Finite Group Databasebeta

← #13147595

Subgroups — #13147595

Order 1728.

878 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88641C211yesmaximal, characteristic
2C2×D3×Dic18C211yesmaximal, characteristic, complemented
3#11721706C211yesmaximal, characteristic, complemented
4C9×#872C211yesmaximal, characteristic, complemented
5C3×#63437C211yesmaximal, characteristic, complemented
6#11721021C214yesmaximal, complemented
7#11720991C214yesmaximal, complemented
8#11720948C211yesmaximal, characteristic, complemented
9#11720932C211yesmaximal, characteristic, complemented
10#10593894—31nomaximal
11#10589628—31nomaximal
12C2×#6226C2211yescharacteristic
13C2×#6226C2211yescharacteristic
14C2×C4×D3×C9C2211yescharacteristic
15C2×C3×D36C2211yescharacteristic
16C2×C3×Dic18C2211yescharacteristic
17C2×#6194C2212yes—
18C2×D3×Dic9C2212yes—
19C2×Dic3×D9C2212yes—
20C2×#6189C2212yes—
21#88641—24no—
22D3×Dic18—24no—
23C3×C42×C9C2211yescharacteristic
24C4⋊(C3⋊Dic9)C2214yes—
25C9×(C2×D3)⋊C4C2212yes—
26C3×(C2×D9)⋊C4C2214yes—
27C9×Dic3⋊C4C2212yes—
28C4×C9×Dic3C2211yescharacteristic
29(C2×D9)⋊Dic3C2214yes—
30Dic3⋊Dic9C2214yes—
31C2×#3795—31no—
32C2×D3×Dic6—31no—
33#64078—31no—
34C3×#875—31no—
35C3×#872D311yescharacteristic
36#63893—34no—
37#63891—34no—
38#63866—31no—
39#63860—31no—
40C2×#3697—31no—
41C22×Dic18—31no—
42C9×#117—31no—
43#63459—34no—
44#63448—34no—
45#63437—31no—
46#63437D311yescharacteristic, complemented
47#63431—31no—
48#63431—31no—
49#63431—61no—
50C22×D3×C9C2311yescharacteristic
51C22×C3×D9C2312yes—
52C2×C3×C4×C9C2312yes—
53C2×C3×C4×C9C2311yescharacteristic
54C2×C3⋊Dic9C2312yes—
55C2×C3⋊Dic9C2312yes—
56#6226—22no—
57#6226—22no—
58C2×C9×Dic3C2312yes—
59C2×C9×Dic3C2311yescharacteristic
60C2×C3×Dic9C2312yes—
61C4×D3×C9D414yes—
62C3×D36—22no—
63C3×Dic18—22no—
64#6194—24no—
65D3×Dic9—24no—
66Dic3×D9—24no—
67#6189—24no—
68#5675—91no—
69#5526—91no—
70C2×#634—31no—
71C2×#634—31no—
72C2×C3×D12—31no—
73C2×C3×C4×D3C2×D311yescharacteristic
74C2×C3×Dic6—31no—
75C2×#627—32no—
76C2×#625—32no—
77C2×D3×Dic3—32no—
78C2×D3×Dic3—32no—
79#3795—64no—
80D3×Dic6—64no—
81C32×C42C2×D311yescharacteristic
82C4⋊(C3⋊Dic3)—34no—
83C3×(C2×D3)⋊C4—34no—
84C3×(C2×D3)⋊C4C2×D312yes—
85C3×Dic3⋊C4C2×D312yes—
86C3×C4×Dic3C2×D311yescharacteristic
87Dic3⋊Dic3—34no—
88(C2×D3)⋊Dic3—34no—
89C22×C4×C9—31no—
90C2×#611—32no—
91C22×Dic9—32no—
92#3697—64no—
93C2×D36C2×D311yescharacteristic
94C2×C4×D9—32no—
95C2×Dic18—31no—
96C2×Dic18—31no—
97C2×Dic18—32no—
98C2×Dic18—64no—
99C2×Dic18—61no—
100C2×Dic18—61no—
101C2×Dic18C2×D311yescharacteristic
102C9×C4⋊C4—32no—
103C9×C22⋊C4—32no—
104C42×C9—21no—
105C42×C9—31no—
106C42×C9C2×D311yescharacteristic
107(C2×D9)⋊C4—34no—
108(C2×D9)⋊C4C2×D314yes—
109C4⋊Dic9—34no—
110C4⋊Dic9—34no—
111C4⋊Dic9—64no—
112C22×C3×C9C2411yescharacteristic
113C2×D3×C9—22no—
114C2×D3×C9C2×D412yes—
115C2×C3×D9—22no—
116C2×C3×D9—24no—
117C3×C4×C9—22no—
118C3×C4×C9C2×D412yes—
119C3⋊Dic9—22no—
120C3⋊Dic9—22no—
121C9×Dic3—22no—
122C9×Dic3C2×D412yes—
123C3×Dic9—22no—
124C2×D3×Q8—91no—
125C2×#237—91no—
126C2×#235—91no—
127C22×Dic6—91no—
