Finite Group Databasebeta

← #13138909

Subgroups — #13138909

Order 1728.

523 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×#88527C211yesmaximal, characteristic
2#11721097C211yesmaximal, characteristic
3C2×#88434C211yesmaximal, characteristic
4#63642⋊C3C211yesmaximal, characteristic, complemented
5C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
6#11720312C211yesmaximal, characteristic, complemented
7#11720310C211yesmaximal, characteristic, complemented
8#10590909—31nomaximal
9#10590908—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C2×#6180C2211yescharacteristic
13#88527C414yescomplemented
14C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18#88451C412yes—
19#88449C412yes—
20#88434C414yescomplemented
21C2×C3×ΓL(1,𝔽25)—31no—
22C2×C3×ΓL(1,𝔽25)—31no—
23C2×C3×ΓL(1,𝔽25)—61no—
24#63855—31no—
25#63855—31no—
26C2×D3×C3⋊C8—31no—
27C2×D3×C3⋊C8—31no—
28#63642D311yescharacteristic, complemented
29C8×C3⋊Dic3D311yescharacteristic, complemented
30C3×#815—31no—
31C3×C8×Dic3—31no—
32#63555—31no—
33#63554—31no—
34#63553—31no—
35Dic3×C3⋊C8—31no—
36C22×#272C2×C411yescharacteristic
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084C2×C411yescharacteristic
39C4×#272C2×C412yes—
40C4×#272C2×C412yes—
41C2×(C3⋊Dic3)⋊C3C2×C412yes—
42C2×(C3⋊Dic3)⋊C3C2×C412yes—
43C8×H(3,𝔽3)C2×C412yes—
44#6180C2×C412yes—
45(C3⋊(C3⋊C8))⋊C3C2×C412yes—
46(C3⋊(C3⋊C8))⋊C3C2×C412yes—
47#5122—91no—
48#4708—91no—
49#4689—91no—
50C2×C3×C4×D3—31no—
51C2×C3×C4×D3—31no—
52C2×C3×C4×D3—61no—
53C2×C32×C8—21no—
54C2×C32×C8C2×D311yescharacteristic
55C2×C32×C8C2×D311yescharacteristic
56C4×C3⋊Dic3C2×D311yescharacteristic
57C4×C3⋊Dic3C2×D311yescharacteristic
58C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
59C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
60C3×C4×Dic3—31no—
61C3×C4×Dic3—31no—
62C2×C3×C3⋊C8—31no—
63C2×C3×C3⋊C8—31no—
64C2×C3×C3⋊C8—31no—
65C2×C3×C3⋊C8—31no—
66C2×C3×C3⋊C8—61no—
67C3×ΓL(1,𝔽25)—34no—
68C3×ΓL(1,𝔽25)—34no—
69C3×ΓL(1,𝔽25)—64no—
70Dic3⋊Dic3—32no—
71Dic3⋊Dic3—32no—
72(C2×D3)⋊Dic3—32no—
73(C2×D3)⋊Dic3—32no—
74D3×C3⋊C8—34no—
75D3×C3⋊C8—34no—
76C22×H(3,𝔽3)C22×C411yescharacteristic
77C2×#272—22no—
78C2×#272C4212yes—
79C4×H(3,𝔽3)C22×C411yescharacteristic
80C4×H(3,𝔽3)C22×C411yescharacteristic
81#1084C4212yes—
82(C3⋊Dic3)⋊C3—22no—
83(C3⋊Dic3)⋊C3—22no—
84C2×C3×M16—91no—
85#926—91no—
86C22×C3⋊C8—91no—
87C2×ΓL(1,𝔽25)—31no—
88C2×ΓL(1,𝔽25)—31no—
89C2×ΓL(1,𝔽25)—61no—
90C2×D3×C8—91no—
91C2×D3×C8—91no—
92#872—91no—
93#872—91no—
94C3×#97—91no—
95C3×C4×C8—91no—
96#815—31no—
97#815—91no—
98#815—91no—
99#815—91no—
100C8×Dic3—31no—
101C8×Dic3—91no—
102C8×Dic3—91no—
103C8×Dic3—91no—
