Finite Group Databasebeta

← #27171328

Subgroups — #27171328

Order 12928.

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

normal subgroups only

This group has 316 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
1C2×#15692213C211yesmaximal, characteristic
2C2×#15692186C211yesmaximal, characteristic, complemented
3C2×#15692167C211yesmaximal, characteristic
4#20265034C211yesmaximal, characteristic
5#20264940C211yesmaximal, characteristic, complemented
6#20264939C211yesmaximal, characteristic, complemented
7C2×#107×C101C211yesmaximal, characteristic, complemented
8C22×C4×D101C2211yescharacteristic
9C22×#11717579C2211yescharacteristic
10#15692213—22no—
11#15692213—22no—
12C2×C4⋊Dic101C2211yescharacteristic
13C2×C4⋊Dic101C2211yescharacteristic
14C2×#13108051—21no—
15C2×#13108051—21no—
16C2×C4×Dic101—21no—
17#15692186—22no—
18#15692167C414yescomplemented
19#15692162—21no—
20C22×C8×C101C2211yescharacteristic
21C2×C4⋊C4×C101C2211yescharacteristic
22C2×C4⋊C4×C101C2211yescharacteristic
23#107×C101—22no—
24C23×D101D411yescharacteristic
25C22×Dic101—21no—
26C22×Dic101—21no—
27C22×Dic101D411yescharacteristic
28C2×C4×D101—21no—
29C2×C4×D101—21no—
30C2×C4×D101—21no—
31C2×C4×D101—21no—
32C2×C4×D101C2×C412yes—
33C2×C4×D101Q812yes—
34C2×#11717579—21no—
35C2×#11717579—21no—
36C2×#11717579C2×C412yes—
37C4⋊Dic101—21no—
38C4⋊Dic101—21no—
39C4⋊Dic101—21no—
40C4⋊Dic101D412yes—
41#13108051—24no—
42#13108051—42no—
43C4×Dic101—24no—
44C22×C4×C101—21no—
45C22×C4×C101—21no—
46C22×C4×C101C2311yescharacteristic
47C2×C8×C101—21no—
48C2×C8×C101—21no—
49C2×C8×C101C2×C412yes—
50C4⋊C4×C101—21no—
51C4⋊C4×C101—21no—
52C4⋊C4×C101—21no—
53C4⋊C4×C101D412yes—
54C23×C101C2×D411yescharacteristic
55C22×D101—21no—
56C22×D101—21no—
57C22×D101—21no—
58C22×D101—21no—
59C22×D101—21no—
60C22×D101—21no—
61C22×D101C4⋊C412yes—
62C2×C4×C101—22no—
63C2×C4×C101—41no—
64C2×C4×C101—41no—
65C2×C4×C101—41no—
66C2×C4×C101—41no—
67C2×C4×C101—41no—
68C2×C4×C101C22×C411yescharacteristic
69C2×C4×C101C2×D411yescharacteristic
70C2×C4×C101C2×D411yescharacteristic
71C2×C4×C101C2×Q811yescharacteristic
72C2×C4×C101#4111yescharacteristic
73C2×C4×C101#4111yescharacteristic
74C2×Dic101—21no—
75C2×Dic101—21no—
76C2×Dic101—22no—
77C2×Dic101—41no—
78C2×Dic101—41no—
79C2×Dic101—41no—
80C2×Dic101—41no—
81C2×Dic101—41no—
82C2×Dic101C4⋊C412yes—
83C4×D101—22no—
84C4×D101—22no—
85C4×D101—22no—
86C4×D101—22no—
87C8×C101—22no—
88#11717579—22no—
89C22×C101C2×C4⋊C411yescharacteristic
90C22×C101#12011yescharacteristic
91C22×C101#12311yescharacteristic
92C22×C101#13511yescharacteristic
93C22×C101#13511yescharacteristic
94C22×C101AGL(1,ℤ/8)11yescharacteristic
95C22×C101#13711yescharacteristic
96C2×D101—22no—
97C2×D101—22no—
98C2×D101—22no—
99C2×D101—41no—
100C2×D101—41no—
101C2×D101—41no—
102C2×D101—41no—
103C4×C101—42no—
104C4×C101—81no—
105C4×C101C4×D411yescharacteristic
106C4×C101C4×D411yescharacteristic
107C4×C101C22⋊Q811yescharacteristic
108C4×C101C22⋊Q811yescharacteristic
109Dic101—22no—
110Dic101—42no—
111Dic101—81no—
112C2×C101#38511yescharacteristic
113C2×C101#42711yescharacteristic
114C2×C101#42811yescharacteristic
115C2×C101#44911yescharacteristic
116C2×C101#45211yescharacteristic
117C2×C101#48211yescharacteristic
