Finite Group Databasebeta

← C3×#10591398 #13158415

Subgroups — C3×#10591398 #13158415

Order 1728.

928 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64330C211yesmaximal, characteristic
2C3×#64138C211yesmaximal, characteristic, complemented
3C32×#929C212yesmaximal, complemented
4C3×#63581C211yesmaximal, characteristic, complemented
5C3×#63580C212yesmaximal, complemented
6C3×#5053—32nomaximal
7#10591398C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic, complemented
9C2×C3×#638C2211yescharacteristic
10C2×C32×#67C2212yes—
11C3×#3811—21no—
12C3×D3×#67—22no—
13C3×#3805—21no—
14C3×#3804—22no—
15C3×#3757C2211yescharacteristic, complemented
16C3×#3752—21no—
17C3×#3750—21no—
18C32×#217C2212yescomplemented
19C32×(C2×D3)⋊C4—22no—
20C32×Dic3⋊C4—22no—
21C3×#1004—32no—
22#64330C2×C311yescharacteristic
23C32×#123—32no—
24#64138C2×C311yescharacteristic, complemented
25C3×#929—22no—
26C3×#929—32no—
27C3×#929—61no—
28C3×#929D312yescomplemented
29C3×#929C2×C312yescomplemented
30C3×#834—32no—
31C3×#806—32no—
32C3×#806—32no—
33#63581C2×C311yescharacteristic, complemented
34#63580C2×C312yescomplemented
35C23×C33C2311yescharacteristic
36C22×C3×C3⋊D3D411yescharacteristic
37C22×C32×D3D412yes—
38C2×C3×D32—21no—
39C2×C33×C4—21no—
40C2×C33×C4—22no—
41C3×#638—22no—
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3D411yescharacteristic
44C32×#67—22no—
45C32×#67—22no—
46C2×C32×Dic3—22no—
47C2×C32×Dic3D412yes—
48C3×#627—21no—
49C3×#626—22no—
50C3×#625—21no—
51C3×#624—21no—
52C3×D3×Dic3—22no—
53C3×#458—91no—
54#5053—32no—
55C2×C32×D4—32no—
56C22×C32×C4—21no—
57C22×C32×C4—22no—
58C22×C32×C4—41no—
59C22×C32×C4C2×D312yescomplemented
60C22×C32×C4C22×C311yescharacteristic, complemented
61C2×#638C22×C311yescharacteristic
62C2×C3×#67—22no—
63C2×C3×#67—32no—
64C2×C3×#67—62no—
65C2×C3×#67—61no—
66C2×C3×#67C2×D312yes—
67C2×C3×#67C22×C312yes—
68C3×#237—62no—
69C3×#237—62no—
70C3×D3×D4—62no—
71C3×D3×D4—62no—
72C3×#235—62no—
73#3811—21no—
74D3×#67—22no—
75#3805—21no—
76#3804—22no—
77C32×C4⋊C4—62no—
78C32×C22⋊C4—32no—
79C32×C22⋊C4—62no—
80#3757C22×C311yescharacteristic, complemented
81#3752—21no—
82#3750—21no—
83C3×#217—22no—
84C3×#217—32no—
85C3×#217—61no—
86C3×#217C2×D312yescomplemented
87C3×#217C22×C312yescomplemented
88C3×(C2×D3)⋊C4—24no—
89C3×(C2×D3)⋊C4—22no—
90C3×(C2×D3)⋊C4—22no—
91C3×(C2×D3)⋊C4—62no—
92C3×(C2×D3)⋊C4—62no—
93C3×Dic3⋊C4—24no—
94C3×Dic3⋊C4—22no—
95C3×Dic3⋊C4—22no—
96C3×Dic3⋊C4—62no—
97C3×Dic3⋊C4—62no—
98C22×C33—41no—
99C22×C33C2×D411yescharacteristic
100C22×C33C2×D412yes—
101C2×C3×C3⋊D3—21no—
102C2×C3×C3⋊D3—41no—
103C2×C32×D3—22no—
104C2×C32×D3—42no—
105C3×D32—81no—
106C33×C4—41no—
107C3×C3⋊Dic3—21no—
108C3×C3⋊Dic3—41no—
109C32×Dic3—22no—
110C32×Dic3—42no—
111C3×H(5,𝔽2)—91no—
112#1004—32no—
113C3×#123—32no—
114C3×#123—32no—
