Finite Group Databasebeta

← #13147561

Subgroups — #13147561

Order 1728.

778 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊(C3⋊D9)C211yesmaximal, characteristic
2C9×#869C211yesmaximal, characteristic, complemented
3C3×C4×Dic18C211yesmaximal, characteristic, complemented
4#11721000C212yesmaximal, complemented
5#11720992C212yesmaximal
6#11720986C214yesmaximal, complemented
7#11720936C212yesmaximal, complemented
8#11720935C212yesmaximal, complemented
9#10593958—31nomaximal
10#10589832—31nomaximal
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C4×C3⋊D9C2212yes—
13C2×C9×Dic6C2212yes—
14C2×C3×Dic18C2211yescharacteristic
15C2×(C3⋊D9)⋊C4C2214yes—
16C3×C42×C9C2211yescharacteristic
17#88536C2212yes—
18C4⋊(C3⋊Dic9)C2211yescharacteristic
19C9×C4⋊Dic3C2212yes—
20C9×C4⋊Dic3C2212yes—
21C3×C4⋊Dic9C2211yescharacteristic
22C3×Dic9⋊C4C2212yes—
23C3×C4×Dic9C2212yes—
24#88445C2214yes—
25Dic3⋊Dic9C2214yes—
26#88442C2214yes—
27C4×C4⋊(C3⋊D3)—31no—
28C3×#869D311yescharacteristic
29C3×C4×Dic6—31no—
30#63900—32no—
31#63894—32no—
32#63888—34no—
33#63871—32no—
34#63869—32no—
35C9×C4⋊Q8—31no—
36#63508—34no—
37#63459—32no—
38#63455—32no—
39C4⋊C4×D9—32no—
40#63452—32no—
41C4×D36—31no—
42C4×D36—61no—
43C4×Dic18D311yescharacteristic, complemented
44C22×C3⋊D9C2312yes—
45C2×C3×C4×C9C2311yescharacteristic
46C2×C3×C4×C9C2311yescharacteristic
47C2×C3×C4×C9C2311yescharacteristic
48C2×C3⋊Dic9C2312yes—
49C4⋊(C3⋊D9)Q814yescomplemented
50C4×C3⋊D9—24no—
51C2×C9×Dic3C2314yes—
52C2×C3×Dic9C2312yes—
53C2×C3×Dic9C2312yes—
54C9×Dic6—42no—
55C3×Dic18—22no—
56(C3⋊D9)⋊C4—44no—
57#5730—91no—
58#5579—91no—
59C2×C4⋊(C3⋊D3)—31no—
60C2×C4×C3⋊D3—32no—
61C2×C3×Dic6—31no—
62C2×C3×Dic6C2×D312yes—
63C2×#624—34no—
64C32×C42C2×D311yescharacteristic
65#3752—32no—
66C4⋊(C3⋊Dic3)—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3C2×D312yes—
69C3×C4⋊Dic3C2×D312yes—
70C3×Dic3⋊C4—32no—
71C3×C4×Dic3—32no—
72#3724—34no—
73Dic3⋊Dic3—34no—
74#3722—34no—
75C2×Q8×C9—32no—
76C2×D36—31no—
77C2×D36—61no—
78C2×C4×D9—32no—
79C2×C4×D9—34no—
80C2×C4×D9—62no—
81C2×Dic18C2×D311yescharacteristic
82C9×C4⋊C4—32no—
83C9×C4⋊C4—32no—
84C42×C9—21no—
85C42×C9C2×D311yescharacteristic
86(C2×D9)⋊C4—32no—
87(C2×D9)⋊C4—34no—
88(C2×D9)⋊C4—62no—
89C4⋊Dic9—31no—
90C4⋊Dic9—34no—
91C4⋊Dic9—61no—
92C4⋊Dic9C2×D311yescharacteristic
93Dic9⋊C4—34no—
94Dic9⋊C4C2×D312yes—
95C4×Dic9C2×D312yes—
96C22×C3×C9C2411yescharacteristic
97C2×C3⋊D9—22no—
98C2×C3⋊D9—22no—
99C2×C3⋊D9C2×Q814yes—
100C3×C4×C9—21no—
101C3×C4×C9C2×Q812yes—
102C3×C4×C9#4112yes—
103C3⋊Dic9—22no—
104C9×Dic3—24no—
105C3×Dic9—22no—
106C3×Dic9—22no—
107C3×C4⋊Q8—31no—
108C3×C4×Q8—91no—
109#946—94no—
110#946—92no—
111#897—92no—
112#896—94no—
113#893—92no—
114D3×C4⋊C4—92no—
115D3×C4⋊C4—92no—
116#890—92no—
117C4×D12—91no—
118C4×D12—91no—
119C4×D12—92no—
120C4×D12—181no—
121#870—92no—
