Finite Group Databasebeta

← #13139023

Subgroups — #13139023

Order 1728.

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

normal subgroups only

This group has 643 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×#88651C211yesmaximal, characteristic
2C2×C8×#272C211yesmaximal, characteristic
3#11721152C211yesmaximal, characteristic
4#63643⋊C3C212yesmaximal, complemented
5#11720313C212yesmaximal, complemented
6D3×#816—32nomaximal
7C2×C4×#272C2211yescharacteristic
8C2×#634⋊C3C2212yes—
9C2×#6199—21no—
10C2×#6199—21no—
11C2×#6196—21no—
12#88651—22no—
13#88651C414yescomplemented
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16C8×#272—22no—
17C4⋊((C3⋊Dic3)⋊C3)C2212yes—
18#88449—21no—
19C2×D3×Dic6—32no—
20C2×C3×D3×C8—32no—
21C2×C3×D3×C8—61no—
22D3×C4⋊Dic3—32no—
23#63643D312yescomplemented
24C3×#816—32no—
25#63567—32no—
26#63566—32no—
27C22×#272D411yescharacteristic
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×#1084D411yescharacteristic
30C4×#272C2×C412yes—
31C4×#272D412yes—
32C2×(C3⋊Dic3)⋊C3—22no—
33C2×(C3⋊Dic3)⋊C3—22no—
34#634⋊C3—22no—
35#634⋊C3C2×C414yescomplemented
36#6199—24no—
37#6199—42no—
38#6196—24no—
39C8×H(3,𝔽3)—21no—
40#6180—21no—
41C2×#816—91no—
42D3×#103—92no—
43C2×#634C2×D312yes—
44C2×C3×C4×D3—32no—
45C2×C3×C4×D3—61no—
46C2×C3×Dic6—32no—
47C2×#627—62no—
48C2×D3×Dic3—62no—
49C2×D3×Dic3—62no—
50D3×Dic6—38no—
51D3×Dic6—64no—
52C2×C32×C8—21no—
53C2×C32×C8C2×D312yes—
54C4⋊(C3⋊Dic3)C2×D312yes—
55C3×C4⋊Dic3—32no—
56C2×C3×C3⋊C8—32no—
57C2×C3×C3⋊C8—61no—
58C3×D3×C8—64no—
59C3×D3×C8—64no—
60Dic3⋊Dic3—62no—
61C22×H(3,𝔽3)C2×D411yescharacteristic
62C2×#272C22⋊C412yes—
63C2×#272SD1612yes—
64C2×#272Q1612yes—
65C4×H(3,𝔽3)C22×C411yescharacteristic
66C4×H(3,𝔽3)C2×D411yescharacteristic
67#1084C22⋊C412yes—
68(C3⋊Dic3)⋊C3—24no—
69(C3⋊Dic3)⋊C3—42no—
70C2×D3×Q8—92no—
71C22×Dic6—91no—
72C22×C3×C8—91no—
73C2×C4⋊Dic3—91no—
74C2×D3×C8—32no—
75C2×D3×C8—61no—
76D3×C4⋊C4—92no—
77C3×#103—92no—
78#835—92no—
79#816—32no—
80#816—92no—
81#816—92no—
82#810—92no—
83C22×C3×D3—32no—
84C22×C3×D3—61no—
85C2×C32×C4—21no—
86C2×C32×C4C22×D312yes—
87C2×C3⋊Dic3—22no—
88C2×C3⋊Dic3—22no—
89#634—22no—
90#634C4×D314yescomplemented
91C2×C3×Dic3—32no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95C3×C4×D3—34no—
96C3×C4×D3—34no—
97C3×C4×D3—62no—
98C3×C4×D3—62no—
99C3×Dic6—34no—
100C3×Dic6—62no—
101#627—68no—
102D3×Dic3—68no—
103D3×Dic3—124no—
104C32×C8—22no—
105C32×C8—22no—
106C3×C3⋊C8—62no—
107C3×C3⋊C8—62no—
108C2×#103—271noSylow
109C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
110C2×H(3,𝔽3)C2×SD1611yescharacteristic
111C2×H(3,𝔽3)C2×Q1611yescharacteristic
112#272#10314yescomplemented
113C2×C3×Q8—92no—
114C22×C3×C4—91no—
115C22×Dic3—181no—
116C22×Dic3—181no—
117D3×Q8—98no—
118D3×Q8—184no—
119C2×C4×D3—32no—
120C2×C4×D3—61no—
121C2×C4×D3—182no—
122C2×C4×D3—182no—
123C2×Dic6—32no—
124C2×Dic6—92no—
125C2×Dic6—94no—
126C2×Dic6—182no—
127C2×Dic6—182no—
128C2×Dic6—181no—
129C2×C3×C8—32no—
130C2×C3×C8—61no—
131C2×C3×C8—91no—
132C2×C3×C8—182no—
133C2×C3×C8D3211yescharacteristic
134C3×C4⋊C4—92no—
135C4⋊Dic3—32no—
136C4⋊Dic3—92no—
137C4⋊Dic3—181no—
138Dic3⋊C4—182no—
139C2×C3⋊C8—32no—
140C2×C3⋊C8—61no—
141D3×C8—64no—
142D3×C8—64no—
143C22×C32—21no—
144C22×C32D3×D412yes—
145C2×C3×D3—34no—
146C2×C3×D3—34no—
147C2×C3×D3—34no—
148C2×C3×D3—62no—
149C2×C3×D3—62no—
