Finite Group Databasebeta

← #13159576

Subgroups — #13159576

Order 1728.

836 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×ΓL(1,𝔽25)C211yesmaximal, characteristic
2#11723423C211yesmaximal, characteristic
3C2×D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
4#11722184C211yesmaximal, characteristic, complemented
5C3×#63641C211yesmaximal, characteristic, complemented
6#11722051C211yesmaximal, characteristic, complemented
7#11722050C211yesmaximal, characteristic, complemented
8#10592818—31nomaximal
9#10590695—34nomaximal
10C2×C32×C4×D3C2211yescharacteristic
11C2×C33×C8C2211yescharacteristic
12C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C32×ΓL(1,𝔽25)—22no—
18Dic3⋊(C3⋊Dic3)C411yescharacteristic
19Dic3⋊(C3⋊Dic3)C411yescharacteristic
20(C2×D3)⋊(C3⋊Dic3)C411yescharacteristic
21(C2×D3)⋊(C3⋊Dic3)C411yescharacteristic
22D3×C3⋊(C3⋊C8)—22no—
23C2×C32×M16—31no—
24#64135—31no—
25C22×C3⋊(C3⋊C8)—31no—
26C2×C3×ΓL(1,𝔽25)D314yes—
27#63855—34no—
28C2×D3×C3⋊C8—34no—
29#63641—31no—
30#63641—31no—
31#63641—31no—
32#63641—34no—
33#63641—64no—
34#63641D311yescharacteristic, complemented
35C3×Dic3⋊C8—34no—
36#63572—34no—
37Dic3⋊(C3⋊C8)—34no—
38C22×C32×D3C2×C411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
41C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
42C2×C3×C3⋊Dic3C2×C411yescharacteristic
43C2×C3×C3⋊Dic3C2×C411yescharacteristic
44C2×C32×Dic3C2×C411yescharacteristic
45C32×C4×D3D412yes—
46C32×C4×D3Q812yes—
47C33×C8—21no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C3×C3⋊(C3⋊C8)—21no—
50C32×C3⋊C8—21no—
51#5119—94no—
52#4839—91no—
53C22×C32×C4—31no—
54C2×C3×C4×D3C2×D314yes—
55C32×M16—62no—
56C2×C32×C8—24no—
57C2×C32×C8—31no—
58C2×C32×C8C2×D314yes—
59C2×C32×C8C2×D311yescharacteristic
60#3757—31no—
61#3757—31no—
62C4⋊(C3⋊Dic3)—31no—
63C4⋊(C3⋊Dic3)—31no—
64C4×C3⋊Dic3—31no—
65C4×C3⋊Dic3—34no—
66C4×C3⋊Dic3—64no—
67C4×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—34no—
70C2×C3⋊(C3⋊C8)—62no—
71C2×C3⋊(C3⋊C8)—64no—
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×C4×Dic3—34no—
74C2×C3×C3⋊C8—34no—
75C2×C3×C3⋊C8C2×D314yes—
76C3×ΓL(1,𝔽25)—28no—
77Dic3⋊Dic3—34no—
78Dic3⋊Dic3—34no—
79(C2×D3)⋊Dic3—34no—
80(C2×D3)⋊Dic3—34no—
81D3×C3⋊C8—68no—
82C22×C33C22×C411yescharacteristic
83C2×C32×D3—21no—
84C2×C32×D3—21no—
85C2×C32×D3C4⋊C412yes—
86C33×C4C2×D411yescharacteristic
87C33×C4C2×Q811yescharacteristic
88C3⋊(C3⋊Dic3)—21no—
89C3⋊(C3⋊Dic3)—21no—
90C3×C3⋊Dic3—21no—
91C3×C3⋊Dic3—21no—
92C32×Dic3C4⋊C412yes—
93C2×C3×M16—34no—
94#926—94no—
95C22×C3⋊C8—94no—
96C2×ΓL(1,𝔽25)C3⋊D311yescharacteristic
97C2×D3×C8—91no—
98#872—91no—
99C3×C4⋊C8—91no—
100Dic3⋊C8—34no—
101Dic3⋊C8—94no—
102Dic3⋊C8—94no—
103Dic3⋊C8—94no—
