Finite Group Databasebeta

← #13147621

Subgroups — #13147621

Order 1728.

740 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721707C211yesmaximal, characteristic, complemented
2C4×C9×D12C211yesmaximal, characteristic
3C3×C4×Dic18C211yesmaximal, characteristic, complemented
4#11721026C212yesmaximal, complemented
5D12×Dic9C211yesmaximal, characteristic
6D12×Dic9C211yesmaximal, characteristic
7#11721018C212yesmaximal, complemented
8D3×Dic9⋊C4C212yesmaximal, complemented
9#11720969C211yesmaximal, characteristic, complemented
10#11720955C212yesmaximal, complemented
11#11720935C211yesmaximal, characteristic, complemented
12#10593929—31nomaximal
13#10589641—31nomaximal
14C2×C9×D12C2211yescharacteristic
15C2×C4×D3×C9C2212yes—
16C2×C3×Dic18C2211yescharacteristic
17C2×D3×Dic9C2212yes—
18C2×D3×Dic9C2212yes—
19C2×#6189C2212yes—
20C3×C42×C9C2211yescharacteristic
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23#88534C2212yes—
24#88534C2212yes—
25C9×(C2×D3)⋊C4C2212yes—
26C9×C4⋊Dic3C2211yescharacteristic
27C3×C4⋊Dic9C2211yescharacteristic
28C3×Dic9⋊C4C2211yescharacteristic
29C3×Dic9⋊C4C2211yescharacteristic
30C3×C4×Dic9C2211yescharacteristic
31C3×C4×Dic9C2211yescharacteristic
32(C2×D3)⋊Dic9C2212yes—
33(C2×D3)⋊Dic9C2212yes—
34Dic3⋊Dic9C2212yes—
35#88442C2212yes—
36Dic9⋊Dic3C2212yes—
37#64079—31no—
38C3×C4×D12D311yescharacteristic
39C3×C4×Dic6—31no—
40#63918—31no—
41#63899—32no—
42Dic3×D12—31no—
43Dic3×D12—31no—
44#63887—32no—
45D3×Dic3⋊C4—32no—
46#63869—31no—
47#63851—32no—
48C4×D4×C9—31no—
49D4×Dic9—31no—
50D4×Dic9—31no—
51#63486—32no—
52C2×Dic9⋊C4—32no—
53#63452—31no—
54#63450—31no—
55#63440—32no—
56#63440—32no—
57#63432—31no—
58#63432—61no—
59C4×Dic18D311yescharacteristic, complemented
60C22×D3×C9C2312yes—
61C2×C3×C4×C9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2312yes—
66C2×C9×Dic3C2312yes—
67C2×C3×Dic9C2311yescharacteristic
68C2×C3×Dic9C2311yescharacteristic
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2311yescharacteristic
71C9×D12Q814yescomplemented
72C4×D3×C9—42no—
73C3×Dic18—22no—
74D3×Dic9—24no—
75D3×Dic9—42no—
76#6189—42no—
77#5579—91no—
78#5539—91no—
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×C4×D3C2×D312yes—
81C2×C3×Dic6—31no—
82C2×#627—32no—
83C2×D3×Dic3—32no—
84C2×D3×Dic3—32no—
85C32×C42C2×D311yescharacteristic
86C4⋊(C3⋊Dic3)—31no—
87C4⋊(C3⋊Dic3)—31no—
88#3750—32no—
89#3750—32no—
90C3×(C2×D3)⋊C4C2×D312yes—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×Dic3⋊C4—31no—
94C3×Dic3⋊C4—31no—
95C3×C4×Dic3—31no—
96C3×C4×Dic3—31no—
97#3724—32no—
98Dic3⋊Dic3—32no—
99Dic3⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102C2×D4×C9—31no—
