Finite Group Databasebeta

← #10595249

Subgroups — #10595249

Order 576.

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

normal subgroups only

This group has 519 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#64147C211yesmaximal, characteristic, complemented
2C3×#938C211yesmaximal, characteristic, complemented
3C3×D4×Dic3C211yesmaximal, characteristic, complemented
4#63978C212yesmaximal, complemented
5#63959C212yesmaximal, complemented
6#63955C212yesmaximal, complemented
7#63953C212yesmaximal, complemented
8#63918C211yesmaximal, characteristic, complemented
9C4×#626C211yesmaximal, characteristic, complemented
10#63849C211yesmaximal, characteristic, complemented
11#63841C211yesmaximal, characteristic, complemented
12#5616—31nomaximal
13#5563—31nomaximal
14C2×C32×D4C2211yescharacteristic
15C2×#638C2212yes—
16C2×C4×C3⋊D3C2211yescharacteristic
17C2×C3×#67C2212yes—
18C22×C3×Dic3C2212yes—
19C2×C3×C4×D3C2211yescharacteristic
20C2×#627C2212yes—
21C2×#626C2211yescharacteristic
22#3757C2212yes—
23C4⋊(C3⋊Dic3)C2211yescharacteristic
24C3×#217C2212yes—
25C3×#217C2212yes—
26C3×C4⋊Dic3C2211yescharacteristic
27C3×C4⋊Dic3C2211yescharacteristic
28C3×C4×Dic3C2211yescharacteristic
29Dic3⋊Dic3C2211yescharacteristic
30Dic3⋊Dic3C2212yes—
31#3722C2212yes—
32#3722C2211yescharacteristic
33(C2×D3)⋊Dic3C2212yes—
34(C2×D3)⋊Dic3C2211yescharacteristic
35Dic32C2212yes—
36Dic32C2212yes—
37C3×#121—31no—
38C3×C4×D4—31no—
39#939—32no—
40#938—31no—
41#938—31no—
42#938—61no—
43#938D311yescharacteristic, complemented
44#936—32no—
45D4×Dic3D311yescharacteristic, complemented
46#930—32no—
47C4×#67—31no—
48#926—32no—
49#926—31no—
50#892—31no—
51#892—31no—
52#892—31no—
53#888—31no—
54#885—32no—
55#885—32no—
56#885—32no—
57#877—32no—
58C4×D12—31no—
59#869—31no—
60C23×C32C2312yes—
61C22×C3⋊D3C2311yescharacteristic
62C22×C3×D3C2311yescharacteristic
63C32×D4—22no—
64C2×C32×C4C2311yescharacteristic
65#638—42no—
66C2×C3⋊Dic3C2311yescharacteristic
67C2×C3⋊Dic3C2312yes—
68C4×C3⋊D3—22no—
69C3×#67—42no—
70C2×C3×Dic3—24no—
71C2×C3×Dic3C2312yes—
72C2×C3×Dic3C2311yescharacteristic
73C2×C3×Dic3C2312yes—
74C2×C3×Dic3C2311yescharacteristic
75C2×C3×Dic3C2311yescharacteristic
76C3×C4×D3—22no—
77#627—42no—
78#626—22no—
79#555—91noSylow
80C2×C3×D4—21no—
81C2×C3×D4—32no—
82C2×C3×D4C2×D311yescharacteristic
83C2×C3×D4C2×D311yescharacteristic
84C22×C3×C4—32no—
85C22×C3×C4—31no—
86C2×#67—32no—
87C2×#67—32no—
88C2×#67—31no—
89C2×#67—62no—
90C2×#67C2×D312yes—
91C22×Dic3C2×D312yes—
92C2×D12—31no—
93C2×C4×D3—31no—
94C2×C4×D3—31no—
95C2×C4×D3—61no—
96C2×C4×D3C2×D311yescharacteristic
