Finite Group Databasebeta

← #13140944

Subgroups — #13140944

Order 1728.

717 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 717 conjugacy classes of subgroups — too many to draw. Only its 91 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×#88748C211yesmaximal, characteristic
2#11721089C211yesmaximal, characteristic, complemented
3C2×#88434C211yesmaximal, characteristic, complemented
4#63662⋊C3C212yesmaximal, complemented
5#11720324C212yesmaximal, complemented
6#10591556—32nomaximal
7C2×Q8×H(3,𝔽3)C2211yescharacteristic
8#88748—22no—
9#88748C414yescomplemented
10C2×C4⋊#272—21no—
11C2×C4⋊#272C2211yescharacteristic
12C2×C4×#272—21no—
13C2×C4×#272C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)C2212yes—
15C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
16#88451—21no—
17#88447—21no—
18#88434—22no—
19C2×C3×#239—32no—
20C2×C3×#239—61no—
21#63841—32no—
22C2×D3×C3⋊C8—32no—
23#63662D312yescomplemented
24C3×#835—32no—
25#63563—32no—
26#63562—32no—
27C22×#272—21no—
28C22×#272D411yescharacteristic
29Q8×H(3,𝔽3)—21no—
30Q8×H(3,𝔽3)C2×C412yes—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×#1084D411yescharacteristic
34C4⋊#272—24no—
35C4⋊#272—21no—
36C4⋊#272C2×C412yes—
37C4×#272—24no—
38C4×#272C2×C412yes—
39C4×#272D412yes—
40C2×(C3⋊Dic3)⋊C3—22no—
41(C3⋊(C3⋊C8))⋊C3—22no—
42#5233—91no—
43#4804—92no—
44C2×C32×Q8—21no—
45C2×C32×Q8C2×D312yes—
46C3×#239—38no—
47C3×#239—64no—
48C3×#239—64no—
49C3×#239—64no—
50C2×C3×D12—32no—
51C2×C3×D12—62no—
52C2×C3×D12—61no—
53C2×C3×D12—62no—
54C2×C3×C4×D3—32no—
55C2×C3×C4×D3—62no—
56C2×C3×C4×D3—61no—
57C2×C3×C4×D3—62no—
58C4⋊(C3⋊Dic3)C2×D312yes—
59C2×C3⋊(C3⋊C8)C2×D312yes—
60C3×C4⋊Dic3—32no—
61C2×C3×C3⋊C8—32no—
62(C2×D3)⋊Dic3—62no—
63Dic32—62no—
64D3×C3⋊C8—64no—
65C22×H(3,𝔽3)C2×D411yescharacteristic
66C2×#272—22no—
67C2×#272—22no—
68C2×#272—42no—
69C2×#272C22⋊C412yes—
70C4×H(3,𝔽3)—22no—
71C4×H(3,𝔽3)C22×C411yescharacteristic
72C4×H(3,𝔽3)C2×D411yescharacteristic
73#1084C22⋊C412yes—
74(C3⋊Dic3)⋊C3—42no—
75C2×C3×#41—91no—
76C2×#239—32no—
77C2×#239—61no—
78#926—91no—
79C22×C3⋊C8—91no—
80C2×D3×C8—92no—
81#892—92no—
82C3×#103—92no—
83#835—32no—
84#835—92no—
85#832—92no—
86#821—92no—
87#809—92no—
88C22×C3×D3—32no—
89C22×C3×D3—61no—
90C22×C3×D3—62no—
91C22×C3×D3—62no—
92C32×Q8—22no—
93C32×Q8—22no—
94C32×Q8—22no—
95C32×Q8C4×D314yes—
96C2×C32×C4—22no—
97C2×C32×C4—22no—
98C2×C32×C4—21no—
99C2×C32×C4C22×D312yes—
100C2×C3⋊Dic3—22no—
101C2×C3×Dic3—32no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—62no—
104C3×D12—34no—
105C3×D12—62no—
106C3×D12—68no—
107C3×D12—68no—
108C3×D12—62no—
109C3×D12—62no—
110C3×C4×D3—34no—
111C3×C4×D3—34no—
112C3×C4×D3—62no—
113C3×C4×D3—62no—
114C3×C4×D3—68no—
115C3×C4×D3—68no—
116C3⋊(C3⋊C8)—22no—
117C3×C3⋊C8—62no—
118#416—271noSylow
119C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
120C2×H(3,𝔽3)#13512yes—
121#272—22no—
122#272—42no—
123C3×#41—94no—
124C3×#41—182no—
125C2×C3×Q8—32no—
126C2×C3×Q8—61no—
127C2×C3×Q8D3211yescharacteristic
128C2×C3×D4—91no—
129C2×C3×D4—181no—
130C22×C3×C4—91no—
131C22×C3×C4—181no—
132#239—38no—
133#239—64no—
134#239—64no—
135#239—64no—
136C2×D12—32no—
137C2×D12—61no—
138C2×D12—62no—
139C2×D12—62no—
140C2×C4×D3—32no—
141C2×C4×D3—61no—
142C2×C4×D3—62no—
143C2×C4×D3—62no—
144C2×C3×C8—92no—
145C3×C4⋊C4—92no—
146#217—181no—
147(C2×D3)⋊C4—182no—
148C4⋊Dic3—32no—
149C4⋊Dic3—92no—
150C4×Dic3—182no—
151C4×Dic3—181no—
152C2×C3⋊C8—32no—
153C2×C3⋊C8—92no—
154C2×C3⋊C8—182no—
155D3×C8—184no—
156C22×C32—21no—
157C22×C32D3×D412yes—
158C2×C3×D3—34no—
159C2×C3×D3—62no—