150C2×C3×D3—62no—
151C32×C4—21no—
152C32×C4—21no—
153C32×C4C2×C4×D312yes—
154C32×C4D3×D412yes—
155C3⋊Dic3—24no—
156C3⋊Dic3—42no—
157C3×Dic3—34no—
158C3×Dic3—62no—
159C3×Dic3—64no—
160C3×Dic3—122no—
161C22×Q8—271no—
162C22×C8—91no—
163C2×C4⋊C4—271no—
164#103—92no—
165#103—272no—
166H(3,𝔽3)C2×#10311yescharacteristic, complemented, Sylow
167C23×C3—91no—
168C22×D3—32no—
169C22×D3—61no—
170C3×Q8—94no—
171C3×Q8—182no—
172C2×C3×C4—32no—
173C2×C3×C4—61no—
174C2×C3×C4—91no—
175C2×C3×C4—92no—
176C2×C3×C4—92no—
177C2×C3×C4—182no—
178C2×C3×C4—182no—
179C2×C3×C4C2×D3211yescharacteristic
180C2×Dic3—32no—
181C2×Dic3—61no—
182C2×Dic3—62no—
183C2×Dic3—62no—
184C2×Dic3—182no—
185C2×Dic3—182no—
186C2×Dic3—184no—
187C2×Dic3—362no—
188C4×D3—34no—
189C4×D3—34no—
190C4×D3—62no—
191C4×D3—62no—
192C4×D3—188no—
193C4×D3—364no—
194Dic6—34no—
195Dic6—62no—
196Dic6—94no—
197Dic6—182no—
198Dic6—188no—
199Dic6—184no—
200C3×C8—21no—
201C3×C8—62no—
202C3×C8—62no—
203C3×C8—181no—
204C3⋊C8—62no—
205C3⋊C8—62no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32D3×C22⋊C412yes—
210C2×C32D3×SD1612yes—
211C2×C32D3×Q1612yes—
212C3×D3—38no—
213C3×D3—64no—
214C2×Q8—92no—
215C2×Q8—274no—
216C2×Q8—541no—
217C2×Q8—542no—
218C22×C4—91no—
219C22×C4—541no—
220C22×C4—541no—
221C2×C8—91no—
222C2×C8—182no—
223C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
224C4⋊C4—92no—
225C4⋊C4—541no—
226C22×C3—32no—
227C22×C3—61no—
228C22×C3—92no—
229C22×C3—92no—
230C22×C3—92no—
231C22×C3#380211yescharacteristic
232C2×D3—34no—
233C2×D3—34no—
234C2×D3—34no—
235C2×D3—62no—
236C2×D3—62no—
237C2×D3—62no—
238C3×C4—32no—
239C3×C4—32no—
240C3×C4—61no—
241C3×C4—61no—
242C3×C4—92no—
243C3×C4—184no—
244C3×C4—362no—
245C3×C4C4×D3211yescharacteristic
246C3×C4#381111yescharacteristic
247Dic3—34no—
248Dic3—62no—
249Dic3—64no—
250Dic3—122no—
251Dic3—184no—
252Dic3—362no—
253C32—21no—
254C32D3×#10312yescomplemented
255C23—91no—
256Q8—94no—
257Q8—182no—
258Q8—544no—
259C2×C4—91no—
260C2×C4—92no—
261C2×C4—92no—
262C2×C4—182no—
263C2×C4—182no—
264C2×C4—544no—
265C2×C4—1082no—
266C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
267C8—21no—
268C8—181no—
269C2×C3—32no—
270C2×C3—32no—
271C2×C3—32no—
272C2×C3—61no—
273C2×C3—61no—
274C2×C3—61no—
275C2×C3—94no—
276C2×C3#6379611yescharacteristic
277C2×C3#6380211yescharacteristic
278C2×C3#6392111yescharacteristic
279D3—38no—
280D3—64no—
281C22—92no—
282C22—92no—
283C22—92no—
284C22#8865511yescharacteristic
285C4—92no—
286C4—184no—
287C4—362no—
288C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
289C4#8867711yescharacteristic
290C3—32no—
291C3—61no—
292C3#1059067211yescharacteristic
293C2—94no—
294C2#1172083311yescharacteristic
295C2#1172083711yescharacteristic
296C2#1172116111yescharacteristic
subgroupG/H
centerC22#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC22×#272D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#103
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. #63643⋊C3C2semidirect
5. #11720313C2semidirect
13. #88651C4semidirect
23. #63643D3semidirect
35. #634⋊C3C2×C4semidirect
90. #634C4×D3semidirect
112. #272#103semidirect
166. H(3,𝔽3)C2×#103semidirect
254. C32D3×#103semidirect

Maximal subgroups

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

subgroupindex
C2×#886512
C2×C8×#2722
#117211522
#63643⋊C32
#117203132
D3×#8163