Finite Group Databasebeta

← #13150668

Subgroups — #13150668

Order 1728.

813 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 813 conjugacy classes of subgroups — too many to draw. Only its 147 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×#88540C211yesmaximal, characteristic
2#64133⋊C3C212yesmaximal
3#11721309C212yesmaximal
4#11721298C212yesmaximal
5#11721086C214yesmaximal, complemented
6#11721079C214yesmaximal, complemented
7#10594812—32nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11C2×#634⋊C3C2212yes—
12C2×#6196C2212yes—
13#88540C2214yes—
14#3757⋊C3C2214yes—
15C4⋊((C3⋊Dic3)⋊C3)C2212yes—
16#3750⋊C3C2214yes—
17#88449C2212yes—
18#88449C2212yes—
19#88449C2212yes—
20#88448C2214yes—
21#88447C2214yes—
22#64133D312yes—
23C3×#924—32no—
24C2×C3×Dic3⋊C4—32no—
25C2×C3×Dic3⋊C4—61no—
26#63964—34no—
27#63955—34no—
28#63846—34no—
29#63845—34no—
30#63840—34no—
31#63839—34no—
32C23×H(3,𝔽3)C2311yescharacteristic
33C2×C4×H(3,𝔽3)—41no—
34C2×C4×H(3,𝔽3)C2312yes—
35C2×#1084—41no—
36C2×#1084Q814yescomplemented
37C2×#1084C2312yes—
38C2×#1084C2312yes—
39C2×(C3⋊Dic3)⋊C3C2314yes—
40C2×(C3⋊Dic3)⋊C3C2314yes—
41#634⋊C3—42no—
42#6196—42no—
43#5623—92no—
44#5500—91no—
45C22×C32×C4—21no—
46C22×C32×C4C2×D312yes—
47C2×#634C2×D312yes—
48C22×C3×Dic3—32no—
49C22×C3×Dic3—32no—
50C22×C3×Dic3—61no—
51C22×C3×Dic3—61no—
52C2×C3×Dic6—32no—
53C2×#627—34no—
54#3757C2×D314yes—
55C4⋊(C3⋊Dic3)C2×D312yes—
56#3750C2×D314yes—
57C3×#217—34no—
58C3×C4⋊Dic3—32no—
59C3×Dic3⋊C4—34no—
60C3×Dic3⋊C4—34no—
61C3×Dic3⋊C4—34no—
62C3×Dic3⋊C4—64no—
63#3724—34no—
64Dic3⋊Dic3—34no—
65Dic3⋊Dic3—34no—
66Dic3⋊Dic3—34no—
67Dic3⋊Dic3—34no—
68Dic32—34no—
69Dic32—34no—
70C22×H(3,𝔽3)—21no—
71C22×H(3,𝔽3)—21no—
72C22×H(3,𝔽3)C2×Q812yes—
73C22×H(3,𝔽3)C2411yescharacteristic
74C4×H(3,𝔽3)—22no—
75#1084—22no—
76#1084C2×Q814yes—
77(C3⋊Dic3)⋊C3—24no—
78(C3⋊Dic3)⋊C3—24no—
79C3×C22⋊Q8—92no—
80C2×C3×C4⋊C4—91no—
81#944—94no—
82#926—92no—
83#924—32no—
84#924—92no—
85#924—92no—
86C2×Dic3⋊C4—32no—
87C2×Dic3⋊C4—61no—
88#890—94no—
89#889—94no—
90#889—94no—
91#888—94no—
92#888—94no—
93#878—94no—
94#877—94no—
95C23×C32—21no—
96C23×C32C22×D312yes—
97C2×C32×C4—22no—
98C2×C32×C4—42no—
99C2×C32×C4—42no—
100C2×C32×C4C22×D312yes—
101C2×C32×C4C22×D312yes—
102C2×C3⋊Dic3C22×D314yes—
103C2×C3⋊Dic3C22×D314yes—
104#634—42no—
105C2×C3×Dic3—38no—
106C2×C3×Dic3—34no—
107C2×C3×Dic3—34no—
108C2×C3×Dic3—34no—
109C2×C3×Dic3—34no—
110C2×C3×Dic3—64no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—122no—
114C2×C3×Dic3—122no—
115C3×Dic6—122no—
116#627—124no—
117#544—271noSylow
118C2×H(3,𝔽3)—22no—
119C2×H(3,𝔽3)C22×Q811yescharacteristic
120C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
121C2×H(3,𝔽3)2−1+411yescharacteristic
122C2×C3×Q8—92no—
123C22×C3×C4—32no—
124C22×C3×C4—61no—
125C22×C3×C4—91no—
126C22×C3×C4—91no—
127C22×C3×C4D3211yescharacteristic
128C22×Dic3—32no—
129C22×Dic3—32no—
130C22×Dic3—61no—
131C22×Dic3—61no—
132C2×Dic6—32no—
133C2×Dic6—92no—
134C2×Dic6—92no—
135C2×Dic6—94no—
136C3×C4⋊C4—94no—
137C3×C4⋊C4—92no—
138C3×C4⋊C4—94no—
139C3×C22⋊C4—94no—
140#217—34no—
141#217—94no—
142C4⋊Dic3—32no—
143C4⋊Dic3—92no—
144C4⋊Dic3—92no—
145C4⋊Dic3—92no—
146C4⋊Dic3—92no—
147C4⋊Dic3—94no—
148Dic3⋊C4—34no—
149Dic3⋊C4—34no—
150Dic3⋊C4—34no—
151Dic3⋊C4—64no—
152Dic3⋊C4—94no—
153Dic3⋊C4—94no—
154Dic3⋊C4—94no—
155Dic3⋊C4—94no—
