Finite Group Databasebeta

← #27187300

Subgroups — #27187300

Order 12928.

344 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 78 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
1D101⋊(C4⋊C8)C211yesmaximal, characteristic, complemented
2C2×#15692318C212yesmaximal, complemented
3C2×#15692317C212yesmaximal
4C4×C4⋊#87938C211yesmaximal, characteristic, complemented
5#124×D101C211yesmaximal, characteristic
6C2×D101⋊C8C2212yes—
7C2×C4⋊#87938C2212yes—
8C2×C4×#87938—21no—
9#15692322—21no—
10C4⋊#11717581—21no—
11#15692318—24no—
12#15692317—24no—
13(C2×#87938)⋊C4—21no—
14C2×Q8×D101C2211yescharacteristic
15C2×D4×D101C2211yescharacteristic
16#15692244C411yescharacteristic, complemented
17#15692243C411yescharacteristic, complemented
18#15692242C411yescharacteristic, complemented
19#15692241—21no—
20#15692241—21no—
21C22⋊C4×D101—21no—
22C22⋊C4×D101—21no—
23C42×D101C2211yescharacteristic
24#124×C101C411yescharacteristic, complemented
25C22×#87938—22no—
26D101⋊C8—22no—
27D101⋊C8—22no—
28C2×#11717581—22no—
29C4⋊#87938—22no—
30C4⋊#87938D414yes—
31C4×#87938—42no—
32C23×D101—21no—
33C2×Dic202—21no—
34C2×Dic202C2×C411yescharacteristic
35Q8×D101—22no—
36Q8×D101—22no—
37C2×D404C2×C411yescharacteristic
38C2×C22⋊D101—21no—
39D4×D101—22no—
40D4×D101—22no—
41C2×C4×D101—21no—
42C2×C4×D101D411yescharacteristic
43C2×C4×D101D411yescharacteristic
44C2×C4×D101C2311yescharacteristic
45C22⋊Dic101—21no—
46C22⋊Dic101—21no—
47#13108051—21no—
48#13108051—21no—
49#13108051—21no—
50#13108051—21no—
51C4×Dic101C2×C411yescharacteristic
52C4×Dic101D411yescharacteristic
53C4×Dic101D411yescharacteristic
54C2×Q8×C101C2×C411yescharacteristic
55C2×D4×C101C2×C411yescharacteristic
56C22⋊C4×C101—21no—
57C22⋊C4×C101—21no—
58C42×C101C2×C411yescharacteristic
59C23×C101—21no—
60C22×D101—21no—
61C22×D101—42no—
62C22×D101—42no—
63C22×D101—41no—
64C22×D101—41no—
65C22×D101C2×D411yescharacteristic
66C2×#87938—24no—
67C2×#87938—42no—
68C2×#87938—42no—
69Q8×C101—22no—
70D4×C101—22no—
71C2×C4×C101—21no—
72C2×C4×C101C22⋊C411yescharacteristic
73C2×C4×C101C22⋊C411yescharacteristic
74C2×C4×C101C22×C411yescharacteristic
75C22⋊D101—41no—
76C22⋊D101—41no—
77C2×Dic101—21no—
78C2×Dic101C22⋊C411yescharacteristic
79C2×Dic101C22⋊C411yescharacteristic
80C2×Dic101C2×D411yescharacteristic
81D404—22no—
82C4×D101—22no—
83C4×D101—21no—
84C4×D101—21no—
85C4×D101—42no—
86C4×D101C2×D412yes—
87C4×D101#4112yes—
88Dic202—22no—
89Dic202—41no—
90Dic202—41no—
91#11717581—42no—
92C22×C101—41no—
93C22×C101—41no—
94C22×C101—41no—
95C22×C101C2×C22⋊C411yescharacteristic
96C2×D101—41no—
97C2×D101—41no—
98C2×D101—41no—
99C2×D101—82no—
100C2×D101—82no—
101C2×D101#12112yes—
102C2×D101#13512yes—
103C2×D101AGL(1,ℤ/8)11yescharacteristic
104C2×D101#13711yescharacteristic
105#87938—44no—
106C4×C101—21no—
107C4×C101—21no—
108C4×C101—41no—
109C4×C101C4×D412yes—
110Dic101—21no—
111Dic101—21no—
112Dic101—41no—
113Dic101#12411yescharacteristic
114Dic101#12711yescharacteristic
115C2×C101—81no—
116C2×C101#39011yescharacteristic
