Finite Group Databasebeta

← #13153911

Subgroups — #13153911

Order 1728.

1180 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
1C3×C4×D4×C9C211yesmaximal, characteristic, complemented
2#11721840C211yesmaximal, characteristic
3#11721836C211yesmaximal, characteristic, complemented
4#11721818C212yesmaximal, complemented
5C4×#6233C212yesmaximal, complemented
6#11721749C211yesmaximal, characteristic, complemented
7#11721743C211yesmaximal, characteristic, complemented
8#11721731C212yesmaximal, complemented
9#11721729C212yesmaximal, complemented
10#11721712C211yesmaximal, characteristic, complemented
11C4×#6226C211yesmaximal, characteristic, complemented
12#10596030—31nomaximal
13#10589661—33nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C22×C3×C4×C9C2212yes—
16C2×#6233C2212yes—
17C2×#6233C2212yes—
18C2×C4⋊(C3⋊D9)C2211yescharacteristic
19C2×C4×C3⋊D9C2212yes—
20C2×#6226C2211yescharacteristic
21C2×#6226C2211yescharacteristic
22C3×C9×C4⋊C4C2211yescharacteristic
23C3×C9×C22⋊C4C2212yes—
24C3×C42×C9C2211yescharacteristic
25#88551C2212yes—
26#88551C2212yes—
27#88536C2212yes—
28#88536C2212yes—
29#88536C2212yes—
30C4⋊(C3⋊Dic9)C2211yescharacteristic
31#88534C2212yes—
32#88534C2211yescharacteristic
33#88534C2211yescharacteristic
34C4×C3⋊Dic9C2212yes—
35C4×C3⋊Dic9C2211yescharacteristic
36C4×C3⋊Dic9C2211yescharacteristic
37C32×C4×D4D311yescharacteristic
38#64149—31no—
39#64145—31no—
40#64138—32no—
41C4×#638—32no—
42#64102—31no—
43#64096—31no—
44#64094—32no—
45#64092—32no—
46#64084—31no—
47C4×#634—31no—
48C4×D4×C9D313yescomplemented
49#63502—33no—
50#63498—33no—
51#63491—36no—
52C4×#611—36no—
53#63455—33no—
54#63449—33no—
55#63447—36no—
56#63445—36no—
57#63437—33no—
58C4×Dic18—33no—
59C23×C3×C9C2312yes—
60C22×C3⋊D9C2312yes—
61C3×D4×C9—22no—
62C2×C3×C4×C9—42no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2312yes—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2311yescharacteristic
67#6233—24no—
68#6233—42no—
69C2×C3⋊Dic9C2311yescharacteristic
70C2×C3⋊Dic9C2312yes—
71C2×C3⋊Dic9C2311yescharacteristic
72C2×C3⋊Dic9C2311yescharacteristic
73C2×C3⋊Dic9C2311yescharacteristic
74C4⋊(C3⋊D9)—22no—
75C4×C3⋊D9—42no—
76#6226—22no—
77#6226—22no—
78#5559—93no—
79#5559—91no—
80C2×C32×D4C2×D311yescharacteristic
81C22×C32×C4C2×D312yes—
82C2×#638—32no—
83C2×#638—32no—
84C2×C4⋊(C3⋊D3)—31no—
85C2×C4×C3⋊D3—32no—
86C2×#634—31no—
87C2×#634—31no—
88C32×C4⋊C4C2×D311yescharacteristic
89C32×C22⋊C4C2×D312yes—
90C32×C42C2×D311yescharacteristic
91#3757—32no—
92#3757—32no—
93#3752—32no—
94#3752—32no—
95#3752—32no—
96C4⋊(C3⋊Dic3)—31no—
