Finite Group Databasebeta

← #27172141

Subgroups — #27172141

Order 12928.

392 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 91 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×#13108064C211yesmaximal, characteristic, complemented
2C2×#15692235C211yesmaximal, characteristic
3C2×#15692231C211yesmaximal, characteristic
4C2×#15692180C211yesmaximal, characteristic, complemented
5C2×#15692167C211yesmaximal, characteristic
6#20264939C211yesmaximal, characteristic, complemented
7C2×#103×C101C211yesmaximal, characteristic, complemented
8C22×Dic202C2211yescharacteristic
9C22×C4×D101C2211yescharacteristic
10C2×#13108064—22no—
11C2×#13108064—22no—
12C2×#13108064—22no—
13C22×#11717579C2211yescharacteristic
14#15692235—22no—
15#15692235—22no—
16#15692231—22no—
17#15692231—22no—
18C2×C4⋊Dic101—21no—
19C2×C4⋊Dic101C2211yescharacteristic
20C2×#13108053—21no—
21C2×#13108051—21no—
22C2×#13108051—21no—
23#15692180—22no—
24#15692167—22no—
25C22×Q8×C101C2211yescharacteristic
26C22×C8×C101C2211yescharacteristic
27C2×C4⋊C4×C101C2211yescharacteristic
28#103×C101—22no—
29C23×D101D411yescharacteristic
30C2×Dic202—21no—
31C2×Dic202—21no—
32C2×Dic202—21no—
33C2×Dic202—21no—
34C2×Dic202—21no—
35C2×Dic202D412yes—
36C22×Dic101—21no—
37C22×Dic101—21no—
38C22×Dic101D411yescharacteristic
39C2×C4×D101—21no—
40C2×C4×D101—21no—
41C2×C4×D101—21no—
42C2×C4×D101—21no—
43C2×C4×D101—21no—
44C2×C4×D101—21no—
45#13108064—28no—
46C2×#11717579—21no—
47C2×#11717579—21no—
48C2×#11717579D412yes—
49C4⋊Dic101—24no—
50C4⋊Dic101—21no—
51C4⋊Dic101D412yes—
52#13108053—24no—
53#13108051—24no—
54#13108051—24no—
55C2×Q8×C101—21no—
56C2×Q8×C101—21no—
57C2×Q8×C101—21no—
58C2×Q8×C101—21no—
59C2×Q8×C101—21no—
60C2×Q8×C101D412yes—
61C22×C4×C101—21no—
62C22×C4×C101—21no—
63C22×C4×C101C2311yescharacteristic
64C2×C8×C101—21no—
65C2×C8×C101—21no—
66C2×C8×C101—21no—
67C4⋊C4×C101—21no—
68C4⋊C4×C101D412yes—
69C23×C101C2×D411yescharacteristic
70C22×D101—21no—
71C22×D101—21no—
72C22×D101—21no—
73C22×D101—21no—
74C22×D101—21no—
75C22×D101—21no—
76C22×D101Q1612yescomplemented
77Q8×C101—22no—
78Q8×C101—22no—
79Q8×C101—41no—
80Q8×C101—41no—
81C2×C4×C101—22no—
82C2×C4×C101—22no—
83C2×C4×C101—41no—
84C2×C4×C101—41no—
85C2×C4×C101—41no—
86C2×C4×C101—41no—
87C2×C4×C101C2×D411yescharacteristic
88C2×C4×C101C2×D411yescharacteristic
89C2×C4×C101C2×D411yescharacteristic
90C2×C4×C101C2×D411yescharacteristic
91C2×C4×C101C2×D411yescharacteristic
92C2×C4×C101#4111yescharacteristic
93C2×Dic101—22no—
94C2×Dic101—22no—
95C2×Dic101—21no—
96C2×Dic101—21no—
97C2×Dic101—21no—
98C2×Dic101—41no—
99C2×Dic101—41no—
100C2×Dic101—41no—
101C2×Dic101—41no—
102C4×D101—41no—
103C4×D101—41no—
104C4×D101—41no—
105C4×D101—41no—
106Dic202—22no—
107Dic202—22no—
108Dic202—41no—
109Dic202—41no—
110C8×C101—41no—
111#11717579—22no—
112C22×C101#12011yescharacteristic
113C22×C101#12011yescharacteristic
114C22×C101#12111yescharacteristic
115C22×C101C2×Q1611yescharacteristic
116C22×C101#13511yescharacteristic
117C22×C101#13711yescharacteristic
118C22×C101#13711yescharacteristic
119C2×D101—22no—
120C2×D101—22no—
121C2×D101—22no—
122C2×D101—41no—
123C2×D101—41no—
124C2×D101—41no—
125C2×D101—41no—
126C4×C101—42no—
127C4×C101—42no—
128C4×C101#12011yescharacteristic
129C4×C101#12111yescharacteristic
130C4×C101#12111yescharacteristic
131C4×C101#12711yescharacteristic
132Dic101—42no—
133Dic101—41no—
134Dic101—42no—
135C2×C101#39211yescharacteristic
136C2×C101#45111yescharacteristic
137C2×C101#45111yescharacteristic
138C2×C101#45211yescharacteristic
139C2×C101#45211yescharacteristic
140C2×C101#46711yescharacteristic
