Finite Group Databasebeta

← #13145727

Subgroups — #13145727

Order 1728.

678 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 103 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×C3×#3752C211yesmaximal, characteristic
2C3×#63657C212yesmaximal, complemented
3#10589146—31nomaximal
4#10589145—32nomaximal
5C23×C3×C3⋊D3C411yescharacteristic
6C22×C33×C4C2211yescharacteristic
7C22×C3×C3⋊Dic3C411yescharacteristic
8C3×#3752—22no—
9C2×C3×C3⋊(C3⋊C8)—22no—
10C2×#3752D311yescharacteristic
11C2×C3×(C2×D3)⋊C4—32no—
12C2×C3×(C2×D3)⋊C4—61no—
13#63657D312yescomplemented
14C3×#830—32no—
15C3×#830—32no—
16C3×#830—62no—
17C23×C33C2×C411yescharacteristic
18C22×C3×C3⋊D3—21no—
19C22×C3×C3⋊D3—21no—
20C22×C3×C3⋊D3—41no—
21C22×C3×C3⋊D3—41no—
22C22×C3×C3⋊D3C812yescomplemented
23C2×C33×C4—41no—
24C2×C33×C4D412yes—
25C2×C3×C3⋊Dic3—21no—
26C2×C3×C3⋊Dic3—41no—
27C3×C3⋊(C3⋊C8)—42no—
28#4525—91no—
29#4468—92no—
30C23×C3⋊D3Dic311yescharacteristic
31C23×C3×D3—32no—
32C23×C3×D3—61no—
33C22×C32×C4—21no—
34C22×C32×C4—22no—
35C22×C32×C4—41no—
36C22×C32×C4C2×D312yes—
37C22×C32×C4C2×D311yescharacteristic
38C22×C3⋊Dic3Dic311yescharacteristic
39C22×C3×Dic3—32no—
40C22×C3×Dic3—61no—
41#3752—22no—
42C2×C3⋊(C3⋊C8)—22no—
43C3×(C2×D3)⋊C4—64no—
44C3×(C2×D3)⋊C4—62no—
45C3×(C2×D3)⋊C4—62no—
46C2×C3×C3⋊C8—64no—
47C2×C3×C3⋊C8—62no—
48C2×C3×C3⋊C8—62no—
49C22×C33—21no—
50C22×C33—21no—
51C22×C33C22⋊C411yescharacteristic
52C22×C33C2×C811yescharacteristic
53C22×C33M1611yescharacteristic
54C2×C3×C3⋊D3—22no—
55C2×C3×C3⋊D3—42no—
56C2×C3×C3⋊D3—41no—
57C2×C3×C3⋊D3—41no—
58C2×C3×C3⋊D3—42no—
59C33×C4—22no—
60C3×C3⋊Dic3—41no—
61C2×C3×C22⋊C4—91no—
62C2×(C2×D3)⋊C4—32no—
63C2×(C2×D3)⋊C4—61no—
64C3×C22⋊C8—92no—
65#830—32no—
66#830—32no—
67#830—62no—
68C23×C32—22no—
69C23×C32—21no—
70C23×C32—41no—
71C23×C32C4×D312yes—
72C23×C32C2×Dic311yescharacteristic
73C22×C3⋊D3—21no—
74C22×C3⋊D3—21no—
75C22×C3⋊D3—41no—
76C22×C3⋊D3—41no—
77C22×C3⋊D3C3⋊C812yescomplemented
78C22×C3×D3—34no—
79C22×C3×D3—62no—
80C22×C3×D3—62no—
81C22×C3×D3—62no—
82C22×C3×D3—62no—
83C22×C3×D3—62no—
84C22×C3×D3—122no—
85C22×C3×D3—122no—
86C22×C3×D3—122no—
87C22×C3×D3—122no—
88C2×C32×C4—22no—
89C2×C32×C4—22no—
90C2×C32×C4—22no—
91C2×C32×C4—42no—
92C2×C32×C4—42no—
93C2×C32×C4—41no—
94C2×C32×C4—44no—
95C2×C32×C4—42no—
96C2×C32×C4—44no—
97C2×C32×C4D1212yes—
98C2×C32×C4#6712yes—
99C2×C32×C4#6712yes—
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3—41no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—62no—
104C2×C3×Dic3—122no—
105C2×C3×Dic3—122no—
106C3⋊(C3⋊C8)—42no—
107C3×C3⋊C8—124no—
108C3×C3⋊C8—122no—
109C3×C3⋊C8—122no—
110#323—271noSylow
111C2×C33—21no—
112C2×C33—21no—
113C2×C33C22⋊C811yescharacteristic
114C2×C33#9911yescharacteristic
115C2×C33#10011yescharacteristic