122#869D911yescharacteristic, complemented
123C4×Dic6—31no—
124C22×C3⋊D3—32no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—32no—
129C4⋊(C3⋊D3)—34no—
130C4×C3⋊D3—64no—
131C2×C3×Dic3—32no—
132C2×C3×Dic3—32no—
133C2×C3×Dic3C22×D314yes—
134C3×Dic6—42no—
135C3×Dic6—62no—
136#624—124no—
137C22×D9—32no—
138C22×D9—62no—
139Q8×C9—122no—
140C2×C4×C9—21no—
141C2×C4×C9—21no—
142C2×C4×C9—21no—
143C2×C4×C9—34no—
144C2×C4×C9C22×D311yescharacteristic
145C2×C4×C9C22×D311yescharacteristic
146C2×C4×C9C22×D311yescharacteristic
147C2×Dic9—32no—
148C2×Dic9—62no—
149C2×Dic9C22×D312yes—
150C2×Dic9C22×D312yes—
151D36—34no—
152D36—64no—
153C4×D9—64no—
154C4×D9—124no—
155C4×D9—124no—
156Dic18—22no—
157#554—271noSylow
158C2×C3×C9C22×Q811yescharacteristic
159C2×C3×C9C2×#4111yescharacteristic
160C2×C3×C9H(5,𝔽2)11yescharacteristic
161C3⋊D9—24no—
162C2×C3×Q8—32no—
163C2×C3×Q8—91no—
164C2×D12—91no—
165C2×D12—91no—
166C2×D12—181no—
167C2×C4×D3—92no—
168C2×C4×D3—92no—
169C2×C4×D3—94no—
170C2×C4×D3—94no—
171C2×C4×D3—182no—
172C2×Dic6—31no—
173C2×Dic6C2×D912yes—
174C3×C4⋊C4—32no—
175C3×C4⋊C4—32no—
176C3×C4⋊C4—91no—
177C3×C4⋊C4—92no—
178C3×C42—21no—
179C3×C42—92no—
180C3×C42C2×D911yescharacteristic
181C3×C42D3211yescharacteristic
182(C2×D3)⋊C4—94no—
183(C2×D3)⋊C4—92no—
184(C2×D3)⋊C4—92no—
185(C2×D3)⋊C4—94no—
186(C2×D3)⋊C4—182no—
187C4⋊Dic3—31no—
188C4⋊Dic3—94no—
189C4⋊Dic3—91no—
190C4⋊Dic3—91no—
191C4⋊Dic3—181no—
192C4⋊Dic3C2×D912yes—
193C4⋊Dic3C2×D912yes—
194Dic3⋊C4—32no—
195Dic3⋊C4—94no—
196Dic3⋊C4—94no—
197Dic3⋊C4—94no—
198C4×Dic3—32no—
199C22×C32C23×D311yescharacteristic
200C2×C3⋊D3—34no—
201C2×C3⋊D3—62no—
202C2×C3⋊D3—62no—
203C32×C4—21no—
204C32×C4D3×Q812yes—
205C32×C4#23912yes—
206C3⋊Dic3—62no—
207C3×Dic3—24no—
208C3×Dic3—62no—
209C3×Dic3—62no—
210C22×C9—21no—
211C22×C9C23×D311yescharacteristic
212C2×D9—34no—
213C2×D9—64no—
214C2×D9—62no—
215C2×D9—62no—
216C2×D9—122no—
217C2×D9—122no—
218C4×C9—21no—
219C4×C9—22no—
220C4×C9—22no—
221C4×C9—41no—
222C4×C9—64no—
223C4×C9#23512yes—
224C4×C9D3×Q812yes—
225Dic9—22no—
226Dic9—22no—
227Dic9—62no—
228Dic9—122no—
229C4⋊Q8—31no—
230#125—272no—
231C22⋊Q8—272no—
232C22⋊Q8—274no—
233C4×Q8—91no—
234C4×D4—272no—
235C4×D4—271no—
236C2×C4⋊C4—272no—
237C3×C9#55411yescharacteristic, complemented, Sylow
238C22×D3—92no—
239C22×D3—92no—
240C22×D3—182no—
241C3×Q8—122no—
242C3×Q8—182no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—34no—
247C2×C3×C4—92no—
248C2×C3×C4—92no—
249C2×C3×C4C22×D911yescharacteristic
250C2×C3×C4C22×D911yescharacteristic
251C2×C3×C4C22×D911yescharacteristic
252C2×C3×C4C2×D3211yescharacteristic
253C2×C3×C4C2×D3211yescharacteristic
254C2×C3×C4C2×D3211yescharacteristic
255C2×Dic3—32no—
256C2×Dic3—32no—
257C2×Dic3—92no—
258C2×Dic3—92no—
259C2×Dic3—182no—
260C2×Dic3C22×D914yes—
261D12—94no—
262D12—94no—
263D12—184no—
