Finite Group Databasebeta

← #13148796

Subgroups — #13148796

Order 1728.

674 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721746C211yesmaximal, characteristic, complemented
2D3×C9×C4⋊C4C211yesmaximal, characteristic
3C3×#63450C211yesmaximal, characteristic, complemented
4#11720971C211yesmaximal, characteristic, complemented
5#11720960C212yesmaximal
6#11720959C211yesmaximal, characteristic
7#11720955C212yesmaximal
8#11720935C211yesmaximal, characteristic, complemented
9#11720914C212yesmaximal, complemented
10#11720906C211yesmaximal, characteristic
11#11720904C212yesmaximal, complemented
12#10594342—31nomaximal
13#10589602—31nomaximal
14C2×C4×D3×C9C2212yes—
15C2×C4×D3×C9C2211yescharacteristic
16C2×C3×Dic18C2212yes—
17C2×#6189C2212yes—
18C3×C9×C4⋊C4C2211yescharacteristic
19C4⋊(C3⋊Dic9)C2212yes—
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21#88534C2212yes—
22C4×C3⋊Dic9C2211yescharacteristic
23C9×C4⋊Dic3C2211yescharacteristic
24C9×Dic3⋊C4C2212yes—
25C3×C4⋊Dic9C2211yescharacteristic
26C3×Dic9⋊C4C2212yes—
27C3×C4×Dic9C2211yescharacteristic
28(C2×D3)⋊Dic9C2211yescharacteristic
29(C2×D3)⋊Dic9C2212yes—
30(C2×D3)⋊Dic9C2211yescharacteristic
31Dic3⋊Dic9C2211yescharacteristic
32Dic3⋊Dic9C2211yescharacteristic
33Dic3⋊Dic9C2212yes—
34#88442C2212yes—
35#88442C2212yes—
36Dic3×Dic9C2212yes—
37#64099—31no—
38C3×D3×C4⋊C4D311yescharacteristic
39C3×#888—31no—
40#63919—31no—
41#63869—31no—
42#63856—32no—
43#63855—31no—
44#63851—32no—
45#63845—32no—
46#63841—31no—
47#63839—32no—
48C2×C9×C4⋊C4—31no—
49#63488—31no—
50#63488—31no—
51#63486—32no—
52#63486—32no—
53#63452—31no—
54#63452—32no—
55#63452—31no—
56#63452—61no—
57#63450—32no—
58#63450—31no—
59#63450D311yescharacteristic, complemented
60C22×D3×C9C2311yescharacteristic
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3⋊Dic9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C9×Dic3C2312yes—
67C2×C9×Dic3C2311yescharacteristic
68C2×C3×Dic9C2312yes—
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2311yescharacteristic
71C4×D3×C9—42no—
72C4×D3×C9Q814yescomplemented
73C3×Dic18—42no—
74#6189—42no—
75#5729—91no—
76#5500—91no—
77C2×C3×C4×D3C2×D312yes—
78C2×C3×C4×D3C2×D311yescharacteristic
79C2×C3×Dic6—32no—
80C2×#627—32no—
81C32×C4⋊C4C2×D311yescharacteristic
82C4⋊(C3⋊Dic3)—31no—
83C4⋊(C3⋊Dic3)—32no—
84#3750—32no—
85C4×C3⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×Dic3⋊C4—32no—
89C3×Dic3⋊C4C2×D312yes—
90C3×C4×Dic3—31no—
91#3724—32no—
92#3724—32no—
93Dic3⋊Dic3—31no—
94Dic3⋊Dic3—31no—
95Dic3⋊Dic3—32no—
96(C2×D3)⋊Dic3—32no—
