Finite Group Databasebeta

← #13143706

Subgroups — #13143706

Order 1728.

850 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88536C211yesmaximal, characteristic
2C2×#88536C211yesmaximal, characteristic
3C2×#88536C211yesmaximal, characteristic
4C2×#88534C211yesmaximal, characteristic, complemented
5C3×C9×#95C211yesmaximal, characteristic, complemented
6#11720572C211yesmaximal, characteristic, complemented
7#11720572C211yesmaximal, characteristic, complemented
8#10592374—31nomaximal
9#10588355—33nomaximal
10C23×C3⋊D9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C22×C3⋊Dic9C2211yescharacteristic
16C22×C3⋊Dic9C2211yescharacteristic
17#88536—22no—
18#88536—22no—
19#88536—22no—
20#88534—22no—
21C2×#3752—31no—
22C2×#3752—31no—
23C2×#3752—31no—
24C2×#3750—31no—
25C32×#95D311yescharacteristic
26#63658—31no—
27#63658—31no—
28C2×(C2×D9)⋊C4—33no—
29C2×(C2×D9)⋊C4—33no—
30C2×(C2×D9)⋊C4—33no—
31C2×Dic9⋊C4—33no—
32C9×#95D313yescomplemented
33C9⋊#95—33no—
34C9⋊#95—33no—
35C23×C3×C9C2311yescharacteristic
36C22×C3⋊D9—21no—
37C22×C3⋊D9—21no—
38C22×C3⋊D9—21no—
39C22×C3⋊D9—21no—
40C22×C3⋊D9—21no—
41C22×C3⋊D9—21no—
42C22×C3⋊D9D412yes—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9—21no—
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9—21no—
50C2×C3×C4×C9—21no—
51C2×C3×C4×C9—21no—
52C2×C3⋊Dic9—21no—
53C2×C3⋊Dic9—21no—
54C2×C3⋊Dic9—21no—
55C2×C3⋊Dic9—21no—
56C2×C3⋊Dic9—21no—
57C2×C3⋊Dic9—21no—
58C2×C3⋊Dic9—21no—
59C2×C3⋊Dic9—21no—
60C2×C3⋊Dic9D412yes—
61#4675—93no—
62#4675—91no—
63C23×C3⋊D3—31no—
64C22×C32×C4C2×D311yescharacteristic
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C2×D311yescharacteristic
67C22×C3⋊Dic3—31no—
68C22×C3⋊Dic3—31no—
69C22×C3⋊Dic3—31no—
70C23×D9—33no—
71#3752—62no—
72#3752—62no—
73#3752—62no—
74#3750—62no—
75C22×C4×C9C2×D313yes—
76C22×C4×C9C2×D313yes—
77C22×C4×C9C2×D313yes—
78C22×Dic9—33no—
79C22×Dic9—33no—
80C22×Dic9—33no—
81(C2×D9)⋊C4—66no—
82(C2×D9)⋊C4—66no—
83(C2×D9)⋊C4—66no—
84Dic9⋊C4—66no—
85C22×C3×C9C2×D411yescharacteristic
86C22×C3×C9C2×D411yescharacteristic
87C22×C3×C9#4111yescharacteristic
88C22×C3×C9#4111yescharacteristic
89C22×C3×C9#4111yescharacteristic
90C22×C3×C9#4111yescharacteristic
91C22×C3×C9#4111yescharacteristic
92C2×C3⋊D9—22no—
93C2×C3⋊D9—22no—
94C2×C3⋊D9—22no—
95C2×C3⋊D9—41no—
96C2×C3⋊D9—41no—
97C2×C3⋊D9—41no—
98C2×C3⋊D9—41no—
99C3×C4×C9—41no—
100C3×C4×C9—41no—
101C3×C4×C9—41no—
102C3⋊Dic9—22no—
103C3⋊Dic9—41no—
104C3⋊Dic9—41no—
105C2×(C2×D3)⋊C4—93no—
