Finite Group Databasebeta

← #10594426

Subgroups — #10594426

Order 576.

496 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 136 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#64099C211yesmaximal, characteristic, complemented
2C3×#895C211yesmaximal, characteristic
3C3×#887C211yesmaximal, characteristic, complemented
4Dic3×D12C212yesmaximal, complemented
5Dic3×D12C211yesmaximal, characteristic, complemented
6#63884C212yesmaximal, complemented
7#63877C212yesmaximal, complemented
8C4×D3×Dic3C211yesmaximal, characteristic, complemented
9#63863C211yesmaximal, characteristic, complemented
10#63861C212yesmaximal, complemented
11#63856C211yesmaximal, characteristic, complemented
12#5600—31nomaximal
13#5580—31nomaximal
14C2×C3×D12C2211yescharacteristic
15C2×C3×D12C2212yes—
16C2×C3×C4×D3C2211yescharacteristic
17C2×C3×Dic6C2211yescharacteristic
18C2×D3×Dic3C2211yescharacteristic
19C2×D3×Dic3C2211yescharacteristic
20C2×D3×Dic3C2212yes—
21C32×C4⋊C4C2211yescharacteristic
22C4⋊(C3⋊Dic3)C2211yescharacteristic
23C4⋊(C3⋊Dic3)C2212yes—
24#3750C2212yes—
25C4×C3⋊Dic3C2211yescharacteristic
26C3×(C2×D3)⋊C4C2212yes—
27C3×Dic3⋊C4C2212yes—
28C3×C4×Dic3C2211yescharacteristic
29C3×C4×Dic3C2212yes—
30#3724C2212yes—
31(C2×D3)⋊Dic3C2212yes—
32(C2×D3)⋊Dic3C2212yes—
33(C2×D3)⋊Dic3C2212yes—
34(C2×D3)⋊Dic3C2212yes—
35Dic32C2211yescharacteristic
36Dic32C2211yescharacteristic
37C3×#121—31no—
38C3×C4×Q8—31no—
39#947—31no—
40#946—31no—
41#936—31no—
42#935—32no—
43D4×Dic3—32no—
44D4×Dic3—31no—
45#924—31no—
46C2×C4×Dic3—31no—
47#898—32no—
48#895D311yescharacteristic
49#894—32no—
50#890—31no—
51#890—31no—
52#890—61no—
53#887D311yescharacteristic, complemented
54#879—32no—
55#877—32no—
56C4×D12—32no—
57C4×D12—31no—
58#872—32no—
59C42×D3—31no—
60C22×C3×D3C2312yes—
61C22×C3×D3C2311yescharacteristic
62C2×C32×C4C2312yes—
63C2×C32×C4C2311yescharacteristic
64C2×C3⋊Dic3C2311yescharacteristic
65C2×C3⋊Dic3C2311yescharacteristic
66C2×C3⋊Dic3C2312yes—
67C2×C3×Dic3C2311yescharacteristic
68C2×C3×Dic3C2311yescharacteristic
69C2×C3×Dic3C2312yes—
70C2×C3×Dic3C2311yescharacteristic
71C3×D12—24no—
72C3×D12—22no—
73C3×C4×D3—22no—
74C3×Dic6—22no—
75D3×Dic3—22no—
76D3×Dic3—22no—
77D3×Dic3—24no—
78#529—91noSylow
79C2×C3×Q8—31no—
80C2×C3×D4—32no—
81C2×C3×D4—31no—
82C22×C3×C4—31no—
83C22×Dic3—32no—
84C22×Dic3—31no—
85C22×Dic3—31no—
86C2×D12C2×D312yes—
87C2×D12C2×D311yescharacteristic
88C2×C4×D3—31no—
89C2×C4×D3—31no—
90C2×C4×D3—32no—
