Finite Group Databasebeta

← #13161207

Subgroups — #13161207

Order 1728.

900 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 146 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×#88863C211yesmaximal, characteristic, complemented
2C3×#64155C211yesmaximal, characteristic
3#11723431C211yesmaximal, characteristic
4C2×#88787C211yesmaximal, characteristic
5C32×#835C211yesmaximal, characteristic, complemented
6#11722053C211yesmaximal, characteristic, complemented
7#11722042C211yesmaximal, characteristic, complemented
8#10592515—31nomaximal
9#10591598—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13#88863—22no—
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3752—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×#3750—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20Dic3⋊(C3⋊Dic3)—21no—
21#88803—21no—
22#88787—22no—
23#64155D311yescharacteristic
24C2×#3756—31no—
25C2×#3756—34no—
26C2×#3756—64no—
27C2×#3743—31no—
28#64106—31no—
29C3×#946—34no—
30#63857—34no—
31C2×#3711—34no—
32C32×#103—31no—
33#63643—31no—
34#63635—31no—
35C3×#835D314yescomplemented
36#63574—34no—
37#63567—34no—
38C22×C3×C3⋊D3D411yescharacteristic
39C33×Q8—21no—
40C33×Q8—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44#6308—21no—
45#6308—21no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3D411yescharacteristic
48C3×C4×C3⋊D3—21no—
49C3×C4×C3⋊D3—21no—
50C2×C32×Dic3—21no—
51C3⋊(C3⋊(C3⋊C8))D412yes—
52C32×C3⋊C8—21no—
53#5052—91no—
54#4816—94no—
55C2×C32×Q8—24no—
56C2×C32×Q8C2×D314yes—
57C2×C32×Q8C2×D311yescharacteristic
58C2×C4×C3⋊D3C2×D311yescharacteristic
59C2×#634—31no—
60C2×#634—34no—
61C2×#634—64no—
62C2×C3×C4×D3—34no—
63C2×C32×C8—31no—
64C32×C4⋊C4—31no—
65#3756—62no—
66#3756—68no—
67#3756—616no—
68#3752—21no—
69#3752—61no—
70C4⋊(C3⋊Dic3)—61no—
71C4⋊(C3⋊Dic3)C2×D311yescharacteristic
72#3750—21no—
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—34no—
75C2×C3⋊(C3⋊C8)—64no—
76#3743—62no—
77C3×(C2×D3)⋊C4—64no—
78C3×C4⋊Dic3—34no—
79C3×C4⋊Dic3C2×D314yes—
80C3×Dic3⋊C4—64no—
81C2×C3×C3⋊C8C2×D314yes—
82Dic3⋊Dic3—64no—
83(C2×D3)⋊Dic3—64no—
84#3711—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3—21no—
88C2×C3×C3⋊D3—21no—
89C33×C4—41no—
90C33×C4C2×D411yescharacteristic
91C33×C4#4111yescharacteristic
92C3⋊(C3⋊Dic3)—41no—
93C3×C3⋊Dic3—21no—
94C3×C3⋊Dic3—41no—
95C32×Dic3—41no—
96C3×C22⋊Q8—91no—
97#946—34no—
98C2×#216—94no—
99C2×#216—91no—
100C2×#216—184no—
101#924—91no—
102C2×#208—91no—
103C2×ΓL(1,𝔽25)—94no—
104#897—94no—
105C3×#103—34no—
106#835C3⋊D311yescharacteristic, complemented
107#816—94no—
108#810—91no—
109#808—94no—
110#808—91no—
111C22×C3⋊D3#6711yescharacteristic
112C22×C3×D3—34no—