116C3×C3⋊D3—42no—
117C24×C3—91no—
118C23×D3—32no—
119C23×D3—61no—
120C22×C3×C4—22no—
121C22×C3×C4—21no—
122C22×C3×C4—41no—
123C22×C3×C4—91no—
124C22×C3×C4D3212yes—
125C22×C3×C4D3211yescharacteristic
126C22×Dic3—32no—
127C22×Dic3—61no—
128C2×C3×C8—182no—
129C3×C22⋊C4—182no—
130(C2×D3)⋊C4—64no—
131(C2×D3)⋊C4—62no—
132(C2×D3)⋊C4—62no—
133C2×C3⋊C8—64no—
134C2×C3⋊C8—62no—
135C2×C3⋊C8—62no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32—22no—
139C22×C32—22no—
140C22×C32—22no—
141C22×C32—22no—
142C22×C32—22no—
143C22×C32—21no—
144C22×C32—22no—
145C22×C32—21no—
146C22×C32—22no—
147C22×C32—21no—
148C22×C32—41no—
149C22×C32—41no—
150C22×C32—41no—
151C22×C32—42no—
152C22×C32—42no—
153C22×C32—42no—
154C22×C32—42no—
155C22×C32D3×C812yes—
156C22×C32ΓL(1,𝔽25)12yes—
157C22×C32C2×C3⋊C811yescharacteristic
158C22×C32#20811yescharacteristic
159C22×C32(C2×D3)⋊C412yes—
160C22×C32#21711yescharacteristic
161C2×C3⋊D3—22no—
162C2×C3⋊D3—42no—
163C2×C3⋊D3—41no—
164C2×C3⋊D3—41no—
165C2×C3⋊D3—42no—
166C2×C3×D3—64no—
167C2×C3×D3—64no—
168C2×C3×D3—124no—
169C2×C3×D3—124no—
170C2×C3×D3—124no—
171C2×C3×D3—122no—
172C2×C3×D3—122no—
173C2×C3×D3—122no—
174C2×C3×D3—122no—
175C2×C3×D3—124no—
176C32×C4—22no—
177C32×C4—24no—
178C32×C4—22no—
179C32×C4—22no—
180C32×C4—42no—
181C32×C4—42no—
182C32×C4—44no—
183C3⋊Dic3—41no—
184C3×Dic3—122no—
185C3×Dic3—122no—
186C2×C22⋊C4—91no—
187C22⋊C8—92no—
188C33#32311yescharacteristic, complemented, Sylow
189C23×C3—21no—
190C23×C3—22no—
191C23×C3—41no—
192C23×C3—92no—
193C23×C3—181no—
194C23×C3—181no—
195C23×C3—361no—
196C23×C3—361no—
197C23×C3D3×Dic312yes—
198C23×C3#62411yescharacteristic
199C22×D3—34no—
200C22×D3—62no—
201C22×D3—62no—
202C22×D3—62no—
203C22×D3—62no—
204C22×D3—62no—
205C22×D3—122no—
206C22×D3—122no—
207C22×D3—122no—
208C22×D3—122no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—42no—
213C2×C3×C4—41no—
214C2×C3×C4—44no—
215C2×C3×C4—42no—
216C2×C3×C4—44no—
217C2×C3×C4—42no—
218C2×C3×C4—181no—
219C2×C3×C4—361no—
220C2×C3×C4#62512yes—
221C2×C3×C4#62612yes—
222C2×C3×C4#62612yes—
223C2×Dic3—62no—
224C2×Dic3—62no—
225C2×Dic3—122no—
226C2×Dic3—122no—
227C3×C8—362no—
228C3⋊C8—124no—
229C3⋊C8—122no—
230C3⋊C8—122no—
231C2×C32—21no—
232C2×C32—21no—
233C2×C32—22no—
234C2×C32—22no—
235C2×C32—22no—
236C2×C32—22no—
237C2×C32—22no—
238C2×C32—21no—
239C2×C32—22no—
240C2×C32—21no—
241C2×C32—22no—
242C2×C32—21no—
243C2×C32—41no—
244C2×C32—41no—
245C2×C32—41no—
246C2×C32—42no—
247C2×C32—42no—
248C2×C32—42no—
249C2×C32—42no—
250C2×C32#80612yes—
251C2×C32(C2×D3)⋊C812yes—
252C2×C32#82312yes—
253C2×C32#83011yescharacteristic
254C2×C32#83311yescharacteristic
255C2×C32#83411yescharacteristic
256C3⋊D3—42no—
257C3×D3—124no—
258C3×D3—124no—
259C24—91no—
260C22×C4—91no—
261C22×C4#111311yescharacteristic
