Finite Group Databasebeta

← #13147891

Subgroups — #13147891

Order 1728.

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

normal subgroups only

This group has 948 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#11721720C211yesmaximal, characteristic, complemented
2#64081⋊C3C211yesmaximal, characteristic, complemented
3#64079⋊C3C211yesmaximal, characteristic, complemented
4#11721157C214yesmaximal, complemented
5#11721156C212yesmaximal, complemented
6#11721151C214yesmaximal, complemented
7#11721149C212yesmaximal, complemented
8#10593950—31nomaximal
9#10593949—31nomaximal
10C2×C4⋊#272C2212yes—
11C2×C4⋊#272C2214yes—
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#6200C2214yes—
14C2×#6199C2214yes—
15C42×H(3,𝔽3)C2211yescharacteristic
16#3752⋊C3C2212yes—
17C4⋊((C3⋊Dic3)⋊C3)C2212yes—
18#3750⋊C3C2214yes—
19#3750⋊C3C2212yes—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21#88451C2214yes—
22#88451C2214yes—
23#64081D311yescharacteristic, complemented
24#64079D311yescharacteristic, complemented
25C3×#874—31no—
26C3×#874—31no—
27C3×#874—61no—
28C3×#872—31no—
29C3×#870—31no—
30#63916—32no—
31#63908—32no—
32#63898—32no—
33Dic3×D12—32no—
34#63886—34no—
35#63885—34no—
36#63884—34no—
37#63884—34no—
38C22×#272C2314yes—
39C22×(C3⋊D3)⋊C3C2311yescharacteristic
40C2×C4×H(3,𝔽3)C2312yes—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C4⋊#272—24no—
43C4⋊#272—44no—
44C2×(C3⋊Dic3)⋊C3C2312yes—
45C2×(C3⋊Dic3)⋊C3C2314yes—
46C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
47C4×(C3⋊D3)⋊C3—22no—
48#6200—44no—
49#6199—44no—
50#5719—91no—
51#5697—91no—
52#5535—91no—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×C3×D12—32no—
55C2×C3×D12—32no—
56C2×C3×D12—34no—
57C2×C3×D12—34no—
58C2×C3×D12—62no—
59C2×C3×D12—64no—
60C2×C3×C4×D3—31no—
61C2×#626—34no—
62C2×#625—34no—
63C2×D3×Dic3—34no—
64C2×D3×Dic3—34no—
65C32×C42—21no—
66C32×C42C2×D311yescharacteristic
67C32×C42C2×D311yescharacteristic
68#3752C2×D312yes—
69C4⋊(C3⋊Dic3)C2×D312yes—
70#3750C2×D314yes—
71#3750C2×D312yes—
72C4×C3⋊Dic3C2×D311yescharacteristic
73C3×(C2×D3)⋊C4—32no—
74C3×C4⋊Dic3—32no—
75C3×Dic3⋊C4—32no—
76C3×Dic3⋊C4—34no—
77C3×C4×Dic3—31no—
78(C2×D3)⋊Dic3—34no—
79(C2×D3)⋊Dic3—34no—
80(C2×D3)⋊Dic3—34no—
81(C2×D3)⋊Dic3—34no—
82C22×H(3,𝔽3)C2411yescharacteristic
83C2×#272—24no—
84C2×#272—24no—
85C2×#272—24no—
86C2×(C3⋊D3)⋊C3—21no—
87C2×(C3⋊D3)⋊C3—22no—
88C4×H(3,𝔽3)—22no—
89C4×H(3,𝔽3)#4112yes—
90(C3⋊Dic3)⋊C3—21no—
91(C3⋊Dic3)⋊C3—22no—
92(C3⋊Dic3)⋊C3—24no—
93C3×#127—91no—
94C3×#125—91no—
95C3×#117—91no—
96#939—94no—
97#938—92no—
98#935—94no—
99D4×Dic3—92no—
100#930—92no—
101#895—94no—
102#895—92no—
103#894—94no—
104#894—94no—
105#893—92no—
106#884—94no—
107#874—31no—
108#874—31no—
109#874—61no—
110C4×D12—92no—
111#872—31no—
112#872—91no—
113#870—31no—
