Finite Group Databasebeta

← C3×#10588730 #13142872

Subgroups — C3×#10588730 #13142872

Order 1728.

530 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3697C211yesmaximal, characteristic
2C3×C9×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×C3×#3675C211yesmaximal, characteristic, complemented
4C2×C3×#3674C211yesmaximal, characteristic, complemented
5C3×#63389C211yesmaximal, characteristic, complemented
6C3×#63369C211yesmaximal, characteristic, complemented
7C3×#63368C211yesmaximal, characteristic, complemented
8C3×#5047—31nomaximal
9#10588730C311yesmaximal, characteristic, direct product factor
10C2×C3×Q8×C9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C2×C3×#611—21no—
13C3×#3697—22no—
14C3×#3697—22no—
15C2×C3×D36C2211yescharacteristic
16C2×C3×C4×D9—21no—
17C2×C3×Dic18C2211yescharacteristic
18C3×C9×C4⋊C4—21no—
19C3×C9×C4⋊C4C2211yescharacteristic
20C3×C9×C22⋊C4—21no—
21C3×#3675—22no—
22C3×#3674—22no—
23C2×C3×C9⋊C8C2211yescharacteristic
24C2×C3×C9⋊C8C2211yescharacteristic
25C2×C3×#235—31no—
26C32×C22⋊Q8D311yescharacteristic
27C2×C3×#216—31no—
28C2×C3×#215—31no—
29C2×#3697C2×C311yescharacteristic
30C3×#830—31no—
31C3×#810—31no—
32C3×#809—31no—
33C9×C22⋊Q8—21no—
34C9×C22⋊Q8C2×C311yescharacteristic, complemented
35C2×#3675C2×C311yescharacteristic, complemented
36C2×#3674C2×C311yescharacteristic, complemented
37#63389C2×C311yescharacteristic, complemented
38#63369C2×C311yescharacteristic, complemented
39#63368C2×C311yescharacteristic, complemented
40C23×C3×C9D411yescharacteristic
41C22×C3×D9—21no—
42C3×Q8×C9—21no—
43C3×Q8×C9—21no—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9—21no—
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9D411yescharacteristic
49C2×C3×C4×C9C2311yescharacteristic
50C3×#611—24no—
51C2×C3×Dic9—21no—
52C3×D36—21no—
53C3×D36D412yes—
54C3×C4×D9—24no—
55C3×Dic18—21no—
56C3×Dic18D412yes—
57C3×C9⋊C8—21no—
58C3×C9⋊C8—21no—
59C3×#452—91no—
60#5047—31no—
61C2×C32×Q8C2×D311yescharacteristic
62C22×C32×C4C2×D311yescharacteristic
63C2×C3×#67—61no—
64C3×#235—62no—
65C3×#235—62no—
66C2×C3×D12—31no—
67C2×C3×C4×D3—61no—
68C2×C3×Dic6—31no—
69C32×C4⋊C4—21no—
70C32×C4⋊C4C2×D311yescharacteristic
71C32×C22⋊C4—21no—
72C3×#216—62no—
73C3×#215—62no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×Q8×C9—21no—
77C2×Q8×C9C22×C311yescharacteristic
78C22×C4×C9—21no—
79C22×C4×C9C22×C311yescharacteristic
80C2×#611—21no—
81#3697—22no—
82#3697—22no—
83C2×D36C22×C311yescharacteristic
84C2×C4×D9—21no—
85C2×Dic18C22×C311yescharacteristic
86C9×C4⋊C4—22no—
87C9×C4⋊C4—21no—
88C9×C4⋊C4—21no—
89C9×C4⋊C4C22×C311yescharacteristic
90C9×C22⋊C4—22no—
91C9×C22⋊C4—21no—
92#3675—22no—
93#3674—22no—
94C2×C9⋊C8C22×C311yescharacteristic
95C2×C9⋊C8C22×C311yescharacteristic
96C22×C3×C9—21no—
97C22×C3×C9—21no—
98C22×C3×C9—21no—
99C22×C3×C9C2×D411yescharacteristic
100C2×C3×D9—22no—
101C2×C3×D9—41no—
102C2×C3×D9—41no—
103C3×C4×C9—21no—
104C3×C4×C9—41no—
105C3×C4×C9—41no—
106C3×C4×C9C2×D411yescharacteristic
107C3×C4×C9C2×D411yescharacteristic
108C3×Dic9—22no—
109C2×C3×#41—91no—
110C2×#235—31no—
111C2×C3×Q16—91no—
112C2×C3×SD16—91no—
113C3×C22⋊Q8—21no—
114C3×C22⋊Q8D911yescharacteristic, complemented
115C3×C22⋊Q8C3×D311yescharacteristic
116C2×#216—31no—
117C2×#215—31no—
118C3×#103—91no—
119C3×#102—91no—
120C3×C22⋊C8—91no—