103C22×C4×C9—32no—
104C22×Dic9—32no—
105C22×Dic9—32no—
106C2×Dic18—32no—
107C2×Dic18C2×D311yescharacteristic
108C9×C4⋊C4—31no—
109C9×C22⋊C4—32no—
110C42×C9—21no—
111C42×C9C2×D311yescharacteristic
112#3676—32no—
113#3676—32no—
114C4⋊Dic9—31no—
115C4⋊Dic9—31no—
116C4⋊Dic9—32no—
117C4⋊Dic9—61no—
118C4⋊Dic9—61no—
119C4⋊Dic9C2×D311yescharacteristic
120Dic9⋊C4—32no—
121Dic9⋊C4—32no—
122Dic9⋊C4—32no—
123Dic9⋊C4—32no—
124Dic9⋊C4—62no—
125Dic9⋊C4—62no—
126Dic9⋊C4C2×D311yescharacteristic
127Dic9⋊C4C2×D311yescharacteristic
128C4×Dic9C2×D311yescharacteristic
129C4×Dic9C2×D311yescharacteristic
130C22×C3×C9C2411yescharacteristic
131C2×D3×C9—22no—
132C2×D3×C9—22no—
133C2×D3×C9C2×Q814yes—
134C3×C4×C9—21no—
135C3×C4×C9—21no—
136C3×C4×C9C2×Q812yes—
137C3⋊Dic9—22no—
138C3⋊Dic9—22no—
139C9×Dic3—22no—
140C3×Dic9—21no—
141C3×Dic9—21no—
142C3×Dic9—21no—
143C3×Dic9#4112yes—
144C3×C4×Q8—91no—
145C3×C4×D4—31no—
146#946—92no—
147D4×Dic3—91no—
148D4×Dic3—91no—
149#924—92no—
150C2×Dic3⋊C4—92no—
151#896—92no—
152#896—92no—
153D3×C4⋊C4—92no—
154#890—91no—
155#888—91no—
156#878—92no—
157#878—92no—
158C4×D12—91no—
159C4×D12—91no—
160C4×D12D911yescharacteristic
161#870—91no—
162#870—91no—
163#870—91no—
164#870—181no—
165#869—91no—
166C4×Dic6—31no—
167C22×C3×D3C22×D312yes—
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C3⋊Dic3—32no—
172C2×C3⋊Dic3—32no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3C22×D312yes—
178C3×D12Dic614yes—
179C3×C4×D3—42no—
180C3×Dic6—62no—
181#627—122no—
182D3×Dic3—64no—
183D3×Dic3—122no—
184C23×C9—32no—
185D4×C9—34no—
186C2×C4×C9—21no—
187C2×C4×C9—21no—
188C2×C4×C9—21no—
189C2×C4×C9—32no—
190C2×C4×C9—122no—
191C2×C4×C9C22×D311yescharacteristic
192C2×C4×C9C22×D311yescharacteristic
193C2×C4×C9C22×D311yescharacteristic
194C2×Dic9—32no—
195C2×Dic9—32no—
196C2×Dic9—62no—
197C2×Dic9—62no—
198C2×Dic9—64no—
199C2×Dic9—122no—
200C2×Dic9C22×D311yescharacteristic
201C2×Dic9C22×D311yescharacteristic
202C2×Dic9C22×D311yescharacteristic
203C2×Dic9C22×D311yescharacteristic
204Dic18—22no—
205Dic18—122no—
206#554—271noSylow
207C2×C3×C9C22×Q811yescharacteristic
208C2×C3×C9C2×#4111yescharacteristic
209C2×C3×C9H(5,𝔽2)11yescharacteristic
210D3×C9—24no—
211C2×C3×Q8—91no—
212C2×C3×D4—31no—
213C22×C3×C4—32no—
214C22×Dic3—92no—
215C22×Dic3—92no—
216C2×D12C2×D911yescharacteristic
217C2×C4×D3—92no—
218C2×C4×D3—92no—
219C2×C4×D3C2×D912yes—
220C2×Dic6—31no—
221C2×Dic6—92no—
222C2×Dic6—92no—
223C3×C4⋊C4—31no—
224C3×C4⋊C4—91no—
225C3×C4⋊C4—91no—
226C3×C4⋊C4—91no—
227C3×C22⋊C4—32no—
228C3×C42—21no—
229C3×C42—91no—
230C3×C42—91no—
