Finite Group Databasebeta

← #13150941

Subgroups — #13150941

Order 1728.

816 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721840C211yesmaximal, characteristic
2C9×#936C211yesmaximal, characteristic, complemented
3C3×D4×Dic9C211yesmaximal, characteristic, complemented
4#11721275C212yesmaximal, complemented
5#11721255C212yesmaximal, complemented
6#11721242C212yesmaximal, complemented
7#11721241C212yesmaximal, complemented
8#11720993C211yesmaximal, characteristic, complemented
9C4×#6189C211yesmaximal, characteristic, complemented
10#11720953C211yesmaximal, characteristic, complemented
11#11720911C211yesmaximal, characteristic, complemented
12#10595210—31nomaximal
13#10589783—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×#6233C2212yes—
17C2×C4⋊(C3⋊D9)C2211yescharacteristic
18C22×C3×Dic9C2212yes—
19C2×C9×Dic6C2211yescharacteristic
20C2×(C3⋊D9)⋊C4C2212yes—
21C2×#6189C2211yescharacteristic
22C4×C3⋊Dic9C2211yescharacteristic
23C9×#217C2212yes—
24C9×#217C2212yes—
25C3×#3676C2212yes—
26C4×C9×Dic3C2211yescharacteristic
27C3×C4⋊Dic9C2211yescharacteristic
28C3×C4×Dic9C2211yescharacteristic
29#88445C2212yes—
30#88445C2212yes—
31#88445C2212yes—
32Dic3⋊Dic9C2211yescharacteristic
33Dic3⋊Dic9C2212yes—
34#88442C2211yescharacteristic
35Dic9⋊Dic3C2211yescharacteristic
36Dic3×Dic9C2212yes—
37#64149—31no—
38C3×#936D311yescharacteristic
39C3×D4×Dic3—31no—
40#63975—32no—
41#63966—32no—
42#63953—32no—
43#63952—32no—
44C4×#627—31no—
45#63894—31no—
46#63864—31no—
47#63843—31no—
48C9×#124—31no—
49#63509—31no—
50Q8×Dic9—31no—
51#63502—31no—
52#63502—61no—
53D4×Dic9D311yescharacteristic, complemented
54#63448—32no—
55#63447—32no—
56#63445—32no—
57#611⋊C4—32no—
58C4×D36—31no—
59C4×Dic18—31no—
60C23×C3×C9C2312yes—
61C22×C3⋊D9C2312yes—
62C3×D4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64#6233—24no—
65#6233—42no—
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68C4⋊(C3⋊D9)—22no—
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C9×Dic3C2312yes—
72C2×C3×Dic9—42no—
73C2×C3×Dic9C2311yescharacteristic
74C2×C3×Dic9C2312yes—
75C2×C3×Dic9C2311yescharacteristic
76C9×Dic6—22no—
77(C3⋊D9)⋊C4—42no—
78#6189—22no—
79#5681—91no—
80#5559—91no—
81C2×C32×D4C2×D311yescharacteristic
82C2×#638—32no—
83C2×#638—32no—
84C2×C4⋊(C3⋊D3)—31no—
85C22×C3×Dic3—32no—
86C2×C3×Dic6C2×D311yescharacteristic
87C2×#627—31no—
88C2×#624—32no—
89C4×C3⋊Dic3—31no—
90C3×#217—32no—
91C3×#217C2×D312yes—
92C3×#217C2×D312yes—
93C3×C4⋊Dic3—31no—
94C3×C4×Dic3—31no—
95C3×C4×Dic3C2×D311yescharacteristic
96#3724—31no—
97Dic3⋊Dic3—32no—
98Dic3⋊Dic3—31no—
99Dic3⋊Dic3—31no—
100#3722—32no—
101#3722—32no—
102#3722—32no—
103Dic32—32no—
104C2×Q8×C9—31no—
105C2×D4×C9—21no—
106C2×D4×C9C2×D311yescharacteristic
107C2×#611—32no—
108C2×#611—32no—
109C2×#611—62no—
