Finite Group Databasebeta

← #13159039

Subgroups — #13159039

Order 1728.

812 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 140 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#11723363C211yesmaximal, characteristic
2#11723354C211yesmaximal, characteristic, complemented
3#11723350C211yesmaximal, characteristic, complemented
4C4×C3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
5C3×#63632C211yesmaximal, characteristic, complemented
6C32×#805C211yesmaximal, characteristic, complemented
7#11722059C211yesmaximal, characteristic, complemented
8#10592523—31nomaximal
9#10590389—34nomaximal
10C3×#3828C2211yescharacteristic
11C32×#235C2211yescharacteristic
12C33×C42C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3748C2211yescharacteristic
16C32×#208C2211yescharacteristic
17#88798C411yescharacteristic, complemented
18#88793C411yescharacteristic, complemented
19#88792C411yescharacteristic, complemented
20#88789C411yescharacteristic, complemented
21#88787—21no—
22#88786—21no—
23C3⋊D3×C3⋊C8—21no—
24D3×C3⋊(C3⋊C8)—21no—
25#64159—31no—
26#64157—31no—
27#64116—31no—
28#63822—34no—
29#63815—34no—
30#63815—34no—
31C32×#104—31no—
32#63649—31no—
33#63632D311yescharacteristic, complemented
34C4×C3⋊(C3⋊C8)—31no—
35C4×C3⋊(C3⋊C8)—34no—
36C4×C3⋊(C3⋊C8)—64no—
37C3×#805—34no—
38C3×#805D314yescomplemented
39#63578—34no—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C3×#638—21no—
43C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
44C3×C4×C3⋊D3—21no—
45C3×#634C2×C411yescharacteristic
46C32×#67—21no—
47C32×D12C2×C411yescharacteristic
48C32×C4×D3—21no—
49C32×Dic6C2×C411yescharacteristic
50C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
51C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
52C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
53C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
54C3×C3⋊(C3⋊C8)—21no—
55C32×C3⋊C8—21no—
56#4824—94no—
57#4824—91no—
58C32×#41—31no—
59#3828C2×D311yescharacteristic
60C3×#235—34no—
61C3×#235C2×D314yes—
62C32×M16—31no—
63C32×C42—24no—
64C32×C42C2×D314yes—
65C32×C42C2×D311yescharacteristic
66#3756—31no—
67#3755—31no—
68#3754—31no—
69#3753—31no—
70#3748—61no—
71#3748C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—34no—
75C2×C3⋊(C3⋊C8)—34no—
76C2×C3⋊(C3⋊C8)—61no—
77C2×C3⋊(C3⋊C8)—64no—
78C2×C3⋊(C3⋊C8)—64no—
79#3743—61no—
80C8×C3⋊D3—61no—
81C3×#208—34no—
82C3×#208C2×D314yes—
83#3718—34no—
84#3715—34no—
85#3715—34no—
86#3713—34no—
87#3711—64no—
88#3711—64no—
89D3×C3⋊C8—64no—
90D3×C3⋊C8—64no—
91C22×C33#4111yescharacteristic
92C2×C3×C3⋊D3—21no—
93C2×C32×D3—21no—
94C33×C4—22no—
95C33×C4C22×C411yescharacteristic
96C33×C4C2×D411yescharacteristic
97C3×C3⋊Dic3—21no—
98C32×Dic3—21no—
99#950—94no—
100#950—91no—