97#3750—31no—
98#3750—31no—
99#3750—32no—
100C4×C3⋊Dic3—31no—
101C4×C3⋊Dic3—31no—
102C4×C3⋊Dic3—32no—
103C2×D4×C9C2×D313yes—
104C22×C4×C9C2×D316yes—
105C2×#611—36no—
106C2×#611—36no—
107C2×D36—33no—
108C2×C4×D9—36no—
109C2×Dic18—33no—
110C2×Dic18—33no—
111C9×C4⋊C4C2×D313yes—
112C9×C22⋊C4C2×D316yes—
113C42×C9C2×D313yes—
114#3676—36no—
115#3676—36no—
116(C2×D9)⋊C4—36no—
117(C2×D9)⋊C4—36no—
118(C2×D9)⋊C4—36no—
119C4⋊Dic9—33no—
120Dic9⋊C4—33no—
121Dic9⋊C4—33no—
122Dic9⋊C4—36no—
123C4×Dic9—36no—
124C4×Dic9—33no—
125C4×Dic9—33no—
126C22×C3×C9—22no—
127C22×C3×C9—22no—
128C22×C3×C9—22no—
129C22×C3×C9C2411yescharacteristic
130C2×C3⋊D9—22no—
131C2×C3⋊D9—22no—
132C2×C3⋊D9—22no—
133C3×C4×C9—21no—
134C3×C4×C9—22no—
135C3×C4×C9—21no—
136C3×C4×C9#4112yes—
137C3⋊Dic9—22no—
138C3⋊Dic9—21no—
139C3⋊Dic9—21no—
140C3⋊Dic9—21no—
141C3⋊Dic9#4112yes—
142C3×C4×D4D913yescomplemented
143C3×C4×D4C3⋊D311yescharacteristic
144#940—93no—
145#940—91no—
146#936—93no—
147#936—91no—
148#929—96no—
149#929—92no—
150C4×#67—96no—
151C4×#67—92no—
152#893—93no—
153#893—91no—
154#887—93no—
155#887—91no—
156#885—96no—
157#885—92no—
158#883—96no—
159#883—92no—
160#875—93no—
161#875—91no—
162C4×Dic6—93no—
163C4×Dic6—91no—
164C23×C32C22×D312yes—
165C22×C3⋊D3—32no—
166C32×D4—22no—
167C2×C32×C4—42no—
168C2×C32×C4C22×D312yes—
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172#638—64no—
173#638—122no—
174C2×C3⋊Dic3—31no—
175C2×C3⋊Dic3—31no—
176C2×C3⋊Dic3—31no—
177C2×C3⋊Dic3—32no—
178C2×C3⋊Dic3—31no—
179C4⋊(C3⋊D3)—62no—
180C4×C3⋊D3—122no—
181#634—62no—
182#634—62no—
183C23×C9C22×D316yes—
184C22×D9—36no—
185D4×C9—26no—
186C2×C4×C9—46no—
187C2×C4×C9C22×D316yes—
188C2×C4×C9C22×D313yes—
189C2×C4×C9C22×D313yes—
190C2×C4×C9C22×D313yes—
191#611—612no—
192#611—126no—
193C2×Dic9—33no—
194C2×Dic9—33no—
195C2×Dic9—33no—
196C2×Dic9—36no—
197C2×Dic9—33no—
198D36—66no—
199C4×D9—126no—
200Dic18—66no—
201Dic18—66no—
202#559—271noSylow
203C2×C3×C9—42no—
204C2×C3×C9C2×#4111yescharacteristic
205C2×C3×C9C2×#4111yescharacteristic
206C2×C3×C9H(5,𝔽2)11yescharacteristic
207C3⋊D9—42no—
208C2×C3×D4C2×D913yes—
209C2×C3×D4C2×C3⋊D311yescharacteristic
210C22×C3×C4C2×D916yes—
211C22×C3×C4C2×C3⋊D312yes—
212C2×#67—96no—
213C2×#67—92no—
214C2×#67—96no—
215C2×#67—92no—
216C2×D12—93no—
217C2×D12—91no—
218C2×C4×D3—96no—
219C2×C4×D3—92no—
220C2×Dic6—93no—
221C2×Dic6—93no—
222C2×Dic6—91no—
223C2×Dic6—91no—
224C3×C4⋊C4C2×D913yes—
225C3×C4⋊C4C2×C3⋊D311yescharacteristic
226C3×C22⋊C4C2×D916yes—
227C3×C22⋊C4C2×C3⋊D312yes—
228C3×C42C2×D913yes—
229C3×C42C2×C3⋊D311yescharacteristic
230#217—96no—
231#217—96no—
