Finite Group Databasebeta

← #13154283

Subgroups — #13154283

Order 1728.

1030 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×C4⋊Q8C211yesmaximal, characteristic, complemented
2#11721856C212yesmaximal
3#11721854C212yesmaximal, complemented
4#11721752C212yesmaximal
5#11721748C211yesmaximal, characteristic
6C4⋊C4×C3⋊D9C211yesmaximal, characteristic
7#11721746C211yesmaximal, characteristic, complemented
8#11721745C212yesmaximal, complemented
9#11721744C211yesmaximal, characteristic, complemented
10#11721709C211yesmaximal, characteristic
11#11721707C211yesmaximal, characteristic, complemented
12#10596200—31nomaximal
13#10589831—33nomaximal
14C2×C3×Q8×C9C2212yes—
15C2×C4×C3⋊D9C2212yes—
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×#6226C2212yes—
18C3×C9×C4⋊C4C2211yescharacteristic
19C3×C9×C4⋊C4C2212yes—
20C3×C9×C4⋊C4C2211yescharacteristic
21C3×C42×C9C2211yescharacteristic
22#88536C2212yes—
23#88536C2211yescharacteristic
24#88536C2211yescharacteristic
25C4⋊(C3⋊Dic9)C2211yescharacteristic
26C4⋊(C3⋊Dic9)C2212yes—
27C4⋊(C3⋊Dic9)C2211yescharacteristic
28#88534C2212yes—
29#88534C2211yescharacteristic
30#88534C2212yes—
31#88534C2212yes—
32#88534C2212yes—
33#88534C2211yescharacteristic
34#88534C2212yes—
35C4×C3⋊Dic9C2211yescharacteristic
36C4×C3⋊Dic9C2212yes—
37C32×C4⋊Q8D311yescharacteristic
38#64155—32no—
39#64153—32no—
40#64105—32no—
41#64101—31no—
42C4⋊C4×C3⋊D3—31no—
43#64099—31no—
44#64098—32no—
45#64097—31no—
46#64081—31no—
47#64079—31no—
48C9×C4⋊Q8D313yescomplemented
49#63508—36no—
50#63506—36no—
51#63458—36no—
52#63454—33no—
53C4⋊C4×D9—33no—
54#63452—33no—
55#63451—36no—
56#63450—33no—
57#63434—33no—
58#63432—33no—
59C22×C3⋊D9C2311yescharacteristic
60C3×Q8×C9—42no—
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2312yes—
68C2×C3⋊Dic9C2311yescharacteristic
69C2×C3⋊Dic9C2312yes—
70C2×C3⋊Dic9C2311yescharacteristic
71C4×C3⋊D9—42no—
72C4×C3⋊D9Q814yescomplemented
73#6226—42no—
74#5729—93no—
75#5729—91no—
76C2×C32×Q8C2×D312yes—
77C2×C4×C3⋊D3—31no—
78C2×C4×C3⋊D3—32no—
79C2×#634—32no—
80C32×C4⋊C4C2×D311yescharacteristic
81C32×C4⋊C4C2×D312yes—
82C32×C4⋊C4C2×D311yescharacteristic
83C32×C42C2×D311yescharacteristic
84#3752—31no—
85#3752—31no—
86#3752—32no—
87C4⋊(C3⋊Dic3)—31no—
88C4⋊(C3⋊Dic3)—31no—
89C4⋊(C3⋊Dic3)—32no—
90#3750—32no—
91#3750—32no—
92#3750—31no—
93#3750—31no—
94#3750—32no—
