Finite Group Databasebeta

← #13159369

Subgroups — #13159369

Order 1728.

864 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 166 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×#64102C211yesmaximal, characteristic
2Dic6×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#88793C211yesmaximal, characteristic
4#11722177C211yesmaximal, characteristic, complemented
5C32×#810C211yesmaximal, characteristic, complemented
6#11722034C211yesmaximal, characteristic
7#11722020C211yesmaximal, characteristic, complemented
8#10592516—31nomaximal
9#10590478—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4×C3⋊D3—21no—
12C2×C32×Dic6C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3752—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19Dic3⋊(C3⋊Dic3)—21no—
20Dic3×C3⋊Dic3—21no—
21#88793C414yescomplemented
22Q8×C3⋊Dic3—31no—
23C2×#3755—31no—
24#64102D311yescharacteristic
25C3×#893—34no—
26Dic3×Dic6—34no—
27C2×#3715—34no—
28C32×#103—31no—
29#63648—31no—
30#63642—31no—
31#63634—31no—
32#63634—34no—
33#63634—64no—
34C3×#810D314yescomplemented
35#63563—34no—
36#63554—34no—
37C22×C3×C3⋊D3—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3D411yescharacteristic
42C2×C3×C3⋊Dic3D411yescharacteristic
43C3×C4⋊(C3⋊D3)—21no—
44C3×C4⋊(C3⋊D3)C2×C412yes—
45C3×C4×C3⋊D3—42no—
46C2×C32×Dic3—21no—
47C32×Dic6—21no—
48C32×Dic6C2×C412yescomplemented
49C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
50C32×C3⋊C8—21no—
51#5018—91no—
52#4817—94no—
53C2×C32×Q8—31no—
54C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
55C2×C4×C3⋊D3—21no—
56C2×C3×D12—34no—
57C2×C3×C4×D3—64no—
58C2×C3×Dic6C2×D314yes—
59C2×C32×C8—31no—
60C32×C4⋊C4—24no—
61C32×C4⋊C4C2×D314yes—
62C32×C4⋊C4C2×D311yescharacteristic
63#3755—34no—
64#3752—21no—
65C4⋊(C3⋊Dic3)—31no—
66C4⋊(C3⋊Dic3)—34no—
67C4⋊(C3⋊Dic3)—61no—
68C4⋊(C3⋊Dic3)—64no—
69C4×C3⋊Dic3—61no—
70C4×C3⋊Dic3C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)—31no—
72C2×C3⋊(C3⋊C8)—34no—
73C2×C3⋊(C3⋊C8)—64no—
74C3×(C2×D3)⋊C4—64no—
75C3×C4×Dic3—34no—
76C2×C3×C3⋊C8C2×D314yes—
77Dic3⋊Dic3—64no—
78Dic32—64no—
79#3715—316no—
80C22×C33C2×D411yescharacteristic
81C2×C3×C3⋊D3—22no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—41no—
84C33×C4—41no—
85C33×C4C22×C411yescharacteristic
86C33×C4#4111yescharacteristic
87C3⋊(C3⋊Dic3)—41no—
88C3×C3⋊Dic3—21no—
89C3×C3⋊Dic3—21no—
90C32×Dic3—22no—
91C3×C4×D4—91no—
92Q8×Dic3—94no—
93C2×#215—94no—
94C2×#214—91no—
95#893—34no—
96C4×Dic6—91no—
97C3×#103—34no—
98#832—91no—
99#821—94no—
100#815—94no—
101#810C3⋊D311yescharacteristic, complemented
102#807—94no—
103#807—91no—
