Finite Group Databasebeta

← #13161961

Subgroups — #13161961

Order 1728.

702 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 99 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×#64149C211yesmaximal, characteristic
2#11723680C211yesmaximal, characteristic, complemented
3C3×#63659C212yesmaximal, complemented
4#11722075C212yesmaximal, complemented
5#11722071C211yesmaximal, characteristic, complemented
6#10591480—31nomaximal
7#10591477—32nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×C3×#638—22no—
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2212yes—
14#88815—22no—
15#64149D311yescharacteristic
16C3×#940—32no—
17C3×#940—61no—
18#63872—31no—
19#63869—32no—
20#63659D312yescomplemented
21C3×#832—32no—
22C3×#832—32no—
23C3×#832—62no—
24#63564—32no—
25#63563—32no—
26#63562—32no—
27#63555—31no—
28#63554—32no—
29C23×C33—21no—
30C22×C3×C3⋊D3—21no—
31C33×D4—22no—
32C2×C33×C4C2311yescharacteristic
33C3×#638—44no—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3D412yes—
36C3×C4⋊(C3⋊D3)—22no—
37C3×C3⋊(C3⋊C8)—22no—
38#5071—91no—
39#4764—92no—
40C2×C32×D4—22no—
41C2×C32×D4—21no—
42C2×C32×D4—41no—
43C2×C32×D4C2×D312yes—
44C2×C32×D4C2×D311yescharacteristic
45C2×#638—22no—
46C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
47C2×C3×#67—62no—
48C2×C3×#67—62no—
49C2×C3×#67—64no—
50C2×C3×D12—32no—
51C2×C3×D12—61no—
52C4⋊(C3⋊Dic3)C2×D312yes—
53C4×C3⋊Dic3C2×D311yescharacteristic
54C2×C3⋊(C3⋊C8)C2×D312yes—
55C3×C4⋊Dic3—32no—
56C3×C4⋊Dic3—32no—
57C3×C4⋊Dic3—62no—
58C3×C4×Dic3—32no—
59C3×C4×Dic3—61no—
60C2×C3×C3⋊C8—32no—
61C2×C3×C3⋊C8—32no—
62C2×C3×C3⋊C8—62no—
63#3724—62no—
64Dic3⋊Dic3—62no—
65Dic3⋊Dic3—62no—
66C22×C33—41no—
67C22×C33—42no—
68C22×C33C2×D411yescharacteristic
69C2×C3×C3⋊D3—41no—
70C2×C3×C3⋊D3—42no—
71C33×C4#4112yes—
72C3×C3⋊Dic3—22no—
73C3×C3⋊Dic3—42no—
74C3×#127—91no—
75#940—32no—
76#940—61no—
77#890—92no—
78#870—91no—
79C3×#102—92no—
80#832—32no—
81#832—32no—
82#832—62no—
83#832—92no—
84#821—92no—
85#815—92no—
86#809—92no—
87#803—91no—
88C23×C32—21no—
89C23×C32—22no—
90C23×C32—22no—
91C23×C32—42no—
92C23×C32—42no—
93C22×C3⋊D3—21no—
94C22×C3×D3—62no—
95C22×C3×D3—62no—
96C32×D4—22no—
97C32×D4—24no—
98C32×D4—22no—
99C32×D4—22no—
100C32×D4—42no—
101C32×D4—42no—
102C32×D4—44no—
103C2×C32×C4—22no—
104C2×C32×C4—21no—
105C2×C32×C4—41no—
106C2×C32×C4C22×D312yes—
107C2×C32×C4C22×D311yescharacteristic
108#638—44no—
109C2×C3⋊Dic3—22no—
110C2×C3⋊Dic3#6712yes—
111C4⋊(C3⋊D3)—22no—
112C3×#67—128no—
