Finite Group Databasebeta

← #10595328

Subgroups — #10595328

Order 576.

526 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#1007C211yesmaximal, characteristic, complemented
2#64298C211yesmaximal, characteristic, complemented
3C2×#3756C211yesmaximal, characteristic, complemented
4C3×#942C211yesmaximal, characteristic, complemented
5#63947C212yesmaximal, complemented
6#63946C212yesmaximal, complemented
7#63943C212yesmaximal, complemented
8D3×#216C212yesmaximal, complemented
9#63831C211yesmaximal, characteristic, complemented
10#63822C211yesmaximal, characteristic, complemented
11#63815C211yesmaximal, characteristic, complemented
12#5838—31nomaximal
13#5779—31nomaximal
14C2×C32×Q8C2211yescharacteristic
15C2×#634C2211yescharacteristic
16C3×#239—21no—
17C3×#239C2212yes—
18C3×D3×Q8—21no—
19C3×D3×Q8C2212yes—
20C3×#235—22no—
21C3×#235C2211yescharacteristic
22C3×#235C2211yescharacteristic
23C3×#235C2211yescharacteristic
24#3805—22no—
25#3795—21no—
26#3795C2212yes—
27D3×Dic6—21no—
28D3×Dic6C2212yes—
29#3756C2212yes—
30#3756C2212yes—
31C2×C3⋊(C3⋊C8)C2211yescharacteristic
32C3×#216C2212yes—
33C3×#215C2212yes—
34C3×#208C2211yescharacteristic
35#3719C2212yes—
36#3718C2211yescharacteristic
37#3716C2212yes—
38#3715C2211yescharacteristic
39#3715C2211yescharacteristic
40#3713C2211yescharacteristic
41#3711C2212yes—
42D3×C3⋊C8C2212yes—
43C3×2−1+4—31no—
44#1010—31no—
45#1010—31no—
46#1007D311yescharacteristic, complemented
47C3×#137—31no—
48#951—32no—
49#951—31no—
50#950—31no—
51#950—31no—
52#950—32no—
53#948—31no—
54C2×#216—31no—
55C2×#216—31no—
56C2×#216—32no—
57C2×#216—61no—
58#942—32no—
59#942D311yescharacteristic, complemented
60#918—32no—
61D3×Q16—32no—
62#916—32no—
63#915—32no—
64#909—31no—
65#907—31no—
66C32×Q8—21no—
67C32×Q8C2312yes—
68C2×C32×C4—21no—
69C2×C32×C4C2311yescharacteristic
70C2×C3⋊Dic3—21no—
71#634—21no—
72#634C2312yes—
73C3×#67—21no—
74C3×#67—21no—
75C3×#67D412yes—
76C3×D12—22no—
77C3×D12D411yescharacteristic
78C3×D12C2311yescharacteristic
79C3×D12C2311yescharacteristic
80C3×C4×D3—22no—
81C3×C4×D3—21no—
82C3×C4×D3—21no—
83C3×C4×D3—22no—
84C3×C4×D3C2312yes—
85C3×Dic6—22no—
86C3×Dic6D411yescharacteristic
87C3×Dic6C2311yescharacteristic
88C3×Dic6C2311yescharacteristic
89#627—22no—
90#625—22no—
91D3×Dic3—22no—
92D3×Dic3—22no—
93C3⋊(C3⋊C8)C2311yescharacteristic
94C3⋊(C3⋊C8)C2311yescharacteristic
95C3×C3⋊C8C2312yes—
96#578—91noSylow
97C3×#41—31no—
98C3×#41—32no—
99C3×#41—31no—
100C3×#41—31no—
101C3×#41—62no—
102C3×#41—61no—
103C2×C3×Q8—21no—
104C2×C3×Q8—32no—
105C2×C3×Q8—61no—
106C2×C3×Q8C2×D311yescharacteristic
107C2×C3×Q8C2×D311yescharacteristic
108#239—21no—
109#239C2×D312yes—
110D3×Q8—21no—
111D3×Q8—32no—
112D3×Q8—61no—
113D3×Q8C2×D312yes—
114#237—32no—
115#237—61no—
116#237—62no—
117#237—62no—
118#235—22no—
119#235—32no—
120#235—61no—
121#235C2×D311yescharacteristic
122#235C2×D311yescharacteristic
123#235C2×D311yescharacteristic
124C2×Dic6—31no—
125C2×Dic6—31no—
126C2×Dic6—32no—
127C2×Dic6—61no—
128C2×Dic6—61no—
129C3×Q16—32no—
130C3×SD16—32no—
131C3×M16—31no—
132#216—32no—
133#216—32no—
134#216—32no—
135#216—32no—
136#216—31no—
137#216—31no—
138#216—32no—
139#216—62no—
140#216—62no—
141#216C2×D312yes—
142#215—31no—
143#215—31no—
144#215C2×D312yes—
145#214—31no—
146#214—31no—
147#214—32no—
148#213—31no—
149#213—31no—
150#208—32no—
151#208C2×D311yescharacteristic
152C2×C3⋊C8—31no—
153C2×C3⋊C8—31no—
154C2×C3⋊C8—32no—
155C2×C3⋊C8—61no—
156Dic12—32no—
157#204—32no—
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—22no—
168C3×Dic3—21no—
169C3×Dic3—21no—
170C3×Dic3C2×D412yes—
1712−1+4—31no—
1722−1+4—91no—
173#137—31no—
174#137—91no—
175#137—92no—
176#137—92no—
177#135—92no—
178#135—91no—
179C2×Q16—92no—
180C2×Q16—91no—
181#131—91no—
182C3×Q8—21no—
183C3×Q8—21no—
184C3×Q8—22no—
185C3×Q8—31no—
186C3×Q8—31no—
187C3×Q8—31no—
