Finite Group Databasebeta

← #69992

Subgroups — #69992

Order 384.

342 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 79 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#5791C211yesmaximal, characteristic
2#5745C211yesmaximal, characteristic, complemented
3C3×#407C211yesmaximal, characteristic, complemented
4#5207C211yesmaximal, characteristic, complemented
5#5132C211yesmaximal, characteristic, complemented
6#4559C211yesmaximal, characteristic, complemented
7#4556C211yesmaximal, characteristic, complemented
8#1891—31nomaximal, Sylow
9C23×C3×C4C2211yescharacteristic
10C2×#235C2211yescharacteristic
11C2×C3×M16C2211yescharacteristic
12C2×C3×M16C2211yescharacteristic
13#953C412yes—
14#930—21no—
15#929C412yes—
16C4×#67—22no—
17#926C2211yescharacteristic
18#924—21no—
19C2×#208C2211yescharacteristic
20C2×#208C2211yescharacteristic
21#907—22no—
22D3×M16—22no—
23#901—22no—
24C3×C22⋊C8—22no—
25#830—22no—
26#824—21no—
27#823—21no—
28#568—31no—
29#525—31no—
30#413—31no—
31#407—31no—
32#407D311yescharacteristic, complemented
33#344—31no—
34#344—31no—
35C24×C3C2×C411yescharacteristic
36C22×C3×C4—21no—
37C22×C3×C4—21no—
38C22×C3×C4—21no—
39C22×C3×C4—21no—
40C22×C3×C4—21no—
41C22×C3×C4—21no—
42C22×C3×C4C2×C411yescharacteristic
43C22×C3×C4C2311yescharacteristic
44C2×#67—22no—
45C2×#67C2×C412yes—
46#235—21no—
47#235—21no—
48#235—21no—
49#235—21no—
50C2×D12D411yescharacteristic
51C2×C4×D3D412yes—
52C2×Dic6D411yescharacteristic
53C3×M16—21no—
54C3×M16—21no—
55C3×M16—21no—
56C3×M16—21no—
57C2×C3×C8—21no—
58C2×C3×C8D412yescomplemented
59#217—22no—
60#217C2×C412yes—
61(C2×D3)⋊C4—22no—
62C4⋊Dic3—21no—
63Dic3⋊C4—22no—
64C4×Dic3—21no—
65#208—21no—
66#208—21no—
67#208—21no—
68#208—21no—
69C2×C3⋊C8—21no—
70C2×C3⋊C8D412yescomplemented
71ΓL(1,𝔽25)—24no—
72D3×C8—24no—
73C2×#41—31no—
74C23×C4C2×D311yescharacteristic
75#131—62no—
76#131—62no—
77C2×M16—31no—
78C2×M16—31no—
79C2×M16—62no—
80C2×M16C2×D311yescharacteristic
81C2×M16C2×D311yescharacteristic
82#123—32no—
83C22⋊Q8—61no—
84#121—61no—
85#120—32no—
86C4×D4—62no—
87#117—31no—
88#101—61no—
89#100—61no—
90C22⋊C8—22no—
91C22⋊C8—62no—
92C23×C3—21no—
93C23×C3—21no—
94C23×C3—21no—
95C23×C3—42no—
96C23×C3C22×C411yescharacteristic
97C22×D3C22⋊C412yes—
98C2×C3×C4—21no—
99C2×C3×C4—21no—
100C2×C3×C4—21no—
101C2×C3×C4—41no—
102C2×C3×C4—41no—
103C2×C3×C4—41no—
104C2×C3×C4—41no—
105C2×C3×C4C2×D411yescharacteristic
106C2×C3×C4C2×D411yescharacteristic
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2×D411yescharacteristic
109C2×C3×C4#4111yescharacteristic
110C2×C3×C4#4111yescharacteristic
111#67—22no—
112#67—22no—
113#67—44no—
114C2×Dic3—22no—
115C2×Dic3C22⋊C412yes—
116D12—21no—
117D12—21no—
118C4×D3—22no—
119C4×D3—22no—
120Dic6—21no—
121Dic6—21no—
122C3×C8—22no—
123C3×C8—41no—
124C3⋊C8—22no—
125C3⋊C8—41no—
