Finite Group Databasebeta

← #69665

Subgroups — #69665

Order 384.

326 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 67 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#5805C211yesmaximal, characteristic
2C3×#390C211yesmaximal, characteristic, complemented
3C2×#833C211yesmaximal, characteristic, complemented
4#4966C211yesmaximal, characteristic, complemented
5#4964C211yesmaximal, characteristic, complemented
6C2×#805C211yesmaximal, characteristic, complemented
7#4523C211yesmaximal, characteristic, complemented
8#2037—31nomaximal, Sylow
9C22×C3×D4C2211yescharacteristic
10C2×#235C2211yescharacteristic
11C2×C3×C4⋊C4C2211yescharacteristic
12C2×C3×C22⋊C4—22no—
13C2×C3×C42C2211yescharacteristic
14#953—22no—
15#940—21no—
16#938—21no—
17#938—21no—
18#936—21no—
19#926C2211yescharacteristic
20C2×#208C2211yescharacteristic
21C2×#208C2211yescharacteristic
22#833—22no—
23#810—21no—
24#809—21no—
25#808—21no—
26#807—21no—
27#805—22no—
28#535—31no—
29#428—31no—
30C2×#104—31no—
31#417—31no—
32C2×#100—31no—
33#390D311yescharacteristic, complemented
34#339—31no—
35C24×C3D412yes—
36C2×C3×D4—21no—
37C2×C3×D4—21no—
38C2×C3×D4—21no—
39C2×C3×D4—21no—
40C2×C3×D4—21no—
41C2×C3×D4—21no—
42C22×C3×C4—21no—
43C22×C3×C4—21no—
44C22×C3×C4C2311yescharacteristic
45C2×#67—21no—
46C2×#67—22no—
47C2×#67—22no—
48#235—21no—
49#235—21no—
50#235—21no—
51#235—21no—
52C2×D12D411yescharacteristic
53C2×C4×D3—21no—
54C2×Dic6D411yescharacteristic
55C3×C4⋊C4—21no—
56C3×C4⋊C4—21no—
57C3×C22⋊C4—44no—
58C3×C42—21no—
59C3×C42—21no—
60#217—22no—
61#217—22no—
62#217—21no—
63C4⋊Dic3D411yescharacteristic
64C4⋊Dic3D411yescharacteristic
65C4×Dic3—21no—
66#208—21no—
67#208—21no—
68#208—21no—
69#208D412yescomplemented
70C2×C3⋊C8—21no—
71C2×C3⋊C8—21no—
72C2×#41—31no—
73C22×D4C2×D311yescharacteristic
74C2×M16—31no—
75C2×M16—31no—
76#127—61no—
77#124—61no—
78#121—61no—
79#121—61no—
80#120—62no—
81#117—31no—
82C2×C4⋊C4C2×D311yescharacteristic
83C2×C22⋊C4—22no—
84C2×C42C2×D311yescharacteristic
85#107—61no—
86#106—61no—
87#104—62no—
88#103—61no—
89#102—61no—
90#100—62no—
91C23×C3—22no—
92C23×C3—22no—
93C23×C3—22no—
94C23×C3—42no—
95C23×C3—42no—
96C23×C3C2×D411yescharacteristic
97C22×D3—21no—
98C3×D4—41no—
99C3×D4—41no—
100C3×D4—41no—
101C3×D4—41no—
102C2×C3×C4—22no—
103C2×C3×C4—41no—
104C2×C3×C4—41no—
105C2×C3×C4—42no—
106C2×C3×C4—41no—
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2×D411yescharacteristic
109C2×C3×C4C2×D411yescharacteristic
110C2×C3×C4#4111yescharacteristic
111C2×C3×C4#4111yescharacteristic
112C2×C3×C4#4111yescharacteristic
113#67—41no—
114#67—41no—
115#67—82no—
116#67—82no—
117C2×Dic3—21no—
118C2×Dic3—21no—
119C2×Dic3—21no—
120D12—21no—
121D12—21no—
122C4×D3—41no—
123C4×D3—41no—
124Dic6—21no—
125Dic6—21no—
126C3⋊C8—22no—
127C3⋊C8—41no—
128C24#6712yes—
129#41—61no—
130#41—61no—
131#41—61no—
