Finite Group Databasebeta

← #13147623

Subgroups — #13147623

Order 1728.

706 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6226C211yesmaximal, characteristic, complemented
2C9×#875C211yesmaximal, characteristic
3C3×C4×Dic18C211yesmaximal, characteristic, complemented
4D12×Dic9C211yesmaximal, characteristic
5#11721018C212yesmaximal, complemented
6#11721011C212yesmaximal, complemented
7#11720966C212yesmaximal, complemented
8#11720964C211yesmaximal, characteristic
9#11720962C212yesmaximal, complemented
10#11720924C211yesmaximal, characteristic, complemented
11Dic6×Dic9C211yesmaximal, characteristic, complemented
12#10593932—31nomaximal
13#10589773—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C9×D12C2211yescharacteristic
16C2×C9×Dic6C2211yescharacteristic
17C2×C3×Dic18C2211yescharacteristic
18C2×D3×Dic9C2212yes—
19C3×C42×C9C2211yescharacteristic
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21#88534C2212yes—
22C4×C3⋊Dic9C2212yes—
23C9×(C2×D3)⋊C4C2212yes—
24C9×(C2×D3)⋊C4C2212yes—
25C3×C4⋊Dic9C2211yescharacteristic
26C3×Dic9⋊C4C2211yescharacteristic
27C3×Dic9⋊C4C2211yescharacteristic
28C3×C4×Dic9C2211yescharacteristic
29C3×C4×Dic9C2211yescharacteristic
30(C2×D3)⋊Dic9C2212yes—
31(C2×D3)⋊Dic9C2212yes—
32(C2×D3)⋊Dic9C2212yes—
33Dic3⋊Dic9C2212yes—
34Dic9⋊Dic3C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37C4×#634—31no—
38C3×#875D311yescharacteristic
39C3×C4×Dic6—31no—
40Dic3×D12—31no—
41#63887—32no—
42#63878—32no—
43#63866—31no—
44#63862—32no—
45#63860—31no—
46#63858—32no—
47Dic3×Dic6—31no—
48C9×#124—31no—
49Q8×Dic9—31no—
50#63506—31no—
51#63498—31no—
52D4×Dic9—31no—
53#63441—32no—
54#63441—32no—
55#63440—32no—
56#63439—32no—
57C4×Dic18—31no—
58C4×Dic18—61no—
59C4×Dic18D311yescharacteristic, complemented
60C22×D3×C9C2312yes—
61C2×C3×C4×C9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2312yes—
66#6226—22no—
67C2×C9×Dic3C2312yes—
68C2×C3×Dic9C2311yescharacteristic
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C9×D12—22no—
73C9×Dic6—22no—
74C3×Dic18—22no—
75D3×Dic9—24no—
76#5671—91no—
77#5584—91no—
78C2×#634—31no—
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×Dic6—31no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×D3×Dic3—32no—
83C32×C42C2×D311yescharacteristic
84C4⋊(C3⋊Dic3)—31no—
85#3750—32no—
86C4×C3⋊Dic3—32no—
87C3×(C2×D3)⋊C4C2×D312yes—
88C3×(C2×D3)⋊C4C2×D312yes—
89C3×C4⋊Dic3—31no—
90C3×Dic3⋊C4—31no—
91C3×Dic3⋊C4—31no—
92C3×C4×Dic3—31no—
93C3×C4×Dic3—31no—
94Dic3⋊Dic3—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97(C2×D3)⋊Dic3—32no—
