Finite Group Databasebeta

← #13159697

Subgroups — #13159697

Order 1728.

880 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 182 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×#3744C211yesmaximal, characteristic
2#11723501C211yesmaximal, characteristic
3#11723410C211yesmaximal, characteristic, complemented
4#11722185C211yesmaximal, characteristic, complemented
5C32×#818C211yesmaximal, characteristic, complemented
6#11722039C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic
8#10592588—31nomaximal
9#10590779—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×Dic3×C3⋊D3—21no—
13C2×C33×C8C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3744C414yescomplemented
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19#88807—21no—
20#88803—21no—
21Dic3×C3⋊Dic3—21no—
22C2×#3744D311yescharacteristic
23#64102—31no—
24#64096—31no—
25C2×C3×#204—34no—
26#63898—34no—
27#63843—34no—
28C32×#107—31no—
29#63642—31no—
30#63642—34no—
31#63642—64no—
32#63637—31no—
33#63636—31no—
34C3×#818D314yescomplemented
35#63565—34no—
36#63562—34no—
37C22×C3×C3⋊D3—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)D411yescharacteristic
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41C2×C3×C3⋊Dic3—21no—
42C3×C4⋊(C3⋊D3)—21no—
43C3×C4⋊(C3⋊D3)C2×C412yes—
44C3×#634—21no—
45C3×#634C2×C412yescomplemented
46C2×C32×Dic3—21no—
47C2×C32×Dic3—21no—
48Dic3×C3⋊D3—42no—
49C33×C8C2×C412yescomplemented
50C3⋊(C3⋊(C3⋊C8))—21no—
51#5164—91no—
52#4889—94no—
53C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
54C2×C4×C3⋊D3—61no—
55C2×#634C2×D311yescharacteristic
56C2×C3×D12—34no—
57C2×C3×Dic6—34no—
58C2×D3×Dic3—64no—
59C2×C32×C8—24no—
60C2×C32×C8C2×D314yes—
61C2×C32×C8C2×D311yescharacteristic
62C32×C4⋊C4—31no—
63C32×C4⋊C4—31no—
64#3752—61no—
65#3750—61no—
66C4×C3⋊Dic3—31no—
67C4×C3⋊Dic3—34no—
68C4×C3⋊Dic3—61no—
69C4×C3⋊Dic3—64no—
70C2×C3⋊(C3⋊C8)—31no—
71C2×C3⋊(C3⋊C8)—34no—
72C2×C3⋊(C3⋊C8)—64no—
73#3744Dic314yescomplemented
74C3×C4⋊Dic3C2×D314yes—
75C3×C4⋊Dic3C2×D314yes—
76C3×#204—316no—
77Dic3⋊Dic3—64no—
78(C2×D3)⋊Dic3—64no—
79Dic32—64no—
80C22×C33C2×D411yescharacteristic
81C2×C3×C3⋊D3—22no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—41no—
84C33×C4C22×C411yescharacteristic
85C33×C4#4111yescharacteristic
86C3⋊(C3⋊Dic3)—21no—
87C3⋊(C3⋊Dic3)—21no—
88C3×C3⋊Dic3—22no—
89C32×Dic3—41no—
90C32×Dic3—41no—
91C2×C3×SD16—91no—
92Q8×Dic3—91no—
93D4×Dic3—91no—
94C2×#204—34no—
95#893—94no—
96#887—94no—
97C3×#107—34no—
98#835—91no—
99#832—91no—
100#818C3⋊D311yescharacteristic, complemented
101#815—94no—
102#815—91no—
103#815—184no—
104#810—94no—
105#809—94no—
106C22×C3⋊D3—21no—
107C22×C3×D3—64no—
108C2×C32×C4—24no—
109C2×C32×C4—61no—
110C2×C32×C4—61no—