104#807—184no—
105#803—91no—
106C22×C3⋊D3—21no—
107C22×C3×D3—64no—
108C32×Q8—32no—
109C32×Q8—61no—
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
121C2×C3⋊Dic3#6711yescharacteristic
122C4⋊(C3⋊D3)—21no—
123C4⋊(C3⋊D3)C4×D312yes—
124C4×C3⋊D3—42no—
125C2×C3×Dic3—24no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—34no—
128C3×D12—38no—
129C3×D12—64no—
130C3×C4×D3—128no—
131C3×Dic6—24no—
132C3×Dic6C4×D318yescomplemented
133C32×C8—61no—
134C3⋊(C3⋊C8)—32no—
135C3⋊(C3⋊C8)—38no—
136C3⋊(C3⋊C8)—68no—
137C3×C3⋊C8—24no—
138#440—271noSylow
139C2×C33C4×D411yescharacteristic
140C2×C33AGL(1,ℤ/8)11yescharacteristic
141C2×C33#13711yescharacteristic
142C3×C3⋊D3—42no—
143C2×C3×Q8—34no—
144C2×C3×D4—91no—
145C22×C3×C4—181no—
146C2×D12—34no—
147C2×C4×D3—64no—
148C2×Dic6C2×C3⋊D311yescharacteristic
149C2×C3×C8—34no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4D3214yes—
152C3×C4⋊C4C2×C3⋊D311yescharacteristic
153C3×C22⋊C4—181no—
154C3×C42—91no—
155#215—916no—
156#214—94no—
157(C2×D3)⋊C4—64no—
158C4⋊Dic3—94no—
159C4⋊Dic3—91no—
160C4⋊Dic3—184no—
161C4⋊Dic3—184no—
162Dic3⋊C4—181no—
163C4×Dic3—34no—
164C4×Dic3—181no—
165C4×Dic3—184no—
166C2×C3⋊C8—94no—
167C2×C3⋊C8—91no—
168C2×C3⋊C8—184no—
169C2×C3⋊C8C2×C3⋊D311yescharacteristic
170C22×C32—24no—
171C22×C32D3×D414yes—
172C22×C32C2×#6711yescharacteristic
173C2×C3⋊D3—22no—
174C2×C3⋊D3—41no—
175C2×C3⋊D3—41no—
176C2×C3×D3—68no—
177C2×C3×D3—124no—
178C2×C3×D3—124no—
179C32×C4—24no—
180C32×C4—24no—
181C32×C4—41no—
182C32×C4—44no—
183C32×C4—62no—
184C32×C4—84no—
185C32×C4C2×C4×D314yes—
186C32×C4C2×C4×D311yescharacteristic
187C32×C4#23511yescharacteristic
188C32×C4#23714yes—
189C3⋊Dic3—21no—
190C3⋊Dic3—21no—
191C3⋊Dic3—121no—
192C3⋊Dic3—124no—
193C3⋊Dic3—244no—
194C3×Dic3—28no—
195C3×Dic3—64no—
196C3×Dic3—64no—
197C2×SD16—271no—
198C4×Q8—271no—
199C4×D4—91no—
200#107—271no—
201#103—31no—
202#102—271no—
203#97—271no—
204C33#44011yescharacteristic, complemented, Sylow
205C23×C3—181no—
206C22×D3—64no—
207C3×Q8—38no—
208C3×Q8—64no—
209C3×D4—92no—
210C3×D4—181no—
211C2×C3×C4—24no—
212C2×C3×C4—21no—
213C2×C3×C4—24no—
214C2×C3×C4—44no—
215C2×C3×C4—64no—
216C2×C3×C4—91no—
217C2×C3×C4—91no—
218C2×C3×C4—362no—
219C2×C3×C4C2×D3214yes—
220C2×C3×C4C22×C3⋊D311yescharacteristic
221C2×Dic3—21no—
222C2×Dic3—34no—
223C2×Dic3—34no—
224C2×Dic3—184no—
225C2×Dic3—181no—
226C2×Dic3—364no—
227D12—38no—
228D12—64no—
229C4×D3—128no—
230Dic6—21no—
231Dic6C4×C3⋊D312yescomplemented
232C3×C8—64no—
233C3⋊C8—21no—
234C3⋊C8—98no—
235C3⋊C8—92no—
236C3⋊C8—188no—
237C2×C32—24no—
238C2×C32—24no—
239C2×C32—24no—
240C2×C32#67⋊C414yes—