113C32×Q8—28no—
114C32×Q8—24no—
115C32×Q8—21no—
116C32×Q8—21no—
117C32×Q8—24no—
118C32×Q8—44no—
119C2×C32×C4—21no—
120C2×C32×C4—24no—
121C2×C32×C4—24no—
122C2×C32×C4—44no—
123C2×C32×C4—61no—
124C2×C32×C4C22×D314yes—
125C2×C32×C4C22×D311yescharacteristic
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—61no—
128C2×C3⋊Dic3—64no—
129C2×C3⋊Dic3—124no—
130C2×C3⋊Dic3#6711yescharacteristic
131C4×C3⋊D3—21no—
132C4×C3⋊D3—21no—
133#634—68no—
134#634—64no—
135#634—61no—
136#634—61no—
137#634—64no—
138#634—124no—
139C2×C3×Dic3—24no—
140C2×C3×Dic3—34no—
141C2×C3×Dic3—64no—
142C3×C4×D3—64no—
143C3×C4×D3—64no—
144C32×C8—61no—
145C3⋊(C3⋊C8)—32no—
146C3⋊(C3⋊C8)—38no—
147C3⋊(C3⋊C8)—68no—
148C3×C3⋊C8—24no—
149#470—271noSylow
150C2×C33#12111yescharacteristic
151C2×C33#13711yescharacteristic
152C2×C33#13711yescharacteristic
153C3×C3⋊D3—41no—
154C2×C3×Q8—24no—
155C2×C3×Q8D3214yes—
156C2×C3×Q8C2×C3⋊D311yescharacteristic
157C22×C3×C4—91no—
158C2×C4×D3—34no—
159C2×Dic6—94no—
160C2×Dic6—91no—
161C2×Dic6—184no—
162C2×C3×C8—34no—
163C3×C4⋊C4—34no—
164C3×C4⋊C4—91no—
165C3×C4⋊C4—181no—
166C3×C22⋊C4—181no—
167#217—181no—
168#216—188no—
169#216—182no—
170#216—1816no—
171(C2×D3)⋊C4—64no—
172(C2×D3)⋊C4—184no—
173C4⋊Dic3—34no—
174C4⋊Dic3—184no—
175C4⋊Dic3C2×C3⋊D311yescharacteristic
176Dic3⋊C4—64no—
177Dic3⋊C4—181no—
178#208—182no—
179C2×C3⋊C8—94no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—184no—
182C2×C3⋊C8C2×C3⋊D311yescharacteristic
183ΓL(1,𝔽25)—188no—
184C22×C32—24no—
185C22×C32D3×D414yes—
186C22×C32C2×#6711yescharacteristic
187C2×C3⋊D3—21no—
188C2×C3⋊D3—21no—
189C2×C3⋊D3—21no—
190C2×C3×D3—64no—
191C2×C3×D3—64no—
192C2×C3×D3—64no—
193C32×C4—24no—
194C32×C4—24no—
195C32×C4—48no—
196C32×C4—44no—
197C32×C4—41no—
198C32×C4—121no—
199C32×C4#23514yes—
200C32×C4D3×D414yes—
201C32×C4D3×D411yescharacteristic
202C32×C4#23711yescharacteristic
203C3⋊Dic3—21no—
204C3⋊Dic3—41no—
205C3⋊Dic3—128no—
206C3⋊Dic3—124no—
207C3⋊Dic3—121no—
208C3×Dic3—44no—
209C3×Dic3—64no—
210C3×Dic3—124no—
211C2×Q16—271no—
212C2×M16—271no—
213C22⋊Q8—91no—
214C22⋊Q8—271no—
215#106—271no—
216#103—31no—
217#103—271no—
218C33#47011yescharacteristic, complemented, Sylow
219C23×C3—91no—
220C22×D3—34no—
221C3×Q8—28no—
222C3×Q8—24no—
223C3×Q8—24no—
224C3×Q8—21no—
225C3×Q8—21no—
226C3×Q8—44no—
227C2×C3×C4—24no—
228C2×C3×C4—21no—
229C2×C3×C4—24no—
230C2×C3×C4—44no—
231C2×C3×C4—64no—
232C2×C3×C4—91no—
233C2×C3×C4—181no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4C2×D3214yes—
237C2×C3×C4C22×C3⋊D311yescharacteristic
238C2×Dic3—21no—
239C2×Dic3—34no—
240C2×Dic3—64no—
241C2×Dic3—184no—
242C2×Dic3—181no—
243C2×Dic3—364no—
244C4×D3—64no—
245C4×D3—64no—
