Finite Group Databasebeta

← #13154287

Subgroups — #13154287

Order 1728.

1080 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
1C3×C9×C4⋊Q8C211yesmaximal, characteristic, complemented
2#11721857C212yesmaximal, complemented
3#11721856C212yesmaximal, complemented
4#11721754C212yesmaximal, complemented
5#11721753C211yesmaximal, characteristic, complemented
6#11721751C211yesmaximal, characteristic
7#11721750C212yesmaximal, complemented
8#11721749C211yesmaximal, characteristic, complemented
9#11721743C211yesmaximal, characteristic, complemented
10#11721712C211yesmaximal, characteristic, complemented
11#11721709C211yesmaximal, characteristic, complemented
12#10596204—31nomaximal
13#10589835—33nomaximal
14C2×C3×Q8×C9C2212yes—
15C2×C4⋊(C3⋊D9)C2211yescharacteristic
16C2×C4⋊(C3⋊D9)C2212yes—
17C2×C4×C3⋊D9C2211yescharacteristic
18C2×C4×C3⋊D9C2212yes—
19C2×#6226C2211yescharacteristic
20C3×C9×C4⋊C4C2211yescharacteristic
21C3×C9×C4⋊C4C2211yescharacteristic
22C3×C9×C4⋊C4C2212yes—
23C3×C42×C9C2211yescharacteristic
24#88536C2212yes—
25#88536C2211yescharacteristic
26#88536C2212yes—
27#88536C2212yes—
28#88536C2212yes—
29#88536C2212yes—
30#88536C2211yescharacteristic
31C4⋊(C3⋊Dic9)C2212yes—
32#88534C2212yes—
33#88534C2211yescharacteristic
34#88534C2211yescharacteristic
35C4×C3⋊Dic9C2211yescharacteristic
36C4×C3⋊Dic9C2212yes—
37C32×C4⋊Q8D311yescharacteristic
38#64156—32no—
39#64155—32no—
40#64107—32no—
41#64106—31no—
42#64104—31no—
43#64103—32no—
44#64102—31no—
45#64096—31no—
46#64084—31no—
47#64081—31no—
48C9×C4⋊Q8D313yescomplemented
49#63509—36no—
50#63508—36no—
51#63460—36no—
52#63459—33no—
53#63457—33no—
54#63456—36no—
55#63455—33no—
56#63449—33no—
57#63437—33no—
58#63434—33no—
59C22×C3⋊D9C2311yescharacteristic
60C22×C3⋊D9C2312yes—
61C3×Q8×C9—42no—
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2312yes—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3⋊Dic9C2312yes—
68C2×C3⋊Dic9C2311yescharacteristic
69C2×C3⋊Dic9C2311yescharacteristic
70C2×C3⋊Dic9C2311yescharacteristic
71C4⋊(C3⋊D9)—22no—
72C4⋊(C3⋊D9)—42no—
73C4×C3⋊D9—22no—
74C4×C3⋊D9—42no—
75#6226—22no—
76#5733—93no—
77#5733—91no—
78C2×C32×Q8C2×D312yes—
79C2×C4⋊(C3⋊D3)—31no—
80C2×C4⋊(C3⋊D3)—32no—
81C2×C4×C3⋊D3—31no—
82C2×C4×C3⋊D3—32no—
83C2×#634—31no—
84C32×C4⋊C4C2×D311yescharacteristic
85C32×C4⋊C4C2×D311yescharacteristic
86C32×C4⋊C4C2×D312yes—
87C32×C42C2×D311yescharacteristic
88#3752—31no—
89#3752—31no—
90#3752—32no—
91#3752—32no—
92#3752—32no—
93#3752—32no—
94#3752—32no—
95C4⋊(C3⋊Dic3)—32no—
96#3750—31no—
97#3750—31no—