121#830—31no—
122#810—31no—
123#809—31no—
124C23×C32#6711yescharacteristic
125C22×C3×D3—61no—
126C32×Q8—21no—
127C32×Q8—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4#6711yescharacteristic
133C2×C32×C4C22×D311yescharacteristic
134C3×#67—64no—
135C2×C3×Dic3—61no—
136C3×D12—32no—
137C3×D12—61no—
138C3×C4×D3—64no—
139C3×Dic6—32no—
140C3×Dic6—61no—
141C23×C9—21no—
142C23×C9C3×D411yescharacteristic
143C22×D9—21no—
144C3×C3⋊C8—61no—
145C3×C3⋊C8—61no—
146Q8×C9—21no—
147Q8×C9—21no—
148Q8×C9—41no—
149Q8×C9—41no—
150C2×C4×C9—21no—
151C2×C4×C9—21no—
152C2×C4×C9—22no—
153C2×C4×C9—22no—
154C2×C4×C9—21no—
155C2×C4×C9—21no—
156C2×C4×C9—21no—
157C2×C4×C9—21no—
158C2×C4×C9—41no—
159C2×C4×C9—41no—
160C2×C4×C9C3×D411yescharacteristic
161C2×C4×C9C23×C311yescharacteristic
162#611—24no—
163C2×Dic9—21no—
164D36—21no—
165D36C3×D412yes—
166C4×D9—24no—
167Dic18—21no—
168Dic18C3×D412yes—
169C9⋊C8—21no—
170C9⋊C8—21no—
171#452—91noSylow
172C2×C3×C9—41no—
173C2×C3×C9#12011yescharacteristic
174C2×C3×C9#13511yescharacteristic
175C2×C3×C9#13711yescharacteristic
176C3×D9—42no—
177C3×#41—182no—
178C3×#41—182no—
179C2×C3×Q8—21no—
180C2×C3×Q8—91no—
181C2×C3×Q8C2×D911yescharacteristic
182C2×C3×Q8C2×C3×D311yescharacteristic
183C2×C3×D4—91no—
184C2×C3×D4—181no—
185C22×C3×C4—21no—
186C22×C3×C4—181no—
187C22×C3×C4C2×D911yescharacteristic
188C22×C3×C4C2×C3×D311yescharacteristic
189C2×#67—61no—
190#235—62no—
191#235—62no—
192C2×D12—31no—
193C2×C4×D3—61no—
194C2×Dic6—31no—
195C3×Q16—182no—
196C3×SD16—182no—
197C2×C3×C8—91no—
198C2×C3×C8—91no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4C2×D911yescharacteristic
204C3×C4⋊C4C2×C3×D311yescharacteristic
205C3×C22⋊C4—22no—
206C3×C22⋊C4—21no—
207C3×C22⋊C4—21no—
208#216—62no—
209#215—62no—
210C2×C3⋊C8—31no—
211C2×C3⋊C8—31no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32C2×#6711yescharacteristic
216C2×C3×D3—62no—
217C2×C3×D3—121no—
218C2×C3×D3—121no—
219C32×C4—21no—
220C32×C4—41no—
221C32×C4—41no—
222C32×C4D3×D411yescharacteristic
223C32×C4D3×D411yescharacteristic
224C3×Dic3—62no—
225C22×C9—22no—
226C22×C9—21no—
227C22×C9—21no—
228C22×C9—21no—
229C22×C9—21no—
230C22×C9—41no—
231C22×C9—41no—
232C22×C9C2×C3×D411yescharacteristic
233C2×D9—22no—
234C2×D9—41no—
235C2×D9—41no—
236C4×C9—21no—
237C4×C9—21no—
238C4×C9—21no—
239C4×C9—41no—
240C4×C9—42no—
241C4×C9—42no—
242C4×C9—41no—
243C4×C9—41no—
244C4×C9C2×C3×D411yescharacteristic
245C4×C9C2×C3×D411yescharacteristic
246Dic9—22no—
247C2×#41—91no—
248C2×Q16—91no—
249C2×SD16—91no—
250C22⋊Q8C3×D911yescharacteristic, complemented
251#103—91no—
252#102—91no—
253C22⋊C8—91no—
254C3×C9#45211yescharacteristic, complemented, Sylow
255C23×C3—21no—
256C23×C3—181no—
257C23×C3#61111yescharacteristic
258C23×C3C3×#6711yescharacteristic
259C22×D3—61no—
260C3×Q8—21no—
261C3×Q8—21no—
262C3×Q8—21no—
263C3×Q8—21no—
264C3×Q8—41no—
265C3×Q8—41no—
266C3×Q8—92no—
267C3×Q8—181no—
268C3×D4—92no—
269C3×D4—184no—
270C3×D4—181no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—184no—
286C2×C3×C4—181no—
287C2×C3×C4#61111yescharacteristic
288C2×C3×C4C22×D911yescharacteristic
289C2×C3×C4C3×#6711yescharacteristic
290C2×C3×C4C22×C3×D311yescharacteristic
291#67—64no—
292C2×Dic3—61no—
293D12—32no—
294D12—61no—
295C4×D3—64no—
296Dic6—32no—
297Dic6—61no—
298C3×C8—181no—
