Finite Group Databasebeta

← #13154297

Subgroups — #13154297

Order 1728.

1221 conjugacy classes of subgroups; only the 241 normal subgroups are shown.

normal subgroups only

This group has 1221 conjugacy classes of subgroups — too many to draw. Only its 241 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
1H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
2#11721863C212yesmaximal
3Q8×#1084C212yesmaximal, complemented
4#11721765C212yesmaximal
5#11721760C212yesmaximal
6#11721758C212yesmaximal, complemented
7#11721755C212yesmaximal, complemented
8C42×#272C211yesmaximal, characteristic
9#11721715C211yesmaximal, characteristic, complemented
10C3×#5728—34nomaximal
11C2×Q8×H(3,𝔽3)C2212yes—
12C2×C4×#272C2211yescharacteristic
13C2×C4×#272C2211yescharacteristic
14C2×C4×#272C2211yescharacteristic
15C2×#6229C2212yes—
16C4⋊C4×H(3,𝔽3)C2212yes—
17C4⋊C4×H(3,𝔽3)C2212yes—
18C42×H(3,𝔽3)C2211yescharacteristic
19#88542C2214yes—
20C4⋊#1084C2212yes—
21C4⋊#1084C2214yes—
22C4⋊#1084C2212yes—
23#88540C2214yes—
24C4×#1084C2211yescharacteristic
25C4×#1084C2214yes—
26C4×#1084C2211yescharacteristic
27C4×#1084C2211yescharacteristic
28C32×C4⋊Q8D314yescomplemented
29C3×#946—38no—
30C3×Q8×Dic3—38no—
31C3×#897—38no—
32C3×#892—38no—
33C3×#890—38no—
34C3×#887—38no—
35C3×C42×D3—34no—
36C3×#869—34no—
37C22×#272C2311yescharacteristic
38Q8×H(3,𝔽3)—24no—
39C2×C4×H(3,𝔽3)C2314yes—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084C2314yes—
44C2×#1084C2311yescharacteristic
45C2×#1084C2311yescharacteristic
46C2×#1084C2311yescharacteristic
47C4×#272—22no—
48C4×#272—22no—
49C4×#272Q814yescomplemented
50#6229—24no—
51C3×#533—91no—
52#5728—34no—
53C2×C32×Q8C2×D318yes—
54C2×C3×C4×D3—34no—
55C2×C3×C4×D3—34no—
56C2×C3×C4×D3—34no—
57C2×C3×Dic6—38no—
58C32×C4⋊C4C2×D318yes—
59C32×C4⋊C4C2×D318yes—
60C32×C42C2×D314yes—
61C3×(C2×D3)⋊C4—316no—
62C3×C4⋊Dic3—38no—
63C3×C4⋊Dic3—316no—
64C3×C4⋊Dic3—38no—
65C3×Dic3⋊C4—316no—
66C3×C4×Dic3—34no—
67C3×C4×Dic3—316no—
68C3×C4×Dic3—34no—
69C3×C4×Dic3—34no—
70C22×H(3,𝔽3)C2411yescharacteristic
71C2×#272—21no—
72C2×#272—21no—
73C2×#272C2×Q812yes—
74C4×H(3,𝔽3)—24no—
75C4×H(3,𝔽3)C2×Q812yes—
76C4×H(3,𝔽3)#4112yes—
77C4×H(3,𝔽3)#4112yes—
78#1084—24no—
79#1084—21no—
80#1084—21no—
81#1084C2×Q812yes—
82C3×C4⋊Q8—34no—
83C3×C4⋊Q8—91no—
84C3×C4⋊Q8C3⋊D311yescharacteristic
85C3×#125—92no—
86C3×C22⋊Q8—92no—
87C3×C22⋊Q8—92no—
88C3×C4×Q8—92no—
89C3×C4×Q8—92no—
90C3×#117—92no—
91C2×C3×C42—91no—
92#946—38no—
93Q8×Dic3—38no—
94#897—38no—
95#892—38no—
96#890—38no—
97#887—38no—
98C42×D3—34no—
99#869—34no—
100C22×C3×D3—34no—
101C32×Q8—216no—
102C2×C32×C4C22×D3116yes—
103C2×C32×C4C22×D314yes—
104C2×C32×C4C22×D314yes—
105C2×C32×C4C22×D314yes—
106C2×C3×Dic3—34no—
107C2×C3×Dic3—34no—
108C2×C3×Dic3—34no—
109C2×C3×Dic3—316no—
110C3×C4×D3—316no—
111C3×C4×D3—68no—
112C3×C4×D3—68no—
113C3×Dic6—616no—
114#533—91noSylow
115C2×H(3,𝔽3)C22×Q811yescharacteristic
116C2×H(3,𝔽3)C2×#4111yescharacteristic
117C2×H(3,𝔽3)C2×#4111yescharacteristic
118#272—22no—
119C2×C3×Q8—38no—
120C2×C3×Q8—92no—
121C2×C3×Q8C2×C3⋊D312yes—
122C22×C3×C4—91no—
123C22×C3×C4—91no—
124C22×C3×C4—91no—
125C2×C4×D3—34no—
126C2×C4×D3—34no—
127C2×C4×D3—34no—
128C2×Dic6—38no—
129C3×C4⋊C4—38no—
130C3×C4⋊C4—38no—
131C3×C4⋊C4—94no—
132C3×C4⋊C4—92no—
133C3×C4⋊C4—94no—
134C3×C4⋊C4—92no—
135C3×C4⋊C4C2×C3⋊D312yes—
136C3×C4⋊C4C2×C3⋊D312yes—
137C3×C22⋊C4—94no—
138C3×C42—34no—
139C3×C42—91no—
140C3×C42—91no—
141C3×C42—91no—
142C3×C42—94no—
143C3×C42C2×C3⋊D311yescharacteristic