188C3×Q8—41no—
189C3×Q8—62no—
190C3×Q8C22×D312yes—
191C3×Q8C22×D312yes—
192C3×D4—32no—
193C3×D4—31no—
194C3×D4—31no—
195C3×D4—31no—
196C3×D4—62no—
197C3×D4—61no—
198C3×D4—61no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—32no—
203C2×C3×C4—41no—
204C2×C3×C4—62no—
205C2×C3×C4—62no—
206C2×C3×C4—61no—
207C2×C3×C4—61no—
208C2×C3×C4C22×D311yescharacteristic
209C2×C3×C4C22×D311yescharacteristic
210#67—21no—
211#67—21no—
212#67—62no—
213#67—62no—
214#67#6712yes—
215C2×Dic3—62no—
216C2×Dic3—61no—
217C2×Dic3—61no—
218C2×Dic3—62no—
219C2×Dic3—121no—
220D12—22no—
221D12#6711yescharacteristic
222D12C22×D311yescharacteristic
223D12C22×D311yescharacteristic
224C4×D3—22no—
225C4×D3—21no—
226C4×D3—21no—
227C4×D3—22no—
228C4×D3—62no—
229C4×D3—62no—
230C4×D3C22×D312yes—
231Dic6—22no—
232Dic6—32no—
233Dic6—32no—
234Dic6—62no—
235Dic6—62no—
236Dic6—61no—
237Dic6—61no—
238Dic6—62no—
239Dic6—121no—
240Dic6#6711yescharacteristic
241Dic6C22×D311yescharacteristic
242Dic6C22×D311yescharacteristic
243C3×C8—32no—
244C3⋊C8—31no—
245C3⋊C8—31no—
246C3⋊C8—31no—
247C3⋊C8—31no—
248C3⋊C8—61no—
249C3⋊C8—61no—
250C3⋊C8C22×D312yes—
251C2×C32—21no—
252C2×C32C22×D411yescharacteristic
253C3×D3—22no—
254C3×D3—41no—
255C3×D3—41no—
256#41—31no—
257#41—32no—
258#41—31no—
259#41—31no—
260#41—62no—
261#41—61no—
262#41—92no—
263#41—181no—
264#41—182no—
265C2×Q8—32no—
266C2×Q8—61no—
267C2×Q8—92no—
268C2×Q8—91no—
269C2×Q8—181no—
270C2×Q8D3211yescharacteristic
271Q16—32no—
272Q16—92no—
273Q16—91no—
274Q16—92no—
275Q16—92no—
276SD16—32no—
277SD16—92no—
278SD16—91no—
279SD16—91no—
280D8—91no—
281M16—31no—
282M16—92no—
283C2×C8—92no—
284C2×C8—91no—
285C22×C3—21no—
286C22×C3—32no—
287C22×C3—61no—
288C22×C3—61no—
289C22×C3D3×D411yescharacteristic
290C22×C3C2×#6711yescharacteristic
291C2×D3—21no—
292C2×D3—21no—
293C2×D3C2×#6712yes—
294C3×C4—22no—
295C3×C4—22no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—32no—
299C3×C4—42no—
300C3×C4—61no—
301C3×C4—61no—
302C3×C4D3×D411yescharacteristic
303C3×C4C2×#6711yescharacteristic
304C3×C4C23×D311yescharacteristic
305C3×C4C23×D311yescharacteristic
306Dic3—21no—
307Dic3—21no—
308Dic3—62no—
309Dic3—62no—
310Dic3—122no—
311Dic3C2×#6712yes—
312C32#57811yescharacteristic, complemented, Sylow
313Q8—21no—
314Q8—31no—
315Q8—31no—
316Q8—31no—
317Q8—62no—
318Q8—92no—
319Q8—182no—
320Q8—181no—
321Q8C2×D3212yes—
322D4—32no—
323D4—31no—
324D4—31no—
325D4—31no—
326D4—62no—
327D4—61no—
328D4—61no—
329D4—182no—
330C2×C4—21no—
331C2×C4—32no—
332C2×C4—62no—
333C2×C4—62no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—182no—
337C2×C4—182no—
338C2×C4—181no—
339C2×C4C2×D3211yescharacteristic
340C8—32no—
341C8—91no—
342C8—91no—
343C2×C3—22no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—62no—
348C2×C3—121no—
349C2×C3—121no—
350C2×C3C2×D3×D411yescharacteristic
351C2×C3C22×#6711yescharacteristic
352D3—22no—
353D3—41no—
354D3—41no—
355C22—32no—
356C22—61no—
357C22—61no—
358C22D3×#6711yescharacteristic
359C4—22no—
360C4—32no—
361C4—61no—
362C4—61no—
363C4—182no—
364C4D3×#6711yescharacteristic
365C4C22×D3211yescharacteristic
366C3—21no—
367C3#577911yescharacteristic, complemented
368C3#583811yescharacteristic, complemented
369C2—21no—
370C2—62no—
371C2—121no—
372C2—121no—
373C2C2×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. #64298C2semidirect
3. C2×#3756C2semidirect
4. C3×#942C2semidirect
5. #63947C2semidirect
6. #63946C2semidirect
7. #63943C2semidirect
8. D3×#216C2semidirect
9. #63831C2semidirect
10. #63822C2semidirect
11. #63815C2semidirect
46. #1007D3semidirect
59. #942D3semidirect
312. C32#578semidirect
367. C3#5779semidirect
368. C3#5838semidirect

Maximal subgroups

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

subgroupindex
C3×#10072
#642982
C2×#37562
C3×#9422
#639472
#639462
#639432
D3×#2162
#638312
#638222
#638152
#58383
#57793