264C4×D3—184no—
265C4×D3—184no—
266C4×D3—364no—
267C4×D3—364no—
268C4×D3—364no—
269Dic6—42no—
270Dic6—62no—
271C2×C32#100411yescharacteristic
272C2×C32C2×D3×Q811yescharacteristic
273C2×C32C2×#23911yescharacteristic
274C3⋊D3—64no—
275C2×C9—21no—
276C2×C9—21no—
277C2×C9—21no—
278C2×C9C2×#23511yescharacteristic
279C2×C9C2×D3×Q811yescharacteristic
280C2×C9#100911yescharacteristic
281D9—64no—
282D9—124no—
283C2×Q8—32no—
284C2×Q8—91no—
285C2×D4—271no—
286C22×C4—274no—
287C22×C4—272no—
288C4⋊C4—32no—
289C4⋊C4—32no—
290C4⋊C4—91no—
291C4⋊C4—92no—
292C4⋊C4—274no—
293C4⋊C4—274no—
294C4⋊C4—271no—
295C22⋊C4—274no—
296C22⋊C4—272no—
297C42—92no—
298C42D3×D911yescharacteristic
299C22×C3—21no—
300C22×C3C23×D911yescharacteristic
301C22×C3C22×D3211yescharacteristic
302C2×D3—94no—
303C2×D3—94no—
304C2×D3—184no—
305C2×D3—182no—
306C2×D3—182no—
307C2×D3—182no—
308C2×D3—182no—
309C2×D3—362no—
310C2×D3—362no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—22no—
314C3×C4—22no—
315C3×C4—41no—
316C3×C4—64no—
317C3×C4—182no—
318C3×C4—182no—
319C3×C4Q8×D912yes—
320C3×C4#370112yes—
321C3×C4#379712yes—
322C3×C4#379912yes—
323Dic3—24no—
324Dic3—62no—
325Dic3—62no—
326Dic3—182no—
327Dic3—182no—
328Dic3—362no—
329C32#573011yescharacteristic
330C9—21no—
331C9#557911yescharacteristic, complemented
332C23—272no—
333Q8—122no—
334Q8—182no—
335D4—274no—
336C2×C4—34no—
337C2×C4—92no—
338C2×C4—92no—
339C2×C4—272no—
340C2×C4—544no—
341C2×C4—1084no—
342C2×C4C2×D3×D911yescharacteristic
343C2×C4C2×D3×D911yescharacteristic
344C2×C4C2×D3×D911yescharacteristic
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3#6371011yescharacteristic
349C2×C3C2×Q8×D911yescharacteristic
350C2×C3C2×#370111yescharacteristic
351C2×C3C2×#379711yescharacteristic
352C2×C3C2×#379911yescharacteristic
353C2×C3#6430711yescharacteristic
354D3—184no—
355D3—184no—
356D3—364no—
357C22—274no—
358C22—542no—
359C22—542no—
360C22C22×D3×D911yescharacteristic
361C4—21no—
362C4—64no—
363C4—182no—
364C4—182no—
365C4—542no—
366C4#8863512yes—
367C4(C3⋊D9)⋊Q812yes—
368C3—21no—
369C3#1058983211yescharacteristic, complemented
370C3#1059395811yescharacteristic
371C2—544no—
372C2C2×#8863511yescharacteristic
373C2C2×(C3⋊D9)⋊Q811yescharacteristic
374C2#1172247611yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenterC42D3×D9
hypocenterC3×C9#554
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C9×#869C2semidirect
3. C3×C4×Dic18C2semidirect
4. #11721000C2semidirect
6. #11720986C2semidirect
7. #11720936C2semidirect
8. #11720935C2semidirect
43. C4×Dic18D3semidirect
49. C4⋊(C3⋊D9)Q8semidirect
122. #869D9semidirect
237. C3×C9#554semidirect
331. C9#5579semidirect
369. C3#10589832semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊(C3⋊D9)2
C9×#8692
C3×C4×Dic182
#117210002
#117209922
#117209862
#117209362
#117209352
#105939583
#105898323