Finite Group Databasebeta

← #13147812

Subgroups — #13147812

Order 1728.

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

normal subgroups only

This group has 896 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#11721716C211yesmaximal, characteristic, complemented
2#11721385C211yesmaximal, characteristic, complemented
3#64081⋊C3C211yesmaximal, characteristic, complemented
4#11721141C212yesmaximal, complemented
5#11721132C212yesmaximal, complemented
6#11721126C214yesmaximal, complemented
7#11721125C214yesmaximal, complemented
8#10593989—31nomaximal
9#10593950—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
11C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#6198C2214yes—
14C2×((C3⋊D3)⋊C3)⋊C4C2214yes—
15C42×H(3,𝔽3)C2211yescharacteristic
16C4⋊#1084C2212yes—
17#88540C2214yes—
18#3752⋊C3C2212yes—
19#3750⋊C3C2212yes—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21#88450C2214yes—
22#88450C2214yes—
23#64083D311yescharacteristic, complemented
24#64081D311yescharacteristic, complemented
25C3×#874—31no—
26C3×#872—31no—
27C3×#870—31no—
28C3×#870—31no—
29C3×#870—61no—
30#63916—32no—
31#63910—34no—
32#63909—32no—
33#63898—32no—
34#63897—34no—
35#63894—32no—
36#63886—34no—
37#63884—34no—
38C22×(C3⋊D3)⋊C3C2314yes—
39C22×(C3⋊D3)⋊C3C2311yescharacteristic
40C2×C4×H(3,𝔽3)C2312yes—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×#1084C2314yes—
43C2×(C3⋊Dic3)⋊C3C2312yes—
44C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
45C4⋊((C3⋊D3)⋊C3)—24no—
46C4⋊((C3⋊D3)⋊C3)—44no—
47C4×(C3⋊D3)⋊C3—22no—
48#6198—44no—
49((C3⋊D3)⋊C3)⋊C4—44no—
50#5719—91no—
51#5697—91no—
52#5535—91no—
53C2×C4⋊(C3⋊D3)C2×D312yes—
54C2×C4⋊(C3⋊D3)C2×D314yes—
55C2×C4×C3⋊D3C2×D311yescharacteristic
56C2×C3×D12—32no—
57C2×C3×D12—34no—
58C2×C3×C4×D3—31no—
59C2×#626—34no—
60C2×#626—34no—
61C2×#624—34no—
62C2×D3×Dic3—34no—
63C32×C42—21no—
64C32×C42C2×D311yescharacteristic
65C32×C42C2×D311yescharacteristic
66#3752C2×D312yes—
67#3750C2×D312yes—
68C4×C3⋊Dic3C2×D311yescharacteristic
69C3×(C2×D3)⋊C4—32no—
70C3×C4⋊Dic3—32no—
71C3×C4⋊Dic3—32no—
72C3×C4⋊Dic3—62no—
73C3×Dic3⋊C4—34no—
74C3×Dic3⋊C4—32no—
75C3×Dic3⋊C4—34no—
76C3×Dic3⋊C4—64no—
77C3×C4×Dic3—31no—
78#3722—34no—
79#3722—34no—
80(C2×D3)⋊Dic3—34no—
81(C2×D3)⋊Dic3—34no—
82C22×H(3,𝔽3)C2411yescharacteristic
83C2×(C3⋊D3)⋊C3—21no—
84C2×(C3⋊D3)⋊C3—24no—
85C2×(C3⋊D3)⋊C3—24no—
86C2×(C3⋊D3)⋊C3—22no—
87C2×(C3⋊D3)⋊C3—24no—
88C4×H(3,𝔽3)—22no—
89C4×H(3,𝔽3)#4112yes—
90#1084—24no—
91(C3⋊Dic3)⋊C3—21no—
92(C3⋊Dic3)⋊C3—22no—
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—91no—
109C4×D12—92no—
110#872—31no—
111#872—91no—
112#870—31no—
113#870—31no—
114#870—61no—
