Finite Group Databasebeta

← #13147569

Subgroups — #13147569

Order 1728.

976 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88635C211yesmaximal, characteristic
2C4×C4⋊(C3⋊D9)C211yesmaximal, characteristic, complemented
3C3×#63434C211yesmaximal, characteristic, complemented
4C4×C9×Dic6C211yesmaximal, characteristic, complemented
5#11720992C211yesmaximal, characteristic, complemented
6D9×C4⋊Dic3C211yesmaximal, characteristic, complemented
7#11720980C212yesmaximal, complemented
8#11720978C212yesmaximal, complemented
9#11720977C212yesmaximal, complemented
10Dic6×Dic9C211yesmaximal, characteristic, complemented
11#11720898C212yesmaximal, complemented
12#10593890—31nomaximal
13#10589676—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9C2212yes—
16C2×C3×C4×D9C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic
18C2×#6192C2212yes—
19C2×(C3⋊D9)⋊C4C2212yes—
20C2×(C3⋊D9)⋊C4C2212yes—
21C2×Dic3×D9C2212yes—
22#88635C418yescomplemented
23C3×C42×C9C2211yescharacteristic
24#88536C2212yes—
25C4⋊(C3⋊Dic9)C2211yescharacteristic
26C3×(C2×D9)⋊C4C2211yescharacteristic
27C3×(C2×D9)⋊C4C2211yescharacteristic
28C9×C4⋊Dic3C2211yescharacteristic
29C9×Dic3⋊C4C2212yes—
30C4×C9×Dic3C2212yes—
31C3×Dic9⋊C4C2211yescharacteristic
32C3×Dic9⋊C4C2211yescharacteristic
33C3×C4×Dic9C2211yescharacteristic
34#88445C2212yes—
35Dic3⋊Dic9C2212yes—
36Dic9⋊Dic3C2212yes—
37Dic3×Dic9C2212yes—
38C2×#3799—31no—
39C4×C4⋊(C3⋊D3)—31no—
40C3×#872—31no—
41C3×C4×Dic6D311yescharacteristic
42#63894—31no—
43D3×C4⋊Dic3—31no—
44#63883—32no—
45#63881—32no—
46#63880—32no—
47Dic3×Dic6—31no—
48#63836—32no—
49C2×#3701—31no—
50C4×Q8×C9—31no—
51Q8×Dic9—31no—
52#63455—32no—
53#63455—31no—
54C4⋊C4×D9—31no—
55C4⋊C4×D9—32no—
56C4×D36—31no—
57C4×D36—32no—
58C4×D36—61no—
59#63434—32no—
60#63434D311yescharacteristic, complemented
61C22×C3⋊D9C2312yes—
62C22×C3×D9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2312yes—
67C4⋊(C3⋊D9)C2×C414yescomplemented
68C4×C3⋊D9—42no—
69C2×C9×Dic3C2312yes—
70C2×C9×Dic3C2312yes—
71C2×C3×Dic9C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C3×C4×D9C2×C414yescomplemented
75C9×Dic6C2×C414yescomplemented
76#6192C2×C418yescomplemented
77(C3⋊D9)⋊C4—42no—
78(C3⋊D9)⋊C4C2×C418yescomplemented
79Dic3×D9—42no—
80#5574—91no—
81#5523—91no—
82C2×C4⋊(C3⋊D3)—31no—
83C2×C4×C3⋊D3—32no—
84C2×C3×C4×D3—31no—
85C2×C3×Dic6C2×D311yescharacteristic
86C2×#626—32no—
87C2×#624—32no—
88C2×#624—32no—
89C2×D3×Dic3—32no—
90#3799—38no—
91C32×C42C2×D311yescharacteristic
92#3752—32no—
93C4⋊(C3⋊Dic3)—31no—
94C3×(C2×D3)⋊C4—31no—
95C3×(C2×D3)⋊C4—31no—
96C3×C4⋊Dic3C2×D311yescharacteristic
97C3×Dic3⋊C4—31no—
98C3×Dic3⋊C4—31no—
99C3×Dic3⋊C4C2×D312yes—
100C3×C4×Dic3—31no—
101C3×C4×Dic3C2×D312yes—
102Dic3⋊Dic3—32no—
103Dic3⋊Dic3—32no—
104#3722—32no—
105Dic32—32no—
106C2×Q8×C9—31no—
107#3701—38no—
108C2×D36—32no—
109C2×D36—31no—
110C2×D36—61no—
111C2×C4×D9—32no—
112C2×C4×D9—32no—
113C2×C4×D9—32no—
