Finite Group Databasebeta

← C3×#10590551 #13157534

Subgroups — C3×#10590551 #13157534

Order 1728.

780 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 124 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×#64102C211yesmaximal, characteristic, complemented
2C3×#63914C211yesmaximal, characteristic, complemented
3C2×C3×#3714C212yesmaximal, complemented
4C32×#809C212yesmaximal, complemented
5C3×#63572C211yesmaximal, characteristic, complemented
6C3×#4786—32nomaximal
7#10590551C311yesmaximal, characteristic, direct product factor
8C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
9C2×C3×C4×C3⋊D3—21no—
10C2×C32×D12C2212yes—
11C2×C3×#625—21no—
12C2×C3×#625—21no—
13C33×C4⋊C4C2211yescharacteristic
14C3×#3752—21no—
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2212yes—
17C3×#3714—24no—
18#64102C2×C311yescharacteristic, complemented
19C3×#940—32no—
20C2×C3×#213—32no—
21C2×C3×D24—32no—
22C3×#893—32no—
23C3×#893—61no—
24#63914C2×C311yescharacteristic, complemented
25C2×#3714C2×C312yescomplemented
26C32×#102—32no—
27C3×Dic3⋊C8—32no—
28C3×#809—22no—
29C3×#809D312yescomplemented
30C3×#809C2×C312yescomplemented
31#63572C2×C311yescharacteristic, complemented
32C22×C3×C3⋊D3—21no—
33C22×C32×D3—22no—
34C2×C33×C4—21no—
35C2×C33×C4C2311yescharacteristic
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C3×C3⋊Dic3D411yescharacteristic
38C3×C4⋊(C3⋊D3)—21no—
39C3×C4⋊(C3⋊D3)D412yescomplemented
40C3×C4×C3⋊D3—42no—
41C32×D12—22no—
42C32×D12—22no—
43C3×#625—42no—
44C3×#625—42no—
45C32×C3⋊C8—22no—
46C3×#459—91no—
47#4786—32no—
48C2×C32×D4—32no—
49C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
50C2×C4×C3⋊D3—21no—
51C2×C3×#67—62no—
52C2×C3×#67—62no—
53C2×C3×D12—22no—
54C2×C3×D12—32no—
55C2×C3×D12—61no—
56C2×C3×D12C2×D312yes—
57C2×C3×D12C22×C312yes—
58C2×C3×C4×D3—62no—
59C2×C3×C4×D3—62no—
60C2×#625—21no—
61C2×#625—21no—
62C2×C32×C8—32no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4—22no—
65C32×C4⋊C4—41no—
66C32×C4⋊C4C2×D312yes—
67C32×C4⋊C4C22×C311yescharacteristic
68#3752—21no—
69C4×C3⋊Dic3C22×C311yescharacteristic
70C3×#213—64no—
71C3×(C2×D3)⋊C4—62no—
72C3×(C2×D3)⋊C4—62no—
73C3×C4×Dic3—32no—
74C3×C4×Dic3—61no—
75C2×C3×C3⋊C8—22no—
76C2×C3×C3⋊C8C2×D312yes—
77C2×C3×C3⋊C8C22×C312yes—
78C3×D24—64no—
79#3714—24no—
80C22×C33C2×D411yescharacteristic
81C2×C3×C3⋊D3—22no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—41no—
84C2×C32×D3—42no—
85C2×C32×D3—42no—
86C2×C32×D3—42no—
87C33×C4—41no—
88C33×C4C2×D411yescharacteristic
89C33×C4#4111yescharacteristic
90C3×C3⋊Dic3—21no—
91C3×C3⋊Dic3D812yescomplemented
92C2×C3×D8—92no—
93C3×#127—91no—
94C3×C4×D4—91no—
95#940—32no—
96C2×#213—32no—
97C2×D24—32no—
98#893—32no—
99#893—61no—
100C3×C4⋊C8—91no—
101C3×#102—32no—
102C3×#102—32no—
103C3×#102—62no—
104Dic3⋊C8—32no—
105#809C3×D312yescomplemented
106C23×C32—62no—
107C22×C3⋊D3—21no—
108C22×C3×D3—22no—
109C22×C3×D3—22no—
110C22×C3×D3—42no—
111C22×C3×D3—62no—
112C22×C3×D3—62no—
113C32×D4—62no—
114C32×D4—62no—
115C2×C32×C4—21no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—21no—
119C2×C32×C4—22no—
120C2×C32×C4—41no—
121C2×C32×C4—42no—
122C2×C32×C4—42no—
123C2×C32×C4C22×D312yes—
124C2×C32×C4C23×C311yescharacteristic
125C2×C3⋊Dic3C3×D411yescharacteristic
126C2×C3⋊Dic3C3×D411yescharacteristic
127C4⋊(C3⋊D3)—21no—