98#3750—32no—
99C4×C3⋊Dic3—31no—
100C4×C3⋊Dic3—32no—
101C2×Q8×C9C2×D316yes—
102C2×D36—33no—
103C2×D36—36no—
104C2×C4×D9—33no—
105C2×C4×D9—36no—
106C2×Dic18—33no—
107C9×C4⋊C4C2×D313yes—
108C9×C4⋊C4C2×D313yes—
109C9×C4⋊C4C2×D316yes—
110C42×C9C2×D313yes—
111(C2×D9)⋊C4—36no—
112(C2×D9)⋊C4—36no—
113(C2×D9)⋊C4—33no—
114(C2×D9)⋊C4—36no—
115(C2×D9)⋊C4—36no—
116(C2×D9)⋊C4—33no—
117(C2×D9)⋊C4—36no—
118C4⋊Dic9—36no—
119Dic9⋊C4—33no—
120Dic9⋊C4—36no—
121Dic9⋊C4—33no—
122C4×Dic9—36no—
123C4×Dic9—33no—
124C22×C3×C9C2411yescharacteristic
125C2×C3⋊D9—21no—
126C2×C3⋊D9—22no—
127C2×C3⋊D9—22no—
128C2×C3⋊D9—21no—
129C2×C3⋊D9—21no—
130C2×C3⋊D9—22no—
131C3×C4×C9—22no—
132C3×C4×C9—22no—
133C3×C4×C9—21no—
134C3×C4×C9—21no—
135C3×C4×C9#4112yes—
136C3⋊Dic9—22no—
137C3⋊Dic9—21no—
138C3⋊Dic9—21no—
139C3⋊Dic9—21no—
140C3×C4⋊Q8D913yescomplemented
141C3×C4⋊Q8C3⋊D311yescharacteristic
142#947—96no—
143#947—92no—
144#946—96no—
145#946—92no—
146#898—96no—
147#898—92no—
148#897—93no—
149#897—91no—
150#895—93no—
151#895—91no—
152#894—96no—
153#894—92no—
154#893—93no—
155#893—91no—
156#887—93no—
157#887—91no—
158#875—93no—
159#875—91no—
160#872—93no—
161#872—91no—
162C22×C3⋊D3—32no—
163C22×C3⋊D3—31no—
164C32×Q8—42no—
165C2×C32×C4C22×D312yes—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D312yes—
169C2×C32×C4C22×D311yescharacteristic
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—31no—
172C2×C3⋊Dic3—32no—
173C2×C3⋊Dic3—31no—
174C4⋊(C3⋊D3)—62no—
175C4⋊(C3⋊D3)—122no—
176C4×C3⋊D3—62no—
177C4×C3⋊D3—122no—
178#634—62no—
179C22×D9—36no—
180C22×D9—33no—
181Q8×C9—46no—
182C2×C4×C9C22×D316yes—
183C2×C4×C9C22×D313yes—
184C2×C4×C9C22×D313yes—
185C2×C4×C9C22×D316yes—
186C2×C4×C9C22×D313yes—
187C2×Dic9—33no—
188C2×Dic9—33no—
189C2×Dic9—36no—
190C2×Dic9—33no—
191D36—66no—
192D36—126no—
193C4×D9—66no—
194C4×D9—126no—
195Dic18—66no—
196#542—271noSylow
197C2×C3×C9C2×#4111yescharacteristic
198C2×C3×C9H(5,𝔽2)11yescharacteristic
199C2×C3×C92−1+411yescharacteristic
200C3⋊D9—41no—
201C3⋊D9—42no—
202C2×C3×Q8C2×D916yes—
203C2×C3×Q8C2×C3⋊D312yes—
204C2×D12—93no—
205C2×D12—91no—
206C2×D12—96no—
207C2×D12—92no—
208C2×C4×D3—93no—
209C2×C4×D3—91no—
210C2×C4×D3—96no—
211C2×C4×D3—92no—
212C2×Dic6—93no—
213C2×Dic6—91no—
214C3×C4⋊C4C2×D913yes—
215C3×C4⋊C4C2×D913yes—
216C3×C4⋊C4C2×D916yes—
217C3×C4⋊C4C2×C3⋊D311yescharacteristic
218C3×C4⋊C4C2×C3⋊D311yescharacteristic
219C3×C4⋊C4C2×C3⋊D312yes—
220C3×C42C2×D913yes—
221C3×C42C2×C3⋊D311yescharacteristic
222(C2×D3)⋊C4—93no—
223(C2×D3)⋊C4—93no—
224(C2×D3)⋊C4—96no—
225(C2×D3)⋊C4—96no—
