Finite Group Databasebeta

← #13154296

Subgroups — #13154296

Order 1728.

1606 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 1606 conjugacy classes of subgroups — too many to draw. Only its 227 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×#88748C212yesmaximal, complemented
2H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
3#11721863C212yesmaximal, complemented
4#11721763C212yesmaximal, complemented
5#11721762C214yesmaximal, complemented
6C4⋊C4×#272C212yesmaximal, complemented
7C4×C4⋊#272C211yesmaximal, characteristic, complemented
8C4×C4⋊#272C211yesmaximal, characteristic, complemented
9C3×#5726—34nomaximal
10C2×Q8×H(3,𝔽3)C2212yes—
11#88748—28no—
12C2×C4⋊#272C2214yes—
13C2×C4⋊#272C2211yescharacteristic
14C2×C4⋊#272C2211yescharacteristic
15C2×C4×#272C2214yes—
16C2×C4×#272C2212yes—
17C2×C4×#272C2212yes—
18C4⋊C4×H(3,𝔽3)C2212yes—
19C4⋊C4×H(3,𝔽3)C2212yes—
20C42×H(3,𝔽3)C2211yescharacteristic
21#88542C2212yes—
22#88542C2212yes—
23#88542C2214yes—
24C4⋊#1084C2211yescharacteristic
25C4⋊#1084C2211yescharacteristic
26#88540C2214yes—
27C2×C3×#239—38no—
28C32×C4⋊Q8D314yescomplemented
29C3×#946—38no—
30C3×#895—38no—
31C3×#894—316no—
32C3×D3×C4⋊C4—38no—
33C3×C4×D12—34no—
34C3×C4×D12—34no—
35C22×#272C2312yes—
36C22×#272C2312yes—
37Q8×H(3,𝔽3)—24no—
38C2×C4×H(3,𝔽3)C2314yes—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084C2312yes—
43C2×#1084C2312yes—
44C4⋊#272—28no—
45C4⋊#272—22no—
46C4⋊#272D414yescomplemented
47C4×#272—28no—
48C4×#272—24no—
49C4×#272—24no—
50C3×#547—91no—
51#5726—34no—
52C2×C32×Q8C2×D318yes—
53C3×#239—632no—
54C2×C3×D12—34no—
55C2×C3×D12—316no—
56C2×C3×D12—34no—
57C2×C3×C4×D3—316no—
58C2×C3×C4×D3—38no—
59C2×C3×C4×D3—38no—
60C32×C4⋊C4C2×D318yes—
61C32×C4⋊C4C2×D318yes—
62C32×C42C2×D314yes—
63C3×(C2×D3)⋊C4—38no—
64C3×(C2×D3)⋊C4—38no—
65C3×(C2×D3)⋊C4—316no—
66C3×C4⋊Dic3—34no—
67C3×C4⋊Dic3—34no—
68C3×Dic3⋊C4—316no—
69C22×H(3,𝔽3)C2411yescharacteristic
70C2×#272—22no—
71C2×#272—22no—
72C2×#272—22no—
73C2×#272—22no—
74C2×#272—22no—
75C2×#272C2×D414yes—
76C4×H(3,𝔽3)—24no—
77C4×H(3,𝔽3)—21no—
78C4×H(3,𝔽3)C2×D412yes—
79C4×H(3,𝔽3)#4112yescomplemented
80#1084—22no—
81#1084—22no—
82C2×C3×#41—92no—
83C2×#239—38no—
84C3×C4⋊Q8—34no—
85C3×C4⋊Q8C3⋊D311yescharacteristic
86C3×#123—94no—
87C3×C22⋊Q8—92no—
88C3×#121—92no—
89C3×C4×D4—91no—
90C3×C4×D4—91no—
91C2×C3×C4⋊C4—92no—
92#946—38no—
93#895—38no—
94#894—316no—
