Finite Group Databasebeta

← #13159952

Subgroups — #13159952

Order 1728.

920 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 174 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#11723484C211yesmaximal, characteristic
2Dic3×#634C211yesmaximal, characteristic, complemented
3C2×#88797C211yesmaximal, characteristic
4#11722191C211yesmaximal, characteristic
5C3×#63643C211yesmaximal, characteristic, complemented
6C32×#815C211yesmaximal, characteristic, complemented
7#11722055C211yesmaximal, characteristic, complemented
8#10592409—31nomaximal
9#10590795—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×#6289—21no—
13C2×C33×C8C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88807—21no—
19#88804—21no—
20Dic3×C3⋊Dic3—21no—
21#88797C414yescomplemented
22C2×#3744—31no—
23C4×C4⋊(C3⋊D3)—31no—
24C4×#634—31no—
25#63894—34no—
26Dic3×Dic6—34no—
27C2×#3717—34no—
28C32×#97—31no—
29#63648—31no—
30#63648—34no—
31#63648—64no—
32#63645—31no—
33#63643D311yescharacteristic, complemented
34C3×#816—34no—
35C3×#815D314yescomplemented
36#63575—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⋊Dic3—21no—
43C2×C3×C3⋊Dic3—21no—
44C3×#634—21no—
45C3×#634C2×C412yescomplemented
46C2×C32×Dic3D411yescharacteristic
47C2×C32×Dic3D411yescharacteristic
48#6289—42no—
49C33×C8—21no—
50C32×C3⋊C8C2×C412yescomplemented
51#4817—91no—
52#4710—94no—
53C2×C4⋊(C3⋊D3)—31no—
54C2×C4⋊(C3⋊D3)—34no—
55C2×C4⋊(C3⋊D3)—64no—
56C2×C4×C3⋊D3—61no—
57C2×#634C2×D311yescharacteristic
58C2×C3×Dic6—34no—
59C2×#624—64no—
60C2×C32×C8—24no—
61C2×C32×C8—31no—
62C2×C32×C8C2×D314yes—
63C2×C32×C8C2×D311yescharacteristic
64C32×C42—31no—
65#3752—61no—
66C4⋊(C3⋊Dic3)—31no—
67C4⋊(C3⋊Dic3)—34no—
68C4⋊(C3⋊Dic3)—64no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70#3750—61no—
71C4×C3⋊Dic3—61no—
72#3744—34no—
73C3×C4⋊Dic3—34no—
74C3×C4×Dic3C2×D314yes—
75C2×C3×C3⋊C8C2×D314yes—
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—22no—
88C3×C3⋊Dic3—41no—
89C32×Dic3—21no—
90C32×Dic3—21no—
91Q8×Dic3—91no—
92C2×#215—91no—
93C2×#204—94no—
94#893—91no—
95C4×D12—94no—
96C4×Dic6—94no—
97C3×#103—91no—
98C3×#97—34no—
99#821—94no—
100#821—91no—
101#821—184no—
102#818—94no—
103#816—34no—
104#815C3⋊D311yescharacteristic, complemented
105#807—91no—
106C22×C3⋊D3—61no—
107C22×C3⋊D3—64no—
108C22×C3⋊D3—124no—
109C2×C32×C4—24no—
110C2×C32×C4—31no—
111C2×C32×C4—31no—
112C2×C32×C4C22×D314yes—
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3—61no—
117C2×C3⋊Dic3—64no—
118C2×C3⋊Dic3—124no—
119C4⋊(C3⋊D3)—32no—
120C4⋊(C3⋊D3)—38no—
121C4⋊(C3⋊D3)—68no—
122C4⋊(C3⋊D3)—61no—
123C4⋊(C3⋊D3)—64no—
124C4⋊(C3⋊D3)—124no—
125C4×C3⋊D3—122no—
126#634—21no—
127#634C4×D312yescomplemented
128C2×C3×Dic3—64no—
129C2×C3×Dic3—64no—
130C2×C3×Dic3D1214yes—
131C2×C3×Dic3D1214yes—
132C3×Dic6—38no—
133C3×Dic6—64no—
134#624—128no—
135C32×C8—21no—
136C32×C8—24no—
137C32×C8—32no—
138C32×C8—44no—
139C3×C3⋊C8C4×D318yescomplemented
140#440—271noSylow
141C2×C33C4×D411yescharacteristic
142C2×C33AGL(1,ℤ/8)11yescharacteristic
143C2×C33#13711yescharacteristic
144C3⋊(C3⋊D3)—42no—
145C2×C3×Q8—91no—
146C2×D12—94no—
147C2×D12—91no—
148C2×D12—184no—
149C2×C4×D3—184no—
150C2×C4×D3—181no—
151C2×Dic6—34no—
152C2×C3×C8—24no—
153C2×C3×C8—34no—
154C2×C3×C8D3214yes—
155C2×C3×C8C2×C3⋊D311yescharacteristic
156C3×C4⋊C4—91no—
157C3×C42—34no—
158#215—94no—
159(C2×D3)⋊C4—184no—
160(C2×D3)⋊C4—181no—
161C4⋊Dic3—34no—
162C4⋊Dic3—94no—
163C4⋊Dic3—91no—
164C4⋊Dic3—184no—
165C4⋊Dic3—181no—
166Dic3⋊C4—184no—
167C4×Dic3—184no—
168C4×Dic3—181no—
169C4×Dic3C2×C3⋊D311yescharacteristic
170C2×C3⋊C8C2×C3⋊D311yescharacteristic
171#204—916no—
172C22×C32—24no—
173C22×C32C2×D1214yes—
174C22×C32D3×D411yescharacteristic
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—61no—
187C32×C4—61no—
188C32×C4C2×C4×D314yes—
