Finite Group Databasebeta

← #13147755

Subgroups — #13147755

Order 1728.

748 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 748 conjugacy classes of subgroups — too many to draw. Only its 139 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
1#11721718C211yesmaximal, characteristic
2#64079⋊C3C211yesmaximal, characteristic, complemented
3#64078⋊C3C211yesmaximal, characteristic, complemented
4#11721094C212yesmaximal
5#11721088C214yesmaximal
6#11721087C214yesmaximal, complemented
7#11721084C212yesmaximal, complemented
8#10593881—31nomaximal
9#10593862—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×#634⋊C3C2212yes—
12C42×H(3,𝔽3)C2211yescharacteristic
13#88542C2212yes—
14#88540C2212yes—
15C4×#1084C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2212yes—
17C4⋊((C3⋊Dic3)⋊C3)C2214yes—
18#3750⋊C3C2214yes—
19#88451C2214yes—
20#88448C2214yes—
21#88448C2214yes—
22#88447C2214yes—
23#64079D311yescharacteristic, complemented
24#64078D311yescharacteristic, complemented
25C3×#872—31no—
26C3×#872—31no—
27C3×#872—61no—
28C3×#870—31no—
29C3×#869—31no—
30#63861—34no—
31#63858—34no—
32#63857—32no—
33#63856—32no—
34#63847—34no—
35#63845—34no—
36#63844—32no—
37Dic3×Dic6—32no—
38C22×#272C2311yescharacteristic
39C2×C4×H(3,𝔽3)C2312yes—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084C2312yes—
42C2×#1084C2311yescharacteristic
43C4×#272—22no—
44C2×(C3⋊Dic3)⋊C3C2314yes—
45C2×(C3⋊Dic3)⋊C3C2314yes—
46#634⋊C3—24no—
47#5731—91no—
48#5698—91no—
49#5518—91no—
50C2×#634C2×D312yes—
51C2×C3×C4×D3—31no—
52C2×C3×C4×D3—31no—
53C2×C3×C4×D3—61no—
54C2×C3×Dic6—32no—
55C32×C42—21no—
56C32×C42C2×D311yescharacteristic
57C32×C42C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)C2×D314yes—
59C4⋊(C3⋊Dic3)C2×D312yes—
60#3750C2×D314yes—
61C3×(C2×D3)⋊C4—32no—
62C3×(C2×D3)⋊C4—32no—
63C3×(C2×D3)⋊C4—62no—
64C3×C4⋊Dic3—32no—
65C3×C4⋊Dic3—34no—
66C3×Dic3⋊C4—32no—
67C3×Dic3⋊C4—34no—
68C3×Dic3⋊C4—32no—
69C3×Dic3⋊C4—62no—
70C3×C4×Dic3—31no—
71C3×C4×Dic3—31no—
72C3×C4×Dic3—61no—
73#3724—34no—
74#3724—34no—
75Dic3⋊Dic3—34no—
76Dic3⋊Dic3—34no—
77(C2×D3)⋊Dic3—34no—
78(C2×D3)⋊Dic3—34no—
79Dic32—34no—
80Dic32—34no—
81C22×H(3,𝔽3)C2411yescharacteristic
82C2×#272—21no—
83C2×#272—22no—
84C4×H(3,𝔽3)—22no—
85C4×H(3,𝔽3)#4112yes—
86#1084—21no—
87#1084—22no—
88(C3⋊Dic3)⋊C3—24no—
89(C3⋊Dic3)⋊C3—24no—
90C3×C4⋊Q8—91no—
91C3×#125—91no—
92C3×#117—91no—
93#946—92no—
94Q8×Dic3—92no—
95#924—92no—
96#898—94no—
97#898—94no—
98#897—92no—
99#890—94no—
100#889—94no—
101#889—94no—
102#887—92no—
103#879—94no—
104#872—31no—
105#872—31no—