231C3×C42C2×D911yescharacteristic
232C3×C42D3211yescharacteristic
233#217—92no—
234#217—92no—
235(C2×D3)⋊C4—92no—
236(C2×D3)⋊C4—92no—
237(C2×D3)⋊C4C2×D912yes—
238C4⋊Dic3—31no—
239C4⋊Dic3—92no—
240C4⋊Dic3—91no—
241C4⋊Dic3—91no—
242C4⋊Dic3—91no—
243C4⋊Dic3—91no—
244C4⋊Dic3—92no—
245C4⋊Dic3—181no—
246C4⋊Dic3—181no—
247C4⋊Dic3C2×D911yescharacteristic
248Dic3⋊C4—31no—
249Dic3⋊C4—31no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4—92no—
258Dic3⋊C4—182no—
259Dic3⋊C4—182no—
260C4×Dic3—31no—
261C4×Dic3—31no—
262C22×C32C23×D311yescharacteristic
263C2×C3×D3—22no—
264C2×C3×D3—22no—
265C2×C3×D3C2×Dic614yes—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4C2×Dic612yes—
269C3⋊Dic3—62no—
270C3⋊Dic3—62no—
271C3×Dic3—22no—
272C3×Dic3—32no—
273C3×Dic3—61no—
274C3×Dic3—61no—
275C3×Dic3—61no—
276C22×C9—21no—
277C22×C9—34no—
278C22×C9—62no—
279C22×C9—62no—
280C22×C9C23×D311yescharacteristic
281C4×C9—21no—
282C4×C9—21no—
283C4×C9—22no—
284C4×C9—41no—
285C4×C9—41no—
286C4×C9—62no—
287C4×C9D3×Q812yes—
288Dic9—21no—
289Dic9—21no—
290Dic9—21no—
291Dic9—62no—
292Dic9—62no—
293Dic9—122no—
294Dic9—122no—
295Dic9#23512yes—
296C4⋊Q8—271no—
297#125—271no—
298#125—271no—
299C22⋊Q8—272no—
300C22⋊Q8—272no—
301C22⋊Q8—272no—
302C4×Q8—91no—
303C4×D4—31no—
304C4×D4—271no—
305C4×D4—271no—
306C2×C4⋊C4—272no—
307C3×C9#55411yescharacteristic, complemented, Sylow
308C23×C3—32no—
309C22×D3C22×D912yes—
310C3×Q8—182no—
311C3×D4—34no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—32no—
316C2×C3×C4—91no—
317C2×C3×C4—91no—
318C2×C3×C4—91no—
319C2×C3×C4—91no—
320C2×C3×C4—122no—
321C2×C3×C4C22×D911yescharacteristic
322C2×C3×C4C22×D911yescharacteristic
323C2×C3×C4C22×D911yescharacteristic
324C2×C3×C4C2×D3211yescharacteristic
325C2×C3×C4C2×D3211yescharacteristic
326C2×C3×C4C2×D3211yescharacteristic
327C2×Dic3—31no—
328C2×Dic3—31no—
329C2×Dic3—31no—
330C2×Dic3—31no—
331C2×Dic3—92no—
332C2×Dic3—92no—
333C2×Dic3—92no—
334C2×Dic3—92no—
335C2×Dic3—182no—
336C2×Dic3—182no—
337C2×Dic3—184no—
338C2×Dic3—362no—
339C2×Dic3C22×D912yes—
340D12Dic1814yescomplemented
341C4×D3—42no—
342C4×D3—184no—
343C4×D3—362no—
344Dic6—62no—
345Dic6—362no—
346Dic6—362no—
347C2×C32C22×Dic611yescharacteristic
348C2×C32#100411yescharacteristic
349C2×C32D3×#4111yescharacteristic
350C3×D3—24no—
351C2×C9—21no—
352C2×C9—21no—
353C2×C9—21no—
354C2×C9—64no—
355C2×C9C2×#23511yescharacteristic
356C2×C9C2×D3×Q811yescharacteristic
357C2×C9#100911yescharacteristic
358C2×Q8—91no—
359C2×Q8—272no—
360C2×D4—31no—
361C22×C4—32no—
362C22×C4—272no—
363C22×C4—272no—
364C4⋊C4—31no—
365C4⋊C4—91no—
366C4⋊C4—91no—
367C4⋊C4—91no—