115C3×#123—62no—
116C3×#123—91no—
117#929—32no—
118#929—61no—
119#929C3×D312yescomplemented
120C3×#99—91no—
121C3×#99—92no—
122#834—32no—
123#806—32no—
124#806—32no—
125C23×C32—21no—
126C23×C32—22no—
127C23×C32—32no—
128C23×C32—41no—
129C23×C32C22×D312yes—
130C23×C32C23×C311yescharacteristic
131C22×C3⋊D3C3×D411yescharacteristic
132C22×C3×D3—22no—
133C22×C3×D3—32no—
134C22×C3×D3—61no—
135C22×C3×D3—62no—
136C22×C3×D3#6712yes—
137C22×C3×D3C3×D412yes—
138C2×D32—21no—
139C32×D4—62no—
140C32×D4—62no—
141C2×C32×C4—22no—
142C2×C32×C4—21no—
143C2×C32×C4—24no—
144C2×C32×C4—22no—
145C2×C32×C4—22no—
146C2×C32×C4—22no—
147C2×C32×C4—22no—
148C2×C32×C4—32no—
149C2×C32×C4—42no—
150C2×C32×C4—42no—
151C2×C32×C4—42no—
152C2×C32×C4—42no—
153C2×C32×C4—44no—
154C2×C32×C4—62no—
155#638—22no—
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3C3×D411yescharacteristic
158C3×#67—22no—
159C3×#67—22no—
160C3×#67—22no—
161C3×#67—22no—
162C3×#67—42no—
163C3×#67—42no—
164C3×#67—62no—
165C3×#67—62no—
166C3×#67—62no—
167C3×#67—62no—
168C3×#67—122no—
169C2×C3×Dic3—24no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—22no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3—62no—
177C2×C3×Dic3—62no—
178C2×C3×Dic3#6712yes—
179C2×C3×Dic3C3×D412yes—
180C3×D12—62no—
181C3×C4×D3—62no—
182C3×C4×D3—62no—
183C3×Dic6—62no—
184#627—21no—
185#626—22no—
186#625—21no—
187#624—21no—
188D3×Dic3—22no—
189#458—91noSylow
190C2×C33—22no—
191C2×C33—21no—
192C2×C33#12011yescharacteristic
193C3×C3⋊D3—41no—
194C32×D3—42no—
195C3×#41—182no—
196C3×#41—181no—
197C2×C3×D4—32no—
198C2×C3×D4—32no—
199C2×C3×D4—62no—
200C2×C3×D4—91no—
201C2×C3×D4—181no—
202C2×C3×D4—182no—
203C22×C3×C4—21no—
204C22×C3×C4—22no—
205C22×C3×C4—41no—
206C22×C3×C4D3211yescharacteristic, complemented
207C22×C3×C4C2×C3×D312yescomplemented
208C2×#67—32no—
209C2×#67—61no—
210C2×#67—62no—
211C2×#67C2×C3×D312yes—
212#237—62no—
213#237—62no—
214D3×D4—62no—
215D3×D4—62no—
216#235—62no—
217C3×C4⋊C4—62no—
218C3×C4⋊C4—62no—
219C3×C4⋊C4—64no—
220C3×C4⋊C4—181no—
221C3×C22⋊C4—32no—
222C3×C22⋊C4—32no—
223C3×C22⋊C4—62no—
224C3×C22⋊C4—62no—
225C3×C22⋊C4—62no—
226C3×C22⋊C4—64no—
227C3×C22⋊C4—91no—
228C3×C22⋊C4—181no—
229#217—32no—
230#217—61no—
231#217C2×C3×D312yescomplemented
232(C2×D3)⋊C4—22no—
233(C2×D3)⋊C4—62no—
234(C2×D3)⋊C4—62no—
235Dic3⋊C4—22no—
236Dic3⋊C4—62no—
237Dic3⋊C4—62no—
238C22×C32—22no—
239C22×C32—21no—
240C22×C32—22no—
241C22×C32—22no—
242C22×C32—22no—
243C22×C32—42no—
244C22×C32—41no—
245C22×C32—42no—
246C22×C32—41no—
247C22×C32—42no—
248C22×C32—44no—
249C22×C32—44no—
250C22×C32—62no—
251C22×C32—122no—
252C22×C32D3×D412yes—
253C22×C32D3×D412yes—
254C22×C32C2×#6712yes—
255C22×C32C2×C3×D411yescharacteristic
256C22×C32C2×C3×D412yes—
257C2×C3⋊D3—21no—
258C2×C3⋊D3—41no—
259C2×C3×D3—22no—
