Finite Group Databasebeta

← #13139030

Subgroups — #13139030

Order 1728.

733 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 733 conjugacy classes of subgroups — too many to draw. Only its 95 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×C4×((C3⋊D3)⋊C3)⋊C2C211yesmaximal, characteristic
2C2×#88527C211yesmaximal, characteristic, complemented
3C2×#88435C211yesmaximal, characteristic, complemented
4#63647⋊C3C212yesmaximal, complemented
5#11720317C212yesmaximal, complemented
6#10590936—32nomaximal
7C22×((C3⋊D3)⋊C3)⋊C2C411yescharacteristic
8C2×C4×#272C2211yescharacteristic
9C2×C4×(C3⋊D3)⋊C3C2212yes—
10C2×#6199C411yescharacteristic
11C2×((C3⋊D3)⋊C3)⋊C4C412yes—
12C4×((C3⋊D3)⋊C3)⋊C2—22no—
13C4×((C3⋊D3)⋊C3)⋊C2—22no—
14C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
15C2×#6180C2211yescharacteristic, complemented
16#88527—22no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
18#88435—22no—
19C2×C4×D32—32no—
20C2×C3×ΓL(1,𝔽25)—32no—
21C2×C3×ΓL(1,𝔽25)—61no—
22C2×#3711—32no—
23#63647D312yescomplemented
24C3×(C2×D3)⋊C8—32no—
25#63557—32no—
26(C2×D3)⋊(C3⋊C8)—32no—
27C22×#272C2×C411yescharacteristic
28C22×(C3⋊D3)⋊C3C2×C412yes—
29C2×((C3⋊D3)⋊C3)⋊C2—22no—
30C2×((C3⋊D3)⋊C3)⋊C2—24no—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084C2×C411yescharacteristic
33C4×#272D412yes—
34C4×#272D412yes—
35C2×(C3⋊Dic3)⋊C3C2×C412yes—
36C4×(C3⋊D3)⋊C3—22no—
37C4×(C3⋊D3)⋊C3—22no—
38#6199—22no—
39((C3⋊D3)⋊C3)⋊C4—24no—
40C8×H(3,𝔽3)—21no—
41#6180—21no—
42(C3⋊(C3⋊C8))⋊C3—22no—
43#5126—91no—
44#4726—92no—
45C22×D32—32no—
46C2×C4×C3⋊D3C2×D312yes—
47C2×C3×C4×D3—32no—
48C2×C3×C4×D3—32no—
49C2×C3×C4×D3—61no—
50C2×#624—32no—
51C2×D3×Dic3—32no—
52C2×D3×Dic3—32no—
53C4×D32—64no—
54C4×D32—64no—
55C2×C32×C8—21no—
56C2×C32×C8C2×D312yescomplemented
57C2×C3⋊(C3⋊C8)C2×D312yescomplemented
58C2×C3×C3⋊C8—32no—
59C2×C3×C3⋊C8—32no—
60C2×C3×C3⋊C8—61no—
61C3×ΓL(1,𝔽25)—64no—
62C3×ΓL(1,𝔽25)—64no—
63#3711—64no—
64C22×H(3,𝔽3)C22×C411yescharacteristic
65C2×#272—22no—
66C2×#272C22⋊C412yes—
67C2×(C3⋊D3)⋊C3—22no—
68C2×(C3⋊D3)⋊C3—22no—
69C2×(C3⋊D3)⋊C3M1614yescomplemented
70((C3⋊D3)⋊C3)⋊C2—28no—
71C4×H(3,𝔽3)C2×D411yescharacteristic
72C4×H(3,𝔽3)C2×D411yescharacteristic
73#1084C22⋊C412yes—
74(C3⋊Dic3)⋊C3—22no—
75C22×C4×D3—91no—
76C22×C4×D3—92no—
77C2×C3×M16—91no—
78C2×#208—91no—
79C2×ΓL(1,𝔽25)—32no—
80C2×ΓL(1,𝔽25)—61no—
81C2×ΓL(1,𝔽25)—92no—
82C3×C22⋊C8—92no—
83#830—92no—
84(C2×D3)⋊C8—32no—
85(C2×D3)⋊C8—92no—
86(C2×D3)⋊C8—92no—
87(C2×D3)⋊C8—92no—
88C22×C3⋊D3C4×D312yes—
89C22×C3×D3—32no—
90C22×C3×D3—32no—
91C22×C3×D3—61no—
92C2×D32—64no—
93C2×D32—64no—
94C2×D32—64no—
