Finite Group Databasebeta

← #10595467

Subgroups — #10595467

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×#1010C211yesmaximal, characteristic, complemented
2#64298C211yesmaximal, characteristic, complemented
3#64160C211yesmaximal, characteristic, complemented
4C3×#950C211yesmaximal, characteristic, complemented
5#63946C212yesmaximal, complemented
6D3×#216C212yesmaximal, complemented
7#63933C212yesmaximal, complemented
8#63929C212yesmaximal, complemented
9C2×#3719C211yesmaximal, characteristic, complemented
10#63827C211yesmaximal, characteristic, complemented
11#63814C211yesmaximal, characteristic, complemented
12#5838—31nomaximal
13#5771—31nomaximal
14C32×#41C2211yescharacteristic
15C2×#634C2211yescharacteristic
16C3×D3×Q8C2212yes—
17C3×#237—22no—
18C3×#237C2212yes—
19C3×#235—21no—
20C3×#235C2211yescharacteristic
21C3×#235C2211yescharacteristic
22C2×C3×Dic6—21no—
23C2×C3×Dic6C2211yescharacteristic
24#3805—22no—
25#3795—21no—
26#3795C2212yes—
27D3×Dic6—21no—
28D3×Dic6C2212yes—
29#3756C2212yes—
30#3754C2212yes—
31#3748C2211yescharacteristic
32C3×#216C2211yescharacteristic
33C3×#215C2211yescharacteristic
34C3×#214C2211yescharacteristic
35C3×#213C2211yescharacteristic
36C2×C3×C3⋊C8C2211yescharacteristic
37#3719C2212yes—
38#3719C2212yes—
39#3718C2212yes—
40#3715C2212yes—
41#3711C2212yes—
42D3×C3⋊C8C2212yes—
43C3×2−1+4—31no—
44#1010—31no—
45#1010—31no—
46#1010D311yescharacteristic, complemented
47C3×#135—31no—
48#951—31no—
49#951—31no—
50#951—31no—
51#951—32no—
52#951—61no—
53#950—32no—
54#950D311yescharacteristic, complemented
55#948—31no—
56C2×#216—32no—
57C2×#216—31no—
58#942—31no—
59#942—32no—
60D3×Q16—32no—
61#915—32no—
62#915—32no—
63#912—32no—
64C2×Dic12—31no—
65#901—31no—
66C32×Q8C2311yescharacteristic
67C32×D4—21no—
68C32×D4C2311yescharacteristic
69C2×C32×C4—21no—
70C2×C32×C4C2311yescharacteristic
71C2×C3⋊Dic3—21no—
72#634—21no—
73#634C2312yes—
74C3×#67—21no—
75C3×#67—22no—
76C3×#67D412yes—
77C2×C3×Dic3—21no—
78C2×C3×Dic3—22no—
79C3×D12D411yescharacteristic
80C3×D12C2311yescharacteristic
81C3×C4×D3—22no—
82C3×C4×D3—21no—
83C3×C4×D3C2312yes—
84C3×Dic6—21no—
85C3×Dic6—22no—
86C3×Dic6D411yescharacteristic
87C3×Dic6C2312yes—
88C3×Dic6C2311yescharacteristic
89#627—22no—
90#625—22no—
91D3×Dic3—22no—
92D3×Dic3—22no—
93C3⋊(C3⋊C8)C2312yes—
94C3×C3⋊C8C2311yescharacteristic
95C3×C3⋊C8C2311yescharacteristic
96#578—91noSylow
97C3×#41—21no—
98C3×#41—31no—
99C3×#41—32no—
100C3×#41—31no—
101C3×#41—61no—
102C3×#41—62no—
103C3×#41C2×D311yescharacteristic
104C3×#41C2×D311yescharacteristic
105C2×C3×Q8—32no—
106C2×C3×Q8—31no—
107C2×C3×Q8—61no—
108D3×Q8—32no—
109D3×Q8—61no—
110D3×Q8C2×D312yes—
111#237—22no—
112#237—32no—
113#237—61no—
114#237—62no—
115#237—62no—
116#237C2×D312yes—
117#235—21no—
118#235—32no—
119#235—61no—
120#235C2×D311yescharacteristic
121#235C2×D311yescharacteristic
122C2×Dic6—21no—
123C2×Dic6—31no—
124C2×Dic6—31no—
125C2×Dic6—32no—
126C2×Dic6—61no—
127C2×Dic6—61no—
128C2×Dic6C2×D311yescharacteristic
129C3×Q16—31no—
130C3×SD16—31no—
131C3×SD16—31no—
132C3×D8—31no—
133C2×C3×C8—31no—
134#216—32no—
135#216—32no—
136#216—32no—
137#216—32no—
138#216—32no—
139#216—32no—
140#216—62no—
141#216C2×D311yescharacteristic
142#215—32no—
143#215C2×D311yescharacteristic
144#214—32no—
145#214—32no—
146#214—32no—
147#214—62no—
148#214C2×D311yescharacteristic
149#213C2×D311yescharacteristic
150#208—31no—
151#208—31no—
152#208—32no—
153#208—61no—
154C2×C3⋊C8—32no—
155C2×C3⋊C8C2×D311yescharacteristic
156Dic12—32no—
157Dic12—32no—
158ΓL(1,𝔽25)—32no—
159D3×C8—32no—
160C22×C32—21no—
161C22×C32C2×D411yescharacteristic
162C2×C3×D3—21no—
163C2×C3×D3C2×D412yes—
164C32×C4—21no—
165C32×C4C2×D411yescharacteristic
166C32×C4C2411yescharacteristic
167C3⋊Dic3—22no—
168C3×Dic3—21no—
169C3×Dic3—22no—
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—31no—
178#135—92no—
179C2×Q16—91no—
180C2×Q16—92no—
181#131—91no—
