Finite Group Databasebeta

← #13159075

Subgroups — #13159075

Order 1728.

828 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 148 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×#64104C211yesmaximal, characteristic
2C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
3#11722177C211yesmaximal, characteristic, complemented
4C32×#807C211yesmaximal, characteristic, complemented
5#11722034C211yesmaximal, characteristic, complemented
6#11722033C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592581—31nomaximal
9#10590438—34nomaximal
10C2×C3×C4⋊(C3⋊D3)—21no—
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×Dic3×C3⋊D3—21no—
14C33×C4⋊C4C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#3752—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88807—21no—
21#64104D311yescharacteristic
22C4⋊C4×C3⋊D3—31no—
23C3×#895—34no—
24D3×C4⋊Dic3—34no—
25C32×#107—31no—
26#63648—31no—
27#63647—31no—
28#63636—31no—
29#63634—31no—
30#63634—34no—
31#63634—64no—
32C3×#807D314yescomplemented
33#63563—34no—
34#63562—34no—
35(C2×D3)⋊(C3⋊C8)—34no—
36C22×C3×C3⋊D3—21no—
37C22×C3×C3⋊D3D411yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3D411yescharacteristic
42C3×C4⋊(C3⋊D3)—21no—
43C3×C4⋊(C3⋊D3)—21no—
44C3×C4⋊(C3⋊D3)—42no—
45C3×C4×C3⋊D3—21no—
46C3×C4×C3⋊D3—21no—
47C2×C32×Dic3—21no—
48Dic3×C3⋊D3—42no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50C32×C3⋊C8—21no—
51#5033—91no—
52#4882—94no—
53C2×C4⋊(C3⋊D3)—21no—
54C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
55C2×C4×C3⋊D3—61no—
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C2×C3×D12—34no—
58C2×C3×D12—64no—
59C2×C3×C4×D3—34no—
60C2×D3×Dic3—64no—
61C2×C32×C8—31no—
62C32×C4⋊C4—24no—
63C32×C4⋊C4—31no—
64C32×C4⋊C4C2×D314yes—
65C32×C4⋊C4C2×D311yescharacteristic
66#3752—21no—
67C4⋊(C3⋊Dic3)—31no—
68C4⋊(C3⋊Dic3)—34no—
69C4⋊(C3⋊Dic3)—64no—
70#3750—61no—
71C2×C3⋊(C3⋊C8)—31no—
72C2×C3⋊(C3⋊C8)—34no—
73C2×C3⋊(C3⋊C8)—64no—
74C3×(C2×D3)⋊C4—64no—
75C3×C4⋊Dic3C2×D314yes—
76C2×C3×C3⋊C8C2×D314yes—
77Dic3⋊Dic3—64no—
78C22×C33C2×D411yescharacteristic
79C2×C3×C3⋊D3—21no—
80C2×C3×C3⋊D3—21no—
81C2×C3×C3⋊D3—41no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—41no—
84C2×C3×C3⋊D3D812yes—
85C33×C4—41no—
86C33×C4#4111yescharacteristic
87C33×C4#4111yescharacteristic
88C3⋊(C3⋊Dic3)—41no—
89C3×C3⋊Dic3—21no—
90C32×Dic3—41no—
91C3×#121—91no—
92C2×C4⋊Dic3—91no—
93#895—34no—
94D3×C4⋊C4—94no—
95C3×#107—34no—
96#832—91no—
97#832—91no—
98#830—91no—
99#821—94no—
100(C2×D3)⋊C8—94no—
101#809—94no—
102#807—94no—
103#807—91no—
104#807—184no—
105#807C3⋊D311yescharacteristic, complemented
106C22×C3⋊D3—21no—
107C22×C3⋊D3#6711yescharacteristic
108C22×C3×D3—34no—
109C22×C3×D3—64no—
110C2×C32×C4—21no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—44no—
