Finite Group Databasebeta

← #13144331

Subgroups — #13144331

Order 1728.

930 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
1C2×C3×C9×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#88536C211yesmaximal, characteristic
3C2×#88536C211yesmaximal, characteristic
4C2×#88536C211yesmaximal, characteristic
5C2×C4⋊(C3⋊Dic9)C211yesmaximal, characteristic, complemented
6C2×#88534C211yesmaximal, characteristic, complemented
7#11720572C211yesmaximal, characteristic, complemented
8#10592693—31nomaximal
9#10588674—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
17C3×C9×C4⋊C4—22no—
18#88536—22no—
19#88536—22no—
20#88536—22no—
21C4⋊(C3⋊Dic9)—22no—
22#88534—22no—
23C2×C32×C4⋊C4D311yescharacteristic
24C2×#3752—31no—
25C2×#3752—31no—
26C2×#3752—31no—
27C2×C4⋊(C3⋊Dic3)—31no—
28C2×#3750—31no—
29#63658—31no—
30C2×C9×C4⋊C4D313yescomplemented
31C2×(C2×D9)⋊C4—33no—
32C2×(C2×D9)⋊C4—33no—
33C2×(C2×D9)⋊C4—33no—
34C2×C4⋊Dic9—33no—
35C2×Dic9⋊C4—33no—
36C9⋊#95—33no—
37C23×C3×C9C2311yescharacteristic
38C22×C3⋊D9—21no—
39C22×C3⋊D9—21no—
40C22×C3⋊D9—21no—
41C22×C3⋊D9—21no—
42C22×C3⋊D9—21no—
43C22×C3⋊D9—21no—
44C22×C3⋊D9Q812yes—
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×C4×C9D412yes—
53C2×C3×C4×C9D412yes—
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⋊Dic9—21no—
61C2×C3⋊Dic9—21no—
62C2×C3⋊Dic9D412yes—
63#4994—93no—
64#4994—91no—
65C23×C3⋊D3—31no—
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C22×C3⋊Dic3—31no—
70C22×C3⋊Dic3—31no—
71C22×C3⋊Dic3—31no—
72C23×D9—33no—
73C32×C4⋊C4—22no—
74#3752—62no—
75#3752—62no—
76#3752—62no—
77C4⋊(C3⋊Dic3)—62no—
78#3750—62no—
79C22×C4×C9C2×D313yes—
80C22×C4×C9C2×D313yes—
81C22×C4×C9C2×D313yes—
82C22×Dic9—33no—
83C22×Dic9—33no—
84C22×Dic9—33no—
85C9×C4⋊C4—26no—
86(C2×D9)⋊C4—66no—
87(C2×D9)⋊C4—66no—
88(C2×D9)⋊C4—66no—
89C4⋊Dic9—66no—
90Dic9⋊C4—66no—
91C22×C3×C9C2×D411yescharacteristic
92C22×C3×C9C2×D411yescharacteristic
93C22×C3×C9C2×D411yescharacteristic
94C22×C3×C9C2×Q811yescharacteristic
95C22×C3×C9#4111yescharacteristic
96C22×C3×C9#4111yescharacteristic
97C22×C3×C9#4111yescharacteristic
98C2×C3⋊D9—22no—
99C2×C3⋊D9—22no—
100C2×C3⋊D9—22no—
101C2×C3⋊D9—41no—
102C2×C3⋊D9—41no—
103C2×C3⋊D9—41no—
104C2×C3⋊D9—41no—
105C3×C4×C9—22no—
106C3×C4×C9—22no—
107C3×C4×C9—41no—
108C3⋊Dic9—22no—
109C3⋊Dic9—41no—
110C3⋊Dic9—41no—
111C2×C3×C4⋊C4D913yescomplemented
112C2×C3×C4⋊C4C3⋊D311yescharacteristic
113C2×(C2×D3)⋊C4—93no—
114C2×(C2×D3)⋊C4—93no—
115C2×(C2×D3)⋊C4—93no—
116C2×(C2×D3)⋊C4—91no—
117C2×(C2×D3)⋊C4—91no—
118C2×(C2×D3)⋊C4—91no—