189C32×C4C2×C4×D311yescharacteristic
190C32×C4#23514yes—
191C32×C4#23711yescharacteristic
192C3⋊Dic3—22no—
193C3⋊Dic3—41no—
194C3⋊Dic3—121no—
195C3⋊Dic3—124no—
196C3⋊Dic3—244no—
197C3×Dic3—24no—
198C3×Dic3—24no—
199C3×Dic3—68no—
200C3×Dic3—124no—
201C2×SD16—271no—
202C4×Q8—271no—
203C4×D4—271no—
204#107—271no—
205#103—91no—
206#102—271no—
207#97—31no—
208C33#44011yescharacteristic, complemented, Sylow
209C22×D3—184no—
210C22×D3—181no—
211C22×D3—364no—
212C3×Q8—92no—
213C3×Q8—181no—
214C2×C3×C4—24no—
215C2×C3×C4—34no—
216C2×C3×C4—34no—
217C2×C3×C4—181no—
218C2×C3×C4—181no—
219C2×C3×C4C2×D3214yes—
220C2×C3×C4C22×C3⋊D311yescharacteristic
221C2×Dic3—64no—
222C2×Dic3—64no—
223C2×Dic3—184no—
224C2×Dic3—181no—
225C2×Dic3—364no—
226C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
227C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
228D12—98no—
229D12—92no—
230D12—188no—
231D12—184no—
232D12—181no—
233D12—364no—
234C4×D3—362no—
235C4×D3—368no—
236Dic6—38no—
237Dic6—64no—
238C3×C8—24no—
239C3×C8—21no—
240C3×C8—38no—
241C3×C8—44no—
242C3⋊C8C4×C3⋊D312yescomplemented
243C2×C32—24no—
244C2×C32—24no—
245C2×C32—24no—
246C2×C32C4×D1214yes—
247C2×C32#67⋊C411yescharacteristic
248C2×C32#90814yes—
249C2×C32#90914yes—
250C2×C32#91411yescharacteristic
251C2×C32#91811yescharacteristic
252C3⋊D3—122no—
253C3⋊D3—128no—
254C3⋊D3—248no—
255C2×Q8—91no—
256C2×D4—271no—
257C22×C4—541no—
258SD16—274no—
259C2×C8—31no—
260C2×C8D3×C3⋊D311yescharacteristic
261C4⋊C4—91no—
262C4⋊C4—271no—
263C4⋊C4—541no—
264C22⋊C4—541no—
265C42—31no—
266C42—541no—
267C22×C3—24no—
268C22×C3D3×D1214yes—
269C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
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—64no—
282C3×C4—64no—
283C3×C4—182no—
284C3×C4—361no—
285C3×C4C4×D3214yes—
286C3×C4#380414yes—
287C3×C4C2×C4×C3⋊D311yescharacteristic
288C3×C4#382811yescharacteristic
289Dic3—21no—
290Dic3—21no—
291Dic3—68no—
292Dic3—124no—
293Dic3—364no—
294Dic3—361no—
295Dic3—724no—
296C32—24no—
297C32#471014yescomplemented
298C32#481711yescharacteristic, complemented
299C23—541no—
300Q8—92no—
301Q8—181no—
302D4—272no—
303D4—541no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—181no—
307C2×C4—181no—
308C2×C4—541no—
309C2×C4—1082no—
310C2×C4C2×D3×C3⋊D311yescharacteristic
311C8—21no—
312C8—32no—
313C2×C3—24no—
314C2×C3—24no—
315C2×C3—24no—
316C2×C3#6379414yes—
317C2×C3#6380114yes—
318C2×C3#6388014yes—
319C2×C3C4×C4⋊(C3⋊D3)11yescharacteristic
320C2×C3#6411711yescharacteristic
321C2×C3#6411811yescharacteristic
322D3—368no—
323D3—362no—
324D3—728no—
325C22—542no—
326C22—1081no—
327C22—1081no—
328C22D3×C4⋊(C3⋊D3)11yescharacteristic
329C4—61no—
330C4—61no—
331C4—182no—
332C4—361no—
333C4—1081no—
334C4C4×D3×C3⋊D311yescharacteristic
335C4#8903111yescharacteristic
336C3—24no—
337C3#1059079514yescomplemented
338C3#1059240911yescharacteristic, complemented
339C2—1082no—
340C2#1172331511yescharacteristic
341C2#1172332711yescharacteristic
342C2#1172347111yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
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
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
2. Dic3×#634C2semidirect
5. C3×#63643C2semidirect
6. C32×#815C2semidirect
7. #11722055C2semidirect
21. #88797C4semidirect
33. #63643D3semidirect
35. C3×#815D3semidirect
45. C3×#634C2×C4semidirect
50. C32×C3⋊C8C2×C4semidirect
104. #815C3⋊D3semidirect
127. #634C4×D3semidirect
139. C3×C3⋊C8C4×D3semidirect
208. C33#440semidirect
242. C3⋊C8C4×C3⋊D3semidirect
297. C32#4710semidirect
298. C32#4817semidirect
337. C3#10590795semidirect
338. C3#10592409semidirect

Maximal subgroups

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

subgroupindex
#117234842
Dic3×#6342
C2×#887972
#117221912
C3×#636432
C32×#8152
#117220552
#105924093
#105907953