260C2×C3×D3—22no—
261C2×C3×D3—42no—
262C2×C3×D3—42no—
263C2×C3×D3—42no—
264C2×C3×D3—44no—
265C2×C3×D3—62no—
266C2×C3×D3—121no—
267C2×C3×D3—122no—
268C2×C3×D3—122no—
269C2×C3×D3—242no—
270D32—81no—
271C32×C4—44no—
272C32×C4—42no—
273C32×C4—44no—
274C32×C4—42no—
275C32×C4—41no—
276C32×C4—62no—
277C32×C4—122no—
278C3⋊Dic3—21no—
279C3⋊Dic3—41no—
280C3×Dic3—22no—
281C3×Dic3—22no—
282C3×Dic3—42no—
283C3×Dic3—44no—
284C3×Dic3—42no—
285C3×Dic3—42no—
286C3×Dic3—62no—
287C3×Dic3—121no—
288C3×Dic3—122no—
289C3×Dic3—122no—
290H(5,𝔽2)—91no—
291#123—32no—
292#123—91no—
293#99—91no—
294#99—92no—
295C33#45811yescharacteristic, complemented, Sylow
296C23×C3—22no—
297C23×C3—21no—
298C23×C3—32no—
299C23×C3—32no—
300C23×C3—41no—
301C23×C3—62no—
302C23×C3—91no—
303C23×C3—181no—
304C23×C3C2×D3211yescharacteristic
305C23×C3C22×C3×D312yes—
306C22×D3—32no—
307C22×D3—61no—
308C22×D3—62no—
309C22×D3C3×#6712yes—
310C3×Q8—181no—
311C3×D4—62no—
312C3×D4—62no—
313C3×D4—62no—
314C3×D4—62no—
315C3×D4—122no—
316C3×D4—122no—
317C3×D4—182no—
318C3×D4—181no—
319C3×D4—182no—
320C2×C3×C4—22no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—32no—
328C2×C3×C4—32no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—44no—
333C2×C3×C4—42no—
334C2×C3×C4—62no—
335C2×C3×C4—64no—
336C2×C3×C4—62no—
337C2×C3×C4—62no—
338C2×C3×C4—91no—
339C2×C3×C4—181no—
340C2×C3×C4—182no—
341C2×C3×C4—181no—
342#67—22no—
343#67—22no—
344#67—62no—
345#67—62no—
346#67—62no—
347#67—62no—
348#67—122no—
349C2×Dic3—22no—
350C2×Dic3—32no—
351C2×Dic3—61no—
352C2×Dic3—62no—
353C2×Dic3—62no—
354C2×Dic3—62no—
355C2×Dic3C3×#6712yes—
356D12—62no—
357C4×D3—62no—
358C4×D3—62no—
359Dic6—62no—
360C2×C32—22no—
361C2×C32—21no—
362C2×C32—24no—
363C2×C32—22no—
364C2×C32—22no—
365C2×C32—22no—
366C2×C32—22no—
367C2×C32—22no—
368C2×C32—21no—
369C2×C32—41no—
370C2×C32—42no—
371C2×C32—42no—
372C2×C32—42no—
373C2×C32—42no—
374C2×C32—44no—
375C2×C32—122no—
376C2×C32#93712yes—
377C2×C32C3×#12011yescharacteristic
378C3⋊D3—41no—
379C3×D3—42no—
380C3×D3—42no—
381C3×D3—82no—
382C3×D3—122no—
383C3×D3—241no—
384#41—182no—
385#41—181no—
386C2×D4—32no—
387C2×D4—91no—
388C2×D4—181no—
389C2×D4—182no—
390C22×C4C3×D3211yescharacteristic, complemented
391C4⋊C4—62no—
392C4⋊C4—181no—
393C22⋊C4—32no—
394C22⋊C4—62no—
395C22⋊C4—91no—
396C22⋊C4—181no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—21no—
401C22×C3—22no—
402C22×C3—41no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3—44no—
406C22×C3—44no—
407C22×C3—42no—
408C22×C3—41no—
409C22×C3—62no—
410C22×C3—62no—
411C22×C3—122no—
412C22×C3—122no—
413C22×C3—122no—
414C22×C3—124no—
415C22×C3—181no—
416C22×C3—361no—
417C22×C3—721no—
418C22×C3D3×#6712yes—
419C22×C3#381111yescharacteristic
420C22×C3C3×D3×D412yes—
421C22×C3C3×D3×D412yes—
422C22×C3C2×C3×#6712yes—