114#870—91no—
115C22×C3⋊D3C22×D311yescharacteristic
116C22×C3×D3—31no—
117C22×C3×D3—34no—
118C22×C3×D3—34no—
119C22×C3×D3—64no—
120C2×C32×C4—22no—
121C2×C32×C4—21no—
122C2×C32×C4C22×D312yes—
123C2×C32×C4C22×D312yes—
124C2×C32×C4C22×D311yescharacteristic
125C2×C32×C4C22×D311yescharacteristic
126C2×C3⋊Dic3C22×D311yescharacteristic
127C2×C3⋊Dic3C22×D312yes—
128C2×C3⋊Dic3C22×D314yes—
129C4×C3⋊D3—22no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—32no—
132C2×C3×Dic3—34no—
133C3×D12—64no—
134C3×D12—64no—
135C3×D12—124no—
136C3×D12—124no—
137C3×D12—124no—
138C3×D12—128no—
139C3×C4×D3—62no—
140#626—124no—
141#625—124no—
142D3×Dic3—124no—
143D3×Dic3—124no—
144#540—271noSylow
145C2×H(3,𝔽3)C2×#4111yescharacteristic
146C2×H(3,𝔽3)H(5,𝔽2)12yes—
147#272—44no—
148(C3⋊D3)⋊C3—41no—
149C2×C3×D4—92no—
150C2×C3×D4—94no—
151C22×C3×C4—91no—
152C2×#67—94no—
153C2×#67—94no—
154C22×Dic3—94no—
155C2×D12—32no—
156C2×D12—32no—
157C2×D12—34no—
158C2×D12—34no—
159C2×D12—62no—
160C2×D12—64no—
161C2×D12—94no—
162C2×C4×D3—31no—
163C2×C4×D3—91no—
164C2×C4×D3—94no—
165C2×C4×D3—94no—
166C3×C4⋊C4—92no—
167C3×C4⋊C4—92no—
168C3×C4⋊C4—94no—
169C3×C22⋊C4—92no—
170C3×C42—31no—
171C3×C42—31no—
172C3×C42—61no—
173C3×C42—91no—
174C3×C42D3211yescharacteristic
175#217—94no—
176#217—94no—
177(C2×D3)⋊C4—32no—
178(C2×D3)⋊C4—92no—
179(C2×D3)⋊C4—94no—
180(C2×D3)⋊C4—94no—
181(C2×D3)⋊C4—94no—
182(C2×D3)⋊C4—94no—
183C4⋊Dic3—32no—
184C4⋊Dic3—92no—
185Dic3⋊C4—34no—
186Dic3⋊C4—32no—
187Dic3⋊C4—94no—
188Dic3⋊C4—92no—
189C4×Dic3—31no—
190C4×Dic3—91no—
191C22×C32—21no—
192C22×C32C23×D311yescharacteristic
193C22×C32C23×D311yescharacteristic
194C2×C3⋊D3—21no—
195C2×C3⋊D3—22no—
196C2×C3×D3—64no—
197C2×C3×D3—61no—
198C2×C3×D3—64no—
199C2×C3×D3—62no—
200C2×C3×D3—64no—
201C2×C3×D3—64no—
202C2×C3×D3—64no—
203C2×C3×D3—64no—
204C2×C3×D3—124no—
205C2×C3×D3—124no—
206C2×C3×D3—124no—
207C32×C4—24no—
208C32×C4—22no—
209C32×C4—22no—
210C32×C4—22no—
211C32×C4#23512yes—
212C32×C4#23912yes—
213C3⋊Dic3—21no—
214C3⋊Dic3—22no—
215C3⋊Dic3—24no—
216C3×Dic3—61no—
217C3×Dic3—62no—
218C3×Dic3—64no—
219#127—91no—
220#125—91no—
221#123—274no—
222#121—274no—
223#121—272no—
224C4×D4—272no—
225#117—91no—
226H(3,𝔽3)#54011yescharacteristic, complemented, Sylow
227C23×C3—91no—
228C23×C3—94no—
229C22×D3—34no—
230C22×D3—34no—
231C22×D3—31no—
232C22×D3—64no—
233C22×D3—91no—
234C3×D4—184no—
235C3×D4—364no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—32no—
239C2×C3×C4—32no—
240C2×C3×C4—62no—
241C2×C3×C4—61no—
242C2×C3×C4—91no—
243C2×C3×C4—92no—
244C2×C3×C4—94no—
245C2×C3×C4—182no—
246C2×C3×C4C2×D3212yes—
247C2×C3×C4C2×D3211yescharacteristic
248#67—364no—
249#67—364no—
250C2×Dic3—32no—
251C2×Dic3—34no—
252C2×Dic3—31no—
253C2×Dic3—92no—
254C2×Dic3—94no—
255C2×Dic3—91no—
256C2×Dic3—364no—
257D12—64no—
