Finite Group Databasebeta

← #13159945

Subgroups — #13159945

Order 1728.

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

normal subgroups only

This group has 908 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
1#11723501C211yesmaximal, characteristic
2#11723397C211yesmaximal, characteristic, complemented
3C2×#88793C211yesmaximal, characteristic
4C8×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
5C3×#63648C211yesmaximal, characteristic
6C32×#816C211yesmaximal, characteristic, complemented
7#11722053C211yesmaximal, characteristic, complemented
8#10592487—31nomaximal
9#10590742—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C2×Dic3×C3⋊D3—21no—
13C2×C33×C8C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18Dic3⋊(C3⋊Dic3)—21no—
19#88803—21no—
20Dic3×C3⋊Dic3—21no—
21#88793C414yescomplemented
22Q8×C3⋊Dic3—31no—
23C2×#3755—31no—
24#64102—31no—
25#63898—34no—
26#63843—34no—
27C2×#3715—34no—
28C32×#103—31no—
29#63648D311yescharacteristic
30C8×C3⋊Dic3—31no—
31C8×C3⋊Dic3—34no—
32C8×C3⋊Dic3—64no—
33#63635—31no—
34C3×#821—34no—
35C3×#816D314yescomplemented
36#63574—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—
44C2×C32×Dic3—21no—
45C2×C32×Dic3—21no—
46C32×Dic6—21no—
47C32×Dic6C2×C412yescomplemented
48Dic3×C3⋊D3—42no—
49C33×C8—21no—
50C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
51#4788—94no—
52#4758—91no—
53C2×C32×Q8—31no—
54C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
55C2×C4×C3⋊D3—61no—
56C2×C3×D12—34no—
57C2×C3×Dic6C2×D314yes—
58C2×D3×Dic3—64no—
59C2×C32×C8—24no—
60C2×C32×C8C2×D314yes—
61C2×C32×C8C2×D311yescharacteristic
62C32×C4⋊C4—31no—
63#3755—34no—
64#3752—61no—
65C4⋊(C3⋊Dic3)—61no—
66C4⋊(C3⋊Dic3)C2×D311yescharacteristic
67C4×C3⋊Dic3—31no—
68C4×C3⋊Dic3—34no—
69C4×C3⋊Dic3—61no—
70C4×C3⋊Dic3—64no—
71C2×C3⋊(C3⋊C8)—31no—
72C2×C3⋊(C3⋊C8)—34no—
73C2×C3⋊(C3⋊C8)—64no—
74C3×C4⋊Dic3—34no—
75C3×C4⋊Dic3C2×D314yes—
76Dic3⋊Dic3—64no—
77(C2×D3)⋊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×C4C22×C411yescharacteristic
85C33×C4#4111yescharacteristic
86C3⋊(C3⋊Dic3)—21no—
87C3⋊(C3⋊Dic3)SD1612yescomplemented
88C3×C3⋊Dic3—41no—
89C32×Dic3—22no—
90C32×Dic3—41no—
91Q8×Dic3—94no—
92C2×#215—94no—
93D4×Dic3—91no—
94C2×#214—91no—
95#893—94no—
96#887—91no—
97C3×#103—34no—
98C3×#102—91no—
99#821—34no—
100#816C3⋊D311yescharacteristic, complemented
101C8×Dic3—94no—
102C8×Dic3—91no—
103C8×Dic3—184no—
104#808—94no—
105#808—91no—
106C22×C3⋊D3—21no—
107C22×C3×D3—64no—
108C32×Q8—32no—
109C32×Q8—61no—
110C2×C32×C4—24no—
111C2×C32×C4—61no—
112C2×C32×C4—61no—
113C2×C32×C4C22×D314yes—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3—31no—
117C2×C3⋊Dic3—31no—
118C2×C3⋊Dic3—34no—
119C2×C3⋊Dic3—34no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3—64no—
122C4⋊(C3⋊D3)—21no—
123C4⋊(C3⋊D3)C4×D312yes—
124C4×C3⋊D3—122no—
125C2×C3×Dic3—24no—
126C2×C3×Dic3—24no—
127C2×C3×Dic3—64no—
128C3×D12—38no—
129C3×D12—64no—
130C3×Dic6—24no—
131C3×Dic6C4×D318yescomplemented
132D3×Dic3—128no—
133C32×C8—28no—
134C32×C8—24no—
135C32×C8—21no—
136C3⋊(C3⋊C8)—32no—
137C3⋊(C3⋊C8)—38no—
138C3⋊(C3⋊C8)—68no—
139C4×SD16—271noSylow
140C2×C33C4×D411yescharacteristic
141C2×C33C2×SD1611yescharacteristic
142C2×C33#13511yescharacteristic
143C3×C3⋊D3—42no—
144C2×C3×Q8—34no—
145C2×C3×D4—91no—
146C22×Dic3—181no—
147C2×D12—34no—
148C2×C4×D3—184no—
149C2×Dic6C2×C3⋊D311yescharacteristic
150C2×C3×C8—24no—
151C2×C3×C8D3214yes—
152C2×C3×C8C2×C3⋊D311yescharacteristic
153C3×C4⋊C4—34no—
154C3×C4⋊C4—91no—
155#217—181no—
156#215—916no—
157#214—94no—
158(C2×D3)⋊C4—184no—