226(C2×D3)⋊C4—96no—
227(C2×D3)⋊C4—96no—
228(C2×D3)⋊C4—96no—
229(C2×D3)⋊C4—91no—
230(C2×D3)⋊C4—91no—
231(C2×D3)⋊C4—92no—
232(C2×D3)⋊C4—92no—
233(C2×D3)⋊C4—92no—
234(C2×D3)⋊C4—92no—
235(C2×D3)⋊C4—92no—
236C4⋊Dic3—96no—
237C4⋊Dic3—92no—
238Dic3⋊C4—93no—
239Dic3⋊C4—93no—
240Dic3⋊C4—96no—
241Dic3⋊C4—91no—
242Dic3⋊C4—91no—
243Dic3⋊C4—92no—
244C4×Dic3—93no—
245C4×Dic3—96no—
246C4×Dic3—91no—
247C4×Dic3—92no—
248C22×C32C23×D311yescharacteristic
249C2×C3⋊D3—62no—
250C2×C3⋊D3—61no—
251C2×C3⋊D3—61no—
252C2×C3⋊D3—62no—
253C2×C3⋊D3—62no—
254C2×C3⋊D3—61no—
255C32×C4—22no—
256C32×C4—22no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4#23912yes—
260C3⋊Dic3—61no—
261C3⋊Dic3—62no—
262C3⋊Dic3—61no—
263C3⋊Dic3—61no—
264C22×C9C23×D313yes—
265C2×D9—66no—
266C2×D9—63no—
267C2×D9—63no—
268C2×D9—66no—
269C2×D9—66no—
270C2×D9—63no—
271C4×C9—26no—
272C4×C9—26no—
273C4×C9—23no—
274C4×C9—23no—
275C4×C9#23916yes—
276Dic9—63no—
277Dic9—66no—
278Dic9—63no—
279Dic9—63no—
280C4⋊Q8C3⋊D911yescharacteristic, complemented
281#126—272no—
282#124—272no—
283#124—271no—
284#123—272no—
285C22⋊Q8—272no—
286C22⋊Q8—271no—
287#121—271no—
288C4×Q8—271no—
289C4×D4—271no—
290#117—271no—
291C3×C9#54211yescharacteristic, complemented, Sylow
292C22×D3—92no—
293C22×D3—91no—
294C22×D3—96no—
295C22×D3—93no—
296C3×Q8—42no—
297C3×Q8—46no—
298C2×C3×C4C22×D916yes—
299C2×C3×C4C22×D913yes—
300C2×C3×C4C22×D913yes—
301C2×C3×C4C22×D916yes—
302C2×C3×C4C22×D913yes—
303C2×C3×C4C22×C3⋊D312yes—
304C2×C3×C4C22×C3⋊D311yescharacteristic
305C2×C3×C4C22×C3⋊D311yescharacteristic
306C2×C3×C4C22×C3⋊D312yes—
307C2×C3×C4C22×C3⋊D311yescharacteristic
308C2×Dic3—91no—
309C2×Dic3—91no—
310C2×Dic3—92no—
311C2×Dic3—91no—
312C2×Dic3—93no—
313C2×Dic3—93no—
314C2×Dic3—96no—
315C2×Dic3—93no—
316D12—186no—
317D12—182no—
318D12—362no—
319D12—366no—
320C4×D3—186no—
321C4×D3—182no—
322C4×D3—362no—
323C4×D3—366no—
324Dic6—186no—
325Dic6—182no—
326C2×C32#100411yescharacteristic
327C2×C32C2×#23911yescharacteristic
328C2×C32#100711yescharacteristic
329C3⋊D3—122no—
330C3⋊D3—121no—
331C2×C9#100413yes—
332C2×C9C2×#23913yes—
333C2×C9#100713yes—
334D9—126no—
335D9—123no—
336C2×Q8—271no—
337C2×Q8C2×C3⋊D912yes—
338C2×D4—271no—
339C2×D4—272no—
340C22×C4—271no—
341C22×C4—272no—
342C4⋊C4—271no—
343C4⋊C4—271no—
344C4⋊C4—272no—
345C4⋊C4—272no—
346C4⋊C4C2×C3⋊D911yescharacteristic
347C4⋊C4C2×C3⋊D911yescharacteristic
348C4⋊C4C2×C3⋊D912yes—
349C22⋊C4—271no—
350C22⋊C4—271no—
351C22⋊C4—272no—
352C22⋊C4—272no—
353C22⋊C4—272no—
354C22⋊C4—272no—
355C22⋊C4—272no—
356C42—271no—
357C42—272no—