118C2×C101#48511yescharacteristic
119D101—42no—
120#1854—1011nomaximal, Sylow
121C101#185411yescharacteristic, complemented, Sylow
122C2×#117—1011no—
123C2×#107D10111yescharacteristic, complemented
124C2×#106—1011no—
125C2×C22⋊C8—1011no—
126#380—1011no—
127#340—1011no—
128#340—1011no—
129C23×C4—1011no—
130C22×C8—1011no—
131C22×C8C2×D10111yescharacteristic
132#117—2022no—
133#117—2022no—
134C2×C4⋊C4—1011no—
135C2×C4⋊C4—1011no—
136C2×C4⋊C4C2×D10111yescharacteristic
137C2×C4⋊C4C2×D10111yescharacteristic
138C2×C22⋊C4—2021no—
139C2×C22⋊C4—2021no—
140C2×C42—2021no—
141#107—22no—
142#106—2022no—
143C22⋊C8—1014no—
144#95—2021no—
145C24—1011no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—1011no—
149C22×C4—1012no—
150C22×C4—1012no—
151C22×C4—2021no—
152C22×C4—2021no—
153C22×C4—2021no—
154C22×C4—2021no—
155C22×C4—2021no—
156C22×C4—2021no—
157C22×C4C22×D10111yescharacteristic
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—1012no—
161C2×C8—2021no—
162C2×C8—2021no—
163C2×C8C4×D10112yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—1012no—
168C4⋊C4—2021no—
169C4⋊C4—2021no—
170C4⋊C4—2021no—
171C4⋊C4C22⋊D10112yes—
172C22⋊C4—4042no—
173C22⋊C4—2024no—
174C42—2024no—
175C23—1012no—
176C23—2021no—
177C23—2021no—
178C23—2021no—
179C23—2021no—
180C23—2021no—
181C23—2021no—
182C23D4×D10111yescharacteristic
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—1012no—
190C2×C4—4041no—
191C2×C4—4041no—
192C2×C4—4041no—
193C2×C4—4041no—
194C2×C4—4041no—
195C2×C4—2022no—
196C2×C4—2021no—
197C2×C4—2021no—
198C2×C4—2022no—
199C2×C4—2022no—
200C2×C4—2022no—
201C2×C4—2022no—
202C2×C4C2×C4×D10111yescharacteristic
203C2×C4D4×D10111yescharacteristic
204C2×C4C2×C22⋊D10111yescharacteristic
205C2×C4Q8×D10111yescharacteristic
206C2×C4#1310807411yescharacteristic
207C2×C4#1310807511yescharacteristic
208C8—22no—
209C8—2022no—
210C22—4041no—
211C22—4041no—
212C22—4041no—
213C22—4041no—
214C22—2022no—
215C22—2022no—
216C22—2022no—
217C22C4⋊C4×D10111yescharacteristic
218C22#1569222311yescharacteristic
219C22#1569224011yescharacteristic
220C22#1569227411yescharacteristic
221C22#1569227811yescharacteristic
222C22#1569227911yescharacteristic
223C22#1569228711yescharacteristic
224C4—42no—
225C4—81no—
226C4—8081no—
227C4—4042no—
228C4—2022no—
229C4C4×C22⋊D10111yescharacteristic
230C4#1569221711yescharacteristic
231C4#1569223111yescharacteristic
232C4#1569223311yescharacteristic
233C2—4042no—
234C2#2026504811yescharacteristic
235C2#2026521411yescharacteristic
236C2#2026521811yescharacteristic
237C2#2026531911yescharacteristic
238C2#2026533811yescharacteristic
239C2#2026550411yescharacteristic
240C2#2026552011yescharacteristic
subgroupG/H
centerC23D4×D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
FittingC2×#107×C101C2
socleC23×C101C2×D4
prime radicalC23×D101D4
cosocleC22×C4×C101C23
hypercenterC2×#107D101
hypocenterC101#1854
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
2. C2×#15692186C2semidirect
5. #20264940C2semidirect
6. #20264939C2semidirect
7. C2×#107×C101C2semidirect
18. #15692167C4semidirect
121. C101#1854semidirect
123. C2×#107D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922132
C2×#156921862
C2×#156921672
#202650342
#202649402
#202649392
C2×#107×C1012
#1854101