Finite Group Databasebeta

← #13148898

Subgroups — #13148898

Order 1728.

742 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721748C211yesmaximal, characteristic
2C9×#888C211yesmaximal, characteristic, complemented
3C3×#63452C211yesmaximal, characteristic, complemented
4#11720995C212yesmaximal
5#11720986C212yesmaximal
6C4×(C3⋊D9)⋊C4C211yesmaximal, characteristic
7#11720970C211yesmaximal, characteristic, complemented
8#11720935C211yesmaximal, characteristic, complemented
9#11720922C211yesmaximal, characteristic
10#11720911C212yesmaximal, complemented
11#11720902C212yesmaximal, complemented
12#10594448—31nomaximal
13#10589830—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C9×Dic6C2212yes—
16C2×(C3⋊D9)⋊C4C2211yescharacteristic
17C2×(C3⋊D9)⋊C4C2211yescharacteristic
18C2×#6189C2212yes—
19C3×C9×C4⋊C4C2211yescharacteristic
20#88536C2212yes—
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22C4×C3⋊Dic9C2212yes—
23C9×C4⋊Dic3C2211yescharacteristic
24C9×Dic3⋊C4C2212yes—
25C4×C9×Dic3C2211yescharacteristic
26C3×C4⋊Dic9C2211yescharacteristic
27C3×C4⋊Dic9C2212yes—
28C3×Dic9⋊C4C2212yes—
29C3×C4×Dic9C2211yescharacteristic
30#88445C2212yes—
31Dic3⋊Dic9C2212yes—
32Dic3⋊Dic9C2212yes—
33#88442C2212yes—
34Dic3×Dic9C2211yescharacteristic
35Dic3×Dic9C2211yescharacteristic
36Dic3×Dic9C2212yes—
37#64101—31no—
38C3×#890—31no—
39C3×#888D311yescharacteristic
40#63918—31no—
41#63895—32no—
42#63888—32no—
43C4×#624—31no—
44#63869—31no—
45#63849—31no—
46#63843—32no—
47#63838—32no—
48C9×C4⋊Q8—31no—
49#63508—32no—
50Q8×Dic9—32no—
51#63459—32no—
52#63454—31no—
53#63454—31no—
54#63454—61no—
55#63452—31no—
56#63452D311yescharacteristic, complemented
57#63449—32no—
58C42×D9—31no—
59#63431—31no—
60C22×C3⋊D9C2311yescharacteristic
61C2×C3×C4×C9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3⋊Dic9C2312yes—
64C2×C3⋊Dic9C2311yescharacteristic
65C4×C3⋊D9Q814yescomplemented
66C2×C9×Dic3C2312yes—
67C2×C9×Dic3C2311yescharacteristic
68C2×C9×Dic3C2311yescharacteristic
69C2×C3×Dic9C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C9×Dic6—24no—
73(C3⋊D9)⋊C4—22no—
74(C3⋊D9)⋊C4—22no—
75#6189—24no—
76#5728—91no—
77#5688—91no—
78C2×C4×C3⋊D3—31no—
79C2×C3×Dic6C2×D312yes—
80C2×#627—32no—
81C2×#624—31no—
82C2×#624—31no—
83C32×C4⋊C4C2×D311yescharacteristic
84#3752—32no—
85C4⋊(C3⋊Dic3)—31no—
86C4×C3⋊Dic3—32no—
87C3×C4⋊Dic3—32no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×Dic3⋊C4—32no—
91C3×Dic3⋊C4C2×D312yes—
92C3×C4×Dic3—31no—
93C3×C4×Dic3C2×D311yescharacteristic
94#3724—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97#3722—32no—
98Dic32—31no—
99Dic32—32no—
