Finite Group Databasebeta

← #27181662

Subgroups — #27181662

Order 12928.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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
1#20265808C211yesmaximal, characteristic
2#20265805C211yesmaximal, characteristic
3#20265585C212yesmaximal, complemented
4#20265584C212yesmaximal, complemented
5#20265578C212yesmaximal, complemented
6#20265574C212yesmaximal, complemented
7#20265219C211yesmaximal, characteristic
8#20265213C211yesmaximal, characteristic
9C2×#15692186C211yesmaximal, characteristic
10#20265135C211yesmaximal, characteristic
11#428×C101C211yesmaximal, characteristic
12C22×#11717579C2211yescharacteristic
13#15692255—21no—
14#15692254—21no—
15#15692254C2212yes—
16#15692254C2212yes—
17#15692247—21no—
18#15692246—21no—
19#15692246C2212yes—
20#15692246C2212yes—
21#15692234—22no—
22#15692232—22no—
23#15692214C2211yescharacteristic
24#15692214C2211yescharacteristic
25C2×C4⋊Dic101C2211yescharacteristic
26#15692189C2212yes—
27#15692188C2211yescharacteristic
28#15692188C2211yescharacteristic
29#15692187C2212yes—
30#15692186C2211yescharacteristic
31#15692186C2212yes—
32#15692186C2211yescharacteristic
33#15692186C2212yes—
34#15692165C2212yes—
35C8×Dic101C2212yes—
36C2×M16×C101C2211yescharacteristic
37#117×C101C2211yescharacteristic
38C2×C4⋊C4×C101C2211yescharacteristic
39#107×C101C2212yes—
40#106×C101C2212yes—
41C2×Dic202—22no—
42C2×Dic202—22no—
43C22×Dic101—21no—
44C2×#11717579Q814yescomplemented
45C2×#11717579C2311yescharacteristic
46C2×#11717579C2311yescharacteristic
47C4⋊Dic101—21no—
48C4⋊Dic101—22no—
49C4⋊Dic101—22no—
50C4⋊Dic101D411yescharacteristic
51C4⋊Dic101D411yescharacteristic
52C4⋊Dic101C2312yes—
53C4⋊Dic101C2311yescharacteristic
54C4⋊Dic101C2311yescharacteristic
55#13108053—22no—
56#13108053—22no—
57#13108053—22no—
58#13108053—22no—
59C22⋊Dic101—21no—
60C22⋊Dic101D412yes—
61C4×Dic101—21no—
62C4×Dic101C2312yes—
63C22×C4×C101—21no—
64C22×C4×C101C2311yescharacteristic
65M16×C101Q814yescomplemented
66C2×C8×C101C2312yes—
67C4⋊C4×C101—21no—
68C4⋊C4×C101C2311yescharacteristic
69C4⋊C4×C101C2312yes—
70C4⋊C4×C101C2311yescharacteristic
71C22⋊C4×C101—21no—
72C42×C101—21no—
73C23×C101C2×D411yescharacteristic
74C2×C4×C101—21no—
75C2×C4×C101—21no—
76C2×C4×C101—22no—
77C2×C4×C101—81no—
78C2×C4×C101C2×D411yescharacteristic
79C2×C4×C101C2×Q812yes—
80C2×C4×C101C2×Q812yes—
81C2×C4×C101C2411yescharacteristic
82C2×Dic101—21no—
83C2×Dic101—21no—
84C2×Dic101—22no—
85C2×Dic101—81no—
86C2×Dic101C2×D412yes—
87C2×Dic101C2×D412yes—
88Dic202—82no—
89Dic202—82no—
90C8×C101C2×Q814yes—
91#11717579C2×Q812yes—
92#11717579C2×Q812yes—
93C22×C101—21no—
94C22×C101—21no—
95C22×C101C4⋊Q812yes—
96C22×C101C22×D411yescharacteristic
97C4×C101—41no—
98C4×C101—42no—
99C4×C101—41no—
100C4×C101C4⋊Q812yes—
101C4×C101C22×Q811yescharacteristic
102C4×C101C22×Q811yescharacteristic
103Dic101—22no—
104Dic101—22no—
105Dic101—41no—
106Dic101—42no—
107Dic101—41no—
108C2×C101—22no—
