Finite Group Databasebeta

← #10591297

Subgroups — #10591297

Order 576.

378 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 96 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×C3×(C2×D3)⋊C4C211yesmaximal, characteristic
2C2×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×(C2×D3)⋊Dic3C211yesmaximal, characteristic
4C2×(C2×D3)⋊Dic3C211yesmaximal, characteristic
5#63658C211yesmaximal, characteristic, complemented
6C3×C3⋊#95C211yesmaximal, characteristic, complemented
7#63579C211yesmaximal, characteristic, complemented
8#4953—31nomaximal
9#4675—31nomaximal
10C23×C3×D3C2211yescharacteristic
11C22×C32×C4C2211yescharacteristic
12C22×C3⋊Dic3C2211yescharacteristic
13C22×C3⋊Dic3C2211yescharacteristic
14C22×C3×Dic3C2211yescharacteristic
15C22×C3×Dic3C2211yescharacteristic
16C22×C3×Dic3C2211yescharacteristic
17C3×(C2×D3)⋊C4—22no—
18Dic3⋊Dic3—22no—
19(C2×D3)⋊Dic3—22no—
20(C2×D3)⋊Dic3—22no—
21C2×C3×C22⋊C4—31no—
22C2×#217—31no—
23C2×#217—31no—
24C2×(C2×D3)⋊C4—31no—
25C2×(C2×D3)⋊C4—31no—
26C2×(C2×D3)⋊C4D311yescharacteristic
27C2×C4⋊Dic3—31no—
28C2×Dic3⋊C4—31no—
29C3×#95—31no—
30C3⋊#95—31no—
31C3⋊#95—31no—
32C3⋊#95—31no—
33C3⋊#95—31no—
34C3⋊#95—61no—
35C3⋊#95D311yescharacteristic, complemented
36C23×C32C2311yescharacteristic
37C22×C3×D3—21no—
38C22×C3×D3—21no—
39C22×C3×D3—21no—
40C22×C3×D3—21no—
41C22×C3×D3—21no—
42C22×C3×D3—21no—
43C22×C3×D3D412yes—
44C2×C32×C4—21no—
45C2×C32×C4—21no—
46C2×C32×C4—21no—
47C2×C3⋊Dic3—21no—
48C2×C3⋊Dic3—21no—
49C2×C3⋊Dic3—21no—
50C2×C3⋊Dic3—21no—
51C2×C3⋊Dic3—21no—
52C2×C3⋊Dic3—21no—
53C2×C3×Dic3—21no—
54C2×C3×Dic3—21no—
55C2×C3×Dic3—21no—
56C2×C3×Dic3—21no—
57C2×C3×Dic3—21no—
58C2×C3×Dic3—21no—
59C2×C3×Dic3—21no—
60C2×C3×Dic3—21no—
61C2×C3×Dic3D412yes—
62#397—91noSylow
63C24×C3—31no—
64C23×D3C2×D311yescharacteristic
65C22×C3×C4—21no—
66C22×C3×C4—31no—
67C22×C3×C4—31no—
68C22×C3×C4—31no—
69C22×C3×C4C2×D311yescharacteristic
70C22×C3×C4C2×D311yescharacteristic
71C22×Dic3—31no—
72C22×Dic3—31no—
73C22×Dic3—31no—
74C22×Dic3—31no—
75C22×Dic3—61no—
76C22×Dic3—61no—
77C22×Dic3C2×D311yescharacteristic
78C22×Dic3C2×D311yescharacteristic
79C22×Dic3C2×D311yescharacteristic
80C3×C22⋊C4—62no—
81#217—62no—
82#217—62no—
83(C2×D3)⋊C4—22no—
84(C2×D3)⋊C4—62no—
85(C2×D3)⋊C4—62no—
86C4⋊Dic3—62no—
87Dic3⋊C4—62no—
88C22×C32C2×D411yescharacteristic
89C22×C32C2×D411yescharacteristic
90C22×C32#4111yescharacteristic
91C22×C32#4111yescharacteristic
92C22×C32#4111yescharacteristic
93C22×C32#4111yescharacteristic
94C22×C32#4111yescharacteristic
95C2×C3×D3—22no—
96C2×C3×D3—22no—
97C2×C3×D3—22no—