299C3×C8—181no—
300C3⋊C8—61no—
301C3⋊C8—61no—
302C2×C32—41no—
303C2×C32#93711yescharacteristic
304C2×C32#94211yescharacteristic
305C2×C32#95011yescharacteristic
306C3×D3—122no—
307C2×C9—21no—
308C2×C9—21no—
309C2×C9—21no—
310C2×C9—42no—
311C2×C9—41no—
312C2×C9C3×#12011yescharacteristic
313C2×C9C3×#13511yescharacteristic
314C2×C9C3×#13711yescharacteristic
315D9—42no—
316#41—182no—
317#41—182no—
318C2×Q8—91no—
319C2×Q8C2×C3×D911yescharacteristic
320C2×D4—91no—
321C2×D4—181no—
322C22×C4—181no—
323C22×C4C2×C3×D911yescharacteristic
324Q16—182no—
325SD16—182no—
326C2×C8—91no—
327C2×C8—91no—
328C4⋊C4—21no—
329C4⋊C4C2×C3×D911yescharacteristic
330C22⋊C4—21no—
331C22×C3—22no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—182no—
342C22×C3—361no—
343C22×C3—361no—
344C22×C3C2×#61111yescharacteristic
345C22×C3C2×C3×#6711yescharacteristic
346C2×D3—62no—
347C2×D3—121no—
348C2×D3—121no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—41no—
354C3×C4—42no—
355C3×C4—42no—
356C3×C4—41no—
357C3×C4—41no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—182no—
361C3×C4D4×D911yescharacteristic
362C3×C4D4×D911yescharacteristic
363C3×C4C3×D3×D411yescharacteristic
364C3×C4C3×D3×D411yescharacteristic
365Dic3—62no—
366C32#504711yescharacteristic
367C9—21no—
368C9C3×#45211yescharacteristic, complemented
369C23—181no—
370C23C3×#61111yescharacteristic
371Q8—21no—
372Q8—21no—
373Q8—92no—
374Q8—181no—
375D4—92no—
376D4—184no—
377D4—181no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—184no—
383C2×C4—181no—
384C2×C4C3×#61111yescharacteristic
385C2×C4C22×C3×D911yescharacteristic
386C8—181no—
387C8—181no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—42no—
392C2×C3—41no—
393C2×C3—41no—
394C2×C3—362no—
395C2×C3#6349911yescharacteristic
396C2×C3#6350511yescharacteristic
397C2×C3#6351311yescharacteristic
398C2×C3C3×#93711yescharacteristic
399C2×C3C3×#94211yescharacteristic
400C2×C3C3×#95011yescharacteristic
401D3—122no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—182no—
406C22—361no—
407C22—361no—
408C22C2×C3×#61111yescharacteristic
409C4—21no—
410C4—41no—
411C4—41no—
412C4—182no—
413C4C3×D4×D911yescharacteristic
414C4C3×D4×D911yescharacteristic
415C3—21no—
416C3#1058873011yescharacteristic, direct product factor
417C3C3×#504711yescharacteristic
418C2—41no—
419C2—362no—
420C2C3×#6349911yescharacteristic
421C2C3×#6350511yescharacteristic
422C2C3×#6351311yescharacteristic
subgroupG/H
centerC22×C3C2×#611
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×C22⋊Q8C2
socleC22×C32C2×#67
prime radicalC2×C3×#3697C2
hypercenterC3×C22⋊Q8D9
hypocenterC9C3×#452
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C3×C9×C22⋊Q8C2semidirect
3. C2×C3×#3675C2semidirect
4. C2×C3×#3674C2semidirect
5. C3×#63389C2semidirect
6. C3×#63369C2semidirect
7. C3×#63368C2semidirect
9. #10588730C3direct (N × H)
34. C9×C22⋊Q8C2×C3semidirect
35. C2×#3675C2×C3semidirect
36. C2×#3674C2×C3semidirect
37. #63389C2×C3semidirect
38. #63369C2×C3semidirect
39. #63368C2×C3semidirect
114. C3×C22⋊Q8D9semidirect
250. C22⋊Q8C3×D9semidirect
254. C3×C9#452semidirect
368. C9C3×#452semidirect
416. C3#10588730direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#36972
C3×C9×C22⋊Q82
C2×C3×#36752
C2×C3×#36742
C3×#633892
C3×#633692
C3×#633682
C3×#50473
#105887303