95D3×C4⋊C4—38no—
96C4×D12—34no—
97C4×D12—34no—
98C22×C3×D3—38no—
99C22×C3×D3—38no—
100C32×Q8—216no—
101C2×C32×C4C22×D3116yes—
102C2×C32×C4C22×D314yes—
103C2×C32×C4C22×D314yes—
104C2×C32×C4C22×D314yes—
105C2×C3×Dic3—38no—
106C2×C3×Dic3—38no—
107C3×D12—316no—
108C3×D12—632no—
109C3×D12—68no—
110C3×C4×D3—632no—
111C3×C4×D3—616no—
112C3×C4×D3—616no—
113#547—91noSylow
114C2×H(3,𝔽3)C22×D411yescharacteristic
115C2×H(3,𝔽3)C2×#4111yescharacteristic
116C2×H(3,𝔽3)2−1+411yescharacteristic
117#272—24no—
118#272—42no—
119C3×#41—188no—
120C2×C3×Q8—38no—
121C2×C3×Q8C2×C3⋊D312yes—
122C2×C3×D4—91no—
123C2×C3×D4—91no—
124C2×C3×D4—94no—
125C22×C3×C4—92no—
126C22×C3×C4—94no—
127C22×C3×C4—92no—
128#239—632no—
129C2×D12—34no—
130C2×D12—34no—
131C2×D12—316no—
132C2×C4×D3—38no—
133C2×C4×D3—38no—
134C2×C4×D3—316no—
135C3×C4⋊C4—38no—
136C3×C4⋊C4—38no—
137C3×C4⋊C4—94no—
138C3×C4⋊C4—91no—
139C3×C4⋊C4—91no—
140C3×C4⋊C4C2×C3⋊D312yes—
141C3×C4⋊C4C2×C3⋊D312yes—
142C3×C22⋊C4—92no—
143C3×C22⋊C4—92no—
144C3×C22⋊C4—94no—
145C3×C42—34no—
146C3×C42C2×C3⋊D311yescharacteristic
147(C2×D3)⋊C4—38no—
148(C2×D3)⋊C4—38no—
149(C2×D3)⋊C4—316no—
150C4⋊Dic3—34no—
151C4⋊Dic3—34no—
152Dic3⋊C4—316no—
153C22×C32C23×D314yes—
154C2×C3×D3—316no—
155C2×C3×D3—68no—
156C2×C3×D3—68no—
157C2×C3×D3—68no—
158C2×C3×D3—68no—
159C2×C3×D3—68no—
160C32×C4—216no—
161C32×C4—24no—
162C32×C4D3×D418yes—
163C32×C4#23918yescomplemented
164C3×Dic3—68no—
165C3×Dic3—68no—
166C2×#41—92no—
167C4⋊Q8#27211yescharacteristic, complemented
168#123—94no—
169C22⋊Q8—92no—
170#121—92no—
171C4×D4—91no—
172C4×D4—91no—
173C2×C4⋊C4—92no—
174H(3,𝔽3)#54711yescharacteristic, complemented, Sylow
175C23×C3—92no—
176C23×C3—92no—
177C22×D3—38no—
178C22×D3—38no—
179C3×Q8—24no—
180C3×Q8—616no—
181C3×D4—94no—
182C3×D4—188no—
183C3×D4—182no—
184C2×C3×C4—34no—
185C2×C3×C4—34no—
186C2×C3×C4—316no—
187C2×C3×C4—34no—
188C2×C3×C4—92no—
189C2×C3×C4—92no—
190C2×C3×C4—188no—
191C2×C3×C4—184no—
192C2×C3×C4—184no—
193C2×C3×C4C22×C3⋊D311yescharacteristic
194C2×C3×C4C22×C3⋊D314yes—
195C2×C3×C4C22×C3⋊D311yescharacteristic
196C2×C3×C4C22×C3⋊D311yescharacteristic
197C2×Dic3—38no—
198C2×Dic3—38no—
199D12—316no—
200D12—68no—
201D12—632no—
202C4×D3—616no—
203C4×D3—616no—
204C4×D3—632no—
205C2×C32C2×D3×D414yes—
206C2×C32C2×#23914yes—
207C2×C32#100714yes—
208C3×D3—616no—
209C3×D3—128no—
210#41—188no—
211C2×Q8C2×#27212yes—
212C2×D4—91no—