141C2×C101#47011yescharacteristic
142D101—42no—
143#2001—1011nomaximal, Sylow
144C101#200111yescharacteristic, complemented, Sylow
145C22×Q16—1011no—
146C2×C22⋊Q8—1011no—
147C2×C22⋊Q8—1011no—
148C2×#103—1011no—
149C2×#103D10111yescharacteristic, complemented
150C2×C22⋊C8—1011no—
151#340—1011no—
152C22×Q8—1011no—
153C22×Q8C2×D10111yescharacteristic
154C23×C4—1011no—
155C2×Q16—2022no—
156C2×Q16—2022no—
157C2×Q16—2022no—
158C22×C8—1011no—
159C22×C8C2×D10111yescharacteristic
160C22⋊Q8—2022no—
161C22⋊Q8—2022no—
162C22⋊Q8—2022no—
163C22⋊Q8—2022no—
164C2×C4⋊C4—1011no—
165C2×C4⋊C4—2021no—
166C2×C4⋊C4—2021no—
167C2×C4⋊C4C2×D10111yescharacteristic
168C2×C22⋊C4—2021no—
169C2×C22⋊C4—2021no—
170#103—22no—
171#103—2022no—
172C22⋊C8—2022no—
173C24—1011no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—1012no—
180C2×Q8—2021no—
181C2×Q8—2021no—
182C2×Q8—2021no—
183C2×Q8—2021no—
184C2×Q8—2021no—
185C2×Q8D40412yes—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—1011no—
189C22×C4—2021no—
190C22×C4—2021no—
191C22×C4—2021no—
192C22×C4—2021no—
193C22×C4—2021no—
194C22×C4—2021no—
195C22×C4—2021no—
196C22×C4—2021no—
197C22×C4C22×D10111yescharacteristic
198Q16—2028no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—1012no—
203C2×C8—2021no—
204C2×C8—2021no—
205C4⋊C4—21no—
206C4⋊C4—1012no—
207C4⋊C4—2024no—
208C4⋊C4—2024no—
209C4⋊C4—2021no—
210C4⋊C4C22⋊D10112yes—
211C22⋊C4—2024no—
212C22⋊C4—2024no—
213C23—1012no—
214C23—2021no—
215C23—2021no—
216C23—2021no—
217C23—2021no—
218C23—2021no—
219C23—2021no—
220C23D4×D10111yescharacteristic
221Q8—22no—
222Q8—22no—
223Q8—41no—
224Q8—41no—
225Q8—4041no—
226Q8—4041no—
227Q8—2022no—
228Q8—2022no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—4041no—
236C2×C4—4041no—
237C2×C4—4041no—
238C2×C4—4041no—
239C2×C4—4041no—
240C2×C4—4041no—
241C2×C4—4041no—
242C2×C4—4041no—
243C2×C4—2022no—
244C2×C4—2021no—
245C2×C4—2022no—
246C2×C4—2021no—
247C2×C4—2021no—
248C2×C4D4×D10111yescharacteristic
249C2×C4D4×D10111yescharacteristic
250C2×C4D4×D10111yescharacteristic
251C2×C4C2×C22⋊D10111yescharacteristic
252C2×C4C2×D40411yescharacteristic
253C2×C4#1310807411yescharacteristic
254C8—41no—
255C8—2022no—
256C22—4041no—
257C22—4041no—
258C22—4041no—
259C22—4041no—
260C22—2022no—
261C22—2022no—
262C22—2022no—
263C22#1569222311yescharacteristic
264C22#1569222511yescharacteristic
265C22#1569222811yescharacteristic
266C22Q16×D10111yescharacteristic
267C22#1569227611yescharacteristic
268C22#1569228411yescharacteristic
269C22#1569228711yescharacteristic
270C4—42no—
271C4—42no—
272C4—4041no—
273C4—4042no—
274C4—4042no—
275C4#1569222511yescharacteristic
276C4#1569222811yescharacteristic
277C4#1569222911yescharacteristic
278C4#1569225111yescharacteristic
279C2—4042no—
280C2#2026508611yescharacteristic
281C2#2026533111yescharacteristic
282C2#2026533411yescharacteristic
283C2#2026533611yescharacteristic
284C2#2026534211yescharacteristic
285C2#2026541711yescharacteristic
286C2#2026543411yescharacteristic
subgroupG/H
centerC23D4×D101
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
FittingC2×#103×C101C2
socleC23×C101C2×D4
prime radicalC23×D101D4
hypercenterC2×#103D101
hypocenterC101#2001
LCSG ▷ C22×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C22×#13108064C2semidirect
4. C2×#15692180C2semidirect
6. #20264939C2semidirect
7. C2×#103×C101C2semidirect
76. C22×D101Q16semidirect
144. C101#2001semidirect
149. C2×#103D101semidirect

Maximal subgroups

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

subgroupindex
C22×#131080642
C2×#156922352
C2×#156922312
C2×#156921802
C2×#156921672
#202649392
C2×#103×C1012
#2001101