91C2×C4×D3C2×D311yescharacteristic
92C2×Dic6C2×D311yescharacteristic
93C3×C4⋊C4—21no—
94C3×C4⋊C4—32no—
95C3×C4⋊C4C2×D311yescharacteristic
96C3×C4⋊C4C2×D311yescharacteristic
97C3×C22⋊C4—32no—
98C3×C42—31no—
99C3×C42—32no—
100#217—32no—
101#217—32no—
102#217—32no—
103#217—32no—
104(C2×D3)⋊C4—32no—
105(C2×D3)⋊C4—32no—
106(C2×D3)⋊C4—32no—
107(C2×D3)⋊C4—32no—
108(C2×D3)⋊C4C2×D312yes—
109C4⋊Dic3—32no—
110C4⋊Dic3—31no—
111C4⋊Dic3—32no—
112C4⋊Dic3—31no—
113C4⋊Dic3—61no—
114C4⋊Dic3—62no—
115Dic3⋊C4—32no—
116Dic3⋊C4—32no—
117Dic3⋊C4—32no—
118Dic3⋊C4—32no—
119Dic3⋊C4—62no—
120Dic3⋊C4C2×D312yes—
121C4×Dic3—31no—
122C4×Dic3—31no—
123C4×Dic3—31no—
124C4×Dic3—31no—
125C4×Dic3—31no—
126C4×Dic3—31no—
127C4×Dic3—61no—
128C4×Dic3C2×D312yes—
129C4×Dic3C2×D311yescharacteristic
130C22×C32C2411yescharacteristic
131C2×C3×D3—21no—
132C2×C3×D3—21no—
133C2×C3×D3—22no—
134C2×C3×D3—22no—
135C2×C3×D3—22no—
136C2×C3×D3#4112yes—
137C32×C4—22no—
138C32×C4#4112yes—
139C3⋊Dic3—21no—
140C3⋊Dic3—21no—
141C3⋊Dic3—22no—
142C3×Dic3—21no—
143C3×Dic3—22no—
144C3×Dic3—21no—
145C3×Dic3#4112yes—
146#126—92no—
147#125—91no—
148#124—91no—
149#123—92no—
150C22⋊Q8—91no—
151#121—31no—
152C4×Q8—31no—
153C4×D4—91no—
154C4×D4—92no—
155#117—92no—
156C2×C42—91no—
157C23×C3—31no—
158C23×C3—32no—
159C22×D3C22×D312yes—
160C22×D3C22×D311yescharacteristic
161C3×Q8—62no—
162C3×D4—64no—
163C3×D4—62no—
164C2×C3×C4—21no—
165C2×C3×C4—22no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—32no—
170C2×C3×C4—62no—
171C2×C3×C4C22×D312yes—
172C2×C3×C4C22×D312yes—
173C2×C3×C4C22×D311yescharacteristic
174C2×C3×C4C22×D311yescharacteristic
175C2×Dic3—31no—
176C2×Dic3—31no—
177C2×Dic3—31no—
178C2×Dic3—31no—
179C2×Dic3—32no—
180C2×Dic3—32no—
181C2×Dic3—61no—
182C2×Dic3—61no—
183C2×Dic3—62no—
184C2×Dic3—62no—
185C2×Dic3—62no—
186C2×Dic3—64no—
187C2×Dic3C22×D311yescharacteristic
188C2×Dic3C22×D311yescharacteristic
189C2×Dic3C22×D312yes—
190C2×Dic3C22×D311yescharacteristic
191D12—24no—
192D12—22no—
193C4×D3—22no—
194C4×D3—62no—
195C4×D3—64no—
196C4×D3—62no—
197Dic6—22no—
198C2×C32C2×#4111yescharacteristic
199C2×C32C2×#4111yescharacteristic
200C2×C32C2×#4111yescharacteristic
201C3×D3—22no—
202C3×D3—42no—
203C2×Q8—31no—
204C2×D4—32no—
205C2×D4—31no—
206C22×C4—31no—
207C22×C4—91no—
208C22×C4—91no—
209C22×C4—92no—
210C4⋊C4—32no—
211C4⋊C4—92no—
212C4⋊C4—92no—
213C4⋊C4—91no—
214C4⋊C4—92no—
215C4⋊C4D3211yescharacteristic
216C22⋊C4—32no—
217C22⋊C4—92no—
218C22⋊C4—92no—
219C22⋊C4—92no—
