Finite Group Databasebeta

← #10594599

Subgroups — #10594599

Order 576.

422 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 122 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#64101C211yesmaximal, characteristic
2C3×#889C212yesmaximal, complemented
3#63919C211yesmaximal, characteristic, complemented
4C4×#627C211yesmaximal, characteristic, complemented
5#63870C211yesmaximal, characteristic, complemented
6#63868C212yesmaximal
7#63865C211yesmaximal, characteristic
8#63865C211yesmaximal, characteristic
9#63848C212yesmaximal
10#63845C212yesmaximal, complemented
11#63838C211yesmaximal, characteristic, complemented
12#5696—32nomaximal
13C2×C4×C3⋊D3C2211yescharacteristic
14C2×#627C2211yescharacteristic
15C2×#627C2211yescharacteristic
16C32×C4⋊C4C2211yescharacteristic
17#3752C2211yescharacteristic
18#3752C2211yescharacteristic
19C4⋊(C3⋊Dic3)C2211yescharacteristic
20C4×C3⋊Dic3C2211yescharacteristic
21C4×C3⋊Dic3C2211yescharacteristic
22C3×C4⋊Dic3C2212yes—
23C3×Dic3⋊C4C2212yes—
24C3×Dic3⋊C4C2212yes—
25C3×Dic3⋊C4C2212yes—
26C3×Dic3⋊C4C2212yes—
27C3×C4×Dic3C2212yes—
28#3724C2211yescharacteristic
29#3724C2212yes—
30#3724C2211yescharacteristic
31Dic3⋊Dic3C2212yes—
32Dic3⋊Dic3C2212yes—
33#3722C2212yes—
34#3722C2212yes—
35Dic32C2212yes—
36C3×#125—32no—
37#898—32no—
38#898—32no—
39#898—32no—
40#896—32no—
41#896—32no—
42#892—32no—
43#892—61no—
44#890—32no—
45#889—32no—
46#889—32no—
47#889D312yescomplemented
48#888—32no—
49#887—32no—
50#876—32no—
51C4×Dic6—32no—
52C22×C3⋊D3C2311yescharacteristic
53C2×C32×C4C2311yescharacteristic
54C2×C32×C4C2311yescharacteristic
55C2×C32×C4C2311yescharacteristic
56C2×C3⋊Dic3C2311yescharacteristic
57C2×C3⋊Dic3C2311yescharacteristic
58C2×C3⋊Dic3C2311yescharacteristic
59C4×C3⋊D3—22no—
60C2×C3×Dic3C2312yes—
61C2×C3×Dic3C2312yes—
62C2×C3×Dic3C2312yes—
63C2×C3×Dic3C2312yes—
64#627—22no—
65#627—22no—
66#541—91noSylow
67C2×C4×D3—32no—
68C2×C4×D3—61no—
69C2×Dic6—32no—
70C2×Dic6—32no—
71C3×C4⋊C4—21no—
72C3×C4⋊C4—32no—
73C3×C4⋊C4—32no—
74C3×C4⋊C4—32no—
75C3×C4⋊C4—32no—
76C3×C4⋊C4—32no—
77C3×C4⋊C4C2×D312yes—
78C3×C42—32no—
79(C2×D3)⋊C4—32no—
80(C2×D3)⋊C4—32no—
81(C2×D3)⋊C4—32no—
82(C2×D3)⋊C4—32no—
83(C2×D3)⋊C4—32no—
84(C2×D3)⋊C4—32no—
85(C2×D3)⋊C4—61no—
86(C2×D3)⋊C4—61no—
87C4⋊Dic3—32no—
88C4⋊Dic3—32no—
89C4⋊Dic3—32no—
90C4⋊Dic3—61no—
91C4⋊Dic3C2×D312yes—
92Dic3⋊C4—32no—
93Dic3⋊C4—32no—
94Dic3⋊C4—32no—
95Dic3⋊C4—32no—
96Dic3⋊C4—32no—
97Dic3⋊C4—32no—
98Dic3⋊C4C2×D312yes—
99Dic3⋊C4C2×D312yes—
100Dic3⋊C4C2×D312yes—
101Dic3⋊C4C2×D312yes—
102C4×Dic3—32no—
103C4×Dic3—32no—
104C4×Dic3—32no—
105C4×Dic3—32no—
106C4×Dic3—61no—
107C4×Dic3—61no—
108C4×Dic3C2×D312yes—
109C22×C32C2411yescharacteristic
110C2×C3⋊D3—21no—
111C2×C3⋊D3—21no—
112C2×C3⋊D3—21no—
113C32×C4—21no—
114C32×C4—21no—
115C32×C4—21no—
116C3⋊Dic3—21no—
117C3⋊Dic3—21no—
118C3⋊Dic3#4112yes—
119C3×Dic3—22no—
120C3×Dic3—22no—
121C3×Dic3—22no—