119C2×C4⋊Dic3—93no—
120C2×C4⋊Dic3—91no—
121C2×Dic3⋊C4—93no—
122C2×Dic3⋊C4—91no—
123C3⋊#95—93no—
124C3⋊#95—91no—
125C23×C32C22×D311yescharacteristic
126C22×C3⋊D3—32no—
127C22×C3⋊D3—61no—
128C22×C3⋊D3—61no—
129C22×C3⋊D3—61no—
130C22×C3⋊D3—61no—
131C22×C3⋊D3—61no—
132C22×C3⋊D3—61no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4D1212yes—
141C2×C32×C4#6712yes—
142C2×C3⋊Dic3—32no—
143C2×C3⋊Dic3—61no—
144C2×C3⋊Dic3—61no—
145C2×C3⋊Dic3—61no—
146C2×C3⋊Dic3—61no—
147C2×C3⋊Dic3—61no—
148C2×C3⋊Dic3—61no—
149C2×C3⋊Dic3—61no—
150C2×C3⋊Dic3—61no—
151C23×C9C22×D313yes—
152C22×D9—36no—
153C22×D9—63no—
154C22×D9—63no—
155C22×D9—63no—
156C22×D9—63no—
157C22×D9—63no—
158C22×D9—63no—
159C2×C4×C9—23no—
160C2×C4×C9—23no—
161C2×C4×C9—23no—
162C2×C4×C9—23no—
163C2×C4×C9—23no—
164C2×C4×C9—23no—
165C2×C4×C9—23no—
166C2×C4×C9D1216yes—
167C2×C4×C9#6716yes—
168C2×Dic9—36no—
169C2×Dic9—63no—
170C2×Dic9—63no—
171C2×Dic9—63no—
172C2×Dic9—63no—
173C2×Dic9—63no—
174C2×Dic9—63no—
175C2×Dic9—63no—
176C2×Dic9—63no—
177#396—271noSylow
178C2×C3×C9#12111yescharacteristic
179C2×C3×C9#12111yescharacteristic
180C2×C3×C9#12111yescharacteristic
181C2×C3×C9C22⋊Q811yescharacteristic
182C2×C3×C9C22⋊Q811yescharacteristic
183C2×C3×C9C22⋊Q811yescharacteristic
184C2×C3×C9#12611yescharacteristic
185C3⋊D9—42no—
186C23×D3—93no—
187C23×D3—91no—
188C22×C3×C4C2×D913yes—
189C22×C3×C4C2×D913yes—
190C22×C3×C4C2×D913yes—
191C22×C3×C4C2×C3⋊D311yescharacteristic
192C22×C3×C4C2×C3⋊D311yescharacteristic
193C22×C3×C4C2×C3⋊D311yescharacteristic
194C22×Dic3—93no—
195C22×Dic3—91no—
196C22×Dic3—93no—
197C22×Dic3—93no—
198C22×Dic3—91no—
199C22×Dic3—91no—
200C3×C4⋊C4—26no—
201C3×C4⋊C4—22no—
202(C2×D3)⋊C4—182no—
203(C2×D3)⋊C4—186no—
204(C2×D3)⋊C4—186no—
205(C2×D3)⋊C4—186no—
206(C2×D3)⋊C4—182no—
207(C2×D3)⋊C4—182no—
208C4⋊Dic3—182no—
209C4⋊Dic3—186no—
210Dic3⋊C4—186no—
211Dic3⋊C4—182no—
212C22×C32C2×D1211yescharacteristic
213C22×C32#23511yescharacteristic
214C22×C32D3×D411yescharacteristic
215C22×C32#23711yescharacteristic
216C22×C32D3×Q811yescharacteristic
217C22×C32#23911yescharacteristic
218C22×C32C2×#6711yescharacteristic
219C2×C3⋊D3—62no—
220C2×C3⋊D3—62no—
221C2×C3⋊D3—62no—
222C2×C3⋊D3—121no—
223C2×C3⋊D3—121no—
224C2×C3⋊D3—121no—
225C2×C3⋊D3—121no—
226C32×C4—22no—
227C32×C4—22no—
228C32×C4—41no—
229C3⋊Dic3—62no—
230C3⋊Dic3—121no—
231C3⋊Dic3—121no—
232C22×C9C2×D1213yes—
233C22×C9#23513yes—
234C22×C9D3×D413yes—
235C22×C9#23713yes—
236C22×C9D3×Q813yes—
237C22×C9#23913yes—
238C22×C9C2×#6713yes—
239C2×D9—66no—
240C2×D9—66no—