114C2×C4×D9—32no—
115C2×C4×D9—62no—
116C2×C4×D9C2×D311yescharacteristic
117C9×C4⋊C4—32no—
118C9×C4⋊C4—31no—
119C42×C9—21no—
120C42×C9—32no—
121C42×C9C2×D311yescharacteristic
122(C2×D9)⋊C4—32no—
123(C2×D9)⋊C4—32no—
124(C2×D9)⋊C4—62no—
125(C2×D9)⋊C4C2×D311yescharacteristic
126(C2×D9)⋊C4C2×D311yescharacteristic
127C4⋊Dic9—31no—
128C4⋊Dic9—32no—
129C4⋊Dic9—61no—
130Dic9⋊C4—32no—
131Dic9⋊C4C2×D311yescharacteristic
132Dic9⋊C4C2×D311yescharacteristic
133C4×Dic9—32no—
134C4×Dic9C2×D311yescharacteristic
135C22×C3×C9C2411yescharacteristic
136C2×C3⋊D9—22no—
137C2×C3⋊D9—22no—
138C2×C3⋊D9C22×C414yes—
139C2×C3×D9—21no—
140C2×C3×D9—21no—
141C2×C3×D9C22×C412yes—
142C3×C4×C9—21no—
143C3×C4×C9—21no—
144C3×C4×C9C22×C412yes—
145C3⋊Dic9—22no—
146C9×Dic3—22no—
147C9×Dic3C22×C414yes—
148C3×Dic9—21no—
149C3×Dic9—21no—
150C3×Dic9C22×C412yes—
151C2×#239—91no—
152C2×#235—91no—
153C3×C4×Q8—31no—
154C3×#117—91no—
155Q8×Dic3—91no—
156C2×C4⋊Dic3—91no—
157#893—91no—
158#893—92no—
159#892—92no—
160D3×C4⋊C4—92no—
161D3×C4⋊C4—91no—
162D3×C4⋊C4—92no—
163#67⋊C4—92no—
164#67⋊C4—92no—
165C4×D12—92no—
166C4×D12—91no—
167C4×D12—91no—
168C4×D12—91no—
169C4×D12—181no—
170#872—31no—
171#872—92no—
172C4×Dic6—91no—
173C4×Dic6D911yescharacteristic, complemented
174C22×C3⋊D3—32no—
175C22×C3×D3—31no—
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C3⋊Dic3—32no—
180C4⋊(C3⋊D3)—34no—
181C4×C3⋊D3—122no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3C22×D312yes—
186C2×C3×Dic3C22×D312yes—
187C3×C4×D3—34no—
188C3×Dic6C4×D314yes—
189#626—38no—
190#624—38no—
191#624—122no—
192D3×Dic3—122no—
193C22×D9—32no—
194C22×D9—62no—
195C22×D9C22×D311yescharacteristic
196Q8×C9—34no—
197C2×C4×C9—21no—
198C2×C4×C9—21no—
199C2×C4×C9—21no—
200C2×C4×C9—32no—
201C2×C4×C9—32no—
202C2×C4×C9C22×D311yescharacteristic
203C2×C4×C9C22×D311yescharacteristic
204C2×C4×C9C22×D311yescharacteristic
205C2×Dic9—32no—
206C2×Dic9—62no—
207C2×Dic9C22×D311yescharacteristic
208C2×Dic9C22×D311yescharacteristic
209C2×Dic9C22×D311yescharacteristic
210D36—38no—
211D36—34no—
212D36—64no—
213C4×D9—38no—
214C4×D9—122no—
215C4×D9—122no—
216C4×D9—122no—
217C4×D9—124no—
218C4×D9C4×D314yescomplemented
219#520—271noSylow
220C2×C3×C9C23×C411yescharacteristic
221C2×C3×C9H(5,𝔽2)11yescharacteristic
222C2×C3×C92−1+411yescharacteristic
223C3⋊D9—24no—
224C3×D9—22no—
225C2×C3×Q8—31no—
226C22×C3×C4—91no—
227C2×#67—92no—
228C22×Dic3—92no—
229#239—98no—
230#235—98no—
231C2×D12—92no—
232C2×D12—91no—
233C2×D12—91no—
234C2×D12—181no—
235C2×C4×D3—31no—
236C2×C4×D3—92no—
237C2×C4×D3—92no—
238C2×C4×D3—92no—
239C2×C4×D3—92no—
240C2×C4×D3—92no—
241C2×C4×D3—92no—
242C2×C4×D3—92no—
243C2×C4×D3—182no—
244C2×Dic6C2×D911yescharacteristic
245C3×C4⋊C4—32no—
246C3×C4⋊C4—31no—
247C3×C4⋊C4—91no—
248C3×C4⋊C4—91no—
249C3×C22⋊C4—91no—
250C3×C22⋊C4—91no—
251C3×C42—21no—
252C3×C42—32no—
253C3×C42—91no—
254C3×C42C2×D911yescharacteristic
255C3×C42D3211yescharacteristic
