Finite Group Databasebeta

← #10595327

Subgroups — #10595327

Order 576.

526 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#1007C211yesmaximal, characteristic, complemented
2#64297C211yesmaximal, characteristic, complemented
3#64151C211yesmaximal, characteristic, complemented
4C2×C3×#216C211yesmaximal, characteristic, complemented
5#63949C212yesmaximal, complemented
6#63948C212yesmaximal, complemented
7#63943C212yesmaximal, complemented
8D3×#216C212yesmaximal, complemented
9#63829C211yesmaximal, characteristic, complemented
10#63827C211yesmaximal, characteristic, complemented
11#63814C211yesmaximal, characteristic, complemented
12#5838—31nomaximal
13#5766—31nomaximal
14C2×C32×Q8C2211yescharacteristic
15#3828C2211yescharacteristic
16C3×#239—21no—
17C3×#239C2212yes—
18C3×D3×Q8—21no—
19C3×D3×Q8C2212yes—
20C3×#235—22no—
21C3×#235C2211yescharacteristic
22C3×#235C2211yescharacteristic
23C2×C3×Dic6C2211yescharacteristic
24#3804—22no—
25#3799C2212yes—
26#3797—21no—
27#3796—21no—
28D3×Dic6C2212yes—
29#3756C2212yes—
30#3755C2212yes—
31#3748C2211yescharacteristic
32C3×#216C2212yes—
33C3×#216C2212yes—
34C2×C3×C3⋊C8C2211yescharacteristic
35#3719C2211yescharacteristic
36#3718C2212yes—
37#3717C2211yescharacteristic
38#3716C2211yescharacteristic
39#3715C2212yes—
40#3714C2211yescharacteristic
41#3711C2212yes—
42D3×C3⋊C8C2212yes—
43C3×2−1+4—31no—
44#1010—31no—
45#1007—31no—
46#1007D311yescharacteristic, complemented
47C2×C3×Q16—31no—
48#951—31no—
49#951—32no—
50#950—32no—
51#950—31no—
52#948—31no—
53C2×#216—32no—
54C2×#216D311yescharacteristic, complemented
55#942—31no—
56#942—31no—
57#942—31no—
58#942—32no—
59#942—61no—
60#919—32no—
61#918—32no—
62#918—32no—
63D3×Q16—32no—
64#904—31no—
65#901—31no—
66C32×Q8—21no—
67C32×Q8C2312yes—
68C2×C32×C4—21no—
69C2×C32×C4C2311yescharacteristic
70#638—21no—
71C4⋊(C3⋊D3)C2311yescharacteristic
72C4×C3⋊D3—21no—
73#634C2311yescharacteristic
74C3×#67—21no—
75C3×#67D412yes—
76C2×C3×Dic3—21no—
77C3×D12—22no—
78C3×D12D411yescharacteristic
79C3×D12C2311yescharacteristic
80C3×C4×D3—22no—
81C3×C4×D3—21no—
82C3×C4×D3—22no—
83C3×C4×D3C2312yes—
84C3×Dic6—21no—
85C3×Dic6—22no—
86C3×Dic6D411yescharacteristic
87C3×Dic6C2312yes—
88C3×Dic6C2311yescharacteristic
89#627—22no—
90#626—22no—
91#624—22no—
92D3×Dic3—22no—
93C3⋊(C3⋊C8)C2312yes—
94C3×C3⋊C8C2311yescharacteristic
95C3×C3⋊C8C2311yescharacteristic
96#578—91noSylow
97C3×#41—31no—
98C3×#41—32no—
99C3×#41—31no—
100C3×#41—62no—
101C3×#41—61no—
102C2×C3×Q8—21no—
103C2×C3×Q8—32no—
104C2×C3×Q8—31no—
105C2×C3×Q8—61no—
106C2×C3×Q8C2×D311yescharacteristic
107C2×C3×Q8C2×D311yescharacteristic
108#239—21no—
109#239—32no—
110#239—61no—
111#239C2×D312yes—
