Finite Group Databasebeta

← #13147673

Subgroups — #13147673

Order 1728.

808 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
2C4×C9×D12C211yesmaximal, characteristic, complemented
3C3×#63432C211yesmaximal, characteristic, complemented
4#11721038C211yesmaximal, characteristic, complemented
5#11721038C211yesmaximal, characteristic, complemented
6D3×C4⋊Dic9C211yesmaximal, characteristic, complemented
7#11721017C212yesmaximal, complemented
8#11721014C212yesmaximal, complemented
9#11720957C212yesmaximal, complemented
10#11720920C212yesmaximal, complemented
11#11720906C211yesmaximal, characteristic, complemented
12#10593925—31nomaximal
13#10589667—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×C9×D12C2211yescharacteristic
17C2×C4×D3×C9C2211yescharacteristic
18C2×C4×D3×C9C2211yescharacteristic
19C2×#6195C2212yes—
20C2×#6195C2212yes—
21C2×D3×Dic9C2212yes—
22C3×C42×C9C2211yescharacteristic
23#88536C2212yes—
24C4⋊(C3⋊Dic9)C2211yescharacteristic
25C9×(C2×D3)⋊C4C2211yescharacteristic
26C9×(C2×D3)⋊C4C2211yescharacteristic
27C9×C4⋊Dic3C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×C4⋊Dic9C2211yescharacteristic
30C3×Dic9⋊C4C2212yes—
31C3×Dic9⋊C4C2212yes—
32(C2×D3)⋊Dic9C2212yes—
33#88445C2212yes—
34#88445C2212yes—
35Dic3⋊Dic9C2212yes—
36Dic3×Dic9C2212yes—
37C4×C4⋊(C3⋊D3)—31no—
38C3×C4×D12D311yescharacteristic
39C3×#870—31no—
40#63909—31no—
41#63909—31no—
42D3×C4⋊Dic3—31no—
43#63886—32no—
44#63881—32no—
45#63853—32no—
46#63848—32no—
47#63841—31no—
48C4×D4×C9—31no—
49#63501—32no—
50#63492—31no—
51#63492—31no—
52#63488—31no—
53C2×C4⋊Dic9—31no—
54#63460—32no—
55#63445—32no—
56#611⋊C4—32no—
57C4×D36—31no—
58C4×D36—61no—
59#63432D311yescharacteristic, complemented
60C22×C3⋊D9C2312yes—
61C22×D3×C9C2311yescharacteristic
62C22×D3×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2312yes—
67C4⋊(C3⋊D9)—22no—
68C4×C3⋊D9—42no—
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2312yes—
73C9×D12—22no—
74C4×D3×C9—22no—
75C4×D3×C9—22no—
76#6195—24no—
77#6195—42no—
78D3×Dic9—42no—
79#5695—91no—
80#5565—91no—
81C2×C4⋊(C3⋊D3)—31no—
82C2×C4×C3⋊D3—32no—
83C2×C3×D12C2×D311yescharacteristic
84C2×C3×C4×D3C2×D311yescharacteristic
85C2×C3×C4×D3C2×D311yescharacteristic
86C2×#626—32no—
87C2×#626—32no—
88C2×D3×Dic3—32no—
89C32×C42C2×D311yescharacteristic
90#3752—32no—
91C4⋊(C3⋊Dic3)—31no—
92C3×(C2×D3)⋊C4C2×D311yescharacteristic
93C3×(C2×D3)⋊C4C2×D311yescharacteristic
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3C2×D311yescharacteristic
97C3×Dic3⋊C4—32no—
98C3×Dic3⋊C4—32no—
99Dic3⋊Dic3—32no—
100#3722—32no—
101#3722—32no—
102(C2×D3)⋊Dic3—32no—
103Dic32—32no—
104C2×D4×C9—31no—
105C22×C4×C9—31no—
106C22×C4×C9—31no—
107C2×#611—32no—
108C2×#611—32no—
109C22×Dic9—32no—
110C2×D36—31no—
111C2×D36—61no—
112C2×C4×D9—32no—
113C2×C4×D9—62no—
114C9×C4⋊C4—31no—
115C9×C22⋊C4—31no—
