Finite Group Databasebeta

← #13147594

Subgroups — #13147594

Order 1728.

886 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88642C211yesmaximal, characteristic
2C2×D3×Dic18C211yesmaximal, characteristic, complemented
3#11721712C211yesmaximal, characteristic, complemented
4C9×#872C211yesmaximal, characteristic, complemented
5C3×#63431C211yesmaximal, characteristic, complemented
6#11721020C214yesmaximal, complemented
7#11720995C214yesmaximal, complemented
8#11720952C211yesmaximal, characteristic, complemented
9#11720932C211yesmaximal, characteristic, complemented
10#10593893—31nomaximal
11#10589628—31nomaximal
12C2×C4⋊(C3⋊D9)C2211yescharacteristic
13C2×#6226C2211yescharacteristic
14C2×C4×D3×C9C2211yescharacteristic
15C2×C3×Dic18C2211yescharacteristic
16C2×C3×Dic18C2211yescharacteristic
17C2×#6195C2212yes—
18C2×D3×Dic9C2212yes—
19C2×(C3⋊D9)⋊C4C2212yes—
20C2×#6189C2212yes—
21#88642—24no—
22D3×Dic18—24no—
23C3×C42×C9C2211yescharacteristic
24#88536C2214yes—
25C9×(C2×D3)⋊C4C2212yes—
26C9×Dic3⋊C4C2212yes—
27C4×C9×Dic3C2211yescharacteristic
28C3×C4⋊Dic9C2214yes—
29#88445C2214yes—
30Dic3⋊Dic9C2214yes—
31C2×#3799—31no—
32C2×D3×Dic6—31no—
33#64084—31no—
34C3×#872D311yescharacteristic
35C3×#869—31no—
36#63895—34no—
37#63890—34no—
38#63866—31no—
39#63864—31no—
40C2×#3697—31no—
41C22×Dic18—31no—
42C9×#117—31no—
43#63459—34no—
44#63448—34no—
45#63437—31no—
46#63437—31no—
47#63437—61no—
48#63431—31no—
49#63431D311yescharacteristic, complemented
50C22×C3⋊D9C2312yes—
51C22×D3×C9C2311yescharacteristic
52C2×C3×C4×C9C2312yes—
53C2×C3×C4×C9C2311yescharacteristic
54C2×C3⋊Dic9C2312yes—
55C4⋊(C3⋊D9)—22no—
56#6226—22no—
57C2×C9×Dic3C2312yes—
58C2×C9×Dic3C2311yescharacteristic
59C2×C3×Dic9C2312yes—
60C2×C3×Dic9C2312yes—
61C4×D3×C9D414yes—
62C3×Dic18—22no—
63C3×Dic18—22no—
64#6195—24no—
65D3×Dic9—24no—
66(C3⋊D9)⋊C4—24no—
67#6189—24no—
68#5724—91no—
69#5526—91no—
70C2×C4⋊(C3⋊D3)—31no—
71C2×#634—31no—
72C2×C3×C4×D3C2×D311yescharacteristic
73C2×C3×Dic6—31no—
74C2×C3×Dic6—31no—
75C2×#627—32no—
76C2×#626—32no—
77C2×#624—32no—
78C2×D3×Dic3—32no—
79#3799—64no—
80D3×Dic6—64no—
81C32×C42C2×D311yescharacteristic
82#3752—34no—
83C3×(C2×D3)⋊C4C2×D312yes—
84C3×C4⋊Dic3—34no—
85C3×Dic3⋊C4C2×D312yes—
86C3×C4×Dic3C2×D311yescharacteristic
87Dic3⋊Dic3—34no—
88#3722—34no—
89C22×C4×C9—31no—
90C2×#611—32no—
91C22×Dic9—32no—
92#3697—64no—
93C2×D36—31no—
94C2×D36—61no—
95C2×C4×D9—32no—
96C2×Dic18—31no—
97C2×Dic18—32no—
98C2×Dic18—64no—
99C2×Dic18—61no—
100C2×Dic18C2×D311yescharacteristic
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)⋊C4—34no—
109(C2×D9)⋊C4—64no—
110C4⋊Dic9—34no—
111C4⋊Dic9C2×D314yes—
112C22×C3×C9C2411yescharacteristic
113C2×C3⋊D9—22no—
114C2×C3⋊D9—24no—
115C2×D3×C9—22no—
116C2×D3×C9C2×D412yes—
117C3×C4×C9—22no—
118C3×C4×C9C2×D412yes—
119C3⋊Dic9—22no—
120C9×Dic3—22no—
121C9×Dic3C2×D412yes—
122C3×Dic9—22no—
123C3×Dic9—22no—
124C2×#239—91no—
125C2×D3×Q8—91no—
126C2×#235—91no—
127C22×Dic6—91no—
128C3×C4⋊Q8—91no—
129C3×#117—31no—
130#947—91no—
131#944—91no—
132#897—94no—
133#896—94no—
