Finite Group Databasebeta

← #13147684

Subgroups — #13147684

Order 1728.

718 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6226C211yesmaximal, characteristic, complemented
2C4×C9×D12C211yesmaximal, characteristic, complemented
3C3×#63434C211yesmaximal, characteristic, complemented
4#11721044C212yesmaximal, complemented
5#11721036C211yesmaximal, characteristic
6#11721032C212yesmaximal, complemented
7#11720969C211yesmaximal, characteristic, complemented
8#11720966C212yesmaximal, complemented
9#11720964C211yesmaximal, characteristic, complemented
10#11720955C212yesmaximal, complemented
11#11720945C211yesmaximal, characteristic, complemented
12#10593983—31nomaximal
13#10589663—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C9×D12C2211yescharacteristic
16C2×C4×D3×C9C2212yes—
17C2×C3×C4×D9C2211yescharacteristic
18C2×#6194C2212yes—
19C2×#6189C2212yes—
20C3×C42×C9C2211yescharacteristic
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22#88534C2212yes—
23C4×C3⋊Dic9C2212yes—
24C9×(C2×D3)⋊C4C2212yes—
25C3×(C2×D9)⋊C4C2211yescharacteristic
26C3×(C2×D9)⋊C4C2211yescharacteristic
27C9×C4⋊Dic3C2211yescharacteristic
28C3×Dic9⋊C4C2211yescharacteristic
29C3×Dic9⋊C4C2211yescharacteristic
30C3×C4×Dic9C2211yescharacteristic
31(C2×D3)⋊Dic9C2212yes—
32(C2×D3)⋊Dic9C2212yes—
33(C2×D3)⋊Dic9C2212yes—
34(C2×D9)⋊Dic3C2212yes—
35Dic3⋊Dic9C2212yes—
36Dic9⋊Dic3C2212yes—
37C4×#634—31no—
38C3×C4×D12D311yescharacteristic
39C3×#872—31no—
40#63918—31no—
41#63915—32no—
42#63908—31no—
43#63905—32no—
44#63862—32no—
45#63860—31no—
46#63857—31no—
47#63851—32no—
48C4×D4×C9—31no—
49#63500—31no—
50#63498—31no—
51#63491—32no—
52#63486—32no—
53#63459—31no—
54#63450—31no—
55#63447—32no—
56#63441—32no—
57#63434D311yescharacteristic, complemented
58C4×Dic18—31no—
59C4×Dic18—61no—
60C22×D3×C9C2312yes—
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2312yes—
67#6226—22no—
68C2×C9×Dic3C2312yes—
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C9×D12—22no—
73C4×D3×C9—42no—
74C3×C4×D9—22no—
75#6194—42no—
76#6189—42no—
77#5561—91no—
78#5534—91no—
79C2×#634—31no—
80C2×C3×D12C2×D311yescharacteristic
81C2×C3×C4×D3—31no—
82C2×C3×C4×D3C2×D312yes—
83C2×#627—32no—
84C2×#625—32no—
85C32×C42C2×D311yescharacteristic
86C4⋊(C3⋊Dic3)—31no—
87#3750—32no—
88C4×C3⋊Dic3—32no—
89C3×(C2×D3)⋊C4—31no—
90C3×(C2×D3)⋊C4—31no—
91C3×(C2×D3)⋊C4C2×D312yes—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×Dic3⋊C4—31no—
94C3×Dic3⋊C4—31no—
95C3×C4×Dic3—31no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98(C2×D3)⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102C2×D4×C9—31no—
103C22×C4×C9—32no—
104C2×#611—32no—
105C2×C4×D9C2×D311yescharacteristic
