Finite Group Databasebeta

← #13150062

Subgroups — #13150062

Order 1728.

652 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721815C211yesmaximal, characteristic
2C9×#924C211yesmaximal, characteristic
3C3×#63486C211yesmaximal, characteristic
4#11721247C212yesmaximal
5#11721246C212yesmaximal
6#11720937C211yesmaximal, characteristic, complemented
7#11720930C211yesmaximal, characteristic, complemented
8#11720915C212yesmaximal, complemented
9#11720914C212yesmaximal, complemented
10#11720911C211yesmaximal, characteristic, complemented
11#11720910C211yesmaximal, characteristic, complemented
12#10594826—31nomaximal
13#10589727—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×C9×Dic6C2211yescharacteristic
16C2×C3×Dic18C2211yescharacteristic
17#88551C2211yescharacteristic
18#88551C2211yescharacteristic
19C4⋊(C3⋊Dic9)C2211yescharacteristic
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21C4×C3⋊Dic9C2211yescharacteristic
22C4×C3⋊Dic9C2211yescharacteristic
23C9×#217C2212yes—
24C3×#3676C2212yes—
25C9×C4⋊Dic3C2211yescharacteristic
26C9×Dic3⋊C4C2212yes—
27C3×C4⋊Dic9C2211yescharacteristic
28C3×Dic9⋊C4C2212yes—
29Dic3⋊Dic9C2212yes—
30Dic3⋊Dic9C2212yes—
31#88442C2212yes—
32#88442C2212yes—
33Dic9⋊Dic3C2212yes—
34Dic9⋊Dic3C2212yes—
35Dic3×Dic9C2212yes—
36Dic3×Dic9C2212yes—
37#64135—31no—
38C3×#924—31no—
39C3×#924D311yescharacteristic
40#63956—32no—
41#63956—32no—
42#63873—31no—
43#63872—31no—
44#63845—32no—
45#63845—32no—
46#63843—31no—
47#63843—31no—
48C9×C22⋊Q8—31no—
49Q8×Dic9—31no—
50#63506—31no—
51#63488—31no—
52#63488—61no—
53#63486D311yescharacteristic
54#63452—31no—
55#63452—32no—
56#63451—32no—
57#63449—31no—
58#63441—32no—
59#63441—32no—
60C23×C3×C9C2311yescharacteristic
61C2×C3×C4×C9—22no—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68C2×C9×Dic3C2312yes—
69C2×C9×Dic3C2312yes—
70C2×C3×Dic9C2312yes—
71C2×C3×Dic9C2312yes—
72C9×Dic6—22no—
73C3×Dic18—22no—
74#5625—91no—
75#5625—91no—
76C22×C32×C4C2×D311yescharacteristic
77C2×C3×Dic6—31no—
78C2×C3×Dic6C2×D311yescharacteristic
79#3757—31no—
80#3757—31no—
81C4⋊(C3⋊Dic3)—31no—
82C4⋊(C3⋊Dic3)—31no—
83C4×C3⋊Dic3—31no—
84C4×C3⋊Dic3—31no—
85C3×#217—32no—
86C3×#217C2×D312yes—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×Dic3⋊C4—32no—
90C3×Dic3⋊C4C2×D312yes—
91#3724—32no—
92#3724—32no—
93Dic3⋊Dic3—32no—
94Dic3⋊Dic3—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97Dic32—32no—
98Dic32—32no—
99C2×Q8×C9—31no—
100C22×C4×C9—21no—
101C22×C4×C9C2×D311yescharacteristic
102C2×Dic18C2×D311yescharacteristic
103C9×C4⋊C4—32no—
104C9×C4⋊C4—31no—
105C9×C22⋊C4—32no—
106#3676—31no—
107#3676—31no—
108#3676—61no—
109#3676—61no—
110#3676C2×D312yes—
111C4⋊Dic9—31no—
112C4⋊Dic9—31no—
113C4⋊Dic9—32no—
114C4⋊Dic9—32no—
115C4⋊Dic9—61no—
116C4⋊Dic9—61no—
117C4⋊Dic9C2×D311yescharacteristic
118Dic9⋊C4—32no—
119Dic9⋊C4—32no—
120Dic9⋊C4—32no—
121Dic9⋊C4—32no—
122Dic9⋊C4C2×D312yes—
123C4×Dic9—31no—
124C4×Dic9—31no—
125C4×Dic9—32no—
126C4×Dic9—32no—
127C4×Dic9—61no—
128C4×Dic9—61no—
129C22×C3×C9—21no—
130C22×C3×C9—21no—
131C22×C3×C9—21no—
132C22×C3×C9C2411yescharacteristic
133C3×C4×C9—21no—
134C3×C4×C9#4112yes—
135C3⋊Dic9—21no—
136C3⋊Dic9—21no—
137C3⋊Dic9—21no—
138C3⋊Dic9—21no—
139C9×Dic3—22no—
140C9×Dic3—22no—
141C3×Dic9—22no—
142C3×Dic9—22no—
143C3×C22⋊Q8—31no—
144C3×C22⋊Q8—91no—
145Q8×Dic3—91no—
146Q8×Dic3—91no—