156Dic3⋊C4—94no—
157Dic3⋊C4—94no—
158C4×Dic3—94no—
159C4×Dic3—94no—
160C4×Dic3—94no—
161C22×C32—22no—
162C22×C32—22no—
163C22×C32—22no—
164C22×C32—22no—
165C22×C32—21no—
166C22×C32—22no—
167C22×C32D3×Q814yes—
168C22×C32C23×D312yes—
169C32×C4—22no—
170C32×C4—22no—
171C32×C4—42no—
172C3⋊Dic3—24no—
173C3⋊Dic3—24no—
174C3×Dic3—38no—
175C3×Dic3—64no—
176C3×Dic3—64no—
177C3×Dic3—64no—
178C3×Dic3—64no—
179C3×Dic3—122no—
180C4⋊Q8—274no—
181#125—274no—
182C22⋊Q8—92no—
183C22⋊Q8—272no—
184#117—272no—
185C2×C4⋊C4—91no—
186H(3,𝔽3)#54411yescharacteristic, complemented, Sylow
187C23×C3—32no—
188C23×C3—61no—
189C23×C3C2×D3211yescharacteristic
190C3×Q8—362no—
191C2×C3×C4—32no—
192C2×C3×C4—32no—
193C2×C3×C4—41no—
194C2×C3×C4—62no—
195C2×C3×C4—92no—
196C2×C3×C4—94no—
197C2×C3×C4—94no—
198C2×C3×C4—94no—
199C2×C3×C4—92no—
200C2×C3×C4—122no—
201C2×C3×C4—122no—
202C2×C3×C4—361no—
203C2×C3×C4C2×D3212yes—
204C2×Dic3—34no—
205C2×Dic3—38no—
206C2×Dic3—34no—
207C2×Dic3—34no—
208C2×Dic3—34no—
209C2×Dic3—62no—
210C2×Dic3—64no—
211C2×Dic3—62no—
212C2×Dic3—94no—
213C2×Dic3—94no—
214C2×Dic3—122no—
215C2×Dic3—122no—
216Dic6—122no—
217Dic6—362no—
218Dic6—362no—
219Dic6—364no—
220C2×C32—24no—
221C2×C32—24no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32#100412yes—
226C2×C32C2×D3×Q812yes—
227C2×C32#101012yes—
228C2×Q8—92no—
229C2×Q8—272no—
230C22×C4—91no—
231C22×C4—91no—
232C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
233C4⋊C4—94no—
234C4⋊C4—92no—
235C4⋊C4—94no—
236C4⋊C4—272no—
237C4⋊C4—272no—
238C4⋊C4—272no—
239C4⋊C4—274no—
240C22⋊C4—94no—
241C42—274no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—32no—
245C22×C3—34no—
246C22×C3—62no—
247C22×C3—62no—
248C22×C3—62no—
249C22×C3—62no—
250C22×C3—61no—
251C22×C3—62no—
252C22×C3#379712yes—
253C22×C3C22×D3211yescharacteristic
254C3×C4—22no—
255C3×C4—62no—
256C3×C4—62no—
257C3×C4—94no—
258C3×C4—122no—
259C3×C4—184no—
260C3×C4—184no—
261C3×C4—182no—
262Dic3—38no—
263Dic3—64no—
264Dic3—64no—
265Dic3—64no—
266Dic3—64no—
267Dic3—122no—
268Dic3—184no—
269Dic3—184no—
270C32—21no—
271C32#562312yescomplemented
272C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
273Q8—362no—
274Q8—1082no—
275C2×C4—41no—
276C2×C4—92no—
277C2×C4—94no—
278C2×C4—94no—
279C2×C4—94no—
280C2×C4—92no—
281C2×C4—361no—
282C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
283C2×C3—22no—
284C2×C3—32no—
285C2×C3—32no—
286C2×C3—32no—
287C2×C3—64no—
288C2×C3—64no—
289C2×C3—61no—
290C2×C3—61no—
291C2×C3—61no—
292C2×C3#6429811yescharacteristic
293C2×C3C2×#379711yescharacteristic
294C2×C3#6433011yescharacteristic
295C22—21no—
296C22—21no—
297C22#8865112yes—
298C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
299C4—22no—
300C4—94no—
301C4—182no—
302C4—184no—
303C4—184no—
304C3—32no—
305C3—61no—
306C3#1059481311yescharacteristic
307C2—22no—
308C2#1172248311yescharacteristic
309C2C2×#8865111yescharacteristic
310C2#1172254511yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#544
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
5. #11721086C2semidirect
6. #11721079C2semidirect
36. C2×#1084Q8semidirect
186. H(3,𝔽3)#544semidirect
271. C32#5623semidirect

Maximal subgroups

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

subgroupindex
C2×#885402
#64133⋊C32
#117213092
#117212982
#117210862
#117210792
#105948123