmaximal, characteristic
3C2×C4⋊C4×D101C212yesmaximal
4#20265111C212yesmaximal, complemented
5#20265110C212yesmaximal, complemented
6#20265109C211yesmaximal, characteristic, complemented
7#20265108C211yesmaximal, characteristic, complemented
8#20265107C211yesmaximal, characteristic, complemented
9#95×D101C212yesmaximal
10#95×D101C211yesmaximal, characteristic
11#398×C101C211yesmaximal, characteristic, complemented
12C22×C4×D101C2211yescharacteristic
13C22×C4×D101C2212yes—
14C22×C4×D101C2211yescharacteristic
15C22×C4×D101C2211yescharacteristic
16C22×C4×D101C2212yes—
17C2×C4⋊Dic101C2211yescharacteristic
18C2×C4⋊Dic101C2212yes—
19C2×C4⋊Dic101C2211yescharacteristic
20C2×#13108053C2212yes—
21C2×#13108053C2211yescharacteristic
22C2×#13108053C2212yes—
23C2×#13108053C2212yes—
24C2×#13108053C2211yescharacteristic
25C4⋊C4×D101—28no—
26C4⋊C4×D101—24no—
27C4⋊C4×D101—24no—
28#15692162C2211yescharacteristic
29#15692162C2212yes—
30#15692162C2212yes—
31#15692162C2212yes—
32#15692162C2212yes—
33C2×C4⋊C4×C101C2211yescharacteristic
34C2×C4⋊C4×C101C2211yescharacteristic
35C2×C4⋊C4×C101C2212yes—
36#95×C101C2211yescharacteristic
37#95×C101C2212yes—
38C23×D101C2311yescharacteristic
39C22×Dic101C2311yescharacteristic
40C22×Dic101C2312yes—
41C22×Dic101C2311yescharacteristic
42C22×Dic101C2311yescharacteristic
43C22×Dic101C2312yes—
44C2×C4×D101—22no—
45C2×C4×D101—24no—
46C2×C4×D101—22no—
47C2×C4×D101—24no—
48C2×C4×D101—24no—
49C2×C4×D101—24no—
50C2×C4×D101—24no—
51C2×C4×D101—22no—
52C2×C4×D101—24no—
53C2×C4×D101—24no—
54C2×C4×D101D414yes—
55C2×C4×D101D414yes—
56C2×C4×D101Q818yes—
57C4⋊Dic101—22no—
58C4⋊Dic101—22no—
59C4⋊Dic101—24no—
60#13108053—24no—
61#13108053—22no—
62#13108053—22no—
63#13108053—24no—
64#13108053—24no—
65C22×C4×C101C2311yescharacteristic
66C22×C4×C101C2312yes—
67C22×C4×C101C2311yescharacteristic
68C22×C4×C101C2311yescharacteristic
69C22×C4×C101C2312yes—
70C4⋊C4×C101—24no—
71C4⋊C4×C101—22no—
72C4⋊C4×C101—22no—
73C23×C101C2411yescharacteristic
74C22×D101C2×D412yes—
75C22×D101C2×D412yes—
76C22×D101C2×Q814yes—
77C22×D101#4112yes—
78C22×D101#4114yes—
79C2×C4×C101—21no—
80C2×C4×C101—22no—
81C2×C4×C101—21no—
82C2×C4×C101—22no—
83C2×C4×C101—22no—
84C2×C4×C101—22no—
85C2×C4×C101—22no—
86C2×C4×C101—21no—
87C2×C4×C101—22no—
88C2×C4×C101—22no—
89C2×C4×C101C2×D412yes—
90C2×C4×C101C2×D412yes—
91C2×C4×C101C2×Q814yes—
92C2×Dic101—21no—
93C2×Dic101—22no—
94C2×Dic101—21no—
95C2×Dic101—22no—
96C2×Dic101—22no—
97C2×Dic101—22no—
98C2×Dic101—22no—
99C2×Dic101—21no—
100C2×Dic101—22no—
101C2×Dic101—22no—
102C2×Dic101C2×D412yes—
103C2×Dic101C2×D412yes—
104C2×Dic101C2×Q814yes—
105C4×D101—28no—
106C4×D101—216no—
107C4×D101—28no—
108C4×D101—48no—
109C4×D101—44no—
110C22×C101C22×D411yescharacteristic
111C22×C101C22×D411yescharacteristic
112C22×C101C22×Q812yes—
113C22×C101C2×#4111yescharacteristic
114C22×C101C2×#4112yes—
115C2×D101#12114yes—
116C2×D101C22⋊Q818yes—
117C2×D101#12314yes—
118C2×D101#12518yes—
119C2×D101C4⋊Q814yes—
120C4×C101—22no—
121C4×C101—24no—
122C4×C101—22no—
123C4×C101—42no—
124C4×C101—41no—
125Dic101—22no—
126Dic101—24no—
127Dic101—22no—
128Dic101—42no—
129Dic101—41no—
130C2×C101C2×#12111yescharacteristic
131C2×C101C2×C22⋊Q812yes—
132C2×C101C2×#12311yescharacteristic
