Finite Group Databasebeta

← #13148806

Subgroups — #13148806

Order 1728.

974 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×#88634C211yesmaximal, characteristic
2C4⋊C4×C3⋊D9C211yesmaximal, characteristic, complemented
3C3×#63455C211yesmaximal, characteristic, complemented
4C9×#887C211yesmaximal, characteristic, complemented
5C4×#6192C212yesmaximal, complemented
6Dic3×D36C211yesmaximal, characteristic, complemented
7#11720978C212yesmaximal, complemented
8#11720974C212yesmaximal, complemented
9D9×Dic3⋊C4C212yesmaximal, complemented
10#11720931C211yesmaximal, characteristic, complemented
11Dic6×Dic9C211yesmaximal, characteristic, complemented
12#10594361—31nomaximal
13#10589676—31nomaximal
14C2×C4×C3⋊D9C2212yes—
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×C3×D36C2211yescharacteristic
17C2×C3×C4×D9C2212yes—
18C2×C9×Dic6C2211yescharacteristic
19C2×#6192C2212yes—
20C2×Dic3×D9C2212yes—
21C2×Dic3×D9C2212yes—
22#88634C418yescomplemented
23C3×C9×C4⋊C4C2211yescharacteristic
24C4⋊(C3⋊Dic9)C2211yescharacteristic
25#88534C2212yes—
26C3×(C2×D9)⋊C4C2212yes—
27C9×Dic3⋊C4C2212yes—
28C4×C9×Dic3C2212yes—
29C4×C9×Dic3C2211yescharacteristic
30C3×C4×Dic9C2211yescharacteristic
31#88445C2211yescharacteristic
32#88445C2211yescharacteristic
33(C2×D9)⋊Dic3C2212yes—
34Dic3⋊Dic9C2212yes—
35#88442C2211yescharacteristic
36#88442C2211yescharacteristic
37Dic3×Dic9C2212yes—
38C2×#3796—31no—
39C4⋊C4×C3⋊D3—31no—
40C3×#893—31no—
41C3×#887D311yescharacteristic
42C4×#626—32no—
43Dic3×D12—31no—
44#63881—32no—
45#63877—32no—
46D3×Dic3⋊C4—32no—
47#63874—31no—
48Dic3×Dic6—31no—
49C2×#3701—31no—
50C4×Q8×C9—31no—
51Q8×Dic9—31no—
52#63455—32no—
53#63455D311yescharacteristic, complemented
54C4⋊C4×D9—32no—
55C4⋊C4×D9—31no—
56C4⋊C4×D9—61no—
57C4×D36—31no—
58C4×D36—32no—
59#63434—32no—
60#63434—31no—
61C22×C3⋊D9C2311yescharacteristic
62C22×C3×D9C2312yes—
63C2×C3×C4×C9C2312yes—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2311yescharacteristic
67C4×C3⋊D9—42no—
68C4×C3⋊D9C2×C414yescomplemented
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C9×Dic3C2312yes—
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C3×D36C2×C414yescomplemented
75C3×C4×D9—42no—
76C9×Dic6C2×C414yescomplemented
77#6192C2×C418yescomplemented
78Dic3×D9—42no—
79Dic3×D9C2×C418yescomplemented
80#5574—91no—
81#5546—91no—
82C2×C4×C3⋊D3—31no—
83C2×C4×C3⋊D3—32no—
84C2×C3×D12—31no—
85C2×C3×C4×D3—32no—
86C2×C3×Dic6C2×D311yescharacteristic
87C2×#626—32no—
88C2×D3×Dic3—32no—
89C2×D3×Dic3—32no—
90#3796—38no—
91C32×C4⋊C4C2×D311yescharacteristic
92C4⋊(C3⋊Dic3)—31no—
93#3750—32no—
94C3×(C2×D3)⋊C4—32no—
95C3×Dic3⋊C4C2×D312yes—
96C3×C4×Dic3—31no—
97C3×C4×Dic3C2×D311yescharacteristic
98C3×C4×Dic3C2×D312yes—
99#3724—31no—
100#3724—31no—
101Dic3⋊Dic3—32no—
102#3722—31no—
103#3722—31no—
104(C2×D3)⋊Dic3—32no—
105Dic32—32no—
106C2×Q8×C9—31no—
107#3701—38no—
108C2×D36—32no—
