Finite Group Databasebeta

← #27171471

Subgroups — #27171471

Order 12928.

338 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 107 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
2C2×Q8×Dic101C211yesmaximal, characteristic
3C2×#15692180C211yesmaximal, characteristic
4#20265082C211yesmaximal, characteristic
5#20264939C211yesmaximal, characteristic
6#20264931C211yesmaximal, characteristic
7C2×#103×C101C211yesmaximal, characteristic
8C22×Dic202C2211yescharacteristic
9C2×#13108064—22no—
10C2×#13108064—22no—
11C2×#13108064C414yescomplemented
12C22×#11717579C2211yescharacteristic
13Q8×Dic101—22no—
14Q8×Dic101—22no—
15C2×C4⋊Dic101—21no—
16C2×C4⋊Dic101C2211yescharacteristic
17C2×C4×Dic101—21no—
18C2×C4×Dic101C2211yescharacteristic
19#15692180—22no—
20#15692162—21no—
21#15692162—21no—
22C22×Q8×C101C2211yescharacteristic
23C22×C8×C101C2211yescharacteristic
24C2×C4⋊C4×C101C2211yescharacteristic
25#103×C101—22no—
26C2×Dic202—21no—
27C2×Dic202—21no—
28C2×Dic202—21no—
29C2×Dic202—21no—
30C2×Dic202—21no—
31C2×Dic202C2×C412yes—
32C22×Dic101—21no—
33C22×Dic101—21no—
34C22×Dic101D411yescharacteristic
35C22×Dic101D411yescharacteristic
36#13108064—28no—
37C2×#11717579—21no—
38C2×#11717579—21no—
39C2×#11717579C2×C412yes—
40C4⋊Dic101—24no—
41C4⋊Dic101—21no—
42C4⋊Dic101D412yes—
43C4×Dic101—24no—
44C4×Dic101—21no—
45C4×Dic101—21no—
46C2×Q8×C101—21no—
47C2×Q8×C101—21no—
48C2×Q8×C101—21no—
49C2×Q8×C101C2×C412yes—
50C2×Q8×C101D412yes—
51C2×Q8×C101D412yes—
52C22×C4×C101—21no—
53C22×C4×C101—21no—
54C22×C4×C101C2311yescharacteristic
55C2×C8×C101—21no—
56C2×C8×C101—21no—
57C2×C8×C101—21no—
58C4⋊C4×C101—21no—
59C4⋊C4×C101—21no—
60C23×C101C2×D411yescharacteristic
61Q8×C101—22no—
62Q8×C101—22no—
63Q8×C101—22no—
64Q8×C101C22⋊C414yes—
65C2×C4×C101—22no—
66C2×C4×C101—22no—
67C2×C4×C101—22no—
68C2×C4×C101—41no—
69C2×C4×C101—41no—
70C2×C4×C101—41no—
71C2×C4×C101C22×C411yescharacteristic
72C2×C4×C101C2×D411yescharacteristic
73C2×C4×C101C2×D411yescharacteristic
74C2×C4×C101C2×D411yescharacteristic
75C2×C4×C101#4111yescharacteristic
76C2×C4×C101#4111yescharacteristic
77C2×Dic101—21no—
78C2×Dic101—21no—
79C2×Dic101—22no—
80C2×Dic101—21no—
81C2×Dic101—22no—
82C2×Dic101—21no—
83C2×Dic101—21no—
84C2×Dic101—41no—
85C2×Dic101—41no—
86C2×Dic101—41no—
87C2×Dic101—41no—
88C2×Dic101Q1612yescomplemented
89Dic202—22no—
90Dic202—22no—
91Dic202—41no—
92Dic202—41no—
93C8×C101—41no—
94#11717579—22no—
95C22×C101C4×D411yescharacteristic
96C22×C101#12011yescharacteristic
97C22×C101#12111yescharacteristic
98C22×C101C2×Q1611yescharacteristic
99C22×C101#13511yescharacteristic
100C22×C101#13711yescharacteristic
101C22×C101#13711yescharacteristic
102C4×C101—24no—
103C4×C101—81no—
104C4×C101C2×C22⋊C411yescharacteristic
105C4×C101C4×D411yescharacteristic
106C4×C101#12111yescharacteristic
107C4×C101#12311yescharacteristic
108Dic101—22no—
109Dic101—42no—
110Dic101—41no—
111Dic101—42no—
112C2×C101#38611yescharacteristic
113C2×C101C4×Q1611yescharacteristic
114C2×C101#44111yescharacteristic
115C2×C101#45111yescharacteristic
116C2×C101#45211yescharacteristic
117C2×C101#46211yescharacteristic
