Finite Group Databasebeta

← #13178141

Subgroups — #13178141

Order 1728.

1222 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1222 conjugacy classes of subgroups — too many to draw. Only its 199 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
1C3×#64083C211yesmaximal, characteristic
2C3×C4×#634C212yesmaximal, complemented
3#11723691C214yesmaximal, complemented
4#11723686C214yesmaximal, complemented
5#11723677C214yesmaximal, complemented
6#10593963—31nomaximal
7#10593951—32nomaximal
8C2×C3×C4⋊(C3⋊D3)C2214yes—
9C2×C3×C4⋊(C3⋊D3)C2212yes—
10C2×C3×#634C2212yes—
11C2×#6294C2214yes—
12C33×C42C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2212yes—
14C3×#3750C2214yes—
15C3×C4×C3⋊Dic3C2214yes—
16#88814C2214yes—
17#88814C2218yes—
18#64083D311yescharacteristic
19C4×#634D312yescomplemented
20C3×#874—32no—
21C3×#874—61no—
22C3×C4×Dic6—32no—
23C3×C4×Dic6—32no—
24C3×C4×Dic6—62no—
25#63898—34no—
26#63897—34no—
27#63894—34no—
28Dic3×D12—34no—
29#63884—38no—
30#63864—34no—
31#63863—34no—
32#63860—34no—
33C22×C3×C3⋊D3C2314yes—
34C2×C33×C4C2311yescharacteristic
35C2×C33×C4C2312yes—
36C2×C3×C3⋊Dic3C2314yes—
37C2×C3×C3⋊Dic3C2314yes—
38C3×C4⋊(C3⋊D3)—28no—
39C3×C4⋊(C3⋊D3)—42no—
40C3×#634—24no—
41#6294—44no—
42#5713—91no—
43#5585—92no—
44C2×C4⋊(C3⋊D3)C2×D314yes—
45C2×C4⋊(C3⋊D3)C2×D312yes—
46C2×#634C2×D312yes—
47C2×C3×D12—34no—
48C2×C3×D12—34no—
49C2×C3×D12—34no—
50C2×C3×D12—64no—
51C2×C3×D12—62no—
52C2×C3×Dic6—32no—
53C2×C3×Dic6—32no—
54C2×C3×Dic6—62no—
55C2×#624—34no—
56C2×D3×Dic3—38no—
57C32×C42—22no—
58C32×C42—21no—
59C32×C42—41no—
60C32×C42C2×D312yes—
61C32×C42C2×D311yescharacteristic
62C4⋊(C3⋊Dic3)C2×D312yes—
63#3750C2×D314yes—
64C4×C3⋊Dic3C2×D314yes—
65C3×C4⋊Dic3—32no—
66C3×C4⋊Dic3—32no—
67C3×C4⋊Dic3—62no—
68C3×Dic3⋊C4—34no—
69C3×Dic3⋊C4—34no—
70C3×Dic3⋊C4—64no—
71C3×C4×Dic3—34no—
72C3×C4×Dic3—34no—
73C3×C4×Dic3—64no—
74#3722—34no—
75#3722—38no—
76(C2×D3)⋊Dic3—38no—
77(C2×D3)⋊Dic3—38no—
78(C2×D3)⋊Dic3—38no—
79C22×C33C2411yescharacteristic
80C2×C3×C3⋊D3—28no—
81C2×C3×C3⋊D3—24no—
82C33×C4—21no—
83C33×C4#4114yes—
84C3×C3⋊Dic3—24no—
85C3×C3⋊Dic3—24no—
86C3×#127—91no—
87C3×C4×Q8—92no—
88#947—94no—
89#936—94no—
90#935—94no—
91D4×Dic3—94no—
92#894—98no—
93#893—94no—
94#875—94no—
95#874—32no—
96#874—61no—
97C4×D12—94no—
98C4×Dic6—32no—
99C4×Dic6—32no—
100C4×Dic6—62no—
101C22×C3⋊D3C22×D314yes—
102C22×C3×D3—38no—
103C22×C3×D3—64no—
104C2×C32×C4—22no—
105C2×C32×C4—22no—
106C2×C32×C4—21no—
107C2×C32×C4—22no—
108C2×C32×C4—22no—
109C2×C32×C4—42no—