241C2×C32#91414yes—
242C2×C32#91814yes—
243C2×C32C4×#6711yescharacteristic
244C2×C32#93211yescharacteristic
245C2×C32#95111yescharacteristic
246C3⋊D3—42no—
247C3×D3—128no—
248C2×Q8—31no—
249C2×D4—91no—
250C22×C4—181no—
251SD16—274no—
252C2×C8—31no—
253C2×C8—271no—
254C4⋊C4—271no—
255C4⋊C4—541no—
256C4⋊C4D3×C3⋊D311yescharacteristic
257C22⋊C4—181no—
258C42—91no—
259C42—541no—
260C22×C3—24no—
261C22×C3—182no—
262C22×C3—361no—
263C22×C3—361no—
264C22×C3D3×#6714yes—
265C22×C3D4×C3⋊D311yescharacteristic
266C2×D3—68no—
267C2×D3—124no—
268C2×D3—124no—
269C3×C4—24no—
270C3×C4—24no—
271C3×C4—41no—
272C3×C4—44no—
273C3×C4—68no—
274C3×C4—84no—
275C3×C4—181no—
276C3×C4—181no—
277C3×C4#379514yes—
278C3×C4C4×D3214yes—
279C3×C4C2×C4×C3⋊D311yescharacteristic
280C3×C4#383011yescharacteristic
281Dic3—22no—
282Dic3—64no—
283Dic3—64no—
284Dic3—361no—
285Dic3—364no—
286Dic3—724no—
287C32—24no—
288C32#481714yescomplemented
289C32#501811yescharacteristic, complemented
290C23—181no—
291Q8—32no—
292Q8—61no—
293D4—92no—
294D4—181no—
295C2×C4—21no—
296C2×C4—61no—
297C2×C4—91no—
298C2×C4—91no—
299C2×C4—362no—
300C2×C4—541no—
301C2×C4C2×D3×C3⋊D311yescharacteristic
302C8—61no—
303C8—272no—
304C2×C3—24no—
305C2×C3—24no—
306C2×C3—24no—
307C2×C3—362no—
308C2×C3#6387914yes—
309C2×C3#6393414yes—
310C2×C3#6394314yes—
311C2×C3#6409011yescharacteristic
312C2×C3#6412311yescharacteristic
313C2×C3#6412711yescharacteristic
314D3—128no—
315C22—182no—
316C22—361no—
317C22—361no—
318C22C3⋊D3×#6711yescharacteristic
319C4—41no—
320C4—62no—
321C4—181no—
322C4—181no—
323C4—1081no—
324C4#8901411yescharacteristic
325C4C4×D3×C3⋊D311yescharacteristic
326C3—24no—
327C3#1059047814yescomplemented
328C3#1059251611yescharacteristic, complemented
329C2—362no—
330C2#1172350711yescharacteristic
331C2#1172356211yescharacteristic
332C2#1172357611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#440
LCSG ▷ 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. Dic6×C3⋊Dic3C2semidirect
4. #11722177C2semidirect
5. C32×#810C2semidirect
7. #11722020C2semidirect
21. #88793C4semidirect
34. C3×#810D3semidirect
48. C32×Dic6C2×C4semidirect
49. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
101. #810C3⋊D3semidirect
132. C3×Dic6C4×D3semidirect
204. C33#440semidirect
231. Dic6C4×C3⋊D3semidirect
288. C32#4817semidirect
289. C32#5018semidirect
327. C3#10590478semidirect
328. C3#10592516semidirect

Maximal subgroups

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

subgroupindex
C3×#641022
Dic6×C3⋊Dic32
C2×#887932
#117221772
C32×#8102
#117220342
#117220202
#105925163
#105904783