Finite Group Databasebeta

← #13140226

Subgroups — #13140226

Order 1728.

729 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 729 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C22×#6196C211yesmaximal, characteristic
2C2×#88540C211yesmaximal, characteristic
3C2×#3750⋊C3C212yesmaximal
4#11720334C212yesmaximal
5#11720334C211yesmaximal, characteristic
6#10591195—32nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3C2212yes—
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C2×#6196—22no—
13C2×#6196—24no—
14#88540—22no—
15#3750⋊C3—24no—
16C22×#627—32no—
17C2×#3750D312yes—
18C2×C3×Dic3⋊C4—32no—
19C2×C3×Dic3⋊C4—32no—
20C2×C3×Dic3⋊C4—61no—
21#63579—32no—
22#63579—32no—
23#63579—32no—
24C23×H(3,𝔽3)C2311yescharacteristic
25C2×C4×H(3,𝔽3)—21no—
26C2×C4×H(3,𝔽3)—22no—
27C2×#1084—21no—
28C2×#1084—22no—
29C2×#1084—21no—
30C2×#1084D412yes—
31C2×#1084Q812yescomplemented
32C2×(C3⋊Dic3)⋊C3—22no—
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3—22no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3D414yes—
37C2×(C3⋊Dic3)⋊C3Q814yescomplemented
38#6196—28no—
39#4979—91no—
40#4979—92no—
41C22×C32×C4—21no—
42C22×C32×C4C2×D312yes—
43C22×C3⋊Dic3C2×D312yes—
44C22×C3⋊Dic3C2×D312yes—
45C22×C3×Dic3—32no—
46C22×C3×Dic3—32no—
47C22×C3×Dic3—32no—
48C22×C3×Dic3—32no—
49C22×C3×Dic3—61no—
50C22×C3×Dic3—61no—
51C2×#627—64no—
52C2×#627—64no—
53C2×#627—64no—
54#3750—24no—
55C3×Dic3⋊C4—64no—
56C3×Dic3⋊C4—64no—
57C3×Dic3⋊C4—64no—
58C22×H(3,𝔽3)C2×D411yescharacteristic
59C22×H(3,𝔽3)C2×D412yes—
60C22×H(3,𝔽3)C2×Q812yes—
61C22×H(3,𝔽3)C2×Q811yescharacteristic
62C22×H(3,𝔽3)#4111yescharacteristic
63C4×H(3,𝔽3)—41no—
64#1084—22no—
65#1084—22no—
66(C3⋊Dic3)⋊C3—24no—
67(C3⋊Dic3)⋊C3—24no—
68C22×Dic6—91no—
69C22×Dic6—92no—
70C2×C3×C4⋊C4—92no—
71C2×C3×C4⋊C4—91no—
72C2×Dic3⋊C4—32no—
73C2×Dic3⋊C4—32no—
74C2×Dic3⋊C4—61no—
75C2×Dic3⋊C4—92no—
76C3⋊#95—91no—
77C3⋊#95—92no—
78C3⋊#95—92no—
79C3⋊#95—92no—
80C3⋊#95—92no—
81C23×C32—21no—
82C23×C32C22×D312yes—
83C2×C32×C4—22no—
84C2×C32×C4—22no—
85C2×C32×C4—22no—
86C2×C32×C4—22no—
87C2×C32×C4—42no—
88C2×C3⋊Dic3—22no—
89C2×C3⋊Dic3—22no—
90C2×C3⋊Dic3—22no—
91C2×C3⋊Dic3—22no—
92C2×C3⋊Dic3Dic614yescomplemented
93C2×C3⋊Dic3#6714yes—
94C2×C3×Dic3—34no—
95C2×C3×Dic3—34no—
96C2×C3×Dic3—34no—
97C2×C3×Dic3—34no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—62no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—62no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—122no—
111#627—616no—
112#395—271noSylow
113C2×H(3,𝔽3)#12011yescharacteristic
114C2×H(3,𝔽3)C22⋊Q811yescharacteristic
115C2×H(3,𝔽3)C22⋊Q812yes—
116C2×H(3,𝔽3)C4⋊Q811yescharacteristic
117C2×H(3,𝔽3)C4⋊Q812yes—
118C22×C3×C4—32no—
119C22×C3×C4—61no—
120C22×C3×C4—92no—
121C22×C3×C4—92no—
122C22×C3×C4—91no—
123C22×C3×C4—91no—
124C22×C3×C4D3211yescharacteristic
125C22×Dic3—32no—
126C22×Dic3—32no—
127C22×Dic3—32no—
128C22×Dic3—32no—
129C22×Dic3—61no—
130C22×Dic3—61no—
131C22×Dic3—92no—
132C22×Dic3—92no—
133C2×Dic6—184no—
134C2×Dic6—184no—
135C2×Dic6—182no—
136C2×Dic6—184no—
137C2×Dic6—184no—
138C3×C4⋊C4—182no—
139C3×C4⋊C4—184no—
140Dic3⋊C4—64no—
141Dic3⋊C4—64no—
142Dic3⋊C4—64no—
143Dic3⋊C4—184no—
144C22×C32—22no—
145C22×C32—21no—
146C22×C32—21no—
147C22×C32—22no—
148C22×C32—21no—
149C22×C32C2×Dic612yes—
150C22×C32#23512yes—
151C22×C32D3×D412yes—
152C22×C32D3×D412yes—
153C22×C32D3×Q812yes—
154C22×C32D3×Q812yes—
155C22×C32C2×#6712yes—
156C32×C4—42no—
157C32×C4—42no—
158C3⋊Dic3—24no—
159C3⋊Dic3—24no—
160C3×Dic3—64no—
161C3×Dic3—64no—
162C3×Dic3—64no—
163C3×Dic3—64no—
