Finite Group Databasebeta

← #10594991

Subgroups — #10594991

Order 576.

503 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#638C211yesmaximal, characteristic
2C2×C3×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C3×#924C211yesmaximal, characteristic, complemented
4#63964C212yesmaximal, complemented
5#63952C212yesmaximal
6#63919C211yesmaximal, characteristic, complemented
7#63900C211yesmaximal, characteristic, complemented
8#63895C212yesmaximal, complemented
9#63871C211yesmaximal, characteristic, complemented
10#63847C212yesmaximal, complemented
11#63843C211yesmaximal, characteristic, complemented
12#5635—31nomaximal
13#5511—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C2×#638C2211yescharacteristic
16C2×C4×C3⋊D3C2211yescharacteristic
17C22×C3×Dic3C2212yes—
18C2×C3×Dic6C2211yescharacteristic
19C2×#627C2212yes—
20C2×#624C2212yes—
21#3757C2211yescharacteristic
22#3752C2211yescharacteristic
23#3750C2211yescharacteristic
24C4×C3⋊Dic3C2211yescharacteristic
25C3×#217C2212yes—
26C3×C4⋊Dic3C2211yescharacteristic
27C3×C4⋊Dic3C2211yescharacteristic
28C3×C4⋊Dic3C2212yes—
29C3×C4⋊Dic3C2211yescharacteristic
30C3×Dic3⋊C4C2212yes—
31#3724C2212yes—
32#3724C2212yes—
33Dic3⋊Dic3C2212yes—
34Dic3⋊Dic3C2212yes—
35#3722C2212yes—
36Dic32C2212yes—
37C3×C22⋊Q8—31no—
38C2×C3×C4⋊C4—31no—
39#946—31no—
40Q8×Dic3—31no—
41C4×#67—31no—
42C4×#67—31no—
43C4×#67—32no—
44C4×#67—61no—
45C2×C4⋊Dic3D311yescharacteristic, complemented
46#924—32no—
47#924D311yescharacteristic, complemented
48#897—31no—
49#897—32no—
50#896—32no—
51D3×C4⋊C4—31no—
52D3×C4⋊C4—31no—
53#889—32no—
54#889—32no—
55#888—31no—
56#888—31no—
57#887—31no—
58#67⋊C4—32no—
59#878—32no—
60C23×C32C2311yescharacteristic
61C22×C3⋊D3C2311yescharacteristic
62C2×C32×C4—22no—
63C2×C32×C4C2311yescharacteristic
64C2×C32×C4C2311yescharacteristic
65#638Q814yescomplemented
66C2×C3⋊Dic3C2311yescharacteristic
67C2×C3⋊Dic3C2311yescharacteristic
68C2×C3⋊Dic3C2311yescharacteristic
69C4×C3⋊D3—22no—
70C2×C3×Dic3—42no—
71C2×C3×Dic3C2312yes—
72C2×C3×Dic3C2312yes—
73C2×C3×Dic3C2312yes—
74C2×C3×Dic3C2312yes—
75C3×Dic6—22no—
76#627—42no—
77#624—42no—
78#554—91noSylow
79C2×C3×Q8—31no—
80C22×C3×C4—21no—
81C22×C3×C4—32no—
82C22×C3×C4C2×D311yescharacteristic
83C22×C3×C4C2×D311yescharacteristic
84C2×#67—31no—
85C2×#67—31no—
86C2×#67—61no—
87C22×Dic3C2×D312yes—
88C2×C4×D3—31no—
89C2×C4×D3—31no—
90C2×C4×D3—32no—
91C2×C4×D3—32no—
92C2×C4×D3—61no—
93C2×Dic6—32no—
94C2×Dic6—32no—
95C2×Dic6C2×D311yescharacteristic
96C3×C4⋊C4—31no—
97C3×C4⋊C4—31no—
98C3×C4⋊C4—31no—
