Finite Group Databasebeta

← #10594515

Subgroups — #10594515

Order 576.

630 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3798C212yesmaximal, complemented
2C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
3C3×#893C211yesmaximal, characteristic, complemented
4C3×D3×C4⋊C4C211yesmaximal, characteristic, complemented
5#63918C211yesmaximal, characteristic, complemented
6#63916C211yesmaximal, characteristic, complemented
7#63908C211yesmaximal, characteristic, complemented
8#63905C212yesmaximal, complemented
9C4×#626C211yesmaximal, characteristic, complemented
10#63853C212yesmaximal, complemented
11#63851C212yesmaximal, complemented
12#5551—31nomaximal
13#5507—31nomaximal
14C2×C4×C3⋊D3C2212yes—
15C2×C4×C3⋊D3C2211yescharacteristic
16C2×C3×D12C2211yescharacteristic
17C2×C3×C4×D3C2212yes—
18C2×C3×C4×D3C2211yescharacteristic
19C2×C3×C4×D3C2212yes—
20C2×#627C2212yes—
21C2×#626C2212yes—
22C2×#626C2211yescharacteristic
23C2×#625C2212yes—
24#3798—28no—
25C32×C4⋊C4C2211yescharacteristic
26C4⋊(C3⋊Dic3)C2211yescharacteristic
27#3750C2212yes—
28C3×(C2×D3)⋊C4C2212yes—
29C3×C4⋊Dic3C2211yescharacteristic
30C3×Dic3⋊C4C2212yes—
31C3×C4×Dic3C2211yescharacteristic
32Dic3⋊Dic3C2211yescharacteristic
33Dic3⋊Dic3C2212yes—
34#3722C2211yescharacteristic
35(C2×D3)⋊Dic3C2212yes—
36(C2×D3)⋊Dic3C2212yes—
37(C2×D3)⋊Dic3C2211yescharacteristic
38C2×#235—32no—
39C2×#235—32no—
40C3×C4×D4—31no—
41C2×C3×C4⋊C4—31no—
42#938—31no—
43#938—31no—
44#930—31no—
45#929—32no—
46C4×#67—31no—
47#924—32no—
48#896—32no—
49#895—31no—
50#894—32no—
51#893D311yescharacteristic, complemented
52D3×C4⋊C4—31no—
53D3×C4⋊C4—31no—
54D3×C4⋊C4—61no—
55D3×C4⋊C4D311yescharacteristic, complemented
56#888—31no—
57#883—32no—
58#883—32no—
59C4×D12—31no—
60#869—31no—
61C22×C3⋊D3C2311yescharacteristic
62C22×C3×D3C2312yes—
63C22×C3×D3C2311yescharacteristic
64C2×C32×C4C2312yes—
65C2×C32×C4C2311yescharacteristic
66C2×C3⋊Dic3C2311yescharacteristic
67C2×C3⋊Dic3C2312yes—
68C4×C3⋊D3—22no—
69C4×C3⋊D3—24no—
70C2×C3×Dic3C2311yescharacteristic
71C2×C3×Dic3C2311yescharacteristic
72C2×C3×Dic3C2312yes—
73C2×C3×Dic3C2311yescharacteristic
74C3×D12—22no—
75C3×C4×D3—24no—
76C3×C4×D3—22no—
77C3×C4×D3—24no—
78#627—24no—
79#626—24no—
80#626D414yescomplemented
81#625—24no—
82#547—91noSylow
83C2×C3×D4—31no—
84C22×C3×C4—32no—
85C22×C3×C4—31no—
86C22×C3×C4—32no—
87C2×#67—32no—
88C2×#67—32no—
89C2×#67—31no—
90C2×#67—32no—
91#235—68no—
92#235—68no—
93C2×D12—31no—
94C2×D12—32no—
95C2×D12C2×D311yescharacteristic
96C2×C4×D3—31no—
97C2×C4×D3—32no—
98C2×C4×D3—32no—
99C2×C4×D3—31no—
100C2×C4×D3—61no—
101C2×C4×D3—62no—
102C2×C4×D3C2×D311yescharacteristic
103C2×C4×D3C2×D312yes—
104C2×C4×D3C2×D312yes—
105C2×Dic6—32no—
106C2×Dic6—32no—