112D3×Q8—21no—
113D3×Q8—32no—
114D3×Q8—61no—
115D3×Q8—61no—
116D3×Q8C2×D312yes—
117#237—61no—
118#237—62no—
119#235—22no—
120#235—31no—
121#235—31no—
122#235—32no—
123#235—61no—
124#235—62no—
125#235C2×D311yescharacteristic
126#235C2×D311yescharacteristic
127C2×Dic6—32no—
128C2×Dic6C2×D311yescharacteristic
129C3×Q16—32no—
130C3×Q16—32no—
131C2×C3×C8—31no—
132#216—32no—
133#216—32no—
134#216—32no—
135#216—32no—
136#216—31no—
137#216—62no—
138#216C2×D312yes—
139#216C2×D312yes—
140#215—32no—
141#215—32no—
142#215—32no—
143#215—31no—
144#215—62no—
145#214—32no—
146#214—31no—
147#213—31no—
148#208—31no—
149#208—31no—
150#208—32no—
151#208—61no—
152C2×C3⋊C8—32no—
153C2×C3⋊C8C2×D311yescharacteristic
154Dic12—31no—
155D24—31no—
156#204—31no—
157#204—31no—
158ΓL(1,𝔽25)—32no—
159D3×C8—32no—
160C22×C32C2×D411yescharacteristic
161C2×C3⋊D3—21no—
162C2×C3×D3—21no—
163C2×C3×D3C2×D412yes—
164C32×C4—22no—
165C32×C4C2×D411yescharacteristic
166C32×C4C2411yescharacteristic
167C3⋊Dic3—21no—
168C3×Dic3—22no—
169C3×Dic3—21no—
170C3×Dic3C2×D412yes—
1712−1+4—31no—
1722−1+4—91no—
173#137—92no—
174#137—92no—
175#137—91no—
176#137—91no—
177#135—91no—
178#135—92no—
179C2×Q16—31no—
180C2×Q16—92no—
181#131—91no—
182C3×Q8—21no—
183C3×Q8—21no—
184C3×Q8—22no—
185C3×Q8—31no—
186C3×Q8—31no—
187C3×Q8—32no—
188C3×Q8—41no—
189C3×Q8—62no—
190C3×Q8—61no—
191C3×Q8C22×D312yes—
192C3×Q8C22×D312yes—
193C3×D4—32no—
194C3×D4—31no—
195C3×D4—31no—
196C3×D4—62no—
197C3×D4—61no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—32no—
202C2×C3×C4—41no—
203C2×C3×C4—62no—
204C2×C3×C4—62no—
205C2×C3×C4—61no—
206C2×C3×C4—61no—
207C2×C3×C4C22×D311yescharacteristic
208C2×C3×C4C22×D311yescharacteristic
209#67—21no—
210#67—62no—
211#67—61no—
212#67—61no—
213#67—121no—
214#67#6712yes—
215C2×Dic3—21no—
216C2×Dic3—62no—
217D12—22no—
218D12—31no—
219D12—31no—
220D12—61no—
221D12—62no—
222D12#6711yescharacteristic
223D12C22×D311yescharacteristic
224C4×D3—22no—
225C4×D3—21no—
226C4×D3—22no—
227C4×D3—62no—
228C4×D3—62no—
229C4×D3—61no—
230C4×D3—61no—
231C4×D3—62no—
232C4×D3—121no—
233C4×D3C22×D312yes—
234Dic6—21no—
235Dic6—22no—
236Dic6—31no—
237Dic6—31no—
238Dic6—61no—
239Dic6—62no—
240Dic6—62no—
241Dic6#6711yescharacteristic
242Dic6C22×D311yescharacteristic
243Dic6C22×D312yes—
244C3×C8—31no—
245C3×C8—31no—
246C3⋊C8—32no—
247C3⋊C8—32no—
248C3⋊C8—62no—
249C3⋊C8C22×D311yescharacteristic
250C3⋊C8C22×D311yescharacteristic
251C2×C32—21no—
252C2×C32C22×D411yescharacteristic
253C3⋊D3—41no—
254C3×D3—22no—
255C3×D3—41no—
256#41—31no—
257#41—32no—
258#41—31no—
259#41—62no—
260#41—61no—
261#41—92no—
262#41—91no—
263#41—182no—
264#41—181no—
265C2×Q8—32no—
266C2×Q8—31no—
267C2×Q8—61no—