101#948—94no—
102#948—91no—
103#907—94no—
104#907—91no—
105C3×#104—34no—
106C3×#104—91no—
107#822—94no—
108#822—91no—
109#805—34no—
110#805C3⋊D311yescharacteristic, complemented
111C4×C3⋊C8—94no—
112C4×C3⋊C8—91no—
113C4×C3⋊C8—184no—
114C32×Q8—31no—
115C32×D4—31no—
116C32×D4—61no—
117C2×C32×C4—28no—
118C2×C32×C4—24no—
119C2×C32×C4—21no—
120C2×C32×C4—24no—
121C2×C32×C4—61no—
122C2×C32×C4C22×D314yes—
123C2×C32×C4C22×D311yescharacteristic
124#638—21no—
125C4⋊(C3⋊D3)C4×D311yescharacteristic
126C4×C3⋊D3—21no—
127#634C4×D311yescharacteristic
128C3×#67—24no—
129C3×#67—64no—
130C3×D12—34no—
131C3×D12C4×D314yes—
132C3×C4×D3—24no—
133C3×C4×D3—64no—
134C3×Dic6—34no—
135C3×Dic6C4×D314yes—
136C32×C8—61no—
137C3⋊(C3⋊C8)—21no—
138C3⋊(C3⋊C8)—31no—
139C3⋊(C3⋊C8)—31no—
140C3⋊(C3⋊C8)—31no—
141C3⋊(C3⋊C8)—31no—
142C3⋊(C3⋊C8)—34no—
143C3⋊(C3⋊C8)—34no—
144C3⋊(C3⋊C8)—34no—
145C3⋊(C3⋊C8)—34no—
146C3⋊(C3⋊C8)—64no—
147C3⋊(C3⋊C8)—64no—
148C3⋊(C3⋊C8)—64no—
149C3⋊(C3⋊C8)—64no—
150C3×C3⋊C8—24no—
151C3×C3⋊C8—64no—
152#443—271noSylow
153C2×C33—21no—
154C2×C33C4×D411yescharacteristic
155C3×C3⋊D3—41no—
156C32×D3—41no—
157C3×#41—34no—
158C3×#41—91no—
159#235—34no—
160#235C2×C3⋊D311yescharacteristic
161C3×M16—34no—
162C3×M16—91no—
163C3×C42—24no—
164C3×C42D3214yes—
165C3×C42C2×C3⋊D311yescharacteristic
166#216—94no—
167#216—91no—
168#215—94no—
169#215—91no—
170#214—94no—
171#214—91no—
172#213—94no—
173#213—91no—
174#208—34no—
175#208—181no—
176#208—184no—
177#208C2×C3⋊D311yescharacteristic
178C2×C3⋊C8—94no—
179C2×C3⋊C8—94no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—91no—
182C2×C3⋊C8—184no—
183C2×C3⋊C8—184no—
184C2×C3⋊C8—181no—
185C2×C3⋊C8—184no—
186ΓL(1,𝔽25)—184no—
187ΓL(1,𝔽25)—181no—
188D3×C8—184no—
189D3×C8—181no—
190C22×C32—24no—
191C22×C32—61no—
192C22×C32#23714yes—
193C22×C32#23711yescharacteristic
194C2×C3⋊D3—21no—
195C2×C3×D3—24no—
196C2×C3×D3—64no—
197C32×C4—216no—
198C32×C4—28no—
199C32×C4—22no—
200C32×C4—24no—
201C32×C4—24no—
202C32×C4—61no—
203C32×C4C2×C4×D314yes—
204C32×C4C2×C4×D311yescharacteristic
205C32×C4D3×D414yes—
206C32×C4D3×D411yescharacteristic
207C3⋊Dic3—21no—
208C3×Dic3—24no—
209C3×Dic3—64no—
210#135—271no—
211#131—271no—
212#131—271no—
213#108—271no—
214#104—31no—
215#104—91no—
216C4×C8—271no—
217C33#44311yescharacteristic, complemented, Sylow
218C3×Q8—34no—
219C3×Q8—91no—
220C3×D4—34no—
221C3×D4—64no—
222C3×D4—91no—
223C3×D4—181no—
224C2×C3×C4—28no—
225C2×C3×C4—24no—
226C2×C3×C4—21no—
227C2×C3×C4—24no—
228C2×C3×C4—64no—
229C2×C3×C4—181no—
230C2×C3×C4C2×D3214yes—
231C2×C3×C4C22×C3⋊D311yescharacteristic
232#67—21no—
233#67—64no—
234D12—34no—
235D12C4×C3⋊D311yescharacteristic
236C4×D3—21no—
237C4×D3—64no—
238Dic6—34no—
239Dic6C4×C3⋊D311yescharacteristic
240C3×C8—64no—
