Finite Group Databasebeta

← #10595465

Subgroups — #10595465

Order 576.

527 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#1010C211yesmaximal, characteristic, complemented
2#64297C211yesmaximal, characteristic, complemented
3#64159C211yesmaximal, characteristic, complemented
4C3×#951C211yesmaximal, characteristic, complemented
5#63948C212yesmaximal, complemented
6D3×#216C212yesmaximal, complemented
7#63935C212yesmaximal, complemented
8#63929C212yesmaximal, complemented
9C2×#3718C211yesmaximal, characteristic, complemented
10#63833C211yesmaximal, characteristic, complemented
11#63815C211yesmaximal, characteristic, complemented
12#5838—31nomaximal
13#5779—31nomaximal
14C32×#41C2211yescharacteristic
15#3828C2211yescharacteristic
16C3×D3×Q8C2212yes—
17C3×#237—22no—
18C3×#237C2212yes—
19C3×#235—21no—
20C3×#235C2211yescharacteristic
21C2×C3×Dic6—21no—
22C2×C3×Dic6C2211yescharacteristic
23C2×C3×Dic6C2211yescharacteristic
24#3804—22no—
25#3799C2212yes—
26#3797—21no—
27#3796—21no—
28D3×Dic6C2212yes—
29#3756C2211yescharacteristic
30#3755C2211yescharacteristic
31#3754C2211yescharacteristic
32#3753C2211yescharacteristic
33C2×C3⋊(C3⋊C8)C2211yescharacteristic
34C3×#216C2212yes—
35C3×#214C2212yes—
36C3×#208C2211yescharacteristic
37#3719C2212yes—
38#3718C2212yes—
39#3718C2212yes—
40#3717C2212yes—
41#3711C2212yes—
42D3×C3⋊C8C2212yes—
43C3×2−1+4—31no—
44#1010—31no—
45#1010D311yescharacteristic, complemented
46#1007—31no—
47C3×#137—31no—
48#951—32no—
49#951D311yescharacteristic, complemented
50#950—31no—
51#950—31no—
52#950—32no—
53#950—61no—
54#948—31no—
55C2×#216—31no—
56C2×#216—31no—
57C2×#216—32no—
58#942—32no—
59#942—31no—
60#918—32no—
61D3×Q16—32no—
62#916—32no—
63#915—32no—
64#909—31no—
65#907—31no—
66C32×Q8C2311yescharacteristic
67C32×D4—21no—
68C32×D4C2311yescharacteristic
69C2×C32×C4—21no—
70C2×C32×C4C2311yescharacteristic
71#638—21no—
72C4⋊(C3⋊D3)C2311yescharacteristic
73C4×C3⋊D3—21no—
74#634C2311yescharacteristic
75C3×#67—22no—
76C3×#67D412yes—
77C2×C3×Dic3—21no—
78C2×C3×Dic3—21no—
79C2×C3×Dic3—22no—
80C3×D12D411yescharacteristic
81C3×C4×D3—22no—
82C3×C4×D3C2312yes—
83C3×Dic6—21no—
84C3×Dic6—21no—
85C3×Dic6—22no—
86C3×Dic6D411yescharacteristic
87C3×Dic6C2312yes—
88C3×Dic6C2312yes—
89#627—22no—
90#626—22no—
91#624—22no—
92D3×Dic3—22no—
93C3⋊(C3⋊C8)C2311yescharacteristic
94C3⋊(C3⋊C8)C2311yescharacteristic
95C3×C3⋊C8C2312yes—
96#578—91noSylow
97C3×#41—21no—
98C3×#41—31no—
99C3×#41—32no—
100C3×#41—61no—
101C3×#41—62no—
102C3×#41C2×D311yescharacteristic
103C3×#41C2×D311yescharacteristic
104C2×C3×Q8—32no—
105C2×C3×Q8—31no—
106C2×C3×Q8—31no—
107C2×C3×Q8—61no—
108#239—32no—
109#239—61no—