106#872—61no—
107#870—31no—
108#870—91no—
109#869—31no—
110#869—91no—
111C4×Dic6—92no—
112C22×C3×D3—31no—
113C22×C3×D3—31no—
114C22×C3×D3—61no—
115C2×C32×C4—21no—
116C2×C32×C4—22no—
117C2×C32×C4C22×D312yes—
118C2×C32×C4C22×D312yes—
119C2×C32×C4C22×D311yescharacteristic
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3C22×D314yes—
122C2×C3⋊Dic3C22×D314yes—
123#634—24no—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—32no—
127C2×C3×Dic3—32no—
128C2×C3×Dic3—31no—
129C2×C3×Dic3—31no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—62no—
132C3×C4×D3—64no—
133C3×C4×D3—62no—
134C3×C4×D3—62no—
135C3×Dic6—64no—
136#541—271noSylow
137C2×H(3,𝔽3)C2×#4111yescharacteristic
138C2×H(3,𝔽3)2−1+412yes—
139#272—41no—
140C2×C3×Q8—92no—
141C22×C3×C4—91no—
142C2×C4×D3—31no—
143C2×C4×D3—31no—
144C2×C4×D3—61no—
145C2×Dic6—32no—
146C2×Dic6—92no—
147C3×C4⋊C4—92no—
148C3×C4⋊C4—94no—
149C3×C4⋊C4—94no—
150C3×C4⋊C4—92no—
151C3×C22⋊C4—92no—
152C3×C42—31no—
153C3×C42—31no—
154C3×C42—61no—
155C3×C42—91no—
156C3×C42D3211yescharacteristic
157#217—94no—
158(C2×D3)⋊C4—32no—
159(C2×D3)⋊C4—32no—
160(C2×D3)⋊C4—62no—
161(C2×D3)⋊C4—94no—
162(C2×D3)⋊C4—94no—
163C4⋊Dic3—32no—
164C4⋊Dic3—34no—
165C4⋊Dic3—94no—
166C4⋊Dic3—92no—
167C4⋊Dic3—94no—
168C4⋊Dic3—94no—
169Dic3⋊C4—32no—
170Dic3⋊C4—32no—
171Dic3⋊C4—34no—
172Dic3⋊C4—62no—
173Dic3⋊C4—94no—
174Dic3⋊C4—94no—
175Dic3⋊C4—94no—
176Dic3⋊C4—94no—
177Dic3⋊C4—94no—
178C4×Dic3—31no—
179C4×Dic3—31no—
180C4×Dic3—61no—
181C4×Dic3—94no—
182C4×Dic3—94no—
183C4×Dic3—94no—
184C22×C32—21no—
185C22×C32C23×D311yescharacteristic
186C22×C32C23×D311yescharacteristic
187C2×C3×D3—62no—
188C2×C3×D3—61no—
189C2×C3×D3—61no—
190C2×C3×D3—62no—
191C2×C3×D3—62no—
192C2×C3×D3—122no—
193C32×C4—22no—
194C32×C4—22no—
195C32×C4—22no—
196C32×C4—42no—
197C32×C4#23712yes—
198C32×C4#23912yes—
199C3⋊Dic3—24no—
200C3⋊Dic3—24no—
201C3×Dic3—62no—
202C3×Dic3—61no—
203C3×Dic3—64no—
204C3×Dic3—61no—
205C3×Dic3—64no—
206C3×Dic3—62no—
207C3×Dic3—62no—
208C3×Dic3—122no—
209C4⋊Q8—91no—
210#126—274no—
211#125—91no—
212#125—274no—
213C22⋊Q8—272no—
214C4×Q8—272no—
215#117—91no—
216H(3,𝔽3)#54111yescharacteristic, complemented, Sylow
217C23×C3—91no—
218C22×D3—31no—
219C22×D3—31no—
220C22×D3—61no—
221C3×Q8—184no—
222C2×C3×C4—31no—
223C2×C3×C4—31no—
224C2×C3×C4—32no—
225C2×C3×C4—32no—
226C2×C3×C4—61no—
227C2×C3×C4—62no—
228C2×C3×C4—94no—
229C2×C3×C4—94no—
230C2×C3×C4—92no—
231C2×C3×C4—91no—
232C2×C3×C4—182no—
233C2×C3×C4C2×D3212yes—
234C2×C3×C4C2×D3211yescharacteristic
235C2×Dic3—32no—
236C2×Dic3—32no—
237C2×Dic3—31no—