113C3×#67—124no—
114C3×#67—124no—
115C2×C3×Dic3—32no—
116C2×C3×Dic3—32no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—122no—
121C3×D12—62no—
122C3×D12—62no—
123C3×D12—122no—
124C3⋊(C3⋊C8)—22no—
125C3×C3⋊C8—62no—
126C3×C3⋊C8—62no—
127C3×C3⋊C8—122no—
128#490—271noSylow
129C2×C33—81no—
130C2×C33#12411yescharacteristic
131C2×C33AGL(1,ℤ/8)12yes—
132C3×C3⋊D3—81no—
133C2×C3×D4—22no—
134C2×C3×D4—21no—
135C2×C3×D4—41no—
136C2×C3×D4—91no—
137C2×C3×D4—182no—
138C2×C3×D4D3211yescharacteristic
139C2×C3×D4D3212yes—
140C2×#67—62no—
141C2×#67—62no—
142C2×#67—64no—
143C2×D12—32no—
144C2×D12—61no—
145C2×C3×C8—92no—
146C3×C4⋊C4—92no—
147C3×C42—91no—
148C4⋊Dic3—32no—
149C4⋊Dic3—32no—
150C4⋊Dic3—62no—
151C4⋊Dic3—182no—
152Dic3⋊C4—182no—
153Dic3⋊C4—182no—
154C4×Dic3—32no—
155C4×Dic3—61no—
156C2×C3⋊C8—32no—
157C2×C3⋊C8—32no—
158C2×C3⋊C8—62no—
159C22×C32—22no—
160C22×C32—21no—
161C22×C32—41no—
162C22×C32—44no—
163C22×C32—42no—
164C22×C32—44no—
165C22×C32—42no—
166C22×C32—42no—
167C22×C32—41no—
168C22×C32—44no—
169C22×C32—42no—
170C22×C32—42no—
171C22×C32—82no—
172C22×C32—82no—
173C22×C32—84no—
174C22×C32D3×D412yes—
175C22×C32C2×#6711yescharacteristic
176C2×C3⋊D3—42no—
177C2×C3⋊D3—41no—
178C2×C3×D3—122no—
179C2×C3×D3—124no—
180C2×C3×D3—122no—
181C2×C3×D3—122no—
182C2×C3×D3—122no—
183C32×C4—22no—
184C32×C4—22no—
185C32×C4—22no—
186C32×C4—42no—
187C32×C4#23512yes—
188C32×C4#23712yes—
189C32×C4#23712yes—
190C3⋊Dic3—22no—
191C3⋊Dic3—42no—
192C3×Dic3—64no—
193C3×Dic3—62no—
194C3×Dic3—62no—
195C3×Dic3—122no—
196C3×Dic3—122no—
197C3×Dic3—242no—
198#127—91no—
199#125—271no—
200#102—92no—
201#102—272no—
202#97—271no—
203C33#49011yescharacteristic, complemented, Sylow
204C23×C3—22no—
205C23×C3—22no—
206C23×C3—21no—
207C23×C3—42no—
208C23×C3—42no—
209C23×C3—181no—
210C22×D3—62no—
211C22×D3—62no—
212C3×D4—24no—
213C3×D4—22no—
214C3×D4—22no—
215C3×D4—22no—
216C3×D4—42no—
217C3×D4—42no—
218C3×D4—44no—
219C3×D4—182no—
220C3×D4—364no—
221C2×C3×C4—22no—
222C2×C3×C4—21no—
223C2×C3×C4—41no—
224C2×C3×C4—92no—
225C2×C3×C4—182no—
226C2×C3×C4C2×D3211yescharacteristic
227C2×C3×C4C2×D3212yes—
228#67—128no—
229#67—124no—
230#67—124no—
231C2×Dic3—32no—
232C2×Dic3—32no—
233C2×Dic3—62no—
234C2×Dic3—62no—
235C2×Dic3—62no—
236C2×Dic3—122no—
237D12—62no—
238D12—62no—
239D12—122no—
240C3×C8—182no—
241C3⋊C8—62no—
242C3⋊C8—62no—
243C3⋊C8—122no—
244C2×C32—22no—
245C2×C32—22no—
246C2×C32—22no—
247C2×C32—21no—
248C2×C32—22no—
249C2×C32—41no—
250C2×C32—42no—
251C2×C32—82no—
