Finite Group Databasebeta

← #27171488

Subgroups — #27171488

Order 12928.

374 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 101 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×#13108063C211yesmaximal, characteristic
2C2×#15692220C211yesmaximal, characteristic, complemented
3C2×#15692178C211yesmaximal, characteristic, complemented
4C2×#15692175C211yesmaximal, characteristic
5#20265078C211yesmaximal, characteristic
6#20264931C211yesmaximal, characteristic, complemented
7#340×C101C211yesmaximal, characteristic, complemented
8C22×Dic202C2211yescharacteristic
9C22×D404C2211yescharacteristic
10C2×#13108063—22no—
11C2×#13108063—22no—
12C2×#13108063C414yescomplemented
13C22×#11717579C2211yescharacteristic
14#15692220—22no—
15#15692220—22no—
16C2×#13108053—21no—
17C2×#13108051—21no—
18C2×#13108051—21no—
19C2×C4×Dic101—21no—
20C2×C4×Dic101C2211yescharacteristic
21#15692178—22no—
22#15692175—22no—
23C22×C8×C101C2211yescharacteristic
24C2×C4⋊C4×C101C2211yescharacteristic
25C2×C4⋊C4×C101C2211yescharacteristic
26C23×D101—21no—
27C2×Dic202—21no—
28C2×Dic202—21no—
29C2×Dic202—21no—
30C2×Dic202C2×C412yes—
31C2×Dic202D412yes—
32C2×Dic202D412yes—
33C2×D404—21no—
34C2×D404—21no—
35C2×D404—21no—
36C2×D404—21no—
37C2×D404—21no—
38C2×D404C2×C412yes—
39C22×Dic101—21no—
40C22×Dic101D411yescharacteristic
41C22×Dic101D411yescharacteristic
42#13108063—28no—
43C2×#11717579—21no—
44C2×#11717579—21no—
45C2×#11717579—21no—
46#13108053—24no—
47#13108051—42no—
48#13108051—42no—
49C4×Dic101—24no—
50C4×Dic101—21no—
51C4×Dic101—21no—
52C22×C4×C101—21no—
53C22×C4×C101—21no—
54C22×C4×C101C2311yescharacteristic
55C2×C8×C101—21no—
56C2×C8×C101—21no—
57C2×C8×C101C2×C412yes—
58C4⋊C4×C101—21no—
59C4⋊C4×C101—21no—
60C4⋊C4×C101—21no—
61C4⋊C4×C101D412yes—
62C23×C101C2×D411yescharacteristic
63C22×D101—22no—
64C22×D101—41no—
65C22×D101—41no—
66C22×D101—41no—
67C22×D101—41no—
68C22×D101—41no—
69C22×D101—41no—
70C2×C4×C101—22no—
71C2×C4×C101—41no—
72C2×C4×C101—41no—
73C2×C4×C101—41no—
74C2×C4×C101—41no—
75C2×C4×C101—41no—
76C2×C4×C101C22×C411yescharacteristic
77C2×C4×C101C2×D411yescharacteristic
78C2×C4×C101C2×D411yescharacteristic
79C2×C4×C101C2×D411yescharacteristic
80C2×C4×C101#4111yescharacteristic
81C2×C4×C101#4111yescharacteristic
82C2×Dic101—22no—
83C2×Dic101—21no—
84C2×Dic101—22no—
85C2×Dic101—22no—
86C2×Dic101—21no—
87C2×Dic101—21no—
88C2×Dic101—21no—
89C2×Dic101—21no—
90C2×Dic101SD1612yescomplemented
91D404—22no—
92D404—22no—
93D404—41no—
94D404—41no—
95Dic202—22no—
96Dic202—22no—
97Dic202—22no—
98Dic202C22⋊C414yescomplemented
99C8×C101—22no—
100#11717579—41no—
101C22×C101C4×D411yescharacteristic
102C22×C101#12011yescharacteristic
103C22×C101#12111yescharacteristic
104C22×C101C2×SD1611yescharacteristic
105C22×C101#13511yescharacteristic
106C22×C101AGL(1,ℤ/8)11yescharacteristic
107C22×C101#13711yescharacteristic
108C2×D101—42no—
109C2×D101—42no—
110C2×D101—42no—
111C2×D101—81no—
112C2×D101—81no—
113C2×D101—81no—
114C2×D101—81no—
115C4×C101—42no—
116C4×C101—81no—
117C4×C101C2×C22⋊C411yescharacteristic
118C4×C101C4×D411yescharacteristic
119C4×C101#12111yescharacteristic
120C4×C101#12311yescharacteristic
121Dic101—24no—
122Dic101—22no—
123Dic101—41no—
124C2×C101#38611yescharacteristic
125C2×C101C4×SD1611yescharacteristic
126C2×C101#44011yescharacteristic
127C2×C101#44811yescharacteristic
128C2×C101#44911yescharacteristic
129C2×C101#46011yescharacteristic
130C2×C101#46411yescharacteristic
131D101—82no—