241C2×D9—66no—
242C2×D9—123no—
243C2×D9—123no—
244C2×D9—123no—
245C2×D9—123no—
246C4×C9—26no—
247C4×C9—26no—
248C4×C9—43no—
249Dic9—66no—
250Dic9—123no—
251Dic9—123no—
252C2×C4⋊C4—271no—
253C2×C4⋊C4—271no—
254C2×C4⋊C4C3⋊D911yescharacteristic, complemented
255C2×C22⋊C4—271no—
256C2×C22⋊C4—271no—
257C2×C22⋊C4—271no—
258#95—271no—
259C3×C9#39611yescharacteristic, complemented, Sylow
260C23×C3C22×D913yes—
261C23×C3C22×C3⋊D311yescharacteristic
262C22×D3—96no—
263C22×D3—92no—
264C22×D3—183no—
265C22×D3—183no—
266C22×D3—183no—
267C22×D3—183no—
268C22×D3—183no—
269C22×D3—183no—
270C22×D3—181no—
271C22×D3—181no—
272C22×D3—181no—
273C22×D3—181no—
274C22×D3—181no—
275C22×D3—181no—
276C2×C3×C4—23no—
277C2×C3×C4—23no—
278C2×C3×C4—23no—
279C2×C3×C4—23no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—23no—
285C2×C3×C4—23no—
286C2×C3×C4—23no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4D3616yes—
291C2×C3×C4#61116yes—
292C2×C3×C4C4⋊(C3⋊D3)12yes—
293C2×C3×C4#63812yes—
294C2×Dic3—96no—
295C2×Dic3—92no—
296C2×Dic3—183no—
297C2×Dic3—183no—
298C2×Dic3—181no—
299C2×Dic3—181no—
300C2×Dic3—183no—
301C2×Dic3—183no—
302C2×Dic3—183no—
303C2×Dic3—183no—
304C2×Dic3—183no—
305C2×Dic3—183no—
306C2×Dic3—181no—
307C2×Dic3—181no—
308C2×Dic3—181no—
309C2×Dic3—181no—
310C2×Dic3—181no—
311C2×Dic3—181no—
312C2×C32#89511yescharacteristic
313C2×C32#89611yescharacteristic
314C2×C32#89711yescharacteristic
315C2×C32#89811yescharacteristic
316C2×C32#93011yescharacteristic
317C2×C32#93911yescharacteristic
318C2×C32#94611yescharacteristic
319C3⋊D3—122no—
320C2×C9#89513yes—
321C2×C9#89613yes—
322C2×C9#89713yes—
323C2×C9#89813yes—
324C2×C9#93013yes—
325C2×C9#93913yes—
326C2×C9#94613yes—
327D9—126no—
328C24—271no—
329C22×C4—271no—
330C22×C4—271no—
331C22×C4—271no—
332C22×C4C2×C3⋊D911yescharacteristic
333C22×C4C2×C3⋊D911yescharacteristic
334C22×C4C2×C3⋊D911yescharacteristic
335C4⋊C4—22no—
336C4⋊C4—542no—
337C4⋊C4—542no—
338C22⋊C4—542no—
339C22⋊C4—542no—
340C22⋊C4—542no—
341C22×C3C2×D3613yes—
342C22×C3#369713yes—
343C22×C3D4×D913yes—
344C22×C3#369913yes—
345C22×C3Q8×D913yes—
346C22×C3#370113yes—
347C22×C3C2×#61113yes—
348C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
349C22×C3#382811yescharacteristic
350C22×C3D4×C3⋊D311yescharacteristic
351C22×C3#383011yescharacteristic
352C22×C3Q8×C3⋊D311yescharacteristic
353C22×C3#383211yescharacteristic
354C22×C3C2×#63811yescharacteristic
355C2×D3—186no—
356C2×D3—182no—
357C2×D3—182no—
358C2×D3—182no—
359C2×D3—186no—
360C2×D3—186no—
361C2×D3—363no—
362C2×D3—363no—