144(C2×D3)⋊C4—316no—
145C4⋊Dic3—38no—
146C4⋊Dic3—38no—
147C4⋊Dic3—316no—
148Dic3⋊C4—316no—
149C4×Dic3—34no—
150C4×Dic3—34no—
151C4×Dic3—34no—
152C4×Dic3—316no—
153C22×C32C23×D314yes—
154C2×C3×D3—38no—
155C2×C3×D3—64no—
156C2×C3×D3—64no—
157C32×C4—216no—
158C32×C4#23718yes—
159C32×C4D3×Q818yes—
160C32×C4#23918yes—
161C3×Dic3—38no—
162C3×Dic3—64no—
163C3×Dic3—616no—
164C3×Dic3—64no—
165C4⋊Q8—91no—
166C4⋊Q8#27211yescharacteristic, complemented
167#125—92no—
168C22⋊Q8—92no—
169C22⋊Q8—92no—
170C4×Q8—92no—
171C4×Q8—92no—
172#117—92no—
173C2×C42—91no—
174H(3,𝔽3)#53311yescharacteristic, complemented, Sylow
175C23×C3—91no—
176C22×D3—34no—
177C3×Q8—24no—
178C3×Q8—616no—
179C3×Q8—184no—
180C2×C3×C4—34no—
181C2×C3×C4—34no—
182C2×C3×C4—34no—
183C2×C3×C4—316no—
184C2×C3×C4—91no—
185C2×C3×C4—94no—
186C2×C3×C4—91no—
187C2×C3×C4—91no—
188C2×C3×C4—94no—
189C2×C3×C4—182no—
190C2×C3×C4—182no—
191C2×C3×C4C22×C3⋊D314yes—
192C2×C3×C4C22×C3⋊D311yescharacteristic
193C2×C3×C4C22×C3⋊D311yescharacteristic
194C2×C3×C4C22×C3⋊D311yescharacteristic
195C2×Dic3—34no—
196C2×Dic3—34no—
197C2×Dic3—34no—
198C2×Dic3—316no—
199C4×D3—316no—
200C4×D3—68no—
201C4×D3—68no—
202Dic6—616no—
203C2×C32C2×#23714yes—
204C2×C32C2×D3×Q814yes—
205C2×C32C2×#23914yes—
206C3×D3—68no—
207C2×Q8—92no—
208C2×Q8C2×#27212yes—
209C22×C4—91no—
210C22×C4—91no—
211C22×C4—91no—
212C4⋊C4—92no—
213C4⋊C4—94no—
214C4⋊C4—94no—
215C4⋊C4—92no—
216C4⋊C4C2×#27212yes—
217C4⋊C4C2×#27212yes—
218C22⋊C4—94no—
219C42—91no—
220C42—91no—
221C42—91no—
222C42—94no—
223C42C2×#27211yescharacteristic
224C22×C3—34no—
225C22×C3—92no—
226C22×C3—181no—
227C22×C3—181no—
228C22×C3C23×C3⋊D311yescharacteristic
229C2×D3—38no—
230C2×D3—64no—
231C2×D3—64no—
232C3×C4—24no—
233C3×C4—38no—
234C3×C4—38no—
235C3×C4—38no—
236C3×C4—616no—
237C3×C4—92no—
238C3×C4—181no—
239C3×C4—181no—
240C3×C4—184no—
241C3×C4#383012yes—
242C3×C4Q8×C3⋊D312yes—
243C3×C4#383212yes—
244Dic3—38no—
245Dic3—616no—
246Dic3—64no—
247Dic3—64no—
248C32#572814yescomplemented
249C23—91no—
250Q8—24no—
251Q8—184no—
252C2×C4—91no—
253C2×C4—94no—
254C2×C4—91no—
255C2×C4—91no—
256C2×C4—94no—
257C2×C4—182no—
258C2×C4—182no—
259C2×C4C22×#27214yes—
260C2×C4C22×#27211yescharacteristic
261C2×C4C22×#27211yescharacteristic
262C2×C4C22×#27211yescharacteristic
263C2×C3—34no—
264C2×C3—34no—
265C2×C3—34no—
266C2×C3—182no—
267C2×C3C2×#383011yescharacteristic
268C2×C3C2×Q8×C3⋊D311yescharacteristic
269C2×C3C2×#383211yescharacteristic
270D3—68no—
271C22—92no—
272C22—181no—
273C22—181no—
274C22C23×#27211yescharacteristic
275C4—24no—
276C4—92no—
277C4—181no—
278C4—181no—
279C4—184no—
280C4#8874412yes—
281C4Q8×#27212yes—
282C4#8874812yes—
283C3—34no—
284C3#1059619911yescharacteristic
285C2—182no—
286C2C2×#8874411yescharacteristic
287C2C2×Q8×#27211yescharacteristic
288C2C2×#8874811yescharacteristic
subgroupG/H
center, Frattini, socleC22×C3C23×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FittingH(3,𝔽3)×C4⋊Q8C2
prime radicalC22×#272C23
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterH(3,𝔽3)#533
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. H(3,𝔽3)×C4⋊Q8C2semidirect
3. Q8×#1084C2semidirect
6. #11721758C2semidirect
7. #11721755C2semidirect
9. #11721715C2semidirect
28. C32×C4⋊Q8D3semidirect
49. C4×#272Q8semidirect
166. C4⋊Q8#272semidirect
174. H(3,𝔽3)#533semidirect
248. C32#5728semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C4⋊Q82
#117218632
Q8×#10842
#117217652
#117217602
#117217582
#117217552
C42×#2722
#117217152
C3×#57283