95#3750—32no—
96#3750—32no—
97C4×C3⋊Dic3—31no—
98C4×C3⋊Dic3—32no—
99C2×Q8×C9C2×D316yes—
100C2×C4×D9—33no—
101C2×C4×D9—36no—
102C2×Dic18—36no—
103C9×C4⋊C4C2×D313yes—
104C9×C4⋊C4C2×D316yes—
105C9×C4⋊C4C2×D313yes—
106C42×C9C2×D313yes—
107(C2×D9)⋊C4—33no—
108(C2×D9)⋊C4—36no—
109(C2×D9)⋊C4—33no—
110C4⋊Dic9—36no—
111C4⋊Dic9—33no—
112C4⋊Dic9—33no—
113Dic9⋊C4—33no—
114Dic9⋊C4—36no—
115Dic9⋊C4—36no—
116Dic9⋊C4—36no—
117Dic9⋊C4—36no—
118Dic9⋊C4—36no—
119Dic9⋊C4—33no—
120C4×Dic9—36no—
121C4×Dic9—33no—
122C22×C3×C9C2411yescharacteristic
123C2×C3⋊D9—21no—
124C2×C3⋊D9—21no—
125C2×C3⋊D9C2×Q812yes—
126C3×C4×C9—22no—
127C3×C4×C9—22no—
128C3×C4×C9—21no—
129C3×C4×C9—21no—
130C3×C4×C9C2×Q812yes—
131C3⋊Dic9—21no—
132C3⋊Dic9—22no—
133C3⋊Dic9—21no—
134C3⋊Dic9—22no—
135C3⋊Dic9C2×Q812yes—
136C3×C4⋊Q8D913yescomplemented
137C3×C4⋊Q8C3⋊D311yescharacteristic
138#946—96no—
139#946—92no—
140#944—96no—
141#944—92no—
142#896—96no—
143#896—92no—
144#892—93no—
145#892—91no—
146D3×C4⋊C4—93no—
147D3×C4⋊C4—91no—
148#890—93no—
149#890—91no—
150#889—96no—
151#889—92no—
152#888—93no—
153#888—91no—
154#872—93no—
155#872—91no—
156#870—93no—
157#870—91no—
158C22×C3⋊D3—31no—
159C32×Q8—42no—
160C2×C32×C4C22×D312yes—
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D312yes—
164C2×C32×C4C22×D311yescharacteristic
165C2×C3⋊Dic3—31no—
166C2×C3⋊Dic3—31no—
167C2×C3⋊Dic3—32no—
168C2×C3⋊Dic3—31no—
169C2×C3⋊Dic3—32no—
170C4×C3⋊D3—34no—
171C4×C3⋊D3—122no—
172#634—122no—
173C22×D9—33no—
174Q8×C9—46no—
175C2×C4×C9C22×D316yes—
176C2×C4×C9C22×D313yes—
177C2×C4×C9C22×D313yes—
178C2×C4×C9C22×D316yes—
179C2×C4×C9C22×D313yes—
180C2×Dic9—33no—
181C2×Dic9—36no—
182C2×Dic9—33no—
183C2×Dic9—33no—
184C2×Dic9—36no—
185C4×D9—312no—
186C4×D9—126no—
187Dic18—126no—
188#544—271noSylow
189C2×C3×C9C22×Q811yescharacteristic
190C2×C3×C9H(5,𝔽2)11yescharacteristic
191C2×C3×C92−1+411yescharacteristic
192C3⋊D9—22no—
193C2×C3×Q8C2×D916yes—
194C2×C3×Q8C2×C3⋊D312yes—
195C2×C4×D3—93no—
196C2×C4×D3—91no—
197C2×C4×D3—96no—
198C2×C4×D3—92no—
199C2×Dic6—96no—
200C2×Dic6—92no—
201C3×C4⋊C4C2×D913yes—
202C3×C4⋊C4C2×D916yes—
203C3×C4⋊C4C2×D913yes—
204C3×C4⋊C4C2×C3⋊D311yescharacteristic
205C3×C4⋊C4C2×C3⋊D312yes—
206C3×C4⋊C4C2×C3⋊D311yescharacteristic
207C3×C42C2×D913yes—
208C3×C42C2×C3⋊D311yescharacteristic
209(C2×D3)⋊C4—93no—
210(C2×D3)⋊C4—93no—
211(C2×D3)⋊C4—96no—
