Finite Group Databasebeta

← #13148746

Subgroups — #13148746

Order 1728.

650 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721753C211yesmaximal, characteristic
2C9×#887C211yesmaximal, characteristic, complemented
3C3×#63452C211yesmaximal, characteristic, complemented
4#11720931C211yesmaximal, characteristic
5#11720929C211yesmaximal, characteristic
6#11720927C212yesmaximal, complemented
7#11720926C212yesmaximal
8#11720924C211yesmaximal, characteristic, complemented
9#11720921C212yesmaximal
10#11720915C212yesmaximal, complemented
11#11720912C211yesmaximal, characteristic, complemented
12#10594355—31nomaximal
13#10589685—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C9×Dic6C2211yescharacteristic
17C3×C9×C4⋊C4C2211yescharacteristic
18#88536C2212yes—
19C4⋊(C3⋊Dic9)C2212yes—
20C9×Dic3⋊C4C2212yes—
21C4×C9×Dic3C2211yescharacteristic
22C4×C9×Dic3C2212yes—
23C3×C4⋊Dic9C2211yescharacteristic
24C3×C4⋊Dic9C2212yes—
25C3×Dic9⋊C4C2212yes—
26C3×C4×Dic9C2211yescharacteristic
27#88445C2211yescharacteristic
28#88445C2212yes—
29#88445C2211yescharacteristic
30#88442C2212yes—
31#88442C2211yescharacteristic
32#88442C2211yescharacteristic
33Dic9⋊Dic3C2212yes—
34Dic9⋊Dic3C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37#64106—31no—
38C3×#890—31no—
39C3×#887D311yescharacteristic
40#63874—31no—
41#63871—31no—
42#63869—32no—
43#63868—32no—
44#63866—31no—
45#63848—32no—
46#63845—32no—
47#63844—31no—
48C4×Q8×C9—31no—
49#63508—31no—
50#63506—31no—
51#63460—32no—
52#63459—31no—
53#63459—61no—
54#63452D311yescharacteristic, complemented
55#63451—32no—
56#63438—32no—
57#63434—31no—
58#63432—32no—
59C4×Dic18—31no—
60C22×C3⋊D9C2311yescharacteristic
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3⋊Dic9C2312yes—
64C2×C3⋊Dic9C2311yescharacteristic
65C4×C3⋊D9—22no—
66#6226—22no—
67C2×C9×Dic3C2312yes—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C3×Dic9C2312yes—
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C9×Dic6—22no—
74#5698—91no—
75#5583—91no—
76C2×C4×C3⋊D3—31no—
77C2×#634—31no—
78C2×C3×Dic6C2×D311yescharacteristic
79C32×C4⋊C4C2×D311yescharacteristic
80#3752—32no—
81C4⋊(C3⋊Dic3)—32no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—32no—
84C3×Dic3⋊C4—32no—
85C3×Dic3⋊C4C2×D312yes—
86C3×C4×Dic3—31no—
87C3×C4×Dic3C2×D311yescharacteristic
88C3×C4×Dic3C2×D312yes—
89#3724—32no—
90#3724—31no—
91#3724—31no—
92Dic3⋊Dic3—32no—
93Dic3⋊Dic3—32no—
94Dic3⋊Dic3—32no—
95#3722—31no—
96#3722—32no—
97#3722—31no—
98Dic32—32no—
99C2×Q8×C9—31no—
100C2×C4×D9—31no—