423C2×D3—22no—
424C2×D3—42no—
425C2×D3—62no—
426C2×D3—121no—
427C2×D3—122no—
428C2×D3—122no—
429C2×D3—242no—
430C3×C4—42no—
431C3×C4—44no—
432C3×C4—44no—
433C3×C4—42no—
434C3×C4—41no—
435C3×C4—62no—
436C3×C4—62no—
437C3×C4—122no—
438C3×C4—124no—
439C3×C4—122no—
440C3×C4—122no—
441C3×C4—181no—
442C3×C4—361no—
443Dic3—22no—
444Dic3—42no—
445Dic3—62no—
446Dic3—121no—
447Dic3—122no—
448Dic3—122no—
449C32—22no—
450C32—21no—
451C32—41no—
452C32#505312yescomplemented
453C32C3×#45811yescharacteristic, complemented
454C23—32no—
455C23—91no—
456C23—181no—
457C23C2×C3×D3211yescharacteristic
458Q8—181no—
459D4—62no—
460D4—62no—
461D4—182no—
462D4—181no—
463D4—182no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—32no—
467C2×C4—62no—
468C2×C4—91no—
469C2×C4—181no—
470C2×C4—182no—
471C2×C4—181no—
472C2×C3—22no—
473C2×C3—24no—
474C2×C3—22no—
475C2×C3—22no—
476C2×C3—22no—
477C2×C3—22no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—22no—
481C2×C3—42no—
482C2×C3—42no—
483C2×C3—42no—
484C2×C3—44no—
485C2×C3—42no—
486C2×C3—41no—
487C2×C3—122no—
488C2×C3—122no—
489C2×C3—242no—
490C2×C3—361no—
491C2×C3#6398311yescharacteristic
492C2×C3C3×#93712yes—
493D3—42no—
494D3—122no—
495D3—241no—
496C22—41no—
497C22—62no—
498C22—122no—
499C22—181no—
500C22—361no—
501C22—721no—
502C22C3×D3×#6712yes—
503C22C3×#381111yescharacteristic
504C4—41no—
505C4—62no—
506C4—122no—
507C4—181no—
508C4—361no—
509C3—21no—
510C3—22no—
511C3—41no—
512C3#1059139811yescharacteristic, direct product factor
513C3C3×#505312yescomplemented
514C2—22no—
515C2—21no—
516C2—122no—
517C2—361no—
518C2C3×#6398311yescharacteristic
subgroupG/H
centerC2×C3#63983
commutator, cosocleC23×C32C23×C3
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socleC2×C33#120
prime radicalC3×#64330C2
hypercenterC22×C3×C4D32
hypocenterC32C3×#458
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×#64138C2semidirect
3. C32×#929C2semidirect
4. C3×#63581C2semidirect
5. C3×#63580C2semidirect
7. #10591398C3direct (N × H)
8. C22×C33×C4C22semidirect
15. C3×#3757C22semidirect
18. C32×#217C22semidirect
24. #64138C2×C3semidirect
28. C3×#929D3semidirect
29. C3×#929C2×C3semidirect
33. #63581C2×C3semidirect
34. #63580C2×C3semidirect
59. C22×C32×C4C2×D3semidirect
60. C22×C32×C4C22×C3semidirect
80. #3757C22×C3semidirect
86. C3×#217C2×D3semidirect
87. C3×#217C22×C3semidirect
119. #929C3×D3semidirect
206. C22×C3×C4D32semidirect
207. C22×C3×C4C2×C3×D3semidirect
231. #217C2×C3×D3semidirect
295. C33#458semidirect
390. C22×C4C3×D32semidirect
452. C32#5053semidirect
453. C32C3×#458semidirect
512. C3#10591398direct (N × H)
513. C3C3×#5053semidirect

Maximal subgroups

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

subgroupindex
C3×#643302
C3×#641382
C32×#9292
C3×#635812
C3×#635802
C3×#50533
#105913983