109C2×D36C2×D311yescharacteristic
110C2×C4×D9—32no—
111C2×C4×D9—31no—
112C2×C4×D9—32no—
113C2×C4×D9—32no—
114C2×C4×D9—61no—
115C2×C4×D9—62no—
116C2×C4×D9C2×D312yes—
117C9×C4⋊C4—21no—
118C9×C4⋊C4—32no—
119C9×C4⋊C4C2×D311yescharacteristic
120C42×C9—32no—
121C42×C9—31no—
122(C2×D9)⋊C4—32no—
123(C2×D9)⋊C4—31no—
124(C2×D9)⋊C4—31no—
125(C2×D9)⋊C4C2×D312yes—
126C4⋊Dic9—31no—
127C4⋊Dic9—32no—
128C4⋊Dic9—61no—
129Dic9⋊C4—32no—
130Dic9⋊C4—31no—
131Dic9⋊C4—31no—
132Dic9⋊C4—62no—
133C4×Dic9—32no—
134C4×Dic9C2×D311yescharacteristic
135C22×C3×C9C2411yescharacteristic
136C2×C3⋊D9—21no—
137C2×C3⋊D9—21no—
138C2×C3⋊D9C22×C412yes—
139C2×C3×D9—22no—
140C2×C3×D9—22no—
141C2×C3×D9C22×C414yes—
142C3×C4×C9—22no—
143C3×C4×C9C22×C412yes—
144C3⋊Dic9—22no—
145C3⋊Dic9C22×C412yes—
146C9×Dic3—21no—
147C9×Dic3—21no—
148C9×Dic3C22×C414yes—
149C3×Dic9—21no—
150C3×Dic9—21no—
151C2×#239—91no—
152C2×#237—91no—
153C3×C4×Q8—31no—
154C3×C4×D4—91no—
155Q8×Dic3—91no—
156D4×Dic3—91no—
157C4×#67—92no—
158C2×Dic3⋊C4—92no—
159#893—31no—
160#893—92no—
161D3×C4⋊C4—92no—
162D3×C4⋊C4—91no—
163D3×C4⋊C4—91no—
164D3×C4⋊C4—181no—
165#887D911yescharacteristic, complemented
166#67⋊C4—92no—
167#877—92no—
168C4×D12—92no—
169C4×D12—91no—
170#872—91no—
171#872—92no—
172#872—91no—
173C4×Dic6—91no—
174C22×C3⋊D3—31no—
175C22×C3×D3—32no—
176C2×C32×C4C22×D312yes—
177C2×C32×C4C22×D311yescharacteristic
178C2×C3⋊Dic3—31no—
179C2×C3⋊Dic3—32no—
180C4×C3⋊D3—34no—
181C4×C3⋊D3—122no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3C22×D311yescharacteristic
185C2×C3×Dic3C22×D311yescharacteristic
186C2×C3×Dic3C22×D312yes—
187C3×D12—34no—
188C3×C4×D3—122no—
189C3×Dic6C4×D314yes—
190#626—38no—
191D3×Dic3—38no—
192D3×Dic3—122no—
193C22×D9—31no—
194C22×D9—61no—
195C22×D9C22×D312yes—
196Q8×C9—34no—
197C2×C4×C9—21no—
198C2×C4×C9—22no—
199C2×C4×C9—32no—
200C2×C4×C9—31no—
201C2×C4×C9—31no—
202C2×C4×C9C22×D312yes—
203C2×C4×C9C22×D311yescharacteristic
204C2×Dic9—31no—
205C2×Dic9—32no—
206C2×Dic9—61no—
207C2×Dic9—62no—
208C2×Dic9C22×D311yescharacteristic
209C2×Dic9C22×D311yescharacteristic
210D36—38no—
211D36C4×D314yescomplemented
212C4×D9—38no—
213C4×D9—34no—
214C4×D9—42no—
215C4×D9—64no—
216C4×D9—122no—
217C4×D9—122no—
218C4×D9—124no—
219#520—271noSylow
220C2×C3×C9C23×C411yescharacteristic
221C2×C3×C9H(5,𝔽2)11yescharacteristic
222C2×C3×C92−1+411yescharacteristic
223C3⋊D9—22no—
224C3×D9—24no—
225C2×C3×Q8—31no—
226C2×C3×D4—91no—
227C22×C3×C4—92no—
228C2×#67—92no—
229C22×Dic3—92no—
230C22×Dic3—92no—
231#239—98no—
232#237—98no—
233C2×D12—31no—
234C2×D12—92no—
235C2×C4×D3—32no—
236C2×C4×D3—92no—
237C2×C4×D3—91no—
238C2×C4×D3—91no—
239C2×C4×D3—92no—
240C2×C4×D3—92no—
241C2×C4×D3—92no—
242C2×C4×D3—181no—
243C2×C4×D3—182no—
244C2×Dic6C2×D911yescharacteristic
245C3×C4⋊C4—21no—
246C3×C4⋊C4—32no—
247C3×C4⋊C4C2×D911yescharacteristic
248C3×C4⋊C4D3211yescharacteristic
249C3×C22⋊C4—92no—
