Finite Group Databasebeta

← #13139478

Subgroups — #13139478

Order 1728.

524 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3697C211yesmaximal, characteristic
2C2×#88413C211yesmaximal, characteristic, complemented
3C2×D9×C3⋊C8C211yesmaximal, characteristic, complemented
4#11720571C211yesmaximal, characteristic, complemented
5C9×#830C211yesmaximal, characteristic, complemented
6(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10591145—31nomaximal
9#10588409—31nomaximal
10C22×C3×C4×C9C2211yescharacteristic
11C2×C3×#611C411yescharacteristic
12C2×C3×#611C411yescharacteristic
13C3×#3697—22no—
14C3×#3697—22no—
15C2×C3×D36C411yescharacteristic
16C2×C3×C4×D9C2211yescharacteristic
17C2×C3×C4×D9C2211yescharacteristic
18C2×C3×Dic18C411yescharacteristic
19C2×C3⋊(C9⋊C8)C2211yescharacteristic
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21C2×C9×C3⋊C8C2211yescharacteristic
22C2×C9×C3⋊C8C2211yescharacteristic
23#88413—22no—
24D9×C3⋊C8—22no—
25C2×C3×#235—31no—
26C2×#3711—31no—
27C2×D3×C3⋊C8—31no—
28C2×#3697D311yescharacteristic
29#63657—31no—
30C3×#830D311yescharacteristic
31(C2×D3)⋊(C3⋊C8)—31no—
32(C2×D3)⋊(C3⋊C8)—31no—
33C2×#3663—31no—
34C2×C8×D9—31no—
35C9×C22⋊C8—31no—
36#63389—31no—
37#63389—61no—
38(C2×D9)⋊C8—31no—
39(C2×D9)⋊C8—31no—
40C23×C3×C9C2×C411yescharacteristic
41C22×C3×D9C2×C411yescharacteristic
42C22×C3×D9C2×C411yescharacteristic
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9C2×C411yescharacteristic
46C2×C3×C4×C9C2311yescharacteristic
47C3×#611—22no—
48C3×#611—22no—
49C2×C3×Dic9C2×C411yescharacteristic
50C2×C3×Dic9C2×C411yescharacteristic
51C3×D36—22no—
52C3×C4×D9—21no—
53C3×C4×D9—21no—
54C3×C4×D9D412yes—
55C3×C4×D9D412yes—
56C3×Dic18—22no—
57C3⋊(C9⋊C8)—21no—
58C3⋊(C9⋊C8)—21no—
59C9×C3⋊C8—21no—
60C9×C3⋊C8—21no—
61#5234—91no—
62#4727—91no—
63C22×C32×C4C2×D311yescharacteristic
64C2×C3×#67—31no—
65C2×C3×#67—31no—
66C3×#235—62no—
67C3×#235—62no—
68C2×C3×D12—31no—
69C2×C3×C4×D3—31no—
70C2×C3×C4×D3—31no—
71C2×C3×Dic6—31no—
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3×C3⋊C8C2×D311yescharacteristic
75C2×C3×C3⋊C8C2×D311yescharacteristic
76#3711—62no—
77D3×C3⋊C8—62no—
78C22×C4×C9—21no—
79C22×C4×C9C2×D311yescharacteristic
80C2×#611Dic311yescharacteristic
81C2×#611Dic311yescharacteristic
82#3697—22no—
83#3697—22no—
84C2×D36Dic311yescharacteristic
85C2×C4×D9C2×D311yescharacteristic
86C2×C4×D9C2×D311yescharacteristic
87C2×Dic18Dic311yescharacteristic
88C2×C8×C9—31no—
89C2×C8×C9—31no—
90C2×C9⋊C8—31no—
91C2×C9⋊C8—31no—
92C2×C9⋊C8—61no—
93C2×C9⋊C8—61no—
94#3663—62no—
95C8×D9—62no—
96C22×C3×C9—21no—
97C22×C3×C9—21no—
98C22×C3×C9—21no—
99C22×C3×C9C22×C411yescharacteristic
100C2×C3×D9—21no—
101C2×C3×D9—21no—
102C2×C3×D9—21no—
103C2×C3×D9—21no—
104C2×C3×D9—21no—
105C2×C3×D9C22⋊C412yes—
106C3×C4×C9—21no—
107C3×C4×C9C2×D411yescharacteristic
108C3×C4×C9C2×D411yescharacteristic
109C3×Dic9—21no—
110C3×Dic9C22⋊C412yes—
111C2×C3×#41—91no—
112C2×#235—31no—
113C2×#208—91no—
114C22×C3⋊C8—91no—
115C2×ΓL(1,𝔽25)—91no—
116C2×D3×C8—91no—
117C3×C22⋊C8—31no—
118#830—91no—
119#830—91no—
120#830—91no—
121#830—91no—
122#830—181no—
123#830D911yescharacteristic, complemented
124(C2×D3)⋊C8—91no—
125(C2×D3)⋊C8—91no—
126C23×C32C4×D311yescharacteristic
127C22×C3×D3—31no—
128C22×C3×D3—31no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4C4×D311yescharacteristic
