Finite Group Databasebeta

← #13156534

Subgroups — #13156534

Order 1728.

614 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 98 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#11723440C211yesmaximal, characteristic
2C3×#63872C211yesmaximal, characteristic, complemented
3#11722070C211yesmaximal, characteristic, complemented
4#11722035C212yesmaximal
5C3×#63565C212yesmaximal, complemented
6#10591539—31nomaximal
7#10590537—32nomaximal
8C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
9C2×C32×Dic6C2211yescharacteristic
10C3×C4×C3⋊Dic3C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2212yes—
12C32×C4⋊Dic3C2212yes—
13#88804—22no—
14C3×Dic3⋊Dic3—22no—
15#64156—31no—
16C3×#890—32no—
17#63872D311yescharacteristic, complemented
18#63864—32no—
19#63864—61no—
20#63636—32no—
21C3×#835—32no—
22C3×#810—32no—
23#63565D312yescomplemented
24#63564—32no—
25#63564—32no—
26#63564—62no—
27#63553—31no—
28Dic3×C3⋊C8—32no—
29C22×C3⋊(C3⋊D3)—21no—
30C2×C33×C4C2311yescharacteristic
31C4⋊(C3⋊(C3⋊D3))—22no—
32C2×C3×C3⋊Dic3D412yes—
33C2×C32×Dic3—21no—
34C2×C32×Dic3—22no—
35C32×Dic6—22no—
36C3×C3⋊(C3⋊C8)—22no—
37#5068—91no—
38#4800—92no—
39C2×C32×Q8—31no—
40C2×C4⋊(C3⋊D3)—31no—
41C2×C4⋊(C3⋊D3)—32no—
42C2×C4⋊(C3⋊D3)—62no—
43C2×C4⋊(C3⋊D3)—61no—
44C2×C4⋊(C3⋊D3)—121no—
45C2×C3×Dic6—21no—
46C2×C3×Dic6C2×D312yes—
47C32×C4⋊C4—32no—
48#3752—62no—
49C4×C3⋊Dic3C2×D311yescharacteristic
50C2×C3⋊(C3⋊C8)C2×D312yes—
51C3×C4⋊Dic3—22no—
52C3×C4⋊Dic3—62no—
53C3×C4⋊Dic3C2×D312yes—
54C3×C4⋊Dic3C2×D312yes—
55C3×Dic3⋊C4—62no—
56C3×C4×Dic3—32no—
57C3×C4×Dic3—61no—
58C2×C3×C3⋊C8—32no—
59C2×C3×C3⋊C8—32no—
60C2×C3×C3⋊C8—62no—
61Dic3⋊Dic3—22no—
62#3722—62no—
63#3722—62no—
64#3722—64no—
65C22×C33C2×D411yescharacteristic
66C2×C3⋊(C3⋊D3)—41no—
67C2×C3⋊(C3⋊D3)—42no—
68C33×C4#4112yes—
69C3×C3⋊Dic3—22no—
70C32×Dic3—41no—
71C32×Dic3—42no—
72C3×#125—91no—
73#947—92no—
74#947—181no—
75#890—32no—
76#875—91no—
77C3×#103—92no—
78#835—32no—
79#821—92no—
80C8×Dic3—92no—
81#810—32no—
82#809—92no—
83#809—92no—
84#809—182no—
85C4×C3⋊C8—91no—
86C22×C3⋊D3—61no—
87C22×C3⋊D3—62no—
88C22×C3⋊D3—62no—
89C22×C3⋊D3—122no—
90C22×C3⋊D3—122no—
91C32×Q8—62no—
92C2×C32×C4—22no—
93C2×C32×C4—21no—
94C2×C32×C4—41no—
95C2×C32×C4—61no—
96C2×C32×C4—62no—
97C2×C32×C4C22×D312yes—
98C2×C32×C4C22×D311yescharacteristic
99C2×C3⋊Dic3#6712yes—
100C4⋊(C3⋊D3)—62no—
101C4⋊(C3⋊D3)—64no—
102C4⋊(C3⋊D3)—62no—