358C42C2×C3⋊D911yescharacteristic
359C22×C3C23×D913yes—
360C22×C3C23×C3⋊D311yescharacteristic
361C2×D3—182no—
362C2×D3—181no—
363C2×D3—181no—
364C2×D3—182no—
365C2×D3—182no—
366C2×D3—181no—
367C2×D3—186no—
368C2×D3—183no—
369C2×D3—183no—
370C2×D3—186no—
371C2×D3—186no—
372C2×D3—183no—
373C3×C4—22no—
374C3×C4—22no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—26no—
378C3×C4—26no—
379C3×C4—23no—
380C3×C4—23no—
381C3×C4#370116yes—
382C3×C4#383212yes—
383Dic3—181no—
384Dic3—182no—
385Dic3—181no—
386Dic3—181no—
387Dic3—183no—
388Dic3—186no—
389Dic3—183no—
390Dic3—183no—
391C32#573311yescharacteristic
392C9#573313yescomplemented
393C23—272no—
394C23—271no—
395Q8—42no—
396Q8—542no—
397D4—542no—
398D4—1082no—
399C2×C4—271no—
400C2×C4—272no—
401C2×C4—271no—
402C2×C4—271no—
403C2×C4—542no—
404C2×C4—1082no—
405C2×C4C22×C3⋊D912yes—
406C2×C4C22×C3⋊D911yescharacteristic
407C2×C4C22×C3⋊D911yescharacteristic
408C2×C4C22×C3⋊D912yes—
409C2×C4C22×C3⋊D911yescharacteristic
410C2×C3#6371013yes—
411C2×C3C2×#370113yes—
412C2×C3#6371313yes—
413C2×C3#6436111yescharacteristic
414C2×C3C2×#383211yescharacteristic
415C2×C3#6436411yescharacteristic
416D3—362no—
417D3—361no—
418D3—366no—
419D3—363no—
420C22—542no—
421C22—541no—
422C22—541no—
423C22—542no—
424C22—542no—
425C22—541no—
426C22C23×C3⋊D911yescharacteristic
427C4—22no—
428C4—22no—
429C4—21no—
430C4—21no—
431C4—541no—
432C4—541no—
433C4—542no—
434C4—541no—
435C4#8874612yes—
436C3#1058983513yescomplemented
437C3#1059620411yescharacteristic
438C2—1082no—
439C2—1081no—
440C2#1172266111yescharacteristic
441C2C2×#8874611yescharacteristic
442C2#1172266711yescharacteristic
subgroupG/H
centerC22C23×C3⋊D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C23×C3⋊D3
FittingC3×C9×C4⋊Q8C2
socleC22×C32C23×D3
prime radical#11721751C2
hypercenterC4⋊Q8C3⋊D9
hypocenterC3×C9#542
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×C9×C4⋊Q8C2semidirect
2. #11721857C2semidirect
3. #11721856C2semidirect
4. #11721754C2semidirect
5. #11721753C2semidirect
7. #11721750C2semidirect
8. #11721749C2semidirect
9. #11721743C2semidirect
10. #11721712C2semidirect
11. #11721709C2semidirect
48. C9×C4⋊Q8D3semidirect
140. C3×C4⋊Q8D9semidirect
280. C4⋊Q8C3⋊D9semidirect
291. C3×C9#542semidirect
392. C9#5733semidirect
436. C3#10589835semidirect

Maximal subgroups

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

subgroupindex
C3×C9×C4⋊Q82
#117218572
#117218562
#117217542
#117217532
#117217512
#117217502
#117217492
#117217432
#117217122
#117217092
#105962043
#105898353