Finite Group Databasebeta

← #10594719

Subgroups — #10594719

Order 576.

547 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 547 conjugacy classes of subgroups — too many to draw. Only its 128 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#64307C211yesmaximal, characteristic, complemented
2#64297C211yesmaximal, characteristic, complemented
3#64116C211yesmaximal, characteristic, complemented
4C3×#909C211yesmaximal, characteristic, complemented
5C3×#908C211yesmaximal, characteristic, complemented
6C2×#3715C211yesmaximal, characteristic, complemented
7#63822C211yesmaximal, characteristic, complemented
8#63811C212yesmaximal, complemented
9#63810C212yesmaximal, complemented
10#63798C212yesmaximal, complemented
11#63796C212yesmaximal, complemented
12#5778—31nomaximal
13#5775—31nomaximal
14#3828C2211yescharacteristic
15C3×#235C2211yescharacteristic
16C3×#235C2211yescharacteristic, complemented
17C2×C3×D12C2211yescharacteristic
18C2×C3×Dic6C2211yescharacteristic, complemented
19D3×#67—22no—
20#3804—22no—
21#3802C2212yes—
22D3×D12—21no—
23#3799—21no—
24#3797C2212yescomplemented
25#3796C2212yes—
26#3796C2212yescomplemented
27#3795—21no—
28D3×Dic6—21no—
29C32×M16C2211yescharacteristic, complemented
30C2×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
31#3743C2212yescomplemented
32C8×C3⋊D3C2212yescomplemented
33C3×Dic12C2212yescomplemented
34C3×D24C2212yescomplemented
35C3×#204C2212yescomplemented
36C3×#204C2212yescomplemented
37#3718C2211yescharacteristic, complemented
38#3715C2211yescharacteristic, complemented
39#3715C2212yescomplemented
40#3715C2212yescomplemented
41#3715C2211yescharacteristic, complemented
42#3713C2211yescharacteristic, complemented
43#1010—31no—
44#1009—31no—
45#1007—31no—
46#1004—31no—
47C3×#137—31no—
48C3×AGL(1,ℤ/8)—31no—
49#950—31no—
50#950—31no—
51C2×#215—31no—
52C2×#214—31no—
53#919—32no—
54#918—32no—
55#915—32no—
56#914—32no—
57D3×SD16—32no—
58D3×SD16—32no—
59#912—32no—
60#911—32no—
61#909D311yescharacteristic, complemented
62#908D311yescharacteristic, complemented
63#907—31no—
64#907—31no—
65#907—61no—
66C22×C3×D3—21no—
67C2×D32—22no—
68C2×C32×C4C2311yescharacteristic
69#638D412yes—
70C4⋊(C3⋊D3)D411yescharacteristic
71C4×C3⋊D3C2312yes—
72#634D411yescharacteristic
73C3×#67—21no—
74C3×#67—21no—
75C2×C3×Dic3—21no—
76C3×D12—21no—
77C3×D12C2311yescharacteristic
78C3×D12C2311yescharacteristic
79C3×D12C2312yes—
80C3×C4×D3—21no—
81C3×C4×D3—21no—
82C3×Dic6—21no—
83C3×Dic6C2312yes—
84C3×Dic6C2311yescharacteristic
85C3×Dic6C2311yescharacteristic
86#627—22no—
87#626—22no—
88#626—22no—
89#625—22no—
90#624—22no—
91D3×Dic3—22no—
92D3×Dic3—22no—
93C32×C8C2312yes—
94C3⋊(C3⋊C8)C2311yescharacteristic
95C3⋊(C3⋊C8)C2311yescharacteristic
96#577—91noSylow
97C3×#41—31no—
98C3×#41—31no—
99C2×C3×Q8—31no—
100C2×C3×D4—31no—
101C2×#67—62no—
102#239—32no—
103#239—32no—
104#239—61no—
105D3×Q8—32no—
106D3×Q8—32no—
107D3×Q8—61no—
108#237—32no—
109#237—32no—
110#237—61no—
111#237—62no—
112D3×D4—32no—
113D3×D4—32no—
114D3×D4—61no—
115D3×D4—62no—
116#235—31no—
117#235—31no—
118#235—61no—
119#235—61no—
120#235—61no—