115C22×C3⋊D3C22×D314yes—
116C22×C3⋊D3C22×D311yescharacteristic
117C22×C3×D3—34no—
118C22×C3×D3—31no—
119C2×C32×C4—21no—
120C2×C32×C4—22no—
121C2×C32×C4C22×D312yes—
122C2×C32×C4C22×D312yes—
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3C22×D311yescharacteristic
126C2×C3⋊Dic3C22×D312yes—
127C4⋊(C3⋊D3)—24no—
128C4⋊(C3⋊D3)—44no—
129C4×C3⋊D3—22no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—32no—
132C2×C3×Dic3—34no—
133C2×C3×Dic3—34no—
134C2×C3×Dic3—64no—
135C3×D12—64no—
136C3×D12—124no—
137C3×C4×D3—62no—
138#626—124no—
139#626—124no—
140#624—124no—
141D3×Dic3—124no—
142#540—271noSylow
143C2×H(3,𝔽3)C2×#4111yescharacteristic
144C2×H(3,𝔽3)H(5,𝔽2)12yes—
145(C3⋊D3)⋊C3—41no—
146(C3⋊D3)⋊C3—44no—
147C2×C3×D4—92no—
148C2×C3×D4—94no—
149C22×C3×C4—91no—
150C2×#67—94no—
151C2×#67—94no—
152C22×Dic3—94no—
153C2×D12—32no—
154C2×D12—34no—
155C2×D12—92no—
156C2×D12—94no—
157C2×D12—94no—
158C2×C4×D3—31no—
159C2×C4×D3—91no—
160C2×C4×D3—94no—
161C2×C4×D3—94no—
162C3×C4⋊C4—92no—
163C3×C4⋊C4—92no—
164C3×C4⋊C4—94no—
165C3×C22⋊C4—92no—
166C3×C42—31no—
167C3×C42—31no—
168C3×C42—61no—
169C3×C42—91no—
170C3×C42D3211yescharacteristic
171#217—94no—
172#217—94no—
173(C2×D3)⋊C4—32no—
174(C2×D3)⋊C4—92no—
175(C2×D3)⋊C4—94no—
176(C2×D3)⋊C4—94no—
177(C2×D3)⋊C4—94no—
178(C2×D3)⋊C4—94no—
179C4⋊Dic3—32no—
180C4⋊Dic3—32no—
181C4⋊Dic3—62no—
182Dic3⋊C4—34no—
183Dic3⋊C4—34no—
184Dic3⋊C4—32no—
185Dic3⋊C4—64no—
186Dic3⋊C4—92no—
187C4×Dic3—31no—
188C4×Dic3—91no—
189C22×C32—21no—
190C22×C32C23×D311yescharacteristic
191C22×C32C23×D311yescharacteristic
192C2×C3⋊D3—21no—
193C2×C3⋊D3—24no—
194C2×C3⋊D3—24no—
195C2×C3⋊D3—24no—
196C2×C3⋊D3—22no—
197C2×C3×D3—64no—
198C2×C3×D3—61no—
199C2×C3×D3—62no—
200C2×C3×D3—64no—
201C2×C3×D3—64no—
202C32×C4—22no—
203C32×C4—22no—
204C32×C4—22no—
205C32×C4—42no—
206C32×C4#23512yes—
207C32×C4#23712yes—
208C3⋊Dic3—21no—
209C3⋊Dic3—22no—
210C3×Dic3—61no—
211C3×Dic3—62no—
212C3×Dic3—64no—
213C3×Dic3—64no—
214C3×Dic3—124no—
215#127—91no—
216#125—91no—
217#123—274no—
218#121—274no—
219#121—272no—
220C4×D4—272no—
221#117—91no—
222H(3,𝔽3)#54011yescharacteristic, complemented, Sylow
223C23×C3—94no—
224C23×C3—91no—
225C22×D3—31no—
226C22×D3—34no—
227C22×D3—94no—
228C22×D3—91no—
229C3×D4—184no—
230C3×D4—364no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—32no—
234C2×C3×C4—32no—
235C2×C3×C4—61no—
236C2×C3×C4—62no—
237C2×C3×C4—91no—
238C2×C3×C4—92no—
239C2×C3×C4—94no—
240C2×C3×C4—182no—
241C2×C3×C4C2×D3212yes—
242C2×C3×C4C2×D3211yescharacteristic
243#67—364no—
244#67—364no—
245C2×Dic3—34no—
246C2×Dic3—34no—
247C2×Dic3—32no—
248C2×Dic3—31no—
249C2×Dic3—64no—
250C2×Dic3—92no—
251C2×Dic3—91no—
252C2×Dic3—364no—