97(C2×D3)⋊Dic3—31no—
98(C2×D3)⋊Dic3—31no—
99Dic32—32no—
100C22×C4×C9—31no—
101C22×C4×C9—32no—
102C2×Dic18—32no—
103C2×Dic18C2×D312yes—
104C9×C4⋊C4—21no—
105C9×C4⋊C4—32no—
106C9×C4⋊C4—31no—
107C9×C4⋊C4C2×D311yescharacteristic
108#3676—32no—
109#3676—31no—
110#3676—31no—
111C4⋊Dic9—31no—
112C4⋊Dic9—32no—
113C4⋊Dic9—31no—
114C4⋊Dic9—31no—
115C4⋊Dic9—32no—
116C4⋊Dic9—62no—
117C4⋊Dic9—61no—
118C4⋊Dic9C2×D311yescharacteristic
119Dic9⋊C4—32no—
120Dic9⋊C4—32no—
121Dic9⋊C4—32no—
122Dic9⋊C4—62no—
123Dic9⋊C4C2×D312yes—
124C4×Dic9—31no—
125C4×Dic9—32no—
126C4×Dic9—61no—
127C4×Dic9C2×D311yescharacteristic
128C22×C3×C9C2411yescharacteristic
129C2×D3×C9—21no—
130C2×D3×C9—21no—
131C2×D3×C9C2×Q812yes—
132C3×C4×C9—22no—
133C3×C4×C9C2×Q812yes—
134C3⋊Dic9—21no—
135C3⋊Dic9—22no—
136C3⋊Dic9—21no—
137C9×Dic3—22no—
138C9×Dic3C2×Q812yes—
139C3×Dic9—22no—
140C3×Dic9—21no—
141C3×Dic9—21no—
142C3×C4⋊Q8—91no—
143C2×C3×C4⋊C4—31no—
144#946—92no—
145#944—92no—
146#926—91no—
147#926—91no—
148#924—92no—
149#924—92no—
150#896—92no—
151#892—91no—
152D3×C4⋊C4D911yescharacteristic
153#890—91no—
154#890—92no—
155#890—91no—
156#890—91no—
157#890—181no—
158#889—92no—
159#888—31no—
160#888—91no—
161#888—92no—
162#888—91no—
163#872—91no—
164#870—91no—
165C22×C3×D3C22×D311yescharacteristic
166C2×C32×C4C22×D312yes—
167C2×C32×C4C22×D311yescharacteristic
168C2×C3⋊Dic3—32no—
169C2×C3⋊Dic3—31no—
170C2×C3⋊Dic3—31no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3C22×D312yes—
175C2×C3×Dic3C22×D311yescharacteristic
176C3×C4×D3—42no—
177C3×C4×D3Dic614yes—
178C3×Dic6—122no—
179#627—122no—
180C23×C9—31no—
181C2×C4×C9—21no—
182C2×C4×C9—22no—
183C2×C4×C9—31no—
184C2×C4×C9—34no—
185C2×C4×C9—32no—
186C2×C4×C9—122no—
187C2×C4×C9C22×D312yes—
188C2×C4×C9C22×D311yescharacteristic
189C2×Dic9—31no—
190C2×Dic9—32no—
191C2×Dic9—31no—
192C2×Dic9—61no—
193C2×Dic9—61no—
194C2×Dic9—62no—
195C2×Dic9C22×D312yes—
196C2×Dic9C22×D311yescharacteristic
197C2×Dic9C22×D311yescharacteristic
198Dic18—42no—
199Dic18—122no—
200#544—271noSylow
201C2×C3×C9C22×Q811yescharacteristic
202C2×C3×C9H(5,𝔽2)11yescharacteristic
203C2×C3×C92−1+411yescharacteristic
204D3×C9—22no—
205C2×C3×Q8—92no—
206C22×C3×C4—31no—
207C22×C3×C4—32no—
208C2×C4×D3C2×D912yes—
209C2×C4×D3C2×D911yescharacteristic
210C2×Dic6—32no—
211C2×Dic6—92no—
212C2×Dic6—92no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—32no—
215C3×C4⋊C4—31no—
216C3×C4⋊C4—91no—
217C3×C4⋊C4—92no—
218C3×C4⋊C4C2×D911yescharacteristic
219C3×C4⋊C4D3211yescharacteristic
220C3×C42—91no—
221#217—91no—
222#217—91no—
223#217—92no—
224(C2×D3)⋊C4—91no—