98(C2×D3)⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100Dic32—32no—
101C2×Q8×C9—31no—
102C2×D4×C9—31no—
103C22×Dic9—32no—
104C2×Dic18—31no—
105C2×Dic18—61no—
106C2×Dic18C2×D311yescharacteristic
107C9×C22⋊C4—32no—
108C9×C22⋊C4—32no—
109C42×C9—21no—
110C42×C9C2×D311yescharacteristic
111#3676—32no—
112#3676—32no—
113#3676—32no—
114C4⋊Dic9—31no—
115C4⋊Dic9—32no—
116C4⋊Dic9—61no—
117C4⋊Dic9C2×D311yescharacteristic
118Dic9⋊C4—32no—
119Dic9⋊C4—32no—
120Dic9⋊C4—32no—
121Dic9⋊C4—62no—
122Dic9⋊C4C2×D311yescharacteristic
123Dic9⋊C4C2×D311yescharacteristic
124C4×Dic9—32no—
125C4×Dic9—32no—
126C4×Dic9—62no—
127C4×Dic9C2×D311yescharacteristic
128C4×Dic9C2×D311yescharacteristic
129C22×C3×C9C2411yescharacteristic
130C2×D3×C9—22no—
131C2×D3×C9—22no—
132C2×D3×C9—22no—
133C3×C4×C9—21no—
134C3×C4×C9—21no—
135C3×C4×C9#4112yes—
136C3⋊Dic9—22no—
137C3⋊Dic9—22no—
138C9×Dic3—22no—
139C3×Dic9—21no—
140C3×Dic9—21no—
141C3×Dic9—21no—
142C3×Dic9#4112yes—
143C3×#124—31no—
144C3×C4×Q8—91no—
145#946—92no—
146Q8×Dic3—91no—
147#944—91no—
148#936—91no—
149D4×Dic3—91no—
150#898—92no—
151#898—92no—
152#892—92no—
153#879—92no—
154#879—92no—
155#878—92no—
156#877—92no—
157#875—91no—
158#875D911yescharacteristic
159C4×D12—91no—
160#869—91no—
161C4×Dic6—31no—
162C4×Dic6—91no—
163C4×Dic6—91no—
164C4×Dic6—91no—
165C4×Dic6—181no—
166C22×C3×D3C22×D312yes—
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C3⋊Dic3—32no—
171C2×C3⋊Dic3—32no—
172#634—62no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3C22×D312yes—
178C3×D12—22no—
179C3×Dic6—22no—
180C3×Dic6—62no—
181D3×Dic3—64no—
182C23×C9—32no—
183Q8×C9—62no—
184D4×C9—62no—
185C2×C4×C9—21no—
186C2×C4×C9—21no—
187C2×C4×C9—21no—
188C2×C4×C9—32no—
189C2×C4×C9C22×D311yescharacteristic
190C2×C4×C9C22×D311yescharacteristic
191C2×C4×C9C22×D311yescharacteristic
192C2×Dic9—32no—
193C2×Dic9—32no—
194C2×Dic9—62no—
195C2×Dic9—62no—
196C2×Dic9—64no—
197C2×Dic9C22×D311yescharacteristic
198C2×Dic9C22×D311yescharacteristic
199C2×Dic9C22×D311yescharacteristic
200C2×Dic9C22×D311yescharacteristic
201Dic18—22no—
202Dic18—64no—
203Dic18—62no—
204#556—271noSylow
205C2×C3×C9C2×#4111yescharacteristic
206C2×C3×C9C2×#4111yescharacteristic
207C2×C3×C92−1+411yescharacteristic
208D3×C9—42no—
209C2×C3×Q8—31no—
210C2×C3×Q8—91no—
211C2×C3×D4—31no—
212C22×Dic3—92no—
213C2×D12C2×D911yescharacteristic
214C2×C4×D3—92no—
215C2×Dic6—31no—
216C2×Dic6—91no—
217C2×Dic6—91no—
218C2×Dic6—181no—
219C2×Dic6C2×D911yescharacteristic
220C3×C4⋊C4—91no—
221C3×C4⋊C4—91no—
222C3×C4⋊C4—91no—
223C3×C22⋊C4—32no—
224C3×C22⋊C4—32no—
225C3×C42—21no—
226C3×C42—91no—
