Finite Group Databasebeta

← #13137959

Subgroups — #13137959

Order 1728.

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

normal subgroups only

This group has 547 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
1C4⋊C4×#272C211yesmaximal, characteristic
2#11721152C211yesmaximal, characteristic
3C2×#88434C211yesmaximal, characteristic
4#63635⋊C3C212yesmaximal, complemented
5#11720307C212yesmaximal, complemented
6D3×#808—32nomaximal
7C2×C4×#272—21no—
8C2×C4×#272C2211yescharacteristic
9C2×#6199—21no—
10C4⋊C4×H(3,𝔽3)C2211yescharacteristic
11C4⋊#1084C2211yescharacteristic
12#88540—21no—
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88449—21no—
16#88434C414yescomplemented
17C3×D3×C4⋊C4—32no—
18C3×D3×C4⋊C4—61no—
19D3×C4⋊Dic3—32no—
20C2×D3×C3⋊C8—32no—
21#63635D312yescomplemented
22C3×#808—32no—
23#63574—32no—
24#63573—32no—
25C22×#272D411yescharacteristic
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×#1084—21no—
29C2×#1084D411yescharacteristic
30C4×#272—42no—
31C4×#272C2×C412yes—
32C4×#272Q812yes—
33C2×(C3⋊Dic3)⋊C3—22no—
34#6199—42no—
35(C3⋊(C3⋊C8))⋊C3C2×C414yescomplemented
36C2×#808—91no—
37D3×#106—92no—
38C2×C3×C4×D3—32no—
39C2×C3×C4×D3—61no—
40C2×C3×C4×D3—62no—
41C2×C3×C4×D3—62no—
42C2×D3×Dic3—62no—
43C32×C4⋊C4—21no—
44C32×C4⋊C4C2×D312yes—
45C4⋊(C3⋊Dic3)C2×D312yes—
46C2×C3⋊(C3⋊C8)C2×D312yes—
47C3×C4⋊Dic3—32no—
48C3×C4⋊Dic3—32no—
49C3×C4⋊Dic3—61no—
50C3×Dic3⋊C4—62no—
51C3×Dic3⋊C4—62no—
52C2×C3×C3⋊C8—32no—
53Dic3⋊Dic3—62no—
54D3×C3⋊C8—38no—
55C22×H(3,𝔽3)C2×D411yescharacteristic
56C2×#272C4⋊C412yes—
57C2×#272SD1612yes—
58C2×#272SD1612yes—
59C4×H(3,𝔽3)—41no—
60C4×H(3,𝔽3)C22×C411yescharacteristic
61C4×H(3,𝔽3)C2×Q811yescharacteristic
62#1084—41no—
63#1084C4⋊C412yes—
64(C3⋊Dic3)⋊C3—42no—
65C2×C3×C4⋊C4—91no—
66C2×C4⋊Dic3—91no—
67C22×C3⋊C8—91no—
68C2×D3×C8—92no—
69D3×C4⋊C4—32no—
70D3×C4⋊C4—61no—
71D3×C4⋊C4—92no—
72C3×#106—92no—
73#817—92no—
74#808—32no—
75#808—92no—
76#808—92no—
77#808—92no—
78C22×C3×D3—32no—
79C22×C3×D3—61no—
80C2×C32×C4—22no—
81C2×C32×C4—22no—
82C2×C32×C4—21no—
83C2×C32×C4C22×D312yes—
84C2×C3⋊Dic3—22no—
85C2×C3×Dic3—32no—
86C2×C3×Dic3—61no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—62no—
90C3×C4×D3—34no—
91C3×C4×D3—34no—
92C3×C4×D3—62no—
93C3×C4×D3—62no—
94C3×C4×D3—124no—
95C3×C4×D3—124no—
96D3×Dic3—124no—
97C3⋊(C3⋊C8)C4×D314yescomplemented
98C3×C3⋊C8—34no—
99C2×#106—271noSylow
100C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
101C2×H(3,𝔽3)C2×SD1611yescharacteristic
102C2×H(3,𝔽3)C2×SD1611yescharacteristic
103#272#10614yescomplemented
104C22×C3×C4—91no—
105C22×C3×C4—181no—
106C22×Dic3—181no—
107C2×C4×D3—32no—
108C2×C4×D3—61no—
109C2×C4×D3—62no—
110C2×C4×D3—62no—
111C2×C4×D3—182no—
112C2×C3×C8—92no—
113C3×C4⋊C4—32no—
114C3×C4⋊C4—61no—
115C3×C4⋊C4—92no—
116C3×C4⋊C4—91no—
117C3×C4⋊C4—181no—
118C3×C4⋊C4D3211yescharacteristic
119C4⋊Dic3—32no—
120C4⋊Dic3—32no—
121C4⋊Dic3—61no—
122C4⋊Dic3—92no—
123C4⋊Dic3—181no—
124Dic3⋊C4—62no—
125Dic3⋊C4—62no—
126Dic3⋊C4—182no—
127C2×C3⋊C8—32no—
128C2×C3⋊C8—92no—
129C2×C3⋊C8—94no—
130D3×C8—98no—
131C22×C32—21no—
132C22×C32D3×D412yes—
133C2×C3×D3—34no—
134C2×C3×D3—34no—
135C2×C3×D3—34no—
136C2×C3×D3—62no—
137C2×C3×D3—62no—
138C2×C3×D3—62no—
139C32×C4—21no—
