Finite Group Databasebeta

← #13159955

Subgroups — #13159955

Order 1728.

932 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 932 conjugacy classes of subgroups — too many to draw. Only its 170 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
1#11723494C211yesmaximal, characteristic
2Dic6×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#88795C211yesmaximal, characteristic
4#11722191C211yesmaximal, characteristic
5C3×C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
6C32×#816C211yesmaximal, characteristic, complemented
7#11722054C211yesmaximal, characteristic, complemented
8#10592487—31nomaximal
9#10590792—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C2×#6289—21no—
13C2×C33×C8C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18Dic3⋊(C3⋊Dic3)—21no—
19#88804—21no—
20Dic3×C3⋊Dic3—21no—
21#88795C414yescomplemented
22Q8×C3⋊Dic3—31no—
23C2×#3755—31no—
24#64102—31no—
25#63894—34no—
26Dic3×Dic6—34no—
27C2×#3717—34no—
28C32×#103—31no—
29#63648—31no—
30#63648—34no—
31#63648—64no—
32C8×C3⋊Dic3D311yescharacteristic, complemented
33#63635—31no—
34C3×#816D314yescomplemented
35C3×C8×Dic3—34no—
36#63573—34no—
37C22×C3⋊(C3⋊D3)—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C4⋊(C3⋊(C3⋊D3))—21no—
41C4⋊(C3⋊(C3⋊D3))C2×C412yes—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C2×C3×C3⋊Dic3D411yescharacteristic
44C2×C32×Dic3—21no—
45C2×C32×Dic3—21no—
46C32×Dic6—21no—
47C32×Dic6C2×C412yescomplemented
48#6289—42no—
49C33×C8—21no—
50C3×C3⋊(C3⋊C8)C2×C412yescomplemented
51#4788—94no—
52C4×#204—91no—
53C2×C32×Q8—31no—
54C2×C4⋊(C3⋊D3)—31no—
55C2×C4⋊(C3⋊D3)—34no—
56C2×C4⋊(C3⋊D3)—64no—
57C2×C4×C3⋊D3—61no—
58C2×C3×Dic6C2×D314yes—
59C2×#624—64no—
60C2×C32×C8—24no—
61C2×C32×C8C2×D314yes—
62C2×C32×C8C2×D311yescharacteristic
63C32×C4⋊C4—31no—
64#3755—34no—
65#3752—61no—
66C4⋊(C3⋊Dic3)—31no—
67C4⋊(C3⋊Dic3)—34no—
68C4⋊(C3⋊Dic3)—61no—
69C4⋊(C3⋊Dic3)—64no—
70C4×C3⋊Dic3—61no—
71C4×C3⋊Dic3C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×C4⋊Dic3C2×D314yes—
74C3×C4×Dic3—34no—
75C2×C3×C3⋊C8—34no—
76Dic3⋊Dic3—64no—
77#3722—64no—
78Dic32—64no—
79#3717—316no—
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)—41no—
87C3×C3⋊Dic3—21no—
88C3×C3⋊Dic3SD1612yescomplemented
89C32×Dic3—22no—
90C32×Dic3—41no—
91Q8×Dic3—94no—
92C2×#215—94no—
93C2×#204—91no—
94#893—94no—
95C4×D12—91no—
96C4×Dic6—91no—
97C3×#103—34no—
98C3×C4×C8—91no—
99#821—94no—
100#821—91no—
101#821—184no—
102#817—91no—
103#816C3⋊D311yescharacteristic, complemented
104C8×Dic3—34no—
105#808—94no—
106C22×C3⋊D3—61no—
107C22×C3⋊D3—64no—
108C22×C3⋊D3—124no—
109C32×Q8—32no—
110C32×Q8—61no—
111C2×C32×C4—24no—
112C2×C32×C4—61no—
113C2×C32×C4—61no—
114C2×C32×C4C22×D314yes—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—61no—
