Finite Group Databasebeta

← #10594911

Subgroups — #10594911

Order 576.

635 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 635 conjugacy classes of subgroups — too many to draw. Only its 144 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×#3804C212yesmaximal, complemented
2#64139C211yesmaximal, characteristic, complemented
3C3×#930C211yesmaximal, characteristic, complemented
4C2×C3×Dic3⋊C4C211yesmaximal, characteristic, complemented
5#63978C212yesmaximal, complemented
6#63910C212yesmaximal, complemented
7#63895C212yesmaximal, complemented
8#63893C212yesmaximal, complemented
9#63886C212yesmaximal, complemented
10#5597—31nomaximal
11#5510—31nomaximal
12C22×C32×C4C2211yescharacteristic
13C2×#638C2212yes—
14C2×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×#67C2212yes—
16C22×C3×Dic3C2211yescharacteristic
17C22×C3×Dic3C2211yescharacteristic
18C2×C3×D12C2211yescharacteristic
19C2×#627C2212yes—
20C2×#626C2212yes—
21C2×#626C2212yes—
22C2×#624C2212yes—
23#3804—28no—
24C2×D3×Dic3C2212yes—
25#3752C2212yes—
26C4⋊(C3⋊Dic3)C2211yescharacteristic
27C3×(C2×D3)⋊C4C2212yes—
28C3×C4⋊Dic3C2211yescharacteristic
29C3×Dic3⋊C4C2212yes—
30C3×Dic3⋊C4C2212yes—
31Dic3⋊Dic3C2212yes—
32#3722C2212yes—
33(C2×D3)⋊Dic3C2212yes—
34C2×#237—32no—
35C2×#235—32no—
36C3×#121—31no—
37C2×C3×C4⋊C4—31no—
38#939—32no—
39#939—32no—
40#930—31no—
41#930—31no—
42#930—32no—
43#930—61no—
44#930D311yescharacteristic, complemented
45C2×Dic3⋊C4D311yescharacteristic, complemented
46#897—32no—
47#897—32no—
48#896—32no—
49#895—32no—
50#895—32no—
51#884—32no—
52#878—32no—
53C23×C32C2311yescharacteristic
54C22×C3⋊D3C2312yes—
55C22×C3×D3C2312yes—
56C2×C32×C4—41no—
57C2×C32×C4C2311yescharacteristic
58C2×C32×C4C2311yescharacteristic
59#638—24no—
60C2×C3⋊Dic3C2312yes—
61C4⋊(C3⋊D3)—41no—
62C3×#67—24no—
63C2×C3×Dic3—41no—
64C2×C3×Dic3D414yescomplemented
65C2×C3×Dic3C2312yes—
66C2×C3×Dic3C2312yes—
67C2×C3×Dic3C2312yes—
68C3×D12—41no—
69#627—24no—
70#626—24no—
71#626—42no—
72#624—24no—
73D3×Dic3—24no—
74#537—91noSylow
75C2×C3×D4—32no—
76C2×C3×D4—31no—
77C22×C3×C4—21no—
78C22×C3×C4—31no—
79C22×C3×C4—31no—
80C22×C3×C4C2×D311yescharacteristic
81C22×C3×C4C2×D311yescharacteristic
82C2×#67—32no—
83C2×#67—32no—
84C2×#67—32no—
85C2×#67—32no—
86C2×#67—62no—
87C2×#67C2×D312yes—
88C22×Dic3—32no—
89C22×Dic3C2×D311yescharacteristic
90C22×Dic3C2×D311yescharacteristic
91#237—68no—
92#235—68no—
93C2×D12—32no—
94C2×D12—32no—
95C2×D12—31no—
96C2×D12—31no—
97C2×D12—61no—
98C2×D12C2×D311yescharacteristic
99C2×C4×D3—32no—
100C2×C4×D3—32no—
101C2×C4×D3—32no—
102C2×Dic6—32no—
103C2×Dic6—32no—
104C3×C4⋊C4—32no—
105C3×C4⋊C4—32no—
106C3×C4⋊C4—31no—
107C3×C22⋊C4—32no—
108#217—32no—
109(C2×D3)⋊C4—32no—
110(C2×D3)⋊C4—32no—
111(C2×D3)⋊C4—32no—
112(C2×D3)⋊C4—32no—
113(C2×D3)⋊C4—32no—
114(C2×D3)⋊C4—62no—
115(C2×D3)⋊C4C2×D312yes—
116C4⋊Dic3—31no—
117C4⋊Dic3—31no—
118C4⋊Dic3—32no—
119C4⋊Dic3—61no—
120C4⋊Dic3C2×D311yescharacteristic
121Dic3⋊C4—32no—
122Dic3⋊C4C2×D312yes—
123Dic3⋊C4C2×D312yes—
124C22×C32—21no—
125C22×C32—21no—
126C22×C32C2×D412yes—
127C22×C32C2411yescharacteristic
128C2×C3⋊D3—22no—
129C2×C3⋊D3—22no—
130C2×C3⋊D3—22no—
131C2×C3×D3—22no—
132C2×C3×D3—22no—
133C2×C3×D3—22no—
134C32×C4—21no—
135C32×C4—21no—
136C3⋊Dic3—22no—
137C3×Dic3—22no—
138C3×Dic3—22no—
139C3×Dic3C2×D414yes—
140C2×#41—92no—
141C22⋊Q8—92no—
142C22⋊Q8—92no—
143#121—31no—
144#121—92no—
145#121—92no—
146#121—92no—
147#121—91no—
148C2×C4⋊C4—31no—
149C23×C3—21no—
150C23×C3—32no—
151C23×C3C22×D311yescharacteristic
152C23×C3C22×D311yescharacteristic
153C22×D3—32no—
154C22×D3—32no—
155C22×D3—62no—
156C22×D3C22×D312yes—
157C3×D4—64no—
158C3×D4—121no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—34no—
162C2×C3×C4—32no—
163C2×C3×C4—32no—