250C3×C42—32no—
251C3×C42—31no—
252C3×C42—91no—
253#217—92no—
254(C2×D3)⋊C4—32no—
255(C2×D3)⋊C4—91no—
256(C2×D3)⋊C4—91no—
257(C2×D3)⋊C4—92no—
258(C2×D3)⋊C4—91no—
259(C2×D3)⋊C4—91no—
260C4⋊Dic3—92no—
261C4⋊Dic3—91no—
262C4⋊Dic3—91no—
263C4⋊Dic3—181no—
264Dic3⋊C4—91no—
265Dic3⋊C4—91no—
266Dic3⋊C4—92no—
267Dic3⋊C4—92no—
268Dic3⋊C4—92no—
269Dic3⋊C4—91no—
270Dic3⋊C4—91no—
271Dic3⋊C4—182no—
272Dic3⋊C4C2×D912yes—
273C4×Dic3—31no—
274C4×Dic3—92no—
275C4×Dic3—92no—
276C4×Dic3C2×D911yescharacteristic
277C4×Dic3C2×D912yes—
278C22×C32C23×D311yescharacteristic
279C2×C3⋊D3—32no—
280C2×C3⋊D3—61no—
281C2×C3⋊D3—61no—
282C2×C3×D3—34no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C32×C4—22no—
286C32×C4C2×C4×D312yes—
287C3⋊Dic3—32no—
288C3⋊Dic3—62no—
289C3×Dic3—21no—
290C3×Dic3—21no—
291C3×Dic3—61no—
292C3×Dic3—61no—
293C3×Dic3C2×C4×D314yes—
294C22×C9—21no—
295C22×C9C23×D311yescharacteristic
296C2×D9—22no—
297C2×D9—22no—
298C2×D9—32no—
299C2×D9—62no—
300C2×D9—62no—
301C2×D9—62no—
302C2×D9—61no—
303C2×D9—61no—
304C2×D9C2×C4×D314yes—
305C4×C9—22no—
306C4×C9—22no—
307C4×C9—34no—
308C4×C9—42no—
309C4×C9—61no—
310C4×C9—61no—
311C4×C9C2×C4×D312yes—
312Dic9—21no—
313Dic9—21no—
314Dic9—32no—
315Dic9—62no—
316Dic9—62no—
317Dic9—122no—
318C2×#41—271no—
319C4×Q8—31no—
320C4×Q8—271no—
321C4×D4—91no—
322C4×D4—272no—
323C4×D4—271no—
324C4×D4—272no—
325#117—271no—
326#117—272no—
327C2×C4⋊C4—272no—
328C2×C4⋊C4—271no—
329C3×C9#52011yescharacteristic, complemented, Sylow
330C23×C3—92no—
331C22×D3—32no—
332C22×D3—91no—
333C22×D3—91no—
334C22×D3—181no—
335C3×Q8—34no—
336C3×D4—94no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—32no—
340C2×C3×C4—31no—
341C2×C3×C4—31no—
342C2×C3×C4—91no—
343C2×C3×C4—91no—
344C2×C3×C4—362no—
345C2×C3×C4C22×D912yes—
346C2×C3×C4C22×D911yescharacteristic
347C2×C3×C4C2×D3212yes—
348C2×C3×C4C2×D3211yescharacteristic
349#67—98no—
350C2×Dic3—31no—
351C2×Dic3—31no—
352C2×Dic3—91no—
353C2×Dic3—91no—
354C2×Dic3—98no—
355C2×Dic3—92no—
356C2×Dic3—92no—
357C2×Dic3—181no—
358C2×Dic3—182no—
359C2×Dic3—362no—
360C2×Dic3C22×D911yescharacteristic
361C2×Dic3C22×D911yescharacteristic
362C2×Dic3C22×D912yes—
363D12—34no—
364D12—98no—
365C4×D3—98no—
366C4×D3—94no—
367C4×D3—94no—
368C4×D3—122no—
369C4×D3—184no—
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—62no—
379C3×D3—64no—
380C2×C9—21no—
381C2×C9—21no—
382C2×C9—21no—
383C2×C9C22×C4×D311yescharacteristic
384C2×C9#100411yescharacteristic
385C2×C9#101011yescharacteristic
386D9—24no—
387D9—64no—
388D9—62no—
389#41—278no—
390C2×Q8—31no—
391C2×D4—91no—
392C2×D4—272no—
393C22×C4—92no—
394C22×C4—272no—
395C22×C4—272no—
396C22×C4—271no—
397C22×C4—272no—
398C4⋊C4—32no—
399C4⋊C4—271no—
400C4⋊C4—272no—
401C4⋊C4—271no—
402C4⋊C4—272no—
403C4⋊C4—271no—
404C4⋊C4D3×D911yescharacteristic
405C22⋊C4—92no—
406C22⋊C4—271no—
407C22⋊C4—271no—
408C22⋊C4—272no—
409C42—32no—
410C42—31no—