132C2×C32×C4C22×D311yescharacteristic
133C3×#67—62no—
134C3×#67—62no—
135C2×C3×Dic3—31no—
136C2×C3×Dic3—31no—
137C3×D12—62no—
138C3×C4×D3—32no—
139C3×C4×D3—32no—
140C3×C4×D3—61no—
141C3×C4×D3—61no—
142C3×Dic6—62no—
143C23×C9—21no—
144C23×C9C2×Dic311yescharacteristic
145C22×D9C2×Dic311yescharacteristic
146C22×D9C2×Dic311yescharacteristic
147C3⋊(C3⋊C8)—61no—
148C3⋊(C3⋊C8)—61no—
149C3×C3⋊C8—21no—
150C3×C3⋊C8—21no—
151C2×C4×C9—21no—
152C2×C4×C9—21no—
153C2×C4×C9—21no—
154C2×C4×C9—21no—
155C2×C4×C9—41no—
156C2×C4×C9—41no—
157C2×C4×C9C2×Dic311yescharacteristic
158C2×C4×C9C22×D311yescharacteristic
159#611—22no—
160#611—22no—
161C2×Dic9C2×Dic311yescharacteristic
162C2×Dic9C2×Dic311yescharacteristic
163D36—22no—
164C4×D9—21no—
165C4×D9—21no—
166C4×D9#6712yes—
167C4×D9#6712yes—
168Dic18—22no—
169C8×C9—61no—
170C8×C9—61no—
171C9⋊C8—61no—
172C9⋊C8—61no—
173C9⋊C8—121no—
174C9⋊C8—121no—
175#408—271noSylow
176C2×C3×C9—41no—
177C2×C3×C9C2×C22⋊C411yescharacteristic
178C2×C3×C9#13111yescharacteristic
179C2×C3×C9#13111yescharacteristic
180C3×D9—22no—
181C3×D9—41no—
182C3×#41—182no—
183C3×#41—182no—
184C2×C3×Q8—91no—
185C2×C3×D4—91no—
186C2×C3×D4—91no—
187C2×C3×D4—91no—
188C22×C3×C4—21no—
189C22×C3×C4—91no—
190C22×C3×C4—91no—
191C22×C3×C4C2×D911yescharacteristic
192C22×C3×C4D3211yescharacteristic
193C2×#67—31no—
194C2×#67—31no—
195#235—62no—
196#235—62no—
197C2×D12—31no—
198C2×C4×D3—31no—
199C2×C4×D3—31no—
200C2×Dic6—31no—
201C2×C3×C8—31no—
202C2×C3×C8—31no—
203#208—182no—
204C2×C3⋊C8—91no—
205C2×C3⋊C8—91no—
206C2×C3⋊C8—91no—
207C2×C3⋊C8—91no—
208C2×C3⋊C8—181no—
209C2×C3⋊C8—181no—
210C2×C3⋊C8—182no—
211C2×C3⋊C8C2×D911yescharacteristic
212C2×C3⋊C8C2×D911yescharacteristic
213ΓL(1,𝔽25)—182no—
214D3×C8—182no—
215C22×C32—21no—
216C22×C32—21no—
217C22×C32—21no—
218C22×C32C2×C4×D311yescharacteristic
219C2×C3×D3—32no—
220C2×C3×D3—61no—
221C2×C3×D3—61no—
222C2×C3×D3—61no—
223C2×C3×D3—61no—
224C2×C3×D3—61no—
225C32×C4—21no—
226C32×C4D3×D411yescharacteristic
227C32×C4D3×D411yescharacteristic
228C3×Dic3—32no—
229C3×Dic3—61no—
230C22×C9—22no—
231C22×C9—21no—
232C22×C9—21no—
233C22×C9—21no—
234C22×C9—21no—
235C22×C9—41no—
236C22×C9—41no—
237C22×C9C22×Dic311yescharacteristic
238C2×D9—21no—
239C2×D9—21no—
240C2×D9—21no—
241C2×D9—21no—
242C2×D9—21no—
243C2×D9#21712yes—
244C4×C9—21no—
245C4×C9—21no—
246C4×C9—21no—
247C4×C9—41no—
248C4×C9C2×#6711yescharacteristic
249C4×C9C2×#6711yescharacteristic
250Dic9—21no—
251Dic9#21712yes—
252C2×#41—91no—
253C2×M16—271no—
254C22×C8—271no—
255C22⋊C8—31no—
256C22⋊C8—271no—
257C22⋊C8—271no—
258C22⋊C8—271no—
259C3×C9#40811yescharacteristic, complemented, Sylow
260C23×C3—21no—
261C23×C3—91no—
262C23×C3—91no—
263C23×C3C4×D911yescharacteristic
264C23×C3D3×Dic311yescharacteristic
265C22×D3—31no—
266C22×D3—31no—
267C3×Q8—182no—
268C3×D4—182no—
269C3×D4—182no—
270C3×D4—182no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—91no—
280C2×C3×C4—92no—
281C2×C3×C4—92no—
282C2×C3×C4—91no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4C4×D911yescharacteristic
286C2×C3×C4C22×D911yescharacteristic
287C2×C3×C4D3×Dic311yescharacteristic
288C2×C3×C4C2×D3211yescharacteristic
289#67—62no—
290#67—62no—
291C2×Dic3—31no—
292C2×Dic3—31no—
293D12—62no—
294C4×D3—32no—
295C4×D3—32no—
296C4×D3—61no—
297C4×D3—61no—
298Dic6—62no—