111C2×C32×C4C22×D314yes—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3—31no—
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—34no—
117C2×C3⋊Dic3—34no—
118C2×C3⋊Dic3—64no—
119C2×C3⋊Dic3—64no—
120C4⋊(C3⋊D3)—21no—
121C4⋊(C3⋊D3)C2×Dic312yes—
122C4×C3⋊D3—122no—
123#634—21no—
124#634C2×Dic312yescomplemented
125C2×C3×Dic3—24no—
126C2×C3×Dic3—24no—
127C2×C3×Dic3—64no—
128C3×D12—38no—
129C3×D12—64no—
130C3×Dic6—38no—
131C3×Dic6—64no—
132D3×Dic3—128no—
133C32×C8—28no—
134C32×C8C4×D318yescomplemented
135C32×C8C2×Dic312yescomplemented
136C3⋊(C3⋊C8)—68no—
137C3⋊(C3⋊C8)—61no—
138C3⋊(C3⋊C8)—64no—
139#440—271noSylow
140C2×C33C4×D411yescharacteristic
141C2×C33AGL(1,ℤ/8)11yescharacteristic
142C2×C33#13711yescharacteristic
143C3×C3⋊D3—42no—
144C2×C3×Q8—91no—
145C2×C3×D4—91no—
146C22×Dic3—181no—
147C2×D12—34no—
148C2×C4×D3—184no—
149C2×Dic6—34no—
150C3×SD16—94no—
151C2×C3×C8—24no—
152C2×C3×C8D3214yes—
153C2×C3×C8C2×C3⋊D311yescharacteristic
154C3×C4⋊C4—34no—
155C3×C4⋊C4—34no—
156#217—181no—
157(C2×D3)⋊C4—184no—
158C4⋊Dic3—181no—
159C4⋊Dic3C2×C3⋊D311yescharacteristic
160C4⋊Dic3C2×C3⋊D311yescharacteristic
161Dic3⋊C4—184no—
162C4×Dic3—94no—
163C4×Dic3—91no—
164C4×Dic3—184no—
165C4×Dic3—181no—
166C4×Dic3—184no—
167C2×C3⋊C8—94no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—184no—
170#204—316no—
171C22×C32—24no—
172C22×C32D3×D414yes—
173C22×C32D3×D411yescharacteristic
174C2×C3⋊D3—22no—
175C2×C3⋊D3—41no—
176C2×C3⋊D3—41no—
177C2×C3×D3—68no—
178C2×C3×D3—124no—
179C2×C3×D3—124no—
180C32×C4—24no—
181C32×C4—24no—
182C32×C4—121no—
183C32×C4—121no—
184C32×C4C2×C4×D314yes—
185C32×C4#23711yescharacteristic
186C32×C4#23914yes—
187C32×C4C22×Dic311yescharacteristic
188C3⋊Dic3—22no—
189C3⋊Dic3—61no—
190C3⋊Dic3—61no—
191C3⋊Dic3—64no—
192C3⋊Dic3—64no—
193C3⋊Dic3—124no—
194C3⋊Dic3—124no—
195C3×Dic3—44no—
196C3×Dic3—44no—
197C3×Dic3—68no—
198C2×SD16—91no—
199C4×Q8—271no—
200C4×D4—271no—
201#107—31no—
202#103—271no—
203#102—271no—
204#97—271no—
205C33#44011yescharacteristic, complemented, Sylow
206C23×C3—181no—
207C22×D3—64no—
208C3×Q8—92no—
209C3×Q8—181no—
210C3×D4—92no—
211C3×D4—181no—
212C2×C3×C4—24no—
213C2×C3×C4—64no—
214C2×C3×C4—64no—
215C2×C3×C4—181no—
216C2×C3×C4C2×D3214yes—
217C2×C3×C4C22×C3⋊D311yescharacteristic
218C2×Dic3—21no—
219C2×Dic3—21no—
220C2×Dic3—64no—
221C2×Dic3—94no—
222C2×Dic3—94no—
223C2×Dic3—91no—
224C2×Dic3—91no—
225C2×Dic3—184no—
226C2×Dic3—184no—
227C2×Dic3—362no—
228D12—38no—
229D12—64no—
230C4×D3—368no—
231Dic6—38no—
232Dic6—64no—
233C3×C8—28no—
234C3×C8D3×Dic318yescomplemented
235C3×C8C4×C3⋊D312yescomplemented
236C3⋊C8—188no—
237C3⋊C8—184no—
238C3⋊C8—181no—
239C2×C32—24no—
240C2×C32—24no—
241C2×C32—24no—
242C2×C32#89314yes—