227C3×C42—91no—
228C3×C42C2×D911yescharacteristic
229C3×C42D3211yescharacteristic
230#217—92no—
231#217—92no—
232#217—92no—
233(C2×D3)⋊C4—92no—
234(C2×D3)⋊C4—92no—
235(C2×D3)⋊C4—92no—
236(C2×D3)⋊C4C2×D912yes—
237(C2×D3)⋊C4C2×D912yes—
238C4⋊Dic3—31no—
239C4⋊Dic3—92no—
240C4⋊Dic3—91no—
241C4⋊Dic3—91no—
242C4⋊Dic3—92no—
243C4⋊Dic3—92no—
244C4⋊Dic3—181no—
245Dic3⋊C4—31no—
246Dic3⋊C4—31no—
247Dic3⋊C4—92no—
248Dic3⋊C4—92no—
249Dic3⋊C4—92no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—182no—
253C4×Dic3—31no—
254C4×Dic3—31no—
255C4×Dic3—92no—
256C4×Dic3—92no—
257C4×Dic3—92no—
258C4×Dic3—92no—
259C4×Dic3—182no—
260C22×C32C23×D311yescharacteristic
261C2×C3×D3—22no—
262C2×C3×D3—22no—
263C2×C3×D3—22no—
264C32×C4—21no—
265C32×C4—21no—
266C32×C4#23712yes—
267C3⋊Dic3—62no—
268C3⋊Dic3—62no—
269C3×Dic3—22no—
270C3×Dic3—32no—
271C3×Dic3—61no—
272C3×Dic3—61no—
273C3×Dic3—61no—
274C22×C9—21no—
275C22×C9—62no—
276C22×C9—62no—
277C22×C9—62no—
278C22×C9C23×D311yescharacteristic
279C4×C9—22no—
280C4×C9—21no—
281C4×C9—21no—
282C4×C9—22no—
283C4×C9—41no—
284C4×C9—62no—
285C4×C9#23912yes—
286Dic9—21no—
287Dic9—21no—
288Dic9—21no—
289Dic9—64no—
290Dic9—62no—
291Dic9—62no—
292Dic9—122no—
293Dic9#23512yes—
294C4⋊Q8—271no—
295#126—272no—
296#126—272no—
297#124—31no—
298#124—271no—
299C22⋊Q8—272no—
300C4×Q8—91no—
301C4×Q8—271no—
302C4×Q8—271no—
303C4×D4—271no—
304#117—272no—
305C3×C9#55611yescharacteristic, complemented, Sylow
306C23×C3—32no—
307C22×D3C22×D912yes—
308C3×Q8—62no—
309C3×Q8—182no—
310C3×D4—62no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—32no—
315C2×C3×C4—91no—
316C2×C3×C4—91no—
317C2×C3×C4—91no—
318C2×C3×C4—91no—
319C2×C3×C4C22×D911yescharacteristic
320C2×C3×C4C22×D911yescharacteristic
321C2×C3×C4C22×D911yescharacteristic
322C2×C3×C4C2×D3211yescharacteristic
323C2×C3×C4C2×D3211yescharacteristic
324C2×C3×C4C2×D3211yescharacteristic
325C2×Dic3—31no—
326C2×Dic3—31no—
327C2×Dic3—31no—
328C2×Dic3—31no—
329C2×Dic3—92no—
330C2×Dic3—92no—
331C2×Dic3—92no—
332C2×Dic3—92no—
333C2×Dic3—182no—
334C2×Dic3—182no—
335C2×Dic3—184no—
336C2×Dic3C22×D912yes—
337D12—22no—
338C4×D3—184no—
339Dic6—22no—
340Dic6—62no—
341Dic6—184no—
342Dic6—182no—
343Dic6—182no—
344C2×C32C2×#23711yescharacteristic
345C2×C32D3×#4111yescharacteristic
346C2×C32#101011yescharacteristic
347C3×D3—42no—
348C2×C9—21no—
349C2×C9—21no—
350C2×C9—21no—
351C2×C9—122no—
352C2×C9C2×#23511yescharacteristic
353C2×C9C2×#23911yescharacteristic
354C2×C9#101011yescharacteristic
355C2×Q8—31no—
356C2×Q8—91no—
357C2×Q8—271no—
358C2×D4—31no—
359C22×C4—272no—
360C4⋊C4—91no—
361C4⋊C4—91no—