97C2×Dic6—32no—
98C2×Dic6—32no—
99C3×C4⋊C4—31no—
100C3×C4⋊C4—31no—
101C3×C22⋊C4—32no—
102C3×C22⋊C4—32no—
103C3×C42—31no—
104#217—32no—
105#217—32no—
106#217—32no—
107#217—31no—
108#217—62no—
109#217C2×D312yes—
110#217C2×D312yes—
111(C2×D3)⋊C4—31no—
112(C2×D3)⋊C4—32no—
113(C2×D3)⋊C4—31no—
114(C2×D3)⋊C4—31no—
115(C2×D3)⋊C4—32no—
116(C2×D3)⋊C4—32no—
117C4⋊Dic3—31no—
118C4⋊Dic3—31no—
119C4⋊Dic3—32no—
120C4⋊Dic3—31no—
121C4⋊Dic3—61no—
122C4⋊Dic3C2×D311yescharacteristic
123C4⋊Dic3C2×D311yescharacteristic
124Dic3⋊C4—31no—
125Dic3⋊C4—32no—
126C4×Dic3—32no—
127C4×Dic3—32no—
128C4×Dic3—32no—
129C4×Dic3—32no—
130C4×Dic3C2×D311yescharacteristic
131C22×C32—22no—
132C22×C32—22no—
133C22×C32—22no—
134C22×C32C2411yescharacteristic
135C2×C3⋊D3—21no—
136C2×C3⋊D3—21no—
137C2×C3⋊D3—21no—
138C2×C3×D3—21no—
139C2×C3×D3—21no—
140C2×C3×D3—21no—
141C32×C4#4112yes—
142C3⋊Dic3—21no—
143C3⋊Dic3—22no—
144C3×Dic3—21no—
145C3×Dic3—22no—
146C3×Dic3—21no—
147C3×Dic3—22no—
148C3×Dic3#4112yes—
149C4⋊Q8—91no—
150#124—92no—
151#124—92no—
152#121—31no—
153#121—92no—
154#121—91no—
155C4×D4—31no—
156C4×D4—91no—
157#117—92no—
158#117—91no—
159#117—91no—
160C23×C3—22no—
161C23×C3—31no—
162C23×C3C22×D312yes—
163C23×C3C22×D312yes—
164C22×D3—31no—
165C22×D3—31no—
166C22×D3—61no—
167C22×D3C22×D311yescharacteristic
168C3×D4—22no—
169C3×D4—22no—
170C3×D4—42no—
171C3×D4—122no—
172C2×C3×C4—21no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—32no—
176C2×C3×C4—32no—
177C2×C3×C4—31no—
178C2×C3×C4—64no—
179C2×C3×C4—62no—
180C2×C3×C4C22×D311yescharacteristic
181C2×C3×C4C22×D311yescharacteristic
182#67—42no—
183#67—62no—
184#67—122no—
185#67—122no—
186#67—124no—
187C2×Dic3—24no—
188C2×Dic3—31no—
189C2×Dic3—31no—
190C2×Dic3—32no—
191C2×Dic3—32no—
192C2×Dic3—62no—
193C2×Dic3—61no—
194C2×Dic3C22×D311yescharacteristic
195C2×Dic3C22×D312yes—
196C2×Dic3C22×D312yes—
197C2×Dic3C22×D311yescharacteristic
198C2×Dic3C22×D311yescharacteristic
199D12—62no—
200C4×D3—22no—
201C4×D3—62no—
202C4×D3—62no—
203C4×D3—122no—
204Dic6—122no—
205Dic6—122no—
206C2×C32—42no—
207C2×C32C2×#4111yescharacteristic
208C2×C32C2×#4111yescharacteristic
209C2×C32H(5,𝔽2)11yescharacteristic
210C3⋊D3—41no—
211C3×D3—41no—
212C2×Q8—92no—
213C2×D4—32no—
214C2×D4—91no—
215C2×D4—92no—
216C2×D4D3211yescharacteristic
217C22×C4—32no—
218C22×C4—31no—
219C22×C4—91no—
220C4⋊C4—31no—
221C4⋊C4—31no—
222C4⋊C4—91no—
223C4⋊C4—92no—
224C4⋊C4—91no—
225C22⋊C4—32no—
226C22⋊C4—32no—
227C22⋊C4—92no—
228C22⋊C4—91no—
229C22⋊C4—92no—
230C22⋊C4—91no—
231C22⋊C4—92no—
232C42—31no—
233C42—92no—
234C42—92no—
235C22×C3—22no—
236C22×C3—22no—
237C22×C3—22no—