134#894—94no—
135#886—94no—
136#875—91no—
137#875—91no—
138#875—91no—
139#875—181no—
140#872D911yescharacteristic, complemented
141#869—31no—
142#869—91no—
143C22×C3⋊D3—32no—
144C22×C3×D3C22×D311yescharacteristic
145C2×C32×C4C22×D312yes—
146C2×C32×C4C22×D311yescharacteristic
147C2×C3⋊Dic3—32no—
148C4⋊(C3⋊D3)—62no—
149#634—62no—
150C2×C3×Dic3—32no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3C22×D312yes—
153C2×C3×Dic3C22×D311yescharacteristic
154C3×C4×D3D1214yes—
155C3×Dic6—62no—
156C3×Dic6—62no—
157#627—64no—
158#626—64no—
159#624—64no—
160D3×Dic3—64no—
161C23×C9—31no—
162C22×D9—32no—
163C22×D9—62no—
164C2×C4×C9—21no—
165C2×C4×C9—22no—
166C2×C4×C9—31no—
167C2×C4×C9—34no—
168C2×C4×C9—32no—
169C2×C4×C9C22×D312yes—
170C2×C4×C9C22×D311yescharacteristic
171#611—64no—
172C2×Dic9—32no—
173C2×Dic9—62no—
174C2×Dic9—64no—
175C2×Dic9C22×D312yes—
176C2×Dic9C22×D312yes—
177D36—62no—
178D36—122no—
179C4×D9—64no—
180Dic18—22no—
181Dic18—22no—
182Dic18—64no—
183Dic18—62no—
184Dic18—122no—
185#536—271noSylow
186C2×C3×C9C22×D411yescharacteristic
187C2×C3×C92−1+412yes—
188C3⋊D9—42no—
189D3×C9—22no—
190C2×C3×Q8—91no—
191C2×C3×Q8—91no—
192C22×C3×C4—31no—
193C2×#67—92no—
194C22×Dic3—92no—
195#239—184no—
196D3×Q8—184no—
197#235—184no—
198C2×D12—92no—
199C2×D12—91no—
200C2×D12—91no—
201C2×D12—181no—
202C2×C4×D3—92no—
203C2×C4×D3—92no—
204C2×C4×D3—92no—
205C2×C4×D3C2×D911yescharacteristic
206C2×Dic6—31no—
207C2×Dic6—31no—
208C2×Dic6—92no—
209C2×Dic6—91no—
210C2×Dic6—91no—
211C2×Dic6—92no—
212C2×Dic6—181no—
213C2×Dic6—184no—
214C3×C4⋊C4—32no—
215C3×C4⋊C4—94no—
216C3×C22⋊C4—32no—
217C3×C42—21no—
218C3×C42—31no—
219C3×C42C2×D911yescharacteristic
220C3×C42D3211yescharacteristic
221(C2×D3)⋊C4—94no—
222(C2×D3)⋊C4—94no—
223(C2×D3)⋊C4—94no—
224(C2×D3)⋊C4—94no—
225(C2×D3)⋊C4—184no—
226(C2×D3)⋊C4C2×D912yes—
227C4⋊Dic3—34no—
228C4⋊Dic3—94no—
229Dic3⋊C4—94no—
230Dic3⋊C4C2×D912yes—
231C4×Dic3C2×D911yescharacteristic
232C22×C32C23×D311yescharacteristic
233C2×C3⋊D3—62no—
234C2×C3⋊D3—64no—
235C2×C3×D3—22no—
236C2×C3×D3C2×D1212yes—
237C32×C4—22no—
238C32×C4C2×D1212yes—
239C3⋊Dic3—62no—
240C3×Dic3—22no—
241C3×Dic3—62no—
242C3×Dic3—62no—
243C3×Dic3C2×D1212yes—
244C22×C9—21no—
245C22×C9—32no—
246C22×C9—62no—
247C22×C9C23×D311yescharacteristic
248C2×D9—62no—
249C2×D9—64no—
250C2×D9—122no—
251C2×D9—124no—
252C4×C9—22no—
253C4×C9—22no—
254C4×C9—32no—
255C4×C9—42no—
256C4×C9—62no—
257C4×C9D3×D412yes—
258Dic9—22no—
259Dic9—22no—
260Dic9—62no—
261Dic9—122no—
262C2×#41—271no—
263C22×Q8—271no—
264C4⋊Q8—91no—
265C4⋊Q8—271no—
266#124—271no—
267#124—271no—
268#123—274no—
269C22⋊Q8—274no—
270#117—31no—
271C3×C9#53611yescharacteristic, complemented, Sylow
272C23×C3—31no—
273C22×D3—92no—
274C22×D3—92no—
275C22×D3—182no—
276C22×D3C22×D911yescharacteristic
277C3×Q8—182no—
278C3×Q8—182no—
279C2×C3×C4—21no—
280C2×C3×C4—22no—
281C2×C3×C4—31no—
282C2×C3×C4—34no—
283C2×C3×C4—32no—
284C2×C3×C4—92no—
285C2×C3×C4—92no—
286C2×C3×C4C22×D912yes—
287C2×C3×C4C22×D911yescharacteristic
288C2×C3×C4C2×D3212yes—
