Finite Group Databasebeta

← #10594559

Subgroups — #10594559

Order 576.

523 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 132 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#64102C211yesmaximal, characteristic, complemented
2C3×#892C211yesmaximal, characteristic, complemented
3C3×D3×C4⋊C4C211yesmaximal, characteristic, complemented
4#63909C211yesmaximal, characteristic, complemented
5#63909C211yesmaximal, characteristic, complemented
6#63906C212yesmaximal, complemented
7C4×#626C212yesmaximal, complemented
8C4×#625C211yesmaximal, characteristic, complemented
9#63872C211yesmaximal, characteristic, complemented
10#63868C212yesmaximal, complemented
11#63853C212yesmaximal, complemented
12#5530—31nomaximal
13#5514—31nomaximal
14C2×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C4×C3⋊D3C2212yes—
16C2×C3×C4×D3C2211yescharacteristic
17C2×C3×C4×D3C2211yescharacteristic
18C2×C3×C4×D3C2212yes—
19C2×#626C2212yes—
20C2×#626C2212yes—
21C2×#625C2211yescharacteristic
22C32×C4⋊C4C2211yescharacteristic
23#3752C2212yes—
24C4×C3⋊Dic3C2211yescharacteristic
25C3×(C2×D3)⋊C4C2212yes—
26C3×C4⋊Dic3C2211yescharacteristic
27C3×C4⋊Dic3C2211yescharacteristic
28C3×Dic3⋊C4C2212yes—
29C3×C4×Dic3C2212yes—
30#3724C2211yescharacteristic
31Dic3⋊Dic3C2212yes—
32Dic3⋊Dic3C2212yes—
33#3722C2212yes—
34#3722C2212yes—
35(C2×D3)⋊Dic3C2211yescharacteristic
36(C2×D3)⋊Dic3C2211yescharacteristic
37C3×#117—31no—
38C2×C3×C4⋊C4—31no—
39#930—32no—
40#930—31no—
41#930—31no—
42C4×#67—31no—
43C4×#67—31no—
44C4×#67—32no—
45#898—32no—
46#895—31no—
47#895—31no—
48#894—32no—
49#893—31no—
50#893—31no—
51#893—61no—
52#892D311yescharacteristic, complemented
53D3×C4⋊C4D311yescharacteristic, complemented
54#890—31no—
55#890—31no—
56#884—32no—
57#883—32no—
58#876—32no—
59C4×D12—32no—
60C22×C3⋊D3C2312yes—
61C22×C3×D3C2311yescharacteristic
62C22×C3×D3C2311yescharacteristic
63C2×C32×C4C2312yes—
64C2×C32×C4C2311yescharacteristic
65C2×C3⋊Dic3C2311yescharacteristic
66C2×C3⋊Dic3C2311yescharacteristic
67C4⋊(C3⋊D3)—22no—
68C4×C3⋊D3—42no—
69C2×C3×Dic3C2311yescharacteristic
70C2×C3×Dic3C2311yescharacteristic
71C2×C3×Dic3C2312yes—
72C2×C3×Dic3C2312yes—
73C3×C4×D3—22no—
74C3×C4×D3—22no—
75C3×C4×D3—42no—
76#626—24no—
77#626—42no—
78#625—22no—
79#553—91noSylow
80C22×C3×C4—31no—
81C22×C3×C4—31no—
82C22×C3×C4—32no—
83C2×#67—31no—
84C2×#67—31no—
85C2×#67—32no—
86C2×#67—32no—
87C2×D12—32no—
88C2×D12—31no—
89C2×D12—31no—
90C2×D12—32no—
91C2×D12—61no—
92C2×C4×D3—32no—
93C2×C4×D3—32no—
94C2×C4×D3—62no—
95C2×C4×D3C2×D311yescharacteristic
96C2×C4×D3C2×D311yescharacteristic
97C2×C4×D3C2×D312yes—