140C32×C4—21no—
141C32×C4—42no—
142C32×C4—42no—
143C32×C4C2×C4×D312yes—
144C32×C4D3×Q812yes—
145C3⋊Dic3—42no—
146C3×Dic3—34no—
147C3×Dic3—62no—
148C3×Dic3—122no—
149C3×Dic3—122no—
150C3×Dic3—122no—
151C22×C8—271no—
152C2×C4⋊C4—91no—
153C2×C4⋊C4—271no—
154#106—92no—
155#106—272no—
156H(3,𝔽3)C2×#10611yescharacteristic, complemented, Sylow
157C23×C3—91no—
158C22×D3—32no—
159C22×D3—61no—
160C2×C3×C4—21no—
161C2×C3×C4—32no—
162C2×C3×C4—62no—
163C2×C3×C4—62no—
164C2×C3×C4—61no—
165C2×C3×C4—91no—
166C2×C3×C4—92no—
167C2×C3×C4—92no—
168C2×C3×C4—182no—
169C2×C3×C4—181no—
170C2×C3×C4—362no—
171C2×C3×C4C2×D3211yescharacteristic
172C2×Dic3—32no—
173C2×Dic3—61no—
174C2×Dic3—62no—
175C2×Dic3—62no—
176C2×Dic3—62no—
177C2×Dic3—182no—
178C2×Dic3—362no—
179C4×D3—34no—
180C4×D3—34no—
181C4×D3—62no—
182C4×D3—62no—
183C4×D3—124no—
184C4×D3—124no—
185C4×D3—364no—
186C3×C8—94no—
187C3⋊C8—34no—
188C3⋊C8—94no—
189C2×C32—21no—
190C2×C32—21no—
191C2×C32—21no—
192C2×C32D3×C4⋊C412yes—
193C2×C32D3×SD1612yes—
194C2×C32D3×SD1612yes—
195C3×D3—38no—
196C3×D3—64no—
197C22×C4—91no—
198C22×C4—181no—
199C22×C4—541no—
200C2×C8—92no—
201C2×C8—274no—
202C4⋊C4—92no—
203C4⋊C4—91no—
204C4⋊C4—181no—
205C4⋊C4—541no—
206C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
207C22×C3—32no—
208C22×C3—61no—
209C22×C3—92no—
210C22×C3—92no—
211C22×C3—92no—
212C22×C3#381111yescharacteristic
213C2×D3—34no—
214C2×D3—34no—
215C2×D3—34no—
216C2×D3—62no—
217C2×D3—62no—
218C2×D3—62no—
219C3×C4—32no—
220C3×C4—32no—
221C3×C4—41no—
222C3×C4—61no—
223C3×C4—61no—
224C3×C4—92no—
225C3×C4—122no—
226C3×C4—122no—
227C3×C4—362no—
228C3×C4—361no—
229C3×C4#379711yescharacteristic
230C3×C4C4×D3211yescharacteristic
231Dic3—34no—
232Dic3—62no—
233Dic3—122no—
234Dic3—122no—
235Dic3—122no—
236Dic3—362no—
237C32—21no—
238C32D3×#10612yescomplemented
239C23—91no—
240C2×C4—21no—
241C2×C4—91no—
242C2×C4—92no—
243C2×C4—92no—
244C2×C4—181no—
245C2×C4—182no—
246C2×C4—362no—
247C2×C4—1082no—
248C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
249C8—94no—
250C2×C3—32no—
251C2×C3—32no—
252C2×C3—32no—
253C2×C3—61no—
254C2×C3—61no—
255C2×C3—61no—
256C2×C3—94no—
257C2×C3#6388311yescharacteristic
258C2×C3#6393011yescharacteristic
259C2×C3#6394011yescharacteristic
260D3—38no—
261D3—64no—
262C22—92no—
263C22—92no—
264C22—92no—
265C22#8867711yescharacteristic
266C4—41no—
267C4—92no—
268C4—362no—
269C4—361no—
270C4#8865111yescharacteristic
271C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
272C3—32no—
273C3—61no—
274C3#1059057511yescharacteristic
275C2—94no—
276C2#1172114711yescharacteristic
277C2#1172118811yescharacteristic
278C2#1172122811yescharacteristic
subgroupG/H
centerC22#88677
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×#272D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#106
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #63635⋊C3C2semidirect
5. #11720307C2semidirect
16. #88434C4semidirect
21. #63635D3semidirect
35. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
97. C3⋊(C3⋊C8)C4×D3semidirect
103. #272#106semidirect
156. H(3,𝔽3)C2×#106semidirect
238. C32D3×#106semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×#2722
#117211522
C2×#884342
#63635⋊C32
#117203072
D3×#8083