Finite Group Databasebeta

← #78092

Subgroups — #78092

Order 384.

430 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 125 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×#888C211yesmaximal, characteristic, complemented
2C2×#879C211yesmaximal, characteristic
3C3×#399C211yesmaximal, characteristic
4#4985C211yesmaximal, characteristic, complemented
5#4982C212yesmaximal, complemented
6#4979C212yesmaximal, complemented
7#4952C212yesmaximal
8#4949C211yesmaximal, characteristic
9#4947C212yesmaximal
10#4659C211yesmaximal, characteristic, complemented
11#4651C211yesmaximal, characteristic, complemented
12#2821—31nomaximal, Sylow
13C22×Dic6C2212yes—
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C4⋊C4C2212yes—
16C2×C3×C22⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4C2212yes—
18C2×#217C2211yescharacteristic
19C2×#217C2211yescharacteristic
20C2×#217C2212yes—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×C4⋊Dic3C2211yescharacteristic
23C2×Dic3⋊C4C2211yescharacteristic
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2211yescharacteristic
26C2×Dic3⋊C4C2212yes—
27C2×C4×Dic3C2211yescharacteristic
28#888—24no—
29#879—24no—
30C3×#95C2211yescharacteristic
31C3⋊#95C2212yes—
32C3⋊#95C2211yescharacteristic
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C4⋊Q8—31no—
39C2×#126—31no—
40#400—31no—
41#400—31no—
42#399—31no—
43#399D311yescharacteristic
44#398—32no—
45#397—32no—
46#395—32no—
47#393—32no—
48#383—31no—
49C24×C3C2311yescharacteristic
50C22×C3×C4C2312yes—
51C22×C3×C4C2312yes—
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C22×Dic3C2312yes—
55C22×Dic3C2311yescharacteristic
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C2×Dic6—24no—
61C2×Dic6—42no—
62C2×Dic6—42no—
63C3×C4⋊C4—22no—
64C3×C4⋊C4—42no—
65C3×C22⋊C4—22no—
66C3×C22⋊C4—42no—
67#217—22no—
68#217—22no—
69#217—42no—
70C4⋊Dic3—22no—
71C4⋊Dic3—22no—
72Dic3⋊C4—22no—
73Dic3⋊C4—22no—
74Dic3⋊C4—24no—
75Dic3⋊C4—42no—
76C4×Dic3D414yes—
77C22×Q8—32no—
78C4⋊Q8—64no—
79#126—64no—
80C2×C4⋊C4—31no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—32no—
84C2×C4⋊C4—31no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4C2×D311yescharacteristic
87C2×C4⋊C4C2×D312yes—
88C2×C22⋊C4—31no—
89C2×C22⋊C4—31no—
90C2×C22⋊C4—32no—
91C2×C22⋊C4C2×D311yescharacteristic
92C2×C22⋊C4C2×D312yes—
93C2×C42—31no—
94#95—32no—
95#95—32no—
96#95—31no—
97#95—32no—
98#95—32no—
99#95—32no—
100#95—32no—
101#95C2×D311yescharacteristic
102C23×C3—21no—
103C23×C3—22no—
104C23×C3—21no—
105C23×C3—22no—
106C23×C3—21no—
107C23×C3C2411yescharacteristic
108C2×C3×C4—22no—
109C2×C3×C4—21no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4C2×D412yes—
118C2×Dic3—22no—
119C2×Dic3—21no—
120C2×Dic3—21no—
121C2×Dic3—21no—
122C2×Dic3—21no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3—21no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—21no—
131C2×Dic3—22no—
132C2×Dic3C2×D412yes—
133C2×Dic3C2×D412yes—
134Dic6—48no—
135C24C22×D311yescharacteristic
136C2×Q8—64no—
137C2×Q8—122no—
138C2×Q8—122no—
139C22×C4—31no—
140C22×C4—31no—
141C22×C4—32no—
142C22×C4—31no—
143C22×C4—31no—
144C22×C4—32no—
145C22×C4C22×D312yes—
146C22×C4C22×D312yes—
147C22×C4C22×D311yescharacteristic
148C22×C4C22×D311yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—42no—
151C4⋊C4—62no—
152C4⋊C4—62no—
153C4⋊C4—64no—
154C4⋊C4—62no—
155C4⋊C4—62no—
156C4⋊C4—122no—
157C22⋊C4—22no—
158C22⋊C4—42no—
159C22⋊C4—62no—
160C22⋊C4—62no—
161C22⋊C4—122no—
162C42—34no—
163C22×C3—41no—
164C22×C3—41no—
165C22×C3—41no—
166C22×C3—42no—
167C22×C3—42no—
168C22×C3C22×D411yescharacteristic
169C22×C3H(5,𝔽2)12yes—
170C22×C32−1+411yescharacteristic
171C22×C32−1+412yes—
172C22×C32−1+411yescharacteristic
173C3×C4—22no—
174C3×C4—42no—
175C3×C4—41no—
176C3×C4—42no—
177Dic3—22no—
178Dic3—22no—
179Dic3—42no—
180Dic3—41no—
181Dic3—41no—
182Dic3—42no—
183C23—21no—
184C23—22no—
185C23—21no—
186C23—22no—
187C23—21no—
188C23C23×D311yescharacteristic
189Q8—128no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—32no—
200C2×C4—32no—
201C2×C4—62no—
202C2×C4—61no—
203C2×C4—61no—
204C2×C4—61no—
205C2×C4—61no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—61no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—61no—
215C2×C4D3×D412yes—
216C2×C3—81no—
217C2×C3#53511yescharacteristic
218C2×C3#53611yescharacteristic
219C2×C3#53611yescharacteristic
220C2×C3#56312yes—
221C2×C3#56312yes—
222C22—42no—
223C22—41no—
224C22—42no—
225C22—41no—
226C22—41no—
227C22C2×D3×D411yescharacteristic
228C22#100412yes—
229C22#100711yescharacteristic
230C22#101012yes—
231C22#101011yescharacteristic
232C4—22no—
233C4—42no—
234C4—42no—
235C4—41no—
236C4—62no—
237C4—62no—
238C4—122no—
239C4—121no—
240C4—122no—
241C4—121no—
242C3#282111yescharacteristic, complemented, Sylow
243C2—81no—
244C2#562811yescharacteristic
245C2#564612yes—
246C2#564811yescharacteristic
247C2#565112yes—
248C2#571511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
prime radicalC24×C3C23
hypercenter#399D3
hypocenterC3#2821
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#888C2semidirect
4. #4985C2semidirect
5. #4982C2semidirect
6. #4979C2semidirect
10. #4659C2semidirect
11. #4651C2semidirect
242. C3#2821semidirect

Maximal subgroups

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

subgroupindex
C2×#8882
C2×#8792
C3×#3992
#49852
#49822
#49792
#49522
#49492
#49472
#46592
#46512
#28213