95C2×C32×C4—21no—
96C2×C32×C4C22×D312yes—
97C2×C3⋊Dic3C4×D312yes—
98C4×C3⋊D3—22no—
99C4×C3⋊D3—22no—
100C2×C3×Dic3—32no—
101C2×C3×Dic3—32no—
102C2×C3×Dic3—61no—
103C3×C4×D3—34no—
104C3×C4×D3—34no—
105C3×C4×D3—62no—
106C3×C4×D3—62no—
107C3×C4×D3—62no—
108C3×C4×D3—62no—
109#624—64no—
110D3×Dic3—64no—
111D3×Dic3—64no—
112C32×C8—22no—
113C32×C8—22no—
114C3⋊(C3⋊C8)—22no—
115C3×C3⋊C8—62no—
116C3×C3⋊C8—62no—
117C3×C3⋊C8—62no—
118#407—271noSylow
119C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
120C2×H(3,𝔽3)C2×M1612yes—
121#272—22no—
122(C3⋊D3)⋊C3—24no—
123C23×D3—91no—
124C23×D3—92no—
125C22×C3×C4—92no—
126C22×C3×C4—91no—
127C22×Dic3—91no—
128C22×Dic3—92no—
129C2×C4×D3—32no—
130C2×C4×D3—32no—
131C2×C4×D3—61no—
132C2×C4×D3—92no—
133C2×C4×D3—92no—
134C2×C4×D3—92no—
135C2×C4×D3—92no—
136C2×C4×D3—184no—
137C2×C4×D3—184no—
138C2×C4×D3—182no—
139C2×C4×D3—182no—
140C3×M16—182no—
141C2×C3×C8—32no—
142C2×C3×C8—61no—
143C2×C3×C8—92no—
144C2×C3×C8—91no—
145C2×C3×C8D3211yescharacteristic
146#208—182no—
147C2×C3⋊C8—32no—
148C2×C3⋊C8—32no—
149C2×C3⋊C8—61no—
150C2×C3⋊C8—92no—
151ΓL(1,𝔽25)—64no—
152ΓL(1,𝔽25)—64no—
153ΓL(1,𝔽25)—184no—
154C22×C32—21no—
155C22×C32C2×C4×D312yes—
156C2×C3⋊D3—22no—
157C2×C3⋊D3—22no—
158C2×C3⋊D3ΓL(1,𝔽25)14yescomplemented
159C2×C3×D3—34no—
160C2×C3×D3—34no—
161C2×C3×D3—62no—
162C2×C3×D3—62no—
163C2×C3×D3—62no—
164C2×C3×D3—62no—
165C2×C3×D3—62no—
166C2×C3×D3—122no—
167D32—616no—
168C32×C4—21no—
169C32×C4—21no—
170C32×C4D3×D412yes—
171C32×C4D3×D412yes—
172C3⋊Dic3—22no—
173C3×Dic3—34no—
174C3×Dic3—62no—
175C3×Dic3—62no—
176C23×C4—271no—
177C2×M16—91no—
178C2×M16—271no—
179C22⋊C8—92no—
180C22⋊C8—272no—
181H(3,𝔽3)#40711yescharacteristic, complemented, Sylow
182C23×C3—92no—
183C23×C3—91no—
184C22×D3—32no—
185C22×D3—32no—
186C22×D3—61no—
187C22×D3—92no—
188C22×D3—184no—
189C22×D3—184no—
190C22×D3—182no—
191C22×D3—184no—
192C22×D3—184no—
193C2×C3×C4—32no—
194C2×C3×C4—61no—
195C2×C3×C4—91no—
196C2×C3×C4—92no—
197C2×C3×C4—92no—
198C2×C3×C4—92no—
199C2×C3×C4—182no—
200C2×C3×C4—182no—
201C2×C3×C4C2×D3211yescharacteristic
202C2×Dic3—32no—
203C2×Dic3—32no—
204C2×Dic3—61no—
205C2×Dic3—92no—
206C2×Dic3—184no—
207C2×Dic3—182no—
208C4×D3—34no—
209C4×D3—34no—
210C4×D3—62no—
211C4×D3—62no—
212C4×D3—62no—
213C4×D3—62no—
214C4×D3—182no—
215C4×D3—182no—
216C4×D3—184no—
217C4×D3—184no—
218C4×D3—184no—
219C3×C8—21no—
220C3×C8—62no—
221C3×C8—62no—
222C3×C8—182no—
223C3×C8—181no—
224C3⋊C8—62no—
225C3⋊C8—62no—
226C3⋊C8—62no—
227C3⋊C8—182no—
228C2×C32—22no—
229C2×C32—21no—
230C2×C32D3×C22⋊C412yes—
231C2×C32C2×ΓL(1,𝔽25)12yes—
232C2×C32D3×M1612yes—
