Finite Group Databasebeta

← #13150430

Subgroups — #13150430

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
2C9×#930C211yesmaximal, characteristic
3C3×#63488C211yesmaximal, characteristic, complemented
4#11721280C212yesmaximal, complemented
5#11721263C212yesmaximal, complemented
6#11721026C212yesmaximal, complemented
7D12×Dic9C211yesmaximal, characteristic, complemented
8#11720969C211yesmaximal, characteristic, complemented
9#11720964C211yesmaximal, characteristic, complemented
10#11720962C212yesmaximal, complemented
11#11720913C211yesmaximal, characteristic, complemented
12#10594983—31nomaximal
13#10589705—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C9×#67C2212yes—
17C2×C9×D12C2211yescharacteristic
18C2×D3×Dic9C2212yes—
19C2×#6189C2212yes—
20#88551C2212yes—
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22#88534C2212yes—
23C3×#3676C2211yescharacteristic
24C3×#3676C2211yescharacteristic
25C9×(C2×D3)⋊C4C2212yes—
26C9×C4⋊Dic3C2211yescharacteristic
27C3×C4⋊Dic9C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×C4×Dic9C2211yescharacteristic
30C3×C4×Dic9C2211yescharacteristic
31(C2×D3)⋊Dic9C2212yes—
32(C2×D3)⋊Dic9C2212yes—
33(C2×D3)⋊Dic9C2212yes—
34#88442C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37#64133—31no—
38C3×#930D311yescharacteristic
39C3×#926—31no—
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×#121—31no—
49#63498—32no—
50#63498—31no—
51#63497—32no—
52D4×Dic9—31no—
53#63488D311yescharacteristic, complemented
54#63486—31no—
55#63486—61no—
56#63450—31no—
57#63449—31no—
58#63441—32no—
59#63440—32no—
60C23×C3×C9C2311yescharacteristic
61C22×D3×C9C2312yes—
62C2×C3×C4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2312yes—
67#6226—22no—
68C9×#67—42no—
69C2×C9×Dic3C2312yes—
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C9×D12—22no—
75D3×Dic9—42no—
76#6189—42no—
77#5603—91no—
78#5534—91no—
79C22×C32×C4C2×D311yescharacteristic
80C2×#634—31no—
81C2×C3×#67C2×D312yes—
82C2×C3×D12C2×D311yescharacteristic
83C2×#627—32no—
84C2×D3×Dic3—32no—
85#3757—32no—
86C4⋊(C3⋊Dic3)—31no—
87#3750—32no—
88C3×#217—31no—
89C3×#217—31no—
90C3×(C2×D3)⋊C4C2×D312yes—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4×Dic3—31no—
95C3×C4×Dic3—31no—
96#3724—32no—
97Dic3⋊Dic3—32no—
98(C2×D3)⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101Dic32—32no—
102C2×D4×C9—32no—
103C2×D4×C9—31no—
104C22×C4×C9—21no—
105C22×C4×C9C2×D311yescharacteristic
106C22×Dic9—32no—
107C2×Dic18—31no—
108C2×Dic18—32no—
109C2×Dic18—61no—
110C9×C4⋊C4—31no—
111C9×C22⋊C4—32no—
112#3676—32no—
113#3676—32no—
114#3676—32no—
115#3676—32no—
116#3676—62no—
117#3676C2×D311yescharacteristic
118#3676C2×D311yescharacteristic
119C4⋊Dic9—31no—
120C4⋊Dic9—61no—
121C4⋊Dic9C2×D311yescharacteristic
122C4⋊Dic9C2×D311yescharacteristic
123Dic9⋊C4—32no—
124Dic9⋊C4—32no—
125Dic9⋊C4—32no—
126Dic9⋊C4—62no—
127C4×Dic9—32no—
128C4×Dic9C2×D311yescharacteristic
129C4×Dic9C2×D311yescharacteristic
130C22×C3×C9—21no—
131C22×C3×C9—21no—
132C22×C3×C9—21no—
133C22×C3×C9C2411yescharacteristic
134C2×D3×C9—22no—
135C2×D3×C9—22no—
136C2×D3×C9—22no—
137C3×C4×C9—21no—
138C3×C4×C9#4112yes—
139C3⋊Dic9—22no—
140C3⋊Dic9—22no—
141C9×Dic3—22no—