164C3×Dic3—64no—
165C3×Dic3—122no—
166C22×Q8—271no—
167C2×C4⋊C4—91no—
168C2×C4⋊C4—92no—
169#95—272no—
170#95—271no—
171H(3,𝔽3)#39511yescharacteristic, complemented, Sylow
172C23×C3—32no—
173C23×C3—61no—
174C23×C3C2×D3211yescharacteristic
175C2×C3×C4—22no—
176C2×C3×C4—21no—
177C2×C3×C4—62no—
178C2×C3×C4—62no—
179C2×C3×C4—62no—
180C2×C3×C4—62no—
181C2×C3×C4—94no—
182C2×C3×C4—94no—
183C2×C3×C4—92no—
184C2×C3×C4—92no—
185C2×C3×C4—122no—
186C2×C3×C4—182no—
187C2×C3×C4—182no—
188C2×C3×C4—182no—
189C2×C3×C4—182no—
190C2×C3×C4—181no—
191C2×C3×C4—182no—
192C2×C3×C4—181no—
193C2×Dic3—34no—
194C2×Dic3—34no—
195C2×Dic3—34no—
196C2×Dic3—34no—
197C2×Dic3—62no—
198C2×Dic3—62no—
199C2×Dic3—62no—
200C2×Dic3—62no—
201C2×Dic3—62no—
202C2×Dic3—62no—
203C2×Dic3—62no—
204C2×Dic3—62no—
205C2×Dic3—62no—
206C2×Dic3—62no—
207C2×Dic3—62no—
208C2×Dic3—62no—
209C2×Dic3—94no—
210C2×Dic3—94no—
211C2×Dic3—122no—
212C2×Dic3—182no—
213C2×Dic3—182no—
214C2×Dic3—182no—
215C2×Dic3—182no—
216Dic6—188no—
217Dic6—1816no—
218C2×C32—22no—
219C2×C32—21no—
220C2×C32—22no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32#88812yes—
224C2×C32#88812yes—
225C2×C32#89612yes—
226C2×C32#89612yes—
227C2×C32#92412yes—
228C2×C32#93712yes—
229C2×C32#94412yes—
230C2×Q8—544no—
231C2×Q8—542no—
232C22×C4—91no—
233C22×C4—92no—
234C22×C4—91no—
235C22×C4—92no—
236C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
237C4⋊C4—182no—
238C4⋊C4—184no—
239C22×C3—32no—
240C22×C3—32no—
241C22×C3—32no—
242C22×C3—32no—
243C22×C3—32no—
244C22×C3—32no—
245C22×C3—32no—
246C22×C3—62no—
247C22×C3—61no—
248C22×C3—61no—
249C22×C3—62no—
250C22×C3—61no—
251C22×C3D3×Dic612yes—
252C22×C3#379711yescharacteristic
253C22×C3#379811yescharacteristic
254C22×C3D3×#6712yes—
255C22×C3#381111yescharacteristic
256C3×C4—41no—
257C3×C4—122no—
258C3×C4—122no—
259C3×C4—184no—
260C3×C4—184no—
261C3×C4—182no—
262C3×C4—182no—
263Dic3—64no—
264Dic3—64no—
265Dic3—64no—
266Dic3—64no—
267Dic3—64no—
268Dic3—122no—
269Dic3—184no—
270Dic3—184no—
271C32—21no—
272C32#497912yescomplemented
273C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
274Q8—548no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—94no—
278C2×C4—92no—
279C2×C4—94no—
280C2×C4—92no—
281C2×C4—181no—
282C2×C4—182no—
283C2×C4—182no—
284C2×C4—182no—
285C2×C4—182no—
286C2×C4—181no—
287C2×C4—182no—
288C2×C3—32no—
289C2×C3—32no—
290C2×C3—32no—
291C2×C3—32no—
292C2×C3—32no—
293C2×C3—32no—
294C2×C3—32no—
295C2×C3—61no—
296C2×C3—62no—
297C2×C3—61no—
298C2×C3—61no—
299C2×C3—62no—
300C2×C3#6383912yes—
301C2×C3#6384011yescharacteristic
302C2×C3#6385112yes—
303C2×C3#6386511yescharacteristic
304C2×C3#6398311yescharacteristic
305C22((C3⋊D3)⋊C3)⋊Q812yes—
306C22#8865011yescharacteristic
307C22#8865111yescharacteristic
308C22#8867612yes—
309C22#8867711yescharacteristic
310C4—41no—
311C4—182no—
312C4—184no—
313C4—182no—
314C4—184no—
315C3—32no—
316C3—61no—
317C3#1059119511yescharacteristic
318C2#1172107912yes—
319C2#1172108011yescharacteristic
320C2#1172109611yescharacteristic
321C2#1172110112yes—
322C2#1172132411yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, prime radical, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#395
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
31. C2×#1084Q8semidirect
37. C2×(C3⋊Dic3)⋊C3Q8semidirect
92. C2×C3⋊Dic3Dic6semidirect
171. H(3,𝔽3)#395semidirect
272. C32#4979semidirect

Maximal subgroups

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

subgroupindex
C22×#61962
C2×#885402
C2×#3750⋊C32
#117203342
#105911953