99C3×C4⋊C4—32no—
100C3×C4⋊C4—32no—
101C3×C22⋊C4—32no—
102#217—31no—
103#217—31no—
104#217—61no—
105#217C2×D312yes—
106(C2×D3)⋊C4—31no—
107(C2×D3)⋊C4—31no—
108(C2×D3)⋊C4—32no—
109(C2×D3)⋊C4—32no—
110(C2×D3)⋊C4—61no—
111C4⋊Dic3—32no—
112C4⋊Dic3—32no—
113C4⋊Dic3C2×D311yescharacteristic
114C4⋊Dic3C2×D311yescharacteristic
115C4⋊Dic3C2×D312yes—
116C4⋊Dic3C2×D311yescharacteristic
117Dic3⋊C4—31no—
118Dic3⋊C4—31no—
119Dic3⋊C4—32no—
120Dic3⋊C4—32no—
121Dic3⋊C4—32no—
122Dic3⋊C4—32no—
123Dic3⋊C4—32no—
124Dic3⋊C4—32no—
125Dic3⋊C4—61no—
126Dic3⋊C4C2×D312yes—
127C4×Dic3—31no—
128C4×Dic3—31no—
129C4×Dic3—32no—
130C4×Dic3—32no—
131C4×Dic3—61no—
132C22×C32—21no—
133C22×C32—21no—
134C22×C32C2×Q812yes—
135C22×C32C2411yescharacteristic
136C2×C3⋊D3—21no—
137C2×C3⋊D3—21no—
138C2×C3⋊D3C2×Q812yes—
139C32×C4—21no—
140C32×C4#4112yes—
141C3⋊Dic3—21no—
142C3⋊Dic3—21no—
143C3⋊Dic3C2×Q812yes—
144C3×Dic3—22no—
145C3×Dic3—22no—
146C3×Dic3—22no—
147C3×Dic3—22no—
148C4⋊Q8—91no—
149#125—92no—
150C22⋊Q8—31no—
151C22⋊Q8—92no—
152C22⋊Q8—92no—
153C22⋊Q8—91no—
154C4×Q8—91no—
155C4×D4—92no—
156C4×D4—91no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—91no—
159C23×C3—21no—
160C23×C3C22×D311yescharacteristic
161C23×C3C22×D311yescharacteristic
162C22×D3—31no—
163C22×D3—31no—
164C22×D3—61no—
165C3×Q8—62no—
166C2×C3×C4—22no—
167C2×C3×C4—22no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—32no—
171C2×C3×C4—32no—
172C2×C3×C4—32no—
173C2×C3×C4—32no—
174C2×C3×C4—42no—
175C2×C3×C4—122no—
176C2×C3×C4C22×D311yescharacteristic
177C2×C3×C4C22×D311yescharacteristic
178C2×C3×C4C22×D311yescharacteristic
179C2×C3×C4C22×D311yescharacteristic
180#67—34no—
181#67—34no—
182#67—64no—
183C2×Dic3—31no—
184C2×Dic3—31no—
185C2×Dic3—31no—
186C2×Dic3—31no—
187C2×Dic3—31no—
188C2×Dic3—31no—
189C2×Dic3—42no—
190C2×Dic3—61no—
191C2×Dic3—61no—
192C2×Dic3—61no—
193C2×Dic3C22×D312yes—
194C2×Dic3C22×D312yes—
195C2×Dic3C22×D312yes—
196C2×Dic3C22×D312yes—
197C4×D3—62no—
198C4×D3—62no—
199C4×D3—122no—
200C4×D3—122no—
201C4×D3—122no—
202Dic6—22no—
203Dic6—122no—
204Dic6—122no—
205C2×C32—22no—
206C2×C32C22×Q811yescharacteristic
207C2×C32C2×#4111yescharacteristic
208C2×C32H(5,𝔽2)11yescharacteristic
209C3⋊D3—22no—
210C2×Q8—31no—
211C2×Q8—92no—
212C2×D4—91no—
213C22×C4—32no—
214C22×C4—92no—
215C22×C4—91no—
216C22×C4D3211yescharacteristic
217C4⋊C4—31no—
218C4⋊C4—32no—
219C4⋊C4—31no—
220C4⋊C4—31no—
221C4⋊C4—32no—
222C4⋊C4—92no—
223C4⋊C4—92no—
224C4⋊C4—92no—
225C4⋊C4—92no—
226C4⋊C4—91no—
227C22⋊C4—32no—
228C22⋊C4—92no—
229C22⋊C4—91no—
230C22⋊C4—91no—
231C42—92no—