258D12—64no—
259D12—124no—
260D12—124no—
261D12—124no—
262D12—128no—
263D12—364no—
264C4×D3—62no—
265C4×D3—182no—
266C4×D3—364no—
267C4×D3—364no—
268C2×C32—21no—
269C2×C32—22no—
270C2×C32C2×#23511yescharacteristic
271C2×C32C2×#23911yescharacteristic
272C2×C32#100912yes—
273C2×C32#100912yes—
274C3⋊D3—41no—
275C3×D3—121no—
276C3×D3—124no—
277C3×D3—124no—
278C3×D3—244no—
279C2×D4—92no—
280C2×D4—94no—
281C2×D4—274no—
282C22×C4—91no—
283C22×C4—274no—
284C4⋊C4—92no—
285C4⋊C4—92no—
286C4⋊C4—94no—
287C22⋊C4—92no—
288C22⋊C4—274no—
289C22⋊C4—274no—
290C42—91no—
291C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
292C22×C3—31no—
293C22×C3—31no—
294C22×C3—61no—
295C22×C3—182no—
296C22×C3—181no—
297C22×C3—184no—
298C22×C3—184no—
299C22×C3—184no—
300C22×C3C22×D3211yescharacteristic
301C2×D3—64no—
302C2×D3—64no—
303C2×D3—64no—
304C2×D3—64no—
305C2×D3—64no—
306C2×D3—64no—
307C2×D3—61no—
308C2×D3—62no—
309C2×D3—124no—
310C2×D3—124no—
311C2×D3—124no—
312C2×D3—181no—
313C2×D3—182no—
314C3×C4—22no—
315C3×C4—32no—
316C3×C4—32no—
317C3×C4—62no—
318C3×C4—62no—
319C3×C4—64no—
320C3×C4—62no—
321C3×C4—182no—
322C3×C4—184no—
323C3×C4—181no—
324C3×C4#379912yes—
325Dic3—61no—
326Dic3—62no—
327Dic3—64no—
328Dic3—181no—
329Dic3—182no—
330Dic3—184no—
331C32—21no—
332C32#553511yescharacteristic, complemented
333C32#569711yescharacteristic, complemented
334C23—91no—
335C23—94no—
336D4—184no—
337D4—364no—
338D4—1084no—
339C2×C4—92no—
340C2×C4—91no—
341C2×C4—94no—
342C2×C4—182no—
343C2×C4—1084no—
344C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
345C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—32no—
349C2×C3—32no—
350C2×C3—61no—
351C2×C3—62no—
352C2×C3—361no—
353C2×C3—364no—
354C2×C3C2×#379911yescharacteristic
355C2×C3#6430812yes—
356D3—124no—
357D3—124no—
358D3—121no—
359D3—244no—
360D3—361no—
361C22—181no—
362C22—184no—
363C22—184no—
364C22—184no—
365C22—182no—
366C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
367C4—22no—
368C4—181no—
369C4—182no—
370C4—184no—
371C4#8865212yes—
372C3—31no—
373C3—31no—
374C3—61no—
375C3#1059398911yescharacteristic
376C2—364no—
377C2—361no—
378C2C2×#8865211yescharacteristic
379C2#1172249312yes—
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#540
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
1. #11721720C2semidirect
2. #64081⋊C3C2semidirect
3. #64079⋊C3C2semidirect
4. #11721157C2semidirect
5. #11721156C2semidirect
6. #11721151C2semidirect
7. #11721149C2semidirect
23. #64081D3semidirect
24. #64079D3semidirect
226. H(3,𝔽3)#540semidirect
332. C32#5535semidirect
333. C32#5697semidirect

Maximal subgroups

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

subgroupindex
#117217202
#64081⋊C32
#64079⋊C32
#117211572
#117211562
#117211512
#117211492
#105939503
#105939493