117C2×C101#41611yescharacteristic
118C2×C101#41711yescharacteristic
119D101—81no—
120D101#46312yes—
121D101#46412yes—
122#1983—1011nomaximal, Sylow
123C101#198311yescharacteristic, complemented, Sylow
124C2×#124—1011no—
125C2×C4⋊C8—1011no—
126C2×#103—1012no—
127C2×#102—1012no—
128C4×C4⋊C4—1011no—
129C22×Q8—1011no—
130C22×D4—1011no—
131C22×C8—1012no—
132#124—1011no—
133#124—1011no—
134#124—1011no—
135#124—2021no—
136#124—2021no—
137#124#8793811yescharacteristic, complemented
138C2×C4⋊C4—1012no—
139C2×C22⋊C4—2021no—
140C2×C22⋊C4—2021no—
141C2×C42—1011no—
142C2×C42—2021no—
143C4⋊C8—2021no—
144C4⋊C8—2021no—
145#103—2024no—
146#102—2024no—
147#95—2021no—
148C24—2021no—
149C2×Q8—1011no—
150C2×Q8—2021no—
151C2×Q8—2022no—
152C2×Q8—2022no—
153C2×Q8C2×#8793811yescharacteristic
154C2×D4—1011no—
155C2×D4—2022no—
156C2×D4—2022no—
157C2×D4—2021no—
158C2×D4C2×#8793811yescharacteristic
159C22×C4—1011no—
160C22×C4—1011no—
161C22×C4—1011no—
162C22×C4—2021no—
163C22×C4—2022no—
164C2×C8—2022no—
165C2×C8—2022no—
166C2×C8—2022no—
167C4⋊C4—1014no—
168C4⋊C4—2022no—
169C22⋊C4—21no—
170C22⋊C4—21no—
171C22⋊C4—2021no—
172C22⋊C4—2021no—
173C22⋊C4—2021no—
174C22⋊C4—2021no—
175C22⋊C4—2021no—
176C22⋊C4—2021no—
177C42—1011no—
178C42—1011no—
179C42—1011no—
180C42—4042no—
181C42C2×#8793811yescharacteristic
182C23—21no—
183C23—1011no—
184C23—4042no—
185C23—4042no—
186C23—4041no—
187C23—4041no—
188C23—2021no—
189Q8—22no—
190Q8—4041no—
191Q8—4041no—
192Q8—2022no—
193D4—22no—
194D4—4041no—
195D4—4041no—
196D4—2022no—
197C2×C4—21no—
198C2×C4—1011no—
199C2×C4—1012no—
200C2×C4—1012no—
201C2×C4—1011no—
202C2×C4—1011no—
203C2×C4—4042no—
204C2×C4—4042no—
205C2×C4—4042no—
206C2×C4—2021no—
207C2×C4—2022no—
208C2×C4—2021no—
209C2×C4—2024no—
210C2×C4—2021no—
211C2×C4C22⋊#8793811yescharacteristic
212C2×C4C22⋊#8793811yescharacteristic
213C2×C4C22×#8793811yescharacteristic
214C8—4042no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—8082no—
219C22—8082no—
220C22—1012no—
221C22—1012no—
222C22—1011no—
223C22—1011no—
224C22—4041no—
225C22—4041no—
226C22—4041no—
227C22C2×C22⋊#8793811yescharacteristic
228C4—21no—
229C4—21no—
230C4—41no—
231C4—1011no—
232C4—1011no—
233C4—4044no—
234C4—4041no—
235C4—2021no—
236C4—2021no—
237C4D4×#8793812yes—
238C2—81no—
239C2—8081no—
240C2—1012no—
241C2—1012no—
242C2#2026612911yescharacteristic
243C2#2026616711yescharacteristic
244C2#2026616811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#124×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC2×D4×D101C22
cosocleC2×C4×D101C23
hypercenter#124#87938
hypocenterC101#1983
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. D101⋊(C4⋊C8)C2semidirect
2. C2×#15692318C2semidirect
4. C4×C4⋊#87938C2semidirect
16. #15692244C4semidirect
17. #15692243C4semidirect
18. #15692242C4semidirect
24. #124×C101C4semidirect
123. C101#1983semidirect
137. #124#87938semidirect

Maximal subgroups

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

subgroupindex
D101⋊(C4⋊C8)2
C2×#156923182
C2×#156923172
C4×C4⋊#879382
#124×D1012
#1983101