Finite Group Databasebeta

← #13150296

Subgroups — #13150296

Order 1728.

722 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721813C211yesmaximal, characteristic, complemented
2C3×#63492C211yesmaximal, characteristic
3C9×#926C211yesmaximal, characteristic, complemented
4#11721269C212yesmaximal, complemented
5#11721257C212yesmaximal, complemented
6#11720999C212yesmaximal, complemented
7Dic3×D36C211yesmaximal, characteristic, complemented
8#11720970C211yesmaximal, characteristic, complemented
9#11720948C211yesmaximal, characteristic, complemented
10#11720946C212yesmaximal, complemented
11#11720912C211yesmaximal, characteristic, complemented
12#10594983—31nomaximal
13#10589636—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C3×#611C2212yes—
17C2×C3×D36C2211yescharacteristic
18C2×Dic3×D9C2212yes—
19C2×#6189C2212yes—
20#88551C2212yes—
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22#88534C2212yes—
23C9×#217C2211yescharacteristic
24C9×#217C2211yescharacteristic
25C3×(C2×D9)⋊C4C2212yes—
26C9×C4⋊Dic3C2211yescharacteristic
27C9×C4⋊Dic3C2211yescharacteristic
28C4×C9×Dic3C2211yescharacteristic
29C4×C9×Dic3C2211yescharacteristic
30C3×C4⋊Dic9C2211yescharacteristic
31(C2×D9)⋊Dic3C2212yes—
32(C2×D9)⋊Dic3C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34Dic3⋊Dic9C2212yes—
35#88442C2212yes—
36Dic3×Dic9C2212yes—
37#64133—31no—
38C3×#930—31no—
39C3×#926D311yescharacteristic
40#63969—32no—
41#63959—32no—
42#63918—31no—
43#63899—32no—
44Dic3×D12—31no—
45#63860—31no—
46#63858—32no—
47#63844—31no—
48C9×#117—31no—
49#63492D311yescharacteristic
50#63486—31no—
51#63486—61no—
52#63460—32no—
53#63458—32no—
54#63447—32no—
55#63445—32no—
56#63437—31no—
57C4×D36—31no—
58#63431—31no—
59C4×Dic18—31no—
60C23×C3×C9C2311yescharacteristic
61C22×C3×D9C2312yes—
62C2×C3×C4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2312yes—
67#6226—22no—
68C3×#611—42no—
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C9×Dic3C2311yescharacteristic
72C2×C9×Dic3C2311yescharacteristic
73C2×C3×Dic9C2312yes—
74C3×D36—22no—
75Dic3×D9—42no—
76#6189—42no—
77#5603—91no—
78#5534—91no—
79C22×C32×C4C2×D311yescharacteristic
80C2×#634—31no—
81C2×C3×#67—32no—
82C2×C3×D12—31no—
83C2×#627—32no—
84C2×D3×Dic3—32no—
85#3757—32no—
86C4⋊(C3⋊Dic3)—31no—
87#3750—32no—
88C3×#217C2×D311yescharacteristic
89C3×#217C2×D311yescharacteristic
90C3×(C2×D3)⋊C4—32no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4×Dic3C2×D311yescharacteristic
95C3×C4×Dic3C2×D311yescharacteristic
96#3724—32no—
97Dic3⋊Dic3—32no—
98(C2×D3)⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101Dic32—32no—
102C22×C4×C9—21no—
103C22×C4×C9C2×D311yescharacteristic
104C2×#611C2×D312yes—
105C2×D36C2×D311yescharacteristic
106C2×C4×D9—32no—
