Finite Group Databasebeta

← #13144765

Subgroups — #13144765

Order 1728.

814 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 115 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×Q8×#272C211yesmaximal, characteristic
2C2×Q16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3#11721863C211yesmaximal, characteristic
4C2×#88550C211yesmaximal, characteristic, complemented
5#11720583C211yesmaximal, characteristic, complemented
6#11720553C211yesmaximal, characteristic
7#11720549C211yesmaximal, characteristic, complemented
8C3×#5186—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11Q8×#272—22no—
12Q8×#272—22no—
13C2×C4×#272—21no—
14C2×C4×#272C2211yescharacteristic
15C2×#6229—21no—
16C2×#6229C2211yescharacteristic
17Q16×H(3,𝔽3)—22no—
18C2×C8×H(3,𝔽3)C2211yescharacteristic
19#88550—22no—
20#88542—21no—
21C4⋊#1084C2211yescharacteristic
22#88540—21no—
23C2×#6180C2211yescharacteristic
24C2×C3×D3×Q8—34no—
25C2×C32×Q16D314yescomplemented
26C3×#946—34no—
27C2×C3×#216—34no—
28C3×#835—34no—
29C3×(C2×D3)⋊C8—34no—
30C3×#816—34no—
31C22×#272D411yescharacteristic
32Q8×H(3,𝔽3)—21no—
33Q8×H(3,𝔽3)—21no—
34Q8×H(3,𝔽3)—21no—
35Q8×H(3,𝔽3)D412yes—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C2×#1084—21no—
41C2×#1084D411yescharacteristic
42C4×#272—24no—
43C4×#272—21no—
44C4×#272—21no—
45#6229—24no—
46#6229—21no—
47#6229D412yes—
48C8×H(3,𝔽3)—21no—
49#6180—21no—
50C3×#451—91no—
51#5186—34no—
52C2×C32×Q8C2×D314yes—
53C2×C32×Q8C2×D314yes—
54C3×D3×Q8—68no—
55C3×D3×Q8—68no—
56C2×C3×C4×D3—34no—
57C2×C3×C4×D3—64no—
58C2×C3×Dic6—34no—
59C2×C3×Dic6—64no—
60C32×Q16—28no—
61C2×C32×C8C2×D314yes—
62C3×#216—68no—
63C3×(C2×D3)⋊C4—64no—
64C3×C4⋊Dic3—34no—
65C3×Dic3⋊C4—64no—
66C2×C3×C3⋊C8—34no—
67C22×H(3,𝔽3)C2×D411yescharacteristic
68C2×#272—21no—
69C2×#272—21no—
70C2×#272Q1612yescomplemented
71C4×H(3,𝔽3)—22no—
72C4×H(3,𝔽3)—41no—
73C4×H(3,𝔽3)C2×D411yescharacteristic
74C4×H(3,𝔽3)C2×D411yescharacteristic
75#1084—21no—
76#1084—22no—
77#1084—41no—
78C22×C3×Q8—91no—
79C2×D3×Q8—34no—
80C2×C3×Q16—34no—
81C2×C3×Q16—91no—
82C2×C3×Q16C3⋊D311yescharacteristic
83C3×C22⋊Q8—91no—
84#946—34no—
85C2×#216—34no—
86C3×#103—91no—
87C3×#103—91no—
88C3×C22⋊C8—91no—
89#835—34no—
90(C2×D3)⋊C8—34no—
91#816—34no—
92C22×C3×D3—34no—
93C32×Q8—24no—
94C32×Q8—24no—
95C32×Q8—24no—
96C32×Q8#6718yes—
97C2×C32×C4—24no—
98C2×C32×C4—24no—
99C2×C32×C4C22×D314yes—
100C2×C3×Dic3—34no—
101C2×C3×Dic3—64no—
102C2×C3×Dic3—64no—
103C3×C4×D3—616no—
104C3×C4×D3—64no—
105C3×C4×D3—64no—
106C3×Dic6—38no—
107C3×Dic6—616no—
108C3×Dic6—64no—
109C32×C8—24no—
110C3×C3⋊C8—64no—
111#451—91noSylow
112C2×H(3,𝔽3)#12011yescharacteristic
113C2×H(3,𝔽3)C2×Q1611yescharacteristic
114C2×H(3,𝔽3)#13711yescharacteristic
115#272—22no—
116C2×C3×Q8—34no—
117C2×C3×Q8—34no—
118C2×C3×Q8—91no—
119C2×C3×Q8—181no—
120C2×C3×Q8—182no—
121C2×C3×Q8—182no—
122C2×C3×Q8C2×C3⋊D311yescharacteristic
123C2×C3×Q8C2×C3⋊D311yescharacteristic
124C22×C3×C4—91no—
125C22×C3×C4—181no—
126D3×Q8—68no—
127D3×Q8—68no—
128C2×C4×D3—34no—
129C2×C4×D3—64no—
130C2×Dic6—34no—
131C2×Dic6—64no—
132C3×Q16—22no—
133C3×Q16—68no—
134C3×Q16—182no—
135C2×C3×C8—34no—
136C2×C3×C8—91no—
137C2×C3×C8C2×C3⋊D311yescharacteristic
138C3×C4⋊C4—91no—
139C3×C4⋊C4—181no—
140C3×C22⋊C4—181no—
141#216—68no—
142(C2×D3)⋊C4—64no—
143C4⋊Dic3—34no—
144Dic3⋊C4—64no—
145C2×C3⋊C8—34no—
146C22×C32D3×D414yes—