147#944—91no—
148#944—91no—
149#926—91no—
150#926—91no—
151#926—181no—
152#924—31no—
153#924D911yescharacteristic
154#890—92no—
155#890—91no—
156#890—91no—
157#890—92no—
158#889—92no—
159#889—92no—
160#887—91no—
161#887—91no—
162#879—92no—
163#879—92no—
164#879—92no—
165#879—92no—
166C23×C32C22×D311yescharacteristic
167C2×C32×C4—22no—
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—31no—
172C2×C3⋊Dic3—31no—
173C2×C3⋊Dic3—31no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3C22×D312yes—
177C2×C3×Dic3C22×D312yes—
178C3×Dic6—22no—
179C3×Dic6—62no—
180C23×C9—21no—
181C23×C9C22×D311yescharacteristic
182Q8×C9—62no—
183C2×C4×C9—24no—
184C2×C4×C9—22no—
185C2×C4×C9—21no—
186C2×C4×C9—21no—
187C2×C4×C9—32no—
188C2×C4×C9—32no—
189C2×C4×C9C22×D311yescharacteristic
190C2×C4×C9C22×D311yescharacteristic
191C2×Dic9—31no—
192C2×Dic9—31no—
193C2×Dic9—31no—
194C2×Dic9—31no—
195C2×Dic9—61no—
196C2×Dic9—61no—
197C2×Dic9—61no—
198C2×Dic9—61no—
199C2×Dic9C22×D312yes—
200C2×Dic9C22×D312yes—
201Dic18—22no—
202#541—271noSylow
203C2×C3×C9—41no—
204C2×C3×C9C2×#4111yescharacteristic
205C2×C3×C92−1+411yescharacteristic
206C2×C3×C92−1+411yescharacteristic
207C2×C3×Q8—31no—
208C2×C3×Q8—91no—
209C22×C3×C4—21no—
210C22×C3×C4C2×D911yescharacteristic
211C22×C3×C4D3211yescharacteristic
212C2×Dic6—31no—
213C2×Dic6C2×D911yescharacteristic
214C3×C4⋊C4—31no—
215C3×C4⋊C4—32no—
216C3×C4⋊C4—91no—
217C3×C4⋊C4—92no—
218C3×C22⋊C4—32no—
219C3×C22⋊C4—92no—
220#217—32no—
221#217—91no—
222#217—91no—
223#217—91no—
224#217—91no—
225#217—181no—
226#217—181no—
227#217C2×D912yes—
228C4⋊Dic3—31no—
229C4⋊Dic3—92no—
230C4⋊Dic3—92no—
231C4⋊Dic3—91no—
232C4⋊Dic3—91no—
233C4⋊Dic3—91no—
234C4⋊Dic3—91no—
235C4⋊Dic3—92no—
236C4⋊Dic3—92no—
237C4⋊Dic3—181no—
238C4⋊Dic3—181no—
239C4⋊Dic3C2×D911yescharacteristic
240Dic3⋊C4—32no—
241Dic3⋊C4—92no—
242Dic3⋊C4—92no—
243Dic3⋊C4—92no—
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246Dic3⋊C4—92no—
247Dic3⋊C4—92no—
248Dic3⋊C4—92no—
249Dic3⋊C4C2×D912yes—
250C4×Dic3—92no—
251C4×Dic3—92no—
252C4×Dic3—91no—
253C4×Dic3—91no—
254C4×Dic3—91no—
255C4×Dic3—91no—
256C4×Dic3—92no—
257C4×Dic3—92no—
258C4×Dic3—181no—
259C4×Dic3—181no—
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32C23×D311yescharacteristic
264C32×C4—21no—
265C32×C4#23712yes—
266C3⋊Dic3—61no—
267C3⋊Dic3—61no—
268C3⋊Dic3—61no—
269C3⋊Dic3—61no—
270C3×Dic3—22no—
271C3×Dic3—22no—
272C3×Dic3—62no—
273C3×Dic3—62no—
274C22×C9—22no—
275C22×C9—21no—
276C22×C9—21no—
277C22×C9—21no—
278C22×C9—21no—
279C22×C9—41no—
280C22×C9—41no—
281C22×C9C23×D311yescharacteristic
282C4×C9—22no—
283C4×C9—21no—
284C4×C9—22no—
285C4×C9—62no—
286C4×C9—62no—
287C4×C9#23712yes—
288Dic9—22no—
289Dic9—22no—
290Dic9—61no—
291Dic9—61no—
292Dic9—61no—
293Dic9—61no—
294Dic9—121no—
295Dic9—121no—
296Dic9—121no—
297Dic9—121no—
298C4⋊Q8—271no—
299#126—272no—
300#126—272no—
301#125—272no—
302#125—272no—
303#125—271no—
304C22⋊Q8—31no—
305C22⋊Q8—91no—
306C4×Q8—271no—
307C4×Q8—271no—
308#117—271no—
309C3×C9#54111yescharacteristic, complemented, Sylow
310C23×C3—21no—
311C23×C3C22×D911yescharacteristic
312C23×C3C2×D3211yescharacteristic
313C3×Q8—62no—
314C3×Q8—182no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—32no—
321C2×C3×C4—32no—
322C2×C3×C4—92no—
323C2×C3×C4—92no—
324C2×C3×C4C22×D911yescharacteristic
325C2×C3×C4C22×D911yescharacteristic