128C3×#124—91no—
129C3×#117—31no—
130#944—91no—
131#936—91no—
132#897—94no—
133#886—94no—
134#883—94no—
135#878—94no—
136#875—31no—
137#875—91no—
138#872D911yescharacteristic, complemented
139#869—91no—
140#869—91no—
141#869—91no—
142#869—181no—
143C22×C3×D3—32no—
144C22×C3×D3C22×D311yescharacteristic
145C2×C32×C4C22×D312yes—
146C2×C32×C4C22×D311yescharacteristic
147C2×C3⋊Dic3—32no—
148C2×C3⋊Dic3—32no—
149#634—62no—
150#634—62no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3C22×D312yes—
153C2×C3×Dic3C22×D311yescharacteristic
154C3×D12—62no—
155C3×C4×D3D1214yes—
156C3×Dic6—62no—
157#627—64no—
158#625—64no—
159D3×Dic3—64no—
160D3×Dic3—64no—
161C23×C9—31no—
162C22×D9C22×D312yes—
163C2×C4×C9—22no—
164C2×C4×C9—21no—
165C2×C4×C9—31no—
166C2×C4×C9—34no—
167C2×C4×C9—32no—
168C2×C4×C9C22×D312yes—
169C2×C4×C9C22×D311yescharacteristic
170#611—64no—
171C2×Dic9—32no—
172C2×Dic9—32no—
173C2×Dic9—62no—
174C2×Dic9—62no—
175C2×Dic9—64no—
176C2×Dic9C22×D312yes—
177D36—22no—
178C4×D9—64no—
179Dic18—22no—
180Dic18—64no—
181Dic18—62no—
182Dic18—62no—
183Dic18—122no—
184Dic18—122no—
185#536—271noSylow
186C2×C3×C9C22×D411yescharacteristic
187C2×C3×C92−1+412yes—
188D3×C9—22no—
189C3×D9—42no—
190C2×C3×Q8—91no—
191C2×C3×D4—91no—
192C22×C3×C4—31no—
193C2×#67—92no—
194C2×#67—92no—
195C22×Dic3—92no—
196C22×Dic3—92no—
197D3×Q8—184no—
198#237—184no—
199#235—184no—
200C2×D12—31no—
201C2×C4×D3—92no—
202C2×C4×D3—92no—
203C2×C4×D3C2×D911yescharacteristic
204C2×Dic6—31no—
205C2×Dic6—92no—
206C2×Dic6—91no—
207C2×Dic6—91no—
208C2×Dic6—91no—
209C2×Dic6—91no—
210C2×Dic6—92no—
211C2×Dic6—181no—
212C2×Dic6—181no—
213C2×Dic6—184no—
214C3×C4⋊C4—32no—
215C3×C22⋊C4—32no—
216C3×C22⋊C4—94no—
217C3×C42—21no—
218C3×C42—31no—
219C3×C42C2×D911yescharacteristic
220C3×C42D3211yescharacteristic
221#217—94no—
222(C2×D3)⋊C4—34no—
223(C2×D3)⋊C4—94no—
224(C2×D3)⋊C4C2×D912yes—
225C4⋊Dic3—94no—
226C4⋊Dic3—94no—
227C4⋊Dic3—94no—
228C4⋊Dic3—184no—
229Dic3⋊C4—94no—
230Dic3⋊C4C2×D912yes—
231C4×Dic3C2×D911yescharacteristic
232C22×C32C23×D311yescharacteristic
233C2×C3×D3—22no—
234C2×C3×D3—62no—
235C2×C3×D3—64no—
236C2×C3×D3C2×D1212yes—
237C32×C4—22no—
238C32×C4C2×D1212yes—
239C3⋊Dic3—62no—
240C3⋊Dic3—62no—
241C3×Dic3—22no—
242C3×Dic3—62no—
243C3×Dic3C2×D1212yes—
244C22×C9—21no—
245C22×C9—32no—
246C22×C9—62no—
247C22×C9C23×D311yescharacteristic
248C2×D9—22no—
249C2×D9—24no—
250C4×C9—24no—
251C4×C9—22no—
252C4×C9—22no—
253C4×C9—32no—
254C4×C9—62no—
255C4×C9D3×D412yes—
256Dic9—22no—
257Dic9—62no—
258Dic9—62no—
259Dic9—122no—
260Dic9—122no—
261C2×#41—271no—
262C22×Q8—271no—
263C4⋊Q8—271no—
264C4⋊Q8—271no—
265#124—91no—
266#124—271no—
267#123—274no—
268C22⋊Q8—274no—
269#117—31no—
270C3×C9#53611yescharacteristic, complemented, Sylow
271C23×C3—31no—
272C23×C3—92no—
273C22×D3—32no—
274C22×D3C22×D911yescharacteristic
275C3×Q8—182no—
276C3×D4—182no—
277C2×C3×C4—22no—
278C2×C3×C4—21no—
279C2×C3×C4—31no—
280C2×C3×C4—34no—
281C2×C3×C4—32no—
282C2×C3×C4—92no—
283C2×C3×C4C22×D912yes—
284C2×C3×C4C22×D911yescharacteristic
285C2×C3×C4C2×D3212yes—
286C2×C3×C4C2×D3211yescharacteristic