109C2×C101C2×C4⋊Q811yescharacteristic
110C2×C101#57711yescharacteristic
111C2×C101#57711yescharacteristic
112#3163—1011nomaximal, Sylow
113C101#316311yescharacteristic, complemented, Sylow
114#544—1011no—
115#544—1011no—
116#501—1012no—
117#501—1012no—
118#499—1012no—
119#498—1012no—
120#428—1011no—
121#428D10111yescharacteristic
122#427—1011no—
123C2×#106—1011no—
124#405—1011no—
125C2×M16C2×D10111yescharacteristic
126C22×C8—1011no—
127C4⋊Q8—1012no—
128C4⋊Q8—1012no—
129C4⋊Q8—2021no—
130C4⋊Q8—2021no—
131#125—1012no—
132#125—1012no—
133#125—2021no—
134#125—2021no—
135C22⋊Q8—2022no—
136C22⋊Q8—2022no—
137#117—1011no—
138#117—1011no—
139#117C2×D10111yescharacteristic
140C2×C4⋊C4—1011no—
141C2×C4⋊C4C2×D10111yescharacteristic
142#107—1012no—
143#107—1011no—
144#107—1011no—
145#107C2×D10112yes—
146#106—1012no—
147#106—1012no—
148#106—1012no—
149#106—1011no—
150#106—1011no—
151#106C2×D10112yes—
152#97—1012no—
153C4×C8—1012no—
154C2×Q8—2022no—
155C2×Q8—2022no—
156C22×C4—21no—
157C22×C4—2021no—
158C22×C4C22×D10111yescharacteristic
159M16Dic20214yescomplemented
160C2×C8—1014no—
161C2×C8—1011no—
162C2×C8—1011no—
163C2×C8C22×D10112yes—
164C4⋊C4—21no—
165C4⋊C4—1011no—
166C4⋊C4—1011no—
167C4⋊C4—1012no—
168C4⋊C4—1011no—
169C4⋊C4—1011no—
170C4⋊C4—2022no—
171C4⋊C4—2022no—
172C4⋊C4—2022no—
173C4⋊C4—2022no—
174C4⋊C4—2022no—
175C4⋊C4—2022no—
176C4⋊C4—2021no—
177C4⋊C4C22×D10112yes—
178C4⋊C4C22×D10111yescharacteristic
179C4⋊C4C22×D10111yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—1012no—
182C22⋊C4—2021no—
183C42—21no—
184C42—1012no—
185C42—2021no—
186C23D4×D10111yescharacteristic
187Q8—8082no—
188Q8—8082no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—81no—
193C2×C4—8081no—
194C2×C4—1012no—
195C2×C4—1012no—
196C2×C4—2021no—
197C2×C4—2021no—
198C2×C4—2022no—
199C2×C4D4×D10111yescharacteristic
200C2×C4Q8×D10112yes—
201C2×C4C2×Dic20212yes—
202C2×C4C23×D10111yescharacteristic
203C8—1012no—
204C8—1012no—
205C8C2×Dic20214yes—
206C22—21no—
207C22—21no—
208C22#1569225412yes—
209C22C2×D4×D10111yescharacteristic
210C4—41no—
211C4—42no—
212C4—41no—
213C4—4042no—
214C4—4041no—
215C4—4041no—
216C4—2022no—
217C4—2022no—
218C4#1569225412yes—
219C4C2×Q8×D10111yescharacteristic
220C4C22×Dic20211yescharacteristic
221C2—22no—
222C2C2×#1569225411yescharacteristic
223C2#2026598611yescharacteristic
224C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#428×C101C2
socleC22×C101C22×D4
prime radicalC23×C101C2×D4
hypercenter#428D101
hypocenterC101#3163
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #20265585C2semidirect
4. #20265584C2semidirect
5. #20265578C2semidirect
6. #20265574C2semidirect
44. C2×#11717579Q8semidirect
65. M16×C101Q8semidirect
113. C101#3163semidirect
159. M16Dic202semidirect

Maximal subgroups

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

subgroupindex
#202658082
#202658052
#202655852
#202655842
#202655782
#202655742
#202652192
#202652132
C2×#156921862
#202651352
#428×C1012
#3163101