117C2×C3⋊Dic3—64no—
118C2×C3⋊Dic3—124no—
119C2×C3⋊Dic3D1211yescharacteristic
120C2×C3⋊Dic3D1211yescharacteristic
121C4⋊(C3⋊D3)—32no—
122C4⋊(C3⋊D3)—38no—
123C4⋊(C3⋊D3)—68no—
124C4⋊(C3⋊D3)—61no—
125C4⋊(C3⋊D3)—64no—
126C4⋊(C3⋊D3)—124no—
127C4×C3⋊D3—122no—
128C2×C3×Dic3—24no—
129C2×C3×Dic3—24no—
130C2×C3×Dic3—34no—
131C2×C3×Dic3—34no—
132C3×Dic6—24no—
133C3×Dic6C4×D318yescomplemented
134#624—128no—
135C32×C8—21no—
136C32×C8—24no—
137C32×C8—44no—
138C3⋊(C3⋊C8)C4×D312yescomplemented
139C3×C3⋊C8—38no—
140C4×SD16—271noSylow
141C2×C33C4×D411yescharacteristic
142C2×C33C2×SD1611yescharacteristic
143C2×C33#13511yescharacteristic
144C3⋊(C3⋊D3)—42no—
145C2×C3×Q8—34no—
146C2×D12—94no—
147C2×D12—91no—
148C2×D12—184no—
149C2×C4×D3—181no—
150C2×C4×D3—184no—
151C2×Dic6C2×C3⋊D311yescharacteristic
152C2×C3×C8—24no—
153C2×C3×C8—91no—
154C2×C3×C8D3214yes—
155C2×C3×C8C2×C3⋊D311yescharacteristic
156C3×C4⋊C4—34no—
157C3×C42—91no—
158#215—916no—
159(C2×D3)⋊C4—181no—
160(C2×D3)⋊C4—184no—
161C4⋊Dic3—94no—
162C4⋊Dic3—91no—
163C4⋊Dic3—184no—
164C4⋊Dic3—184no—
165C4⋊Dic3C2×C3⋊D311yescharacteristic
166Dic3⋊C4—181no—
167C4×Dic3—34no—
168C4×Dic3—181no—
169C4×Dic3—184no—
170C2×C3⋊C8—34no—
171#204—94no—
172C22×C32—24no—
173C22×C32C2×D1211yescharacteristic
174C22×C32D3×D414yes—
175C2×C3⋊D3—62no—
176C2×C3⋊D3—68no—
177C2×C3⋊D3—128no—
178C2×C3⋊D3—121no—
179C2×C3⋊D3—121no—
180C2×C3⋊D3—124no—
181C2×C3⋊D3—124no—
182C2×C3⋊D3—244no—
183C2×C3⋊D3—244no—
184C32×C4—24no—
185C32×C4—24no—
186C32×C4—62no—
187C32×C4—121no—
188C32×C4C2×C4×D314yes—
189C32×C4C2×C4×D311yescharacteristic
190C32×C4#23511yescharacteristic
191C32×C4#23714yes—
192C3⋊Dic3—21no—
193C3⋊Dic3—121no—
194C3⋊Dic3—124no—
195C3⋊Dic3—244no—
196C3⋊Dic3#20412yescomplemented
197C3×Dic3—28no—
198C3×Dic3—38no—
199C3×Dic3—44no—
200C3×Dic3—64no—
201C2×SD16—271no—
202C4×Q8—271no—
203C4×D4—271no—
204#106—271no—
205#103—31no—
206#102—271no—
207C4×C8—91no—
208C33C4×SD1611yescharacteristic, complemented, Sylow
209C22×D3—184no—
210C22×D3—181no—
211C22×D3—364no—
212C3×Q8—38no—
213C3×Q8—64no—
214C2×C3×C4—24no—
215C2×C3×C4—64no—
216C2×C3×C4—64no—
217C2×C3×C4—91no—
218C2×C3×C4—91no—
219C2×C3×C4C2×D3214yes—
220C2×C3×C4C22×C3⋊D311yescharacteristic
221C2×Dic3—21no—
222C2×Dic3—21no—
223C2×Dic3—34no—
224C2×Dic3—34no—
225C2×Dic3—184no—
226C2×Dic3—181no—
227C2×Dic3—364no—
228D12—98no—
229D12—92no—
230D12—188no—
231D12—184no—
232D12—181no—
233D12—364no—
234C4×D3—368no—
235C4×D3—362no—
236Dic6—21no—
237Dic6C4×C3⋊D312yescomplemented
238C3×C8—24no—
239C3×C8—21no—
240C3×C8—44no—
241C3×C8—92no—
242C3⋊C8—38no—
243C2×C32—24no—
244C2×C32—24no—
245C2×C32—24no—
246C2×C32C4×D1211yescharacteristic
247C2×C32#67⋊C414yes—
248C2×C32C2×#20411yescharacteristic