225(C2×D3)⋊C4—91no—
226(C2×D3)⋊C4—92no—
227C4⋊Dic3—31no—
228C4⋊Dic3—91no—
229C4⋊Dic3—92no—
230C4⋊Dic3—91no—
231C4⋊Dic3—91no—
232C4⋊Dic3—92no—
233C4⋊Dic3—92no—
234C4⋊Dic3—91no—
235C4⋊Dic3—181no—
236C4⋊Dic3—182no—
237C4⋊Dic3C2×D911yescharacteristic
238Dic3⋊C4—32no—
239Dic3⋊C4—92no—
240Dic3⋊C4—92no—
241Dic3⋊C4—92no—
242Dic3⋊C4—92no—
243Dic3⋊C4—92no—
244Dic3⋊C4—92no—
245Dic3⋊C4—91no—
246Dic3⋊C4—91no—
247Dic3⋊C4—92no—
248Dic3⋊C4—182no—
249Dic3⋊C4C2×D912yes—
250C4×Dic3—31no—
251C4×Dic3—92no—
252C4×Dic3—91no—
253C4×Dic3—91no—
254C4×Dic3—92no—
255C4×Dic3—181no—
256C22×C32C23×D311yescharacteristic
257C2×C3×D3—21no—
258C2×C3×D3—21no—
259C2×C3×D3C2×Dic612yes—
260C32×C4—22no—
261C32×C4C2×Dic612yes—
262C3⋊Dic3—61no—
263C3⋊Dic3—62no—
264C3⋊Dic3—61no—
265C3×Dic3—22no—
266C3×Dic3—62no—
267C3×Dic3—61no—
268C3×Dic3—61no—
269C3×Dic3C2×Dic612yes—
270C22×C9—21no—
271C22×C9—32no—
272C22×C9—61no—
273C22×C9—61no—
274C22×C9C23×D311yescharacteristic
275C4×C9—22no—
276C4×C9—22no—
277C4×C9—32no—
278C4×C9—42no—
279C4×C9—62no—
280C4×C9D3×Q812yes—
281Dic9—21no—
282Dic9—22no—
283Dic9—21no—
284Dic9—61no—
285Dic9—62no—
286Dic9—61no—
287Dic9—121no—
288Dic9—121no—
289Dic9—122no—
290C4⋊Q8—91no—
291C4⋊Q8—271no—
292C4⋊Q8—272no—
293#125—271no—
294#125—272no—
295#125—271no—
296C22⋊Q8—272no—
297C22⋊Q8—272no—
298#117—271no—
299#117—271no—
300C2×C4⋊C4—31no—
301C3×C9#54411yescharacteristic, complemented, Sylow
302C23×C3—31no—
303C22×D3C22×D911yescharacteristic
304C3×Q8—362no—
305C2×C3×C4—21no—
306C2×C3×C4—22no—
307C2×C3×C4—31no—
308C2×C3×C4—34no—
309C2×C3×C4—32no—
310C2×C3×C4—92no—
311C2×C3×C4—91no—
312C2×C3×C4—91no—
313C2×C3×C4—122no—
314C2×C3×C4C22×D912yes—
315C2×C3×C4C22×D911yescharacteristic
316C2×C3×C4C2×D3212yes—
317C2×C3×C4C2×D3211yescharacteristic
318C2×Dic3—31no—
319C2×Dic3—31no—
320C2×Dic3—32no—
321C2×Dic3—91no—
322C2×Dic3—92no—
323C2×Dic3—91no—
324C2×Dic3—92no—
325C2×Dic3—91no—
326C2×Dic3—91no—
327C2×Dic3—181no—
328C2×Dic3—181no—
329C2×Dic3—182no—
330C2×Dic3C22×D912yes—
331C2×Dic3C22×D911yescharacteristic
332C4×D3—42no—
333C4×D3Dic1814yescomplemented
334Dic6—122no—
335Dic6—362no—
336Dic6—362no—
337C2×C32C22×Dic611yescharacteristic
338C2×C32#100411yescharacteristic
339C2×C32#100711yescharacteristic
340C3×D3—22no—
341C2×C9—21no—
342C2×C9—21no—
343C2×C9—21no—
344C2×C9—62no—
345C2×C9#100411yescharacteristic
346C2×C9C2×D3×Q811yescharacteristic
347C2×C9#100711yescharacteristic
348C2×Q8—92no—
349C2×Q8—272no—
350C22×C4—31no—
351C22×C4—32no—
352C4⋊C4—32no—
353C4⋊C4—31no—
354C4⋊C4—91no—
355C4⋊C4—92no—
356C4⋊C4—271no—
357C4⋊C4—271no—