107C3×C4⋊C4—21no—
108C3×C4⋊C4—31no—
109C3×C4⋊C4—32no—
110C3×C4⋊C4C2×D311yescharacteristic
111C3×C4⋊C4C2×D311yescharacteristic
112C3×C22⋊C4—32no—
113C3×C42—31no—
114#217—32no—
115#217—32no—
116#217—31no—
117(C2×D3)⋊C4—31no—
118(C2×D3)⋊C4—32no—
119(C2×D3)⋊C4—32no—
120(C2×D3)⋊C4—31no—
121(C2×D3)⋊C4—31no—
122(C2×D3)⋊C4C2×D312yes—
123C4⋊Dic3—31no—
124C4⋊Dic3—31no—
125C4⋊Dic3—32no—
126C4⋊Dic3—31no—
127C4⋊Dic3—61no—
128C4⋊Dic3C2×D311yescharacteristic
129Dic3⋊C4—32no—
130Dic3⋊C4—32no—
131Dic3⋊C4—32no—
132Dic3⋊C4—31no—
133Dic3⋊C4—62no—
134Dic3⋊C4C2×D312yes—
135C4×Dic3C2×D311yescharacteristic
136C22×C32C2411yescharacteristic
137C2×C3⋊D3—21no—
138C2×C3⋊D3—21no—
139C2×C3⋊D3C2×D412yes—
140C2×C3×D3—21no—
141C2×C3×D3—21no—
142C2×C3×D3—22no—
143C2×C3×D3—22no—
144C2×C3×D3—22no—
145C2×C3×D3C2×D412yes—
146C32×C4—22no—
147C32×C4#4112yescomplemented
148C3⋊Dic3—21no—
149C3⋊Dic3—22no—
150C3×Dic3—22no—
151C3×Dic3—21no—
152C3×Dic3—21no—
153C3×Dic3C2×D412yes—
154C2×#41—92no—
155C4⋊Q8—91no—
156#123—92no—
157#123—92no—
158C22⋊Q8—92no—
159#121—91no—
160#121—91no—
161C4×D4—31no—
162C4×D4—91no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4—91no—
165C23×C3—31no—
166C23×C3—32no—
167C22×D3—31no—
168C22×D3—31no—
169C22×D3—61no—
170C22×D3C22×D312yes—
171C22×D3C22×D311yescharacteristic
172C3×D4—62no—
173C2×C3×C4—21no—
174C2×C3×C4—22no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—32no—
178C2×C3×C4—31no—
179C2×C3×C4—64no—
180C2×C3×C4—64no—
181C2×C3×C4—62no—
182C2×C3×C4C22×D312yes—
183C2×C3×C4C22×D312yes—
184C2×C3×C4C22×D311yescharacteristic
185C2×C3×C4C22×D311yescharacteristic
186#67—34no—
187#67—64no—
188#67—64no—
189#67—64no—
190C2×Dic3—31no—
191C2×Dic3—31no—
192C2×Dic3—32no—
193C2×Dic3—32no—
194C2×Dic3—61no—
195C2×Dic3—62no—
196C2×Dic3C22×D311yescharacteristic
197C2×Dic3C22×D311yescharacteristic
198C2×Dic3C22×D312yes—
199C2×Dic3C22×D311yescharacteristic
200D12—22no—
201D12—34no—
202D12—64no—
203C4×D3—24no—
204C4×D3—22no—
205C4×D3—24no—
206C4×D3—64no—
207C4×D3—64no—
208C4×D3—64no—
209C4×D3—62no—
210C4×D3—62no—
211C4×D3—124no—
212Dic6—64no—
213Dic6—64no—
214C2×C32C22×D411yescharacteristic
215C2×C32C2×#4111yescharacteristic
216C2×C322−1+411yescharacteristic
217C3⋊D3—22no—
218C3×D3—22no—
219C3×D3—42no—
220#41—188no—
221C2×Q8—92no—
222C2×D4—31no—
223C2×D4—91no—
224C2×D4—92no—
225C2×D4—92no—
226C22×C4—31no—
227C22×C4—32no—
228C22×C4—32no—
229C22×C4—91no—
230C22×C4—92no—
231C4⋊C4—31no—
232C4⋊C4—32no—
233C4⋊C4—91no—
234C4⋊C4—92no—
235C4⋊C4—91no—
236C4⋊C4—92no—
237C4⋊C4D3211yescharacteristic
238C22⋊C4—32no—
239C22⋊C4—91no—
240C22⋊C4—91no—
241C22⋊C4—92no—
242C22⋊C4—92no—
243C42—31no—
244C22×C3—21no—
245C22×C3—32no—
246C22×C3—61no—
247C22×C3—61no—
248C22×C3—62no—