110C2×C32×C4—41no—
111C2×C32×C4C22×D311yescharacteristic
112C2×C32×C4C22×D312yes—
113C2×C32×C4C22×D312yes—
114C2×C32×C4C22×D312yes—
115C2×C32×C4C22×D312yes—
116C2×C3⋊Dic3C22×D314yes—
117C2×C3⋊Dic3C22×D314yes—
118C4⋊(C3⋊D3)—28no—
119C4⋊(C3⋊D3)—42no—
120#634—24no—
121C2×C3×Dic3—34no—
122C2×C3×Dic3—34no—
123C2×C3×Dic3—34no—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—64no—
126C2×C3×Dic3—64no—
127C3×D12—68no—
128C3×D12—68no—
129C3×D12—128no—
130C3×D12—124no—
131C3×D12—124no—
132C3×Dic6—68no—
133C3×Dic6—64no—
134C3×Dic6—64no—
135#624—124no—
136D3×Dic3—128no—
137#559—271noSylow
138C2×C33C2×#4112yes—
139C2×C33H(5,𝔽2)11yescharacteristic
140C3×C3⋊D3—44no—
141C2×C3×Q8—92no—
142C2×C3×D4—94no—
143C2×C3×D4—92no—
144C22×Dic3—94no—
145C2×D12—34no—
146C2×D12—34no—
147C2×D12—34no—
148C2×D12—64no—
149C2×D12—62no—
150C2×C4×D3—98no—
151C2×Dic6—32no—
152C2×Dic6—32no—
153C2×Dic6—62no—
154C3×C4⋊C4—92no—
155C3×C4⋊C4—94no—
156C3×C42—22no—
157C3×C42—21no—
158C3×C42—41no—
159C3×C42—94no—
160C3×C42D3211yescharacteristic
161C3×C42D3212yes—
162#217—98no—
163#217—94no—
164(C2×D3)⋊C4—98no—
165(C2×D3)⋊C4—98no—
166(C2×D3)⋊C4—98no—
167C4⋊Dic3—32no—
168C4⋊Dic3—32no—
169C4⋊Dic3—62no—
170Dic3⋊C4—34no—
171Dic3⋊C4—34no—
172Dic3⋊C4—64no—
173C4×Dic3—34no—
174C4×Dic3—34no—
175C4×Dic3—64no—
176C22×C32—22no—
177C22×C32—21no—
178C22×C32—41no—
179C22×C32C23×D312yes—
180C22×C32C23×D311yescharacteristic
181C2×C3⋊D3—28no—
182C2×C3⋊D3—24no—
183C2×C3×D3—68no—
184C2×C3×D3—68no—
185C2×C3×D3—68no—
186C2×C3×D3—128no—
187C2×C3×D3—124no—
188C32×C4—21no—
189C32×C4—22no—
190C32×C4—22no—
191C32×C4—24no—
192C32×C4—24no—
193C32×C4—24no—
194C32×C4—44no—
195C32×C4—42no—
196C32×C4—42no—
197C32×C4#23514yes—
198C32×C4#23714yes—
199C32×C4#23914yes—
200C3⋊Dic3—24no—
201C3⋊Dic3—24no—
202C3×Dic3—68no—
203C3×Dic3—64no—
204C3×Dic3—64no—
205C3×Dic3—64no—
206C3×Dic3—64no—
207C3×Dic3—124no—
208#127—91no—
209#124—274no—
210#123—274no—
211C4×Q8—92no—
212C4×D4—274no—
213C33#55911yescharacteristic, complemented, Sylow
214C23×C3—94no—
215C22×D3—38no—
216C22×D3—64no—
217C3×Q8—184no—
218C3×D4—188no—
219C3×D4—362no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—21no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—94no—
228C2×C3×C4—94no—
229C2×C3×C4C2×D3211yescharacteristic
230C2×C3×C4C2×D3212yes—
231C2×C3×C4C2×D3212yes—
232C2×C3×C4C2×D3212yes—
233C2×C3×C4C2×D3212yes—
234C2×Dic3—34no—
235C2×Dic3—34no—
236C2×Dic3—34no—
237C2×Dic3—34no—
238C2×Dic3—64no—
239C2×Dic3—64no—
240C2×Dic3—364no—
241D12—68no—
242D12—68no—