256(C2×D3)⋊C4—31no—
257(C2×D3)⋊C4—31no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—182no—
263C4⋊Dic3—92no—
264C4⋊Dic3—91no—
265C4⋊Dic3—91no—
266C4⋊Dic3—92no—
267C4⋊Dic3—181no—
268C4⋊Dic3C2×D911yescharacteristic
269Dic3⋊C4—31no—
270Dic3⋊C4—31no—
271Dic3⋊C4—92no—
272Dic3⋊C4—92no—
273Dic3⋊C4C2×D912yes—
274C4×Dic3—31no—
275C4×Dic3—92no—
276C4×Dic3—92no—
277C4×Dic3C2×D912yes—
278C22×C32C23×D311yescharacteristic
279C2×C3⋊D3—34no—
280C2×C3⋊D3—62no—
281C2×C3⋊D3—62no—
282C2×C3×D3—32no—
283C2×C3×D3—61no—
284C2×C3×D3—61no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4C2×C4×D312yes—
288C3⋊Dic3—62no—
289C3×Dic3—22no—
290C3×Dic3—32no—
291C3×Dic3—61no—
292C3×Dic3—61no—
293C3×Dic3C2×C4×D314yes—
294C22×C9—21no—
295C22×C9C23×D311yescharacteristic
296C2×D9—21no—
297C2×D9—21no—
298C2×D9—34no—
299C2×D9—64no—
300C2×D9—62no—
301C2×D9—62no—
302C2×D9—122no—
303C2×D9—122no—
304C2×D9C2×C4×D312yes—
305C4×C9—22no—
306C4×C9—21no—
307C4×C9—21no—
308C4×C9—22no—
309C4×C9—34no—
310C4×C9—41no—
311C4×C9—62no—
312C4×C9C2×C4×D312yes—
313Dic9—21no—
314Dic9—21no—
315Dic9—62no—
316Dic9—122no—
317Dic9C2×C4×D312yes—
318C2×#41—271no—
319C4×Q8—31no—
320C4×Q8—271no—
321C4×D4—272no—
322C4×D4—271no—
323C4×D4—272no—
324C4×D4—271no—
325#117—91no—
326#117—272no—
327C2×C4⋊C4—272no—
328C2×C4⋊C4—271no—
329C3×C9#52011yescharacteristic, complemented, Sylow
330C23×C3—91no—
331C22×D3—31no—
332C22×D3—92no—
333C22×D3—92no—
334C22×D3—182no—
335C3×Q8—34no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—32no—
340C2×C3×C4—32no—
341C2×C3×C4—91no—
342C2×C3×C4—94no—
343C2×C3×C4—91no—
344C2×C3×C4—91no—
345C2×C3×C4C22×D911yescharacteristic
346C2×C3×C4C22×D911yescharacteristic
347C2×C3×C4C22×D911yescharacteristic
348C2×C3×C4C2×D3211yescharacteristic
349C2×C3×C4C2×D3211yescharacteristic
350C2×C3×C4C2×D3211yescharacteristic
351#67—98no—
352C2×Dic3—31no—
353C2×Dic3—31no—
354C2×Dic3—31no—
355C2×Dic3—92no—
356C2×Dic3—92no—
357C2×Dic3—182no—
358C2×Dic3—362no—
359C2×Dic3C22×D912yes—
360C2×Dic3C22×D912yes—
361D12—98no—
362D12—94no—
363D12—94no—
364D12—184no—
365C4×D3—34no—
366C4×D3—98no—
367C4×D3—98no—
368C4×D3—362no—
369C4×D3—362no—
370C4×D3—362no—
371C4×D3—362no—
372C4×D3—362no—
373C4×D3—364no—
374Dic6C4×D914yescomplemented
375C2×C32C22×C4×D311yescharacteristic
376C2×C32#100711yescharacteristic
377C2×C32#100911yescharacteristic
378C3⋊D3—64no—
379C3×D3—62no—
380C2×C9—21no—
381C2×C9—21no—
382C2×C9—21no—
383C2×C9C22×C4×D311yescharacteristic
384C2×C9#100911yescharacteristic
385C2×C9#101011yescharacteristic
386D9—22no—
387D9—64no—
388D9—124no—
389#41—278no—
390C2×Q8—31no—
391C2×D4—272no—
392C2×D4—271no—
393C22×C4—91no—
394C22×C4—272no—
395C22×C4—272no—
396C22×C4—272no—
397C22×C4—272no—
398C4⋊C4—32no—
399C4⋊C4—31no—
400C4⋊C4—91no—
401C4⋊C4—91no—
402C4⋊C4—272no—
403C4⋊C4—272no—
404C4⋊C4—271no—
405C22⋊C4—91no—
406C22⋊C4—91no—
407C22⋊C4—272no—
408C22⋊C4—272no—
409C42—32no—
410C42—91no—
411C42—272no—