122C3×Dic3—22no—
123C4⋊Q8—91no—
124#126—92no—
125#126—92no—
126#125—32no—
127#125—91no—
128#125—92no—
129C22⋊Q8—91no—
130C22⋊Q8—91no—
131C4×Q8—91no—
132C4×Q8—91no—
133#117—91no—
134C22×D3—32no—
135C22×D3—61no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—32no—
140C2×C3×C4—32no—
141C2×C3×C4—32no—
142C2×C3×C4—32no—
143C2×C3×C4C22×D312yes—
144C2×C3×C4C22×D312yes—
145C2×C3×C4C22×D312yes—
146C2×Dic3—32no—
147C2×Dic3—32no—
148C2×Dic3—32no—
149C2×Dic3—61no—
150C2×Dic3—61no—
151C2×Dic3—61no—
152C2×Dic3C22×D312yes—
153C2×Dic3C22×D312yes—
154C2×Dic3C22×D312yes—
155C2×Dic3C22×D312yes—
156C4×D3—64no—
157C4×D3—64no—
158Dic6—64no—
159Dic6—64no—
160C2×C32C2×#4111yescharacteristic
161C2×C322−1+411yescharacteristic
162C2×C322−1+411yescharacteristic
163C3⋊D3—41no—
164C2×Q8—91no—
165C2×Q8—91no—
166C22×C4—91no—
167C4⋊C4—32no—
168C4⋊C4—32no—
169C4⋊C4—32no—
170C4⋊C4—32no—
171C4⋊C4—32no—
172C4⋊C4—91no—
173C4⋊C4—91no—
174C4⋊C4—92no—
175C4⋊C4—92no—
176C4⋊C4—92no—
177C4⋊C4—91no—
178C4⋊C4D3211yescharacteristic
179C22⋊C4—92no—
180C22⋊C4—92no—
181C22⋊C4—91no—
182C22⋊C4—91no—
183C42—32no—
184C42—92no—
185C42—91no—
186C42—91no—
187C22×C3—21no—
188C22×C3C23×D312yes—
189C2×D3—62no—
190C2×D3—62no—
191C2×D3—62no—
192C2×D3—62no—
193C2×D3—121no—
194C2×D3—121no—
195C3×C4—22no—
196C3×C4—22no—
197C3×C4—22no—
198C3×C4—22no—
199C3×C4—41no—
200C3×C4—41no—
201C3×C4—62no—
202C3×C4—62no—
203C3×C4—62no—
204C3×C4—62no—
205Dic3—22no—
206Dic3—22no—
207Dic3—22no—
208Dic3—22no—
209Dic3—34no—
210Dic3—62no—
211Dic3—62no—
212Dic3—62no—
213Dic3—121no—
214Dic3—121no—
215C32#54111yescharacteristic, complemented, Sylow
216C23—91no—
217Q8—182no—
218Q8—182no—
219C2×C4—32no—
220C2×C4—32no—
221C2×C4—32no—
222C2×C4—32no—
223C2×C4—91no—
224C2×C4—91no—
225C2×C4—91no—
226C2×C4—182no—
227C2×C4C2×D3211yescharacteristic
228C2×C4C2×D3211yescharacteristic
229C2×C4C2×D3211yescharacteristic
230C2×C3—21no—
231C2×C3—21no—
232C2×C3—21no—
233C2×C3#100712yes—
234C2×C3D3×#4112yes—
235C2×C3#101012yes—
236D3—122no—
237D3—122no—
238C22—181no—
239C22—181no—
240C22—181no—
241C22C22×D3211yescharacteristic
242C4—21no—
243C4—21no—
244C4—21no—
245C4—62no—
246C4—62no—
247C4—62no—
248C4—62no—
249C4—92no—
250C4—181no—
251C4—181no—
252C3—21no—
253C3#569612yescomplemented
254C2—361no—
255C2#6430611yescharacteristic
256C2#6430911yescharacteristic
257C2#6431611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC22×C3⋊D3C23
hypercenterC4⋊C4D32
hypocenterC32#541
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#889C2semidirect
3. #63919C2semidirect
4. C4×#627C2semidirect
5. #63870C2semidirect
10. #63845C2semidirect
11. #63838C2semidirect
47. #889D3semidirect
215. C32#541semidirect
253. C3#5696semidirect

Maximal subgroups

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

subgroupindex
#641012
C3×#8892
#639192
C4×#6272
#638702
#638682
#638652
#638482
#638452
#638382
#56963