Finite Group Databasebeta

← #79601

Subgroups — #79601

Order 384.

386 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 115 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#5562C212yesmaximal, complemented
2C3×C4×D8C211yesmaximal, characteristic, complemented
3#5157C211yesmaximal, characteristic, complemented
4#5151C211yesmaximal, characteristic, complemented
5#5014C212yesmaximal, complemented
6#4883C211yesmaximal, characteristic, complemented
7#4878C211yesmaximal, characteristic, complemented
8#4782C212yesmaximal, complemented
9#4764C212yesmaximal, complemented
10C4×D24C211yesmaximal, characteristic, complemented
11#4689C211yesmaximal, characteristic, complemented
12#3176—31nomaximal, Sylow
13C2×C3×D8C2211yescharacteristic
14C3×C4×D4C2212yes—
15#940C2212yes—
16#938C2212yes—
17C2×#213C2212yes—
18#930—22no—
19#929—22no—
20C2×D24C2211yescharacteristic
21C2×ΓL(1,𝔽25)C2211yescharacteristic
22C2×D3×C8C2211yescharacteristic
23#895C2212yes—
24#890C2212yes—
25#884—22no—
26#883—22no—
27C4×D12C2212yes—
28#872C2211yescharacteristic
29C3×#107C2211yescharacteristic
30C3×#102C2212yes—
31C3×C4×C8C2211yescharacteristic
32#832C2212yes—
33#821C2212yes—
34#818C2211yescharacteristic
35#815C2211yescharacteristic
36C8×Dic3C2211yescharacteristic
37#809C2212yes—
38#808C2212yes—
39#803C2211yescharacteristic
40#540—32no—
41#499—31no—
42#493—31no—
43#490—32no—
44#469—32no—
45#466—31no—
46#466—31no—
47#442—32no—
48C4×D8—31no—
49C4×D8D311yescharacteristic, complemented
50#405—31no—
51C2×C3×D4C2312yes—
52C22×C3×C4—22no—
53C2×#67—22no—
54C2×#67—22no—
55C2×#67—22no—
56C2×D12—22no—
57C2×D12C2312yes—
58C2×C4×D3—22no—
59C2×C4×D3C2311yescharacteristic
60C3×D8—22no—
61C2×C3×C8C2311yescharacteristic
62C2×C3×C8C2311yescharacteristic
63C3×C4⋊C4C2312yes—
64C3×C22⋊C4—22no—
65C3×C42C2311yescharacteristic
66#217—22no—
67#213—24no—
68(C2×D3)⋊C4—22no—
69(C2×D3)⋊C4—22no—
70(C2×D3)⋊C4D411yescharacteristic
71(C2×D3)⋊C4D411yescharacteristic
72C4⋊Dic3—22no—
73C4⋊Dic3C2312yes—
74Dic3⋊C4—22no—
75Dic3⋊C4D411yescharacteristic
76Dic3⋊C4D411yescharacteristic
77C4×Dic3C2311yescharacteristic
78C2×C3⋊C8C2311yescharacteristic
79C2×C3⋊C8C2311yescharacteristic
80D24—22no—
81ΓL(1,𝔽25)—22no—
82D3×C8—22no—
83C2×D8—32no—
84C2×D8—31no—
85C2×D8C2×D311yescharacteristic
86C2×M16—31no—
87C22×C8—31no—
88#127—32no—
89#125—32no—
90#123—62no—
91#123—62no—
92#121—32no—
93#121—32no—
94#121—62no—
95#121—62no—
96C4×D4—32no—
97C4×D4C2×D312yes—
98#117—31no—
99#107—31no—
100#107C2×D311yescharacteristic
101#106—32no—
102#102—32no—
103#102—32no—
104#102—32no—
105#102C2×D312yes—
106#97—31no—
107#97—31no—
108C4×C8—31no—
109C4×C8C2×D311yescharacteristic
110C23×C3—22no—
111C22×D3—22no—
112C22×D3C2×D411yescharacteristic
113C3×D4—22no—
114C3×D4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—82no—
117C2×C3×C4C2×D411yescharacteristic
118C2×C3×C4C2×D411yescharacteristic
119C2×C3×C4C2411yescharacteristic