118C2×C101#46411yescharacteristic
119#1878—1011nomaximal, Sylow
120C101#187811yescharacteristic, complemented, Sylow
121C22×Q16—1011no—
122C2×C4×Q8—1011no—
123C2×#103—1011no—
124C2×#103D10111yescharacteristic
125#391—1011no—
126#340—1011no—
127#336—1011no—
128C22×Q8—1011no—
129C22×Q8C2×D10111yescharacteristic
130C2×Q16—1014no—
131C2×Q16—2022no—
132C2×Q16—2022no—
133C22×C8—1011no—
134C22×C8C2×D10111yescharacteristic
135C4×Q8—2022no—
136C4×Q8—2022no—
137C2×C4⋊C4—1011no—
138C2×C4⋊C4—2021no—
139C2×C4⋊C4C2×D10111yescharacteristic
140C2×C42—1011no—
141C2×C42—2021no—
142#103—22no—
143#103—2022no—
144#95—2021no—
145#95—2021no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—1012no—
150C2×Q8—2021no—
151C2×Q8—2021no—
152C2×Q8—2021no—
153C2×Q8—2021no—
154C2×Q8—2021no—
155C2×Q8C4×D10112yes—
156C2×Q8D40412yes—
157C2×Q8C22⋊D10112yes—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—1011no—
161C22×C4—1011no—
162C22×C4—2021no—
163C22×C4—2021no—
164C22×C4C22×D10111yescharacteristic
165Q16—2028no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—1012no—
170C2×C8—2021no—
171C2×C8—2021no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—1012no—
175C4⋊C4—2024no—
176C4⋊C4—2021no—
177C42—2024no—
178C42—2021no—
179C42—2021no—
180C23D4×D10111yescharacteristic
181Q8—22no—
182Q8—22no—
183Q8—22no—
184Q8—4041no—
185Q8—4041no—
186Q8—2022no—
187Q8—2022no—
188Q8#1310805114yes—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—1012no—
196C2×C4—4041no—
197C2×C4—4041no—
198C2×C4—4041no—
199C2×C4—4041no—
200C2×C4—2022no—
201C2×C4—2022no—
202C2×C4—2021no—
203C2×C4—2021no—
204C2×C4—2021no—
205C2×C4—2021no—
206C2×C4—2021no—
207C2×C4C2×C4×D10111yescharacteristic
208C2×C4D4×D10111yescharacteristic
209C2×C4C2×C22⋊D10111yescharacteristic
210C2×C4C2×D40411yescharacteristic
211C2×C4#1310807311yescharacteristic
212C2×C4#1310807311yescharacteristic
213C8—41no—
214C8—2022no—
215C22#1569221511yescharacteristic
216C22#1569222511yescharacteristic
217C22#1569222711yescharacteristic
218C22Q16×D10111yescharacteristic
219C22#1569227611yescharacteristic
220C22#1569228411yescharacteristic
221C22#1569228711yescharacteristic
222C4—24no—
223C4—81no—
224C4—4042no—
225C4—4042no—
226C4—4041no—
227C4—2022no—
228C4C2×#1310805111yescharacteristic
229C4#1569221511yescharacteristic
230C4#1569222711yescharacteristic
231C4#1569223911yescharacteristic
232C2#2026505411yescharacteristic
233C2#2026526911yescharacteristic
234C2#2026528111yescharacteristic
235C2#2026533411yescharacteristic
236C2#2026534211yescharacteristic
237C2#2026538511yescharacteristic
238C2#2026539711yescharacteristic
subgroupG/H
centerC23D4×D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
FittingC2×#103×C101C2
socle, prime radicalC23×C101C2×D4
cosocleC22×C4×C101C23
hypercenterC2×#103D101
hypocenterC101#1878
LCSG ▷ C2×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
11. C2×#13108064C4semidirect
88. C2×Dic101Q16semidirect
120. C101#1878semidirect

Maximal subgroups

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

subgroupindex
C22×#131080642
C2×Q8×Dic1012
C2×#156921802
#202650822
#202649392
#202649312
C2×#103×C1012
#1878101