268C2×Q8—92no—
269C2×Q8—181no—
270C2×Q8D3211yescharacteristic
271Q16—32no—
272Q16—32no—
273Q16—91no—
274Q16—92no—
275Q16—92no—
276SD16—91no—
277SD16—91no—
278SD16—92no—
279SD16—92no—
280D8—91no—
281M16—92no—
282M16—91no—
283C2×C8—31no—
284C2×C8—92no—
285C22×C3—21no—
286C22×C3—32no—
287C22×C3—61no—
288C22×C3D3×D411yescharacteristic
289C22×C3C2×#6711yescharacteristic
290C2×D3—21no—
291C2×D3—61no—
292C2×D3—61no—
293C2×D3—121no—
294C2×D3C2×#6712yes—
295C3×C4—22no—
296C3×C4—22no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—32no—
300C3×C4—42no—
301C3×C4—61no—
302C3×C4—62no—
303C3×C4D3×D411yescharacteristic
304C3×C4C2×#6711yescharacteristic
305C3×C4C23×D311yescharacteristic
306C3×C4C23×D311yescharacteristic
307Dic3—22no—
308Dic3—21no—
309Dic3—61no—
310Dic3—61no—
311Dic3—121no—
312Dic3C2×#6712yes—
313C32#57811yescharacteristic, complemented, Sylow
314Q8—21no—
315Q8—31no—
316Q8—32no—
317Q8—31no—
318Q8—62no—
319Q8—61no—
320Q8—91no—
321Q8—182no—
322Q8C2×D3212yes—
323D4—32no—
324D4—31no—
325D4—31no—
326D4—62no—
327D4—61no—
328D4—91no—
329D4—182no—
330D4—181no—
331C2×C4—21no—
332C2×C4—32no—
333C2×C4—62no—
334C2×C4—62no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—182no—
338C2×C4—182no—
339C2×C4—181no—
340C2×C4C2×D3211yescharacteristic
341C8—31no—
342C8—31no—
343C8—92no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—41no—
348C2×C3—62no—
349C2×C3—121no—
350C2×C3C2×D3×D411yescharacteristic
351C2×C3C22×#6711yescharacteristic
352D3—22no—
353D3—41no—
354D3—121no—
355D3—121no—
356D3—241no—
357C22—32no—
358C22—61no—
359C22—181no—
360C22D3×#6711yescharacteristic
361C4—22no—
362C4—32no—
363C4—61no—
364C4—62no—
365C4—181no—
366C4D3×#6711yescharacteristic
367C4C22×D3211yescharacteristic
368C3—21no—
369C3#576611yescharacteristic, complemented
370C3#583811yescharacteristic, complemented
371C2—21no—
372C2—62no—
373C2—121no—
374C2—361no—
375C2C2×D3×#6711yescharacteristic
subgroupG/H
centerC2C2×D3×#67
commutator, cosocleC32×C4C24
FrattiniC4C22×D32
FittingC2×C32×Q8C22
socleC2×C32C22×D4
hypercenterC2×Q8D32
hypocenterC32#578
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#1007C2semidirect
2. #64297C2semidirect
3. #64151C2semidirect
4. C2×C3×#216C2semidirect
5. #63949C2semidirect
6. #63948C2semidirect
7. #63943C2semidirect
8. D3×#216C2semidirect
9. #63829C2semidirect
10. #63827C2semidirect
11. #63814C2semidirect
46. #1007D3semidirect
54. C2×#216D3semidirect
313. C32#578semidirect
369. C3#5766semidirect
370. C3#5838semidirect

Maximal subgroups

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

subgroupindex
C3×#10072
#642972
#641512
C2×C3×#2162
#639492
#639482
#639432
D3×#2162
#638292
#638272
#638142
#58383
#57663