114C2×C32×C4—61no—
115C2×C32×C4C22×D314yes—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—61no—
118C2×C3⋊Dic3—64no—
119C2×C3⋊Dic3—124no—
120C2×C3⋊Dic3#6711yescharacteristic
121C4⋊(C3⋊D3)—21no—
122C4⋊(C3⋊D3)—21no—
123C4⋊(C3⋊D3)—42no—
124C4×C3⋊D3—21no—
125C4×C3⋊D3—21no—
126C4×C3⋊D3—122no—
127C2×C3×Dic3—24no—
128C2×C3×Dic3—34no—
129C3×D12—64no—
130C3×D12—64no—
131C3×D12—128no—
132C3×C4×D3—64no—
133C3×C4×D3—64no—
134D3×Dic3—128no—
135C32×C8—61no—
136C3⋊(C3⋊C8)—68no—
137C3⋊(C3⋊C8)—61no—
138C3⋊(C3⋊C8)—64no—
139C3×C3⋊C8—24no—
140#480—271noSylow
141C2×C33#12311yescharacteristic
142C2×C33C2×D811yescharacteristic
143C2×C33#13711yescharacteristic
144C3×C3⋊D3—22no—
145C3×C3⋊D3—81no—
146C2×C3×D4—91no—
147C2×C3×D4—181no—
148C22×C3×C4—91no—
149C22×Dic3—181no—
150C2×D12—34no—
151C2×D12—64no—
152C2×C4×D3—34no—
153C2×C4×D3—184no—
154C2×C3×C8—34no—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—34no—
157C3×C4⋊C4D3214yes—
158C3×C4⋊C4C2×C3⋊D311yescharacteristic
159C3×C22⋊C4—181no—
160(C2×D3)⋊C4—64no—
161C4⋊Dic3—94no—
162C4⋊Dic3—91no—
163C4⋊Dic3—184no—
164C4⋊Dic3—181no—
165C4⋊Dic3C2×C3⋊D311yescharacteristic
166Dic3⋊C4—184no—
167C2×C3⋊C8—94no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—184no—
170C2×C3⋊C8C2×C3⋊D311yescharacteristic
171C22×C32—24no—
172C22×C32D3×D414yes—
173C22×C32C2×#6711yescharacteristic
174C2×C3⋊D3—21no—
175C2×C3⋊D3—21no—
176C2×C3⋊D3—41no—
177C2×C3⋊D3—41no—
178C2×C3⋊D3—41no—
179C2×C3⋊D3#21312yes—
180C2×C3×D3—38no—
181C2×C3×D3—64no—
182C2×C3×D3—64no—
183C2×C3×D3—124no—
184C2×C3×D3—124no—
185C2×C3×D3—124no—
186C32×C4—24no—
187C32×C4—24no—
188C32×C4—41no—
189C32×C4—44no—
190C32×C4—84no—
191C32×C4—121no—
192C32×C4#23514yes—
193C32×C4#23711yescharacteristic
194C32×C4#23711yescharacteristic
195C32×C4#23914yes—
196C3⋊Dic3—21no—
197C3⋊Dic3—121no—
198C3⋊Dic3—124no—
199C3⋊Dic3—244no—
200C3×Dic3—44no—
201C3×Dic3—64no—
202#121—91no—
203C2×C4⋊C4—271no—
204#107—31no—
205#107—271no—
206#102—271no—
207#102—271no—
208C22⋊C8—271no—
209C33#48011yescharacteristic, complemented, Sylow
210C23×C3—91no—
211C23×C3—181no—
212C22×D3—34no—
213C22×D3—64no—
214C3×D4—181no—
215C3×D4—181no—
216C3×D4—362no—
217C2×C3×C4—24no—
218C2×C3×C4—21no—
219C2×C3×C4—24no—
220C2×C3×C4—44no—
221C2×C3×C4—64no—
222C2×C3×C4—91no—
223C2×C3×C4—181no—
224C2×C3×C4—181no—
225C2×C3×C4C2×D3214yes—
226C2×C3×C4C22×C3⋊D311yescharacteristic
227C2×Dic3—21no—
228C2×Dic3—34no—
229C2×Dic3—184no—
230C2×Dic3—181no—
231C2×Dic3—364no—
232C2×Dic3—362no—
233D12—64no—
234D12—64no—
235D12—128no—
236C4×D3—64no—
237C4×D3—64no—
238C4×D3—368no—
239C3×C8—64no—
240C3⋊C8—21no—
241C3⋊C8—188no—
242C3⋊C8—184no—
243C3⋊C8—181no—
244C2×C32—24no—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32#89414yes—