262C2×C8—182no—
263C22⋊C4—182no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—22no—
267C22×C3—21no—
268C22×C3—22no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—22no—
272C22×C3—21no—
273C22×C3—22no—
274C22×C3—21no—
275C22×C3—22no—
276C22×C3—42no—
277C22×C3—42no—
278C22×C3—42no—
279C22×C3—42no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—182no—
284C22×C3—362no—
285C22×C3—362no—
286C22×C3—361no—
287C22×C3—361no—
288C22×C3D3×C3⋊C812yes—
289C22×C3#371011yescharacteristic
290C22×C3#371112yes—
291C22×C3#371211yescharacteristic
292C22×C3(C2×D3)⋊Dic312yes—
293C22×C3#372211yescharacteristic
294C2×D3—64no—
295C2×D3—64no—
296C2×D3—124no—
297C2×D3—124no—
298C2×D3—124no—
299C2×D3—122no—
300C2×D3—122no—
301C2×D3—122no—
302C2×D3—122no—
303C2×D3—124no—
304C3×C4—24no—
305C3×C4—22no—
306C3×C4—22no—
307C3×C4—22no—
308C3×C4—42no—
309C3×C4—42no—
310C3×C4—44no—
311C3×C4—361no—
312Dic3—122no—
313Dic3—122no—
314C32—22no—
315C32—21no—
316C32—41no—
317C32#446812yescomplemented
318C32#452511yescharacteristic, complemented
319C23—92no—
320C23—181no—
321C23—181no—
322C23—361no—
323C23—361no—
324C23#629411yescharacteristic
325C2×C4—41no—
326C2×C4—181no—
327C2×C4—361no—
328C2×C4#629512yes—
329C8—362no—
330C2×C3—22no—
331C2×C3—22no—
332C2×C3—22no—
333C2×C3—22no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—22no—
338C2×C3—21no—
339C2×C3—22no—
340C2×C3—22no—
341C2×C3—21no—
342C2×C3—42no—
343C2×C3—42no—
344C2×C3—42no—
345C2×C3—42no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—362no—
350C2×C3(C2×D3)⋊(C3⋊C8)12yes—
351C2×C3#6355711yescharacteristic
352C2×C3#6355812yes—
353C2×C3#6355911yescharacteristic
354C2×C3#6358012yes—
355C2×C3#6358111yescharacteristic
356D3—124no—
357D3—124no—
358C22—21no—
359C22—21no—
360C22—182no—
361C22—362no—
362C22—362no—
363C22—361no—
364C22—361no—
365C22#8880811yescharacteristic
366C22#8880911yescharacteristic
367C22#8881411yescharacteristic
368C4—22no—
369C4—361no—
370C3—21no—
371C3—22no—
372C3—41no—
373C3#1058914512yescomplemented
374C3#1058914611yescharacteristic, complemented
375C2—21no—
376C2—21no—
377C2—362no—
378C2#1172207211yescharacteristic
379C2#1172207311yescharacteristic
380C2#1172208311yescharacteristic
subgroupG/H
centerC22#88814
commutatorC22×C33C2×C8
Frattini, hypercenterC22×C4#1113
Fitting, cosocleC22×C33×C4C22
socleC22×C33C22⋊C4
prime radicalC23×C3×C3⋊D3C4
hypocenterC33#323
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C3×#63657C2semidirect
13. #63657D3semidirect
22. C22×C3×C3⋊D3C8semidirect
77. C22×C3⋊D3C3⋊C8semidirect
188. C33#323semidirect
317. C32#4468semidirect
318. C32#4525semidirect
373. C3#10589145semidirect
374. C3#10589146semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37522
C3×#636572
#105891463
#105891453