Finite Group Databasebeta

← #13147567

Subgroups — #13147567

Order 1728.

664 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721712C211yesmaximal, characteristic
2C9×#870C211yesmaximal, characteristic, complemented
3C3×#63432C211yesmaximal, characteristic, complemented
4#11720997C212yesmaximal
5#11720995C212yesmaximal, complemented
6#11720994C212yesmaximal, complemented
7#11720921C212yesmaximal, complemented
8#11720920C212yesmaximal, complemented
9#11720905C212yesmaximal, complemented
10#10593857—31nomaximal
11#10589801—31nomaximal
12C2×C4⋊(C3⋊D9)C2211yescharacteristic
13C2×#6226C2211yescharacteristic
14C2×(C3⋊D9)⋊C4C2212yes—
15C2×#6189C2212yes—
16C3×C42×C9C2211yescharacteristic
17#88536C2212yes—
18#88536C2212yes—
19C9×C4⋊Dic3C2211yescharacteristic
20C9×C4⋊Dic3C2211yescharacteristic
21C9×Dic3⋊C4C2212yes—
22C9×Dic3⋊C4C2212yes—
23C3×C4⋊Dic9C2211yescharacteristic
24C3×C4⋊Dic9C2211yescharacteristic
25C3×Dic9⋊C4C2212yes—
26C3×Dic9⋊C4C2212yes—
27#88445C2212yes—
28#88445C2212yes—
29#88445C2212yes—
30Dic3⋊Dic9C2212yes—
31Dic9⋊Dic3C2212yes—
32Dic3×Dic9C2212yes—
33#64084—31no—
34C3×#870—31no—
35C3×#870D311yescharacteristic
36#63897—32no—
37#63895—32no—
38#63895—32no—
39#63848—32no—
40#63848—32no—
41#63840—32no—
42C9×#125—31no—
43#63460—32no—
44#63460—32no—
45#63459—32no—
46#63458—32no—
47#63456—32no—
48#63450—32no—
49#63437—31no—
50#63437—61no—
51#63432D311yescharacteristic, complemented
52C22×C3⋊D9C2312yes—
53C2×C3×C4×C9C2311yescharacteristic
54C2×C3×C4×C9C2311yescharacteristic
55C2×C3×C4×C9C2311yescharacteristic
56C2×C3⋊Dic9C2312yes—
57C4⋊(C3⋊D9)—41no—
58#6226—41no—
59C2×C9×Dic3C2312yes—
60C2×C9×Dic3C2312yes—
61C2×C3×Dic9C2312yes—
62C2×C3×Dic9C2312yes—
63(C3⋊D9)⋊C4—42no—
64#6189—42no—
65#5699—91no—
66#5699—91no—
67C2×C4⋊(C3⋊D3)—31no—
68C2×#634—31no—
69C2×#627—32no—
70C2×#624—32no—
71C32×C42C2×D311yescharacteristic
72#3752—32no—
73#3752—32no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—31no—
76C3×C4⋊Dic3C2×D311yescharacteristic
77C3×C4⋊Dic3C2×D311yescharacteristic
78C3×Dic3⋊C4—32no—
79C3×Dic3⋊C4—32no—
80C3×Dic3⋊C4C2×D312yes—
81C3×Dic3⋊C4C2×D312yes—
82Dic3⋊Dic3—32no—
83Dic3⋊Dic3—32no—
84#3722—32no—
85#3722—32no—
86#3722—32no—
87Dic32—32no—
88C2×D36—31no—
89C2×D36—61no—
90C2×C4×D9—32no—
91C2×Dic18—31no—
92C2×Dic18—32no—
93C2×Dic18—61no—
94C9×C4⋊C4—32no—
95C9×C4⋊C4—32no—
96C9×C4⋊C4—31no—
97C9×C4⋊C4—31no—
98C42×C9—21no—
99C42×C9C2×D311yescharacteristic
100(C2×D9)⋊C4—32no—
101(C2×D9)⋊C4—32no—
102(C2×D9)⋊C4—32no—
103(C2×D9)⋊C4—32no—
104(C2×D9)⋊C4—32no—
105(C2×D9)⋊C4—62no—
106(C2×D9)⋊C4—62no—
107C4⋊Dic9—32no—
108C4⋊Dic9C2×D311yescharacteristic
109C4⋊Dic9C2×D311yescharacteristic
110Dic9⋊C4—32no—
111Dic9⋊C4C2×D312yes—
112Dic9⋊C4C2×D312yes—
113C4×Dic9—32no—
114C22×C3×C9C2411yescharacteristic
115C2×C3⋊D9—22no—
116C2×C3⋊D9—22no—
117C2×C3⋊D9—22no—
118C3×C4×C9—21no—
119C3×C4×C9—21no—
120C3×C4×C9—21no—
121C3⋊Dic9—22no—
122C9×Dic3—22no—
123C9×Dic3—22no—
124C3×Dic9—22no—
125C3×Dic9—22no—
126C3×#125—31no—
127C3×#125—91no—
128#898—92no—
129#898—92no—
130#898—92no—
131#898—92no—
132#897—92no—