220C22⋊C4—92no—
221C42—31no—
222C42—32no—
223C42—91no—
224C42—91no—
225C42—91no—
226C22×C3—21no—
227C22×C3—32no—
228C22×C3—61no—
229C22×C3—61no—
230C22×C3—62no—
231C22×C3—62no—
232C22×C3—62no—
233C22×C3C23×D311yescharacteristic
234C22×C3C23×D311yescharacteristic
235C2×D3—21no—
236C2×D3—21no—
237C2×D3—22no—
238C2×D3—22no—
239C2×D3—22no—
240C2×D3#23712yes—
241C3×C4—22no—
242C3×C4—22no—
243C3×C4—22no—
244C3×C4—32no—
245C3×C4—42no—
246C3×C4—61no—
247C3×C4—61no—
248C3×C4—62no—
249C3×C4#23512yes—
250C3×C4#23712yes—
251Dic3—22no—
252Dic3—21no—
253Dic3—21no—
254Dic3—61no—
255Dic3—61no—
256Dic3—62no—
257Dic3—61no—
258Dic3—62no—
259Dic3—61no—
260Dic3—121no—
261Dic3—121no—
262Dic3—122no—
263Dic3#23912yes—
264C32#52911yescharacteristic, complemented, Sylow
265C23—31no—
266C23—32no—
267Q8—62no—
268D4—64no—
269D4—62no—
270C2×C4—31no—
271C2×C4—31no—
272C2×C4—32no—
273C2×C4—31no—
274C2×C4—62no—
275C2×C4—91no—
276C2×C4—91no—
277C2×C4—92no—
278C2×C4—182no—
279C2×C4—184no—
280C2×C4—182no—
281C2×C4C2×D3212yes—
282C2×C4C2×D3211yescharacteristic
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—62no—
287C2×C3—122no—
288C2×C3C2×#23511yescharacteristic
289C2×C3C2×#23711yescharacteristic
290C2×C3C2×#23711yescharacteristic
291C2×C3C2×#23911yescharacteristic
292C2×C3D3×#4111yescharacteristic
293C2×C3D3×#4111yescharacteristic
294D3—22no—
295D3—42no—
296C22—32no—
297C22—61no—
298C22—61no—
299C22—62no—
300C22—62no—
301C22—62no—
302C22C22×D3211yescharacteristic
303C4—22no—
304C4—32no—
305C4—61no—
306C4—62no—
307C4—61no—
308C4—181no—
309C4—181no—
310C4—182no—
311C4#379512yes—
312C3—21no—
313C3#558011yescharacteristic, complemented
314C3#560011yescharacteristic, complemented
315C2—62no—
316C2—122no—
317C2C2×#379511yescharacteristic
318C2D3×#23711yescharacteristic
319C2D3×#23911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC3×#895C2
hypercenterC4⋊C4D32
hypocenterC32#529
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64099C2semidirect
3. C3×#887C2semidirect
4. Dic3×D12C2semidirect
5. Dic3×D12C2semidirect
6. #63884C2semidirect
7. #63877C2semidirect
8. C4×D3×Dic3C2semidirect
9. #63863C2semidirect
10. #63861C2semidirect
11. #63856C2semidirect
53. #887D3semidirect
264. C32#529semidirect
313. C3#5580semidirect
314. C3#5600semidirect

Maximal subgroups

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

subgroupindex
#640992
C3×#8952
C3×#8872
Dic3×D122
#638842
#638772
C4×D3×Dic32
#638632
#638612
#638562
#56003
#55803