103C4⋊(C3⋊D3)—62no—
104C4⋊(C3⋊D3)—122no—
105C4⋊(C3⋊D3)—122no—
106C4⋊(C3⋊D3)—124no—
107C2×C3×Dic3—22no—
108C2×C3×Dic3—22no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—22no—
111C2×C3×Dic3—32no—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—42no—
114C2×C3×Dic3—62no—
115C3×Dic6—22no—
116C3×Dic6—22no—
117C3×Dic6—42no—
118C3⋊(C3⋊C8)—22no—
119C3×C3⋊C8—62no—
120C3×C3⋊C8—62no—
121C3×C3⋊C8—122no—
122#488—271noSylow
123C2×C33#12411yescharacteristic
124C2×C33#13512yes—
125C3⋊(C3⋊D3)—81no—
126C2×C3×Q8—32no—
127C2×C3×Q8—61no—
128C2×D12—91no—
129C2×D12—92no—
130C2×D12—182no—
131C2×D12—181no—
132C2×D12—361no—
133C2×Dic6D3211yescharacteristic
134C2×C3×C8—92no—
135C3×C4⋊C4—32no—
136C3×C4⋊C4—32no—
137C3×C4⋊C4—62no—
138C3×C4⋊C4—182no—
139C3×C42—91no—
140(C2×D3)⋊C4—182no—
141(C2×D3)⋊C4—182no—
142(C2×D3)⋊C4—182no—
143(C2×D3)⋊C4—184no—
144C4⋊Dic3—62no—
145C4⋊Dic3D3212yes—
146Dic3⋊C4—62no—
147C4×Dic3—32no—
148C4×Dic3—61no—
149C2×C3⋊C8—32no—
150C2×C3⋊C8—32no—
151C2×C3⋊C8—62no—
152C22×C32—22no—
153C22×C32—21no—
154C22×C32—41no—
155C22×C32D3×D412yes—
156C22×C32C2×#6711yescharacteristic
157C2×C3⋊D3—124no—
158C2×C3⋊D3—122no—
159C2×C3⋊D3—124no—
160C2×C3⋊D3—122no—
161C2×C3⋊D3—121no—
162C2×C3⋊D3—122no—
163C2×C3⋊D3—124no—
164C2×C3⋊D3—122no—
165C2×C3⋊D3—122no—
166C2×C3⋊D3—242no—
167C2×C3⋊D3—242no—
168C2×C3⋊D3—244no—
169C32×C4—22no—
170C32×C4—22no—
171C32×C4—22no—
172C32×C4—42no—
173C32×C4—121no—
174C32×C4—122no—
175C32×C4#23512yes—
176C32×C4#23712yes—
177C32×C4#23912yes—
178C3⋊Dic3—22no—
179C3×Dic3—42no—
180C3×Dic3—42no—
181C3×Dic3—42no—
182C3×Dic3—42no—
183C3×Dic3—62no—
184C3×Dic3—62no—
185C3×Dic3—82no—
186C3×Dic3—122no—
187#125—91no—
188#124—271no—
189#103—92no—
190#102—272no—
191C4×C8—271no—
192C33#48811yescharacteristic, complemented, Sylow
193C22×D3—182no—
194C22×D3—182no—
195C22×D3—181no—
196C22×D3—362no—
197C22×D3—362no—
198C3×Q8—62no—
199C3×Q8—62no—
200C3×Q8—122no—
201C2×C3×C4—22no—
202C2×C3×C4—21no—
203C2×C3×C4—41no—
204C2×C3×C4—62no—
205C2×C3×C4—62no—
206C2×C3×C4—62no—
207C2×C3×C4—62no—
208C2×C3×C4—92no—
209C2×C3×C4—122no—
210C2×C3×C4C2×D3212yes—
211C2×C3×C4C2×D3211yescharacteristic
212C2×Dic3—22no—
213C2×Dic3—21no—
214C2×Dic3—32no—
215C2×Dic3—32no—
216C2×Dic3—62no—
217D12—184no—
218D12—182no—
219D12—182no—
220D12—182no—
221D12—362no—
222D12—362no—
223D12—364no—
224Dic6—22no—
225C3×C8—182no—
226C3⋊C8—62no—
227C3⋊C8—62no—