411C42—91no—
412C42—272no—
413C22×C3—21no—
414C22×C3—94no—
415C22×C3—182no—
416C22×C3—182no—
417C22×C3C23×D911yescharacteristic
418C22×C3C22×D3211yescharacteristic
419C2×D3—34no—
420C2×D3—62no—
421C2×D3—62no—
422C2×D3—92no—
423C2×D3—92no—
424C2×D3—182no—
425C2×D3—182no—
426C2×D3—182no—
427C2×D3—181no—
428C2×D3—181no—
429C2×D3—181no—
430C2×D3—181no—
431C3×C4—22no—
432C3×C4—22no—
433C3×C4—22no—
434C3×C4—34no—
435C3×C4—42no—
436C3×C4—61no—
437C3×C4—61no—
438C3×C4—181no—
439C3×C4—181no—
440C3×C4C2×C4×D912yes—
441C3×C4C4×D3212yes—
442Dic3—21no—
443Dic3—21no—
444Dic3—61no—
445Dic3—61no—
446Dic3—92no—
447Dic3—92no—
448Dic3—182no—
449Dic3—182no—
450Dic3—182no—
451Dic3—362no—
452Dic3C2×C4×D914yes—
453C32#557411yescharacteristic
454C9—21no—
455C9#554611yescharacteristic, complemented
456C23—92no—
457C23—271no—
458Q8—34no—
459D4—94no—
460D4—278no—
461C2×C4—32no—
462C2×C4—31no—
463C2×C4—31no—
464C2×C4—91no—
465C2×C4—91no—
466C2×C4—278no—
467C2×C4—271no—
468C2×C4—274no—
469C2×C4—272no—
470C2×C4—362no—
471C2×C4—1082no—
472C2×C4—1082no—
473C2×C4C2×D3×D912yes—
474C2×C4C2×D3×D911yescharacteristic
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—184no—
479C2×C3C22×C4×D911yescharacteristic
480C2×C3#6371311yescharacteristic
481C2×C3#6371511yescharacteristic
482C2×C3C2×C4×D3211yescharacteristic
483C2×C3#6431411yescharacteristic
484C2×C3#6431511yescharacteristic
485D3—64no—
486D3—184no—
487D3—182no—
488D3—182no—
489C22—94no—
490C22—182no—
491C22—182no—
492C22—272no—
493C22—541no—
494C22—541no—
495C22C22×D3×D911yescharacteristic
496C4—22no—
497C4—34no—
498C4—61no—
499C4—61no—
500C4—181no—
501C4—181no—
502C4—272no—
503C4—542no—
504C4C4×D3×D912yes—
505C3—21no—
506C3#1058967611yescharacteristic, complemented
507C3#1059436111yescharacteristic
508C2—184no—
509C2—542no—
510C2C2×C4×D3×D911yescharacteristic
511C2#1172250111yescharacteristic
512C2#1172251011yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×#88634C2
cosocleC22×C3×C9C24
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#520
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×C3⋊D9C2semidirect
3. C3×#63455C2semidirect
4. C9×#887C2semidirect
5. C4×#6192C2semidirect
6. Dic3×D36C2semidirect
7. #11720978C2semidirect
8. #11720974C2semidirect
9. D9×Dic3⋊C4C2semidirect
10. #11720931C2semidirect
11. Dic6×Dic9C2semidirect
22. #88634C4semidirect
53. #63455D3semidirect
68. C4×C3⋊D9C2×C4semidirect
74. C3×D36C2×C4semidirect
76. C9×Dic6C2×C4semidirect
77. #6192C2×C4semidirect
79. Dic3×D9C2×C4semidirect
165. #887D9semidirect
211. D36C4×D3semidirect
329. C3×C9#520semidirect
374. Dic6C4×D9semidirect
455. C9#5546semidirect
506. C3#10589676semidirect

Maximal subgroups

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

subgroupindex
C2×#886342
C4⋊C4×C3⋊D92
C3×#634552
C9×#8872
C4×#61922
Dic3×D362
#117209782
#117209742
D9×Dic3⋊C42
#117209312
Dic6×Dic92
#105943613
#105896763