107C2×Dic18—31no—
108C2×Dic18—32no—
109C2×Dic18—61no—
110C9×C4⋊C4—31no—
111C9×C4⋊C4—31no—
112C9×C22⋊C4—31no—
113C9×C22⋊C4—31no—
114C42×C9—31no—
115C42×C9—31no—
116#3676—32no—
117#3676—62no—
118(C2×D9)⋊C4—32no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4C2×D312yes—
122C4⋊Dic9—31no—
123C4⋊Dic9—32no—
124C4⋊Dic9—61no—
125C4⋊Dic9C2×D311yescharacteristic
126Dic9⋊C4—32no—
127Dic9⋊C4—32no—
128Dic9⋊C4—62no—
129C4×Dic9—32no—
130C22×C3×C9—21no—
131C22×C3×C9—21no—
132C22×C3×C9—21no—
133C22×C3×C9C2411yescharacteristic
134C2×C3×D9—22no—
135C2×C3×D9—22no—
136C2×C3×D9—22no—
137C3×C4×C9—21no—
138C3×C4×C9#4112yes—
139C3⋊Dic9—22no—
140C3⋊Dic9—22no—
141C9×Dic3—21no—
142C9×Dic3—21no—
143C9×Dic3—21no—
144C9×Dic3—21no—
145C3×Dic9—22no—
146C3×#121—91no—
147C3×#117—31no—
148#936—92no—
149#936—91no—
150#935—92no—
151D4×Dic3—91no—
152#930—31no—
153#926D911yescharacteristic, complemented
154#924—91no—
155#924—91no—
156#924—181no—
157#898—92no—
158#896—92no—
159#888—91no—
160#887—91no—
161#885—92no—
162#883—92no—
163#879—92no—
164#878—92no—
165#875—91no—
166C4×D12—91no—
167#869—91no—
168C4×Dic6—91no—
169C23×C32C22×D311yescharacteristic
170C22×C3×D3—32no—
171C2×C32×C4—22no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3—32no—
175C2×C3⋊Dic3—32no—
176#634—62no—
177C3×#67—122no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3C22×D311yescharacteristic
180C2×C3×Dic3C22×D311yescharacteristic
181C2×C3×Dic3C22×D311yescharacteristic
182C2×C3×Dic3C22×D311yescharacteristic
183C3×D12—62no—
184#627—122no—
185D3×Dic3—122no—
186C23×C9—21no—
187C23×C9C22×D311yescharacteristic
188C22×D9C22×D312yes—
189C2×C4×C9—22no—
190C2×C4×C9—21no—
191C2×C4×C9—21no—
192C2×C4×C9—31no—
193C2×C4×C9—31no—
194C2×C4×C9—31no—
195C2×C4×C9—31no—
196C2×C4×C9—42no—
197C2×C4×C9C22×D311yescharacteristic
198C2×C4×C9C22×D311yescharacteristic
199#611—42no—
200C2×Dic9—32no—
201C2×Dic9—32no—
202C2×Dic9—62no—
203C2×Dic9—62no—
204C2×Dic9C22×D312yes—
205D36—22no—
206C4×D9—122no—
207Dic18—62no—
208Dic18—122no—
209Dic18—122no—
210#542—271noSylow
211C2×C3×C9—41no—
212C2×C3×C9C2×#4111yescharacteristic
213C2×C3×C9H(5,𝔽2)11yescharacteristic
214C2×C3×C92−1+411yescharacteristic
215C3×D9—42no—
216C2×C3×D4—92no—
217C2×C3×D4—91no—
218C22×C3×C4—21no—
219C22×C3×C4C2×D911yescharacteristic
220C22×C3×C4D3211yescharacteristic
221C2×#67—32no—
222C22×Dic3—92no—
223C2×D12—31no—
224C2×C4×D3—92no—
225C2×Dic6—91no—
226C2×Dic6—91no—
227C2×Dic6—92no—
228C2×Dic6—92no—
229C2×Dic6—181no—
230C3×C4⋊C4—31no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—91no—
233C3×C22⋊C4—31no—
234C3×C22⋊C4—31no—
235C3×C22⋊C4—92no—
236C3×C42—31no—
237C3×C42—31no—
238#217—92no—
239#217—92no—
240#217—92no—
241#217—92no—
242#217—92no—
243#217—182no—