358C4⋊C4—272no—
359C4⋊C4—272no—
360C4⋊C4—272no—
361C4⋊C4—271no—
362C4⋊C4—272no—
363C4⋊C4—272no—
364C4⋊C4D3×D911yescharacteristic
365C22⋊C4—271no—
366C22⋊C4—271no—
367C22⋊C4—272no—
368C42—91no—
369C42—272no—
370C42—271no—
371C22×C3—21no—
372C22×C3—32no—
373C22×C3—61no—
374C22×C3—61no—
375C22×C3C23×D911yescharacteristic
376C22×C3C22×D3211yescharacteristic
377C2×D3—21no—
378C2×D3—21no—
379C2×D3C2×Dic1812yes—
380C3×C4—22no—
381C3×C4—22no—
382C3×C4—22no—
383C3×C4—32no—
384C3×C4—42no—
385C3×C4—62no—
386C3×C4—182no—
387C3×C4—181no—
388C3×C4—181no—
389C3×C4C2×Dic1812yes—
390C3×C4D3×Dic612yes—
391Dic3—22no—
392Dic3—61no—
393Dic3—62no—
394Dic3—61no—
395Dic3—181no—
396Dic3—181no—
397Dic3—182no—
398Dic3—181no—
399Dic3—182no—
400Dic3—181no—
401Dic3—361no—
402Dic3—361no—
403Dic3—362no—
404Dic3C2×Dic1812yes—
405C32#550011yescharacteristic
406C9—21no—
407C9#572911yescharacteristic, complemented
408C23—31no—
409Q8—362no—
410Q8—1082no—
411C2×C4—31no—
412C2×C4—34no—
413C2×C4—32no—
414C2×C4—91no—
415C2×C4—91no—
416C2×C4—92no—
417C2×C4—122no—
418C2×C4—272no—
419C2×C4—271no—
420C2×C4—271no—
421C2×C4C2×D3×D912yes—
422C2×C4C2×D3×D911yescharacteristic
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—62no—
427C2×C3C22×Dic1811yescharacteristic
428C2×C3#6371011yescharacteristic
429C2×C3#6371311yescharacteristic
430C2×C3C2×D3×Dic611yescharacteristic
431C2×C3#6431311yescharacteristic
432C2×C3#6431611yescharacteristic
433D3—22no—
434C22—32no—
435C22—61no—
436C22—61no—
437C22C22×D3×D911yescharacteristic
438C4—22no—
439C4—32no—
440C4—62no—
441C4—181no—
442C4—182no—
443C4—181no—
444C4—541no—
445C4—542no—
446C4—541no—
447C4D3×Dic1812yes—
448C3—21no—
449C3#1058960211yescharacteristic, complemented
450C3#1059434211yescharacteristic
451C2—62no—
452C2C2×D3×Dic1811yescharacteristic
453C2#1172249911yescharacteristic
454C2#1172250911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC22×C3×D3C22×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#544
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721746C2semidirect
3. C3×#63450C2semidirect
4. #11720971C2semidirect
8. #11720935C2semidirect
9. #11720914C2semidirect
11. #11720904C2semidirect
59. #63450D3semidirect
72. C4×D3×C9Q8semidirect
301. C3×C9#544semidirect
333. C4×D3Dic18semidirect
407. C9#5729semidirect
449. C3#10589602semidirect

Maximal subgroups

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

subgroupindex
#117217462
D3×C9×C4⋊C42
C3×#634502
#117209712
#117209602
#117209592
#117209552
#117209352
#117209142
#117209062
#117209042
#105943423
#105896023