243D12—128no—
244D12—124no—
245D12—124no—
246C4×D3—368no—
247Dic6—68no—
248Dic6—64no—
249Dic6—64no—
250C2×C32—22no—
251C2×C32—22no—
252C2×C32—22no—
253C2×C32—22no—
254C2×C32—21no—
255C2×C32—42no—
256C2×C32—41no—
257C2×C32C2×#23512yes—
258C2×C32C2×#23712yes—
259C2×C32#100411yescharacteristic
260C2×C32C2×#23912yes—
261C2×C32#100912yes—
262C3⋊D3—44no—
263C3×D3—128no—
264C3×D3—244no—
265C2×Q8—92no—
266C2×D4—94no—
267C2×D4—92no—
268C22×C4—274no—
269C4⋊C4—92no—
270C4⋊C4—94no—
271C22⋊C4—278no—
272C22⋊C4—274no—
273C42—94no—
274C42#111311yescharacteristic
275C22×C3—22no—
276C22×C3—21no—
277C22×C3—41no—
278C22×C3—188no—
279C22×C3—184no—
280C22×C3C22×D3212yes—
281C22×C3C22×D3211yescharacteristic
282C2×D3—68no—
283C2×D3—68no—
284C2×D3—68no—
285C2×D3—128no—
286C2×D3—124no—
287C3×C4—22no—
288C3×C4—22no—
289C3×C4—21no—
290C3×C4—24no—
291C3×C4—24no—
292C3×C4—24no—
293C3×C4—42no—
294C3×C4—42no—
295C3×C4—44no—
296C3×C4—184no—
297C3×C4—184no—
298C3×C4#379514yes—
299C3×C4#379614yes—
300C3×C4#379914yes—
301Dic3—68no—
302Dic3—64no—
303Dic3—64no—
304Dic3—64no—
305Dic3—64no—
306Dic3—124no—
307C32—22no—
308C32—21no—
309C32—41no—
310C32#558512yescomplemented
311C32#571311yescharacteristic, complemented
312C23—94no—
313Q8—184no—
314D4—188no—
315D4—362no—
316C2×C4—94no—
317C2×C4—94no—
318C2×C4—1084no—
319C2×C4C2×#111311yescharacteristic
320C2×C4C2×#111312yes—
321C2×C3—22no—
322C2×C3—22no—
323C2×C3—22no—
324C2×C3—22no—
325C2×C3—21no—
326C2×C3—42no—
327C2×C3—41no—
328C2×C3—364no—
329C2×C3C2×#379512yes—
330C2×C3C2×#379612yes—
331C2×C3C2×#379912yes—
332C2×C3#6430712yes—
333C2×C3#6430811yescharacteristic
334D3—128no—
335D3—244no—
336C22—188no—
337C22—184no—
338C22C22×#111311yescharacteristic
339C4—21no—
340C4—184no—
341C4—184no—
342C4#8904114yes—
343C3—21no—
344C3—22no—
345C3—41no—
346C3#1059395112yescomplemented
347C3#1059396311yescharacteristic, complemented
348C2—364no—
349C2C2×#8904112yes—
350C2#1172461911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radicalC3×#64083C2
hypercenterC42#1113
hypocenterC33#559
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×C4×#634C2semidirect
3. #11723691C2semidirect
4. #11723686C2semidirect
5. #11723677C2semidirect
19. C4×#634D3semidirect
213. C33#559semidirect
310. C32#5585semidirect
311. C32#5713semidirect
346. C3#10593951semidirect
347. C3#10593963semidirect

Maximal subgroups

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

subgroupindex
C3×#640832
C3×C4×#6342
#117236912
#117236862
#117236772
#105939633
#105939513