232#217—92no—
233#217—92no—
234(C2×D3)⋊C4—96no—
235(C2×D3)⋊C4—96no—
236(C2×D3)⋊C4—96no—
237(C2×D3)⋊C4—92no—
238(C2×D3)⋊C4—92no—
239(C2×D3)⋊C4—92no—
240C4⋊Dic3—93no—
241C4⋊Dic3—91no—
242Dic3⋊C4—93no—
243Dic3⋊C4—93no—
244Dic3⋊C4—96no—
245Dic3⋊C4—91no—
246Dic3⋊C4—91no—
247Dic3⋊C4—92no—
248C4×Dic3—93no—
249C4×Dic3—93no—
250C4×Dic3—96no—
251C4×Dic3—91no—
252C4×Dic3—91no—
253C4×Dic3—92no—
254C22×C32—22no—
255C22×C32—22no—
256C22×C32—22no—
257C22×C32C23×D311yescharacteristic
258C2×C3⋊D3—62no—
259C2×C3⋊D3—62no—
260C2×C3⋊D3—62no—
261C32×C4—22no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4#23512yes—
265C3⋊Dic3—32no—
266C3⋊Dic3—62no—
267C3⋊Dic3—61no—
268C3⋊Dic3—61no—
269C3⋊Dic3—61no—
270C22×C9—26no—
271C22×C9—26no—
272C22×C9—26no—
273C22×C9C23×D313yes—
274C2×D9—66no—
275C2×D9—66no—
276C2×D9—66no—
277C4×C9—26no—
278C4×C9—23no—
279C4×C9—23no—
280C4×C9#23516yes—
281Dic9—36no—
282Dic9—66no—
283Dic9—63no—
284Dic9—63no—
285Dic9—63no—
286#127—271no—
287#124—271no—
288#124—272no—
289#124—271no—
290#123—272no—
291#123—272no—
292C4×Q8—271no—
293C4×Q8—271no—
294C4×D4—272no—
295C4×D4—271no—
296C4×D4C3⋊D911yescharacteristic, complemented
297C3×C9#55911yescharacteristic, complemented, Sylow
298C23×C3C22×D916yes—
299C23×C3C22×C3⋊D312yes—
300C22×D3—92no—
301C22×D3—96no—
302C3×D4—26no—
303C3×D4—22no—
304C2×C3×C4—42no—
305C2×C3×C4—46no—
306C2×C3×C4C22×D916yes—
307C2×C3×C4C22×D913yes—
308C2×C3×C4C22×D913yes—
309C2×C3×C4C22×D913yes—
310C2×C3×C4C22×C3⋊D312yes—
311C2×C3×C4C22×C3⋊D311yescharacteristic
312C2×C3×C4C22×C3⋊D311yescharacteristic
313C2×C3×C4C22×C3⋊D311yescharacteristic
314#67—184no—
315#67—1812no—
316#67—362no—
317#67—366no—
318C2×Dic3—91no—
319C2×Dic3—93no—
320C2×Dic3—91no—
321C2×Dic3—91no—
322C2×Dic3—92no—
323C2×Dic3—91no—
324C2×Dic3—93no—
325C2×Dic3—93no—
326C2×Dic3—96no—
327C2×Dic3—93no—
328D12—186no—
329D12—182no—
330C4×D3—362no—
331C4×D3—366no—
332Dic6—182no—
333Dic6—186no—
334Dic6—186no—
335Dic6—182no—
336C2×C32—42no—
337C2×C32C2×#23511yescharacteristic
338C2×C32#100411yescharacteristic
339C2×C32D3×#4111yescharacteristic
340C3⋊D3—122no—
341C2×C9—46no—
342C2×C9C2×#23513yes—
343C2×C9#100413yes—
344C2×C9D3×#4113yes—
345D9—126no—
346C2×Q8—271no—
347C2×Q8—271no—
348C2×D4—272no—
349C2×D4—272no—
350C2×D4—271no—
351C2×D4C2×C3⋊D911yescharacteristic
352C22×C4—272no—
353C22×C4C2×C3⋊D912yes—
354C4⋊C4—271no—
355C4⋊C4—272no—
356C4⋊C4—271no—
357C4⋊C4—271no—
358C4⋊C4C2×C3⋊D911yescharacteristic
359C22⋊C4—272no—
360C22⋊C4—272no—
361C22⋊C4—272no—
362C22⋊C4—272no—
363C22⋊C4—272no—
364C22⋊C4C2×C3⋊D912yes—
365C42—272no—
366C42—271no—