133#897—92no—
134#896—92no—
135#896—92no—
136#894—92no—
137#894—92no—
138#888—92no—
139#888—92no—
140#875—91no—
141#875—91no—
142#875—181no—
143#870—31no—
144#870D911yescharacteristic, complemented
145C22×C3⋊D3—32no—
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C22×D311yescharacteristic
149C2×C3⋊Dic3—32no—
150C4⋊(C3⋊D3)—121no—
151#634—121no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3C22×D312yes—
155C2×C3×Dic3C22×D312yes—
156#627—122no—
157#624—122no—
158C22×D9—32no—
159C22×D9—62no—
160C2×C4×C9—21no—
161C2×C4×C9—21no—
162C2×C4×C9—21no—
163C2×C4×C9—32no—
164C2×C4×C9—32no—
165C2×C4×C9C22×D311yescharacteristic
166C2×C4×C9C22×D311yescharacteristic
167C2×C4×C9C22×D311yescharacteristic
168C2×Dic9—32no—
169C2×Dic9—62no—
170C2×Dic9C22×D312yes—
171C2×Dic9C22×D312yes—
172D36—121no—
173D36—122no—
174C4×D9—122no—
175Dic18—122no—
176Dic18—121no—
177Dic18—122no—
178#563—271noSylow
179C2×C3×C9H(5,𝔽2)11yescharacteristic
180C2×C3×C92−1+411yescharacteristic
181C2×C3×C92−1+411yescharacteristic
182C3⋊D9—42no—
183C2×D12—91no—
184C2×D12—91no—
185C2×D12—181no—
186C2×C4×D3—92no—
187C2×C4×D3—92no—
188C2×Dic6—92no—
189C2×Dic6—91no—
190C2×Dic6—91no—
191C2×Dic6—92no—
192C2×Dic6—181no—
193C3×C4⋊C4—32no—
194C3×C4⋊C4—31no—
195C3×C4⋊C4—32no—
196C3×C4⋊C4—31no—
197C3×C4⋊C4—92no—
198C3×C4⋊C4—91no—
199C3×C4⋊C4—92no—
200C3×C4⋊C4—91no—
201C3×C42—21no—
202C3×C42C2×D911yescharacteristic
203C3×C42D3211yescharacteristic
204(C2×D3)⋊C4—92no—
205(C2×D3)⋊C4—92no—
206(C2×D3)⋊C4—92no—
207(C2×D3)⋊C4—92no—
208(C2×D3)⋊C4—92no—
209(C2×D3)⋊C4—92no—
210(C2×D3)⋊C4—92no—
211(C2×D3)⋊C4—92no—
212(C2×D3)⋊C4—92no—
213(C2×D3)⋊C4—92no—
214(C2×D3)⋊C4—182no—
215(C2×D3)⋊C4—182no—
216C4⋊Dic3—31no—
217C4⋊Dic3—31no—
218C4⋊Dic3—92no—
219C4⋊Dic3—92no—
220C4⋊Dic3C2×D911yescharacteristic
221C4⋊Dic3C2×D911yescharacteristic
222Dic3⋊C4—32no—
223Dic3⋊C4—32no—
224Dic3⋊C4—92no—
225Dic3⋊C4—92no—
226Dic3⋊C4C2×D912yes—
227Dic3⋊C4C2×D912yes—
228C4×Dic3—92no—
229C4×Dic3—92no—
230C22×C32C23×D311yescharacteristic
231C2×C3⋊D3—62no—
232C2×C3⋊D3—62no—
233C2×C3⋊D3—62no—
234C32×C4—21no—
235C32×C4—21no—
236C32×C4—21no—
237C3⋊Dic3—62no—
238C3×Dic3—22no—
239C3×Dic3—22no—
240C3×Dic3—62no—
241C3×Dic3—62no—
242C22×C9—21no—
243C22×C9C23×D311yescharacteristic
244C2×D9—62no—
245C2×D9—62no—
246C2×D9—62no—
247C2×D9—122no—
248C2×D9—122no—
249C2×D9—122no—
250C4×C9—22no—
251C4×C9—21no—
252C4×C9—21no—
253C4×C9—21no—
254C4×C9—41no—
255C4×C9—41no—
256C4×C9—62no—
257C4×C9—62no—
258Dic9—22no—
259Dic9—22no—
260Dic9—62no—
261Dic9—122no—
262C4⋊Q8—272no—
263#126—272no—
264#126—272no—
265#125—31no—
266#125—91no—
267#124—271no—
268#123—272no—
269C22⋊Q8—272no—
270C22⋊Q8—272no—
271C3×C9#56311yescharacteristic, complemented, Sylow
272C22×D3—92no—
273C22×D3—92no—
274C22×D3—182no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—32no—
279C2×C3×C4—32no—
280C2×C3×C4—92no—
281C2×C3×C4—92no—
282C2×C3×C4C22×D911yescharacteristic
283C2×C3×C4C22×D911yescharacteristic
284C2×C3×C4C22×D911yescharacteristic
285C2×C3×C4C2×D3211yescharacteristic
286C2×C3×C4C2×D3211yescharacteristic
