Finite Group Databasebeta

← #10595191

Subgroups — #10595191

Order 576.

515 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 515 conjugacy classes of subgroups — too many to draw. Only its 134 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×#936C211yesmaximal, characteristic, complemented
3C3×D4×Dic3C211yesmaximal, characteristic, complemented
4#63975C212yesmaximal, complemented
5#63956C212yesmaximal, complemented
6#63955C212yesmaximal, complemented
7#63952C212yesmaximal, complemented
8C4×#624C211yesmaximal, characteristic, complemented
9#63871C211yesmaximal, characteristic, complemented
10#63869C211yesmaximal, characteristic, complemented
11Dic3×Dic6C211yesmaximal, characteristic, complemented
12#5680—31nomaximal
13#5557—31nomaximal
14C2×C32×D4C2211yescharacteristic
15C2×#638C2212yes—
16C2×C4×C3⋊D3C2211yescharacteristic
17C22×C3×Dic3C2212yes—
18C2×C3×Dic6C2211yescharacteristic
19C2×#624C2211yescharacteristic
20C2×#624C2211yescharacteristic
21#3757C2212yes—
22C4⋊(C3⋊Dic3)C2211yescharacteristic
23C3×#217C2212yes—
24C3×#217C2212yes—
25C3×#217C2212yes—
26C3×C4⋊Dic3C2211yescharacteristic
27C3×C4×Dic3C2211yescharacteristic
28C3×C4×Dic3C2211yescharacteristic
29#3724C2212yes—
30Dic3⋊Dic3C2212yes—
31Dic3⋊Dic3C2212yes—
32Dic3⋊Dic3C2212yes—
33#3722C2212yes—
34Dic32C2212yes—
35Dic32C2211yescharacteristic
36Dic32C2211yescharacteristic
37C3×#124—31no—
38C3×C4×D4—31no—
39#946—31no—
40Q8×Dic3—31no—
41#938—31no—
42#938—31no—
43#938—61no—
44#936D311yescharacteristic, complemented
45D4×Dic3D311yescharacteristic, complemented
46C4×#67—32no—
47#926—32no—
48#897—31no—
49#890—31no—
50#883—32no—
51#883—32no—
52#67⋊C4—32no—
53#879—32no—
54#879—32no—
55#877—32no—
56C42×D3—31no—
57C42×D3—31no—
58#870—31no—
59C4×Dic6—31no—
60C23×C32C2312yes—
61C22×C3⋊D3C2311yescharacteristic
62C32×D4—22no—
63C2×C32×C4C2311yescharacteristic
64#638—24no—
65C2×C3⋊Dic3C2311yescharacteristic
66C2×C3⋊Dic3C2312yes—
67C4×C3⋊D3—22no—
68C2×C3×Dic3—24no—
69C2×C3×Dic3C2312yes—
70C2×C3×Dic3C2311yescharacteristic
71C2×C3×Dic3C2311yescharacteristic
72C2×C3×Dic3C2311yescharacteristic
73C2×C3×Dic3C2312yes—
74C2×C3×Dic3C2311yescharacteristic
75C3×Dic6—22no—
76#624—22no—
77#624—22no—
78#529—91noSylow
79C2×C3×Q8—31no—
80C2×C3×D4—21no—
81C2×C3×D4C2×D311yescharacteristic
82C2×C3×D4C2×D311yescharacteristic
83C22×C3×C4—32no—
84C2×#67—32no—
85C2×#67—32no—
86C2×#67—62no—
87C22×Dic3C2×D312yes—
88C2×C4×D3—31no—
89C2×C4×D3—31no—
90C2×C4×D3—31no—
91C2×C4×D3—31no—
92C2×C4×D3—31no—
93C2×C4×D3—31no—
94C2×C4×D3—61no—
95C2×Dic6C2×D311yescharacteristic
96C3×C4⋊C4—31no—
97C3×C22⋊C4—32no—
98C3×C22⋊C4—32no—
99C3×C22⋊C4—32no—
100C3×C42—31no—
101C3×C42—31no—
102#217—32no—
103#217—32no—
104#217—62no—
105#217C2×D312yes—
106#217C2×D312yes—
107#217C2×D312yes—
108(C2×D3)⋊C4—32no—
109(C2×D3)⋊C4—32no—
110C4⋊Dic3—31no—
111C4⋊Dic3—31no—
112C4⋊Dic3—32no—
113C4⋊Dic3—32no—
114C4⋊Dic3—32no—
115C4⋊Dic3—61no—
116C4⋊Dic3C2×D311yescharacteristic
117Dic3⋊C4—32no—
118Dic3⋊C4—32no—
119Dic3⋊C4—32no—
120Dic3⋊C4—32no—
121Dic3⋊C4—32no—
122C4×Dic3—32no—
123C4×Dic3—31no—
124C4×Dic3—31no—
125C4×Dic3—31no—
126C4×Dic3—31no—
127C4×Dic3—32no—
128C4×Dic3C2×D311yescharacteristic
129C4×Dic3C2×D311yescharacteristic
130C22×C32—22no—
131C22×C32—22no—
132C22×C32—22no—
133C22×C32C2411yescharacteristic
134C2×C3⋊D3—21no—
135C2×C3⋊D3—21no—
136C2×C3⋊D3#4112yes—
137C32×C4#4112yes—
138C3⋊Dic3—21no—
139C3⋊Dic3—22no—
140C3×Dic3—21no—
141C3×Dic3—21no—
142C3×Dic3—21no—
143C3×Dic3—22no—
144C3×Dic3—22no—
145C3×Dic3#4112yes—
146#126—92no—
147#125—91no—
148#124—31no—
149#123—92no—
150C22⋊Q8—91no—
151#121—91no—
152C4×Q8—91no—
153C4×D4—31no—
154C4×D4—92no—
155#117—92no—
156C2×C42—91no—
157C23×C3—22no—
158C23×C3C22×D312yes—
159C23×C3C22×D312yes—
160C22×D3—31no—
161C22×D3—31no—