252C2×C32—84no—
253C2×C32—84no—
254C2×C32—82no—
255C2×C32—81no—
256C2×C32#88512yes—
257C2×C32#91112yes—
258C2×C32#91412yes—
259C2×C32#93212yes—
260C2×C32#93611yescharacteristic
261C3⋊D3—81no—
262C3×D3—242no—
263C3×D3—242no—
264C2×D4—91no—
265C2×D4—182no—
266C2×D4#111311yescharacteristic
267C2×C8—92no—
268C4⋊C4—92no—
269C4⋊C4—542no—
270C42—91no—
271C22×C3—21no—
272C22×C3—22no—
273C22×C3—44no—
274C22×C3—44no—
275C22×C3—42no—
276C22×C3—44no—
277C22×C3—42no—
278C22×C3—42no—
279C22×C3—42no—
280C22×C3—41no—
281C22×C3—42no—
282C22×C3—41no—
283C22×C3—82no—
284C22×C3—82no—
285C22×C3—84no—
286C22×C3—362no—
287C22×C3—361no—
288C22×C3D3×#6712yes—
289C22×C3#381111yescharacteristic
290C2×D3—122no—
291C2×D3—124no—
292C2×D3—122no—
293C2×D3—122no—
294C2×D3—122no—
295C3×C4—22no—
296C3×C4—22no—
297C3×C4—22no—
298C3×C4—42no—
299C3×C4—182no—
300C3×C4—362no—
301C3×C4#380412yes—
302C3×C4#380412yes—
303C3×C4#380512yes—
304Dic3—64no—
305Dic3—62no—
306Dic3—62no—
307Dic3—122no—
308Dic3—122no—
309Dic3—242no—
310C32—22no—
311C32—21no—
312C32—41no—
313C32#476412yescomplemented
314C32#507111yescharacteristic, complemented
315C23—21no—
316C23—181no—
317D4—22no—
318D4—182no—
319D4—364no—
320C2×C4—92no—
321C2×C4—182no—
322C2×C4C2×#111311yescharacteristic
323C8—182no—
324C2×C3—22no—
325C2×C3—22no—
326C2×C3—22no—
327C2×C3—21no—
328C2×C3—22no—
329C2×C3—42no—
330C2×C3—41no—
331C2×C3—82no—
332C2×C3—84no—
333C2×C3—84no—
334C2×C3—82no—
335C2×C3—81no—
336C2×C3—721no—
337C2×C3#6393212yes—
338C2×C3#6393412yes—
339C2×C3#6393712yes—
340C2×C3#6395311yescharacteristic
341C2×C3#6395912yes—
342D3—242no—
343D3—242no—
344C22—42no—
345C22—41no—
346C22—362no—
347C22—361no—
348C22#8904911yescharacteristic
349C4—182no—
350C4—362no—
351C4#8904612yes—
352C3—21no—
353C3—22no—
354C3—41no—
355C3#1059147712yescomplemented
356C3#1059148011yescharacteristic, complemented
357C2—81no—
358C2—721no—
359C2#1172370612yes—
360C2#1172371811yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC3×#64149C2
hypercenterC2×D4#1113
hypocenterC33#490
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11723680C2semidirect
3. C3×#63659C2semidirect
4. #11722075C2semidirect
5. #11722071C2semidirect
20. #63659D3semidirect
203. C33#490semidirect
313. C32#4764semidirect
314. C32#5071semidirect
355. C3#10591477semidirect
356. C3#10591480semidirect

Maximal subgroups

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

subgroupindex
C3×#641492
#117236802
C3×#636592
#117220752
#117220712
#105914803
#105914773