Finite Group Databasebeta

← #27142772

Subgroups — #27142772

Order 12416.

362 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 75 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×#15690841C211yesmaximal, characteristic
2#20261309C211yesmaximal, characteristic
3C2×#15690738C211yesmaximal, characteristic
4#20260744C211yesmaximal, characteristic, complemented
5C2×#15690729C211yesmaximal, characteristic, complemented
6#20260495C211yesmaximal, characteristic, complemented
7C2×C4⋊Q8×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic
9#15690841—22no—
10#15690841—22no—
11C2×#13095501—21no—
12C2×#13095498—21no—
13C2×#13095495C2211yescharacteristic
14C2×#13095495C2211yescharacteristic
15#15690808—21no—
16#15690799—21no—
17#15690793—22no—
18#15690786—22no—
19#15690769C2211yescharacteristic
20#15690738—22no—
21#15690734—22no—
22#15690729—22no—
23C22×Q8×C97C2211yescharacteristic
24C22×Q8×C97C2211yescharacteristic
25C4⋊Q8×C97—22no—
26C4⋊Q8×C97—22no—
27C2×C4⋊C4×C97—22no—
28C2×C42×C97C2211yescharacteristic
29#13095511—21no—
30#13095511—21no—
31#13095511—21no—
32#13095511—21no—
33C2×Dic194D411yescharacteristic
34C2×D388D411yescharacteristic
35C2×C22⋊D97—21no—
36C2×C4×D97—21no—
37#13095501—24no—
38#13095498—24no—
39#13095495—21no—
40#13095495—21no—
41#13095495—21no—
42#13095495D412yes—
43C2×#11715895—21no—
44C2×#11715895—21no—
45C4⋊Dic97D412yes—
46#13095490—24no—
47C22⋊Dic97—21no—
48#13095488—24no—
49C4×Dic97—21no—
50C2×Q8×C97—21no—
51C2×Q8×C97—21no—
52C2×Q8×C97—21no—
53C2×Q8×C97—21no—
54C2×Q8×C97—21no—
55C2×Q8×C97—21no—
56C2×Q8×C97—21no—
57C2×Q8×C97—21no—
58C2×Q8×C97—21no—
59C2×Q8×C97—21no—
60C2×Q8×C97—21no—
61C2×Q8×C97D412yes—
62C22×C4×C97—21no—
63C22×C4×C97—21no—
64C22×C4×C97D412yes—
65C22×C4×C97C2311yescharacteristic
66C4⋊C4×C97—28no—
67C42×C97—21no—
68C42×C97D412yes—
69C23×C97C2×D411yescharacteristic
70C22×D97—21no—
71Q8×C97—22no—
72Q8×C97—22no—
73Q8×C97—41no—
74Q8×C97—41no—
75Q8×C97—41no—
76Q8×C97—41no—
77Q8×C97—41no—
78Q8×C97—41no—
79C2×C4×C97—24no—
80C2×C4×C97—22no—
81C2×C4×C97—22no—
82C2×C4×C97—22no—
83C2×C4×C97—22no—
84C2×C4×C97—22no—
85C2×C4×C97—41no—
86C2×C4×C97—41no—
87C2×C4×C97C2×D411yescharacteristic
88C2×C4×C97C2×D411yescharacteristic
89C2×C4×C97C2×D411yescharacteristic
90C2×C4×C97C2×D411yescharacteristic
91C2×C4×C97C2×D411yescharacteristic
92C2×C4×C97#4111yescharacteristic
93C22⋊D97—41no—
94C22⋊D97—41no—
95C2×Dic97—22no—
96C2×Dic97—21no—
97D388—21no—
98D388—21no—
99C4×D97—41no—
100C4×D97—41no—
101Dic194—21no—
102Dic194—21no—
103#11715895—22no—
104#11715895—41no—
105C22×C97—22no—
106C22×C97#12011yescharacteristic
107C22×C97#12011yescharacteristic
108C22×C97#12111yescharacteristic
109C2×D97—41no—
110C2×D97—42no—
111C4×C97—24no—
112C4×C97—42no—
113C4×C97—42no—
114C4×C97#12011yescharacteristic
115C4×C97#12111yescharacteristic
116C4×C97#12111yescharacteristic
117C4×C97#12711yescharacteristic