182C3×Q8—21no—
183C3×Q8—31no—
184C3×Q8—32no—
185C3×Q8—31no—
186C3×Q8—62no—
187C3×Q8—61no—
188C3×Q8C22×D311yescharacteristic
189C3×Q8C22×D311yescharacteristic
190C3×D4—21no—
191C3×D4—21no—
192C3×D4—21no—
193C3×D4—32no—
194C3×D4—31no—
195C3×D4—31no—
196C3×D4—41no—
197C3×D4—62no—
198C3×D4—61no—
199C3×D4C22×D311yescharacteristic
200C3×D4C22×D311yescharacteristic
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—32no—
205C2×C3×C4—41no—
206C2×C3×C4—62no—
207C2×C3×C4—62no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4C22×D311yescharacteristic
211C2×C3×C4C22×D311yescharacteristic
212#67—21no—
213#67—22no—
214#67—62no—
215#67—62no—
216#67#6712yes—
217C2×Dic3—21no—
218C2×Dic3—22no—
219C2×Dic3—62no—
220C2×Dic3—61no—
221C2×Dic3—61no—
222C2×Dic3—62no—
223C2×Dic3—121no—
224D12#6711yescharacteristic
225D12C22×D311yescharacteristic
226C4×D3—22no—
227C4×D3—21no—
228C4×D3—62no—
229C4×D3—62no—
230C4×D3C22×D312yes—
231Dic6—21no—
232Dic6—22no—
233Dic6—32no—
234Dic6—32no—
235Dic6—62no—
236Dic6—62no—
237Dic6—61no—
238Dic6—61no—
239Dic6—62no—
240Dic6—121no—
241Dic6#6711yescharacteristic
242Dic6C22×D312yes—
243Dic6C22×D311yescharacteristic
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×C32—41no—
253C2×C32C22×D411yescharacteristic
254C3×D3—22no—
255C3×D3—41no—
256#41—31no—
257#41—32no—
258#41—31no—
259#41—61no—
260#41—62no—
261#41—92no—
262#41—181no—
263#41—182no—
264#41D3211yescharacteristic
265C2×Q8—32no—
266C2×Q8—31no—
267C2×Q8—61no—
268C2×Q8—92no—
269C2×Q8—91no—
270C2×Q8—181no—
271Q16—31no—
272Q16—92no—
273Q16—92no—
274Q16—92no—
275Q16—92no—
276SD16—31no—
277SD16—31no—
278SD16—92no—
279SD16—92no—
280D8—31no—
281M16—92no—
282M16—91no—
283C2×C8—31no—
284C2×C8—92no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—32no—
289C22×C3—41no—
290C22×C3—61no—
291C22×C3D3×D411yescharacteristic
292C22×C3C2×#6711yescharacteristic
293C2×D3—21no—
294C2×D3C2×#6712yes—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—32no—
300C3×C4—41no—
301C3×C4—62no—
302C3×C4—61no—
303C3×C4D3×D411yescharacteristic
304C3×C4C2×#6711yescharacteristic
305C3×C4C23×D311yescharacteristic
306C3×C4C23×D311yescharacteristic
307Dic3—21no—
308Dic3—22no—
309Dic3—62no—
310Dic3—62no—
311Dic3—122no—
312Dic3C2×#6712yes—
313C32#57811yescharacteristic, complemented, Sylow
314Q8—31no—
315Q8—31no—
316Q8—32no—
317Q8—62no—
318Q8—61no—
319Q8—92no—
320Q8—182no—
321Q8—181no—
322Q8C2×D3211yescharacteristic
323D4—21no—
324D4—32no—
325D4—31no—
326D4—31no—
327D4—62no—
328D4—61no—
329D4—182no—
330D4C2×D3211yescharacteristic
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—41no—
349C2×C3—41no—
350C2×C3—62no—
351C2×C3—81no—
352C2×C3—121no—
353C2×C3C2×D3×D411yescharacteristic
354C2×C3C22×#6711yescharacteristic
355D3—22no—
356D3—41no—
357C22—21no—
358C22—32no—
359C22—61no—
360C22D3×#6711yescharacteristic
361C4—21no—
362C4—32no—
363C4—62no—
364C4—61no—
365C4—182no—
366C4D3×#6711yescharacteristic
367C4C22×D3211yescharacteristic
368C3—21no—
369C3#577111yescharacteristic, complemented
370C3#583811yescharacteristic, complemented
371C2—21no—
372C2—41no—
373C2—62no—
374C2—121no—
375C2C2×D3×#6711yescharacteristic
subgroupG/H
centerC2C2×D3×#67
commutator, cosocleC32×C4C24
FrattiniC4C22×D32
FittingC32×#41C22
socleC2×C32C22×D4
hypercenter#41D32
hypocenterC32#578
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. C3×#1010C2semidirect
2. #64298C2semidirect
3. #64160C2semidirect
4. C3×#950C2semidirect
5. #63946C2semidirect
6. D3×#216C2semidirect
7. #63933C2semidirect
8. #63929C2semidirect
9. C2×#3719C2semidirect
10. #63827C2semidirect
11. #63814C2semidirect
46. #1010D3semidirect
54. #950D3semidirect
313. C32#578semidirect
369. C3#5771semidirect
370. C3#5838semidirect

Maximal subgroups

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

subgroupindex
C3×#10102
#642982
#641602
C3×#9502
#639462
D3×#2162
#639332
#639292
C2×#37192
#638272
#638142
#58383
#57713