233C3⋊D3—24no—
234C3×D3—64no—
235C3×D3—64no—
236C3×D3—122no—
237C24—271no—
238C22×C4—91no—
239C22×C4—92no—
240C22×C4—272no—
241C22×C4—271no—
242C22×C4—542no—
243C22×C4—542no—
244M16—182no—
245M16—542no—
246C2×C8—92no—
247C2×C8—91no—
248C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
249C22×C3—32no—
250C22×C3—61no—
251C22×C3—94no—
252C22×C3—92no—
253C22×C3—182no—
254C22×C3—182no—
255C22×C3—182no—
256C22×C3C4×D3211yescharacteristic
257C2×D3—34no—
258C2×D3—34no—
259C2×D3—62no—
260C2×D3—62no—
261C2×D3—62no—
262C2×D3—62no—
263C2×D3—62no—
264C2×D3—94no—
265C2×D3—122no—
266C2×D3—182no—
267C2×D3—182no—
268C2×D3—188no—
269C2×D3—1816no—
270C3×C4—32no—
271C3×C4—32no—
272C3×C4—61no—
273C3×C4—61no—
274C3×C4—92no—
275C3×C4—182no—
276C3×C4#380211yescharacteristic
277C3×C4#381111yescharacteristic
278Dic3—34no—
279Dic3—62no—
280Dic3—62no—
281Dic3—182no—
282C32—21no—
283C32#472612yescomplemented
284C23—91no—
285C23—92no—
286C23—542no—
287C23—544no—
288C2×C4—91no—
289C2×C4—92no—
290C2×C4—92no—
291C2×C4—92no—
292C2×C4—182no—
293C2×C4—182no—
294C2×C4—544no—
295C2×C4—542no—
296C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
297C8—21no—
298C8—182no—
299C8—181no—
300C2×C3—32no—
301C2×C3—32no—
302C2×C3—32no—
303C2×C3—62no—
304C2×C3—61no—
305C2×C3—184no—
306C2×C3—182no—
307C2×C3D3×ΓL(1,𝔽25)12yes—
308C2×C3#6392111yescharacteristic
309D3—64no—
310D3—64no—
311D3—122no—
312D3—184no—
313C22—94no—
314C22—92no—
315C22—182no—
316C22—182no—
317C22—182no—
318C22—548no—
319C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
320C4—92no—
321C4—182no—
322C4#8865511yescharacteristic
323C4#8867711yescharacteristic
324C3—32no—
325C3—61no—
326C3#1059093711yescharacteristic
327C2—182no—
328C2—184no—
329C2#1172083512yes—
330C2#1172116111yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3C4×D32
prime radicalC22×((C3⋊D3)⋊C3)⋊C2C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#407
LCSG ▷ C22×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
2. C2×#88527C2semidirect
3. C2×#88435C2semidirect
4. #63647⋊C3C2semidirect
5. #11720317C2semidirect
14. C2×C8×H(3,𝔽3)C22semidirect
15. C2×#6180C22semidirect
17. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
23. #63647D3semidirect
56. C2×C32×C8C2×D3semidirect
57. C2×C3⋊(C3⋊C8)C2×D3semidirect
69. C2×(C3⋊D3)⋊C3M16semidirect
158. C2×C3⋊D3ΓL(1,𝔽25)semidirect
181. H(3,𝔽3)#407semidirect
248. C2×C8((C3⋊D3)⋊C3)⋊C2semidirect
283. C32#4726semidirect

Maximal subgroups

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

subgroupindex
C2×C4×((C3⋊D3)⋊C3)⋊C22
C2×#885272
C2×#884352
#63647⋊C32
#117203172
#105909363