Finite Group Databasebeta

← #13140224

Subgroups — #13140224

Order 1728.

743 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 743 conjugacy classes of subgroups — too many to draw. Only its 135 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×#88447C211yesmaximal, characteristic
4#63658⋊C3C212yesmaximal
5#11720334C212yesmaximal
6#10591235—32nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×#1084C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3C2212yes—
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C2×#6196—22no—
13C2×#6196—22no—
14C2×#6196C414yescomplemented
15#88540—22no—
16#88447—22no—
17C22×#627—32no—
18C2×C3×Dic3⋊C4—32no—
19C2×C3×Dic3⋊C4—61no—
20C2×Dic32—32no—
21#63658D312yes—
22C3×C3⋊#95—32no—
23#63579—32no—
24#63579—32no—
25C23×H(3,𝔽3)C2311yescharacteristic
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)—22no—
28C2×#1084—21no—
29C2×#1084—22no—
30C2×#1084C2×C412yes—
31C2×#1084D412yes—
32C2×#1084D412yes—
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3—22no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3C2×C414yes—
38C2×(C3⋊Dic3)⋊C3Q814yescomplemented
39#6196—28no—
40#4973—91no—
41#4646—92no—
42C22×C32×C4—21no—
43C22×C32×C4C2×D312yes—
44C22×C3⋊Dic3C2×D312yes—
45C22×C3⋊Dic3C2×D312yes—
46C22×C3×Dic3—32no—
47C22×C3×Dic3—32no—
48C22×C3×Dic3—32no—
49C22×C3×Dic3—32no—
50C22×C3×Dic3—61no—
51C22×C3×Dic3—61no—
52C2×#627—38no—
53C2×#627—64no—
54C2×#627—64no—
55C3×Dic3⋊C4—64no—
56C3×Dic3⋊C4—64no—
57Dic32—64no—
58C22×H(3,𝔽3)C22×C411yescharacteristic
59C22×H(3,𝔽3)C2×D411yescharacteristic
60C22×H(3,𝔽3)C2×D411yescharacteristic
61C22×H(3,𝔽3)C2×Q812yes—
62C22×H(3,𝔽3)#4112yes—
63C4×H(3,𝔽3)—41no—
64#1084—41no—
65#1084C22⋊C414yes—
66(C3⋊Dic3)⋊C3—24no—
67(C3⋊Dic3)⋊C3—24no—
68C22×Dic6—91no—
69C22×Dic6—92no—
70C2×C3×C4⋊C4—91no—
71C2×Dic3⋊C4—32no—
72C2×Dic3⋊C4—61no—
73C2×C4×Dic3—91no—
74C2×C4×Dic3—92no—
75C3×#95—92no—
76C3⋊#95—32no—
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⋊Dic3C4×D314yes—
94C2×C3×Dic3—34no—
95C2×C3×Dic3—34no—
96C2×C3×Dic3—34no—
97C2×C3×Dic3—34no—
98C2×C3×Dic3—34no—
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#391—271noSylow
113C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
114C2×H(3,𝔽3)C4×Q812yes—
115C2×H(3,𝔽3)C22⋊Q812yes—
116C2×H(3,𝔽3)#12411yescharacteristic
117C2×H(3,𝔽3)C4⋊Q811yescharacteristic
118C22×C3×C4—32no—
119C22×C3×C4—61no—
120C22×C3×C4—91no—
121C22×C3×C4—92no—
122C22×C3×C4—92no—
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—94no—
134C2×Dic6—98no—
135C2×Dic6—184no—
136C2×Dic6—184no—
137C2×Dic6—182no—
138C2×Dic6—182no—
139C3×C4⋊C4—182no—
140Dic3⋊C4—64no—
141Dic3⋊C4—64no—
142C4×Dic3—184no—
143C4×Dic3—182no—
144C22×C32—22no—
145C22×C32—21no—
146C22×C32—21no—
147C22×C32—22no—
148C22×C32—21no—
149C22×C32C2×Dic612yes—
150C22×C32C2×C4×D312yes—
151C22×C32#23512yes—
152C22×C32D3×D412yes—
153C22×C32D3×D412yes—
154C22×C32#23712yes—
155C22×C32D3×Q812yes—
156C32×C4—42no—
157C32×C4—42no—
158C3⋊Dic3—24no—
159C3⋊Dic3—24no—
160C3×Dic3—38no—
161C3×Dic3—64no—
162C3×Dic3—64no—
163C3×Dic3—64no—
164C3×Dic3—122no—