104#803—91no—
105C4×C3⋊C8—91no—
106C22×C3×D3—31no—
107C22×C3×D3—31no—
108C22×C3×D3—61no—
109C2×C32×C4—21no—
110C2×C32×C4C22×D311yescharacteristic
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3C4×D312yes—
113C2×C3⋊Dic3C4×D312yes—
114C2×C3×Dic3—31no—
115C2×C3×Dic3—31no—
116C2×C3×Dic3—32no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—61no—
119C3×C4×D3—32no—
120C3×C4×D3—32no—
121C3×C4×D3—32no—
122C3×C4×D3—32no—
123C3×C4×D3—62no—
124C3×C4×D3—62no—
125C32×C8—22no—
126C32×C8C4×D312yes—
127C32×C8C4×D312yes—
128C3⋊(C3⋊C8)C4×D312yes—
129C3⋊(C3⋊C8)C4×D312yes—
130C3×C3⋊C8—32no—
131C3×C3⋊C8—32no—
132C3×C3⋊C8—32no—
133C3×C3⋊C8—32no—
134C3×C3⋊C8—62no—
135#405—271noSylow
136C2×H(3,𝔽3)C2×C4211yescharacteristic
137C2×H(3,𝔽3)#13112yes—
138#272—22no—
139C22×C3×C4—91no—
140C2×C4×D3—31no—
141C2×C4×D3—31no—
142C2×C4×D3—61no—
143C3×M16—94no—
144C2×C3×C8—31no—
145C2×C3×C8—31no—
146C2×C3×C8—61no—
147C2×C3×C8—91no—
148C2×C3×C8—91no—
149C2×C3×C8—91no—
150C2×C3×C8D3211yescharacteristic
151C3×C42—91no—
152C3×C42—91no—
153#217—92no—
154(C2×D3)⋊C4—92no—
155(C2×D3)⋊C4—92no—
156C4⋊Dic3—92no—
157Dic3⋊C4—92no—
158Dic3⋊C4—92no—
159C4×Dic3—31no—
160C4×Dic3—31no—
161C4×Dic3—91no—
162C4×Dic3—91no—
163C2×C3⋊C8—31no—
164C2×C3⋊C8—31no—
165C2×C3⋊C8—31no—
166C2×C3⋊C8—31no—
167C2×C3⋊C8—61no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—91no—
170C2×C3⋊C8—94no—
171ΓL(1,𝔽25)—34no—
172ΓL(1,𝔽25)—34no—
173ΓL(1,𝔽25)—64no—
174D3×C8—94no—
175D3×C8—94no—
176C22×C32—21no—
177C22×C32C2×C4×D311yescharacteristic
178C22×C32C2×C4×D311yescharacteristic
179C2×C3×D3—32no—
180C2×C3×D3—32no—
181C2×C3×D3—62no—
182C2×C3×D3—62no—
183C2×C3×D3—62no—
184C2×C3×D3—122no—
185C32×C4—21no—
186C32×C4—21no—
187C32×C4C2×C4×D311yescharacteristic
188C32×C4C2×C4×D311yescharacteristic
189C32×C4C2×C4×D311yescharacteristic
190C32×C4C2×C4×D311yescharacteristic
191C3⋊Dic3—22no—
192C3⋊Dic3—22no—
193C3×Dic3—32no—
194C3×Dic3—32no—
195C3×Dic3—62no—
196C3×Dic3—62no—
197C3×Dic3—62no—
198C2×M16—91no—
199C22×C8—271no—
200#117—271no—
201#97—91no—
202#97—271no—
203C4×C8—91no—
204C4×C8—271no—
205H(3,𝔽3)#40511yescharacteristic, complemented, Sylow
206C23×C3—91no—
207C22×D3—31no—
208C22×D3—31no—
209C22×D3—61no—
210C2×C3×C4—31no—
211C2×C3×C4—31no—
212C2×C3×C4—61no—
213C2×C3×C4—91no—
214C2×C3×C4—92no—
215C2×C3×C4—92no—
216C2×C3×C4—92no—
217C2×C3×C4—92no—
218C2×C3×C4C2×D3211yescharacteristic
219C2×Dic3—31no—
220C2×Dic3—31no—
221C2×Dic3—32no—
222C2×Dic3—32no—
223C2×Dic3—61no—
224C2×Dic3—92no—
225C2×Dic3—92no—
226C4×D3—32no—
227C4×D3—32no—
228C4×D3—32no—
229C4×D3—32no—
230C4×D3—62no—
231C4×D3—62no—
232C3×C8—32no—
233C3×C8—32no—
234C3×C8—62no—
235C3×C8—92no—