326C2×C3×C4C2×D3211yescharacteristic
327C2×C3×C4C2×D3211yescharacteristic
328C2×Dic3—32no—
329C2×Dic3—32no—
330C2×Dic3—91no—
331C2×Dic3—91no—
332C2×Dic3—91no—
333C2×Dic3—91no—
334C2×Dic3—91no—
335C2×Dic3—91no—
336C2×Dic3—91no—
337C2×Dic3—91no—
338C2×Dic3—181no—
339C2×Dic3—181no—
340C2×Dic3—181no—
341C2×Dic3—181no—
342C2×Dic3C22×D912yes—
343C2×Dic3C22×D912yes—
344Dic6—22no—
345Dic6—62no—
346C2×C32—41no—
347C2×C32C2×#23711yescharacteristic
348C2×C32#100711yescharacteristic
349C2×C32#101011yescharacteristic
350C2×C9—21no—
351C2×C9—21no—
352C2×C9—21no—
353C2×C9—42no—
354C2×C9—41no—
355C2×C9C2×#23711yescharacteristic
356C2×C9#100711yescharacteristic
357C2×C9#101011yescharacteristic
358C2×Q8—31no—
359C2×Q8—91no—
360C22×C4D3×D911yescharacteristic
361C4⋊C4—31no—
362C4⋊C4—32no—
363C4⋊C4—91no—
364C4⋊C4—92no—
365C4⋊C4—272no—
366C4⋊C4—272no—
367C4⋊C4—272no—
368C4⋊C4—272no—
369C4⋊C4—272no—
370C4⋊C4—272no—
371C4⋊C4—271no—
372C4⋊C4—271no—
373C22⋊C4—32no—
374C22⋊C4—92no—
375C22⋊C4—271no—
376C22⋊C4—271no—
377C42—272no—
378C42—272no—
379C42—271no—
380C42—271no—
381C22×C3—22no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—41no—
390C22×C3—41no—
391C22×C3C23×D911yescharacteristic
392C22×C3C22×D3211yescharacteristic
393C3×C4—22no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—22no—
397C3×C4—62no—
398C3×C4—62no—
399C3×C4—182no—
400C3×C4—182no—
401C3×C4#369912yes—
402C3×C4#380512yes—
403Dic3—22no—
404Dic3—22no—
405Dic3—62no—
406Dic3—62no—
407Dic3—181no—
408Dic3—181no—
409Dic3—181no—
410Dic3—181no—
411Dic3—181no—
412Dic3—181no—
413Dic3—181no—
414Dic3—181no—
415Dic3—361no—
416Dic3—361no—
417Dic3—361no—
418Dic3—361no—
419C32#562511yescharacteristic
420C9—21no—
421C9#562511yescharacteristic, complemented
422C23C2×D3×D911yescharacteristic
423Q8—62no—
424Q8—182no—
425C2×C4—22no—
426C2×C4—32no—
427C2×C4—32no—
428C2×C4—92no—
429C2×C4—92no—
430C2×C4—271no—
431C2×C4—271no—
432C2×C4—271no—
433C2×C4—271no—
434C2×C4C2×D3×D911yescharacteristic
435C2×C4C2×D3×D911yescharacteristic
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—42no—
440C2×C3—41no—
441C2×C3—41no—
442C2×C3C2×#369911yescharacteristic
443C2×C3#6371311yescharacteristic
444C2×C3#6371611yescharacteristic
445C2×C3#6429711yescharacteristic
446C2×C3#6429711yescharacteristic
447C2×C3C2×#380511yescharacteristic
448C22—21no—
449C22—21no—
450C22—21no—
451C22C22×D3×D911yescharacteristic
452C4—21no—
453C4—62no—
454C4—62no—
455C4—182no—
456C4—182no—
457C4—541no—
458C4—541no—
459C4—541no—
460C4—541no—
461C4#8866312yes—
462C3—21no—
463C3#1058972711yescharacteristic, complemented
464C3#1059482611yescharacteristic
465C2—41no—
466C2#1172245811yescharacteristic
467C2#1172246411yescharacteristic
468C2C2×#8866311yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC23×C32C22×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#541
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. #11720937C2semidirect
7. #11720930C2semidirect
8. #11720915C2semidirect
9. #11720914C2semidirect
10. #11720911C2semidirect
11. #11720910C2semidirect
309. C3×C9#541semidirect
421. C9#5625semidirect
463. C3#10589727semidirect

Maximal subgroups

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

subgroupindex
#117218152
C9×#9242
C3×#634862
#117212472
#117212462
#117209372
#117209302
#117209152
#117209142
#117209112
#117209102
#105948263
#105897273