116C9×C22⋊C4—31no—
117C42×C9—21no—
118C42×C9C2×D311yescharacteristic
119#3676—32no—
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4—32no—
122(C2×D9)⋊C4—32no—
123(C2×D9)⋊C4—62no—
124C4⋊Dic9—31no—
125C4⋊Dic9—32no—
126C4⋊Dic9—61no—
127C4⋊Dic9C2×D311yescharacteristic
128C4⋊Dic9C2×D311yescharacteristic
129Dic9⋊C4C2×D312yes—
130Dic9⋊C4C2×D312yes—
131C4×Dic9—32no—
132C22×C3×C9C2411yescharacteristic
133C2×C3⋊D9—22no—
134C2×C3⋊D9—22no—
135C2×C3⋊D9—22no—
136C2×D3×C9—21no—
137C2×D3×C9—21no—
138C2×D3×C9—21no—
139C2×D3×C9—21no—
140C2×D3×C9—21no—
141C2×D3×C9#4112yes—
142C3×C4×C9—21no—
143C3×C4×C9—21no—
144C3×C4×C9#4112yes—
145C3⋊Dic9—22no—
146C9×Dic3—21no—
147C9×Dic3—21no—
148C3×Dic9—22no—
149C3×Dic9—22no—
150C3×#125—91no—
151C3×C4×D4—31no—
152#939—92no—
153#930—91no—
154#930—91no—
155#926—91no—
156C2×C4⋊Dic3—91no—
157#898—92no—
158#898—92no—
159#895—92no—
160#895—91no—
161#895—91no—
162#894—92no—
163#893—92no—
164#892—91no—
165D3×C4⋊C4—91no—
166#883—92no—
167#67⋊C4—92no—
168C4×D12—91no—
169C4×D12—91no—
170C4×D12—181no—
171C4×D12D911yescharacteristic, complemented
172#870—31no—
173C22×C3⋊D3—32no—
174C22×C3×D3C22×D311yescharacteristic
175C22×C3×D3C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C3⋊Dic3—32no—
180C4⋊(C3⋊D3)—62no—
181C4×C3⋊D3—122no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3C22×D311yescharacteristic
185C2×C3×Dic3C22×D311yescharacteristic
186C3×D12—22no—
187C3×C4×D3—22no—
188C3×C4×D3—22no—
189#626—64no—
190#626—122no—
191D3×Dic3—122no—
192C23×C9—31no—
193C23×C9—31no—
194C22×D9—32no—
195C22×D9—62no—
196D4×C9—62no—
197C2×C4×C9—21no—
198C2×C4×C9—21no—
199C2×C4×C9—21no—
200C2×C4×C9—31no—
201C2×C4×C9—31no—
202C2×C4×C9—62no—
203C2×C4×C9—62no—
204C2×C4×C9C22×D311yescharacteristic
205C2×C4×C9C22×D311yescharacteristic
206C2×C4×C9C22×D311yescharacteristic
207#611—64no—
208#611—122no—
209C2×Dic9—32no—
210C2×Dic9—62no—
211C2×Dic9—122no—
212C2×Dic9C22×D312yes—
213C2×Dic9C22×D312yes—
214D36—64no—
215D36—62no—
216C4×D9—122no—
217C4×D9—124no—
218#553—271noSylow
219C2×C3×C9C2×#4111yescharacteristic
220C2×C3×C9C2×#4111yescharacteristic
221C2×C3×C9H(5,𝔽2)11yescharacteristic
222C3⋊D9—42no—
223D3×C9—22no—
224D3×C9—41no—
225C2×C3×D4—31no—
226C22×C3×C4—31no—
227C22×C3×C4—31no—
228C2×#67—92no—
229C2×#67—92no—
230C22×Dic3—92no—
231C2×D12—92no—
232C2×D12—91no—
233C2×D12—91no—
234C2×D12—92no—
235C2×D12—181no—
236C2×D12C2×D911yescharacteristic
237C2×C4×D3—92no—
238C2×C4×D3—92no—
239C2×C4×D3—92no—
240C2×C4×D3—182no—
241C2×C4×D3C2×D911yescharacteristic
242C2×C4×D3C2×D911yescharacteristic
243C3×C4⋊C4—31no—
244C3×C4⋊C4—91no—
245C3×C4⋊C4—91no—
246C3×C4⋊C4—92no—
247C3×C4⋊C4—92no—
248C3×C22⋊C4—31no—
249C3×C22⋊C4—31no—
250C3×C42—21no—
251C3×C42C2×D911yescharacteristic
252C3×C42D3211yescharacteristic
253#217—92no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—92no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—182no—