98C2×C3×D3—41no—
99C2×C3×D3—41no—
100C2×C3×D3—41no—
101C2×C3×D3—41no—
102C32×C4—41no—
103C3⋊Dic3—41no—
104C3⋊Dic3—41no—
105C3×Dic3—22no—
106C3×Dic3—41no—
107C3×Dic3—41no—
108C2×C4⋊C4—91no—
109C2×C22⋊C4—31no—
110C2×C22⋊C4—91no—
111C2×C22⋊C4—91no—
112#95—31no—
113#95—91no—
114#95—91no—
115C23×C3—21no—
116C23×C3—32no—
117C23×C3—61no—
118C23×C3—61no—
119C23×C3—61no—
120C23×C3—61no—
121C23×C3—61no—
122C23×C3—61no—
123C23×C3C22×D311yescharacteristic
124C23×C3C22×D311yescharacteristic
125C22×D3—21no—
126C22×D3—21no—
127C22×D3—21no—
128C22×D3—21no—
129C22×D3—21no—
130C22×D3—21no—
131C22×D3D1212yes—
132C2×C3×C4—22no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—32no—
140C2×C3×C4—41no—
141C2×C3×C4—41no—
142C2×C3×C4—61no—
143C2×C3×C4—61no—
144C2×C3×C4—61no—
145C2×C3×C4—61no—
146C2×C3×C4—61no—
147C2×C3×C4—61no—
148C2×C3×C4—61no—
149C2×C3×C4—61no—
150C2×Dic3—21no—
151C2×Dic3—21no—
152C2×Dic3—21no—
153C2×Dic3—21no—
154C2×Dic3—21no—
155C2×Dic3—21no—
156C2×Dic3—21no—
157C2×Dic3—21no—
158C2×Dic3—62no—
159C2×Dic3—62no—
160C2×Dic3—61no—
161C2×Dic3—61no—
162C2×Dic3—61no—
163C2×Dic3—61no—
164C2×Dic3—61no—
165C2×Dic3—61no—
166C2×Dic3—61no—
167C2×Dic3—61no—
168C2×Dic3—61no—
169C2×Dic3—61no—
170C2×Dic3—61no—
171C2×Dic3—61no—
172C2×Dic3—121no—
173C2×Dic3—121no—
174C2×Dic3—121no—
175C2×Dic3—121no—
176C2×Dic3#6712yes—
177C2×C32#12111yescharacteristic
178C2×C32#12311yescharacteristic
179C2×C32#12311yescharacteristic
180C2×C32#12311yescharacteristic
181C2×C32#12411yescharacteristic
182C2×C32#12611yescharacteristic
183C2×C32#12611yescharacteristic
184C3×D3—42no—
185C24—31no—
186C22×C4—31no—
187C22×C4—31no—
188C22×C4—31no—
189C22×C4—91no—
190C22×C4—91no—
191C22×C4D3211yescharacteristic
192C4⋊C4—182no—
193C22⋊C4—62no—
194C22⋊C4—182no—
195C22⋊C4—182no—
196C22×C3—21no—
197C22×C3—21no—
198C22×C3—21no—
199C22×C3—21no—
200C22×C3—21no—
201C22×C3—21no—
202C22×C3—21no—
203C22×C3—62no—
204C22×C3—62no—
205C22×C3—62no—
206C22×C3—121no—
207C22×C3—121no—
208C22×C3—121no—
209C22×C3—121no—
210C22×C3C2×D1211yescharacteristic
211C22×C3#23511yescharacteristic
212C22×C3#23511yescharacteristic
213C22×C3#23511yescharacteristic
214C22×C3#23511yescharacteristic
215C22×C3D3×D411yescharacteristic
216C22×C3D3×D411yescharacteristic
217C22×C3#23711yescharacteristic
218C22×C3#23711yescharacteristic
219C22×C3#23711yescharacteristic
220C22×C3#23711yescharacteristic
221C22×C3#23711yescharacteristic
222C22×C3#23911yescharacteristic
223C22×C3C2×#6711yescharacteristic
224C2×D3—22no—
225C2×D3—22no—
226C2×D3—22no—
227C2×D3—41no—
228C2×D3—41no—
229C2×D3—41no—