100Dic32—31no—
101C2×Q8×C9—32no—
102C2×C4×D9—31no—
103C2×C4×D9—31no—
104C2×C4×D9—31no—
105C2×C4×D9—61no—
106C2×Dic18—32no—
107C9×C4⋊C4—21no—
108C9×C4⋊C4—32no—
109C9×C4⋊C4—31no—
110C9×C4⋊C4C2×D311yescharacteristic
111C42×C9—31no—
112(C2×D9)⋊C4—32no—
113(C2×D9)⋊C4—32no—
114(C2×D9)⋊C4—62no—
115C4⋊Dic9—31no—
116C4⋊Dic9—32no—
117C4⋊Dic9—32no—
118C4⋊Dic9—61no—
119C4⋊Dic9C2×D312yes—
120C4⋊Dic9C2×D311yescharacteristic
121Dic9⋊C4—32no—
122Dic9⋊C4C2×D312yes—
123C4×Dic9—32no—
124C4×Dic9—31no—
125C4×Dic9—32no—
126C4×Dic9—31no—
127C4×Dic9—62no—
128C4×Dic9C2×D311yescharacteristic
129C22×C3×C9C2411yescharacteristic
130C2×C3⋊D9—21no—
131C2×C3⋊D9—21no—
132C2×C3⋊D9C2×Q812yes—
133C3×C4×C9—22no—
134C3×C4×C9C2×Q812yes—
135C3⋊Dic9—22no—
136C3⋊Dic9C2×Q812yes—
137C9×Dic3—22no—
138C9×Dic3#4112yes—
139C9×Dic3#4112yes—
140C3×Dic9—22no—
141C3×Dic9—21no—
142C3×Dic9—21no—
143C3×C4⋊Q8—31no—
144C3×#125—91no—
145#946—92no—
146Q8×Dic3—92no—
147#897—92no—
148#896—92no—
149#896—92no—
150#892—91no—
151#892—91no—
152#892—91no—
153#892—91no—
154#892—181no—
155#890—31no—
156#890—91no—
157#888—91no—
158#888D911yescharacteristic, complemented
159#887—92no—
160#887—92no—
161#887—92no—
162C42×D3—91no—
163C42×D3—91no—
164#870—91no—
165#869—91no—
166C22×C3⋊D3—31no—
167C2×C32×C4C22×D312yes—
168C2×C32×C4C22×D311yescharacteristic
169C2×C3⋊Dic3—31no—
170C2×C3⋊Dic3—32no—
171C4×C3⋊D3—34no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3C22×D312yes—
176C2×C3×Dic3C22×D311yescharacteristic
177C2×C3×Dic3C22×D311yescharacteristic
178C3×Dic6—24no—
179#627—64no—
180#624—62no—
181#624—62no—
182C22×D9—31no—
183C22×D9—61no—
184Q8×C9—64no—
185C2×C4×C9—21no—
186C2×C4×C9—22no—
187C2×C4×C9—31no—
188C2×C4×C9—31no—
189C2×C4×C9—32no—
190C2×C4×C9C22×D312yes—
191C2×C4×C9C22×D311yescharacteristic
192C2×Dic9—31no—
193C2×Dic9—32no—
194C2×Dic9—61no—
195C2×Dic9—62no—
196C2×Dic9C22×D311yescharacteristic
197C2×Dic9C22×D311yescharacteristic
198C2×Dic9C22×D312yes—
199C4×D9—34no—
200C4×D9—64no—
201C4×D9—62no—
202C4×D9—62no—
203Dic18—64no—
204#533—271noSylow
205C2×C3×C9C22×Q811yescharacteristic
206C2×C3×C9C2×#4111yescharacteristic
207C2×C3×C9C2×#4111yescharacteristic
208C3⋊D9—22no—
209C2×C3×Q8—32no—
210C2×C4×D3—91no—
211C2×C4×D3—91no—
212C2×C4×D3—91no—
213C2×C4×D3—91no—
214C2×C4×D3—91no—
215C2×C4×D3—91no—
216C2×C4×D3—181no—
217C2×Dic6—92no—
218C2×Dic6—92no—
219C2×Dic6C2×D912yes—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—32no—
222C3×C4⋊C4—31no—
223C3×C4⋊C4—92no—
224C3×C4⋊C4—92no—
225C3×C4⋊C4—91no—
226C3×C4⋊C4C2×D911yescharacteristic
227C3×C4⋊C4D3211yescharacteristic
228C3×C42—31no—
229C3×C42—91no—