262(C2×D3)⋊C4C2×D911yescharacteristic
263(C2×D3)⋊C4C2×D911yescharacteristic
264C4⋊Dic3—31no—
265C4⋊Dic3—31no—
266C4⋊Dic3—92no—
267C4⋊Dic3—91no—
268C4⋊Dic3—91no—
269C4⋊Dic3—181no—
270C4⋊Dic3C2×D911yescharacteristic
271Dic3⋊C4—32no—
272Dic3⋊C4—32no—
273Dic3⋊C4—92no—
274C4×Dic3—92no—
275C4×Dic3—92no—
276C22×C32C23×D311yescharacteristic
277C2×C3⋊D3—62no—
278C2×C3⋊D3—62no—
279C2×C3⋊D3—62no—
280C2×C3×D3—21no—
281C2×C3×D3—21no—
282C2×C3×D3—21no—
283C2×C3×D3—21no—
284C2×C3×D3—21no—
285C2×C3×D3#23512yes—
286C32×C4—21no—
287C32×C4—21no—
288C32×C4#23712yes—
289C3⋊Dic3—62no—
290C3×Dic3—21no—
291C3×Dic3—21no—
292C3×Dic3—62no—
293C3×Dic3—62no—
294C22×C9—21no—
295C22×C9—32no—
296C22×C9—61no—
297C22×C9—61no—
298C22×C9—61no—
299C22×C9—61no—
300C22×C9—61no—
301C22×C9C23×D311yescharacteristic
302C2×D9—64no—
303C2×D9—62no—
304C2×D9—62no—
305C2×D9—62no—
306C2×D9—122no—
307C2×D9—122no—
308C4×C9—22no—
309C4×C9—21no—
310C4×C9—21no—
311C4×C9—22no—
312C4×C9—41no—
313C4×C9—61no—
314C4×C9—61no—
315C4×C9#23912yes—
316Dic9—22no—
317Dic9—22no—
318Dic9—62no—
319Dic9—122no—
320#126—272no—
321#125—91no—
322#123—272no—
323#121—271no—
324#121—271no—
325#121—272no—
326C4×D4—31no—
327C4×D4—272no—
328C4×D4—271no—
329#117—271no—
330C2×C4⋊C4—271no—
331C3×C9#55311yescharacteristic, complemented, Sylow
332C23×C3—31no—
333C23×C3—31no—
334C22×D3—92no—
335C22×D3—92no—
336C22×D3—182no—
337C22×D3C22×D911yescharacteristic
338C22×D3C22×D911yescharacteristic
339C3×D4—62no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—62no—
346C2×C3×C4—62no—
347C2×C3×C4—92no—
348C2×C3×C4—92no—
349C2×C3×C4C22×D911yescharacteristic
350C2×C3×C4C22×D911yescharacteristic
351C2×C3×C4C22×D911yescharacteristic
352C2×C3×C4C2×D3211yescharacteristic
353C2×C3×C4C2×D3211yescharacteristic
354C2×C3×C4C2×D3211yescharacteristic
355#67—184no—
356#67—362no—
357C2×Dic3—32no—
358C2×Dic3—32no—
359C2×Dic3—92no—
360C2×Dic3—92no—
361C2×Dic3—182no—
362C2×Dic3—362no—
363C2×Dic3C22×D911yescharacteristic
364C2×Dic3C22×D911yescharacteristic
365D12—22no—
366D12—184no—
367D12—184no—
368D12—182no—
369D12—182no—
370D12—362no—
371C4×D3—22no—
372C4×D3—22no—
373C4×D3—362no—
374C4×D3—362no—
375C4×D3—362no—
376C4×D3—364no—
377C2×C32C2×#23511yescharacteristic
378C2×C32C2×#23711yescharacteristic
379C2×C32#100911yescharacteristic
380C3⋊D3—122no—
381C3×D3—22no—
382C3×D3—41no—
383C2×C9—21no—
384C2×C9—21no—
385C2×C9—21no—
386C2×C9—62no—
387C2×C9—121no—
388C2×C9C2×#23911yescharacteristic
389C2×C9D3×#4111yescharacteristic
390C2×C9#100911yescharacteristic
391D9—124no—
392D9—122no—
393C2×D4—31no—
394C2×D4—272no—
395C2×D4—272no—
396C2×D4—271no—
397C22×C4—31no—
398C22×C4—31no—
399C22×C4—272no—
400C22×C4—272no—
401C4⋊C4—31no—
402C4⋊C4—91no—
403C4⋊C4—91no—
404C4⋊C4—92no—
405C4⋊C4—92no—
406C4⋊C4—272no—
407C4⋊C4—271no—