241C3×C8—181no—
242C3⋊C8—21no—
243C3⋊C8—64no—
244C3⋊C8—94no—
245C3⋊C8—94no—
246C3⋊C8—94no—
247C3⋊C8—94no—
248C3⋊C8—91no—
249C3⋊C8—91no—
250C3⋊C8—91no—
251C3⋊C8—91no—
252C3⋊C8—184no—
253C3⋊C8—184no—
254C3⋊C8—184no—
255C3⋊C8—184no—
256C2×C32—28no—
257C2×C32—24no—
258C2×C32—21no—
259C2×C32—24no—
260C2×C32—121no—
261C2×C32#67⋊C414yes—
262C2×C32#67⋊C411yescharacteristic
263C3⋊D3—41no—
264C3×D3—44no—
265C3×D3—124no—
266#41—31no—
267#41—91no—
268Q16—271no—
269SD16—271no—
270SD16—271no—
271D8—271no—
272M16—31no—
273M16—91no—
274M16—541no—
275M16—541no—
276C2×C8—271no—
277C2×C8—271no—
278C2×C8—541no—
279C2×C8—541no—
280C42D3×C3⋊D311yescharacteristic
281C22×C3—24no—
282C22×C3—64no—
283C22×C3—181no—
284C22×C3#380514yes—
285C22×C3#383011yescharacteristic
286C2×D3—21no—
287C2×D3—64no—
288C3×C4—216no—
289C3×C4—22no—
290C3×C4—28no—
291C3×C4—24no—
292C3×C4—24no—
293C3×C4—64no—
294C3×C4—181no—
295C3×C4C4×D3214yes—
296C3×C4#380214yes—
297C3×C4C2×C4×C3⋊D311yescharacteristic
298C3×C4D4×C3⋊D311yescharacteristic
299Dic3—21no—
300Dic3—64no—
301C32—24no—
302C32#482414yescomplemented
303C32#482411yescharacteristic, complemented
304Q8—31no—
305Q8—91no—
306D4—31no—
307D4—61no—
308D4—91no—
309D4—181no—
310C2×C4—21no—
311C2×C4—61no—
312C2×C4—181no—
313C2×C4C2×D3×C3⋊D311yescharacteristic
314C8—61no—
315C8—181no—
316C8—271no—
317C8—271no—
318C8—271no—
319C8—271no—
320C2×C3—28no—
321C2×C3—21no—
322C2×C3—24no—
323C2×C3—24no—
324C2×C3—124no—
325C2×C3—361no—
326C2×C3#6390214yes—
327C2×C3#6409011yescharacteristic
328D3—41no—
329D3—124no—
330C22—61no—
331C22—181no—
332C22#8902911yescharacteristic
333C4—22no—
334C4—61no—
335C4—181no—
336C4C4×D3×C3⋊D311yescharacteristic
337C4#8902611yescharacteristic
338C3—24no—
339C3#1059038914yescomplemented
340C3#1059252311yescharacteristic, complemented
341C2—21no—
342C2—121no—
343C2—361no—
344C2#1172350811yescharacteristic
subgroupG/H
centerC4#89026
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC2×C33C4×D4
prime radical#11723363C2
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33#443
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11723354C2semidirect
3. #11723350C2semidirect
4. C4×C3⋊(C3⋊(C3⋊C8))C2semidirect
5. C3×#63632C2semidirect
6. C32×#805C2semidirect
7. #11722059C2semidirect
17. #88798C4semidirect
18. #88793C4semidirect
19. #88792C4semidirect
20. #88789C4semidirect
33. #63632D3semidirect
38. C3×#805D3semidirect
110. #805C3⋊D3semidirect
217. C33#443semidirect
302. C32#4824semidirect
303. C32#4824semidirect
339. C3#10590389semidirect
340. C3#10592523semidirect

Maximal subgroups

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

subgroupindex
#117233632
#117233542
#117233502
C4×C3⋊(C3⋊(C3⋊C8))2
C3×#636322
C32×#8052
#117220592
#105925233
#105903893