110C2×#611—62no—
111C22×Dic9C2×D312yes—
112C2×D36—31no—
113C2×D36—61no—
114C2×C4×D9—32no—
115C2×Dic18—31no—
116C9×C22⋊C4—32no—
117C9×C22⋊C4—32no—
118C42×C9—31no—
119#3676C2×D312yes—
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4—32no—
122(C2×D9)⋊C4—32no—
123C4⋊Dic9—32no—
124C4⋊Dic9—31no—
125C4⋊Dic9C2×D311yescharacteristic
126Dic9⋊C4—31no—
127Dic9⋊C4—31no—
128C4×Dic9—31no—
129C4×Dic9—32no—
130C4×Dic9—61no—
131C4×Dic9C2×D311yescharacteristic
132C22×C3×C9—22no—
133C22×C3×C9—22no—
134C22×C3×C9—22no—
135C22×C3×C9C2411yescharacteristic
136C2×C3⋊D9—22no—
137C2×C3⋊D9—22no—
138C2×C3⋊D9—22no—
139C3×C4×C9#4112yes—
140C3⋊Dic9—21no—
141C3⋊Dic9#4112yes—
142C9×Dic3—21no—
143C9×Dic3—22no—
144C9×Dic3—21no—
145C3×Dic9—21no—
146C3×Dic9—22no—
147C3×Dic9—21no—
148C3×#124—31no—
149C3×C4×D4—91no—
150#947—91no—
151Q8×Dic3—91no—
152#940—91no—
153#940—91no—
154#940—181no—
155#936D911yescharacteristic, complemented
156D4×Dic3—31no—
157#929—92no—
158C4×#67—92no—
159#893—91no—
160#887—91no—
161#886—92no—
162#885—92no—
163#885—92no—
164#883—92no—
165#883—92no—
166#67⋊C4—92no—
167#875—91no—
168C4×D12—91no—
169C4×Dic6—91no—
170C4×Dic6—91no—
171C23×C32C22×D312yes—
172C22×C3⋊D3—32no—
173C32×D4—22no—
174C2×C32×C4C22×D311yescharacteristic
175#638—64no—
176#638—122no—
177C2×C3⋊Dic3—31no—
178C2×C3⋊Dic3—31no—
179C4⋊(C3⋊D3)—62no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—122no—
184C2×C3×Dic3C22×D312yes—
185C2×C3×Dic3C22×D311yescharacteristic
186C2×C3×Dic3C22×D311yescharacteristic
187C3×Dic6—22no—
188#627—62no—
189#624—122no—
190C23×C9—22no—
191C23×C9C22×D312yes—
192C22×D9—32no—
193C22×D9—62no—
194Q8×C9—62no—
195D4×C9—22no—
196D4×C9—42no—
197C2×C4×C9—21no—
198C2×C4×C9—32no—
199C2×C4×C9—31no—
200C2×C4×C9—31no—
201C2×C4×C9C22×D311yescharacteristic
202#611—64no—
203#611—124no—
204#611—122no—
205#611—124no—
206C2×Dic9—31no—
207C2×Dic9—31no—
208C2×Dic9—42no—
209C2×Dic9—61no—
210C2×Dic9—61no—
211C2×Dic9C22×D312yes—
212C2×Dic9C22×D311yescharacteristic
213C2×Dic9C22×D311yescharacteristic
214D36—62no—
215D36—122no—
216C4×D9—122no—
217Dic18—62no—
218#559—271noSylow
219C2×C3×C9—42no—
220C2×C3×C9C2×#4111yescharacteristic
221C2×C3×C9C2×#4111yescharacteristic
222C2×C3×C9H(5,𝔽2)11yescharacteristic
223C3⋊D9—42no—
224C2×C3×Q8—31no—
225C2×C3×D4—21no—
226C2×C3×D4C2×D911yescharacteristic
227C2×C3×D4D3211yescharacteristic
228C22×C3×C4—92no—
229C2×#67—92no—
230C2×#67—92no—
231C2×#67—92no—
232C2×#67—92no—
233C2×#67—182no—
234C2×#67—182no—
235C22×Dic3—32no—
236C2×D12—91no—
237C2×D12—91no—
238C2×D12—181no—
239C2×C4×D3—92no—
240C2×C4×D3—92no—
241C2×Dic6—91no—
242C2×Dic6—91no—
243C2×Dic6C2×D911yescharacteristic
244C3×C4⋊C4—91no—
245C3×C22⋊C4—32no—
246C3×C22⋊C4—32no—
247C3×C22⋊C4—92no—
248C3×C42—31no—
249C3×C42—91no—
250#217—32no—
251#217C2×D912yes—