238C22×C3—22no—
239C22×C3—22no—
240C22×C3—22no—
241C22×C3—21no—
242C22×C3—42no—
243C22×C3—42no—
244C22×C3—42no—
245C22×C3—61no—
246C22×C3—61no—
247C22×C3—61no—
248C22×C3C23×D311yescharacteristic
249C22×C3C23×D311yescharacteristic
250C2×D3—21no—
251C2×D3—21no—
252C2×D3—21no—
253C2×D3—62no—
254C2×D3—61no—
255C2×D3—61no—
256C2×D3—61no—
257C2×D3—61no—
258C2×D3—61no—
259C2×D3—61no—
260C2×D3—121no—
261C2×D3—121no—
262C3×C4—22no—
263C3×C4—32no—
264C3×C4—61no—
265C3×C4—61no—
266C3×C4—62no—
267C3×C4—62no—
268C3×C4#23512yes—
269C3×C4#23712yes—
270Dic3—21no—
271Dic3—22no—
272Dic3—22no—
273Dic3—21no—
274Dic3—61no—
275Dic3—61no—
276Dic3—62no—
277Dic3—62no—
278Dic3—121no—
279Dic3—122no—
280Dic3#23712yes—
281C32#55511yescharacteristic, complemented, Sylow
282C23—31no—
283C23—91no—
284C23C2×D3212yes—
285Q8—362no—
286D4—22no—
287D4—122no—
288D4—182no—
289D4—362no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—31no—
293C2×C4—32no—
294C2×C4—32no—
295C2×C4—64no—
296C2×C4—62no—
297C2×C4—91no—
298C2×C4—92no—
299C2×C4—182no—
300C2×C4C2×D3211yescharacteristic
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—42no—
305C2×C3—42no—
306C2×C3—82no—
307C2×C3—121no—
308C2×C3C2×#23511yescharacteristic
309C2×C3C2×#23711yescharacteristic
310C2×C3C2×#23711yescharacteristic
311C2×C3#100411yescharacteristic
312C2×C3D3×#4111yescharacteristic
313C2×C3#100911yescharacteristic
314D3—41no—
315D3—122no—
316D3—121no—
317D3—121no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—61no—
322C22—61no—
323C22—61no—
324C22—181no—
325C22—181no—
326C22—181no—
327C22C22×D3211yescharacteristic
328C4—32no—
329C4—62no—
330C4—62no—
331C4—61no—
332C4—61no—
333C4—182no—
334C4—181no—
335C4#380412yes—
336C3—21no—
337C3#556311yescharacteristic, complemented
338C3#561611yescharacteristic, complemented
339C2—42no—
340C2—121no—
341C2—361no—
342C2D3×#23711yescharacteristic
343C2#6431411yescharacteristic
344C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
hypercenterC2×D4D32
hypocenterC32#555
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #64147C2semidirect
2. C3×#938C2semidirect
3. C3×D4×Dic3C2semidirect
4. #63978C2semidirect
5. #63959C2semidirect
6. #63955C2semidirect
7. #63953C2semidirect
8. #63918C2semidirect
9. C4×#626C2semidirect
10. #63849C2semidirect
11. #63841C2semidirect
43. #938D3semidirect
45. D4×Dic3D3semidirect
281. C32#555semidirect
337. C3#5563semidirect
338. C3#5616semidirect

Maximal subgroups

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

subgroupindex
#641472
C3×#9382
C3×D4×Dic32
#639782
#639592
#639552
#639532
#639182
C4×#6262
#638492
#638412
#56163
#55633