142C3×Dic9—21no—
143C3×Dic9—21no—
144C3×Dic9—21no—
145C3×Dic9—21no—
146C3×#121—31no—
147C3×#117—91no—
148#936—91no—
149#936—92no—
150#935—92no—
151D4×Dic3—91no—
152#930D911yescharacteristic
153#926—31no—
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×D3C22×D312yes—
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—42no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3C22×D312yes—
183C3×D12—22no—
184#627—122no—
185D3×Dic3—122no—
186C23×C9—21no—
187C23×C9—32no—
188C23×C9C22×D311yescharacteristic
189D4×C9—62no—
190D4×C9—122no—
191C2×C4×C9—22no—
192C2×C4×C9—21no—
193C2×C4×C9—21no—
194C2×C4×C9—32no—
195C2×C4×C9—42no—
196C2×C4×C9C22×D311yescharacteristic
197C2×C4×C9C22×D311yescharacteristic
198C2×Dic9—32no—
199C2×Dic9—32no—
200C2×Dic9—62no—
201C2×Dic9—62no—
202C2×Dic9—122no—
203C2×Dic9C22×D311yescharacteristic
204C2×Dic9C22×D311yescharacteristic
205C2×Dic9C22×D311yescharacteristic
206C2×Dic9C22×D311yescharacteristic
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
215D3×C9—42no—
216C2×C3×D4—32no—
217C2×C3×D4—31no—
218C22×C3×C4—21no—
219C22×C3×C4C2×D911yescharacteristic
220C22×C3×C4D3211yescharacteristic
221C2×#67C2×D912yes—
222C22×Dic3—92no—
223C2×D12C2×D911yescharacteristic
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—91no—
232C3×C4⋊C4—91no—
233C3×C22⋊C4—32no—
234C3×C22⋊C4—91no—
235C3×C22⋊C4—91no—
236C3×C42—91no—
237C3×C42—91no—
238#217—31no—
239#217—31no—
240#217—92no—
241#217—92no—
242#217—92no—
243#217—92no—
244#217—92no—
245#217—182no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4C2×D912yes—
250C4⋊Dic3—31no—
251C4⋊Dic3—31no—
252C4⋊Dic3—91no—
253C4⋊Dic3—91no—
254C4⋊Dic3—92no—
255C4⋊Dic3—181no—
256C4⋊Dic3C2×D911yescharacteristic
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261Dic3⋊C4—92no—
262Dic3⋊C4—182no—
263C4×Dic3—31no—
264C4×Dic3—31no—
265C4×Dic3—92no—
266C4×Dic3—92no—
267C22×C32—21no—
268C22×C32—21no—
269C22×C32—21no—
270C22×C32C23×D311yescharacteristic
271C2×C3×D3—22no—
272C2×C3×D3—22no—
273C2×C3×D3—22no—
274C32×C4—21no—
275C32×C4#23712yes—
276C3⋊Dic3—62no—
277C3⋊Dic3—62no—
278C3×Dic3—22no—
279C3×Dic3—61no—
280C3×Dic3—61no—
281C3×Dic3—61no—
282C3×Dic3—61no—
283C22×C9—22no—
284C22×C9—21no—
285C22×C9—21no—
286C22×C9—21no—
287C22×C9—21no—
288C22×C9—41no—
289C22×C9—41no—
290C22×C9—62no—
291C22×C9—62no—
292C22×C9—62no—
293C22×C9C23×D311yescharacteristic
294C4×C9—21no—
295C4×C9—22no—
296C4×C9—41no—
297C4×C9—62no—
298C4×C9#23512yes—
299Dic9—21no—
300Dic9—21no—
301Dic9—21no—
302Dic9—21no—
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—31no—
315C4×Q8—271no—
316C4×D4—271no—
317#117—91no—
318C3×C9#54211yescharacteristic, complemented, Sylow
319C23×C3—21no—
320C23×C3—32no—
321C23×C3C22×D911yescharacteristic
322C23×C3C2×D3211yescharacteristic
323C22×D3C22×D912yes—
324C3×D4—62no—
325C3×D4—122no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—32no—
331C2×C3×C4—42no—
332C2×C3×C4—91no—
333C2×C3×C4—91no—
334C2×C3×C4—91no—
335C2×C3×C4—91no—
336C2×C3×C4C22×D911yescharacteristic
337C2×C3×C4C22×D911yescharacteristic
338C2×C3×C4C2×D3211yescharacteristic
339C2×C3×C4C2×D3211yescharacteristic
340#67—42no—
341C2×Dic3—31no—
342C2×Dic3—31no—
343C2×Dic3—31no—
344C2×Dic3—31no—
