Finite Group Databasebeta

← #70417

Subgroups — #70417

Order 384.

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

normal subgroups only

This group has 332 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#5832C211yesmaximal, characteristic
2#5745C211yesmaximal, characteristic, complemented
3C3×#421C211yesmaximal, characteristic, complemented
4#5236C211yesmaximal, characteristic, complemented
5#5006C211yesmaximal, characteristic, complemented
6#4555C211yesmaximal, characteristic, complemented
7#4532C211yesmaximal, characteristic, complemented
8#1884—31nomaximal, Sylow
9C2×C3×#41C2211yescharacteristic
10C2×#235C2211yescharacteristic
11C2×C3×M16C2211yescharacteristic
12C3×#117C2211yescharacteristic
13#947C411yescharacteristic, complemented
14#944C411yescharacteristic, complemented
15#940C411yescharacteristic, complemented
16#939—22no—
17#936C411yescharacteristic, complemented
18C4×#67—22no—
19#926C2211yescharacteristic
20C2×C4×Dic3C2211yescharacteristic
21C2×#208C2211yescharacteristic
22#907—22no—
23D3×M16—22no—
24#901—22no—
25C3×#104—22no—
26#837—22no—
27#806—22no—
28#568—31no—
29#532—31no—
30#421—31no—
31#421D311yescharacteristic, complemented
32#410—31no—
33#339—31no—
34#339—31no—
35C3×#41—21no—
36C3×#41—21no—
37C3×#41—21no—
38C3×#41—21no—
39C2×C3×Q8C2×C411yescharacteristic
40C2×C3×D4—21no—
41C2×C3×D4C2×C411yescharacteristic
42C22×C3×C4—21no—
43C22×C3×C4C2311yescharacteristic
44C2×#67—22no—
45C2×#67D412yes—
46C22×Dic3—21no—
47#235—21no—
48#235—21no—
49#235—21no—
50#235—21no—
51C2×D12C2×C411yescharacteristic
52C2×C4×D3D412yes—
53C2×Dic6C2×C411yescharacteristic
54C3×M16—21no—
55C3×M16D412yescomplemented
56C2×C3×C8—21no—
57C3×C4⋊C4—21no—
58C3×C22⋊C4—21no—
59C3×C42—21no—
60#217—22no—
61#217—21no—
62(C2×D3)⋊C4—22no—
63C4⋊Dic3—21no—
64Dic3⋊C4—22no—
65C4×Dic3—21no—
66C4×Dic3—21no—
67C4×Dic3C2×C411yescharacteristic
68C4×Dic3C2×C411yescharacteristic
69#208—21no—
70#208D412yescomplemented
71C2×C3⋊C8—21no—
72ΓL(1,𝔽25)—24no—
73D3×C8—24no—
74C2×#41—31no—
75C2×#41C2×D311yescharacteristic
76#131—62no—
77#131—62no—
78C2×M16—31no—
79C2×M16—62no—
80C2×M16C2×D311yescharacteristic
81C4⋊Q8—31no—
82#127—31no—
83#124—31no—
84#124—31no—
85#121—62no—
86C4×D4—62no—
87#117—31no—
88#117C2×D311yescharacteristic
89C2×C42—31no—
90#104—22no—
91#104—62no—
92#99—62no—
93C23×C3—21no—
94C23×C3C2×D411yescharacteristic
95C22×D3C22⋊C412yes—
96C3×Q8—21no—
97C3×Q8—21no—
98C3×D4—21no—
99C3×D4—21no—
100C3×D4—41no—
101C3×D4—41no—
102C2×C3×C4—21no—
103C2×C3×C4—41no—
104C2×C3×C4—41no—
105C2×C3×C4—41no—
106C2×C3×C4C22×C411yescharacteristic
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2×D411yescharacteristic
109C2×C3×C4C2×D411yescharacteristic
110C2×C3×C4#4111yescharacteristic
111C2×C3×C4#4111yescharacteristic
112#67—22no—
113#67—22no—
114#67—44no—
115C2×Dic3—21no—
116C2×Dic3—21no—
117C2×Dic3—42no—
118C2×Dic3—41no—
119C2×Dic3C22⋊C412yes—
120D12—21no—
121D12—21no—
122C4×D3—22no—
123C4×D3—22no—
124Dic6—21no—
125Dic6—21no—
126C3×C8—22no—
