Finite Group Databasebeta

← #10593981

Subgroups — #10593981

Order 576.

507 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C4×#634C211yesmaximal, characteristic, complemented
2C3×#875C211yesmaximal, characteristic, complemented
3C3×C4×D12C211yesmaximal, characteristic, complemented
4#63914C211yesmaximal, characteristic
5#63913C212yesmaximal, complemented
6#63905C212yesmaximal, complemented
7#63898C211yesmaximal, characteristic, complemented
8#63891C212yesmaximal, complemented
9#63879C212yesmaximal, complemented
10#63863C211yesmaximal, characteristic, complemented
11#63843C211yesmaximal, characteristic, complemented
12#5681—31nomaximal
13#5559—31nomaximal
14C2×#634C2211yescharacteristic
15C2×C3×D12C2211yescharacteristic
16C2×C3×D12C2211yescharacteristic
17C2×C3×C4×D3C2212yes—
18C2×C3×Dic6C2211yescharacteristic
19C2×#625C2212yes—
20C2×#625C2212yes—
21C2×D3×Dic3C2212yes—
22C32×C42C2211yescharacteristic
23C4⋊(C3⋊Dic3)C2211yescharacteristic
24#3750C2211yescharacteristic
25#3750C2211yescharacteristic
26C4×C3⋊Dic3C2211yescharacteristic
27C4×C3⋊Dic3C2211yescharacteristic
28C3×(C2×D3)⋊C4C2212yes—
29C3×(C2×D3)⋊C4C2212yes—
30C3×(C2×D3)⋊C4C2212yes—
31C3×C4⋊Dic3C2211yescharacteristic
32Dic3⋊Dic3C2212yes—
33(C2×D3)⋊Dic3C2212yes—
34(C2×D3)⋊Dic3C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36Dic32C2212yes—
37C3×#124—31no—
38C3×C4×D4—31no—
39#947—31no—
40Q8×Dic3—31no—
41#940—31no—
42#940—31no—
43#936—31no—
44D4×Dic3—31no—
45#929—32no—
46C4×#67—32no—
47#893—31no—
48#887—31no—
49#886—32no—
50#885—32no—
51#885—32no—
52#883—32no—
53#883—32no—
54#67⋊C4—32no—
55#875D311yescharacteristic, complemented
56C4×D12D311yescharacteristic, complemented
57C4×Dic6—31no—
58C4×Dic6—31no—
59C4×Dic6—61no—
60C22×C3×D3C2312yes—
61C22×C3×D3C2312yes—
62C2×C32×C4C2311yescharacteristic
63C2×C32×C4C2311yescharacteristic
64C2×C32×C4C2311yescharacteristic
65C2×C3⋊Dic3C2311yescharacteristic
66C2×C3⋊Dic3C2311yescharacteristic
67C2×C3⋊Dic3C2311yescharacteristic
68C2×C3⋊Dic3C2311yescharacteristic
69#634—22no—
70C2×C3×Dic3C2312yes—
71C2×C3×Dic3C2312yes—
72C3×D12—22no—
73C3×D12—22no—
74C3×C4×D3—42no—
75C3×Dic6—22no—
76#625—24no—
77#625—42no—
78D3×Dic3—42no—
79#559—91noSylow
80C2×C3×Q8—31no—
81C2×C3×D4—31no—
82C2×C3×D4—31no—
83C22×C3×C4—32no—
84C2×#67—32no—
85C2×#67—32no—
86C2×#67—32no—
87C2×#67—32no—
88C22×Dic3—32no—
89C2×D12C2×D311yescharacteristic
90C2×D12C2×D311yescharacteristic
91C2×C4×D3—32no—
92C2×C4×D3C2×D312yes—
93C2×Dic6—31no—
94C2×Dic6—31no—
95C2×Dic6—61no—