101C2×C4×D9—61no—
102C2×Dic18—31no—
103C2×Dic18—61no—
104C9×C4⋊C4—21no—
105C9×C4⋊C4—32no—
106C9×C4⋊C4C2×D311yescharacteristic
107C42×C9—31no—
108C42×C9—32no—
109(C2×D9)⋊C4—32no—
110(C2×D9)⋊C4—32no—
111(C2×D9)⋊C4—31no—
112(C2×D9)⋊C4—31no—
113(C2×D9)⋊C4—62no—
114C4⋊Dic9—32no—
115C4⋊Dic9—62no—
116C4⋊Dic9C2×D311yescharacteristic
117C4⋊Dic9C2×D312yes—
118Dic9⋊C4—32no—
119Dic9⋊C4—32no—
120Dic9⋊C4—32no—
121Dic9⋊C4—32no—
122Dic9⋊C4—31no—
123Dic9⋊C4—31no—
124Dic9⋊C4C2×D312yes—
125C4×Dic9—32no—
126C4×Dic9C2×D311yescharacteristic
127C22×C3×C9C2411yescharacteristic
128C2×C3⋊D9—21no—
129C2×C3⋊D9—21no—
130C2×C3⋊D9—21no—
131C3×C4×C9—22no—
132C3×C4×C9#4112yes—
133C3⋊Dic9—22no—
134C3⋊Dic9—21no—
135C9×Dic3—22no—
136C9×Dic3—21no—
137C9×Dic3—21no—
138C3×Dic9—22no—
139C3×Dic9—21no—
140C3×Dic9—21no—
141C3×#125—91no—
142C3×C4×Q8—31no—
143#946—91no—
144#944—91no—
145#898—92no—
146#898—92no—
147#898—92no—
148#897—91no—
149#897—91no—
150#897—91no—
151#897—181no—
152#890—31no—
153#890—92no—
154#890—92no—
155#889—92no—
156#887—91no—
157#887D911yescharacteristic, complemented
158#876—92no—
159#872—91no—
160#872—91no—
161#870—92no—
162#869—91no—
163C4×Dic6—91no—
164C22×C3⋊D3—31no—
165C2×C32×C4C22×D312yes—
166C2×C32×C4C22×D311yescharacteristic
167C2×C3⋊Dic3—32no—
168C2×C3⋊Dic3—31no—
169C4×C3⋊D3—62no—
170#634—62no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3C22×D311yescharacteristic
175C2×C3×Dic3C22×D312yes—
176C2×C3×Dic3C22×D311yescharacteristic
177C3×Dic6—22no—
178C22×D9—31no—
179C22×D9—61no—
180Q8×C9—62no—
181C2×C4×C9—21no—
182C2×C4×C9—22no—
183C2×C4×C9—31no—
184C2×C4×C9—31no—
185C2×C4×C9—32no—
186C2×C4×C9C22×D312yes—
187C2×C4×C9C22×D311yescharacteristic
188C2×Dic9—32no—
189C2×Dic9—31no—
190C2×Dic9—61no—
191C2×Dic9—62no—
192C2×Dic9C22×D312yes—
193C2×Dic9C22×D311yescharacteristic
194C2×Dic9C22×D311yescharacteristic
195C4×D9—62no—
196C4×D9—122no—
197Dic18—62no—
198Dic18—122no—
199#541—271noSylow
200C2×C3×C9C2×#4111yescharacteristic
201C2×C3×C92−1+411yescharacteristic
202C2×C3×C92−1+411yescharacteristic
203C3⋊D9—41no—
204C2×C3×Q8—31no—
205C2×C4×D3—91no—
206C2×C4×D3—91no—
207C2×C4×D3—181no—
208C2×Dic6—91no—
209C2×Dic6—91no—
210C2×Dic6—181no—
211C2×Dic6C2×D911yescharacteristic
212C3×C4⋊C4—21no—
213C3×C4⋊C4—32no—
214C3×C4⋊C4—91no—
215C3×C4⋊C4—92no—
216C3×C4⋊C4—92no—
217C3×C4⋊C4C2×D911yescharacteristic
218C3×C4⋊C4D3211yescharacteristic
219C3×C42—31no—
220C3×C42—32no—
221C3×C42—91no—
222(C2×D3)⋊C4—91no—
223(C2×D3)⋊C4—91no—
224(C2×D3)⋊C4—92no—
225(C2×D3)⋊C4—92no—
226(C2×D3)⋊C4—92no—