213C2×D4—91no—
214C2×D4—94no—
215C22×C4—92no—
216C22×C4—94no—
217C22×C4—92no—
218C4⋊C4—91no—
219C4⋊C4—94no—
220C4⋊C4—91no—
221C4⋊C4C2×#27212yes—
222C4⋊C4C2×#27212yes—
223C22⋊C4—92no—
224C22⋊C4—92no—
225C22⋊C4—94no—
226C42C2×#27211yescharacteristic
227C22×C3—34no—
228C22×C3—94no—
229C22×C3—182no—
230C22×C3—182no—
231C22×C3—182no—
232C22×C3—182no—
233C22×C3—182no—
234C22×C3C23×C3⋊D311yescharacteristic
235C2×D3—316no—
236C2×D3—68no—
237C2×D3—68no—
238C2×D3—68no—
239C2×D3—68no—
240C2×D3—68no—
241C3×C4—24no—
242C3×C4—21no—
243C3×C4—38no—
244C3×C4—38no—
245C3×C4—616no—
246C3×C4—64no—
247C3×C4—182no—
248C3×C4—182no—
249C3×C4D4×C3⋊D312yes—
250C3×C4#383212yes—
251Dic3—68no—
252Dic3—68no—
253C32#572614yescomplemented
254C23—92no—
255C23—92no—
256Q8—24no—
257D4—94no—
258D4—188no—
259D4—182no—
260C2×C4—92no—
261C2×C4—92no—
262C2×C4—188no—
263C2×C4—184no—
264C2×C4—184no—
265C2×C4C22×#27211yescharacteristic
266C2×C4C22×#27214yes—
267C2×C4C22×#27211yescharacteristic
268C2×C4C22×#27211yescharacteristic
269C2×C3—34no—
270C2×C3—34no—
271C2×C3—34no—
272C2×C3—184no—
273C2×C3—362no—
274C2×C3C2×D4×C3⋊D311yescharacteristic
275C2×C3C2×#383211yescharacteristic
276C2×C3#6436411yescharacteristic
277D3—616no—
278D3—128no—
279C22—94no—
280C22—182no—
281C22—182no—
282C22—182no—
283C22—182no—
284C22—182no—
285C22C23×#27211yescharacteristic
286C4—24no—
287C4—21no—
288C4—182no—
289C4—182no—
290C4D4×#27212yes—
291C4#8874812yescomplemented
292C3—34no—
293C3#1059619711yescharacteristic
294C2—184no—
295C2—362no—
296C2C2×D4×#27211yescharacteristic
297C2C2×#8874811yescharacteristic
298C2#1172267011yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×C4⋊Q8C2
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterH(3,𝔽3)#547
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88748C2semidirect
2. H(3,𝔽3)×C4⋊Q8C2semidirect
3. #11721863C2semidirect
4. #11721763C2semidirect
5. #11721762C2semidirect
6. C4⋊C4×#272C2semidirect
7. C4×C4⋊#272C2semidirect
8. C4×C4⋊#272C2semidirect
28. C32×C4⋊Q8D3semidirect
46. C4⋊#272D4semidirect
79. C4×H(3,𝔽3)#41semidirect
163. C32×C4#239semidirect
167. C4⋊Q8#272semidirect
174. H(3,𝔽3)#547semidirect
253. C32#5726semidirect
291. C4#88748semidirect

Maximal subgroups

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

subgroupindex
C2×#887482
H(3,𝔽3)×C4⋊Q82
#117218632
#117217632
#117217622
C4⋊C4×#2722
C4×C4⋊#2722
C3×#57263