249C22×C3—62no—
250C22×C3—62no—
251C22×C3C23×D311yescharacteristic
252C22×C3C23×D311yescharacteristic
253C2×D3—21no—
254C2×D3—21no—
255C2×D3—22no—
256C2×D3—22no—
257C2×D3—22no—
258C2×D3—32no—
259C2×D3—32no—
260C2×D3—62no—
261C2×D3—62no—
262C2×D3—62no—
263C2×D3—61no—
264C2×D3—61no—
265C2×D3—61no—
266C2×D3—61no—
267C2×D3D3×D412yes—
268C3×C4—22no—
269C3×C4—22no—
270C3×C4—22no—
271C3×C4—32no—
272C3×C4—42no—
273C3×C4—61no—
274C3×C4—62no—
275C3×C4—61no—
276C3×C4#23512yescomplemented
277C3×C4#23512yescomplemented
278Dic3—21no—
279Dic3—22no—
280Dic3—21no—
281Dic3—62no—
282Dic3—61no—
283Dic3—61no—
284Dic3—62no—
285Dic3—62no—
286Dic3—122no—
287Dic3D3×D412yes—
288C32#54711yescharacteristic, complemented, Sylow
289C23—31no—
290C23—32no—
291C23—91no—
292Q8—184no—
293D4—62no—
294D4—94no—
295D4—184no—
296D4—184no—
297C2×C4—31no—
298C2×C4—31no—
299C2×C4—31no—
300C2×C4—32no—
301C2×C4—64no—
302C2×C4—64no—
303C2×C4—62no—
304C2×C4—91no—
305C2×C4—92no—
306C2×C4—184no—
307C2×C4—182no—
308C2×C4C2×D3212yes—
309C2×C4C2×D3211yescharacteristic
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—62no—
314C2×C3—122no—
315C2×C3C2×#23511yescharacteristic
316C2×C3C2×#23511yescharacteristic
317C2×C3C2×D3×D411yescharacteristic
318C2×C3C2×D3×D411yescharacteristic
319C2×C3#100711yescharacteristic
320C2×C3#101011yescharacteristic
321D3—22no—
322D3—42no—
323D3—64no—
324D3—62no—
325D3—62no—
326C22—32no—
327C22—61no—
328C22—61no—
329C22—62no—
330C22—62no—
331C22—62no—
332C22—92no—
333C22—181no—
334C22—181no—
335C22C22×D3211yescharacteristic
336C4—22no—
337C4—32no—
338C4—61no—
339C4—62no—
340C4—61no—
341C4—181no—
342C4—182no—
343C4#379812yescomplemented
344C3—21no—
345C3#550711yescharacteristic, complemented
346C3#555111yescharacteristic, complemented
347C2—62no—
348C2—122no—
349C2—182no—
350C2C2×#379811yescharacteristic
351C2D32×D411yescharacteristic
352C2#6431511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
hypercenterC4⋊C4D32
hypocenterC32#547
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#3798C2semidirect
2. C4⋊C4×C3⋊D3C2semidirect
3. C3×#893C2semidirect
4. C3×D3×C4⋊C4C2semidirect
5. #63918C2semidirect
6. #63916C2semidirect
7. #63908C2semidirect
8. #63905C2semidirect
9. C4×#626C2semidirect
10. #63853C2semidirect
11. #63851C2semidirect
51. #893D3semidirect
55. D3×C4⋊C4D3semidirect
80. #626D4semidirect
147. C32×C4#41semidirect
276. C3×C4#235semidirect
277. C3×C4#235semidirect
288. C32#547semidirect
343. C4#3798semidirect
345. C3#5507semidirect
346. C3#5551semidirect

Maximal subgroups

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

subgroupindex
C2×#37982
C4⋊C4×C3⋊D32
C3×#8932
C3×D3×C4⋊C42
#639182
#639162
#639082
#639052
C4×#6262
#638532
#638512
#55513
#55073