408C22⋊C4—31no—
409C22⋊C4—31no—
410C22⋊C4—272no—
411C22⋊C4—272no—
412C22⋊C4—272no—
413C22⋊C4—272no—
414C42—272no—
415C42D3×D911yescharacteristic
416C22×C3—21no—
417C22×C3—32no—
418C22×C3—61no—
419C22×C3—61no—
420C22×C3—61no—
421C22×C3—61no—
422C22×C3—61no—
423C22×C3C23×D911yescharacteristic
424C22×C3C22×D3211yescharacteristic
425C2×D3—21no—
426C2×D3—21no—
427C2×D3—21no—
428C2×D3—21no—
429C2×D3—21no—
430C2×D3—184no—
431C2×D3—182no—
432C2×D3—182no—
433C2×D3—182no—
434C2×D3—182no—
435C2×D3—182no—
436C2×D3—182no—
437C2×D3—362no—
438C2×D3—362no—
439C2×D3#369712yes—
440C3×C4—22no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—22no—
446C3×C4—41no—
447C3×C4—61no—
448C3×C4—61no—
449C3×C4—182no—
450C3×C4—182no—
451C3×C4#369912yes—
452C3×C4#379612yes—
453Dic3—21no—
454Dic3—21no—
455Dic3—62no—
456Dic3—62no—
457Dic3—182no—
458Dic3—182no—
459Dic3—362no—
460C32#556511yescharacteristic
461C9—21no—
462C9#569511yescharacteristic, complemented
463C23—31no—
464C23—31no—
465C23—272no—
466D4—62no—
467D4—544no—
468D4—542no—
469D4—1082no—
470C2×C4—31no—
471C2×C4—31no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—92no—
475C2×C4—92no—
476C2×C4—272no—
477C2×C4—1082no—
478C2×C4—1082no—
479C2×C4C2×D3×D911yescharacteristic
480C2×C4C2×D3×D911yescharacteristic
481C2×C4C2×D3×D911yescharacteristic
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—62no—
486C2×C3—121no—
487C2×C3C2×#369711yescharacteristic
488C2×C3C2×#369911yescharacteristic
489C2×C3#6371511yescharacteristic
490C2×C3C2×#379611yescharacteristic
491C2×C3D3×#23511yescharacteristic
492C2×C3#6430811yescharacteristic
493D3—22no—
494D3—41no—
495D3—364no—
496D3—362no—
497D3—362no—
498C22—32no—
499C22—61no—
500C22—61no—
501C22—61no—
502C22—61no—
503C22—61no—
504C22—542no—
505C22—542no—
506C22—542no—
507C22C22×D3×D911yescharacteristic
508C4—21no—
509C4—21no—
510C4—61no—
511C4—61no—
512C4—182no—
513C4—182no—
514C4—542no—
515C4#8863912yes—
516C3—21no—
517C3#1058966711yescharacteristic, complemented
518C3#1059392511yescharacteristic
519C2—62no—
520C2—121no—
521C2—1082no—
522C2C2×#8863911yescharacteristic
523C2D3×#369711yescharacteristic
524C2#1172247811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9#553
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×C4⋊(C3⋊D9)C2semidirect
2. C4×C9×D12C2semidirect
3. C3×#63432C2semidirect
4. #11721038C2semidirect
5. #11721038C2semidirect
6. D3×C4⋊Dic9C2semidirect
7. #11721017C2semidirect
8. #11721014C2semidirect
9. #11720957C2semidirect
10. #11720920C2semidirect
11. #11720906C2semidirect
59. #63432D3semidirect
171. C4×D12D9semidirect
331. C3×C9#553semidirect
462. C9#5695semidirect
517. C3#10589667semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊(C3⋊D9)2
C4×C9×D122
C3×#634322
#117210382
D3×C4⋊Dic92
#117210172
#117210142
#117209572
#117209202
#117209062
#105939253
#105896673