Finite Group Databasebeta

← #27171383

Subgroups — #27171383

Order 12928.

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

normal subgroups only

This group has 298 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
1C2×#15692220C211yesmaximal, characteristic
2C2×#15692180C211yesmaximal, characteristic
3C2×#15692178C211yesmaximal, characteristic
4#20265073C211yesmaximal, characteristic
5#20264939C211yesmaximal, characteristic
6#20264931C211yesmaximal, characteristic
7C2×#107×C101C211yesmaximal, characteristic
8C22×Dic202C2211yescharacteristic
9C22×#11717579C2211yescharacteristic
10#15692220—22no—
11#15692220—22no—
12C2×C4⋊Dic101C2211yescharacteristic
13C2×#13108053—21no—
14C2×#13108053—21no—
15C2×C4×Dic101—21no—
16C2×C4×Dic101C2211yescharacteristic
17#15692180C414yescomplemented
18#15692178—22no—
19#15692162—21no—
20C22×C8×C101C2211yescharacteristic
21C2×C4⋊C4×C101C2211yescharacteristic
22C2×C4⋊C4×C101C2211yescharacteristic
23#107×C101—22no—
24C2×Dic202—21no—
25C2×Dic202—21no—
26C2×Dic202—21no—
27C2×Dic202C2×C412yes—
28C2×Dic202D412yes—
29C2×Dic202Q812yes—
30C22×Dic101—21no—
31C22×Dic101—21no—
32C22×Dic101D411yescharacteristic
33C22×Dic101D411yescharacteristic
34C2×#11717579—21no—
35C2×#11717579—21no—
36C2×#11717579—21no—
37C4⋊Dic101—21no—
38C4⋊Dic101C2×C412yes—
39#13108053—24no—
40#13108053—42no—
41C4×Dic101—24no—
42C4×Dic101—21no—
43C4×Dic101—21no—
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
55C2×C4×C101—22no—
56C2×C4×C101—41no—
57C2×C4×C101—41no—
58C2×C4×C101—41no—
59C2×C4×C101—41no—
60C2×C4×C101—41no—
61C2×C4×C101C22×C411yescharacteristic
62C2×C4×C101C2×D411yescharacteristic
63C2×C4×C101C2×D411yescharacteristic
64C2×C4×C101C2×Q811yescharacteristic
65C2×C4×C101#4111yescharacteristic
66C2×C4×C101#4111yescharacteristic
67C2×Dic101—22no—
68C2×Dic101—21no—
69C2×Dic101—22no—
70C2×Dic101—22no—
71C2×Dic101—21no—
72C2×Dic101—22no—
73C2×Dic101—21no—
74C2×Dic101—21no—
75C2×Dic101—21no—
76C2×Dic101—41no—
77C2×Dic101—41no—
78C2×Dic101Q1612yes—
79Dic202—22no—
80Dic202—22no—
81Dic202—22no—
82Dic202C4⋊C414yescomplemented
83C8×C101—22no—
84#11717579—41no—
85C22×C101C4×D411yescharacteristic
86C22×C101#12011yescharacteristic
87C22×C101C22⋊Q811yescharacteristic
88C22×C101C2×Q1611yescharacteristic
89C22×C101#13511yescharacteristic
90C22×C101AGL(1,ℤ/8)11yescharacteristic
91C22×C101#13711yescharacteristic
92C4×C101—42no—
93C4×C101—81no—
94C4×C101C2×C4⋊C411yescharacteristic
95C4×C101C4×D411yescharacteristic
96C4×C101C22⋊Q811yescharacteristic
97C4×C101#12311yescharacteristic
98Dic101—24no—
99Dic101—22no—
100Dic101—42no—
101Dic101—41no—
102C2×C101#38511yescharacteristic
103C2×C101C4×Q1611yescharacteristic
104C2×C101#44011yescharacteristic
105C2×C101#44911yescharacteristic
106C2×C101#45111yescharacteristic
107C2×C101#47711yescharacteristic
108C2×C101#47911yescharacteristic
109#1863—1011nomaximal, Sylow