252#217C2×D912yes—
253(C2×D3)⋊C4—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—
259C4⋊Dic3—31no—
260C4⋊Dic3—91no—
261C4⋊Dic3—92no—
262C4⋊Dic3—91no—
263Dic3⋊C4—91no—
264Dic3⋊C4—91no—
265Dic3⋊C4—91no—
266Dic3⋊C4—91no—
267Dic3⋊C4—92no—
268C4×Dic3—31no—
269C4×Dic3—92no—
270C4×Dic3—91no—
271C4×Dic3—91no—
272C4×Dic3—92no—
273C4×Dic3—181no—
274C4×Dic3C2×D911yescharacteristic
275C22×C32—22no—
276C22×C32—22no—
277C22×C32—22no—
278C22×C32C23×D311yescharacteristic
279C2×C3⋊D3—62no—
280C2×C3⋊D3—62no—
281C2×C3⋊D3—62no—
282C32×C4#23712yes—
283C3⋊Dic3—32no—
284C3⋊Dic3—61no—
285C3×Dic3—22no—
286C3×Dic3—21no—
287C3×Dic3—21no—
288C3×Dic3—61no—
289C3×Dic3—62no—
290C3×Dic3—61no—
291C22×C9—22no—
292C22×C9—22no—
293C22×C9—22no—
294C22×C9—21no—
295C22×C9—42no—
296C22×C9—42no—
297C22×C9—42no—
298C22×C9C23×D311yescharacteristic
299C2×D9—62no—
300C2×D9—62no—
301C2×D9—62no—
302C2×D9—122no—
303C2×D9—122no—
304C2×D9—122no—
305C4×C9—22no—
306C4×C9—62no—
307C4×C9—61no—
308C4×C9—61no—
309C4×C9#23512yes—
310Dic9—22no—
311Dic9—21no—
312Dic9—21no—
313Dic9—32no—
314Dic9—62no—
315Dic9—62no—
316Dic9—61no—
317#127—271no—
318#124—31no—
319#124—271no—
320#124—272no—
321#123—272no—
322#123—272no—
323C4×Q8—271no—
324C4×Q8—271no—
325C4×D4—91no—
326C4×D4—271no—
327C4×D4—272no—
328C3×C9#55911yescharacteristic, complemented, Sylow
329C23×C3—22no—
330C23×C3C22×D912yes—
331C23×C3C2×D3212yes—
332C22×D3—92no—
333C22×D3—92no—
334C22×D3—182no—
335C3×Q8—62no—
336C3×D4—22no—
337C3×D4—22no—
338C3×D4—42no—
339C2×C3×C4—21no—
340C2×C3×C4—32no—
341C2×C3×C4—31no—
342C2×C3×C4—31no—
343C2×C3×C4—92no—
344C2×C3×C4—91no—
345C2×C3×C4—91no—
346C2×C3×C4—362no—
347C2×C3×C4C22×D911yescharacteristic
348C2×C3×C4C2×D3211yescharacteristic
349#67—184no—
350#67—184no—
351#67—364no—
352#67—362no—
353#67—362no—
354#67—364no—
355C2×Dic3—31no—
356C2×Dic3—31no—
357C2×Dic3—32no—
358C2×Dic3—91no—
359C2×Dic3—91no—
360C2×Dic3—91no—
361C2×Dic3—91no—
362C2×Dic3—122no—
363C2×Dic3—181no—
364C2×Dic3—181no—
365C2×Dic3C22×D912yes—
366C2×Dic3C22×D911yescharacteristic
367C2×Dic3C22×D911yescharacteristic
368D12—182no—
369D12—182no—
370D12—362no—
371C4×D3—362no—
372C4×D3—362no—
373Dic6—22no—
374Dic6—182no—
375Dic6—182no—
376C2×C32—42no—
377C2×C32C2×#23711yescharacteristic
378C2×C32D3×#4111yescharacteristic
379C2×C32#100911yescharacteristic
380C3⋊D3—122no—
381C2×C9—21no—
382C2×C9—21no—
383C2×C9—21no—
384C2×C9—42no—
385C2×C9—82no—
386C2×C9C2×#23511yescharacteristic
387C2×C9#100411yescharacteristic
388C2×C9D3×#4111yescharacteristic
389D9—122no—
390D9—242no—
391C2×Q8—31no—
392C2×Q8—271no—
393C2×D4—272no—
394C2×D4—272no—
395C2×D4—271no—
396C2×D4D3×D911yescharacteristic
397C22×C4—92no—
398C22×C4—272no—
399C4⋊C4—91no—
400C4⋊C4—271no—
401C4⋊C4—271no—
402C4⋊C4—271no—
403C4⋊C4—272no—