96C2×Dic6C2×D311yescharacteristic
97C3×C4⋊C4—31no—
98C3×C22⋊C4—32no—
99C3×C22⋊C4—32no—
100C3×C22⋊C4—32no—
101C3×C42—21no—
102C3×C42C2×D311yescharacteristic
103C3×C42C2×D311yescharacteristic
104#217—32no—
105#217—32no—
106#217—32no—
107(C2×D3)⋊C4—32no—
108(C2×D3)⋊C4—32no—
109(C2×D3)⋊C4—32no—
110(C2×D3)⋊C4C2×D312yes—
111(C2×D3)⋊C4C2×D312yes—
112(C2×D3)⋊C4C2×D312yes—
113C4⋊Dic3—31no—
114C4⋊Dic3—31no—
115C4⋊Dic3—32no—
116C4⋊Dic3—61no—
117C4⋊Dic3C2×D311yescharacteristic
118Dic3⋊C4—31no—
119Dic3⋊C4—31no—
120Dic3⋊C4—31no—
121Dic3⋊C4—31no—
122Dic3⋊C4—32no—
123Dic3⋊C4—61no—
124Dic3⋊C4—61no—
125C4×Dic3—31no—
126C4×Dic3—31no—
127C4×Dic3—31no—
128C4×Dic3—31no—
129C4×Dic3—32no—
130C4×Dic3—32no—
131C4×Dic3—61no—
132C4×Dic3—61no—
133C22×C32C2411yescharacteristic
134C2×C3×D3—22no—
135C2×C3×D3—22no—
136C2×C3×D3—22no—
137C2×C3×D3—22no—
138C2×C3×D3—22no—
139C2×C3×D3—22no—
140C32×C4—21no—
141C32×C4—21no—
142C32×C4#4112yes—
143C3⋊Dic3—21no—
144C3⋊Dic3—21no—
145C3⋊Dic3—21no—
146C3⋊Dic3#4112yes—
147C3×Dic3—22no—
148C3×Dic3—22no—
149#127—91no—
150#124—31no—
151#124—92no—
152#124—91no—
153#123—92no—
154#123—92no—
155C4×Q8—91no—
156C4×Q8—91no—
157C4×D4—31no—
158C4×D4—92no—
159C4×D4—91no—
160C23×C3—32no—
161C23×C3—32no—
162C22×D3C22×D312yes—
163C22×D3C22×D312yes—
164C3×Q8—62no—
165C3×D4—62no—
166C3×D4—62no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—32no—
171C2×C3×C4—32no—
172C2×C3×C4—122no—
173C2×C3×C4C22×D311yescharacteristic
174C2×C3×C4C22×D311yescharacteristic
175C2×C3×C4C22×D311yescharacteristic
176C2×C3×C4C22×D311yescharacteristic
177C2×C3×C4C22×D311yescharacteristic
178C2×C3×C4C22×D311yescharacteristic
179#67—64no—
180#67—64no—
181#67—122no—
182#67—122no—
183C2×Dic3—31no—
184C2×Dic3—31no—
185C2×Dic3—31no—
186C2×Dic3—31no—
187C2×Dic3—31no—
188C2×Dic3—31no—
189C2×Dic3—31no—
190C2×Dic3—31no—
191C2×Dic3—61no—
192C2×Dic3—61no—
193C2×Dic3—61no—
194C2×Dic3—61no—
195C2×Dic3—122no—
196C2×Dic3C22×D312yes—
197C2×Dic3C22×D312yes—
198D12—22no—
199D12—22no—
200C4×D3—42no—
201C4×D3—122no—
202Dic6—22no—
203Dic6—62no—
204Dic6—62no—
205Dic6—64no—
206C2×C32C2×#4111yescharacteristic
207C2×C32C2×#4111yescharacteristic
208C2×C32H(5,𝔽2)11yescharacteristic
209C3×D3—42no—
210C3×D3—42no—
211C2×Q8—31no—
212C2×Q8—91no—
213C2×D4—31no—
214C2×D4—31no—
215C2×D4—92no—
216C2×D4—92no—
217C22×C4—32no—
218C22×C4—92no—
219C4⋊C4—31no—
220C4⋊C4—92no—
221C4⋊C4—91no—
222C4⋊C4—91no—
223C4⋊C4—91no—
224C22⋊C4—32no—
225C22⋊C4—32no—
226C22⋊C4—32no—
227C22⋊C4—92no—
228C22⋊C4—92no—
229C22⋊C4—92no—
230C42—92no—