128C4⋊(C3⋊D3)C3×D412yescomplemented
129C4×C3⋊D3—42no—
130C3×#67—124no—
131C3×#67—124no—
132C2×C3×Dic3—32no—
133C2×C3×Dic3—32no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C3×D12—24no—
137C3×D12—22no—
138C3×D12—22no—
139C3×D12—22no—
140C3×D12—22no—
141C3×D12—34no—
142C3×D12—42no—
143C3×D12—62no—
144C3×D12—62no—
145C3×D12—62no—
146C3×C4×D3—124no—
147C3×C4×D3—124no—
148#625—42no—
149#625—42no—
150C32×C8—62no—
151C3×C3⋊C8—22no—
152C3×C3⋊C8—22no—
153C3×C3⋊C8—42no—
154#459—91noSylow
155C2×C33#12111yescharacteristic
156C2×C33C2×D811yescharacteristic
157C2×C33AGL(1,ℤ/8)11yescharacteristic
158C3×C3⋊D3—42no—
159C32×D3—82no—
160C2×C3×D4—32no—
161C2×C3×D4—32no—
162C2×C3×D4—62no—
163C2×C3×D4—91no—
164C2×C3×D4—181no—
165C2×C3×D4—181no—
166C22×C3×C4—181no—
167C2×#67—62no—
168C2×#67—62no—
169C2×D12—32no—
170C2×D12—61no—
171C2×D12C2×C3×D312yes—
172C2×C4×D3—62no—
173C2×C4×D3—62no—
174C3×D8—184no—
175C2×C3×C8—32no—
176C2×C3×C8—32no—
177C2×C3×C8—62no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—41no—
181C3×C4⋊C4D3211yescharacteristic
182C3×C4⋊C4C2×C3×D312yes—
183C3×C22⋊C4—181no—
184C3×C42—91no—
185#213—64no—
186(C2×D3)⋊C4—62no—
187(C2×D3)⋊C4—62no—
188C4×Dic3—32no—
189C4×Dic3—61no—
190C2×C3⋊C8C2×C3×D312yes—
191D24—64no—
192C22×C32—21no—
193C22×C32—22no—
194C22×C32—41no—
195C22×C32—122no—
196C22×C32—122no—
197C22×C32—122no—
198C22×C32D3×D412yes—
199C22×C32C2×C3×D411yescharacteristic
200C2×C3⋊D3—22no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3×D3—44no—
204C2×C3×D3—42no—
205C2×C3×D3—42no—
206C2×C3×D3—42no—
207C2×C3×D3—42no—
208C2×C3×D3—42no—
209C2×C3×D3—42no—
210C2×C3×D3—64no—
211C2×C3×D3—64no—
212C2×C3×D3—82no—
213C2×C3×D3—82no—
214C2×C3×D3—122no—
215C2×C3×D3—122no—
216C2×C3×D3—122no—
217C2×C3×D3—122no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4—22no—
221C32×C4—22no—
222C32×C4—41no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4—42no—
226C32×C4—42no—
227C32×C4—82no—
228C32×C4—82no—
229C32×C4#23512yes—
230C32×C4D3×D412yes—
231C32×C4C2×C3×D411yescharacteristic
232C32×C4C3×#4111yescharacteristic
233C3⋊Dic3—21no—
234C3⋊Dic3C3×D812yescomplemented
235C3×Dic3—34no—
236C3×Dic3—62no—
237C3×Dic3—62no—
238C3×Dic3—121no—
239C2×D8—92no—
240#127—91no—
241C4×D4—91no—
242C4⋊C8—91no—
243#102—32no—
244C33#45911yescharacteristic, complemented, Sylow
245C23×C3—62no—
246C23×C3—62no—
247C23×C3—122no—
248C23×C3—181no—
249C22×D3—22no—
250C22×D3—62no—
251C22×D3—62no—
252C3×D4—64no—
253C3×D4—62no—
254C3×D4—62no—
255C3×D4—62no—
256C3×D4—62no—
257C3×D4—92no—
258C3×D4—122no—
259C3×D4—181no—
260C3×D4—362no—
261C3×D4—362no—
262C2×C3×C4—22no—
263C2×C3×C4—21no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—21no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—91no—
271C2×C3×C4—91no—
272C2×C3×C4—362no—
273C2×C3×C4C2×D3211yescharacteristic
274C2×C3×C4C22×C3×D312yes—
275#67—124no—
276#67—124no—
277C2×Dic3—32no—
278C2×Dic3—32no—
279C2×Dic3—61no—
280C2×Dic3—61no—
281D12—22no—
282D12—22no—
283D12—34no—
284D12—62no—
285D12—62no—
286D12—62no—
287C4×D3—124no—
288C4×D3—124no—
289C3×C8—62no—
290C3×C8—62no—
291C3×C8—122no—
292C3⋊C8—22no—
293C2×C32—22no—
294C2×C32—22no—
295C2×C32—22no—
296C2×C32—21no—
297C2×C32—21no—
298C2×C32—21no—
299C2×C32—41no—
300C2×C32—41no—
301C2×C32—41no—
302C2×C32—242no—
303C2×C32#88412yes—
304C2×C32D3×D812yes—