127C3⋊C8—22no—
128#41—21no—
129#41—21no—
130#41—21no—
131#41—21no—
132#41—61no—
133#41—61no—
134#41—61no—
135#41—61no—
136C2×Q8—31no—
137C2×Q8C4×D311yescharacteristic
138C2×D4—21no—
139C2×D4—32no—
140C2×D4—31no—
141C2×D4—62no—
142C2×D4C4×D311yescharacteristic
143C22×C4—21no—
144C22×C4—32no—
145C22×C4—61no—
146C22×C4C22×D311yescharacteristic
147M16—21no—
148M16—32no—
149M16—64no—
150M16—61no—
151M16#6712yescomplemented
152C2×C8—21no—
153C2×C8—64no—
154C2×C8—61no—
155C4⋊C4—21no—
156C4⋊C4—62no—
157C4⋊C4—61no—
158C22⋊C4—21no—
159C22⋊C4—62no—
160C22⋊C4—62no—
161C22⋊C4—61no—
162C42—21no—
163C42—31no—
164C42—31no—
165C42—61no—
166C42—61no—
167C22×C3—41no—
168C22×C3—42no—
169C22×C3—41no—
170C22×C3C2×C22⋊C411yescharacteristic
171C22×C3#12111yescharacteristic
172C22×C3#12111yescharacteristic
173C2×D3—22no—
174C2×D3—42no—
175C3×C4—41no—
176C3×C4—81no—
177C3×C4C4×D411yescharacteristic
178C3×C4C4×D411yescharacteristic
179C3×C4#12011yescharacteristic
180C3×C4#12311yescharacteristic
181Dic3—22no—
182Dic3—22no—
183Dic3—41no—
184Dic3—81no—
185C23—21no—
186C23—32no—
187C23D3×D411yescharacteristic
188Q8—21no—
189Q8—21no—
190Q8—61no—
191Q8—61no—
192D4—21no—
193D4—21no—
194D4—41no—
195D4—41no—
196D4—62no—
197D4—62no—
198D4—61no—
199D4—61no—
200D4—124no—
201C2×C4—21no—
202C2×C4—32no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—61no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—61no—
210C2×C4—122no—
211C2×C4—121no—
212C2×C4C2×C4×D311yescharacteristic
213C2×C4#23511yescharacteristic
214C2×C4D3×D411yescharacteristic
215C2×C4D3×D411yescharacteristic
216C2×C4#23711yescharacteristic
217C2×C4C2×#6711yescharacteristic
218C8—22no—
219C8—62no—
220C2×C3—21no—
221C2×C3—21no—
222C2×C3—21no—
223C2×C3—81no—
224C2×C3#38611yescharacteristic
225D3—42no—
226C22—41no—
227C22—42no—
228C22—41no—
229C22—62no—
230C22—122no—
231C22D3×C22⋊C411yescharacteristic
232C22#88411yescharacteristic
233C22#93811yescharacteristic
234C4—41no—
235C4—62no—
236C4—62no—
237C4—81no—
238C4—121no—
239C4—241no—
240C4#67⋊C411yescharacteristic
241C4#88311yescharacteristic
242C4C4×#6711yescharacteristic
243C4#93711yescharacteristic
244C3#188411yescharacteristic, complemented, Sylow
245C2—21no—
246C2—21no—
247C2—21no—
248C2—81no—
249C2—122no—
250C2#495611yescharacteristic
subgroupG/H
centerC4#937
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#421C2
socleC2×C3#386
prime radical#5832C2
cosocleC22×C3×C4C23
hypercenter#421D3
hypocenterC3#1884
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #5745C2semidirect
3. C3×#421C2semidirect
4. #5236C2semidirect
5. #5006C2semidirect
6. #4555C2semidirect
7. #4532C2semidirect
13. #947C4semidirect
14. #944C4semidirect
15. #940C4semidirect
17. #936C4semidirect
31. #421D3semidirect
55. C3×M16D4semidirect
70. #208D4semidirect
151. M16#67semidirect
244. C3#1884semidirect

Maximal subgroups

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

subgroupindex
#58322
#57452
C3×#4212
#52362
#50062
#45552
#45322
#18843