106C2×(C2×D3)⋊C4—93no—
107C2×(C2×D3)⋊C4—93no—
108C2×(C2×D3)⋊C4—91no—
109C2×(C2×D3)⋊C4—91no—
110C2×(C2×D3)⋊C4—91no—
111C2×Dic3⋊C4—93no—
112C2×Dic3⋊C4—91no—
113C3×#95D913yescomplemented
114C3×#95C3⋊D311yescharacteristic
115C3⋊#95—93no—
116C3⋊#95—93no—
117C3⋊#95—91no—
118C3⋊#95—91no—
119C23×C32C22×D311yescharacteristic
120C22×C3⋊D3—32no—
121C22×C3⋊D3—61no—
122C22×C3⋊D3—61no—
123C22×C3⋊D3—61no—
124C22×C3⋊D3—61no—
125C22×C3⋊D3—61no—
126C22×C3⋊D3—61no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C3⋊Dic3—32no—
137C2×C3⋊Dic3—61no—
138C2×C3⋊Dic3—61no—
139C2×C3⋊Dic3—61no—
140C2×C3⋊Dic3—61no—
141C2×C3⋊Dic3—61no—
142C2×C3⋊Dic3—61no—
143C2×C3⋊Dic3—61no—
144C2×C3⋊Dic3—61no—
145C23×C9C22×D313yes—
146C22×D9—36no—
147C22×D9—63no—
148C22×D9—63no—
149C22×D9—63no—
150C22×D9—63no—
151C22×D9—63no—
152C22×D9—63no—
153C2×C4×C9—23no—
154C2×C4×C9—23no—
155C2×C4×C9—23no—
156C2×C4×C9—23no—
157C2×C4×C9—23no—
158C2×C4×C9—23no—
159C2×C4×C9—23no—
160C2×C4×C9—23no—
161C2×C4×C9—23no—
162C2×Dic9—36no—
163C2×Dic9—63no—
164C2×Dic9—63no—
165C2×Dic9—63no—
166C2×Dic9—63no—
167C2×Dic9—63no—
168C2×Dic9—63no—
169C2×Dic9—63no—
170C2×Dic9—63no—
171#397—271noSylow
172C2×C3×C9#12111yescharacteristic
173C2×C3×C9#12311yescharacteristic
174C2×C3×C9#12311yescharacteristic
175C2×C3×C9#12311yescharacteristic
176C2×C3×C9#12411yescharacteristic
177C2×C3×C9#12611yescharacteristic
178C2×C3×C9#12611yescharacteristic
179C3⋊D9—42no—
180C23×D3—93no—
181C23×D3—91no—
182C22×C3×C4C2×D913yes—
183C22×C3×C4C2×D913yes—
184C22×C3×C4C2×D913yes—
185C22×C3×C4C2×C3⋊D311yescharacteristic
186C22×C3×C4C2×C3⋊D311yescharacteristic
187C22×C3×C4C2×C3⋊D311yescharacteristic
188C22×Dic3—93no—
189C22×Dic3—91no—
190C22×Dic3—93no—
191C22×Dic3—93no—
192C22×Dic3—91no—
193C22×Dic3—91no—
194(C2×D3)⋊C4—182no—
195(C2×D3)⋊C4—186no—
196(C2×D3)⋊C4—182no—
197(C2×D3)⋊C4—182no—
198(C2×D3)⋊C4—186no—
199(C2×D3)⋊C4—186no—
200Dic3⋊C4—182no—
201Dic3⋊C4—186no—
202C22×C32#23511yescharacteristic
203C22×C32#23511yescharacteristic
204C22×C32#23511yescharacteristic
205C22×C32D3×D411yescharacteristic
206C22×C32D3×D411yescharacteristic
207C22×C32#23711yescharacteristic
208C22×C32#23911yescharacteristic
209C2×C3⋊D3—62no—
210C2×C3⋊D3—62no—
211C2×C3⋊D3—62no—
212C2×C3⋊D3—121no—
213C2×C3⋊D3—121no—
214C2×C3⋊D3—121no—
215C2×C3⋊D3—121no—
216C32×C4—41no—
217C32×C4—41no—
218C32×C4—41no—
219C3⋊Dic3—62no—
220C3⋊Dic3—121no—
221C3⋊Dic3—121no—
222C22×C9#23513yes—
223C22×C9#23513yes—
224C22×C9#23513yes—
225C22×C9D3×D413yes—