246Dic6—188no—
247Dic6—184no—
248Dic6—184no—
249Dic6—181no—
250Dic6—181no—
251Dic6—364no—
252C3×C8—64no—
253C3⋊C8—21no—
254C3⋊C8—98no—
255C3⋊C8—92no—
256C3⋊C8—188no—
257C2×C32—24no—
258C2×C32—24no—
259C2×C32—24no—
260C2×C32#88414yes—
261C2×C32#91514yes—
262C2×C32#91814yes—
263C2×C32#93911yescharacteristic
264C2×C32#94211yescharacteristic
265C2×C32#95111yescharacteristic
266C3⋊D3—41no—
267C3×D3—124no—
268C2×Q8—271no—
269C2×Q8D3×C3⋊D311yescharacteristic
270C22×C4—91no—
271Q16—542no—
272M16—542no—
273C2×C8—31no—
274C2×C8—271no—
275C4⋊C4—31no—
276C4⋊C4—91no—
277C4⋊C4—181no—
278C4⋊C4—541no—
279C22⋊C4—181no—
280C22⋊C4—541no—
281C22×C3—24no—
282C22×C3—181no—
283C22×C3—181no—
284C22×C3—181no—
285C22×C3D3×#6714yes—
286C22×C3D4×C3⋊D311yescharacteristic
287C2×D3—64no—
288C2×D3—64no—
289C2×D3—64no—
290C3×C4—24no—
291C3×C4—24no—
292C3×C4—48no—
293C3×C4—41no—
294C3×C4—44no—
295C3×C4—124no—
296C3×C4—181no—
297C3×C4—361no—
298C3×C4#380414yes—
299C3×C4#381114yes—
300C3×C4#382811yescharacteristic
301C3×C4D4×C3⋊D311yescharacteristic
302Dic3—41no—
303Dic3—64no—
304Dic3—124no—
305Dic3—368no—
306Dic3—361no—
307Dic3—364no—
308C32—24no—
309C32#481614yescomplemented
310C32#505211yescharacteristic, complemented
311C23—91no—
312Q8—21no—
313Q8—21no—
314Q8—541no—
315Q8—541no—
316C2×C4—21no—
317C2×C4—61no—
318C2×C4—91no—
319C2×C4—181no—
320C2×C4—181no—
321C2×C4—181no—
322C2×C4—541no—
323C2×C4C2×D3×C3⋊D311yescharacteristic
324C8—61no—
325C8—272no—
326C2×C3—24no—
327C2×C3—24no—
328C2×C3—24no—
329C2×C3—361no—
330C2×C3#6394314yes—
331C2×C3#6394614yes—
332C2×C3#6397114yes—
333C2×C3#6409311yescharacteristic
334C2×C3#6412411yescharacteristic
335C2×C3#6412711yescharacteristic
336D3—124no—
337C22—181no—
338C22—181no—
339C22—181no—
340C22C3⋊D3×#6711yescharacteristic
341C4—41no—
342C4—121no—
343C4—181no—
344C4—361no—
345C4—1081no—
346C4#8903111yescharacteristic
347C4#8903911yescharacteristic
348C3—24no—
349C3#1059159814yescomplemented
350C3#1059251511yescharacteristic, complemented
351C2—361no—
352C2#1172357611yescharacteristic
353C2#1172358411yescharacteristic
354C2#1172362411yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#470
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88863C2semidirect
5. C32×#835C2semidirect
6. #11722053C2semidirect
7. #11722042C2semidirect
35. C3×#835D3semidirect
106. #835C3⋊D3semidirect
218. C33#470semidirect
309. C32#4816semidirect
310. C32#5052semidirect
349. C3#10591598semidirect
350. C3#10592515semidirect

Maximal subgroups

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

subgroupindex
C2×#888632
C3×#641552
#117234312
C2×#887872
C32×#8352
#117220532
#117220422
#105925153
#105915983