248C2×C32D3×D814yes—
249C2×C32#91814yes—
250C2×C32C2×#21311yescharacteristic
251C2×C32#93511yescharacteristic
252C2×C32#95111yescharacteristic
253C3⋊D3—22no—
254C3⋊D3—81no—
255C3×D3—68no—
256C3×D3—244no—
257C2×D4—91no—
258C2×D4—181no—
259C22×C4—91no—
260C22×C4—541no—
261C2×C8—31no—
262C2×C8—271no—
263C4⋊C4—31no—
264C4⋊C4—271no—
265C4⋊C4—541no—
266C4⋊C4D3×C3⋊D311yescharacteristic
267C22⋊C4—181no—
268C22×C3—24no—
269C22×C3—92no—
270C22×C3—181no—
271C22×C3—181no—
272C22×C3—361no—
273C22×C3—361no—
274C22×C3—361no—
275C22×C3D3×#6714yes—
276C22×C3D4×C3⋊D311yescharacteristic
277C2×D3—38no—
278C2×D3—64no—
279C2×D3—64no—
280C2×D3—124no—
281C2×D3—124no—
282C2×D3—124no—
283C3×C4—24no—
284C3×C4—24no—
285C3×C4—41no—
286C3×C4—44no—
287C3×C4—84no—
288C3×C4—124no—
289C3×C4—181no—
290C3×C4#379514yes—
291C3×C4#379614yes—
292C3×C4#382811yescharacteristic
293C3×C4#383211yescharacteristic
294Dic3—41no—
295Dic3—64no—
296Dic3—361no—
297Dic3—364no—
298Dic3—724no—
299C32—24no—
300C32#488214yescomplemented
301C32#503311yescharacteristic, complemented
302C23—91no—
303C23—181no—
304D4—181no—
305D4—181no—
306D4—362no—
307C2×C4—21no—
308C2×C4—61no—
309C2×C4—91no—
310C2×C4—181no—
311C2×C4—181no—
312C2×C4—541no—
313C2×C4—1082no—
314C2×C4C2×D3×C3⋊D311yescharacteristic
315C8—61no—
316C8—541no—
317C2×C3—24no—
318C2×C3—24no—
319C2×C3—24no—
320C2×C3—182no—
321C2×C3—721no—
322C2×C3#6388414yes—
323C2×C3D3×#21314yes—
324C2×C3#6394314yes—
325C2×C3#6410311yescharacteristic
326C2×C3D8×C3⋊D311yescharacteristic
327C2×C3#6412711yescharacteristic
328D3—68no—
329D3—244no—
330C22—92no—
331C22—181no—
332C22—181no—
333C22—361no—
334C22—361no—
335C22—361no—
336C22C3⋊D3×#6711yescharacteristic
337C4—41no—
338C4—121no—
339C4—181no—
340C4—1081no—
341C4#8901711yescharacteristic
342C4#8902111yescharacteristic
343C3—24no—
344C3#1059043814yescomplemented
345C3#1059258111yescharacteristic, complemented
346C2—182no—
347C2—721no—
348C2#1172348811yescharacteristic
349C2C3⋊D3×#21311yescharacteristic
350C2#1172357611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC3×#64104C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#480
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3⋊D3×C4⋊Dic3C2semidirect
3. #11722177C2semidirect
4. C32×#807C2semidirect
5. #11722034C2semidirect
6. #11722033C2semidirect
7. #11722023C2semidirect
32. C3×#807D3semidirect
105. #807C3⋊D3semidirect
209. C33#480semidirect
300. C32#4882semidirect
301. C32#5033semidirect
344. C3#10590438semidirect
345. C3#10592581semidirect

Maximal subgroups

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

subgroupindex
C3×#641042
C3⋊D3×C4⋊Dic32
#117221772
C32×#8072
#117220342
#117220332
#117220232
#105925813
#105904383