362C4⋊C4—91no—
363C4⋊C4—272no—
364C4⋊C4—272no—
365C4⋊C4—272no—
366C4⋊C4—271no—
367C4⋊C4—272no—
368C22⋊C4—32no—
369C22⋊C4—32no—
370C22⋊C4—272no—
371C22⋊C4—272no—
372C22⋊C4—272no—
373C42—91no—
374C42—91no—
375C42—272no—
376C42—272no—
377C42D3×D911yescharacteristic
378C22×C3—21no—
379C22×C3—62no—
380C22×C3—62no—
381C22×C3—62no—
382C22×C3C23×D911yescharacteristic
383C22×C3C22×D3211yescharacteristic
384C2×D3—22no—
385C2×D3—22no—
386C2×D3—22no—
387C3×C4—22no—
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—22no—
393C3×C4—41no—
394C3×C4—62no—
395C3×C4—92no—
396C3×C4—181no—
397C3×C4—181no—
398C3×C4—181no—
399C3×C4#369912yes—
400C3×C4#379612yes—
401Dic3—22no—
402Dic3—32no—
403Dic3—61no—
404Dic3—61no—
405Dic3—61no—
406Dic3—184no—
407Dic3—182no—
408Dic3—182no—
409Dic3—182no—
410Dic3—182no—
411Dic3—362no—
412C32#567111yescharacteristic
413C9—21no—
414C9#558411yescharacteristic, complemented
415C23—32no—
416Q8—62no—
417Q8—182no—
418Q8—542no—
419D4—62no—
420C2×C4—32no—
421C2×C4—91no—
422C2×C4—91no—
423C2×C4—91no—
424C2×C4—91no—
425C2×C4—272no—
426C2×C4—272no—
427C2×C4—544no—
428C2×C4C2×D3×D911yescharacteristic
429C2×C4C2×D3×D911yescharacteristic
430C2×C4C2×D3×D911yescharacteristic
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—122no—
435C2×C3C2×#369911yescharacteristic
436C2×C3#41×D911yescharacteristic
437C2×C3#6371611yescharacteristic
438C2×C3C2×#379611yescharacteristic
439C2×C3#6429811yescharacteristic
440C2×C3D3×#23511yescharacteristic
441D3—42no—
442C22—62no—
443C22—62no—
444C22—62no—
445C22C22×D3×D911yescharacteristic
446C4—21no—
447C4—21no—
448C4—62no—
449C4—92no—
450C4—181no—
451C4—181no—
452C4—181no—
453C4—542no—
454C4—542no—
455C4#8863912yes—
456C3—21no—
457C3#1058977311yescharacteristic, complemented
458C3#1059393211yescharacteristic
459C2—122no—
460C2C2×#8863911yescharacteristic
461C2#1172246511yescharacteristic
462C2D9×#23511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×C3×D12C2×D3
hypercenterC42D3×D9
hypocenterC3×C9#556
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#6226C2semidirect
3. C3×C4×Dic18C2semidirect
5. #11721018C2semidirect
6. #11721011C2semidirect
7. #11720966C2semidirect
9. #11720962C2semidirect
10. #11720924C2semidirect
11. Dic6×Dic9C2semidirect
59. C4×Dic18D3semidirect
305. C3×C9#556semidirect
414. C9#5584semidirect
457. C3#10589773semidirect

Maximal subgroups

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

subgroupindex
C4×#62262
C9×#8752
C3×C4×Dic182
D12×Dic92
#117210182
#117210112
#117209662
#117209642
#117209622
#117209242
Dic6×Dic92
#105939323
#105897733