227(C2×D3)⋊C4—91no—
228(C2×D3)⋊C4—91no—
229(C2×D3)⋊C4—92no—
230(C2×D3)⋊C4—182no—
231C4⋊Dic3—31no—
232C4⋊Dic3—32no—
233C4⋊Dic3—92no—
234C4⋊Dic3—92no—
235C4⋊Dic3—92no—
236C4⋊Dic3—92no—
237C4⋊Dic3—92no—
238C4⋊Dic3—182no—
239Dic3⋊C4—32no—
240Dic3⋊C4—92no—
241Dic3⋊C4—91no—
242Dic3⋊C4—92no—
243Dic3⋊C4—92no—
244Dic3⋊C4—91no—
245Dic3⋊C4—92no—
246Dic3⋊C4—91no—
247Dic3⋊C4—91no—
248Dic3⋊C4—92no—
249Dic3⋊C4C2×D912yes—
250C4×Dic3—31no—
251C4×Dic3—92no—
252C4×Dic3—92no—
253C4×Dic3C2×D911yescharacteristic
254C4×Dic3C2×D912yes—
255C22×C32C23×D311yescharacteristic
256C2×C3⋊D3—61no—
257C2×C3⋊D3—61no—
258C2×C3⋊D3—61no—
259C32×C4—22no—
260C32×C4#23512yes—
261C3⋊Dic3—62no—
262C3⋊Dic3—61no—
263C3×Dic3—22no—
264C3×Dic3—21no—
265C3×Dic3—21no—
266C3×Dic3—62no—
267C3×Dic3—61no—
268C3×Dic3—61no—
269C22×C9—21no—
270C22×C9C23×D311yescharacteristic
271C2×D9—62no—
272C2×D9—61no—
273C2×D9—61no—
274C2×D9—61no—
275C2×D9—121no—
276C2×D9—121no—
277C4×C9—22no—
278C4×C9—22no—
279C4×C9—42no—
280C4×C9—62no—
281C4×C9—61no—
282C4×C9—61no—
283C4×C9#23912yes—
284Dic9—21no—
285Dic9—22no—
286Dic9—21no—
287Dic9—62no—
288Dic9—61no—
289Dic9—122no—
290Dic9—121no—
291C4⋊Q8—271no—
292#126—272no—
293#126—272no—
294#125—91no—
295#125—272no—
296#125—272no—
297C22⋊Q8—271no—
298C22⋊Q8—271no—
299C4×Q8—31no—
300C4×Q8—271no—
301#117—271no—
302C3×C9#54111yescharacteristic, complemented, Sylow
303C22×D3—91no—
304C22×D3—91no—
305C22×D3—181no—
306C3×Q8—62no—
307C2×C3×C4—21no—
308C2×C3×C4—22no—
309C2×C3×C4—31no—
310C2×C3×C4—31no—
311C2×C3×C4—32no—
312C2×C3×C4—92no—
313C2×C3×C4—91no—
314C2×C3×C4—91no—
315C2×C3×C4C22×D912yes—
316C2×C3×C4C22×D911yescharacteristic
317C2×C3×C4C2×D3212yes—
318C2×C3×C4C2×D3211yescharacteristic
319C2×Dic3—31no—
320C2×Dic3—31no—
321C2×Dic3—32no—
322C2×Dic3—92no—
323C2×Dic3—91no—
324C2×Dic3—92no—
325C2×Dic3—91no—
326C2×Dic3—181no—
327C2×Dic3—182no—
328C2×Dic3C22×D911yescharacteristic
329C2×Dic3C22×D911yescharacteristic
330C2×Dic3C22×D912yes—
331C4×D3—182no—
332C4×D3—182no—
333C4×D3—362no—
334Dic6—22no—
335Dic6—182no—
336Dic6—182no—
337Dic6—362no—
338C2×C32C2×#23511yescharacteristic
339C2×C32#100711yescharacteristic
340C2×C32#101011yescharacteristic
341C3⋊D3—121no—
342C2×C9—21no—
343C2×C9—21no—
344C2×C9—21no—
345C2×C9C2×#23911yescharacteristic
346C2×C9#101011yescharacteristic
347C2×C9#101011yescharacteristic
348D9—122no—
349D9—121no—
350C2×Q8—31no—
351C2×Q8—271no—
352C22×C4—271no—
353C4⋊C4—32no—
354C4⋊C4—91no—
355C4⋊C4—92no—
356C4⋊C4—92no—
357C4⋊C4—271no—
358C4⋊C4—271no—
359C4⋊C4—272no—