412C42D3×D911yescharacteristic
413C22×C3—21no—
414C22×C3—92no—
415C22×C3—181no—
416C22×C3—181no—
417C22×C3C23×D911yescharacteristic
418C22×C3C22×D3211yescharacteristic
419C2×D3—32no—
420C2×D3—61no—
421C2×D3—61no—
422C2×D3—94no—
423C2×D3—94no—
424C2×D3—184no—
425C2×D3—182no—
426C2×D3—182no—
427C2×D3—182no—
428C2×D3—182no—
429C2×D3—362no—
430C2×D3—362no—
431C3×C4—22no—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—22no—
437C3×C4—34no—
438C3×C4—41no—
439C3×C4—62no—
440C3×C4—92no—
441C3×C4—181no—
442C3×C4—181no—
443C3×C4C2×C4×D912yes—
444C3×C4C4×D3212yes—
445Dic3—22no—
446Dic3—32no—
447Dic3—61no—
448Dic3—61no—
449Dic3—182no—
450Dic3—182no—
451Dic3—362no—
452Dic3C2×C4×D914yes—
453C32#557411yescharacteristic
454C9—21no—
455C9#552311yescharacteristic, complemented
456C23—91no—
457C23—272no—
458Q8—34no—
459D4—278no—
460D4—274no—
461C2×C4—32no—
462C2×C4—32no—
463C2×C4—91no—
464C2×C4—94no—
465C2×C4—91no—
466C2×C4—91no—
467C2×C4—278no—
468C2×C4—272no—
469C2×C4—1082no—
470C2×C4—1082no—
471C2×C4—1082no—
472C2×C4C2×D3×D911yescharacteristic
473C2×C4C2×D3×D911yescharacteristic
474C2×C4C2×D3×D911yescharacteristic
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—182no—
479C2×C3C22×C4×D911yescharacteristic
480C2×C3#6371311yescharacteristic
481C2×C3#6371511yescharacteristic
482C2×C3#6429711yescharacteristic
483C2×C3C2×C4×D3211yescharacteristic
484C2×C3#6430811yescharacteristic
485D3—62no—
486D3—184no—
487D3—184no—
488D3—364no—
489C22—92no—
490C22—181no—
491C22—181no—
492C22—274no—
493C22—542no—
494C22—542no—
495C22C22×D3×D911yescharacteristic
496C4—21no—
497C4—21no—
498C4—34no—
499C4—62no—
500C4—92no—
501C4—181no—
502C4—181no—
503C4—542no—
504C4C4×D3×D912yes—
505C3—21no—
506C3#1058967611yescharacteristic, complemented
507C3#1059389011yescharacteristic
508C2—182no—
509C2—544no—
510C2#1172245811yescharacteristic
511C2C2×C4×D3×D911yescharacteristic
512C2#1172247811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC2×#88635C2
cosocleC22×C3×C9C24
hypercenterC42D3×D9
hypocenterC3×C9#520
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4×C4⋊(C3⋊D9)C2semidirect
3. C3×#63434C2semidirect
4. C4×C9×Dic6C2semidirect
5. #11720992C2semidirect
6. D9×C4⋊Dic3C2semidirect
7. #11720980C2semidirect
8. #11720978C2semidirect
9. #11720977C2semidirect
10. Dic6×Dic9C2semidirect
11. #11720898C2semidirect
22. #88635C4semidirect
60. #63434D3semidirect
67. C4⋊(C3⋊D9)C2×C4semidirect
74. C3×C4×D9C2×C4semidirect
75. C9×Dic6C2×C4semidirect
76. #6192C2×C4semidirect
78. (C3⋊D9)⋊C4C2×C4semidirect
173. C4×Dic6D9semidirect
218. C4×D9C4×D3semidirect
329. C3×C9#520semidirect
374. Dic6C4×D9semidirect
455. C9#5523semidirect
506. C3#10589676semidirect

Maximal subgroups

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

subgroupindex
C2×#886352
C4×C4⋊(C3⋊D9)2
C3×#634342
C4×C9×Dic62
#117209922
D9×C4⋊Dic32
#117209802
#117209782
#117209772
Dic6×Dic92
#117208982
#105938903
#105896763