Finite Group Databasebeta

← #10595209

Subgroups — #10595209

Order 576.

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

normal subgroups only

This group has 483 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#64147C211yesmaximal, characteristic
2C3×#935C212yesmaximal, complemented
3#63975C212yesmaximal, complemented
4#63964C212yesmaximal, complemented
5#63956C212yesmaximal, complemented
6#63952C212yesmaximal, complemented
7#63883C211yesmaximal, characteristic, complemented
8#63865C211yesmaximal, characteristic, complemented
9#63845C212yesmaximal, complemented
10#5660—32nomaximal
11C2×C32×D4C2211yescharacteristic
12C2×#638C2212yes—
13C2×C4×C3⋊D3C2211yescharacteristic
14C22×C3×Dic3C2212yes—
15C2×#627C2211yescharacteristic
16C2×#624C2211yescharacteristic
17C2×#624C2211yescharacteristic
18#3757C2212yes—
19C4⋊(C3⋊Dic3)C2211yescharacteristic
20C3×#217C2212yes—
21C3×#217C2212yes—
22C3×#217C2212yes—
23C3×Dic3⋊C4C2212yes—
24C3×Dic3⋊C4C2212yes—
25#3724C2212yes—
26#3724C2211yescharacteristic
27Dic3⋊Dic3C2212yes—
28Dic3⋊Dic3C2212yes—
29Dic3⋊Dic3C2212yes—
30#3722C2212yes—
31Dic32C2212yes—
32C3×#123—32no—
33#938—32no—
34#938—61no—
35#935D312yescomplemented
36C4×#67—32no—
37#924—32no—
38#896—32no—
39D3×C4⋊C4—32no—
40#890—32no—
41#889—32no—
42#883—32no—
43#883—32no—
44#67⋊C4—32no—
45#879—32no—
46#879—32no—
47#878—32no—
48C23×C32C2312yes—
49C22×C3⋊D3C2311yescharacteristic
50C32×D4—41no—
51C2×C32×C4C2311yescharacteristic
52#638—24no—
53C2×C3⋊Dic3C2312yes—
54C2×C3⋊Dic3C2311yescharacteristic
55C4×C3⋊D3—41no—
56C2×C3×Dic3—42no—
57C2×C3×Dic3C2312yes—
58C2×C3×Dic3C2312yes—
59C2×C3×Dic3C2312yes—
60C2×C3×Dic3C2312yes—
61#627—41no—
62#624—22no—
63#624—41no—
64#539—91noSylow
65C2×C3×D4—21no—
66C2×C3×D4C2×D312yes—
67C22×C3×C4—32no—
68C2×#67—32no—
69C2×#67—32no—
70C2×#67—62no—
71C22×Dic3C2×D312yes—
72C2×C4×D3—32no—
73C2×C4×D3—32no—
74C2×C4×D3—32no—
75C2×C4×D3—61no—
76C2×Dic6—32no—
77C3×C4⋊C4—32no—
78C3×C4⋊C4—32no—
79C3×C22⋊C4—32no—
80C3×C22⋊C4—32no—
81C3×C22⋊C4—32no—
82#217—32no—
83#217—32no—
84#217—62no—
85#217C2×D312yes—
86#217C2×D312yes—
87#217C2×D312yes—
88(C2×D3)⋊C4—32no—
89(C2×D3)⋊C4—32no—
90C4⋊Dic3—32no—
91C4⋊Dic3—32no—
92C4⋊Dic3—32no—
93C4⋊Dic3—32no—
94C4⋊Dic3—61no—
95Dic3⋊C4—32no—
96Dic3⋊C4—32no—
97Dic3⋊C4—32no—
98Dic3⋊C4—32no—
99Dic3⋊C4—32no—
100Dic3⋊C4—32no—
101Dic3⋊C4C2×D312yes—
102Dic3⋊C4C2×D312yes—
103C4×Dic3—32no—
104C4×Dic3—32no—
105C22×C32—22no—
106C22×C32—22no—
107C22×C32—22no—
108C22×C32C2411yescharacteristic
109C2×C3⋊D3—21no—
110C2×C3⋊D3—21no—
111C2×C3⋊D3#4112yes—
112C32×C4—21no—
113C3⋊Dic3—22no—
114C3⋊Dic3—21no—
115C3×Dic3—22no—
116C3×Dic3—22no—
117C3×Dic3—22no—
118C3×Dic3—22no—
119#126—92no—
120#125—92no—
121#123—32no—
122#123—92no—
123C22⋊Q8—92no—
124C22⋊Q8—91no—
125#121—91no—
126C4×D4—92no—
127C2×C4⋊C4—91no—
128C23×C3—22no—
129C23×C3C22×D312yes—
130C23×C3C22×D312yes—
131C22×D3—32no—