363C2×D3—361no—
364C2×D3—361no—
365C2×D3—361no—
366C2×D3—361no—
367C2×D3—363no—
368C2×D3—363no—
369C3×C4—26no—
370C3×C4—26no—
371C3×C4—22no—
372C3×C4—22no—
373C3×C4—43no—
374C3×C4—41no—
375Dic3—182no—
376Dic3—186no—
377Dic3—363no—
378Dic3—361no—
379Dic3—361no—
380Dic3—363no—
381C32#499411yescharacteristic
382C9#499413yescomplemented
383C23—272no—
384C23—541no—
385C23—541no—
386C23—541no—
387C23—541no—
388C23—541no—
389C23—541no—
390C23C22×C3⋊D911yescharacteristic
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—272no—
399C2×C4—541no—
400C2×C4—541no—
401C2×C4—541no—
402C2×C4—541no—
403C2×C4—541no—
404C2×C4—541no—
405C2×C4—541no—
406C2×C4—541no—
407C2×C4C4⋊(C3⋊D9)12yes—
408C2×C4#623312yes—
409C2×C3#6345713yes—
410C2×C3#6345813yes—
411C2×C3#6345913yes—
412C2×C3#6346013yes—
413C2×C3#6349213yes—
414C2×C3#6350113yes—
415C2×C3#6350813yes—
416C2×C3#6410411yescharacteristic
417C2×C3#6410511yescharacteristic
418C2×C3#6410611yescharacteristic
419C2×C3#6410711yescharacteristic
420C2×C3#6413911yescharacteristic
421C2×C3#6414811yescharacteristic
422C2×C3#6415511yescharacteristic
423D3—366no—
424D3—362no—
425C22—542no—
426C22—542no—
427C22—542no—
428C22—1081no—
429C22—1081no—
430C22—1081no—
431C22—1081no—
432C22C2×C4⋊(C3⋊D9)11yescharacteristic
433C22#8873611yescharacteristic
434C22D4×C3⋊D911yescharacteristic
435C22#8874211yescharacteristic
436C22Q8×C3⋊D911yescharacteristic
437C22#8874611yescharacteristic
438C22C2×#623311yescharacteristic
439C4—22no—
440C4—22no—
441C4—41no—
442C4—542no—
443C4—1081no—
444C4—1081no—
445C3#1058867413yescomplemented
446C3#1059269311yescharacteristic
447C2—1082no—
448C2#1172175111yescharacteristic
449C2#1172175211yescharacteristic
450C2#1172175311yescharacteristic
451C2#1172175411yescharacteristic
452C2#1172181911yescharacteristic
453C2#1172183911yescharacteristic
454C2#1172185611yescharacteristic
subgroupG/H
centerC23C22×C3⋊D9
commutator, cosocleC23×C3×C9C23
FrattiniC23×C3C22×C3⋊D3
FittingC2×C3×C9×C4⋊C4C2
socleC23×C32C22×D3
prime radicalC23×C3⋊D9C22
hypercenterC2×C4⋊C4C3⋊D9
hypocenterC3×C9#396
LCSG ▷ C23×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C9×C4⋊C4C2semidirect
5. C2×C4⋊(C3⋊Dic9)C2semidirect
6. C2×#88534C2semidirect
7. #11720572C2semidirect
30. C2×C9×C4⋊C4D3semidirect
111. C2×C3×C4⋊C4D9semidirect
254. C2×C4⋊C4C3⋊D9semidirect
259. C3×C9#396semidirect
382. C9#4994semidirect
445. C3#10588674semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C9×C4⋊C42
C2×#885362
C2×C4⋊(C3⋊Dic9)2
C2×#885342
#117205722
#105926933
#105886743