244#217C2×D911yescharacteristic
245#217C2×D911yescharacteristic
246(C2×D3)⋊C4—32no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250C4⋊Dic3—31no—
251C4⋊Dic3—92no—
252C4⋊Dic3—91no—
253C4⋊Dic3—91no—
254C4⋊Dic3—181no—
255C4⋊Dic3C2×D911yescharacteristic
256C4⋊Dic3C2×D911yescharacteristic
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261Dic3⋊C4—92no—
262Dic3⋊C4—182no—
263C4×Dic3—92no—
264C4×Dic3—92no—
265C4×Dic3C2×D911yescharacteristic
266C4×Dic3C2×D911yescharacteristic
267C22×C32—21no—
268C22×C32—21no—
269C22×C32—21no—
270C22×C32C23×D311yescharacteristic
271C2×C3×D3—62no—
272C2×C3×D3—62no—
273C2×C3×D3—62no—
274C32×C4—21no—
275C32×C4#23512yes—
276C3⋊Dic3—62no—
277C3⋊Dic3—62no—
278C3×Dic3—21no—
279C3×Dic3—21no—
280C3×Dic3—21no—
281C3×Dic3—21no—
282C3×Dic3—62no—
283C22×C9—22no—
284C22×C9—21no—
285C22×C9—21no—
286C22×C9—21no—
287C22×C9—21no—
288C22×C9—41no—
289C22×C9—41no—
290C22×C9C23×D311yescharacteristic
291C2×D9—22no—
292C2×D9—22no—
293C2×D9—22no—
294C4×C9—21no—
295C4×C9—22no—
296C4×C9—41no—
297C4×C9—61no—
298C4×C9—61no—
299C4×C9—61no—
300C4×C9—61no—
301C4×C9#23712yes—
302Dic9—22no—
303Dic9—62no—
304Dic9—62no—
305Dic9—122no—
306Dic9—122no—
307C4⋊Q8—271no—
308#126—272no—
309#124—271no—
310#124—272no—
311#123—272no—
312C22⋊Q8—272no—
313C22⋊Q8—271no—
314#121—91no—
315C4×Q8—271no—
316C4×D4—271no—
317#117—31no—
318C3×C9#54211yescharacteristic, complemented, Sylow
319C23×C3—21no—
320C23×C3—92no—
321C23×C3C22×D911yescharacteristic
322C23×C3C2×D3211yescharacteristic
323C22×D3—32no—
324C3×D4—182no—
325C3×D4—362no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—31no—
331C2×C3×C4—31no—
332C2×C3×C4—31no—
333C2×C3×C4—31no—
334C2×C3×C4—42no—
335C2×C3×C4—92no—
336C2×C3×C4C22×D911yescharacteristic
337C2×C3×C4C22×D911yescharacteristic
338C2×C3×C4C2×D3211yescharacteristic
339C2×C3×C4C2×D3211yescharacteristic
340#67—122no—
341C2×Dic3—32no—
342C2×Dic3—92no—
343C2×Dic3—92no—
344C2×Dic3—92no—
345C2×Dic3—92no—
346C2×Dic3—182no—
347C2×Dic3—182no—
348C2×Dic3—362no—
349C2×Dic3C22×D911yescharacteristic
350C2×Dic3C22×D911yescharacteristic
351C2×Dic3C22×D911yescharacteristic
352C2×Dic3C22×D911yescharacteristic
353D12—62no—
354C4×D3—362no—
355Dic6—182no—
356Dic6—182no—
357Dic6—362no—
358Dic6—362no—
359Dic6—362no—
360C2×C32—41no—
361C2×C32C2×#23511yescharacteristic
362C2×C32#100911yescharacteristic
363C2×C32#101011yescharacteristic
364C3×D3—122no—
365C2×C9—21no—
366C2×C9—21no—
367C2×C9—21no—
368C2×C9—42no—
369C2×C9—41no—
370C2×C9C2×#23711yescharacteristic
371C2×C9#100411yescharacteristic
372C2×C9#101011yescharacteristic
373D9—42no—
374C2×Q8—272no—
375C2×Q8—271no—
376C2×D4—92no—
377C2×D4—91no—
378C22×C4—272no—
379C22×C4D3×D911yescharacteristic
380C4⋊C4—31no—
381C4⋊C4—31no—
382C4⋊C4—91no—