121#235—62no—
122#235C2×D311yescharacteristic
123#235C2×D311yescharacteristic, complemented
124C2×D12—61no—
125C2×D12C2×D311yescharacteristic
126C2×Dic6—61no—
127C2×Dic6C2×D311yescharacteristic, complemented
128C3×Q16—32no—
129C3×SD16—32no—
130C3×SD16—32no—
131C3×D8—32no—
132C3×M16—21no—
133C3×M16C2×D311yescharacteristic, complemented
134C3×M16C2×D311yescharacteristic, complemented
135#216—31no—
136#216—31no—
137#215—31no—
138#215—32no—
139#215—32no—
140#215—31no—
141#214—32no—
142#214—31no—
143#214—32no—
144#214—31no—
145#213—31no—
146#213—31no—
147C2×C3⋊C8—31no—
148C2×C3⋊C8—31no—
149C2×C3⋊C8—61no—
150Dic12C2×D312yescomplemented
151D24C2×D312yescomplemented
152#204C2×D312yescomplemented
153#204C2×D312yescomplemented
154ΓL(1,𝔽25)—32no—
155ΓL(1,𝔽25)—32no—
156ΓL(1,𝔽25)—62no—
157D3×C8—32no—
158D3×C8—32no—
159D3×C8—62no—
160C22×C32C2×D411yescharacteristic
161C2×C3⋊D3C2×D412yes—
162C2×C3×D3—21no—
163C2×C3×D3—22no—
164C2×C3×D3—21no—
165C2×C3×D3—81no—
166D32—82no—
167C32×C4C2×D411yescharacteristic
168C32×C4C2411yescharacteristic
169C3⋊Dic3C2×D412yes—
170C3×Dic3—21no—
171C3×Dic3—22no—
172C3×Dic3—21no—
1732−1+4—91no—
174H(5,𝔽2)—91no—
175#137—31no—
176#137—92no—
177AGL(1,ℤ/8)—31no—
178AGL(1,ℤ/8)—92no—
179#135—92no—
180#135—91no—
181C2×SD16—92no—
182C2×SD16—91no—
183#131—91no—
184C23×C3—61no—
185C22×D3—21no—
186C22×D3—62no—
187C22×D3—62no—
188C3×Q8—32no—
189C3×Q8—31no—
190C3×Q8—31no—
191C3×Q8—61no—
192C3×D4—31no—
193C3×D4—31no—
194C3×D4—32no—
195C3×D4—61no—
196C3×D4—61no—
197C3×D4—61no—
198C2×C3×C4—21no—
199C2×C3×C4—61no—
200C2×C3×C4—61no—
201C2×C3×C4—61no—
202C2×C3×C4C22×D311yescharacteristic
203C2×C3×C4C22×D311yescharacteristic
204#67—21no—
205#67—21no—
206#67—32no—
207#67—32no—
208#67—62no—
209#67—62no—
210#67—62no—
211#67—62no—
212#67—62no—
213C2×Dic3—21no—
214C2×Dic3—62no—
215C2×Dic3—62no—
216D12—21no—
217D12—31no—
218D12—31no—
219D12—61no—
220D12—62no—
221D12—62no—
222D12C22×D311yescharacteristic
223D12C22×D312yes—
224D12C22×D311yescharacteristic
225C4×D3—21no—
226C4×D3—21no—
227C4×D3—32no—
228C4×D3—32no—
229C4×D3—62no—
230C4×D3—62no—
231C4×D3—62no—
232C4×D3—62no—
233C4×D3—62no—
234Dic6—21no—
235Dic6—31no—
236Dic6—31no—
237Dic6—61no—
238Dic6—62no—
239Dic6—62no—
240Dic6C22×D311yescharacteristic
241Dic6C22×D312yes—
242Dic6C22×D311yescharacteristic
243C3×C8—22no—
244C3×C8C22×D312yes—
245C3×C8C22×D312yes—
246C3⋊C8—31no—
247C3⋊C8—31no—
248C3⋊C8—31no—
249C3⋊C8—31no—
250C3⋊C8—61no—
251C3⋊C8—61no—
252C2×C32—21no—
253C2×C32C22×D411yescharacteristic
254C3⋊D3—22no—
255C3×D3—41no—
256C3×D3—42no—
257C3×D3—41no—
258#41—31no—
259#41—31no—
260#41—91no—
261#41—92no—
262#41—92no—
263#41—182no—
264#41—181no—
265#41—181no—
266C2×Q8—31no—
267C2×Q8—92no—
268C2×Q8—181no—
269C2×D4—31no—
270C2×D4—92no—
271C2×D4—181no—
272C2×D4—182no—
273Q16—32no—
274Q16—91no—
275SD16—32no—
276SD16—32no—
277SD16—92no—
278SD16—91no—
279SD16—92no—
280SD16—91no—
281D8—32no—
282D8—91no—
283M16—92no—
284M16D3211yescharacteristic, complemented
285C2×C8—91no—
286C2×C8—92no—
287C22×C3—21no—
288C22×C3—61no—
289C22×C3—62no—
290C22×C3—61no—
291C22×C3—241no—