287#67—184no—
288#67—184no—
289C2×Dic3—32no—
290C2×Dic3—92no—
291C2×Dic3—92no—
292C2×Dic3—92no—
293C2×Dic3—92no—
294C2×Dic3—182no—
295C2×Dic3—182no—
296C2×Dic3—184no—
297C2×Dic3—184no—
298C2×Dic3C22×D912yes—
299C2×Dic3C22×D911yescharacteristic
300D12—62no—
301C4×D3—184no—
302C4×D3—184no—
303C4×D3D3614yes—
304Dic6—62no—
305Dic6—184no—
306Dic6—182no—
307Dic6—182no—
308Dic6—182no—
309Dic6—182no—
310Dic6—184no—
311Dic6—362no—
312Dic6—362no—
313C2×C32C22×D1211yescharacteristic
314C2×C32#101012yes—
315C3×D3—22no—
316C3×D3—122no—
317C2×C9—22no—
318C2×C9—21no—
319C2×C9—62no—
320C2×C9C2×D3×D411yescharacteristic
321C2×C9#101012yes—
322D9—42no—
323#41—544no—
324C2×Q8—91no—
325C2×Q8—272no—
326C2×Q8—271no—
327C2×Q8—271no—
328C2×Q8—544no—
329C2×D4—91no—
330C2×D4—272no—
331C22×C4—31no—
332C22×C4—272no—
333C22×C4—272no—
334C4⋊C4—32no—
335C4⋊C4—274no—
336C4⋊C4—274no—
337C22⋊C4—32no—
338C22⋊C4—94no—
339C22⋊C4—274no—
340C42—31no—
341C42D3×D911yescharacteristic
342C22×C3—21no—
343C22×C3—32no—
344C22×C3—62no—
345C22×C3—182no—
346C22×C3—184no—
347C22×C3C23×D911yescharacteristic
348C22×C3C22×D3211yescharacteristic
349C2×D3—22no—
350C2×D3—64no—
351C2×D3—62no—
352C2×D3C2×D3612yes—
353C3×C4—24no—
354C3×C4—22no—
355C3×C4—22no—
356C3×C4—22no—
357C3×C4—32no—
358C3×C4—62no—
359C3×C4—182no—
360C3×C4C2×D3612yes—
361C3×C4D3×D1212yes—
362Dic3—22no—
363Dic3—62no—
364Dic3—182no—
365Dic3—182no—
366Dic3—182no—
367Dic3—182no—
368Dic3—362no—
369Dic3—362no—
370Dic3C2×D3612yes—
371C32#552611yescharacteristic
372C9—21no—
373C9#567511yescharacteristic, complemented
374C23—31no—
375C23—92no—
376Q8—182no—
377Q8—544no—
378Q8—542no—
379Q8—542no—
380D4—182no—
381D4—544no—
382C2×C4—31no—
383C2×C4—34no—
384C2×C4—32no—
385C2×C4—92no—
386C2×C4—272no—
387C2×C4—272no—
388C2×C4—544no—
389C2×C4—544no—
390C2×C4C2×D3×D912yes—
391C2×C4C2×D3×D911yescharacteristic
392C2×C3—22no—
393C2×C3—21no—
394C2×C3—62no—
395C2×C3—362no—
396C2×C3C22×D3611yescharacteristic
397C2×C3#6371612yes—
398C2×C3#6429812yes—
399C2×C3C2×D3×D1211yescharacteristic
400D3—22no—
401D3—122no—
402C22—32no—
403C22—62no—
404C22—184no—
405C22—182no—
406C22C22×D3×D911yescharacteristic
407C4—22no—
408C4—32no—
409C4—62no—
410C4—182no—
411C4—542no—
412C4—542no—
413C4D3×D3612yes—
414C3—21no—
415C3#1058962811yescharacteristic, complemented
416C3#1059389411yescharacteristic
417C2—62no—
418C2—362no—
419C2#1172246512yes—
420C2C2×D3×D3611yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×#88641C2
hypercenterC42D3×D9
hypocenterC3×C9#536
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C2×D3×Dic18C2semidirect
3. #11721706C2semidirect
4. C9×#872C2semidirect
5. C3×#63437C2semidirect
6. #11721021C2semidirect
7. #11720991C2semidirect
8. #11720948C2semidirect
9. #11720932C2semidirect
46. #63437D3semidirect
138. #872D9semidirect
270. C3×C9#536semidirect
373. C9#5675semidirect
415. C3#10589628semidirect

Maximal subgroups

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

subgroupindex
C2×#886412
C2×D3×Dic182
#117217062
C9×#8722
C3×#634372
#117210212
#117209912
#117209482
#117209322
#105938943
#105896283