360C4⋊C4—272no—
361C4⋊C4—272no—
362C4⋊C4—272no—
363C4⋊C4—272no—
364C4⋊C4D3×D911yescharacteristic
365C22⋊C4—271no—
366C22⋊C4—271no—
367C22⋊C4—272no—
368C22⋊C4—272no—
369C42—31no—
370C42—32no—
371C42—91no—
372C42—272no—
373C22×C3—21no—
374C22×C3C23×D911yescharacteristic
375C22×C3C22×D3211yescharacteristic
376C2×D3—182no—
377C2×D3—181no—
378C2×D3—181no—
379C2×D3—181no—
380C2×D3—181no—
381C2×D3—181no—
382C2×D3—181no—
383C2×D3—361no—
384C2×D3—361no—
385C3×C4—22no—
386C3×C4—22no—
387C3×C4—22no—
388C3×C4—42no—
389C3×C4—62no—
390C3×C4—61no—
391C3×C4—61no—
392C3×C4—182no—
393C3×C4—181no—
394C3×C4—181no—
395C3×C4#369712yes—
396C3×C4#379912yes—
397Dic3—22no—
398Dic3—21no—
399Dic3—21no—
400Dic3—61no—
401Dic3—61no—
402Dic3—62no—
403Dic3—182no—
404Dic3—181no—
405Dic3—182no—
406Dic3—181no—
407Dic3—362no—
408Dic3—361no—
409C32#558311yescharacteristic
410C9—21no—
411C9#569811yescharacteristic, complemented
412C23—271no—
413Q8—62no—
414Q8—542no—
415C2×C4—31no—
416C2×C4—31no—
417C2×C4—32no—
418C2×C4—91no—
419C2×C4—91no—
420C2×C4—92no—
421C2×C4—272no—
422C2×C4—271no—
423C2×C4—542no—
424C2×C4C2×D3×D912yes—
425C2×C4C2×D3×D911yescharacteristic
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3C2×#369711yescharacteristic
430C2×C3#6371311yescharacteristic
431C2×C3#6371611yescharacteristic
432C2×C3C2×#379911yescharacteristic
433C2×C3#6430911yescharacteristic
434C2×C3#6431511yescharacteristic
435D3—362no—
436D3—361no—
437D3—361no—
438C22—541no—
439C22—541no—
440C22—541no—
441C22C22×D3×D911yescharacteristic
442C4—22no—
443C4—61no—
444C4—62no—
445C4—61no—
446C4—181no—
447C4—182no—
448C4—181no—
449C4—541no—
450C4—542no—
451C4#8864212yes—
452C3—21no—
453C3#1058968511yescharacteristic, complemented
454C3#1059435511yescharacteristic
455C2—1081no—
456C2C2×#8864211yescharacteristic
457C2#1172249411yescharacteristic
458C2#1172251011yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC22×C3⋊D9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#541
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C9×#887C2semidirect
3. C3×#63452C2semidirect
6. #11720927C2semidirect
8. #11720924C2semidirect
10. #11720915C2semidirect
11. #11720912C2semidirect
54. #63452D3semidirect
157. #887D9semidirect
302. C3×C9#541semidirect
411. C9#5698semidirect
453. C3#10589685semidirect

Maximal subgroups

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

subgroupindex
#117217532
C9×#8872
C3×#634522
#117209312
#117209292
#117209272
#117209262
#117209242
#117209212
#117209152
#117209122
#105943553
#105896853