120#67—82no—
121#67—82no—
122#67—82no—
123C2×Dic3—22no—
124C2×Dic3C2×D411yescharacteristic
125C2×Dic3C2×D411yescharacteristic
126C2×Dic3C2×D411yescharacteristic
127D12—22no—
128D12—22no—
129D12—82no—
130C4×D3—21no—
131C4×D3—21no—
132C4×D3—82no—
133C3×C8—21no—
134C3×C8#4112yes—
135C3⋊C8—21no—
136C3⋊C8#4112yes—
137C2×D4—32no—
138C2×D4—62no—
139C2×D4—62no—
140C2×D4—62no—
141C2×D4—62no—
142C2×D4C22×D312yes—
143C22×C4—22no—
144C22×C4—31no—
145C22×C4—62no—
146D8—22no—
147D8—64no—
148D8—62no—
149M16—62no—
150C2×C8—31no—
151C2×C8—31no—
152C2×C8—62no—
153C2×C8C22×D311yescharacteristic
154C2×C8C22×D311yescharacteristic
155C4⋊C4—31no—
156C4⋊C4—31no—
157C4⋊C4—32no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4C22×D312yes—
161C22⋊C4—22no—
162C22⋊C4—31no—
163C22⋊C4—31no—
164C22⋊C4—62no—
165C22⋊C4—62no—
166C22⋊C4—62no—
167C42—31no—
168C42C22×D311yescharacteristic
169C22×C3—42no—
170C22×C3—42no—
171C22×C3—42no—
172C22×C3C22×D411yescharacteristic
173C2×D3—21no—
174C2×D3—21no—
175C2×D3—21no—
176C2×D3—42no—
177C2×D3—42no—
178C2×D3—42no—
179C3×C4—21no—
180C3×C4—21no—
181C3×C4—42no—
182C3×C4C2×#4111yescharacteristic
183C3×C4C2×#4111yescharacteristic
184Dic3—21no—
185Dic3—21no—
186Dic3—21no—
187Dic3—42no—
188C23—22no—
189C23—31no—
190C23—62no—
191D4—22no—
192D4—22no—
193D4—62no—
194D4—62no—
195D4—242no—
196D4—242no—
197D4—242no—
198D4—242no—
199C2×C4—22no—
200C2×C4—31no—
201C2×C4—31no—
202C2×C4—31no—
203C2×C4—61no—
204C2×C4—61no—
205C2×C4—62no—
206C2×C4—82no—
207C2×C4—242no—
208C2×C4D3×D411yescharacteristic
209C2×C4D3×D411yescharacteristic
210C2×C4C23×D311yescharacteristic
211C8—21no—
212C8—32no—
213C8—61no—
214C8#23512yes—
215C2×C3—82no—
216C2×C3#53211yescharacteristic
217C2×C3#57611yescharacteristic
218C2×C3#57611yescharacteristic
219D3—41no—
220D3—82no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22—61no—
225C22—61no—
226C22—61no—
227C22—122no—
228C22—122no—
229C22—122no—
230C22C2×D3×D411yescharacteristic
231C4—21no—
232C4—21no—
233C4—42no—
234C4—61no—
235C4—61no—
236C4—61no—
237C4—122no—
238C4C2×#23511yescharacteristic
239C4D3×#4111yescharacteristic
240C3#317611yescharacteristic, complemented, Sylow
241C2—82no—
242C2—121no—
243C2—242no—
244C2#554811yescharacteristic
245C2#575711yescharacteristic
246C2#576911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×D8C2
socleC22×C3C22×D4
hypercenterC4×D8D3
hypocenterC3#3176
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #5562C2semidirect
2. C3×C4×D8C2semidirect
3. #5157C2semidirect
4. #5151C2semidirect
5. #5014C2semidirect
6. #4883C2semidirect
7. #4878C2semidirect
8. #4782C2semidirect
9. #4764C2semidirect
10. C4×D24C2semidirect
11. #4689C2semidirect
49. C4×D8D3semidirect
240. C3#3176semidirect

Maximal subgroups

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

subgroupindex
#55622
C3×C4×D82
#51572
#51512
#50142
#48832
#48782
#47822
#47642
C4×D242
#46892
#31763