118Dic97—41no—
119Dic97—42no—
120C2×C97—21no—
121C2×C97—21no—
122C2×C97#39211yescharacteristic
123C2×C97#45612yes—
124D97—81no—
125#2005—971nomaximal, Sylow
126C97#200511yescharacteristic, complemented, Sylow
127C2×#137—971no—
128#536—971no—
129C2×C4⋊Q8D9711yescharacteristic, complemented
130C2×#104—971no—
131#419—971no—
132C2×#101—971no—
133#339—971no—
134C2×#41—971no—
135C22×Q8C2×D9711yescharacteristic
136C22×Q8C2×D9711yescharacteristic
137#137—1942no—
138#137—1942no—
139C2×Q16—1941no—
140C2×SD16—1941no—
141C2×M16—971no—
142C2×M16—971no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—1941no—
146#124—1941no—
147#123—1942no—
148C22⋊Q8—1942no—
149#117—971no—
150C2×C4⋊C4—22no—
151C2×C42C2×D9711yescharacteristic
152#104—1942no—
153#103—1942no—
154#101—1942no—
155#41—1941no—
156#41—1941no—
157#41—1941no—
158#41—1941no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—971no—
171C2×Q8C22⋊D9712yes—
172C2×D4—971no—
173C2×D4—1941no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—1941no—
177C22×C4C22⋊D9712yes—
178C22×C4C22×D9711yescharacteristic
179Q16—1944no—
180SD16—1944no—
181M16—972no—
182M16—1941no—
183M16—1941no—
184M16—1941no—
185C2×C8—1941no—
186C2×C8—1941no—
187C4⋊C4—28no—
188C4⋊C4—972no—
189C4⋊C4—1944no—
190C22⋊C4—1944no—
191C22⋊C4—1941no—
192C42—21no—
193C42—1941no—
194C42C22⋊D9712yes—
195C23—1941no—
196C23C2×C22⋊D9711yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201Q8—41no—
202Q8—41no—
203Q8—41no—
204Q8—41no—
205Q8—1941no—
206Q8—1941no—
207D4—3881no—
208D4—3881no—
209D4—1941no—
210D4—1941no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—3881no—
220C2×C4—3881no—
221C2×C4—1942no—
222C2×C4—1941no—
223C2×C4D4×D9711yescharacteristic
224C2×C4D4×D9711yescharacteristic
225C2×C4D4×D9711yescharacteristic
226C2×C4C2×C22⋊D9711yescharacteristic
227C2×C4C2×C22⋊D9711yescharacteristic
228C2×C4#1309551011yescharacteristic
229C8—3881no—
230C8—1942no—
231C22—22no—
232C22—3881no—
233C22—3882no—
234C22#1569077811yescharacteristic
235C22#1569077911yescharacteristic
236C22#1569078211yescharacteristic
237C4—24no—
238C4—42no—
239C4—42no—
240C4—3881no—
241C4—3882no—
242C4#1569077811yescharacteristic
243C4#1569078111yescharacteristic
244C4#1569078211yescharacteristic
245C4#1569080611yescharacteristic
246C2—21no—
247C2—21no—
248C2—7761no—
249C2#2026064311yescharacteristic
250C2#2026091712yes—
subgroupG/H
centerC22#15690778
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
FittingC2×C4⋊Q8×C97C2
socleC22×C97#120
prime radicalC2×#13095511C22
hypercenterC2×C4⋊Q8D97
hypocenterC97#2005
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. #20260744C2semidirect
5. C2×#15690729C2semidirect
6. #20260495C2semidirect
7. C2×C4⋊Q8×C97C2semidirect
126. C97#2005semidirect
129. C2×C4⋊Q8D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908412
#202613092
C2×#156907382
#202607442
C2×#156907292
#202604952
C2×C4⋊Q8×C972
#200597