226C22×C9D3×D413yes—
227C22×C9#23713yes—
228C22×C9#23913yes—
229C2×D9—66no—
230C2×D9—66no—
231C2×D9—66no—
232C2×D9—123no—
233C2×D9—123no—
234C2×D9—123no—
235C2×D9—123no—
236C4×C9—43no—
237C4×C9—43no—
238C4×C9—43no—
239Dic9—66no—
240Dic9—123no—
241Dic9—123no—
242C2×C4⋊C4—271no—
243C2×C22⋊C4—271no—
244C2×C22⋊C4—271no—
245C2×C22⋊C4—271no—
246#95—271no—
247#95—271no—
248#95C3⋊D911yescharacteristic, complemented
249C3×C9#39711yescharacteristic, complemented, Sylow
250C23×C3C22×D913yes—
251C23×C3C22×C3⋊D311yescharacteristic
252C22×D3—96no—
253C22×D3—92no—
254C22×D3—183no—
255C22×D3—183no—
256C22×D3—183no—
257C22×D3—183no—
258C22×D3—183no—
259C22×D3—183no—
260C22×D3—181no—
261C22×D3—181no—
262C22×D3—181no—
263C22×D3—181no—
264C22×D3—181no—
265C22×D3—181no—
266C2×C3×C4—23no—
267C2×C3×C4—23no—
268C2×C3×C4—23no—
269C2×C3×C4—23no—
270C2×C3×C4—23no—
271C2×C3×C4—23no—
272C2×C3×C4—23no—
273C2×C3×C4—23no—
274C2×C3×C4—23no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×Dic3—96no—
285C2×Dic3—92no—
286C2×Dic3—183no—
287C2×Dic3—183no—
288C2×Dic3—181no—
289C2×Dic3—181no—
290C2×Dic3—183no—
291C2×Dic3—183no—
292C2×Dic3—183no—
293C2×Dic3—183no—
294C2×Dic3—183no—
295C2×Dic3—183no—
296C2×Dic3—181no—
297C2×Dic3—181no—
298C2×Dic3—181no—
299C2×Dic3—181no—
300C2×Dic3—181no—
301C2×Dic3—181no—
302C2×C32#87611yescharacteristic
303C2×C32#88311yescharacteristic
304C2×C32#88411yescharacteristic
305C2×C32#88511yescharacteristic
306C2×C32#89411yescharacteristic
307C2×C32#89411yescharacteristic
308C2×C32#89811yescharacteristic
309C3⋊D3—122no—
310C2×C9#87613yes—
311C2×C9#88313yes—
312C2×C9#88413yes—
313C2×C9#88513yes—
314C2×C9#89413yes—
315C2×C9#89413yes—
316C2×C9#89813yes—
317D9—126no—
318C24—271no—
319C22×C4—271no—
320C22×C4—271no—
321C22×C4—271no—
322C22×C4C2×C3⋊D911yescharacteristic
323C22×C4C2×C3⋊D911yescharacteristic
324C22×C4C2×C3⋊D911yescharacteristic
325C4⋊C4—542no—
326C22⋊C4—542no—
327C22⋊C4—542no—
328C22⋊C4—542no—
329C22×C3#369713yes—
330C22×C3#369713yes—
331C22×C3#369713yes—
332C22×C3D4×D913yes—
333C22×C3D4×D913yes—
334C22×C3#369913yes—
335C22×C3#370113yes—
336C22×C3#382811yescharacteristic
337C22×C3#382811yescharacteristic
338C22×C3#382811yescharacteristic
339C22×C3D4×C3⋊D311yescharacteristic
340C22×C3D4×C3⋊D311yescharacteristic
341C22×C3#383011yescharacteristic
342C22×C3#383211yescharacteristic
343C2×D3—182no—
344C2×D3—182no—
345C2×D3—182no—
346C2×D3—186no—
347C2×D3—186no—
348C2×D3—186no—
349C2×D3—361no—
350C2×D3—361no—
351C2×D3—361no—