106C2×Dic18—31no—
107C2×Dic18—32no—
108C2×Dic18—61no—
109C9×C4⋊C4—31no—
110C9×C22⋊C4—32no—
111C42×C9—21no—
112C42×C9C2×D311yescharacteristic
113#3676—32no—
114#3676—32no—
115#3676—32no—
116(C2×D9)⋊C4—32no—
117(C2×D9)⋊C4C2×D311yescharacteristic
118(C2×D9)⋊C4C2×D311yescharacteristic
119C4⋊Dic9—31no—
120C4⋊Dic9—32no—
121C4⋊Dic9—61no—
122Dic9⋊C4—32no—
123Dic9⋊C4—32no—
124Dic9⋊C4—62no—
125Dic9⋊C4C2×D311yescharacteristic
126Dic9⋊C4C2×D311yescharacteristic
127C4×Dic9—32no—
128C4×Dic9—62no—
129C4×Dic9C2×D311yescharacteristic
130C22×C3×C9C2411yescharacteristic
131C2×D3×C9—22no—
132C2×D3×C9—22no—
133C2×D3×C9—22no—
134C2×C3×D9—21no—
135C2×C3×D9—21no—
136C2×C3×D9—21no—
137C3×C4×C9—21no—
138C3×C4×C9—21no—
139C3×C4×C9#4112yes—
140C3⋊Dic9—22no—
141C3⋊Dic9—22no—
142C9×Dic3—22no—
143C3×Dic9—21no—
144C3×Dic9—21no—
145C3×Dic9—21no—
146C3×C4×D4—31no—
147C3×#117—91no—
148#938—91no—
149#936—91no—
150#930—91no—
151#929—92no—
152#924—92no—
153#924—91no—
154#898—92no—
155#897—91no—
156#896—92no—
157#888—91no—
158#885—92no—
159#885—92no—
160#883—92no—
161#879—92no—
162#875—91no—
163C4×D12D911yescharacteristic, complemented
164#872—31no—
165#869—91no—
166C4×Dic6—91no—
167C4×Dic6—91no—
168C4×Dic6—181no—
169C22×C3×D3—31no—
170C22×C3×D3C22×D312yes—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3—32no—
175C2×C3⋊Dic3—32no—
176#634—62no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3C22×D312yes—
181C3×D12—22no—
182C3×C4×D3—42no—
183C3×C4×D3—62no—
184#627—122no—
185#625—122no—
186C23×C9—32no—
187C22×D9C22×D311yescharacteristic
188D4×C9—62no—
189C2×C4×C9—21no—
190C2×C4×C9—21no—
191C2×C4×C9—21no—
192C2×C4×C9—32no—
193C2×C4×C9—122no—
194C2×C4×C9C22×D311yescharacteristic
195C2×C4×C9C22×D311yescharacteristic
196C2×C4×C9C22×D311yescharacteristic
197#611—122no—
198C2×Dic9—32no—
199C2×Dic9—32no—
200C2×Dic9—62no—
201C2×Dic9—62no—
202C2×Dic9C22×D311yescharacteristic
203C2×Dic9C22×D311yescharacteristic
204C2×Dic9C22×D311yescharacteristic
205C4×D9—22no—
206Dic18—64no—
207Dic18—62no—
208Dic18—122no—
209#542—271noSylow
210C2×C3×C9C2×#4111yescharacteristic
211C2×C3×C9H(5,𝔽2)11yescharacteristic
212C2×C3×C92−1+411yescharacteristic
213D3×C9—42no—
214C3×D9—41no—
215C2×C3×D4—31no—
216C22×C3×C4—32no—
217C22×C3×C4—91no—
218C2×#67—92no—
219C2×#67—92no—
220C2×D12C2×D911yescharacteristic
221C2×C4×D3—31no—
222C2×C4×D3C2×D912yes—
223C2×Dic6—91no—
224C2×Dic6—91no—
225C2×Dic6—92no—
226C2×Dic6—92no—
227C2×Dic6—181no—
228C3×C4⋊C4—31no—
229C3×C4⋊C4—91no—
230C3×C4⋊C4—91no—
231C3×C22⋊C4—32no—
232C3×C22⋊C4—91no—
233C3×C22⋊C4—91no—
234C3×C42—21no—
235C3×C42—91no—
236C3×C42C2×D911yescharacteristic
237C3×C42D3211yescharacteristic
238#217—92no—
239#217—92no—