230(C2×D3)⋊C4—92no—
231(C2×D3)⋊C4—92no—
232(C2×D3)⋊C4—92no—
233(C2×D3)⋊C4—92no—
234(C2×D3)⋊C4—182no—
235C4⋊Dic3—31no—
236C4⋊Dic3—32no—
237C4⋊Dic3—92no—
238C4⋊Dic3—92no—
239C4⋊Dic3—91no—
240C4⋊Dic3—91no—
241C4⋊Dic3—181no—
242C4⋊Dic3C2×D911yescharacteristic
243Dic3⋊C4—32no—
244Dic3⋊C4—92no—
245Dic3⋊C4—92no—
246Dic3⋊C4—92no—
247Dic3⋊C4—92no—
248Dic3⋊C4C2×D912yes—
249C4×Dic3—31no—
250C4×Dic3—91no—
251C4×Dic3—91no—
252C4×Dic3—92no—
253C4×Dic3—92no—
254C4×Dic3—92no—
255C4×Dic3—91no—
256C4×Dic3—91no—
257C4×Dic3—92no—
258C4×Dic3—182no—
259C4×Dic3C2×D911yescharacteristic
260C22×C32C23×D311yescharacteristic
261C2×C3⋊D3—32no—
262C2×C3⋊D3—61no—
263C2×C3⋊D3—61no—
264C32×C4—22no—
265C32×C4D3×Q812yes—
266C3⋊Dic3—32no—
267C3⋊Dic3—62no—
268C3×Dic3—22no—
269C3×Dic3—61no—
270C3×Dic3—61no—
271C3×Dic3—62no—
272C3×Dic3#23712yes—
273C3×Dic3#23912yes—
274C22×C9—21no—
275C22×C9C23×D311yescharacteristic
276C2×D9—32no—
277C2×D9—62no—
278C2×D9—61no—
279C2×D9—61no—
280C2×D9—121no—
281C2×D9—121no—
282C4×C9—22no—
283C4×C9—22no—
284C4×C9—32no—
285C4×C9—32no—
286C4×C9—42no—
287C4×C9—62no—
288C4×C9D3×Q812yes—
289Dic9—22no—
290Dic9—21no—
291Dic9—21no—
292Dic9—32no—
293Dic9—62no—
294Dic9—62no—
295Dic9—122no—
296C4⋊Q8—31no—
297C4⋊Q8—271no—
298#125—91no—
299#125—271no—
300C22⋊Q8—272no—
301C22⋊Q8—272no—
302C4×Q8—272no—
303C4×Q8—272no—
304#117—271no—
305#117—271no—
306C2×C42—271no—
307C3×C9#53311yescharacteristic, complemented, Sylow
308C22×D3—91no—
309C22×D3—91no—
310C22×D3—181no—
311C3×Q8—64no—
312C2×C3×C4—21no—
313C2×C3×C4—22no—
314C2×C3×C4—31no—
315C2×C3×C4—31no—
316C2×C3×C4—32no—
317C2×C3×C4—91no—
318C2×C3×C4—91no—
319C2×C3×C4—92no—
320C2×C3×C4C22×D912yes—
321C2×C3×C4C22×D911yescharacteristic
322C2×C3×C4C2×D3212yes—
323C2×C3×C4C2×D3211yescharacteristic
324C2×Dic3—31no—
325C2×Dic3—31no—
326C2×Dic3—32no—
327C2×Dic3—91no—
328C2×Dic3—91no—
329C2×Dic3—92no—
330C2×Dic3—92no—
331C2×Dic3—181no—
332C2×Dic3—182no—
333C2×Dic3C22×D912yes—
334C2×Dic3C22×D911yescharacteristic
335C2×Dic3C22×D911yescharacteristic
336C4×D3—94no—
337C4×D3—94no—
338C4×D3—184no—
339C4×D3—182no—
340C4×D3—182no—
341C4×D3—182no—
342C4×D3—182no—
343Dic6—24no—
344Dic6—184no—
345Dic6—184no—
346C2×C32C2×#23711yescharacteristic
347C2×C32C2×D3×Q811yescharacteristic
348C2×C32C2×#23911yescharacteristic
349C3⋊D3—62no—
350C2×C9—21no—
351C2×C9—21no—
352C2×C9—21no—
353C2×C9C2×D3×Q811yescharacteristic
354C2×C9D3×#4111yescharacteristic
355C2×C9D3×#4111yescharacteristic
356D9—62no—
357D9—122no—
358C2×Q8—32no—
359C2×Q8—272no—
360C22×C4—271no—
361C22×C4—271no—
362C22×C4—271no—
363C4⋊C4—31no—
364C4⋊C4—32no—
365C4⋊C4—91no—
366C4⋊C4—92no—