299C3×C8—61no—
300C3×C8—61no—
301C3⋊C8—21no—
302C3⋊C8—21no—
303C3⋊C8—181no—
304C3⋊C8—181no—
305C3⋊C8—181no—
306C3⋊C8—181no—
307C3⋊C8—361no—
308C3⋊C8—361no—
309C2×C32—41no—
310C2×C32D3×C22⋊C411yescharacteristic
311C2×C32#90111yescharacteristic
312C2×C32#90711yescharacteristic
313C3×D3—62no—
314C3×D3—121no—
315C2×C9—21no—
316C2×C9—21no—
317C2×C9—21no—
318C2×C9—42no—
319C2×C9—41no—
320C2×C9#94811yescharacteristic
321C2×C9#94811yescharacteristic
322C2×C9C2×#21711yescharacteristic
323D9—22no—
324D9—41no—
325#41—182no—
326#41—182no—
327C2×Q8—91no—
328C2×D4—91no—
329C2×D4—91no—
330C2×D4—91no—
331C22×C4—91no—
332C22×C4—91no—
333C22×C4D3×D911yescharacteristic
334M16—542no—
335C2×C8—31no—
336C2×C8—31no—
337C2×C8—271no—
338C2×C8—271no—
339C2×C8—542no—
340C22×C3—22no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—92no—
351C22×C3—181no—
352C22×C3—181no—
353C22×C3—181no—
354C22×C3—181no—
355C22×C3—181no—
356C22×C3C2×C4×D911yescharacteristic
357C22×C3C2×D3×Dic311yescharacteristic
358C2×D3—32no—
359C2×D3—61no—
360C2×D3—61no—
361C2×D3—61no—
362C2×D3—61no—
363C2×D3—61no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—41no—
369C3×C4—92no—
370C3×C4—181no—
371C3×C4D4×D911yescharacteristic
372C3×C4D4×D911yescharacteristic
373C3×C4D3×#6711yescharacteristic
374C3×C4D3×#6711yescharacteristic
375Dic3—32no—
376Dic3—61no—
377C32#472711yescharacteristic
378C9—21no—
379C9#523411yescharacteristic, complemented
380C23—91no—
381C23—91no—
382C23Dic3×D911yescharacteristic
383Q8—182no—
384D4—182no—
385D4—182no—
386D4—182no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—91no—
390C2×C4—92no—
391C2×C4—92no—
392C2×C4—91no—
393C2×C4—181no—
394C2×C4—181no—
395C2×C4Dic3×D911yescharacteristic
396C2×C4C2×D3×D911yescharacteristic
397C8—61no—
398C8—61no—
399C8—541no—
400C8—541no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—42no—
405C2×C3—41no—
406C2×C3—41no—
407C2×C3—182no—
408C2×C3—361no—
409C2×C3C22⋊C4×D911yescharacteristic
410C2×C3#6346411yescharacteristic
411C2×C3#6346911yescharacteristic
412C2×C3#6381411yescharacteristic
413C2×C3#6381511yescharacteristic
414C2×C3D3×#21711yescharacteristic
415D3—62no—
416D3—121no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—92no—
421C22—181no—
422C22—181no—
423C22—181no—
424C22—181no—
425C22—181no—
426C22C2×Dic3×D911yescharacteristic
427C4—21no—
428C4—92no—
429C4—181no—
430C4D9×#6711yescharacteristic
431C4D9×#6711yescharacteristic
432C3—21no—
433C3#1058840911yescharacteristic, complemented
434C3#1059114511yescharacteristic
435C2—41no—
436C2—182no—
437C2—361no—
438C2#1172085811yescharacteristic
439C2#1172085911yescharacteristic
440C2D9×#21711yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D9
commutatorC22×C3×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C2×C4×D3
prime radicalC2×C3×#3697C2
cosocleC2×C3×C4×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9#408
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88413C2semidirect
3. C2×D9×C3⋊C8C2semidirect
4. #11720571C2semidirect
5. C9×#830C2semidirect
6. (C2×D9)⋊(C3⋊C8)C2semidirect
7. (C2×D9)⋊(C3⋊C8)C2semidirect
123. #830D9semidirect
259. C3×C9#408semidirect
379. C9#5234semidirect
433. C3#10588409semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36972
C2×#884132
C2×D9×C3⋊C82
#117205712
C9×#8302
(C2×D9)⋊(C3⋊C8)2
#105911453
#105884093