212(C2×D3)⋊C4—91no—
213(C2×D3)⋊C4—91no—
214(C2×D3)⋊C4—92no—
215C4⋊Dic3—93no—
216C4⋊Dic3—93no—
217C4⋊Dic3—96no—
218C4⋊Dic3—91no—
219C4⋊Dic3—91no—
220C4⋊Dic3—92no—
221Dic3⋊C4—96no—
222Dic3⋊C4—96no—
223Dic3⋊C4—93no—
224Dic3⋊C4—93no—
225Dic3⋊C4—96no—
226Dic3⋊C4—96no—
227Dic3⋊C4—96no—
228Dic3⋊C4—92no—
229Dic3⋊C4—92no—
230Dic3⋊C4—91no—
231Dic3⋊C4—91no—
232Dic3⋊C4—92no—
233Dic3⋊C4—92no—
234Dic3⋊C4—92no—
235C4×Dic3—93no—
236C4×Dic3—96no—
237C4×Dic3—91no—
238C4×Dic3—92no—
239C22×C32C23×D311yescharacteristic
240C2×C3⋊D3—32no—
241C2×C3⋊D3—61no—
242C2×C3⋊D3—61no—
243C32×C4—22no—
244C32×C4—22no—
245C32×C4—21no—
246C32×C4—21no—
247C32×C4D3×Q812yes—
248C3⋊Dic3—32no—
249C3⋊Dic3—61no—
250C3⋊Dic3—62no—
251C3⋊Dic3—61no—
252C3⋊Dic3—62no—
253C22×C9C23×D313yes—
254C2×D9—36no—
255C2×D9—63no—
256C2×D9—63no—
257C4×C9—26no—
258C4×C9—26no—
259C4×C9—23no—
260C4×C9—23no—
261C4×C9D3×Q816yes—
262Dic9—36no—
263Dic9—63no—
264Dic9—66no—
265Dic9—66no—
266Dic9—63no—
267C4⋊Q8—272no—
268C4⋊Q8—271no—
269C4⋊Q8C3⋊D911yescharacteristic, complemented
270#125—272no—
271#125—271no—
272#125—271no—
273C22⋊Q8—272no—
274C22⋊Q8—272no—
275#117—271no—
276#117—271no—
277C2×C4⋊C4—271no—
278C3×C9#54411yescharacteristic, complemented, Sylow
279C22×D3—91no—
280C22×D3—93no—
281C3×Q8—42no—
282C3×Q8—46no—
283C2×C3×C4C22×D916yes—
284C2×C3×C4C22×D913yes—
285C2×C3×C4C22×D913yes—
286C2×C3×C4C22×D916yes—
287C2×C3×C4C22×D913yes—
288C2×C3×C4C22×C3⋊D312yes—
289C2×C3×C4C22×C3⋊D311yescharacteristic
290C2×C3×C4C22×C3⋊D311yescharacteristic
291C2×C3×C4C22×C3⋊D312yes—
292C2×C3×C4C22×C3⋊D311yescharacteristic
293C2×Dic3—91no—
294C2×Dic3—93no—
295C2×Dic3—91no—
296C2×Dic3—92no—
297C2×Dic3—91no—
298C2×Dic3—92no—
299C2×Dic3—93no—
300C2×Dic3—96no—
301C2×Dic3—93no—
302C2×Dic3—96no—
303C4×D3—912no—
304C4×D3—94no—
305C4×D3—362no—
306C4×D3—366no—
307Dic6—362no—
308Dic6—366no—
309C2×C32#100411yescharacteristic
310C2×C32C2×D3×Q811yescharacteristic
311C2×C32#100711yescharacteristic
312C3⋊D3—62no—
313C2×C9#100413yes—
314C2×C9C2×D3×Q813yes—
315C2×C9#100713yes—
316D9—66no—
317C2×Q8—272no—
318C2×Q8C2×C3⋊D912yes—
319C22×C4—271no—
320C22×C4—272no—
321C4⋊C4—272no—
322C4⋊C4—271no—
323C4⋊C4—271no—
324C4⋊C4—271no—
325C4⋊C4—271no—
326C4⋊C4—272no—
327C4⋊C4—272no—
328C4⋊C4—272no—
329C4⋊C4—272no—
330C4⋊C4—272no—
331C4⋊C4C2×C3⋊D911yescharacteristic
332C4⋊C4C2×C3⋊D911yescharacteristic
333C4⋊C4C2×C3⋊D912yes—