126C24C4×D311yescharacteristic
127#41—61no—
128#41—61no—
129#41—61no—
130#41—61no—
131C2×Q8—31no—
132C2×D4—32no—
133C2×D4—31no—
134C2×D4—62no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—32no—
142C22×C4C4×D311yescharacteristic
143C22×C4C22×D311yescharacteristic
144M16—21no—
145M16—21no—
146M16—21no—
147M16—21no—
148M16—64no—
149M16—61no—
150M16—61no—
151M16—61no—
152M16—61no—
153C2×C8—21no—
154C2×C8—32no—
155C2×C8—64no—
156C2×C8—61no—
157C2×C8#6712yescomplemented
158C4⋊C4—62no—
159C4⋊C4—61no—
160C22⋊C4—32no—
161C22⋊C4—62no—
162C22⋊C4—62no—
163C42—61no—
164C22×C3—22no—
165C22×C3—22no—
166C22×C3—22no—
167C22×C3—41no—
168C22×C3—41no—
169C22×C3—41no—
170C22×C3—41no—
171C22×C3—41no—
172C22×C3C2×C22⋊C411yescharacteristic
173C22×C3C4×D411yescharacteristic
174C22×C3C4×D411yescharacteristic
175C2×D3—22no—
176C2×D3—42no—
177C3×C4—41no—
178C3×C4#12011yescharacteristic
179C3×C4#12111yescharacteristic
180C3×C4#12111yescharacteristic
181C3×C4#12311yescharacteristic
182Dic3—22no—
183Dic3—42no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—32no—
188C23—42no—
189C23C2×C4×D311yescharacteristic
190Q8—61no—
191Q8—61no—
192D4—62no—
193D4—62no—
194D4—61no—
195D4—61no—
196D4—124no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—32no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—62no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4#23511yescharacteristic
209C2×C4D3×D411yescharacteristic
210C2×C4D3×D411yescharacteristic
211C2×C4D3×D411yescharacteristic
212C2×C4#23711yescharacteristic
213C2×C4C2×#6711yescharacteristic
214C8—22no—
215C8—41no—
216C8—62no—
217C8—121no—
218C2×C3—21no—
219C2×C3—21no—
220C2×C3—21no—
221C2×C3—42no—
222C2×C3#38611yescharacteristic
223D3—42no—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—62no—
233C22—122no—
234C22D3×C22⋊C411yescharacteristic
235C22#67⋊C411yescharacteristic
236C22C4×#6711yescharacteristic
237C4—41no—
238C4—62no—
239C4—122no—
240C4#88311yescharacteristic
241C4#88411yescharacteristic
242C4#93711yescharacteristic
243C4#93811yescharacteristic
244C3#189111yescharacteristic, complemented, Sylow
245C2—21no—
246C2—21no—
247C2—21no—
248C2—42no—
249C2—122no—
250C2#495611yescharacteristic
subgroupG/H
centerC4#937
commutatorC23×C3C22×C4
FrattiniC22×C4C22×D3
FittingC3×#407C2
socleC2×C3#386
prime radical#5791C2
cosocleC22×C3×C4C23
hypercenter#407D3
hypocenterC3#1891
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #5745C2semidirect
3. C3×#407C2semidirect
4. #5207C2semidirect
5. #5132C2semidirect
6. #4559C2semidirect
7. #4556C2semidirect
32. #407D3semidirect
58. C2×C3×C8D4semidirect
70. C2×C3⋊C8D4semidirect
157. C2×C8#67semidirect
244. C3#1891semidirect

Maximal subgroups

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

subgroupindex
#57912
#57452
C3×#4072
#52072
#51322
#45592
#45562
#18913