367C4⋊C4—92no—
368C4⋊C4—272no—
369C4⋊C4—272no—
370C4⋊C4—272no—
371C4⋊C4—271no—
372C4⋊C4D3×D911yescharacteristic
373C22⋊C4—272no—
374C22⋊C4—272no—
375C42—31no—
376C42—91no—
377C42—271no—
378C42—271no—
379C42—272no—
380C42—272no—
381C22×C3—21no—
382C22×C3C23×D911yescharacteristic
383C22×C3C22×D3211yescharacteristic
384C2×D3—92no—
385C2×D3—92no—
386C2×D3—182no—
387C2×D3—181no—
388C2×D3—181no—
389C2×D3—181no—
390C2×D3—181no—
391C2×D3—361no—
392C2×D3—361no—
393C3×C4—22no—
394C3×C4—22no—
395C3×C4—22no—
396C3×C4—32no—
397C3×C4—32no—
398C3×C4—42no—
399C3×C4—62no—
400C3×C4—181no—
401C3×C4—182no—
402C3×C4—181no—
403C3×C4Q8×D912yes—
404C3×C4#379712yes—
405Dic3—22no—
406Dic3—61no—
407Dic3—62no—
408Dic3—61no—
409Dic3—92no—
410Dic3—92no—
411Dic3—182no—
412Dic3—182no—
413Dic3—182no—
414Dic3—362no—
415Dic3#369912yes—
416Dic3#370112yes—
417C32#572811yescharacteristic
418C9—21no—
419C9#568811yescharacteristic, complemented
420C23—271no—
421Q8—64no—
422Q8—544no—
423C2×C4—31no—
424C2×C4—31no—
425C2×C4—32no—
426C2×C4—91no—
427C2×C4—91no—
428C2×C4—92no—
429C2×C4—271no—
430C2×C4—274no—
431C2×C4—272no—
432C2×C4—542no—
433C2×C4—542no—
434C2×C4C2×D3×D912yes—
435C2×C4C2×D3×D911yescharacteristic
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3C2×#369911yescharacteristic
440C2×C3C2×Q8×D911yescharacteristic
441C2×C3C2×#370111yescharacteristic
442C2×C3C2×#379711yescharacteristic
443C2×C3D3×#23711yescharacteristic
444C2×C3D3×#23911yescharacteristic
445D3—182no—
446D3—182no—
447D3—362no—
448C22—272no—
449C22—541no—
450C22—541no—
451C22C22×D3×D911yescharacteristic
452C4—22no—
453C4—32no—
454C4—32no—
455C4—62no—
456C4—181no—
457C4—181no—
458C4—182no—
459C4—272no—
460C4—542no—
461C4(C3⋊D9)⋊Q812yes—
462C3—21no—
463C3#1058983011yescharacteristic, complemented
464C3#1059444811yescharacteristic
465C2—542no—
466C2C2×(C3⋊D9)⋊Q811yescharacteristic
467C2D3×#369911yescharacteristic
468C2D3×#370111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC22×C3⋊D9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#533
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C9×#888C2semidirect
3. C3×#63452C2semidirect
7. #11720970C2semidirect
8. #11720935C2semidirect
10. #11720911C2semidirect
11. #11720902C2semidirect
56. #63452D3semidirect
65. C4×C3⋊D9Q8semidirect
158. #888D9semidirect
307. C3×C9#533semidirect
419. C9#5688semidirect
463. C3#10589830semidirect

Maximal subgroups

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

subgroupindex
#117217482
C9×#8882
C3×#634522
#117209952
#117209862
C4×(C3⋊D9)⋊C42
#117209702
#117209352
#117209222
#117209112
#117209022
#105944483
#105898303