236C3×C8—92no—
237C3×C8—92no—
238C3×C8#62412yes—
239C3⋊C8—32no—
240C3⋊C8—32no—
241C3⋊C8—32no—
242C3⋊C8—32no—
243C3⋊C8—62no—
244C3⋊C8—92no—
245C3⋊C8—92no—
246C2×C32—22no—
247C2×C32—21no—
248C2×C32C42×D311yescharacteristic
249C2×C32C42×D311yescharacteristic
250C2×C32#90112yes—
251C2×C32#90712yes—
252C3×D3—62no—
253C3×D3—62no—
254C3×D3—122no—
255C22×C4—91no—
256M16—94no—
257C2×C8—91no—
258C2×C8—91no—
259C2×C8—91no—
260C2×C8—274no—
261C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
262C4⋊C4—272no—
263C22⋊C4—272no—
264C42—91no—
265C42—91no—
266C22×C3—31no—
267C22×C3—31no—
268C22×C3—61no—
269C22×C3—92no—
270C22×C3—182no—
271C22×C3C4×D3211yescharacteristic
272C2×D3—32no—
273C2×D3—32no—
274C2×D3—62no—
275C2×D3—62no—
276C2×D3—62no—
277C2×D3—122no—
278C3×C4—31no—
279C3×C4—31no—
280C3×C4—31no—
281C3×C4—31no—
282C3×C4—61no—
283C3×C4—61no—
284C3×C4—92no—
285C3×C4—182no—
286C3×C4—182no—
287C3×C4C4×D3211yescharacteristic
288C3×C4C2×#62411yescharacteristic
289Dic3—32no—
290Dic3—32no—
291Dic3—62no—
292Dic3—62no—
293Dic3—62no—
294Dic3—182no—
295Dic3—182no—
296C32—21no—
297C32#468911yescharacteristic, complemented
298C32#470811yescharacteristic, complemented
299C23—91no—
300C2×C4—91no—
301C2×C4—92no—
302C2×C4—92no—
303C2×C4—92no—
304C2×C4—92no—
305C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
306C8—92no—
307C8—92no—
308C8—92no—
309C8#619912yes—
310C2×C3—32no—
311C2×C3—32no—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—62no—
315C2×C3—61no—
316C2×C3—182no—
317C2×C3#6380412yes—
318C2×C3C4×#62411yescharacteristic
319D3—62no—
320D3—62no—
321D3—122no—
322C22—92no—
323C22—182no—
324C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
325C4—92no—
326C4—182no—
327C4—182no—
328C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
329C4C2×#619911yescharacteristic
330C3—31no—
331C3—31no—
332C3—61no—
333C3#1059091011yescharacteristic
334C2—182no—
335C2#1172084112yes—
336C2C4×#619911yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C2×C42
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC22×#272C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#405
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. #63642⋊C3C2semidirect
5. C8×(C3⋊Dic3)⋊C3C2semidirect
6. #11720312C2semidirect
7. #11720310C2semidirect
13. #88527C4semidirect
20. #88434C4semidirect
28. #63642D3semidirect
29. C8×C3⋊Dic3D3semidirect
205. H(3,𝔽3)#405semidirect
297. C32#4689semidirect
298. C32#4708semidirect

Maximal subgroups

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

subgroupindex
C2×#885272
#117210972
C2×#884342
#63642⋊C32
C8×(C3⋊Dic3)⋊C32
#117203122
#117203102
#105909093
#105909083