240#217—92no—
241#217—92no—
242(C2×D3)⋊C4—31no—
243(C2×D3)⋊C4—31no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4C2×D912yes—
249C4⋊Dic3—92no—
250C4⋊Dic3—91no—
251C4⋊Dic3—91no—
252C4⋊Dic3—92no—
253C4⋊Dic3—181no—
254C4⋊Dic3C2×D911yescharacteristic
255Dic3⋊C4—31no—
256Dic3⋊C4—31no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261Dic3⋊C4—182no—
262C4×Dic3—31no—
263C4×Dic3—92no—
264C4×Dic3—92no—
265C4×Dic3—182no—
266C22×C32C23×D311yescharacteristic
267C2×C3×D3—22no—
268C2×C3×D3—22no—
269C2×C3×D3—22no—
270C2×C3×D3—61no—
271C2×C3×D3—61no—
272C2×C3×D3—61no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4#23512yes—
276C3⋊Dic3—62no—
277C3⋊Dic3—62no—
278C3×Dic3—22no—
279C3×Dic3—61no—
280C3×Dic3—61no—
281C3×Dic3—61no—
282C22×C9—21no—
283C22×C9—62no—
284C22×C9—62no—
285C22×C9—62no—
286C22×C9C23×D311yescharacteristic
287C2×D9—21no—
288C2×D9—21no—
289C2×D9—21no—
290C4×C9—22no—
291C4×C9—21no—
292C4×C9—21no—
293C4×C9—22no—
294C4×C9—41no—
295C4×C9—62no—
296C4×C9#23512yes—
297Dic9—21no—
298Dic9—21no—
299Dic9—21no—
300Dic9—64no—
301Dic9—62no—
302Dic9—62no—
303Dic9—122no—
304C4⋊Q8—271no—
305#126—272no—
306#124—271no—
307#124—272no—
308#123—272no—
309C22⋊Q8—271no—
310C22⋊Q8—272no—
311#121—271no—
312C4×Q8—271no—
313C4×D4—31no—
314#117—91no—
315C3×C9#54211yescharacteristic, complemented, Sylow
316C23×C3—32no—
317C23×C3—91no—
318C22×D3—31no—
319C22×D3C22×D912yes—
320C3×D4—62no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—32no—
325C2×C3×C4—91no—
326C2×C3×C4—91no—
327C2×C3×C4—91no—
328C2×C3×C4—122no—
329C2×C3×C4—182no—
330C2×C3×C4C22×D911yescharacteristic
331C2×C3×C4C22×D911yescharacteristic
332C2×C3×C4C22×D911yescharacteristic
333C2×C3×C4C2×D3211yescharacteristic
334C2×C3×C4C2×D3211yescharacteristic
335C2×C3×C4C2×D3211yescharacteristic
336#67—362no—
337#67—362no—
338C2×Dic3—31no—
339C2×Dic3—31no—
340C2×Dic3—31no—
341C2×Dic3—92no—
342C2×Dic3—92no—
343C2×Dic3—92no—
344C2×Dic3—92no—
345C2×Dic3—182no—
346C2×Dic3—182no—
347C2×Dic3C22×D912yes—
348D12—22no—
349C4×D3—42no—
350C4×D3—62no—
351Dic6—184no—
352Dic6—182no—
353Dic6—182no—
354Dic6—362no—
355Dic6—362no—
356C2×C32C2×#23511yescharacteristic
357C2×C32#100411yescharacteristic
358C2×C32#101011yescharacteristic
359C3×D3—42no—
360C3×D3—121no—
361C2×C9—21no—
362C2×C9—21no—
363C2×C9—21no—
364C2×C9—122no—
365C2×C9C2×#23511yescharacteristic
366C2×C9#100911yescharacteristic
367C2×C9#101011yescharacteristic
368D9—41no—
369C2×Q8—272no—
370C2×Q8—271no—
371C2×D4—31no—
372C2×D4—272no—
373C22×C4—32no—
374C22×C4—91no—
375C4⋊C4—31no—
376C4⋊C4—91no—
377C4⋊C4—91no—
378C4⋊C4—272no—
379C4⋊C4—272no—
380C4⋊C4—271no—
381C4⋊C4—272no—
382C22⋊C4—32no—
383C22⋊C4—91no—
384C22⋊C4—91no—