164C2×C3×C4—32no—
165C2×C3×C4—41no—
166C2×C3×C4—41no—
167C2×C3×C4—42no—
168C2×C3×C4—121no—
169C2×C3×C4C22×D311yescharacteristic
170C2×C3×C4C22×D311yescharacteristic
171C2×C3×C4C22×D311yescharacteristic
172C2×C3×C4C22×D311yescharacteristic
173#67—24no—
174#67—64no—
175#67—64no—
176#67—64no—
177#67—124no—
178#67—122no—
179C2×Dic3—32no—
180C2×Dic3—32no—
181C2×Dic3—41no—
182C2×Dic3—62no—
183C2×Dic3—64no—
184C2×Dic3D1214yescomplemented
185C2×Dic3C22×D312yes—
186C2×Dic3C22×D312yes—
187C2×Dic3C22×D312yes—
188D12—41no—
189D12—64no—
190D12—121no—
191D12—121no—
192D12—122no—
193D12—122no—
194C4×D3—64no—
195C4×D3—64no—
196C4×D3—64no—
197Dic6—64no—
198Dic6—64no—
199C2×C32—22no—
200C2×C32C22×D411yescharacteristic
201C2×C32H(5,𝔽2)11yescharacteristic
202C2×C322−1+411yescharacteristic
203C3⋊D3—42no—
204C3×D3—42no—
205#41—188no—
206C2×Q8—92no—
207C2×D4—32no—
208C2×D4—31no—
209C2×D4—92no—
210C2×D4—92no—
211C2×D4—92no—
212C2×D4—91no—
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—92no—
216C22×C4—92no—
217C22×C4D3211yescharacteristic
218C4⋊C4—32no—
219C4⋊C4—32no—
220C4⋊C4—31no—
221C4⋊C4—92no—
222C4⋊C4—91no—
223C22⋊C4—32no—
224C22⋊C4—92no—
225C22⋊C4—92no—
226C22⋊C4—92no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—22no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3—62no—
236C22×C3—62no—
237C22×C3—62no—
238C22×C3C2×D1212yes—
239C22×C3D3×D412yes—
240C22×C3C23×D311yescharacteristic
241C22×C3C23×D311yescharacteristic
242C2×D3—22no—
243C2×D3—22no—
244C2×D3—22no—
245C2×D3—62no—
246C2×D3—62no—
247C2×D3—62no—
248C2×D3—62no—
249C2×D3—62no—
250C2×D3—62no—
251C2×D3—122no—
252C2×D3—122no—
253C2×D3—122no—
254C3×C4—22no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—34no—
260C3×C4—41no—
261C3×C4—62no—
262C3×C4—62no—
263Dic3—22no—
264Dic3—22no—
265Dic3—62no—
266Dic3—62no—
267Dic3—122no—
268Dic3C2×D1214yes—
269C32#53711yescharacteristic, complemented, Sylow
270C23—32no—
271C23—92no—
272C23C2×D3211yescharacteristic
273Q8—184no—
274D4—64no—
275D4—121no—
276D4—184no—
277D4—184no—
278D4—361no—
279D4—362no—
280C2×C4—32no—
281C2×C4—34no—
282C2×C4—32no—
283C2×C4—32no—
284C2×C4—41no—
285C2×C4—92no—
286C2×C4—121no—
287C2×C4—184no—
288C2×C4—184no—
289C2×C4C2×D3211yescharacteristic
290C2×C4C2×D3211yescharacteristic
291C2×C3—22no—
292C2×C3—22no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—42no—
297C2×C3—122no—
298C2×C3C22×D1211yescharacteristic
299C2×C3C2×D3×D411yescharacteristic
300C2×C3#100411yescharacteristic
301C2×C3#100411yescharacteristic
302C2×C3#100711yescharacteristic
303C2×C3#101011yescharacteristic
304D3—42no—
305D3—122no—
306D3—122no—
307D3—242no—
308C22—21no—
309C22—21no—
310C22—62no—
311C22—62no—
312C22—62no—
313C22—182no—
314C22—182no—
315C22—182no—
316C22D3×D1212yes—
317C22C22×D3211yescharacteristic
318C4—21no—
319C4—21no—
320C4—34no—
321C4—62no—
322C4—62no—
323C4—182no—
324C3—21no—
325C3#551011yescharacteristic, complemented
326C3#559711yescharacteristic, complemented
327C2—22no—
328C2—122no—
329C2—362no—
330C2#6429711yescharacteristic
331C2C2×D3×D1211yescharacteristic
332C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#537
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#3804C2semidirect
2. #64139C2semidirect
3. C3×#930C2semidirect
4. C2×C3×Dic3⋊C4C2semidirect
5. #63978C2semidirect
6. #63910C2semidirect
7. #63895C2semidirect
8. #63893C2semidirect
9. #63886C2semidirect
44. #930D3semidirect
45. C2×Dic3⋊C4D3semidirect
64. C2×C3×Dic3D4semidirect
184. C2×Dic3D12semidirect
269. C32#537semidirect
325. C3#5510semidirect
326. C3#5597semidirect

Maximal subgroups

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

subgroupindex
C2×#38042
#641392
C3×#9302
C2×C3×Dic3⋊C42
#639782
#639102
#638952
#638932
#638862
#55973
#55103