238C2×Dic3—31no—
239C2×Dic3—34no—
240C2×Dic3—34no—
241C2×Dic3—61no—
242C2×Dic3—62no—
243C2×Dic3—94no—
244C2×Dic3—94no—
245C4×D3—62no—
246C4×D3—62no—
247C4×D3—64no—
248Dic6—64no—
249Dic6—184no—
250C2×C32—21no—
251C2×C32—22no—
252C2×C32C2×#23711yescharacteristic
253C2×C32C2×#23911yescharacteristic
254C2×C32#100712yes—
255C2×C32#101012yes—
256C3×D3—122no—
257C3×D3—121no—
258C3×D3—121no—
259C2×Q8—92no—
260C22×C4—91no—
261C4⋊C4—92no—
262C4⋊C4—92no—
263C4⋊C4—94no—
264C4⋊C4—94no—
265C4⋊C4—274no—
266C4⋊C4—274no—
267C22⋊C4—92no—
268C22⋊C4—274no—
269C42—91no—
270C42—274no—
271C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
272C22×C3—31no—
273C22×C3—31no—
274C22×C3—61no—
275C22×C3—181no—
276C22×C3—182no—
277C22×C3C22×D3211yescharacteristic
278C2×D3—61no—
279C2×D3—61no—
280C2×D3—62no—
281C2×D3—62no—
282C2×D3—62no—
283C2×D3—122no—
284C3×C4—22no—
285C3×C4—32no—
286C3×C4—32no—
287C3×C4—62no—
288C3×C4—62no—
289C3×C4—62no—
290C3×C4—122no—
291C3×C4—184no—
292C3×C4—184no—
293C3×C4—181no—
294C3×C4—182no—
295C3×C4#379612yes—
296Dic3—61no—
297Dic3—61no—
298Dic3—62no—
299Dic3—62no—
300Dic3—62no—
301Dic3—64no—
302Dic3—64no—
303Dic3—122no—
304Dic3—184no—
305Dic3—184no—
306C32—21no—
307C32#569811yescharacteristic, complemented
308C32#573111yescharacteristic, complemented
309C23—91no—
310Q8—184no—
311C2×C4—92no—
312C2×C4—94no—
313C2×C4—94no—
314C2×C4—91no—
315C2×C4—182no—
316C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
317C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—32no—
321C2×C3—32no—
322C2×C3—61no—
323C2×C3—62no—
324C2×C3—361no—
325C2×C3C2×#379611yescharacteristic
326C2×C3#6429712yes—
327D3—121no—
328D3—121no—
329D3—122no—
330C22—181no—
331C22—182no—
332C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
333C4—22no—
334C4—181no—
335C4—184no—
336C4—182no—
337C4—184no—
338C4#8864912yes—
339C3—31no—
340C3—31no—
341C3—61no—
342C3#1059386711yescharacteristic
343C2—361no—
344C2C2×#8864911yescharacteristic
345C2#1172248212yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×#272C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#541
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #64079⋊C3C2semidirect
3. #64078⋊C3C2semidirect
6. #11721087C2semidirect
7. #11721084C2semidirect
23. #64079D3semidirect
24. #64078D3semidirect
216. H(3,𝔽3)#541semidirect
307. C32#5698semidirect
308. C32#5731semidirect

Maximal subgroups

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

subgroupindex
#117217182
#64079⋊C32
#64078⋊C32
#117210942
#117210882
#117210872
#117210842
#105938813
#105938623