110D3×Q8—32no—
111D3×Q8—61no—
112D3×Q8—61no—
113D3×Q8C2×D312yes—
114#237—22no—
115#237—61no—
116#237—62no—
117#237C2×D312yes—
118#235—21no—
119#235—31no—
120#235—31no—
121#235—32no—
122#235—61no—
123#235—62no—
124#235C2×D311yescharacteristic
125C2×Dic6—21no—
126C2×Dic6—32no—
127C2×Dic6C2×D311yescharacteristic
128C2×Dic6C2×D311yescharacteristic
129C3×Q16—32no—
130C3×SD16—32no—
131C3×M16—31no—
132#216—31no—
133#216—31no—
134#216—32no—
135#216—32no—
136#216—32no—
137#216—32no—
138#216—32no—
139#216—61no—
140#216C2×D312yes—
141#215—31no—
142#215—31no—
143#215—32no—
144#215—61no—
145#214—31no—
146#214—31no—
147#214—61no—
148#214C2×D312yes—
149#213—31no—
150#213—31no—
151#213—61no—
152#208—32no—
153#208C2×D311yescharacteristic
154C2×C3⋊C8—31no—
155C2×C3⋊C8—31no—
156C2×C3⋊C8—32no—
157C2×C3⋊C8—61no—
158Dic12—32no—
159#204—32no—
160ΓL(1,𝔽25)—32no—
161D3×C8—32no—
162C22×C32—21no—
163C22×C32C2×D411yescharacteristic
164C2×C3⋊D3—21no—
165C2×C3×D3C2×D412yes—
166C32×C4—21no—
167C32×C4C2×D411yescharacteristic
168C32×C4C2411yescharacteristic
169C3⋊Dic3—21no—
170C3×Dic3—22no—
171C3×Dic3—22no—
172C3×Dic3C2×D412yes—
1732−1+4—31no—
1742−1+4—91no—
175#137—31no—
176#137—91no—
177#137—92no—
178#137—92no—
179#135—92no—
180#135—91no—
181C2×Q16—92no—
182C2×Q16—91no—
183#131—91no—
184C3×Q8—21no—
185C3×Q8—31no—
186C3×Q8—32no—
187C3×Q8—32no—
188C3×Q8—62no—
189C3×Q8—61no—
190C3×Q8—61no—
191C3×Q8C22×D311yescharacteristic
192C3×Q8C22×D311yescharacteristic
193C3×D4—21no—
194C3×D4—21no—
195C3×D4—21no—
196C3×D4—32no—
197C3×D4—31no—
198C3×D4—41no—
199C3×D4—62no—
200C3×D4C22×D311yescharacteristic
201C3×D4C22×D311yescharacteristic
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—32no—
206C2×C3×C4—41no—
207C2×C3×C4—62no—
208C2×C3×C4—62no—
209C2×C3×C4—61no—
210C2×C3×C4—61no—
211C2×C3×C4C22×D311yescharacteristic
212C2×C3×C4C22×D311yescharacteristic
213#67—22no—
214#67—62no—
215#67—61no—
216#67—61no—
217#67—121no—
218#67#6712yes—
219C2×Dic3—21no—
220C2×Dic3—21no—
221C2×Dic3—22no—
222C2×Dic3—62no—
223D12—31no—
224D12—31no—
225D12—61no—
226D12—62no—
227D12#6711yescharacteristic
228C4×D3—22no—
229C4×D3—62no—
230C4×D3—62no—
231C4×D3—61no—
232C4×D3—61no—
233C4×D3—62no—
234C4×D3—121no—
235C4×D3C22×D312yes—
236Dic6—21no—
237Dic6—21no—
238Dic6—22no—
239Dic6—31no—
240Dic6—31no—
241Dic6—61no—
242Dic6—62no—
243Dic6—62no—
244Dic6#6711yescharacteristic
245Dic6C22×D312yes—
246Dic6C22×D312yes—
247C3×C8—32no—
248C3⋊C8—31no—
249C3⋊C8—31no—
250C3⋊C8—31no—
251C3⋊C8—31no—
252C3⋊C8—61no—
253C3⋊C8—61no—
254C3⋊C8C22×D312yes—
255C2×C32—21no—
256C2×C32—41no—
257C2×C32C22×D411yescharacteristic
258C3⋊D3—41no—
259C3×D3—22no—
260#41—31no—