230C2×D3—41no—
231C3×C4—42no—
232C3×C4—41no—
233C3×C4—41no—
234C3×C4—62no—
235C3×C4—121no—
236C3×C4—121no—
237Dic3—22no—
238Dic3—41no—
239Dic3—41no—
240Dic3—122no—
241Dic3—122no—
242Dic3—121no—
243Dic3—121no—
244Dic3—121no—
245Dic3—121no—
246C32#39711yescharacteristic, complemented, Sylow
247C23—32no—
248C23—61no—
249C23—61no—
250C23—61no—
251C23—61no—
252C23—61no—
253C23—61no—
254C23C2×D3211yescharacteristic
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—32no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—61no—
262C2×C4—61no—
263C2×C4—61no—
264C2×C4—61no—
265C2×C4—61no—
266C2×C4—61no—
267C2×C4—181no—
268C2×C4—181no—
269C2×C4—181no—
270C2×C4—181no—
271C2×C4—181no—
272C2×C4—181no—
273C2×C3—21no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3—122no—
281C2×C3#87611yescharacteristic
282C2×C3#87911yescharacteristic
283C2×C3#87911yescharacteristic
284C2×C3#88311yescharacteristic
285C2×C3#88411yescharacteristic
286C2×C3#88511yescharacteristic
287C2×C3#88611yescharacteristic
288C2×C3#88611yescharacteristic
289C2×C3#89411yescharacteristic
290C2×C3#89411yescharacteristic
291C2×C3#89811yescharacteristic
292C2×C3#93011yescharacteristic
293C2×C3#93511yescharacteristic
294C2×C3#93611yescharacteristic
295D3—42no—
296C22—62no—
297C22—62no—
298C22—62no—
299C22—121no—
300C22—121no—
301C22—121no—
302C22—121no—
303C22#379511yescharacteristic
304C22#379611yescharacteristic
305C22#379811yescharacteristic
306C22D3×D1211yescharacteristic
307C22#380411yescharacteristic
308C22#380511yescharacteristic
309C22D3×#6711yescharacteristic
310C4—41no—
311C4—62no—
312C4—121no—
313C4—121no—
314C4—361no—
315C4—361no—
316C3—21no—
317C3#467511yescharacteristic, complemented
318C3#495311yescharacteristic, complemented
319C2—122no—
320C2#6385911yescharacteristic
321C2#6386211yescharacteristic
322C2#6388411yescharacteristic
323C2#6389011yescharacteristic
324C2#6389111yescharacteristic
325C2#6390611yescharacteristic
326C2#6395911yescharacteristic
subgroupG/H
center, FrattiniC23C2×D32
commutator, socle, cosocleC23×C32C23
FittingC22×C32×C4C22
prime radicalC23×C3×D3C22
hypercenterC22×C4D32
hypocenterC32#397
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×Dic3⋊Dic3C2semidirect
5. #63658C2semidirect
6. C3×C3⋊#95C2semidirect
7. #63579C2semidirect
35. C3⋊#95D3semidirect
246. C32#397semidirect
317. C3#4675semidirect
318. C3#4953semidirect

Maximal subgroups

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

subgroupindex
C2×C3×(C2×D3)⋊C42
C2×Dic3⋊Dic32
C2×(C2×D3)⋊Dic32
#636582
C3×C3⋊#952
#635792
#49533
#46753