162C22×D3—61no—
163C3×Q8—62no—
164C3×D4—22no—
165C3×D4—22no—
166C3×D4—42no—
167C2×C3×C4—21no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—32no—
173C2×C3×C4—32no—
174C2×C3×C4—64no—
175C2×C3×C4C22×D311yescharacteristic
176C2×C3×C4C22×D311yescharacteristic
177#67—64no—
178#67—64no—
179#67—124no—
180C2×Dic3—24no—
181C2×Dic3—31no—
182C2×Dic3—31no—
183C2×Dic3—32no—
184C2×Dic3—32no—
185C2×Dic3—62no—
186C2×Dic3—61no—
187C2×Dic3C22×D311yescharacteristic
188C2×Dic3C22×D311yescharacteristic
189C2×Dic3C22×D311yescharacteristic
190C2×Dic3C22×D312yes—
191C2×Dic3C22×D312yes—
192C2×Dic3C22×D311yescharacteristic
193C4×D3—62no—
194C4×D3—62no—
195C4×D3—62no—
196C4×D3—62no—
197C4×D3—62no—
198C4×D3—62no—
199C4×D3—122no—
200Dic6—22no—
201C2×C32—42no—
202C2×C32C2×#4111yescharacteristic
203C2×C32C2×#4111yescharacteristic
204C2×C32C2×#4111yescharacteristic
205C3⋊D3—22no—
206C2×Q8—31no—
207C2×D4—92no—
208C2×D4D3211yescharacteristic
209C22×C4—32no—
210C22×C4—91no—
211C22×C4—91no—
212C22×C4—91no—
213C4⋊C4—31no—
214C4⋊C4—92no—
215C4⋊C4—92no—
216C4⋊C4—92no—
217C4⋊C4—92no—
218C4⋊C4—91no—
219C22⋊C4—32no—
220C22⋊C4—32no—
221C22⋊C4—32no—
222C22⋊C4—92no—
223C22⋊C4—92no—
224C42—31no—
225C42—31no—
226C42—91no—
227C42—91no—
228C42—92no—
229C22×C3—22no—
230C22×C3—22no—
231C22×C3—22no—
232C22×C3—22no—
233C22×C3—22no—
234C22×C3—22no—
235C22×C3—21no—
236C22×C3—42no—
237C22×C3—42no—
238C22×C3—42no—
239C22×C3C23×D311yescharacteristic
240C22×C3C23×D311yescharacteristic
241C2×D3—32no—
242C2×D3—32no—
243C2×D3—62no—
244C2×D3—61no—
245C2×D3—61no—
246C2×D3—61no—
247C2×D3—61no—
248C2×D3—121no—
249C2×D3—121no—
250C3×C4—22no—
251C3×C4—32no—
252C3×C4—62no—
253C3×C4—61no—
254C3×C4—61no—
255C3×C4—61no—
256C3×C4—62no—
257C3×C4#23512yes—
258C3×C4#23712yes—
259Dic3—22no—
260Dic3—22no—
261Dic3—21no—
262Dic3—21no—
263Dic3—21no—
264Dic3—61no—
265Dic3—61no—
266Dic3—62no—
267Dic3—62no—
268Dic3—121no—
269Dic3—122no—
270Dic3#23712yes—
271C32#52911yescharacteristic, complemented, Sylow
272C23—91no—
273C23C2×D3212yes—
274Q8—62no—
275D4—22no—
276D4—184no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—31no—
281C2×C4—32no—
282C2×C4—32no—
283C2×C4—64no—
284C2×C4—91no—
285C2×C4—92no—
286C2×C4—182no—
287C2×C4—182no—
288C2×C4—182no—
289C2×C4C2×D3211yescharacteristic
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—42no—
294C2×C3—42no—
295C2×C3—82no—
296C2×C3C2×#23511yescharacteristic
297C2×C3C2×#23711yescharacteristic
298C2×C3C2×#23711yescharacteristic
299C2×C3D3×#4111yescharacteristic
300C2×C3D3×#4111yescharacteristic
301C2×C3D3×#4111yescharacteristic
302D3—62no—
303D3—62no—
304D3—122no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—92no—
309C22—181no—
310C22—181no—
311C22C22×D3211yescharacteristic
312C4—32no—
313C4—62no—
314C4—61no—
315C4—61no—
316C4—61no—
317C4—62no—
318C4—182no—
319C4—181no—
320C4#380412yes—
321C3—21no—
322C3#555711yescharacteristic, complemented
323C3#568011yescharacteristic, complemented
324C2—42no—
325C2—182no—
326C2D3×#23711yescharacteristic
327C2#6431211yescharacteristic
328C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
prime radical#64147C2
hypercenterC2×D4D32
hypocenterC32#529
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C3×#936C2semidirect
3. C3×D4×Dic3C2semidirect
4. #63975C2semidirect
5. #63956C2semidirect
6. #63955C2semidirect
7. #63952C2semidirect
8. C4×#624C2semidirect
9. #63871C2semidirect
10. #63869C2semidirect
11. Dic3×Dic6C2semidirect
44. #936D3semidirect
45. D4×Dic3D3semidirect
271. C32#529semidirect
322. C3#5557semidirect
323. C3#5680semidirect

Maximal subgroups

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

subgroupindex
#641472
C3×#9362
C3×D4×Dic32
#639752
#639562
#639552
#639522
C4×#6242
#638712
#638692
Dic3×Dic62
#56803
#55573