243C2×C32#91114yes—
244C2×C32#91411yescharacteristic
245C2×C32#91511yescharacteristic
246C2×C32#91814yes—
247C2×C32D4×Dic311yescharacteristic
248C3⋊D3—42no—
249C3×D3—128no—
250C2×Q8—91no—
251C2×D4—91no—
252C22×C4—541no—
253SD16—94no—
254C2×C8—271no—
255C2×C8D3×C3⋊D311yescharacteristic
256C4⋊C4—31no—
257C4⋊C4—31no—
258C4⋊C4—541no—
259C22⋊C4—541no—
260C42—271no—
261C42—541no—
262C22×C3—24no—
263C22×C3—182no—
264C22×C3—361no—
265C22×C3—361no—
266C22×C3#380214yes—
267C22×C3D4×C3⋊D311yescharacteristic
268C2×D3—68no—
269C2×D3—124no—
270C2×D3—124no—
271C3×C4—24no—
272C3×C4—24no—
273C3×C4—124no—
274C3×C4—124no—
275C3×C4—182no—
276C3×C4#379614yes—
277C3×C4C2×D3×Dic314yes—
278C3×C4C2×C4×C3⋊D311yescharacteristic
279C3×C4#383211yescharacteristic
280Dic3—41no—
281Dic3—41no—
282Dic3—68no—
283Dic3—184no—
284Dic3—184no—
285Dic3—181no—
286Dic3—181no—
287Dic3—364no—
288Dic3—364no—
289C32—24no—
290C32#488914yescomplemented
291C32#516411yescharacteristic, complemented
292C23—181no—
293Q8—92no—
294Q8—181no—
295D4—92no—
296D4—181no—
297C2×C4—61no—
298C2×C4—61no—
299C2×C4—181no—
300C2×C4—271no—
301C2×C4—271no—
302C2×C4—1082no—
303C2×C4C2×D3×C3⋊D311yescharacteristic
304C8—541no—
305C8Dic3×C3⋊D312yescomplemented
306C2×C3—24no—
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—362no—
310C2×C3#6379814yes—
311C2×C3#6381114yes—
312C2×C3#6389814yes—
313C2×C3#6410211yescharacteristic
314C2×C3#6412011yescharacteristic
315C2×C3#6412711yescharacteristic
316D3—128no—
317C22—182no—
318C22—361no—
319C22—361no—
320C22#8902611yescharacteristic
321C4—121no—
322C4—121no—
323C4—182no—
324C4—541no—
325C4—541no—
326C4#8901711yescharacteristic
327C4C2×Dic3×C3⋊D311yescharacteristic
328C3—24no—
329C3#1059077914yescomplemented
330C3#1059258811yescharacteristic, complemented
331C2—362no—
332C2#1172332411yescharacteristic
333C2#1172334611yescharacteristic
334C2#1172350111yescharacteristic
subgroupG/H
centerC22#89026
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#440
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. #11723410C2semidirect
4. #11722185C2semidirect
5. C32×#818C2semidirect
6. #11722039C2semidirect
16. C3×#3744C4semidirect
34. C3×#818D3semidirect
45. C3×#634C2×C4semidirect
49. C33×C8C2×C4semidirect
73. #3744Dic3semidirect
100. #818C3⋊D3semidirect
124. #634C2×Dic3semidirect
134. C32×C8C4×D3semidirect
135. C32×C8C2×Dic3semidirect
205. C33#440semidirect
234. C3×C8D3×Dic3semidirect
235. C3×C8C4×C3⋊D3semidirect
290. C32#4889semidirect
291. C32#5164semidirect
305. C8Dic3×C3⋊D3semidirect
329. C3#10590779semidirect
330. C3#10592588semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37442
#117235012
#117234102
#117221852
C32×#8182
#117220392
#117220332
#105925883
#105907793