334C22⋊C4—272no—
335C22⋊C4—271no—
336C22⋊C4—271no—
337C42—272no—
338C42—271no—
339C42C2×C3⋊D911yescharacteristic
340C22×C3C23×D913yes—
341C22×C3C23×C3⋊D311yescharacteristic
342C2×D3—92no—
343C2×D3—96no—
344C2×D3—181no—
345C2×D3—181no—
346C2×D3—183no—
347C2×D3—183no—
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—26no—
353C3×C4—26no—
354C3×C4—23no—
355C3×C4—23no—
356C3×C4Q8×D916yes—
357C3×C4Q8×C3⋊D312yes—
358Dic3—92no—
359Dic3—96no—
360Dic3—181no—
361Dic3—182no—
362Dic3—181no—
363Dic3—182no—
364Dic3—183no—
365Dic3—186no—
366Dic3—183no—
367Dic3—186no—
368C32#572911yescharacteristic
369C9#572913yescomplemented
370C23—271no—
371Q8—42no—
372Q8—1082no—
373C2×C4—271no—
374C2×C4—274no—
375C2×C4—271no—
376C2×C4—272no—
377C2×C4—271no—
378C2×C4—272no—
379C2×C4—1082no—
380C2×C4C22×C3⋊D912yes—
381C2×C4C22×C3⋊D911yescharacteristic
382C2×C4C22×C3⋊D911yescharacteristic
383C2×C4C22×C3⋊D912yes—
384C2×C4C22×C3⋊D911yescharacteristic
385C2×C3#6371013yes—
386C2×C3C2×Q8×D913yes—
387C2×C3#6371313yes—
388C2×C3#6436111yescharacteristic
389C2×C3C2×Q8×C3⋊D311yescharacteristic
390C2×C3#6436411yescharacteristic
391D3—182no—
392D3—186no—
393C22—272no—
394C22—541no—
395C22—541no—
396C22C23×C3⋊D911yescharacteristic
397C4—22no—
398C4—21no—
399C4—22no—
400C4—21no—
401C4—272no—
402C4—542no—
403C4—541no—
404C4—541no—
405C4—542no—
406C4Q8×C3⋊D912yes—
407C3#1058983113yescomplemented
408C3#1059620011yescharacteristic
409C2—542no—
410C2#1172266111yescharacteristic
411C2C2×Q8×C3⋊D911yescharacteristic
412C2#1172266711yescharacteristic
subgroupG/H
centerC22C23×C3⋊D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C23×C3⋊D3
FittingC3×C9×C4⋊Q8C2
socleC22×C32C23×D3
prime radicalC22×C3⋊D9C23
hypercenterC4⋊Q8C3⋊D9
hypocenterC3×C9#544
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
3. #11721854C2semidirect
7. #11721746C2semidirect
8. #11721745C2semidirect
9. #11721744C2semidirect
11. #11721707C2semidirect
48. C9×C4⋊Q8D3semidirect
72. C4×C3⋊D9Q8semidirect
136. C3×C4⋊Q8D9semidirect
269. C4⋊Q8C3⋊D9semidirect
278. C3×C9#544semidirect
369. C9#5729semidirect
407. C3#10589831semidirect

Maximal subgroups

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

subgroupindex
C3×C9×C4⋊Q82
#117218562
#117218542
#117217522
#117217482
C4⋊C4×C3⋊D92
#117217462
#117217452
#117217442
#117217092
#117217072
#105962003
#105898313