98C3×C4⋊C4—21no—
99C3×C4⋊C4—31no—
100C3×C4⋊C4—31no—
101C3×C4⋊C4—32no—
102C3×C4⋊C4C2×D311yescharacteristic
103C3×C4⋊C4C2×D311yescharacteristic
104C3×C22⋊C4—32no—
105C3×C42—32no—
106#217—31no—
107#217—31no—
108(C2×D3)⋊C4—31no—
109(C2×D3)⋊C4—31no—
110(C2×D3)⋊C4—32no—
111(C2×D3)⋊C4—32no—
112(C2×D3)⋊C4—32no—
113(C2×D3)⋊C4—32no—
114(C2×D3)⋊C4—32no—
115(C2×D3)⋊C4—32no—
116(C2×D3)⋊C4—62no—
117(C2×D3)⋊C4C2×D312yes—
118C4⋊Dic3—32no—
119C4⋊Dic3—32no—
120C4⋊Dic3C2×D311yescharacteristic
121C4⋊Dic3C2×D311yescharacteristic
122Dic3⋊C4—31no—
123Dic3⋊C4—32no—
124Dic3⋊C4—32no—
125Dic3⋊C4—31no—
126Dic3⋊C4C2×D312yes—
127C4×Dic3—31no—
128C4×Dic3—31no—
129C4×Dic3—61no—
130C4×Dic3C2×D312yes—
131C22×C32C2411yescharacteristic
132C2×C3⋊D3—22no—
133C2×C3⋊D3—22no—
134C2×C3⋊D3—22no—
135C2×C3×D3—21no—
136C2×C3×D3—21no—
137C2×C3×D3—21no—
138C2×C3×D3—21no—
139C2×C3×D3—21no—
140C2×C3×D3#4112yes—
141C32×C4—22no—
142C32×C4#4112yes—
143C3⋊Dic3—21no—
144C3⋊Dic3—21no—
145C3×Dic3—22no—
146C3×Dic3—21no—
147C3×Dic3—21no—
148C3×Dic3—22no—
149#126—92no—
150#125—91no—
151#123—92no—
152#121—92no—
153#121—91no—
154#121—91no—
155C4×D4—92no—
156C4×D4—91no—
157C4×D4—91no—
158#117—31no—
159C2×C4⋊C4—31no—
160C23×C3—31no—
161C23×C3—31no—
162C22×D3—32no—
163C22×D3—32no—
164C22×D3—62no—
165C22×D3C22×D311yescharacteristic
166C22×D3C22×D311yescharacteristic
167C2×C3×C4—21no—
168C2×C3×C4—22no—
169C2×C3×C4—32no—
170C2×C3×C4—32no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—62no—
174C2×C3×C4—62no—
175C2×C3×C4—122no—
176C2×C3×C4C22×D312yes—
177C2×C3×C4C22×D312yes—
178C2×C3×C4C22×D311yescharacteristic
179C2×C3×C4C22×D311yescharacteristic
180#67—64no—
181#67—62no—
182#67—62no—
183#67—122no—
184C2×Dic3—31no—
185C2×Dic3—31no—
186C2×Dic3—31no—
187C2×Dic3—31no—
188C2×Dic3—61no—
189C2×Dic3—61no—
190C2×Dic3C22×D311yescharacteristic
191C2×Dic3C22×D312yes—
192C2×Dic3C22×D311yescharacteristic
193C2×Dic3C22×D312yes—
194D12—64no—
195D12—64no—
196D12—62no—
197D12—62no—
198D12—122no—
199C4×D3—22no—
200C4×D3—22no—
201C4×D3—42no—
202C4×D3—122no—
203C4×D3—122no—
204C4×D3—124no—
205C2×C32C2×#4111yescharacteristic
206C2×C32C2×#4111yescharacteristic
207C2×C32H(5,𝔽2)11yescharacteristic
208C3⋊D3—42no—
209C3×D3—22no—
210C3×D3—41no—
211C2×D4—92no—
212C2×D4—92no—
213C2×D4—91no—
214C2×D4—91no—
215C22×C4—31no—
216C22×C4—31no—
217C22×C4—32no—
218C22×C4—92no—
219C4⋊C4—31no—
220C4⋊C4—32no—
221C4⋊C4—31no—
222C4⋊C4—92no—
223C4⋊C4—91no—
224C4⋊C4—92no—
225C4⋊C4D3211yescharacteristic
226C22⋊C4—32no—
227C22⋊C4—92no—
228C22⋊C4—92no—
229C22⋊C4—91no—
230C22⋊C4—91no—
231C22⋊C4—92no—
232C42—32no—
233C42—91no—