368C4⋊C4—272no—
369C4⋊C4—272no—
370C4⋊C4—272no—
371C4⋊C4—272no—
372C4⋊C4—271no—
373C4⋊C4—271no—
374C4⋊C4—272no—
375C22⋊C4—32no—
376C22⋊C4—272no—
377C22⋊C4—272no—
378C42—91no—
379C42—91no—
380C42D3×D911yescharacteristic
381C22×C3—21no—
382C22×C3—34no—
383C22×C3—62no—
384C22×C3—62no—
385C22×C3C23×D911yescharacteristic
386C22×C3C22×D3211yescharacteristic
387C2×D3—22no—
388C2×D3—22no—
389C2×D3C2×Dic1814yes—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—22no—
395C3×C4—41no—
396C3×C4—41no—
397C3×C4—62no—
398C3×C4—92no—
399C3×C4—181no—
400C3×C4—181no—
401C3×C4—181no—
402C3×C4C2×Dic1812yes—
403C3×C4D3×Dic612yes—
404Dic3—22no—
405Dic3—32no—
406Dic3—61no—
407Dic3—61no—
408Dic3—61no—
409Dic3—182no—
410Dic3—182no—
411Dic3—182no—
412Dic3—182no—
413Dic3—362no—
414Dic3—362no—
415C32#553911yescharacteristic
416C9—21no—
417C9#557911yescharacteristic, complemented
418C23—32no—
419Q8—182no—
420Q8—1082no—
421D4—34no—
422C2×C4—32no—
423C2×C4—91no—
424C2×C4—91no—
425C2×C4—91no—
426C2×C4—91no—
427C2×C4—122no—
428C2×C4—272no—
429C2×C4—272no—
430C2×C4—544no—
431C2×C4—1082no—
432C2×C4C2×D3×D911yescharacteristic
433C2×C4C2×D3×D911yescharacteristic
434C2×C4C2×D3×D911yescharacteristic
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—64no—
439C2×C3C22×Dic1811yescharacteristic
440C2×C3#6371011yescharacteristic
441C2×C3#41×D911yescharacteristic
442C2×C3C2×D3×Dic611yescharacteristic
443C2×C3D3×#23511yescharacteristic
444C2×C3#6430711yescharacteristic
445D3—24no—
446C22—34no—
447C22—62no—
448C22—62no—
449C22C22×D3×D911yescharacteristic
450C4—21no—
451C4—21no—
452C4—62no—
453C4—92no—
454C4—181no—
455C4—181no—
456C4—181no—
457C4—542no—
458C4—542no—
459C4D3×Dic1812yes—
460C3—21no—
461C3#1058964111yescharacteristic, complemented
462C3#1059392911yescharacteristic
463C2—64no—
464C2C2×D3×Dic1811yescharacteristic
465C2D9×#23511yescharacteristic
466C2#1172247611yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×C3×D12C2×D3
hypercenterC42D3×D9
hypocenterC3×C9#554
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721707C2semidirect
3. C3×C4×Dic18C2semidirect
4. #11721026C2semidirect
7. #11721018C2semidirect
8. D3×Dic9⋊C4C2semidirect
9. #11720969C2semidirect
10. #11720955C2semidirect
11. #11720935C2semidirect
59. C4×Dic18D3semidirect
71. C9×D12Q8semidirect
307. C3×C9#554semidirect
340. D12Dic18semidirect
417. C9#5579semidirect
461. C3#10589641semidirect

Maximal subgroups

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

subgroupindex
#117217072
C4×C9×D122
C3×C4×Dic182
#117210262
D12×Dic92
#117210182
D3×Dic9⋊C42
#117209692
#117209552
#117209352
#105939293
#105896413