104Dic3⋊C8—91no—
105Dic3⋊C8—91no—
106Dic3⋊C8—184no—
107C4⋊(C3⋊C8)—91no—
108C23×C32—31no—
109C22×C3×D3C4×D314yes—
110C2×C32×C4—24no—
111C2×C32×C4—31no—
112C2×C32×C4—32no—
113C2×C32×C4—32no—
114C2×C32×C4C22×D314yes—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—31no—
117C2×C3⋊Dic3—31no—
118C2×C3⋊Dic3—34no—
119C2×C3⋊Dic3—34no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3C4×D311yescharacteristic
123C2×C3⋊Dic3C4×D311yescharacteristic
124C2×C3×Dic3—34no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3C4×D314yes—
127C3×C4×D3Dic618yes—
128C3×C4×D3#6718yes—
129C32×C8—21no—
130C32×C8—24no—
131C32×C8—44no—
132C32×C8—61no—
133C3⋊(C3⋊C8)—21no—
134C3⋊(C3⋊C8)—61no—
135C3⋊(C3⋊C8)—64no—
136C3⋊(C3⋊C8)—124no—
137C3×C3⋊C8—24no—
138C3×C3⋊C8—64no—
139#424—271noSylow
140C2×C33C2×C4⋊C411yescharacteristic
141C2×C33#13111yescharacteristic
142C2×C33#13111yescharacteristic
143C32×D3—22no—
144C22×C3×C4—34no—
145C2×C4×D3C2×C3⋊D311yescharacteristic
146C3×M16—68no—
147C2×C3×C8—24no—
148C2×C3×C8—34no—
149C2×C3×C8—91no—
150C2×C3×C8D3214yes—
151C2×C3×C8C2×C3⋊D311yescharacteristic
152C3×C42—91no—
153#217—94no—
154#217—94no—
155(C2×D3)⋊C4—91no—
156(C2×D3)⋊C4—91no—
157C4⋊Dic3—94no—
158C4⋊Dic3—94no—
159Dic3⋊C4—91no—
160Dic3⋊C4—91no—
161C4×Dic3—34no—
162C4×Dic3—94no—
163C4×Dic3—91no—
164C4×Dic3—184no—
165C2×C3⋊C8—34no—
166C2×C3⋊C8—94no—
167C2×C3⋊C8—91no—
168C2×C3⋊C8—184no—
169C2×C3⋊C8—188no—
170C2×C3⋊C8C2×C3⋊D311yescharacteristic
171ΓL(1,𝔽25)—22no—
172D3×C8—182no—
173C22×C32—24no—
174C22×C32—32no—
175C22×C32—61no—
176C22×C32—61no—
177C22×C32C2×C4×D314yes—
178C22×C32C2×C4×D311yescharacteristic
179C2×C3×D3—24no—
180C2×C3×D3—24no—
181C2×C3×D3Dic3⋊C418yes—
182C32×C4—24no—
183C32×C4—24no—
184C32×C4—32no—
185C32×C4C2×Dic614yes—
186C32×C4D3×D411yescharacteristic
187C32×C4D3×Q811yescharacteristic
188C32×C4C2×#6714yes—
189C3⋊Dic3—21no—
190C3⋊Dic3—21no—
191C3⋊Dic3—68no—
192C3⋊Dic3—61no—
193C3⋊Dic3—64no—
194C3⋊Dic3—61no—
195C3⋊Dic3—64no—
196C3⋊Dic3—124no—
197C3×Dic3—64no—
198C3×Dic3—64no—
199C3×Dic3Dic3⋊C418yes—
200C2×M16—31no—
201C22×C8—271no—
202#117—271no—
203C4⋊C8—91no—
204C4⋊C8—271no—
205C4⋊C8—271no—
206C4⋊C8—271no—
207C33#42411yescharacteristic, complemented, Sylow
208C23×C3—34no—
209C22×D3C4×C3⋊D311yescharacteristic
210C2×C3×C4—24no—
211C2×C3×C4—34no—
212C2×C3×C4—38no—
213C2×C3×C4—38no—
214C2×C3×C4—91no—
215C2×C3×C4—91no—
216C2×C3×C4C2×D3214yes—
217C2×C3×C4C22×C3⋊D311yescharacteristic
218C2×Dic3—34no—
219C2×Dic3—34no—
220C2×Dic3—94no—
221C2×Dic3—94no—
222C2×Dic3—91no—
223C2×Dic3—91no—
224C2×Dic3—184no—
225C2×Dic3—184no—
226C2×Dic3C4×C3⋊D311yescharacteristic
227C4×D3#63412yes—
228C4×D3#63812yes—
229C3×C8—24no—
230C3×C8—21no—
231C3×C8—44no—
232C3×C8—64no—