234C22×C3—21no—
235C22×C3—32no—
236C22×C3—61no—
237C22×C3—61no—
238C22×C3—61no—
239C22×C3—61no—
240C22×C3—61no—
241C22×C3C23×D311yescharacteristic
242C22×C3C23×D311yescharacteristic
243C2×D3—21no—
244C2×D3—21no—
245C2×D3—21no—
246C2×D3—21no—
247C2×D3—21no—
248C2×D3—64no—
249C2×D3—62no—
250C2×D3—62no—
251C2×D3—62no—
252C2×D3—62no—
253C2×D3—62no—
254C2×D3—62no—
255C2×D3—122no—
256C2×D3—122no—
257C2×D3#23912yes—
258C3×C4—22no—
259C3×C4—22no—
260C3×C4—22no—
261C3×C4—42no—
262C3×C4—61no—
263C3×C4—61no—
264C3×C4—62no—
265C3×C4—62no—
266C3×C4#23512yes—
267C3×C4#23512yes—
268Dic3—21no—
269Dic3—21no—
270Dic3—22no—
271Dic3—22no—
272Dic3—62no—
273Dic3—61no—
274Dic3—61no—
275Dic3—61no—
276Dic3—61no—
277Dic3—121no—
278C32#55311yescharacteristic, complemented, Sylow
279C23—31no—
280C23—31no—
281C23—92no—
282D4—184no—
283D4—182no—
284D4—182no—
285D4—362no—
286C2×C4—31no—
287C2×C4—32no—
288C2×C4—31no—
289C2×C4—32no—
290C2×C4—62no—
291C2×C4—62no—
292C2×C4—91no—
293C2×C4—91no—
294C2×C4—122no—
295C2×C4—362no—
296C2×C4C2×D3212yes—
297C2×C4C2×D3211yescharacteristic
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—62no—
302C2×C3—121no—
303C2×C3C2×#23511yescharacteristic
304C2×C3C2×#23511yescharacteristic
305C2×C3#100411yescharacteristic
306C2×C3C2×#23911yescharacteristic
307C2×C3D3×#4111yescharacteristic
308C2×C3#100911yescharacteristic
309D3—22no—
310D3—41no—
311D3—124no—
312D3—122no—
313D3—122no—
314C22—32no—
315C22—61no—
316C22—61no—
317C22—61no—
318C22—61no—
319C22—61no—
320C22—182no—
321C22—182no—
322C22—182no—
323C22C22×D3211yescharacteristic
324C4—22no—
325C4—61no—
326C4—61no—
327C4—62no—
328C4—62no—
329C4—181no—
330C4—181no—
331C4#379812yes—
332C3—21no—
333C3#551411yescharacteristic, complemented
334C3#553011yescharacteristic, complemented
335C2—62no—
336C2—121no—
337C2—362no—
338C2C2×#379811yescharacteristic
339C2#6431411yescharacteristic
340C2D3×#23911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
hypercenterC4⋊C4D32
hypocenterC32#553
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64102C2semidirect
2. C3×#892C2semidirect
3. C3×D3×C4⋊C4C2semidirect
4. #63909C2semidirect
5. #63909C2semidirect
6. #63906C2semidirect
7. C4×#626C2semidirect
8. C4×#625C2semidirect
9. #63872C2semidirect
10. #63868C2semidirect
11. #63853C2semidirect
52. #892D3semidirect
53. D3×C4⋊C4D3semidirect
278. C32#553semidirect
333. C3#5514semidirect
334. C3#5530semidirect

Maximal subgroups

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

subgroupindex
#641022
C3×#8922
C3×D3×C4⋊C42
#639092
#639062
C4×#6262
C4×#6252
#638722
#638682
#638532
#55303
#55143