Finite Group Databasebeta

← #27171564

Subgroups — #27171564

Order 12928.

360 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 99 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×#13108057C211yesmaximal, characteristic
2C2×#15692235C211yesmaximal, characteristic
3C2×#15692180C211yesmaximal, characteristic, complemented
4C2×#15692178C211yesmaximal, characteristic, complemented
5#20265034C211yesmaximal, characteristic
6#20264939C211yesmaximal, characteristic, complemented
7#340×C101C211yesmaximal, characteristic, complemented
8C22×Dic202C2211yescharacteristic
9C22×C4×D101C2211yescharacteristic
10C2×#13108057—22no—
11C2×#13108057—22no—
12C2×#13108057—22no—
13C22×#11717579C2211yescharacteristic
14#15692235—22no—
15#15692235C414yescomplemented
16C2×C4⋊Dic101—21no—
17C2×C4⋊Dic101C2211yescharacteristic
18C2×#13108051—21no—
19C2×#13108051—21no—
20#15692180—22no—
21#15692178—22no—
22#15692162—21no—
23C22×C8×C101C2211yescharacteristic
24C2×C4⋊C4×C101C2211yescharacteristic
25C2×C4⋊C4×C101C2211yescharacteristic
26C23×D101D411yescharacteristic
27C2×Dic202—21no—
28C2×Dic202—21no—
29C2×Dic202—21no—
30C2×Dic202—21no—
31C2×Dic202—21no—
32C2×Dic202C2×C412yes—
33C22×Dic101—21no—
34C22×Dic101—21no—
35C22×Dic101D411yescharacteristic
36C2×C4×D101—21no—
37C2×C4×D101—21no—
38C2×C4×D101—21no—
39C2×C4×D101—21no—
40C2×C4×D101C2×C412yes—
41C2×C4×D101D412yes—
42#13108057—28no—
43C2×#11717579—21no—
44C2×#11717579—21no—
45C2×#11717579D412yes—
46C4⋊Dic101—24no—
47C4⋊Dic101—21no—
48C4⋊Dic101—21no—
49#13108051—24no—
50#13108051—42no—
51C22×C4×C101—21no—
52C22×C4×C101—21no—
53C22×C4×C101C2311yescharacteristic
54C2×C8×C101—21no—
55C2×C8×C101—21no—
56C2×C8×C101D412yes—
57C4⋊C4×C101—21no—
58C4⋊C4×C101—21no—
59C4⋊C4×C101—21no—
60C4⋊C4×C101C2×C412yes—
61C23×C101C2×D411yescharacteristic
62C22×D101—21no—
63C22×D101—21no—
64C22×D101—21no—
65C22×D101—21no—
66C22×D101—21no—
67C22×D101—21no—
68C22×D101C22⋊C412yes—
69C2×C4×C101—22no—
70C2×C4×C101—41no—
71C2×C4×C101—41no—
72C2×C4×C101—41no—
73C2×C4×C101—41no—
74C2×C4×C101—41no—
75C2×C4×C101C22×C411yescharacteristic
76C2×C4×C101C2×D411yescharacteristic
77C2×C4×C101C2×D411yescharacteristic
78C2×C4×C101C2×D411yescharacteristic
79C2×C4×C101#4111yescharacteristic
80C2×C4×C101#4111yescharacteristic
81C2×Dic101—21no—
82C2×Dic101—21no—
83C2×Dic101—22no—
84C2×Dic101—41no—
85C2×Dic101—41no—
86C2×Dic101—41no—
87C2×Dic101—41no—
88C2×Dic101—41no—
89C2×Dic101C22⋊C412yes—
90C4×D101—22no—
91C4×D101—22no—
92C4×D101—22no—
93C4×D101—22no—
94Dic202—22no—
95Dic202—22no—
96Dic202—41no—
97Dic202—41no—
98C8×C101—22no—
99#11717579—22no—
100C22×C101C2×C22⋊C411yescharacteristic
101C22×C101#12111yescharacteristic
102C22×C101#12111yescharacteristic
103C22×C101AGL(1,ℤ/8)11yescharacteristic
104C22×C101#13711yescharacteristic
105C22×C101#13711yescharacteristic
106C22×C101#13711yescharacteristic
107C2×D101—22no—
108C2×D101—22no—
109C2×D101—22no—
110C2×D101—41no—
111C2×D101—41no—
112C2×D101—41no—
113C2×D101—41no—
114C4×C101—42no—
115C4×C101—81no—
116C4×C101C4×D411yescharacteristic
117C4×C101C4×D411yescharacteristic
118C4×C101#12011yescharacteristic
119C4×C101#12311yescharacteristic
120Dic101—22no—
121Dic101—42no—
122Dic101—81no—
123C2×C101#38611yescharacteristic
124C2×C101#41711yescharacteristic
125C2×C101#41911yescharacteristic
126C2×C101#46811yescharacteristic
127C2×C101#46811yescharacteristic
128C2×C101#47011yescharacteristic
129C2×C101#47011yescharacteristic