385C22⋊C4—272no—
386C22⋊C4—272no—
387C22⋊C4—272no—
388C22⋊C4—272no—
389C42—91no—
390C42—272no—
391C42D3×D911yescharacteristic
392C22×C3—21no—
393C22×C3—62no—
394C22×C3—62no—
395C22×C3—62no—
396C22×C3—181no—
397C22×C3—181no—
398C22×C3—181no—
399C22×C3C23×D911yescharacteristic
400C22×C3C22×D3211yescharacteristic
401C2×D3—22no—
402C2×D3—22no—
403C2×D3—22no—
404C2×D3—61no—
405C2×D3—61no—
406C2×D3—61no—
407C3×C4—22no—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—22no—
413C3×C4—41no—
414C3×C4—62no—
415C3×C4—181no—
416C3×C4—181no—
417C3×C4—181no—
418C3×C4#369712yes—
419C3×C4#379812yes—
420Dic3—22no—
421Dic3—61no—
422Dic3—61no—
423Dic3—61no—
424Dic3—184no—
425Dic3—182no—
426Dic3—182no—
427Dic3—182no—
428Dic3—182no—
429Dic3—362no—
430C32#556111yescharacteristic
431C9—21no—
432C9#553411yescharacteristic, complemented
433C23—32no—
434C23—91no—
435Q8—542no—
436Q8—1082no—
437D4—62no—
438D4—1082no—
439C2×C4—32no—
440C2×C4—91no—
441C2×C4—91no—
442C2×C4—91no—
443C2×C4—122no—
444C2×C4—182no—
445C2×C4—272no—
446C2×C4—272no—
447C2×C4C2×D3×D911yescharacteristic
448C2×C4C2×D3×D911yescharacteristic
449C2×C4C2×D3×D911yescharacteristic
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—122no—
454C2×C3—361no—
455C2×C3C2×#369711yescharacteristic
456C2×C3#6371011yescharacteristic
457C2×C3#6371611yescharacteristic
458C2×C3#6429811yescharacteristic
459C2×C3C2×#379811yescharacteristic
460C2×C3#6430711yescharacteristic
461D3—42no—
462D3—121no—
463C22—62no—
464C22—62no—
465C22—62no—
466C22—181no—
467C22—181no—
468C22—181no—
469C22C22×D3×D911yescharacteristic
470C4—21no—
471C4—21no—
472C4—62no—
473C4—181no—
474C4—181no—
475C4—181no—
476C4—542no—
477C4—542no—
478C4#8864012yes—
479C3—21no—
480C3#1058966311yescharacteristic, complemented
481C3#1059398311yescharacteristic
482C2—122no—
483C2—361no—
484C2#1172246511yescharacteristic
485C2C2×#8864011yescharacteristic
486C2#1172247611yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radical#11721036C2
hypercenterC42D3×D9
hypocenterC3×C9#542
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×#6226C2semidirect
2. C4×C9×D12C2semidirect
3. C3×#63434C2semidirect
4. #11721044C2semidirect
6. #11721032C2semidirect
7. #11720969C2semidirect
8. #11720966C2semidirect
9. #11720964C2semidirect
10. #11720955C2semidirect
11. #11720945C2semidirect
57. #63434D3semidirect
163. C4×D12D9semidirect
315. C3×C9#542semidirect
432. C9#5534semidirect
480. C3#10589663semidirect

Maximal subgroups

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

subgroupindex
C4×#62262
C4×C9×D122
C3×#634342
#117210442
#117210362
#117210322
#117209692
#117209662
#117209642
#117209552
#117209452
#105939833
#105896633