165C3×Dic3—122no—
166C22×Q8—271no—
167C2×C4⋊C4—91no—
168C2×C42—271no—
169#95—92no—
170#95—272no—
171H(3,𝔽3)#39111yescharacteristic, 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—92no—
182C2×C3×C4—92no—
183C2×C3×C4—92no—
184C2×C3×C4—94no—
185C2×C3×C4—94no—
186C2×C3×C4—122no—
187C2×C3×C4—182no—
188C2×C3×C4—182no—
189C2×C3×C4—182no—
190C2×C3×C4—182no—
191C2×C3×C4—182no—
192C2×C3×C4—181no—
193C2×Dic3—34no—
194C2×Dic3—34no—
195C2×Dic3—34no—
196C2×Dic3—34no—
197C2×Dic3—34no—
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—1816no—
217Dic6—188no—
218C2×C32—22no—
219C2×C32—21no—
220C2×C32—22no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32C4×Dic612yes—
224C2×C32#87812yes—
225C2×C32D3×C22⋊C412yes—
226C2×C32#88512yes—
227C2×C32#88712yes—
228C2×C32#88812yes—
229C2×C32#89612yes—
230C2×Q8—274no—
231C2×Q8—542no—
232C2×Q8—542no—
233C22×C4—91no—
234C22×C4—92no—
235C22×C4—92no—
236C22×C4—91no—
237C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
238C4⋊C4—182no—
239C42—542no—
240C22×C3—32no—
241C22×C3—32no—
242C22×C3—32no—
243C22×C3—32no—
244C22×C3—32no—
245C22×C3—32no—
246C22×C3—32no—
247C22×C3—62no—
248C22×C3—61no—
249C22×C3—61no—
250C22×C3—62no—
251C22×C3—61no—
252C22×C3D3×Dic612yes—
253C22×C3C4×D3211yescharacteristic
254C22×C3#380211yescharacteristic
255C22×C3#380412yes—
256C22×C3#381111yescharacteristic
257C3×C4—41no—
258C3×C4—94no—
259C3×C4—122no—
260C3×C4—122no—
261C3×C4—184no—
262C3×C4—184no—
263C3×C4—361no—
264Dic3—38no—
265Dic3—64no—
266Dic3—64no—
267Dic3—64no—
268Dic3—122no—
269Dic3—122no—
270Dic3—184no—
271Dic3—184no—
272C32—21no—
273C32#464612yescomplemented
274C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
275Q8—548no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—92no—
279C2×C4—92no—
280C2×C4—92no—
281C2×C4—94no—
282C2×C4—94no—
283C2×C4—182no—
284C2×C4—182no—
285C2×C4—182no—
286C2×C4—182no—
287C2×C4—182no—
288C2×C4—181no—
289C2×C3—32no—
290C2×C3—32no—
291C2×C3—32no—
292C2×C3—32no—
293C2×C3—32no—
294C2×C3—32no—
295C2×C3—32no—
296C2×C3—61no—
297C2×C3—62no—
298C2×C3—61no—
299C2×C3—62no—
300C2×C3—61no—
301C2×C3#6383712yes—
302C2×C3#6384011yescharacteristic
303C2×C3#6389912yes—
304C2×C3#6392111yescharacteristic
305C2×C3#6395311yescharacteristic
306C22((C3⋊D3)⋊C3)⋊Q812yes—
307C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
308C22#8865511yescharacteristic
309C22#8867112yes—
310C22#8867711yescharacteristic
311C4—41no—
312C4—94no—
313C4—184no—
314C4—184no—
315C4—361no—
316C3—32no—
317C3—61no—
318C3#1059123611yescharacteristic
319C2#1172107712yes—
320C2#1172108011yescharacteristic
321C2#1172113412yes—
322C2#1172116111yescharacteristic
323C2#1172129411yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radical, cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#391
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
14. C2×#6196C4semidirect
38. C2×(C3⋊Dic3)⋊C3Q8semidirect
92. C2×C3⋊Dic3Dic6semidirect
171. H(3,𝔽3)#391semidirect
273. C32#4646semidirect

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×#884472
#63658⋊C32
#117203342
#105912353