130D101—42no—
131#1894—1011nomaximal, Sylow
132C101#189411yescharacteristic, complemented, Sylow
133C22×M16—1011no—
134C2×C22⋊Q8—1011no—
135C2×#103—1011no—
136C2×#103—1011no—
137#380—1011no—
138#340—1011no—
139#340D10111yescharacteristic, complemented
140C22×Q8—1011no—
141C23×C4—1011no—
142C2×M16—2022no—
143C2×M16—2022no—
144C2×M16—2022no—
145C22×C8—1011no—
146C22×C8C2×D10111yescharacteristic
147C22⋊Q8—1014no—
148C22⋊Q8—2022no—
149C2×C4⋊C4—1011no—
150C2×C4⋊C4—2021no—
151C2×C4⋊C4C2×D10111yescharacteristic
152C2×C4⋊C4C2×D10111yescharacteristic
153C2×C22⋊C4—2021no—
154C2×C22⋊C4—2021no—
155#103—2022no—
156#103—2022no—
157#95—2021no—
158C24—1011no—
159C2×Q8—1012no—
160C2×Q8—2021no—
161C2×Q8—2021no—
162C2×Q8—2021no—
163C2×Q8—2021no—
164C2×Q8—2021no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—1011no—
168C22×C4—1012no—
169C22×C4—1012no—
170C22×C4—2021no—
171C22×C4—2021no—
172C22×C4—2021no—
173C22×C4—2021no—
174C22×C4—2021no—
175C22×C4—2021no—
176C22×C4C22×D10111yescharacteristic
177M16—2028no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—1012no—
181C2×C8—2021no—
182C2×C8—2021no—
183C2×C8D40412yes—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—2024no—
188C4⋊C4—2021no—
189C4⋊C4—2021no—
190C4⋊C4C4×D10112yes—
191C22⋊C4—4042no—
192C22⋊C4—2024no—
193C23—1012no—
194C23—2021no—
195C23—2021no—
196C23—2021no—
197C23—2021no—
198C23—2021no—
199C23—2021no—
200C23D4×D10111yescharacteristic
201Q8—4041no—
202Q8—4041no—
203Q8—2022no—
204Q8—2022no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—1012no—
212C2×C4—4041no—
213C2×C4—4041no—
214C2×C4—4041no—
215C2×C4—4041no—
216C2×C4—4041no—
217C2×C4—2022no—
218C2×C4—2021no—
219C2×C4—2021no—
220C2×C4—2022no—
221C2×C4—2022no—
222C2×C4—2022no—
223C2×C4—2022no—
224C2×C4C2×C4×D10111yescharacteristic
225C2×C4D4×D10111yescharacteristic
226C2×C4D4×D10111yescharacteristic
227C2×C4C2×D40411yescharacteristic
228C2×C4#1310807411yescharacteristic
229C2×C4#1310807511yescharacteristic
230C8—22no—
231C8—2022no—
232C22—4041no—
233C22—4041no—
234C22—4041no—
235C22—4041no—
236C22—2022no—
237C22—2022no—
238C22—2022no—
239C22C22⋊C4×D10111yescharacteristic
240C22#1569222811yescharacteristic
241C22#1569223011yescharacteristic
242C22#1569227911yescharacteristic
243C22#1569228411yescharacteristic
244C22#1569228411yescharacteristic
245C22#1569228711yescharacteristic
246C4—42no—
247C4—81no—
248C4—8081no—
249C4—4042no—
250C4—2022no—
251C4#1569221711yescharacteristic
252C4C4×D40411yescharacteristic
253C4#1569222511yescharacteristic
254C4#1569224011yescharacteristic
255C2—4042no—
256C2#2026505511yescharacteristic
257C2#2026517511yescharacteristic
258C2#2026518411yescharacteristic
259C2#2026542211yescharacteristic
260C2#2026542411yescharacteristic
261C2#2026543411yescharacteristic
262C2#2026543611yescharacteristic
subgroupG/H
centerC23D4×D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#340×C101C2
socleC23×C101C2×D4
prime radicalC23×D101D4
cosocleC22×C4×C101C23
hypercenter#340D101
hypocenterC101#1894
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
3. C2×#15692180C2semidirect
4. C2×#15692178C2semidirect
6. #20264939C2semidirect
7. #340×C101C2semidirect
15. #15692235C4semidirect
132. C101#1894semidirect
139. #340D101semidirect

Maximal subgroups

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

subgroupindex
C22×#131080572
C2×#156922352
C2×#156921802
C2×#156921782
#202650342
#202649392
#340×C1012
#1894101