367C42—271no—
368C42C2×C3⋊D911yescharacteristic
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—26no—
373C22×C3—26no—
374C22×C3—26no—
375C22×C3C23×D913yes—
376C22×C3C23×C3⋊D311yescharacteristic
377C2×D3—182no—
378C2×D3—182no—
379C2×D3—182no—
380C2×D3—186no—
381C2×D3—186no—
382C2×D3—186no—
383C3×C4—22no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—26no—
387C3×C4—23no—
388C3×C4—23no—
389C3×C4#369716yes—
390C3×C4#382812yes—
391Dic3—92no—
392Dic3—96no—
393Dic3—182no—
394Dic3—181no—
395Dic3—181no—
396Dic3—181no—
397Dic3—186no—
398Dic3—183no—
399Dic3—183no—
400Dic3—183no—
401C32#555911yescharacteristic
402C9#555913yescomplemented
403C23—272no—
404C23C22×C3⋊D912yes—
405Q8—542no—
406Q8—542no—
407D4—22no—
408D4—544no—
409D4—542no—
410D4—1082no—
411C2×C4—42no—
412C2×C4—271no—
413C2×C4—272no—
414C2×C4—271no—
415C2×C4—271no—
416C2×C4—271no—
417C2×C4—1082no—
418C2×C4C22×C3⋊D912yes—
419C2×C4C22×C3⋊D911yescharacteristic
420C2×C4C22×C3⋊D911yescharacteristic
421C2×C4C22×C3⋊D911yescharacteristic
422C2×C3—42no—
423C2×C3—46no—
424C2×C3C2×#369713yes—
425C2×C3#6371013yes—
426C2×C3#41×D913yes—
427C2×C3C2×#382811yescharacteristic
428C2×C3#6436111yescharacteristic
429C2×C3#41×C3⋊D311yescharacteristic
430D3—362no—
431D3—366no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—542no—
436C22—542no—
437C22—542no—
438C22C23×C3⋊D911yescharacteristic
439C4—21no—
440C4—22no—
441C4—21no—
442C4—272no—
443C4—542no—
444C4—541no—
445C4—541no—
446C4—541no—
447C4#8873612yes—
448C3#1058966113yescomplemented
449C3#1059603011yescharacteristic
450C2—42no—
451C2—1082no—
452C2C2×#8873611yescharacteristic
453C2#1172266111yescharacteristic
454C2#41×C3⋊D911yescharacteristic
subgroupG/H
centerC22C23×C3⋊D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C23×C3⋊D3
FittingC3×C4×D4×C9C2
socleC22×C32C23×D3
prime radical#11721840C2
hypercenterC4×D4C3⋊D9
hypocenterC3×C9#559
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C3×C4×D4×C9C2semidirect
3. #11721836C2semidirect
4. #11721818C2semidirect
5. C4×#6233C2semidirect
6. #11721749C2semidirect
7. #11721743C2semidirect
8. #11721731C2semidirect
9. #11721729C2semidirect
10. #11721712C2semidirect
11. C4×#6226C2semidirect
48. C4×D4×C9D3semidirect
142. C3×C4×D4D9semidirect
296. C4×D4C3⋊D9semidirect
297. C3×C9#559semidirect
402. C9#5559semidirect
448. C3#10589661semidirect

Maximal subgroups

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

subgroupindex
C3×C4×D4×C92
#117218402
#117218362
#117218182
C4×#62332
#117217492
#117217432
#117217312
#117217292
#117217122
C4×#62262
#105960303
#105896613