305C2×C32#91412yes—
306C2×C32C3×#12111yescharacteristic
307C2×C32C2×C3×D811yescharacteristic
308C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
309C3⋊D3—42no—
310C3×D3—84no—
311C3×D3—82no—
312C3×D3—82no—
313C3×D3—124no—
314C3×D3—124no—
315C2×D4—32no—
316C2×D4—91no—
317C2×D4—181no—
318C2×D4—181no—
319C22×C4—181no—
320D8—184no—
321C2×C8—32no—
322C4⋊C4C3×D3211yescharacteristic
323C22⋊C4—181no—
324C42—91no—
325C22×C3—21no—
326C22×C3—22no—
327C22×C3—41no—
328C22×C3—124no—
329C22×C3—122no—
330C22×C3—122no—
331C22×C3—122no—
332C22×C3—122no—
333C22×C3—122no—
334C22×C3—122no—
335C22×C3—182no—
336C22×C3—242no—
337C22×C3—242no—
338C22×C3—361no—
339C22×C3—361no—
340C22×C3#381111yescharacteristic
341C22×C3C3×D3×D412yes—
342C2×D3—42no—
343C2×D3—42no—
344C2×D3—42no—
345C2×D3—64no—
346C2×D3—64no—
347C2×D3—122no—
348C2×D3—122no—
349C2×D3—122no—
350C2×D3—122no—
351C3×C4—21no—
352C3×C4—22no—
353C3×C4—22no—
354C3×C4—21no—
355C3×C4—42no—
356C3×C4—42no—
357C3×C4—41no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—82no—
361C3×C4—82no—
362C3×C4—92no—
363C3×C4—181no—
364C3×C4#379811yescharacteristic
365C3×C4#380211yescharacteristic
366C3×C4C3×#23512yes—
367C3×C4C3×D3×D412yes—
368Dic3—34no—
369Dic3—62no—
370Dic3—62no—
371Dic3—121no—
372C32—21no—
373C32—22no—
374C32—41no—
375C32#478612yescomplemented
376C32C3×#45911yescharacteristic, complemented
377C23—62no—
378C23—181no—
379D4—62no—
380D4—62no—
381D4—92no—
382D4—181no—
383D4—362no—
384D4—362no—
385C2×C4—21no—
386C2×C4—91no—
387C2×C4—91no—
388C2×C4—362no—
389C2×C4C2×C3×D3211yescharacteristic
390C8—62no—
391C2×C3—21no—
392C2×C3—22no—
393C2×C3—21no—
394C2×C3—22no—
395C2×C3—21no—
396C2×C3—22no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—244no—
401C2×C3—242no—
402C2×C3—242no—
403C2×C3—362no—
404C2×C3#6391211yescharacteristic
405C2×C3#6392611yescharacteristic
406C2×C3#6394111yescharacteristic
407C2×C3C3×#88412yes—
408C2×C3C3×D3×D812yes—
409C2×C3C3×#91412yes—
410D3—82no—
411D3—124no—
412D3—124no—
413C22—122no—
414C22—122no—
415C22—122no—
416C22—182no—
417C22—361no—
418C22—361no—
419C22C3×#381111yescharacteristic
420C4—41no—
421C4—92no—
422C4—181no—
423C4C3×#379811yescharacteristic
424C4C3×#380211yescharacteristic
425C3—21no—
426C3—22no—
427C3—41no—
428C3#1059055111yescharacteristic, direct product factor
429C3C3×#478612yescomplemented
430C2—242no—
431C2—362no—
432C2C3×#6391211yescharacteristic
433C2C3×#6392611yescharacteristic
434C2C3×#6394111yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#459
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#64102C2semidirect
2. C3×#63914C2semidirect
3. C2×C3×#3714C2semidirect
4. C32×#809C2semidirect
5. C3×#63572C2semidirect
7. #10590551C3direct (N × H)
18. #64102C2×C3semidirect
24. #63914C2×C3semidirect
25. C2×#3714C2×C3semidirect
29. C3×#809D3semidirect
30. C3×#809C2×C3semidirect
31. #63572C2×C3semidirect
39. C3×C4⋊(C3⋊D3)D4semidirect
91. C3×C3⋊Dic3D8semidirect
105. #809C3×D3semidirect
128. C4⋊(C3⋊D3)C3×D4semidirect
234. C3⋊Dic3C3×D8semidirect
244. C33#459semidirect
375. C32#4786semidirect
376. C32C3×#459semidirect
428. C3#10590551direct (N × H)
429. C3C3×#4786semidirect

Maximal subgroups

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

subgroupindex
C3×#641022
C3×#639142
C2×C3×#37142
C32×#8092
C3×#635722
C3×#47863
#105905513