160C2×C3×D3—62no—
161C2×C3×D3—62no—
162C2×C3×D3—64no—
163C2×C3×D3—64no—
164C2×C3×D3—122no—
165C2×C3×D3—124no—
166C2×C3×D3—122no—
167C2×C3×D3—122no—
168C32×C4—24no—
169C32×C4—24no—
170C32×C4—21no—
171C32×C4—21no—
172C32×C4C2×C4×D312yes—
173C32×C4D3×D412yes—
174C3⋊Dic3—42no—
175C3×Dic3—34no—
176C3×Dic3—62no—
177C3×Dic3—122no—
178C2×#41—91no—
179C22×C8—271no—
180#117—271no—
181#103—92no—
182#102—272no—
183H(3,𝔽3)#41611yescharacteristic, complemented, Sylow
184C23×C3—91no—
185C23×C3—181no—
186C22×D3—32no—
187C22×D3—61no—
188C22×D3—62no—
189C22×D3—62no—
190C3×Q8—21no—
191C3×Q8—34no—
192C3×Q8—62no—
193C3×Q8—62no—
194C3×Q8—62no—
195C3×Q8#62412yes—
196C3×D4—92no—
197C3×D4—184no—
198C3×D4—181no—
199C2×C3×C4—21no—
200C2×C3×C4—32no—
201C2×C3×C4—62no—
202C2×C3×C4—62no—
203C2×C3×C4—61no—
204C2×C3×C4—91no—
205C2×C3×C4—92no—
206C2×C3×C4—92no—
207C2×C3×C4—184no—
208C2×C3×C4—182no—
209C2×C3×C4C2×D3211yescharacteristic
210C2×Dic3—32no—
211C2×Dic3—61no—
212C2×Dic3—62no—
213C2×Dic3—182no—
214D12—34no—
215D12—62no—
216D12—62no—
217D12—68no—
218D12—68no—
219D12—62no—
220C4×D3—34no—
221C4×D3—34no—
222C4×D3—62no—
223C4×D3—62no—
224C4×D3—68no—
225C4×D3—68no—
226C3×C8—182no—
227C3⋊C8—62no—
228C3⋊C8—182no—
229C2×C32—22no—
230C2×C32—21no—
231C2×C32D3×C22⋊C412yes—
232C2×C32#91612yes—
233C2×C32#91912yes—
234C3×D3—64no—
235C3×D3—122no—
236C3×D3—124no—
237C3×D3—124no—
238#41—94no—
239#41—182no—
240C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
241C2×D4—91no—
242C2×D4—181no—
243C22×C4—91no—
244C22×C4—181no—
245C2×C8—92no—
246C2×C8—542no—
247C4⋊C4—92no—
248C22⋊C4—541no—
249C42—541no—
250C22×C3—32no—
251C22×C3—61no—
252C22×C3—92no—
253C22×C3—182no—
254C22×C3—182no—
255C22×C3—362no—
256C22×C3#381111yescharacteristic
257C2×D3—34no—
258C2×D3—62no—
259C2×D3—62no—
260C2×D3—62no—
261C2×D3—64no—
262C2×D3—64no—
263C2×D3—122no—
264C2×D3—122no—
265C2×D3—122no—
266C2×D3—124no—
267C3×C4—22no—
268C3×C4—32no—
269C3×C4—32no—
270C3×C4—64no—
271C3×C4—64no—
272C3×C4—61no—
273C3×C4—61no—
274C3×C4—92no—
275C3×C4—362no—
276C3×C4C2×#62411yescharacteristic
277C3×C4#381111yescharacteristic
278Dic3—34no—
279Dic3—62no—
280Dic3—122no—
281Dic3—362no—
282C32—21no—
283C32#480412yescomplemented
284C23—91no—
285C23—181no—
286Q8—21no—
287Q8#619912yes—
288D4—92no—
289D4—184no—
290D4—181no—
291C2×C4—21no—
292C2×C4—91no—
293C2×C4—92no—
294C2×C4—92no—
295C2×C4—184no—
296C2×C4—182no—
297C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
298C8—182no—
299C2×C3—32no—
300C2×C3—32no—
301C2×C3—32no—
302C2×C3—62no—
303C2×C3—61no—
304C2×C3—182no—
305C2×C3—362no—
306C2×C3#6395012yes—
307C2×C3#6397411yescharacteristic
308D3—64no—
309D3—122no—
310D3—124no—
311D3—124no—
312C22—92no—
313C22—182no—
314C22—182no—
315C22—362no—
316C22#8867711yescharacteristic
317C4—22no—
318C4—92no—
319C4—362no—
320C4C2×#619911yescharacteristic
321C4#8867711yescharacteristic
322C3—32no—
323C3—61no—
324C3#1059156011yescharacteristic
325C2—182no—
326C2—362no—
327C2#1172123012yes—
328C2#1172132011yescharacteristic
subgroupG/H
centerC22#88677
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC2×#88748C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#416
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11721089C2semidirect
3. C2×#88434C2semidirect
4. #63662⋊C3C2semidirect
5. #11720324C2semidirect
9. #88748C4semidirect
23. #63662D3semidirect
183. H(3,𝔽3)#416semidirect
283. C32#4804semidirect

Maximal subgroups

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

subgroupindex
C2×#887482
#117210892
C2×#884342
#63662⋊C32
#117203242
#105915563