132#41—61no—
133C2×Q8—31no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—31no—
141C2×D4—61no—
142C2×D4—62no—
143C2×D4—62no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—61no—
147C22×C4C22×D311yescharacteristic
148M16—32no—
149M16—61no—
150M16—61no—
151M16—61no—
152C2×C8—61no—
153C2×C8—61no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—31no—
157C4⋊C4—31no—
158C22⋊C4—44no—
159C22⋊C4—62no—
160C22⋊C4—62no—
161C22⋊C4—61no—
162C42—21no—
163C42—21no—
164C42—61no—
165C22×C3—21no—
166C22×C3—21no—
167C22×C3—44no—
168C22×C3—42no—
169C22×C3—42no—
170C22×C3—42no—
171C22×C3—82no—
172C22×C3#12011yescharacteristic
173C22×C3#12111yescharacteristic
174C22×C3#12311yescharacteristic
175C2×D3—41no—
176C2×D3—41no—
177C2×D3—41no—
178C3×C4—42no—
179C3×C4—81no—
180C3×C4#12111yescharacteristic
181C3×C4#12111yescharacteristic
182C3×C4#12311yescharacteristic
183C3×C4#12411yescharacteristic
184Dic3—41no—
185Dic3—41no—
186Dic3—41no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23—42no—
191C23—42no—
192C23—61no—
193C23C2×#6711yescharacteristic
194Q8—61no—
195Q8—61no—
196D4—41no—
197D4—41no—
198D4—41no—
199D4—41no—
200D4—61no—
201D4—61no—
202D4—121no—
203D4—121no—
204D4—242no—
205D4—242no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—61no—
212C2×C4—61no—
213C2×C4—61no—
214C2×C4—121no—
215C2×C4—121no—
216C2×C4#23511yescharacteristic
217C2×C4#23511yescharacteristic
218C2×C4D3×D411yescharacteristic
219C2×C4D3×D411yescharacteristic
220C2×C4D3×D411yescharacteristic
221C2×C4#23711yescharacteristic
222C8—62no—
223C8—121no—
224C2×C3—21no—
225C2×C3—21no—
226C2×C3—82no—
227C2×C3#39411yescharacteristic
228C2×C3#45311yescharacteristic
229C2×C3#45511yescharacteristic
230D3—81no—
231C22—21no—
232C22—21no—
233C22—44no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—82no—
238C22—121no—
239C22—121no—
240C22—121no—
241C22#92911yescharacteristic
242C22#93711yescharacteristic
243C22#93911yescharacteristic
244C4—42no—
245C4—81no—
246C4—121no—
247C4—121no—
248C4—121no—
249C4#88311yescharacteristic
250C4#88411yescharacteristic
251C4#88411yescharacteristic
252C4#88511yescharacteristic
253C3#203711yescharacteristic, complemented, Sylow
254C2—21no—
255C2—21no—
256C2—82no—
257C2—241no—
258C2#495911yescharacteristic
259C2#506111yescharacteristic
260C2#507711yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#390C2
socleC22×C3#120
prime radical#5805C2
hypercenter#390D3
hypocenterC3#2037
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C3×#390C2semidirect
3. C2×#833C2semidirect
4. #4966C2semidirect
5. #4964C2semidirect
6. C2×#805C2semidirect
7. #4523C2semidirect
33. #390D3semidirect
69. #208D4semidirect
253. C3#2037semidirect

Maximal subgroups

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

subgroupindex
#58052
C3×#3902
C2×#8332
#49662
#49642
C2×#8052
#45232
#20373