233C3×C8—181no—
234C3⋊C8—21no—
235C3⋊C8—64no—
236C3⋊C8—184no—
237C3⋊C8—181no—
238C3⋊C8—364no—
239C2×C32—24no—
240C2×C32—24no—
241C2×C32—24no—
242C2×C32—62no—
243C2×C32D3×C4⋊C411yescharacteristic
244C2×C32#90111yescharacteristic
245C2×C32#90714yes—
246C2×C32#90714yes—
247C2×C32#90711yescharacteristic
248C2×C32C2×Dic3⋊C414yes—
249C3×D3—28no—
250C22×C4—31no—
251M16—62no—
252C2×C8—31no—
253C2×C8—91no—
254C2×C8—271no—
255C2×C8—542no—
256C2×C8D3×C3⋊D311yescharacteristic
257C4⋊C4—271no—
258C4⋊C4—271no—
259C22⋊C4—271no—
260C22⋊C4—271no—
261C42—91no—
262C42—271no—
263C22×C3—24no—
264C22×C3—38no—
265C22×C3—64no—
266C22×C3—64no—
267C22×C3C4×D3214yes—
268C22×C3C2×C4×C3⋊D311yescharacteristic
269C2×D3—21no—
270C2×D3—21no—
271C2×D3#375012yes—
272C3×C4—24no—
273C3×C4—24no—
274C3×C4—38no—
275C3×C4—181no—
276C3×C4—181no—
277C3×C4D3×Dic614yes—
278C3×C4D3×#6714yes—
279C3×C4C2×#63411yescharacteristic
280C3×C4C2×#63811yescharacteristic
281Dic3—64no—
282Dic3—64no—
283Dic3—188no—
284Dic3—184no—
285Dic3—184no—
286Dic3—181no—
287Dic3—181no—
288Dic3—364no—
289Dic3#375012yes—
290C32—24no—
291C32#483911yescharacteristic, complemented
292C32#511914yescomplemented
293C23—31no—
294C2×C4—31no—
295C2×C4—32no—
296C2×C4—32no—
297C2×C4—91no—
298C2×C4—91no—
299C2×C4—271no—
300C2×C4—271no—
301C2×C4C2×D3×C3⋊D311yescharacteristic
302C8—21no—
303C8—61no—
304C8—181no—
305C8—541no—
306C2×C3—24no—
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—68no—
310C2×C3#6380414yes—
311C2×C3#6380514yes—
312C2×C3D3×Dic3⋊C414yes—
313C2×C3#6411611yescharacteristic
314C2×C3#6411611yescharacteristic
315C2×C3C2×#375011yescharacteristic
316D3—22no—
317C22—32no—
318C22—61no—
319C22—61no—
320C22C4×D3×C3⋊D311yescharacteristic
321C4—32no—
322C4—181no—
323C4—181no—
324C4—541no—
325C4—541no—
326C4D3×#63411yescharacteristic
327C4D3×#63811yescharacteristic
328C3—24no—
329C3#1059069514yescomplemented
330C3#1059281811yescharacteristic, complemented
331C2—62no—
332C2#1172333311yescharacteristic
333C2#1172333411yescharacteristic
334C2D3×#375011yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC2×C33×C8C22
prime radicalC22×C32×D3C2×C4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#424
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. #11722184C2semidirect
5. C3×#63641C2semidirect
6. #11722051C2semidirect
7. #11722050C2semidirect
34. #63641D3semidirect
207. C33#424semidirect
291. C32#4839semidirect
292. C32#5119semidirect
329. C3#10590695semidirect
330. C3#10592818semidirect

Maximal subgroups

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

subgroupindex
C2×C32×ΓL(1,𝔽25)2
#117234232
C2×D3×C3⋊(C3⋊C8)2
#117221842
C3×#636412
#117220512
#117220502
#105928183
#105906953