132C22×D3—61no—
133C3×D4—42no—
134C3×D4—42no—
135C2×C3×C4—21no—
136C2×C3×C4—32no—
137C2×C3×C4—32no—
138C2×C3×C4—32no—
139C2×C3×C4—32no—
140C2×C3×C4—122no—
141C2×C3×C4C22×D312yes—
142#67—64no—
143#67—64no—
144#67—124no—
145C2×Dic3—32no—
146C2×Dic3—32no—
147C2×Dic3—32no—
148C2×Dic3—42no—
149C2×Dic3—62no—
150C2×Dic3—61no—
151C2×Dic3C22×D312yes—
152C2×Dic3C22×D312yes—
153C2×Dic3C22×D312yes—
154C2×Dic3C22×D312yes—
155C4×D3—64no—
156C4×D3—122no—
157C4×D3—122no—
158C4×D3—122no—
159Dic6—122no—
160C2×C32—42no—
161C2×C32C2×#4111yescharacteristic
162C2×C32H(5,𝔽2)11yescharacteristic
163C2×C322−1+411yescharacteristic
164C3⋊D3—22no—
165C2×Q8—91no—
166C2×D4—92no—
167C2×D4D3211yescharacteristic
168C22×C4—32no—
169C22×C4—91no—
170C22×C4—91no—
171C22×C4—91no—
172C4⋊C4—32no—
173C4⋊C4—32no—
174C4⋊C4—92no—
175C4⋊C4—92no—
176C4⋊C4—92no—
177C4⋊C4—91no—
178C4⋊C4—92no—
179C4⋊C4—91no—
180C22⋊C4—32no—
181C22⋊C4—32no—
182C22⋊C4—32no—
183C22⋊C4—92no—
184C22⋊C4—92no—
185C42—92no—
186C22×C3—22no—
187C22×C3—22no—
188C22×C3—22no—
189C22×C3—22no—
190C22×C3—22no—
191C22×C3—22no—
192C22×C3—21no—
193C22×C3—42no—
194C22×C3—42no—
195C22×C3—42no—
196C22×C3C23×D312yes—
197C2×D3—34no—
198C2×D3—62no—
199C2×D3—62no—
200C2×D3—62no—
201C2×D3—121no—
202C2×D3—121no—
203C3×C4—22no—
204C3×C4—22no—
205C3×C4—62no—
206C3×C4—62no—
207C3×C4—62no—
208C3×C4—62no—
209Dic3—22no—
210Dic3—22no—
211Dic3—22no—
212Dic3—22no—
213Dic3—62no—
214Dic3—62no—
215Dic3—62no—
216Dic3—121no—
217Dic3—122no—
218C32#53911yescharacteristic, complemented, Sylow
219C23—91no—
220C23C2×D3212yes—
221Q8—361no—
222D4—41no—
223D4—184no—
224C2×C4—32no—
225C2×C4—32no—
226C2×C4—32no—
227C2×C4—32no—
228C2×C4—92no—
229C2×C4—91no—
230C2×C4—122no—
231C2×C4—182no—
232C2×C4—361no—
233C2×C4—361no—
234C2×C4C2×D3211yescharacteristic
235C2×C3—21no—
236C2×C3—21no—
237C2×C3—21no—
238C2×C3—42no—
239C2×C3—42no—
240C2×C3—82no—
241C2×C3#100412yes—
242C2×C3D3×#4112yes—
243C2×C3#101012yes—
244D3—64no—
245D3—122no—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—92no—
250C22—181no—
251C22—181no—
252C22C22×D3211yescharacteristic
253C4—21no—
254C4—62no—
255C4—62no—
256C4—62no—
257C4—62no—
258C4—182no—
259C4—181no—
260C3—21no—
261C3#566012yescomplemented
262C2—42no—
263C2—182no—
264C2#6430911yescharacteristic
265C2#6431211yescharacteristic
266C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
prime radical#64147C2
hypercenterC2×D4D32
hypocenterC32#539
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C3×#935C2semidirect
3. #63975C2semidirect
4. #63964C2semidirect
5. #63956C2semidirect
6. #63952C2semidirect
7. #63883C2semidirect
8. #63865C2semidirect
9. #63845C2semidirect
35. #935D3semidirect
218. C32#539semidirect
261. C3#5660semidirect

Maximal subgroups

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

subgroupindex
#641472
C3×#9352
#639752
#639642
#639562
#639522
#638832
#638652
#638452
#56603