232C42—91no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—22no—
239C22×C3—41no—
240C22×C3—41no—
241C22×C3D3×Q812yes—
242C22×C3D3×Q812yes—
243C22×C3C23×D311yescharacteristic
244C22×C3C23×D311yescharacteristic
245C2×D3—32no—
246C2×D3—32no—
247C2×D3—62no—
248C2×D3—61no—
249C2×D3—61no—
250C2×D3—61no—
251C2×D3—61no—
252C2×D3—121no—
253C2×D3—121no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—22no—
257C3×C4—41no—
258C3×C4—62no—
259C3×C4—62no—
260C3×C4—62no—
261C3×C4—62no—
262C3×C4#23512yes—
263C3×C4#23712yes—
264Dic3—22no—
265Dic3—22no—
266Dic3—22no—
267Dic3—22no—
268Dic3—32no—
269Dic3—32no—
270Dic3—62no—
271Dic3—61no—
272Dic3—61no—
273Dic3—61no—
274Dic3—61no—
275Dic3—121no—
276Dic3—121no—
277C32#55411yescharacteristic, complemented, Sylow
278C23—91no—
279C23C2×D3211yescharacteristic
280Q8—62no—
281Q8—362no—
282D4—94no—
283C2×C4—22no—
284C2×C4—32no—
285C2×C4—32no—
286C2×C4—32no—
287C2×C4—32no—
288C2×C4—91no—
289C2×C4—91no—
290C2×C4—91no—
291C2×C4—122no—
292C2×C4—182no—
293C2×C4—362no—
294C2×C4C2×D3211yescharacteristic
295C2×C4C2×D3211yescharacteristic
296C2×C3—22no—
297C2×C3—22no—
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—42no—
302C2×C3C2×#23511yescharacteristic
303C2×C3C2×#23711yescharacteristic
304C2×C3#100411yescharacteristic
305C2×C3C2×D3×Q811yescharacteristic
306C2×C3C2×D3×Q811yescharacteristic
307C2×C3#100911yescharacteristic
308D3—62no—
309D3—62no—
310D3—122no—
311C22—21no—
312C22—21no—
313C22—92no—
314C22—181no—
315C22—181no—
316C22#379712yes—
317C22C22×D3211yescharacteristic
318C4—21no—
319C4—62no—
320C4—62no—
321C4—62no—
322C4—62no—
323C4—92no—
324C4—181no—
325C4—181no—
326C4#380412yes—
327C3—21no—
328C3#551111yescharacteristic, complemented
329C3#563511yescharacteristic, complemented
330C2—22no—
331C2—182no—
332C2C2×#379711yescharacteristic
333C2#6430711yescharacteristic
334C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radicalC2×#638C22
hypercenterC22×C4D32
hypocenterC32#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊Dic3C2semidirect
3. C3×#924C2semidirect
4. #63964C2semidirect
6. #63919C2semidirect
7. #63900C2semidirect
8. #63895C2semidirect
9. #63871C2semidirect
10. #63847C2semidirect
11. #63843C2semidirect
45. C2×C4⋊Dic3D3semidirect
47. #924D3semidirect
65. #638Q8semidirect
277. C32#554semidirect
328. C3#5511semidirect
329. C3#5635semidirect

Maximal subgroups

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

subgroupindex
C4×#6382
C2×C3×C4⋊Dic32
C3×#9242
#639642
#639522
#639192
#639002
#638952
#638712
#638472
#638432
#56353
#55113