159C4⋊Dic3—34no—
160C4⋊Dic3—184no—
161C4⋊Dic3C2×C3⋊D311yescharacteristic
162Dic3⋊C4—181no—
163C4×Dic3—94no—
164C4×Dic3—91no—
165C4×Dic3—184no—
166C4×Dic3—181no—
167C4×Dic3—184no—
168C2×C3⋊C8—94no—
169C2×C3⋊C8—91no—
170C2×C3⋊C8—184no—
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—62no—
183C32×C4—121no—
184C32×C4C2×C4×D314yes—
185C32×C4C2×C4×D311yescharacteristic
186C32×C4#23714yes—
187C32×C4#23711yescharacteristic
188C3⋊Dic3—32no—
189C3⋊Dic3—38no—
190C3⋊Dic3—41no—
191C3⋊Dic3—68no—
192C3⋊Dic3—68no—
193C3⋊Dic3—61no—
194C3⋊Dic3—64no—
195C3×Dic3—28no—
196C3×Dic3—44no—
197C3×Dic3—124no—
198C2×SD16—271no—
199C4×Q8—271no—
200C4×D4—271no—
201#106—271no—
202#103—31no—
203#102—91no—
204C4×C8—271no—
205C33C4×SD1611yescharacteristic, complemented, Sylow
206C23×C3—181no—
207C22×D3—64no—
208C3×Q8—38no—
209C3×Q8—64no—
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—91no—
223C2×Dic3—94no—
224C2×Dic3—91no—
225C2×Dic3—184no—
226C2×Dic3—184no—
227C2×Dic3—362no—
228D12—38no—
229D12—64no—
230C4×D3—368no—
231Dic6—21no—
232Dic6C4×C3⋊D312yescomplemented
233C3×C8—28no—
234C3×C8—24no—
235C3×C8—21no—
236C3⋊C8—98no—
237C3⋊C8—92no—
238C3⋊C8—188no—
239C2×C32—24no—
240C2×C32—24no—
241C2×C32—24no—
242C2×C32#67⋊C414yes—
243C2×C32#67⋊C411yescharacteristic
244C2×C32#91211yescharacteristic
245C2×C32D3×SD1614yes—
246C2×C32D3×SD1611yescharacteristic
247C2×C32#91914yes—
248C3⋊D3—42no—
249C3×D3—128no—
250C2×Q8—31no—
251C2×D4—91no—
252C22×C4—541no—
253SD16—274no—
254C2×C8—271no—
255C2×C8D3×C3⋊D311yescharacteristic
256C4⋊C4—31no—
257C4⋊C4—91no—
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—68no—
274C3×C4—124no—
275C3×C4—361no—
276C3×C4C4×D3214yes—
277C3×C4#380514yes—
278C3×C4C2×C4×C3⋊D311yescharacteristic
279C3×C4#383011yescharacteristic
280Dic3—22no—
281Dic3—41no—
282Dic3—98no—
283Dic3—92no—
284Dic3—124no—
285Dic3—188no—
286Dic3—188no—
287Dic3—184no—
288Dic3—181no—
289C32—24no—
290C32#475811yescharacteristic, complemented
291C32#478814yescomplemented
292C23—181no—
293Q8—32no—
294Q8—61no—
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—21no—
305C8—272no—
306C2×C3—24no—
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—362no—
310C2×C3#6379614yes—
311C2×C3#6381014yes—
312C2×C3#6390214yes—
313C2×C3#6409011yescharacteristic
314C2×C3SD16×C3⋊D311yescharacteristic
315C2×C3#6412811yescharacteristic
316D3—128no—
317C22—182no—
318C22—361no—
319C22—361no—
320C22#8902611yescharacteristic
321C4—62no—
322C4—121no—
323C4—272no—
324C4—361no—
325C4—541no—
326C4C4×D3×C3⋊D311yescharacteristic
327C4#8902911yescharacteristic
328C3—24no—
329C3#1059074214yescomplemented
330C3#1059248711yescharacteristic, complemented
331C2—362no—
332C2#1172332111yescharacteristic
333C2#1172334511yescharacteristic
334C2#1172350811yescharacteristic
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
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. #11723397C2semidirect
4. C8×C3⋊(C3⋊Dic3)C2semidirect
6. C32×#816C2semidirect
7. #11722053C2semidirect
21. #88793C4semidirect
35. C3×#816D3semidirect
47. C32×Dic6C2×C4semidirect
50. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
87. C3⋊(C3⋊Dic3)SD16semidirect
100. #816C3⋊D3semidirect
131. C3×Dic6C4×D3semidirect
205. C33C4×SD16semidirect
232. Dic6C4×C3⋊D3semidirect
290. C32#4758semidirect
291. C32#4788semidirect
329. C3#10590742semidirect
330. C3#10592487semidirect

Maximal subgroups

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

subgroupindex
#117235012
#117233972
C2×#887932
C8×C3⋊(C3⋊Dic3)2
C3×#636482
C32×#8162
#117220532
#105924873
#105907423