404C22⋊C4—32no—
405C22⋊C4—32no—
406C22⋊C4—92no—
407C22⋊C4—272no—
408C22⋊C4—272no—
409C22⋊C4—272no—
410C42—31no—
411C42—91no—
412C42—272no—
413C42—271no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—21no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3C23×D911yescharacteristic
425C22×C3C22×D3211yescharacteristic
426C2×D3—182no—
427C2×D3—182no—
428C2×D3—182no—
429C2×D3—182no—
430C2×D3—182no—
431C2×D3—182no—
432C2×D3—362no—
433C2×D3—362no—
434C2×D3—362no—
435C3×C4—22no—
436C3×C4—61no—
437C3×C4—61no—
438C3×C4—62no—
439C3×C4—181no—
440C3×C4—182no—
441C3×C4—181no—
442C3×C4#369912yes—
443C3×C4#380412yes—
444Dic3—22no—
445Dic3—21no—
446Dic3—21no—
447Dic3—61no—
448Dic3—62no—
449Dic3—61no—
450Dic3—92no—
451Dic3—92no—
452Dic3—182no—
453Dic3—182no—
454Dic3—181no—
455Dic3—181no—
456C32#568111yescharacteristic
457C9—21no—
458C9#555911yescharacteristic, complemented
459C23—272no—
460C23C2×D3×D912yes—
461Q8—62no—
462Q8—542no—
463D4—22no—
464D4—544no—
465D4—542no—
466D4—1082no—
467C2×C4—31no—
468C2×C4—31no—
469C2×C4—32no—
470C2×C4—91no—
471C2×C4—91no—
472C2×C4—92no—
473C2×C4—271no—
474C2×C4—271no—
475C2×C4—362no—
476C2×C4—1082no—
477C2×C4C2×D3×D911yescharacteristic
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—42no—
482C2×C3—42no—
483C2×C3—82no—
484C2×C3C2×#369911yescharacteristic
485C2×C3#41×D911yescharacteristic
486C2×C3#6371511yescharacteristic
487C2×C3#6431211yescharacteristic
488C2×C3#6431411yescharacteristic
489C2×C3C2×#380411yescharacteristic
490D3—362no—
491D3—362no—
492D3—722no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—542no—
497C22—542no—
498C22—542no—
499C22C22×D3×D911yescharacteristic
500C4—61no—
501C4—61no—
502C4—62no—
503C4—181no—
504C4—181no—
505C4—182no—
506C4—272no—
507C4—541no—
508C4#8866212yes—
509C3—21no—
510C3#1058978311yescharacteristic, complemented
511C3#1059521011yescharacteristic
512C2—42no—
513C2—1082no—
514C2#1172249811yescharacteristic
515C2#1172250111yescharacteristic
516C2C2×#8866211yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
prime radical#11721840C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#559
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C9×#936C2semidirect
3. C3×D4×Dic9C2semidirect
4. #11721275C2semidirect
5. #11721255C2semidirect
6. #11721242C2semidirect
7. #11721241C2semidirect
8. #11720993C2semidirect
9. C4×#6189C2semidirect
10. #11720953C2semidirect
11. #11720911C2semidirect
53. D4×Dic9D3semidirect
155. #936D9semidirect
328. C3×C9#559semidirect
458. C9#5559semidirect
510. C3#10589783semidirect

Maximal subgroups

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

subgroupindex
#117218402
C9×#9362
C3×D4×Dic92
#117212752
#117212552
#117212422
#117212412
#117209932
C4×#61892
#117209532
#117209112
#105952103
#105897833