147C2×C3×D3—38no—
148C2×C3×D3—64no—
149C2×C3×D3—64no—
150C32×C4—28no—
151C32×C4—44no—
152C32×C4D3×D414yes—
153C32×C4C2×#6714yes—
154C3×Dic3—68no—
155C3×Dic3—64no—
156C3×Dic3—124no—
157C22×Q8—91no—
158C2×Q16—91no—
159C2×Q16#27211yescharacteristic, complemented
160C22⋊Q8—91no—
161#103—91no—
162#103—91no—
163C22⋊C8—91no—
164H(3,𝔽3)#45111yescharacteristic, complemented, Sylow
165C23×C3—91no—
166C22×D3—34no—
167C3×Q8—21no—
168C3×Q8—21no—
169C3×Q8—21no—
170C3×Q8—38no—
171C3×Q8—64no—
172C3×Q8—64no—
173C3×Q8—64no—
174C3×Q8—92no—
175C3×Q8—184no—
176C3×Q8—181no—
177C3×Q8#63812yes—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—34no—
181C2×C3×C4—64no—
182C2×C3×C4—64no—
183C2×C3×C4—91no—
184C2×C3×C4—184no—
185C2×C3×C4—181no—
186C2×C3×C4—181no—
187C2×C3×C4—181no—
188C2×C3×C4—181no—
189C2×C3×C4C22×C3⋊D311yescharacteristic
190C2×Dic3—34no—
191C2×Dic3—64no—
192C2×Dic3—64no—
193C4×D3—64no—
194C4×D3—64no—
195C4×D3—616no—
196Dic6—38no—
197Dic6—64no—
198Dic6—616no—
199C3×C8—21no—
200C3×C8—64no—
201C3×C8—181no—
202C3⋊C8—64no—
203C2×C32D3×Q1614yes—
204C2×C32#91814yes—
205C2×C32#93714yes—
206C3×D3—68no—
207C2×Q8—91no—
208C2×Q8—181no—
209C2×Q8—182no—
210C2×Q8—182no—
211C2×Q8C2×#27211yescharacteristic
212C2×Q8C2×#27211yescharacteristic
213C22×C4—91no—
214C22×C4—181no—
215Q16—22no—
216Q16—182no—
217C2×C8—91no—
218C2×C8C2×#27211yescharacteristic
219C4⋊C4—91no—
220C4⋊C4—181no—
221C22⋊C4—181no—
222C22×C3—34no—
223C22×C3—92no—
224C22×C3—181no—
225C22×C3—181no—
226C22×C3D4×C3⋊D311yescharacteristic
227C2×D3—38no—
228C2×D3—64no—
229C2×D3—64no—
230C3×C4—22no—
231C3×C4—34no—
232C3×C4—34no—
233C3×C4—41no—
234C3×C4—68no—
235C3×C4—124no—
236C3×C4—181no—
237C3×C4—182no—
238C3×C4—361no—
239C3×C4D4×C3⋊D311yescharacteristic
240C3×C4C2×#63811yescharacteristic
241Dic3—68no—
242Dic3—64no—
243Dic3—124no—
244C32#518614yescomplemented
245C23—91no—
246Q8—21no—
247Q8—21no—
248Q8—21no—
249Q8—92no—
250Q8—184no—
251Q8—181no—
252Q8#623512yes—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—91no—
256C2×C4—184no—
257C2×C4—181no—
258C2×C4—181no—
259C2×C4—181no—
260C2×C4—181no—
261C2×C4C22×#27211yescharacteristic
262C8—21no—
263C8—181no—
264C2×C3—34no—
265C2×C3—34no—
266C2×C3—34no—
267C2×C3—182no—
268C2×C3Q16×C3⋊D311yescharacteristic
269C2×C3#6412711yescharacteristic
270C2×C3#6414611yescharacteristic
271D3—68no—
272C22—92no—
273C22—181no—
274C22—181no—
275C22D4×#27211yescharacteristic
276C4—22no—
277C4—41no—
278C4—181no—
279C4—182no—
280C4—361no—
281C4D4×#27211yescharacteristic
282C4C2×#623511yescharacteristic
283C3—34no—
284C3#1059288511yescharacteristic
285C2—182no—
286C2Q16×#27211yescharacteristic
287C2#1172180711yescharacteristic
288C2#1172184711yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×Q16×H(3,𝔽3)C2
prime radicalC22×#272D4
hypercenterC2×C3×Q16C3⋊D3
hypocenterH(3,𝔽3)#451
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q16

Complements

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

NH ≅ G/Ntype
2. C2×Q16×H(3,𝔽3)C2semidirect
4. C2×#88550C2semidirect
5. #11720583C2semidirect
7. #11720549C2semidirect
25. C2×C32×Q16D3semidirect
70. C2×#272Q16semidirect
159. C2×Q16#272semidirect
164. H(3,𝔽3)#451semidirect
244. C32#5186semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#2722
C2×Q16×H(3,𝔽3)2
#117218632
C2×#885502
#117205832
#117205532
#117205492
C3×#51863