110C101#186311yescharacteristic, complemented, Sylow
111C2×C4×Q8—1011no—
112C2×#107D10111yescharacteristic
113C2×#103—1011no—
114C2×#103—1011no—
115#389—1011no—
116#340—1011no—
117#336—1011no—
118C22×Q8—1011no—
119C22×C8—1011no—
120C22×C8C2×D10111yescharacteristic
121C4×Q8—2022no—
122C4×Q8—2022no—
123C2×C4⋊C4—1011no—
124C2×C4⋊C4—2021no—
125C2×C4⋊C4—2021no—
126C2×C4⋊C4C2×D10111yescharacteristic
127C2×C4⋊C4C2×D10111yescharacteristic
128C2×C42—1011no—
129C2×C42—2021no—
130#107—22no—
131#103—1014no—
132#103—2022no—
133#95—2021no—
134C2×Q8—1012no—
135C2×Q8—1012no—
136C2×Q8—1012no—
137C2×Q8—2021no—
138C2×Q8—2021no—
139C2×Q8—2021no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—1011no—
143C22×C4—1011no—
144C22×C4—2021no—
145C22×C4—2021no—
146C22×C4C22×D10111yescharacteristic
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—2021no—
150C2×C8—2021no—
151C2×C8—2021no—
152C2×C8C4×D10112yes—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—1012no—
157C4⋊C4—4042no—
158C4⋊C4—2024no—
159C4⋊C4—2021no—
160C4⋊C4C22⋊D10112yes—
161C42—2024no—
162C42—2021no—
163C42—2021no—
164C23D4×D10111yescharacteristic
165Q8—1014no—
166Q8—2022no—
167Q8—2022no—
168Q8—2022no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—1012no—
176C2×C4—4041no—
177C2×C4—4041no—
178C2×C4—2022no—
179C2×C4—2021no—
180C2×C4—2021no—
181C2×C4—2021no—
182C2×C4—2022no—
183C2×C4—2021no—
184C2×C4—2022no—
185C2×C4—2022no—
186C2×C4—2021no—
187C2×C4C2×C4×D10111yescharacteristic
188C2×C4D4×D10111yescharacteristic
189C2×C4C2×C22⋊D10111yescharacteristic
190C2×C4Q8×D10111yescharacteristic
191C2×C4#1310807411yescharacteristic
192C2×C4#1310807511yescharacteristic
193C8—22no—
194C8—4041no—
195C22#1569221711yescharacteristic
196C22#1569222311yescharacteristic
197C22#1569223311yescharacteristic
198C22Q16×D10111yescharacteristic
199C22#1569227411yescharacteristic
200C22#1569227911yescharacteristic
201C22#1569228711yescharacteristic
202C4—42no—
203C4—81no—
204C4—4042no—
205C4—4041no—
206C4—2024no—
207C4—2022no—
208C4C4⋊C4×D10111yescharacteristic
209C4C4×C22⋊D10111yescharacteristic
210C4#1569223111yescharacteristic
211C4#1569224011yescharacteristic
212C2#2026504811yescharacteristic
213C2#2026527111yescharacteristic
214C2#2026527611yescharacteristic
215C2#2026531911yescharacteristic
216C2#2026533111yescharacteristic
217C2#2026547111yescharacteristic
218C2#2026548311yescharacteristic
subgroupG/H
centerC23D4×D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
FittingC2×#107×C101C2
socle, prime radicalC23×C101C2×D4
cosocleC22×C4×C101C23
hypercenterC2×#107D101
hypocenterC101#1863
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
17. #15692180C4semidirect
82. Dic202C4⋊C4semidirect
110. C101#1863semidirect

Maximal subgroups

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

subgroupindex
C2×#156922202
C2×#156921802
C2×#156921782
#202650732
#202649392
#202649312
C2×#107×C1012
#1863101