383C4⋊C4—272no—
384C4⋊C4—272no—
385C4⋊C4—271no—
386C4⋊C4—272no—
387C22⋊C4—31no—
388C22⋊C4—31no—
389C22⋊C4—92no—
390C22⋊C4—272no—
391C22⋊C4—272no—
392C22⋊C4—272no—
393C22⋊C4—272no—
394C42—31no—
395C42—31no—
396C42—272no—
397C22×C3—22no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3—182no—
408C22×C3—182no—
409C22×C3—182no—
410C22×C3C23×D911yescharacteristic
411C22×C3C22×D3211yescharacteristic
412C2×D3—62no—
413C2×D3—62no—
414C2×D3—62no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—22no—
418C3×C4—41no—
419C3×C4—61no—
420C3×C4—61no—
421C3×C4—61no—
422C3×C4—61no—
423C3×C4—182no—
424C3×C4#369712yes—
425C3×C4#380412yes—
426Dic3—21no—
427Dic3—21no—
428Dic3—21no—
429Dic3—21no—
430Dic3—62no—
431Dic3—182no—
432Dic3—182no—
433Dic3—182no—
434Dic3—182no—
435Dic3—362no—
436Dic3—362no—
437C32#553411yescharacteristic
438C9—21no—
439C9#560311yescharacteristic, complemented
440C23—92no—
441C23C2×D3×D911yescharacteristic
442Q8—542no—
443Q8—1082no—
444D4—182no—
445D4—362no—
446C2×C4—22no—
447C2×C4—31no—
448C2×C4—31no—
449C2×C4—31no—
450C2×C4—31no—
451C2×C4—92no—
452C2×C4—272no—
453C2×C4—272no—
454C2×C4—1082no—
455C2×C4C2×D3×D911yescharacteristic
456C2×C4C2×D3×D911yescharacteristic
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—42no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3—362no—
464C2×C3C2×#369711yescharacteristic
465C2×C3#6371511yescharacteristic
466C2×C3#6371611yescharacteristic
467C2×C3#6429811yescharacteristic
468C2×C3#6430711yescharacteristic
469C2×C3C2×#380411yescharacteristic
470D3—122no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—182no—
475C22—182no—
476C22—182no—
477C22C22×D3×D911yescharacteristic
478C4—21no—
479C4—61no—
480C4—61no—
481C4—61no—
482C4—61no—
483C4—182no—
484C4—542no—
485C4—542no—
486C4#8865812yes—
487C3—21no—
488C3#1058963611yescharacteristic, complemented
489C3#1059498311yescharacteristic
490C2—41no—
491C2—362no—
492C2#1172246511yescharacteristic
493C2#1172247711yescharacteristic
494C2C2×#8865811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC3×#63492C2
hypercenterC22×C4D3×D9
hypocenterC3×C9#542
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11721813C2semidirect
3. C9×#926C2semidirect
4. #11721269C2semidirect
5. #11721257C2semidirect
6. #11720999C2semidirect
7. Dic3×D36C2semidirect
8. #11720970C2semidirect
9. #11720948C2semidirect
10. #11720946C2semidirect
11. #11720912C2semidirect
153. #926D9semidirect
318. C3×C9#542semidirect
439. C9#5603semidirect
488. C3#10589636semidirect

Maximal subgroups

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

subgroupindex
#117218132
C3×#634922
C9×#9262
#117212692
#117212572
#117209992
Dic3×D362
#117209702
#117209482
#117209462
#117209122
#105949833
#105896363