352C2×D3—361no—
353C2×D3—363no—
354C2×D3—363no—
355C2×D3—363no—
356C2×D3—363no—
357C3×C4—41no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—43no—
361C3×C4—43no—
362C3×C4—43no—
363Dic3—182no—
364Dic3—186no—
365Dic3—361no—
366Dic3—361no—
367Dic3—363no—
368Dic3—363no—
369C32#467511yescharacteristic
370C9#467513yescomplemented
371C23—272no—
372C23—541no—
373C23—541no—
374C23—541no—
375C23—541no—
376C23—541no—
377C23—541no—
378C23C22×C3⋊D911yescharacteristic
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—272no—
389C2×C4—541no—
390C2×C4—541no—
391C2×C4—541no—
392C2×C4—541no—
393C2×C4—541no—
394C2×C4—541no—
395C2×C4—541no—
396C2×C4—541no—
397C2×C3#6343813yes—
398C2×C3#6344513yes—
399C2×C3#6344613yes—
400C2×C3#6344713yes—
401C2×C3#6345613yes—
402C2×C3#6345613yes—
403C2×C3#6346013yes—
404C2×C3#6408511yescharacteristic
405C2×C3#6409211yescharacteristic
406C2×C3#6409311yescharacteristic
407C2×C3#6409411yescharacteristic
408C2×C3#6410311yescharacteristic
409C2×C3#6410311yescharacteristic
410C2×C3#6410711yescharacteristic
411D3—362no—
412D3—366no—
413C22—542no—
414C22—542no—
415C22—542no—
416C22—1081no—
417C22—1081no—
418C22—1081no—
419C22—1081no—
420C22#8873611yescharacteristic
421C22#8873611yescharacteristic
422C22#8873611yescharacteristic
423C22D4×C3⋊D911yescharacteristic
424C22D4×C3⋊D911yescharacteristic
425C22#8874211yescharacteristic
426C22#8874611yescharacteristic
427C4—41no—
428C4—41no—
429C4—41no—
430C4—542no—
431C4—1081no—
432C4—1081no—
433C3#1058835513yescomplemented
434C3#1059237411yescharacteristic
435C2—1082no—
436C2#1172171311yescharacteristic
437C2#1172172911yescharacteristic
438C2#1172173011yescharacteristic
439C2#1172173111yescharacteristic
440C2#1172175011yescharacteristic
441C2#1172175011yescharacteristic
442C2#1172175411yescharacteristic
subgroupG/H
centerC23C22×C3⋊D9
commutator, cosocleC23×C3×C9C23
FrattiniC23×C3C22×C3⋊D3
FittingC3×C9×#95C2
socleC23×C32C22×D3
prime radicalC23×C3⋊D9C22
hypercenter#95C3⋊D9
hypocenterC3×C9#397
LCSG ▷ C23×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
4. C2×#88534C2semidirect
5. C3×C9×#95C2semidirect
6. #11720572C2semidirect
7. #11720572C2semidirect
32. C9×#95D3semidirect
113. C3×#95D9semidirect
248. #95C3⋊D9semidirect
249. C3×C9#397semidirect
370. C9#4675semidirect
433. C3#10588355semidirect

Maximal subgroups

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

subgroupindex
C2×#885362
C2×#885342
C3×C9×#952
#117205722
#105923743
#105883553