261#41—32no—
262#41—61no—
263#41—62no—
264#41—92no—
265#41—91no—
266#41—182no—
267#41—181no—
268#41D3211yescharacteristic
269C2×Q8—32no—
270C2×Q8—31no—
271C2×Q8—31no—
272C2×Q8—61no—
273C2×Q8—92no—
274C2×Q8—181no—
275Q16—32no—
276Q16—92no—
277Q16—92no—
278Q16—92no—
279Q16—91no—
280SD16—32no—
281SD16—92no—
282SD16—91no—
283SD16—91no—
284D8—91no—
285M16—31no—
286M16—92no—
287C2×C8—92no—
288C2×C8—91no—
289C22×C3—21no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—32no—
293C22×C3—41no—
294C22×C3D3×D411yescharacteristic
295C22×C3C2×#6711yescharacteristic
296C2×D3—61no—
297C2×D3—61no—
298C2×D3—121no—
299C2×D3C2×#6712yes—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—32no—
305C3×C4—41no—
306C3×C4—62no—
307C3×C4—62no—
308C3×C4D3×D411yescharacteristic
309C3×C4C2×#6711yescharacteristic
310C3×C4C23×D311yescharacteristic
311C3×C4C23×D311yescharacteristic
312Dic3—22no—
313Dic3—22no—
314Dic3—61no—
315Dic3—61no—
316Dic3—121no—
317Dic3C2×#6712yes—
318C32#57811yescharacteristic, complemented, Sylow
319Q8—31no—
320Q8—32no—
321Q8—32no—
322Q8—62no—
323Q8—61no—
324Q8—61no—
325Q8—91no—
326Q8—182no—
327Q8C2×D3211yescharacteristic
328D4—21no—
329D4—32no—
330D4—31no—
331D4—62no—
332D4—91no—
333D4—182no—
334D4—181no—
335D4C2×D3211yescharacteristic
336C2×C4—21no—
337C2×C4—32no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—182no—
343C2×C4—182no—
344C2×C4—181no—
345C2×C4C2×D3211yescharacteristic
346C8—32no—
347C8—91no—
348C8—91no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—41no—
353C2×C3—41no—
354C2×C3—41no—
355C2×C3—62no—
356C2×C3—81no—
357C2×C3C2×D3×D411yescharacteristic
358C2×C3C22×#6711yescharacteristic
359D3—22no—
360D3—121no—
361D3—121no—
362D3—241no—
363C22—21no—
364C22—32no—
365C22—181no—
366C22D3×#6711yescharacteristic
367C4—21no—
368C4—32no—
369C4—62no—
370C4—62no—
371C4—181no—
372C4D3×#6711yescharacteristic
373C4C22×D3211yescharacteristic
374C3—21no—
375C3#577911yescharacteristic, complemented
376C3#583811yescharacteristic, complemented
377C2—21no—
378C2—41no—
379C2—62no—
380C2—361no—
381C2C2×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. #64297C2semidirect
3. #64159C2semidirect
4. C3×#951C2semidirect
5. #63948C2semidirect
6. D3×#216C2semidirect
7. #63935C2semidirect
8. #63929C2semidirect
9. C2×#3718C2semidirect
10. #63833C2semidirect
11. #63815C2semidirect
45. #1010D3semidirect
49. #951D3semidirect
318. C32#578semidirect
375. C3#5779semidirect
376. C3#5838semidirect

Maximal subgroups

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

subgroupindex
C3×#10102
#642972
#641592
C3×#9512
#639482
D3×#2162
#639352
#639292
C2×#37182
#638332
#638152
#58383
#57793