287C2×C3×C4C2×D3211yescharacteristic
288C2×Dic3—32no—
289C2×Dic3—32no—
290C2×Dic3—92no—
291C2×Dic3—92no—
292C2×Dic3—182no—
293C2×Dic3C22×D912yes—
294C2×Dic3C22×D912yes—
295D12—361no—
296D12—361no—
297D12—362no—
298C4×D3—362no—
299C4×D3—362no—
300Dic6—362no—
301Dic6—362no—
302Dic6—361no—
303Dic6—361no—
304Dic6—362no—
305C2×C32#100711yescharacteristic
306C2×C32#100911yescharacteristic
307C2×C32#101011yescharacteristic
308C3⋊D3—122no—
309C2×C9—21no—
310C2×C9—21no—
311C2×C9—21no—
312C2×C9#100711yescharacteristic
313C2×C9#100911yescharacteristic
314C2×C9#101011yescharacteristic
315D9—122no—
316D9—242no—
317C2×Q8—272no—
318C2×Q8—271no—
319C2×D4—271no—
320C22×C4—272no—
321C4⋊C4—31no—
322C4⋊C4—32no—
323C4⋊C4—32no—
324C4⋊C4—31no—
325C4⋊C4—91no—
326C4⋊C4—92no—
327C4⋊C4—92no—
328C4⋊C4—91no—
329C4⋊C4—272no—
330C4⋊C4—272no—
331C22⋊C4—272no—
332C22⋊C4—272no—
333C22⋊C4—272no—
334C22⋊C4—272no—
335C22⋊C4—272no—
336C42—272no—
337C42D3×D911yescharacteristic
338C22×C3—21no—
339C22×C3C23×D911yescharacteristic
340C22×C3C22×D3211yescharacteristic
341C2×D3—182no—
342C2×D3—182no—
343C2×D3—182no—
344C2×D3—182no—
345C2×D3—182no—
346C2×D3—182no—
347C2×D3—362no—
348C2×D3—362no—
349C2×D3—362no—
350C3×C4—22no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—41no—
358C3×C4—41no—
359C3×C4—62no—
360C3×C4—62no—
361C3×C4—182no—
362C3×C4—182no—
363Dic3—22no—
364Dic3—22no—
365Dic3—62no—
366Dic3—62no—
367Dic3—182no—
368Dic3—182no—
369Dic3—362no—
370C32#569911yescharacteristic
371C9—21no—
372C9#569911yescharacteristic, complemented
373C23—272no—
374Q8—1081no—
375Q8—1082no—
376D4—1081no—
377C2×C4—32no—
378C2×C4—32no—
379C2×C4—92no—
380C2×C4—92no—
381C2×C4—272no—
382C2×C4—1082no—
383C2×C4C2×D3×D911yescharacteristic
384C2×C4C2×D3×D911yescharacteristic
385C2×C4C2×D3×D911yescharacteristic
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3#6371311yescharacteristic
390C2×C3#6371511yescharacteristic
391C2×C3#6371611yescharacteristic
392C2×C3#6429711yescharacteristic
393C2×C3#6429711yescharacteristic
394C2×C3#6430811yescharacteristic
395D3—362no—
396D3—362no—
397D3—722no—
398C22—542no—
399C22—542no—
400C22—542no—
401C22C22×D3×D911yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—62no—
406C4—62no—
407C4—182no—
408C4—182no—
409C4—542no—
410C3—21no—
411C3#1058980111yescharacteristic, complemented
412C3#1059385711yescharacteristic
413C2—1082no—
414C2#1172245811yescharacteristic
415C2#1172246411yescharacteristic
416C2#1172247811yescharacteristic
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#563
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×#870C2semidirect
3. C3×#63432C2semidirect
5. #11720995C2semidirect
6. #11720994C2semidirect
7. #11720921C2semidirect
8. #11720920C2semidirect
9. #11720905C2semidirect
51. #63432D3semidirect
144. #870D9semidirect
271. C3×C9#563semidirect
372. C9#5699semidirect
411. C3#10589801semidirect

Maximal subgroups

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

subgroupindex
#117217122
C9×#8702
C3×#634322
#117209972
#117209952
#117209942
#117209212
#117209202
#117209052
#105938573
#105898013