289C2×C3×C4C2×D3211yescharacteristic
290#67—184no—
291C2×Dic3—32no—
292C2×Dic3—32no—
293C2×Dic3—92no—
294C2×Dic3—92no—
295C2×Dic3—182no—
296C2×Dic3—184no—
297C2×Dic3C22×D912yes—
298C2×Dic3C22×D911yescharacteristic
299D12—184no—
300D12—182no—
301D12—182no—
302D12—362no—
303C4×D3—184no—
304C4×D3—184no—
305C4×D3—184no—
306C4×D3D3614yes—
307Dic6—62no—
308Dic6—62no—
309Dic6—184no—
310Dic6—182no—
311Dic6—182no—
312Dic6—184no—
313Dic6—362no—
314C2×C32C22×D1211yescharacteristic
315C2×C32#101012yes—
316C3⋊D3—122no—
317C3×D3—22no—
318C2×C9—22no—
319C2×C9—21no—
320C2×C9—62no—
321C2×C9C2×D3×D411yescharacteristic
322C2×C9#100712yes—
323D9—122no—
324D9—242no—
325#41—544no—
326C2×Q8—91no—
327C2×Q8—91no—
328C2×Q8—272no—
329C2×Q8—271no—
330C2×Q8—544no—
331C2×D4—272no—
332C2×D4—271no—
333C22×C4—31no—
334C22×C4—272no—
335C22×C4—272no—
336C4⋊C4—32no—
337C4⋊C4—94no—
338C4⋊C4—274no—
339C22⋊C4—32no—
340C22⋊C4—274no—
341C22⋊C4—274no—
342C42—31no—
343C42D3×D911yescharacteristic
344C22×C3—21no—
345C22×C3—32no—
346C22×C3—62no—
347C22×C3C23×D911yescharacteristic
348C22×C3C22×D3211yescharacteristic
349C2×D3—22no—
350C2×D3—184no—
351C2×D3—182no—
352C2×D3—182no—
353C2×D3—184no—
354C2×D3—362no—
355C2×D3—364no—
356C2×D3C2×D3612yes—
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—32no—
361C3×C4—42no—
362C3×C4—62no—
363C3×C4—182no—
364C3×C4—182no—
365C3×C4C2×D3612yes—
366C3×C4D3×D1212yes—
367Dic3—22no—
368Dic3—62no—
369Dic3—62no—
370Dic3—182no—
371Dic3—182no—
372Dic3—362no—
373Dic3C2×D3612yes—
374C32#552611yescharacteristic
375C9—21no—
376C9#572411yescharacteristic, complemented
377C23—31no—
378C23—272no—
379Q8—182no—
380Q8—182no—
381Q8—544no—
382Q8—542no—
383D4—544no—
384D4—542no—
385C2×C4—31no—
386C2×C4—34no—
387C2×C4—32no—
388C2×C4—92no—
389C2×C4—92no—
390C2×C4—272no—
391C2×C4—544no—
392C2×C4—544no—
393C2×C4C2×D3×D912yes—
394C2×C4C2×D3×D911yescharacteristic
395C2×C3—22no—
396C2×C3—21no—
397C2×C3—62no—
398C2×C3C22×D3611yescharacteristic
399C2×C3#6371612yes—
400C2×C3#6429712yes—
401C2×C3C2×D3×D1211yescharacteristic
402D3—22no—
403D3—362no—
404D3—362no—
405D3—722no—
406C22—32no—
407C22—62no—
408C22—544no—
409C22—542no—
410C22C22×D3×D911yescharacteristic
411C4—22no—
412C4—32no—
413C4—62no—
414C4—182no—
415C4—182no—
416C4—542no—
417C4D3×D3612yes—
418C3—21no—
419C3#1058962811yescharacteristic, complemented
420C3#1059389311yescharacteristic
421C2—62no—
422C2—1082no—
423C2#1172246412yes—
424C2C2×D3×D3611yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×#88642C2
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. #11721712C2semidirect
4. C9×#872C2semidirect
5. C3×#63431C2semidirect
6. #11721020C2semidirect
7. #11720995C2semidirect
8. #11720952C2semidirect
9. #11720932C2semidirect
49. #63431D3semidirect
140. #872D9semidirect
271. C3×C9#536semidirect
376. C9#5724semidirect
419. C3#10589628semidirect

Maximal subgroups

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

subgroupindex
C2×#886422
C2×D3×Dic182
#117217122
C9×#8722
C3×#634312
#117210202
#117209952
#117209522
#117209322
#105938933
#105896283