228C3⋊C8—122no—
229C2×C32—22no—
230C2×C32—22no—
231C2×C32—22no—
232C2×C32—22no—
233C2×C32—21no—
234C2×C32—41no—
235C2×C32—42no—
236C2×C32#88512yes—
237C2×C32#91612yes—
238C2×C32#91912yes—
239C2×C32#94711yescharacteristic
240C2×C32#95012yes—
241C3⋊D3—242no—
242C3⋊D3—244no—
243C3⋊D3—244no—
244C3⋊D3—242no—
245C3⋊D3—241no—
246C2×Q8—31no—
247C2×D4—271no—
248C2×C8—92no—
249C4⋊C4—32no—
250C4⋊C4—182no—
251C22⋊C4—542no—
252C42—91no—
253C22×C3—21no—
254C22×C3—22no—
255C22×C3—41no—
256C22×C3D3×#6712yes—
257C22×C3#381111yescharacteristic
258C2×D3—364no—
259C2×D3—362no—
260C2×D3—364no—
261C2×D3—364no—
262C2×D3—362no—
263C2×D3—362no—
264C2×D3—362no—
265C2×D3—362no—
266C2×D3—361no—
267C2×D3—722no—
268C2×D3—722no—
269C2×D3—724no—
270C3×C4—22no—
271C3×C4—22no—
272C3×C4—22no—
273C3×C4—42no—
274C3×C4—122no—
275C3×C4—122no—
276C3×C4—122no—
277C3×C4—122no—
278C3×C4—182no—
279C3×C4—242no—
280C3×C4#379612yes—
281C3×C4#379912yes—
282C3×C4#380412yes—
283Dic3—42no—
284Dic3—41no—
285Dic3—62no—
286Dic3—62no—
287Dic3—122no—
288C32—21no—
289C32—22no—
290C32—41no—
291C32#480012yescomplemented
292C32#506811yescharacteristic, complemented
293C23—541no—
294Q8—62no—
295D4—542no—
296C2×C4—61no—
297C2×C4—62no—
298C2×C4—92no—
299C2×C4D3311yescharacteristic
300C8—182no—
301C2×C3—22no—
302C2×C3—22no—
303C2×C3—22no—
304C2×C3—21no—
305C2×C3—22no—
306C2×C3—42no—
307C2×C3—41no—
308C2×C3#6390712yes—
309C2×C3#6393512yes—
310C2×C3#6394912yes—
311C2×C3#6395012yes—
312C2×C3#6395311yescharacteristic
313D3—722no—
314D3—724no—
315D3—724no—
316D3—721no—
317D3—722no—
318C22—1082no—
319C22—1081no—
320C22#8895211yescharacteristic
321C4—122no—
322C4—121no—
323C4—182no—
324C4#8896212yes—
325C3—22no—
326C3—21no—
327C3—41no—
328C3#1059053712yescomplemented
329C3#1059153911yescharacteristic, complemented
330C2—2161no—
331C2#1172285812yes—
332C2#1172300511yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypocenterC33#488
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#63872C2semidirect
3. #11722070C2semidirect
5. C3×#63565C2semidirect
17. #63872D3semidirect
23. #63565D3semidirect
192. C33#488semidirect
291. C32#4800semidirect
292. C32#5068semidirect
328. C3#10590537semidirect
329. C3#10591539semidirect

Maximal subgroups

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

subgroupindex
#117234402
C3×#638722
#117220702
#117220352
C3×#635652
#105915393
#105905373