231C42—91no—
232C42—91no—
233C42D3211yescharacteristic
234C22×C3—21no—
235C22×C3—62no—
236C22×C3—62no—
237C22×C3—62no—
238C22×C3—62no—
239C22×C3—62no—
240C22×C3—62no—
241C22×C3C23×D311yescharacteristic
242C22×C3C23×D311yescharacteristic
243C2×D3—22no—
244C2×D3—22no—
245C2×D3—22no—
246C2×D3—22no—
247C2×D3—22no—
248C2×D3—22no—
249C3×C4—22no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—22no—
255C3×C4—41no—
256C3×C4—62no—
257C3×C4—62no—
258C3×C4#23512yes—
259C3×C4#23712yes—
260Dic3—22no—
261Dic3—22no—
262Dic3—32no—
263Dic3—32no—
264Dic3—62no—
265Dic3—62no—
266Dic3—61no—
267Dic3—61no—
268Dic3—61no—
269Dic3—61no—
270Dic3—61no—
271Dic3—61no—
272Dic3—121no—
273Dic3—121no—
274C32#55911yescharacteristic, complemented, Sylow
275C23—32no—
276C23—32no—
277Q8—62no—
278Q8—182no—
279D4—62no—
280D4—62no—
281D4—184no—
282D4—362no—
283C2×C4—32no—
284C2×C4—32no—
285C2×C4—91no—
286C2×C4—91no—
287C2×C4—91no—
288C2×C4—91no—
289C2×C4—122no—
290C2×C4—362no—
291C2×C4C2×D3211yescharacteristic
292C2×C4C2×D3211yescharacteristic
293C2×C4C2×D3211yescharacteristic
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—122no—
298C2×C3—122no—
299C2×C3C2×#23511yescharacteristic
300C2×C3C2×#23711yescharacteristic
301C2×C3#100411yescharacteristic
302C2×C3D3×#4111yescharacteristic
303C2×C3D3×#4111yescharacteristic
304C2×C3#100911yescharacteristic
305D3—42no—
306D3—42no—
307C22—62no—
308C22—62no—
309C22—62no—
310C22—62no—
311C22—62no—
312C22—62no—
313C22C22×D3211yescharacteristic
314C4—21no—
315C4—21no—
316C4—62no—
317C4—62no—
318C4—92no—
319C4—181no—
320C4—181no—
321C4—181no—
322C4#379512yes—
323C3—21no—
324C3#555911yescharacteristic, complemented
325C3#568111yescharacteristic, complemented
326C2—122no—
327C2—122no—
328C2C2×#379511yescharacteristic
329C2#6430611yescharacteristic
330C2#6430711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C42C22
prime radical#63914C2
hypercenterC42D32
hypocenterC32#559
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#634C2semidirect
2. C3×#875C2semidirect
3. C3×C4×D12C2semidirect
5. #63913C2semidirect
6. #63905C2semidirect
7. #63898C2semidirect
8. #63891C2semidirect
9. #63879C2semidirect
10. #63863C2semidirect
11. #63843C2semidirect
55. #875D3semidirect
56. C4×D12D3semidirect
274. C32#559semidirect
324. C3#5559semidirect
325. C3#5681semidirect

Maximal subgroups

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

subgroupindex
C4×#6342
C3×#8752
C3×C4×D122
#639142
#639132
#639052
#638982
#638912
#638792
#638632
#638432
#56813
#55593