133C2×C101C2×#12512yes—
134C2×C101C2×C4⋊Q811yescharacteristic
135D101#39818yesdirect product factor
136C2×#398—1011nomaximal, Sylow
137C101C2×#39811yescharacteristic, complemented, Sylow
138C22×C4⋊C4—1011no—
139C22×C4⋊C4—1012no—
140C22×C4⋊C4—1011no—
141#398—1011no—
142#398—1011no—
143#398—1012no—
144#398—1011no—
145#398—1012no—
146#398D10111yescharacteristic, direct product factor
147C2×#95—1011no—
148C2×#95—1012no—
149C23×C4—1011no—
150C23×C4—1011no—
151C23×C4—1012no—
152C23×C4—1012no—
153C23×C4—1011no—
154C2×C4⋊C4—1012no—
155C2×C4⋊C4—1011no—
156C2×C4⋊C4—1012no—
157C2×C4⋊C4—1012no—
158C2×C4⋊C4—1011no—
159C2×C4⋊C4—1011no—
160C2×C4⋊C4—1012no—
161C2×C4⋊C4—1011no—
162C2×C4⋊C4—2024no—
163C2×C4⋊C4—2028no—
164C2×C4⋊C4—2024no—
165C2×C4⋊C4C2×D10111yescharacteristic
166C2×C4⋊C4C2×D10111yescharacteristic
167C2×C4⋊C4C2×D10112yes—
168#95—1011no—
169#95—1012no—
170#95—1012no—
171#95—1012no—
172#95—1012no—
173#95C2×D10112yes—
174#95C2×D10111yescharacteristic
175C24—1011no—
176C22×C4—1012no—
177C22×C4—1011no—
178C22×C4—1011no—
179C22×C4—1014no—
180C22×C4—1014no—
181C22×C4—1018no—
182C22×C4—1011no—
183C22×C4—1012no—
184C22×C4—2024no—
185C22×C4—2024no—
186C22×C4—2022no—
187C22×C4—2024no—
188C22×C4—2024no—
189C22×C4—2022no—
190C22×C4—2024no—
191C22×C4—2022no—
192C22×C4—2024no—
193C22×C4—2024no—
194C22×C4C22×D10111yescharacteristic
195C22×C4C22×D10112yes—
196C22×C4C22×D10111yescharacteristic
197C22×C4C22×D10111yescharacteristic
198C22×C4C22×D10112yes—
199C4⋊C4—22no—
200C4⋊C4—24no—
201C4⋊C4—22no—
202C4⋊C4—2024no—
203C4⋊C4—2024no—
204C4⋊C4—2022no—
205C4⋊C4—2022no—
206C4⋊C4—2024no—
207C4⋊C4—2024no—
208C4⋊C4—2022no—
209C4⋊C4—2022no—
210C23—1012no—
211C23—1012no—
212C23—1014no—
213C23—1012no—
214C23—1014no—
215C23C23×D10111yescharacteristic
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—1012no—
227C2×C4—1012no—
228C2×C4—1014no—
229C2×C4—4048no—
230C2×C4—4044no—
231C2×C4—2028no—
232C2×C4—20216no—
233C2×C4—2028no—
234C2×C4—2022no—
235C2×C4—2022no—
236C2×C4—2021no—
237C2×C4—2022no—
238C2×C4—2022no—
239C2×C4—2021no—
240C2×C4—2022no—
241C2×C4—2021no—
242C2×C4—2022no—
243C2×C4—2022no—
244C2×C4D4×D10112yes—
245C2×C4D4×D10112yes—
246C2×C4Q8×D10114yes—
247C22—1014no—
248C22—1018no—
249C22—1014no—
250C22—1018no—
251C22—1014no—
252C22C2×D4×D10111yescharacteristic
253C22C2×D4×D10111yescharacteristic
254C22C2×Q8×D10112yes—
255C22#41×D10111yescharacteristic
256C22#41×D10112yes—
257C4—22no—
258C4—24no—
259C4—22no—
260C4—42no—
261C4—41no—
262C4—4042no—
263C4—4041no—
264C4—2022no—
265C4—2024no—
266C4—2022no—
267C2—1018no—
268C2#121×D10111yescharacteristic
269C2C22⋊Q8×D10112yes—
270C2#123×D10111yescharacteristic
271C2#125×D10112yes—
272C2C4⋊Q8×D10111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#398×C101C2
prime radicalC23×D101C23
hypercenter#398D101
hypocenterC101C2×#398
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
4. #20265111C2semidirect
5. #20265110C2semidirect
6. #20265109C2semidirect
7. #20265108C2semidirect
8. #20265107C2semidirect
11. #398×C101C2semidirect
135. D101#398direct (N × H)
137. C101C2×#398semidirect
146. #398D101direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D1012
#202651112
#202651102
#202651092
#202651082
#202651072
#95×D1012
#398×C1012
C2×#398101