345C2×Dic3—92no—
346C2×Dic3—92no—
347C2×Dic3—92no—
348C2×Dic3—92no—
349C2×Dic3—182no—
350C2×Dic3—182no—
351C2×Dic3—362no—
352C2×Dic3C22×D912yes—
353D12—22no—
354C4×D3—362no—
355Dic6—182no—
356Dic6—182no—
357Dic6—362no—
358Dic6—362no—
359Dic6—362no—
360C2×C32—41no—
361C2×C32C2×#23711yescharacteristic
362C2×C32#100411yescharacteristic
363C2×C32#101011yescharacteristic
364C3×D3—42no—
365C2×C9—21no—
366C2×C9—21no—
367C2×C9—21no—
368C2×C9—42no—
369C2×C9—41no—
370C2×C9—122no—
371C2×C9C2×#23511yescharacteristic
372C2×C9#100911yescharacteristic
373C2×C9#101011yescharacteristic
374C2×Q8—272no—
375C2×Q8—271no—
376C2×D4—32no—
377C2×D4—31no—
378C22×C4—272no—
379C22×C4D3×D911yescharacteristic
380C4⋊C4—31no—
381C4⋊C4—91no—
382C4⋊C4—91no—
383C4⋊C4—272no—
384C4⋊C4—272no—
385C4⋊C4—271no—
386C4⋊C4—272no—
387C22⋊C4—32no—
388C22⋊C4—91no—
389C22⋊C4—91no—
390C22⋊C4—272no—
391C22⋊C4—272no—
392C22⋊C4—272no—
393C22⋊C4—272no—
394C42—91no—
395C42—91no—
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—62no—
408C22×C3—62no—
409C22×C3—62no—
410C22×C3C23×D911yescharacteristic
411C22×C3C22×D3211yescharacteristic
412C2×D3—22no—
413C2×D3—22no—
414C2×D3—22no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—22no—
418C3×C4—41no—
419C3×C4—62no—
420C3×C4—181no—
421C3×C4—181no—
422C3×C4—181no—
423C3×C4—181no—
424C3×C4#369912yes—
425C3×C4#380412yes—
426Dic3—22no—
427Dic3—61no—
428Dic3—61no—
429Dic3—61no—
430Dic3—61no—
431Dic3—182no—
432Dic3—182no—
433Dic3—182no—
434Dic3—182no—
435Dic3—362no—
436Dic3—362no—
437C32#560311yescharacteristic
438C9—21no—
439C9#553411yescharacteristic, complemented
440C23—32no—
441C23C2×D3×D911yescharacteristic
442Q8—542no—
443Q8—1082no—
444D4—62no—
445D4—122no—
446C2×C4—22no—
447C2×C4—32no—
448C2×C4—91no—
449C2×C4—91no—
450C2×C4—91no—
451C2×C4—91no—
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—122no—
464C2×C3C2×#369911yescharacteristic
465C2×C3#6371011yescharacteristic
466C2×C3#6371611yescharacteristic
467C2×C3#6429811yescharacteristic
468C2×C3#6430711yescharacteristic
469C2×C3C2×#380411yescharacteristic
470D3—42no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—62no—
475C22—62no—
476C22—62no—
477C22C22×D3×D911yescharacteristic
478C4—21no—
479C4—62no—
480C4—181no—
481C4—181no—
482C4—181no—
483C4—181no—
484C4—542no—
485C4—542no—
486C4#8866212yes—
487C3—21no—
488C3#1058970511yescharacteristic, complemented
489C3#1059498311yescharacteristic
490C2—41no—
491C2—122no—
492C2#1172246511yescharacteristic
493C2#1172247611yescharacteristic
494C2C2×#8866211yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC3×#930D3
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. C3×#63488C2semidirect
4. #11721280C2semidirect
5. #11721263C2semidirect
6. #11721026C2semidirect
7. D12×Dic9C2semidirect
8. #11720969C2semidirect
9. #11720964C2semidirect
10. #11720962C2semidirect
11. #11720913C2semidirect
53. #63488D3semidirect
318. C3×C9#542semidirect
439. C9#5534semidirect
488. C3#10589705semidirect

Maximal subgroups

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

subgroupindex
#117218132
C9×#9302
C3×#634882
#117212802
#117212632
#117210262
D12×Dic92
#117209692
#117209642
#117209622
#117209132
#105949833
#105897053