249C2×C32#90411yescharacteristic
250C2×C32D3×SD1614yes—
251C2×C32#91914yes—
252C3⋊D3—122no—
253C3⋊D3—128no—
254C3⋊D3—248no—
255C2×Q8—31no—
256C2×D4—271no—
257C22×C4—541no—
258SD16—274no—
259C2×C8—91no—
260C2×C8D3×C3⋊D311yescharacteristic
261C4⋊C4—31no—
262C4⋊C4—271no—
263C4⋊C4—541no—
264C22⋊C4—541no—
265C42—91no—
266C42—541no—
267C22×C3—24no—
268C22×C3D3×D1214yes—
269C22×C3D4×C3⋊D311yescharacteristic
270C2×D3—182no—
271C2×D3—188no—
272C2×D3—368no—
273C2×D3—364no—
274C2×D3—364no—
275C2×D3—361no—
276C2×D3—361no—
277C2×D3—724no—
278C2×D3—724no—
279C3×C4—24no—
280C3×C4—24no—
281C3×C4—68no—
282C3×C4—92no—
283C3×C4—124no—
284C3×C4—181no—
285C3×C4C4×D3214yes—
286C3×C4#380414yes—
287C3×C4C2×C4×C3⋊D311yescharacteristic
288C3×C4#383011yescharacteristic
289Dic3—22no—
290Dic3—38no—
291Dic3—41no—
292Dic3—64no—
293Dic3—364no—
294Dic3—361no—
295Dic3—724no—
296C32—24no—
297C32C4×#20411yescharacteristic, complemented
298C32#478814yescomplemented
299C23—541no—
300Q8—32no—
301Q8—61no—
302D4—272no—
303D4—541no—
304C2×C4—61no—
305C2×C4—61no—
306C2×C4—91no—
307C2×C4—91no—
308C2×C4—541no—
309C2×C4—1082no—
310C2×C4C2×D3×C3⋊D311yescharacteristic
311C8—21no—
312C8—92no—
313C2×C3—24no—
314C2×C3—24no—
315C2×C3—24no—
316C2×C3D3×#20414yes—
317C2×C3#6380814yes—
318C2×C3#6388014yes—
319C2×C3#6409011yescharacteristic
320C2×C3SD16×C3⋊D311yescharacteristic
321C2×C3#6412811yescharacteristic
322D3—368no—
323D3—362no—
324D3—728no—
325C22—542no—
326C22—1081no—
327C22—1081no—
328C22C3⋊D3×D1211yescharacteristic
329C4—62no—
330C4—92no—
331C4—121no—
332C4—181no—
333C4—1081no—
334C4C4×D3×C3⋊D311yescharacteristic
335C4#8902811yescharacteristic
336C3—24no—
337C3#1059079214yescomplemented
338C3#1059248711yescharacteristic, complemented
339C2—1082no—
340C2C3⋊D3×#20411yescharacteristic
341C2#1172334111yescharacteristic
342C2#1172346111yescharacteristic
subgroupG/H
centerC22C3⋊D3×D12
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33C4×SD16
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
2. Dic6×C3⋊Dic3C2semidirect
5. C3×C8×C3⋊Dic3C2semidirect
6. C32×#816C2semidirect
7. #11722054C2semidirect
21. #88795C4semidirect
32. C8×C3⋊Dic3D3semidirect
34. C3×#816D3semidirect
47. C32×Dic6C2×C4semidirect
50. C3×C3⋊(C3⋊C8)C2×C4semidirect
88. C3×C3⋊Dic3SD16semidirect
103. #816C3⋊D3semidirect
133. C3×Dic6C4×D3semidirect
138. C3⋊(C3⋊C8)C4×D3semidirect
196. C3⋊Dic3#204semidirect
208. C33C4×SD16semidirect
237. Dic6C4×C3⋊D3semidirect
297. C32C4×#204semidirect
298. C32#4788semidirect
337. C3#10590792semidirect
338. C3#10592487semidirect

Maximal subgroups

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

subgroupindex
#117234942
Dic6×C3⋊Dic32
C2×#887952
#117221912
C3×C8×C3⋊Dic32
C32×#8162
#117220542
#105924873
#105907923