132#1880—1011nomaximal, Sylow
133C101#188011yescharacteristic, complemented, Sylow
134C22×SD16—1011no—
135C2×C4×Q8—1011no—
136C2×#103—1011no—
137C2×#102—1011no—
138#390—1011no—
139#340D10111yescharacteristic, complemented
140#336—1011no—
141C22×Q8—1011no—
142C22×D4—1011no—
143C2×SD16—1014no—
144C2×SD16—2022no—
145C2×SD16—2022no—
146C22×C8—1011no—
147C22×C8C2×D10111yescharacteristic
148C4×Q8—2022no—
149C4×Q8—2022no—
150C2×C4⋊C4—2021no—
151C2×C4⋊C4C2×D10111yescharacteristic
152C2×C4⋊C4C2×D10111yescharacteristic
153C2×C22⋊C4—2021no—
154C2×C22⋊C4—2021no—
155C2×C42—1011no—
156C2×C42—2021no—
157#103—2022no—
158#102—2022no—
159C24—2021no—
160C2×Q8—1012no—
161C2×Q8—1012no—
162C2×Q8—1012no—
163C2×Q8—2021no—
164C2×Q8—2021no—
165C2×Q8—2021no—
166C2×D4—1012no—
167C2×D4—2021no—
168C2×D4—2021no—
169C2×D4—2021no—
170C2×D4—2021no—
171C2×D4—2021no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—1011no—
175C22×C4—1011no—
176C22×C4—2021no—
177C22×C4C22×D10111yescharacteristic
178SD16—2028no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—2021no—
182C2×C8—2021no—
183C2×C8—2021no—
184C2×C8C4×D10112yes—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—2024no—
189C4⋊C4D40412yes—
190C22⋊C4—4042no—
191C22⋊C4—4042no—
192C42—2024no—
193C42—2021no—
194C42—2021no—
195C23—4041no—
196C23—4041no—
197C23—4041no—
198C23—4041no—
199C23—4041no—
200C23—4041no—
201C23—2022no—
202C23D4×D10111yescharacteristic
203Q8—1014no—
204Q8—2022no—
205Q8—2022no—
206Q8—2022no—
207D4—4041no—
208D4—4041no—
209D4—2022no—
210D4—2022no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—1012no—
218C2×C4—2021no—
219C2×C4—2021no—
220C2×C4—2021no—
221C2×C4—2022no—
222C2×C4—2021no—
223C2×C4—2022no—
224C2×C4—2022no—
225C2×C4—2021no—
226C2×C4C2×C4×D10111yescharacteristic
227C2×C4D4×D10111yescharacteristic
228C2×C4D4×D10111yescharacteristic
229C2×C4C2×D40411yescharacteristic
230C2×C4#1310807411yescharacteristic
231C2×C4#1310807511yescharacteristic
232C8—22no—
233C8—4041no—
234C22—8081no—
235C22—8081no—
236C22—8081no—
237C22—8081no—
238C22—4042no—
239C22—4042no—
240C22—4042no—
241C22#1569221711yescharacteristic
242C22#1569222511yescharacteristic
243C22#1569222811yescharacteristic
244C22SD16×D10111yescharacteristic
245C22#1569227611yescharacteristic
246C22#1569227911yescharacteristic
247C22#1569228711yescharacteristic
248C4—42no—
249C4—81no—
250C4—4041no—
251C4—2024no—
252C4—2022no—
253C4C22⋊C4×D10111yescharacteristic
254C4C4×D40411yescharacteristic
255C4#1569223011yescharacteristic
256C4#1569224011yescharacteristic
257C2—8082no—
258C2#2026505511yescharacteristic
259C2#2026526411yescharacteristic
260C2#2026527611yescharacteristic
261C2#2026531711yescharacteristic
262C2#2026532311yescharacteristic
263C2#2026537611yescharacteristic
264C2#2026540211yescharacteristic
subgroupG/H
centerC23D4×D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#340×C101C2
socleC23×C101C2×D4
prime radicalC22×D404C22
cosocleC22×C4×C101C23
hypercenter#340D101
hypocenterC101#1880
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
2. C2×#15692220C2semidirect
3. C2×#15692178C2semidirect
6. #20264931C2semidirect
7. #340×C101C2semidirect
12. C2×#13108063C4semidirect
90. C2×Dic101SD16semidirect
98. Dic202C22⋊C4semidirect
133. C101#1880semidirect
139. #340D101semidirect

Maximal subgroups

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

subgroupindex
C22×#131080632
C2×#156922202
C2×#156921782
C2×#156921752
#202650782
#202649312
#340×C1012
#1880101