292C22×C3D3×D411yescharacteristic
293C22×C3D3×D411yescharacteristic
294C2×D3—21no—
295C2×D3—21no—
296C2×D3—22no—
297C2×D3—32no—
298C2×D3—32no—
299C2×D3—62no—
300C2×D3—81no—
301C2×D3—242no—
302C2×D3—242no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—61no—
306C3×C4—62no—
307C3×C4—61no—
308C3×C4D3×D411yescharacteristic
309C3×C4D3×D411yescharacteristic
310C3×C4C23×D311yescharacteristic
311C3×C4C23×D311yescharacteristic
312Dic3—22no—
313Dic3—21no—
314Dic3—21no—
315Dic3—32no—
316Dic3—32no—
317Dic3—62no—
318C32#57711yescharacteristic, complemented, Sylow
319C23—61no—
320C23—182no—
321Q8—31no—
322Q8—32no—
323Q8—31no—
324Q8—61no—
325Q8—91no—
326Q8—182no—
327D4—31no—
328D4—32no—
329D4—31no—
330D4—61no—
331D4—61no—
332D4—61no—
333D4—92no—
334D4—91no—
335D4—182no—
336D4—182no—
337D4—182no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—92no—
342C2×C4—182no—
343C2×C4—182no—
344C2×C4—182no—
345C2×C4C2×D3211yescharacteristic
346C8—91no—
347C8—91no—
348C8C2×D3212yes—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—41no—
353C2×C3—121no—
354C2×C3—122no—
355C2×C3—121no—
356C2×C3C2×D3×D411yescharacteristic
357C2×C3C2×D3×D411yescharacteristic
358D3—42no—
359D3—41no—
360D3—41no—
361D3—62no—
362D3—62no—
363D3—122no—
364C22—61no—
365C22—61no—
366C22—62no—
367C22—92no—
368C22—241no—
369C22—722no—
370C22#380211yescharacteristic
371C4—61no—
372C4—61no—
373C4—62no—
374C4—92no—
375C4#380211yescharacteristic
376C4C22×D3211yescharacteristic
377C3—21no—
378C3#577511yescharacteristic, complemented
379C3#577811yescharacteristic, complemented
380C2—21no—
381C2—122no—
382C2—121no—
383C2—121no—
384C2—182no—
385C2C2×#380211yescharacteristic
subgroupG/H
centerC2C2×#3802
commutator, cosocleC32×C4C24
FrattiniC4C22×D32
FittingC32×M16C22
socleC2×C32C22×D4
hypercenterM16D32
hypocenterC32#577
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. #64307C2semidirect
2. #64297C2semidirect
3. #64116C2semidirect
4. C3×#909C2semidirect
5. C3×#908C2semidirect
6. C2×#3715C2semidirect
7. #63822C2semidirect
8. #63811C2semidirect
9. #63810C2semidirect
10. #63798C2semidirect
11. #63796C2semidirect
16. C3×#235C22semidirect
18. C2×C3×Dic6C22semidirect
24. #3797C22semidirect
26. #3796C22semidirect
29. C32×M16C22semidirect
30. C2×C3⋊(C3⋊C8)C22semidirect
31. #3743C22semidirect
32. C8×C3⋊D3C22semidirect
33. C3×Dic12C22semidirect
34. C3×D24C22semidirect
35. C3×#204C22semidirect
36. C3×#204C22semidirect
37. #3718C22semidirect
38. #3715C22semidirect
39. #3715C22semidirect
40. #3715C22semidirect
41. #3715C22semidirect
42. #3713C22semidirect
61. #909D3semidirect
62. #908D3semidirect
123. #235C2×D3semidirect
127. C2×Dic6C2×D3semidirect
133. C3×M16C2×D3semidirect
134. C3×M16C2×D3semidirect
150. Dic12C2×D3semidirect
151. D24C2×D3semidirect
152. #204C2×D3semidirect
153. #204C2×D3semidirect
284. M16D32semidirect
318. C32#577semidirect
378. C3#5775semidirect
379. C3#5778semidirect

Maximal subgroups

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

subgroupindex
#643072
#642972
#641162
C3×#9092
C3×#9082
C2×#37152
#638222
#638112
#638102
#637982
#637962
#57783
#57753