253D12—64no—
254D12—124no—
255D12—184no—
256D12—364no—
257D12—364no—
258C4×D3—62no—
259C4×D3—182no—
260C4×D3—364no—
261C4×D3—364no—
262C2×C32—21no—
263C2×C32—22no—
264C2×C32C2×#23511yescharacteristic
265C2×C32C2×#23711yescharacteristic
266C2×C32#100412yes—
267C2×C32#100912yes—
268C3⋊D3—44no—
269C3⋊D3—41no—
270C3×D3—121no—
271C3×D3—124no—
272C2×D4—92no—
273C2×D4—94no—
274C2×D4—274no—
275C22×C4—91no—
276C22×C4—274no—
277C4⋊C4—92no—
278C4⋊C4—92no—
279C4⋊C4—94no—
280C22⋊C4—92no—
281C22⋊C4—274no—
282C22⋊C4—274no—
283C42—91no—
284C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
285C22×C3—31no—
286C22×C3—31no—
287C22×C3—61no—
288C22×C3—184no—
289C22×C3—182no—
290C22×C3—184no—
291C22×C3—181no—
292C22×C3—184no—
293C22×C3C22×D3211yescharacteristic
294C2×D3—61no—
295C2×D3—64no—
296C2×D3—64no—
297C2×D3—64no—
298C2×D3—62no—
299C2×D3—181no—
300C2×D3—184no—
301C2×D3—184no—
302C2×D3—184no—
303C2×D3—182no—
304C3×C4—22no—
305C3×C4—32no—
306C3×C4—32no—
307C3×C4—62no—
308C3×C4—62no—
309C3×C4—62no—
310C3×C4—122no—
311C3×C4—182no—
312C3×C4—181no—
313C3×C4—184no—
314C3×C4#379512yes—
315Dic3—64no—
316Dic3—64no—
317Dic3—61no—
318Dic3—62no—
319Dic3—124no—
320Dic3—181no—
321Dic3—182no—
322C32—21no—
323C32#553511yescharacteristic, complemented
324C32#571911yescharacteristic, complemented
325C23—91no—
326C23—94no—
327D4—184no—
328D4—364no—
329D4—1084no—
330C2×C4—92no—
331C2×C4—91no—
332C2×C4—94no—
333C2×C4—182no—
334C2×C4—1084no—
335C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
336C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—32no—
340C2×C3—32no—
341C2×C3—61no—
342C2×C3—62no—
343C2×C3—361no—
344C2×C3—364no—
345C2×C3C2×#379511yescharacteristic
346C2×C3#6430712yes—
347D3—121no—
348D3—124no—
349D3—364no—
350D3—361no—
351C22—181no—
352C22—184no—
353C22—184no—
354C22—184no—
355C22—182no—
356C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
357C4—22no—
358C4—181no—
359C4—182no—
360C4—184no—
361C4#8864812yes—
362C3—31no—
363C3—31no—
364C3—61no—
365C3#1059394911yescharacteristic
366C2—361no—
367C2—364no—
368C2C2×#8864811yescharacteristic
369C2#1172249212yes—
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. #11721716C2semidirect
2. #11721385C2semidirect
3. #64081⋊C3C2semidirect
4. #11721141C2semidirect
5. #11721132C2semidirect
6. #11721126C2semidirect
7. #11721125C2semidirect
23. #64083D3semidirect
24. #64081D3semidirect
222. H(3,𝔽3)#540semidirect
323. C32#5535semidirect
324. C32#5719semidirect

Maximal subgroups

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

subgroupindex
#117217162
#117213852
#64081⋊C32
#117211412
#117211322
#117211262
#117211252
#105939893
#105939503