Finite Group Databasebeta

← C3×#10593425 #13166784

Subgroups — C3×#10593425 #13166784

Order 1728.

814 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C2×C3×#3802C211yesmaximal, characteristic, complemented
2C2×C3×#3774C212yesmaximal, complemented
3C3×#63912C211yesmaximal, characteristic, complemented
4C3×#63778C211yesmaximal, characteristic, complemented
5C3×#63739C212yesmaximal, complemented
6#10593425C311yesmaximal, characteristic, direct product factor
7C22×C3×D32—21no—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×D12—21no—
10C2×C3×#626—21no—
11C2×C3×#625C2212yes—
12C2×C3×#625C2211yescharacteristic, complemented
13C3×#3802—24no—
14C2×C3×#622C2212yescomplemented
15C3×#3774—24no—
16C32×(C2×D3)⋊C4—21no—
17C3×#3724C2211yescharacteristic, complemented
18C2×C3×D3×D4—61no—
19C2×#3802C2×C311yescharacteristic, complemented
20C2×#3774C2×C312yescomplemented
21C3×#884—61no—
22#63912C2×C311yescharacteristic, complemented
23#63778C2×C311yescharacteristic, complemented
24#63739C2×C312yescomplemented
25C22×C3×C3⋊D3D411yescharacteristic
26C22×C32×D3—22no—
27C22×C32×D3—21no—
28C2×C3×D32—24no—
29C2×C3×D32—42no—
30C2×C3×D32—42no—
31C2×C33×C4D411yescharacteristic
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C3×C4×C3⋊D3—22no—
34C2×C32×Dic3—21no—
35C32×D12—24no—
36C3×#626—42no—
37C3×#625—22no—
38C3×#625—22no—
39C3×#625D414yescomplemented
40C3×#622—22no—
41C3×#447—91nomaximal
42C23×C3×D3—62no—
43C22×D32—21no—
44C2×C32×D4—61no—
45C2×C4×C3⋊D3C22×C311yescharacteristic
46C2×C3×#67—61no—
47C2×C3×#67—62no—
48C2×C3×#67—61no—
49C3×D3×D4—68no—
50C2×C3×D12—21no—
51C2×C3×D12—21no—
52C2×C3×D12—41no—
53C2×C3×D12—61no—
54C2×C3×C4×D3—61no—
55C2×C3×C4×D3—61no—
56C2×#626—21no—
57C2×#625C22×C312yes—
58C2×#625C22×C311yescharacteristic, complemented
59#3802—24no—
60C2×#622C22×C312yescomplemented
61#3774—24no—
62C32×C22⋊C4—61no—
63C3×(C2×D3)⋊C4—21no—
64C3×(C2×D3)⋊C4—21no—
65C3×(C2×D3)⋊C4—41no—
66C3×Dic3⋊C4—61no—
67#3724C22×C311yescharacteristic, complemented
68C22×C33C2×D411yescharacteristic
69C2×C3×C3⋊D3—21no—
70C2×C3×C3⋊D3—21no—
71C2×C3×C3⋊D3D812yescomplemented
72C2×C32×D3—24no—
73C2×C32×D3—42no—
74C2×C32×D3—42no—
75C2×C32×D3—41no—
76C2×C32×D3—41no—
77C2×C32×D3—41no—
78C3×D32—48no—
79C33×C4—21no—
80C3×C3⋊Dic3C2×D412yes—
81C32×Dic3—41no—
82C22×C3×D4—91no—
83C2×D3×D4—61no—
84C2×C3×D8—92no—
85C3×#121—91no—
86#884—61no—
87C3×#102—92no—
88C3×C22⋊C8—91no—
89C23×C32—62no—
90C23×C32—61no—
91C22×C3⋊D3C3×D411yescharacteristic
92C22×C3×D3—22no—
93C22×C3×D3—22no—
94C22×C3×D3—21no—
95C22×C3×D3—21no—
96C22×C3×D3—41no—
97C22×C3×D3—42no—
98C22×C3×D3—61no—
99C22×C3×D3—61no—
100C22×C3×D3—68no—
101C22×C3×D3—124no—
102C22×C3×D3—124no—
103C2×D32—24no—
104C2×D32—42no—
105C2×D32—42no—
106C32×D4—64no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—41no—
110C2×C32×C4—41no—
111C2×C32×C4—61no—
112C2×C32×C4C3×D411yescharacteristic
113C2×C3⋊Dic3C23×C311yescharacteristic
114C4×C3⋊D3—22no—
115C3×#67—64no—
116C3×#67—64no—
117C3×#67—122no—
118C3×#67—122no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—41no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C3×D12—24no—
125C3×D12—24no—
126C3×D12—44no—
127C3×D12—122no—
128C3×C4×D3—64no—
129C3×C4×D3—122no—
130#626—42no—
131#625—22no—
132#625—22no—
133#625C3×D414yescomplemented
134#622—22no—
135#447—91noSylow
136C2×C33#12011yescharacteristic
137C2×C33C2×D811yescharacteristic
138C2×C33AGL(1,ℤ/8)11yescharacteristic
139C3×C3⋊D3—22no—
140C32×D3—44no—
141C32×D3—81no—
142C24×C3—181no—
143C23×D3—62no—
144C2×C3×D4—61no—
145C2×C3×D4—61no—
146C2×C3×D4—92no—
147C2×C3×D4—91no—
148C2×C3×D4—121no—
149C2×C3×D4—181no—
150C2×C3×D4—184no—
151C22×C3×C4—91no—
152C2×#67—62no—
153C2×#67—61no—
154C2×#67—61no—
155D3×D4—68no—
156C2×D12—21no—
157C2×D12—61no—
158C2×C4×D3—61no—
159C2×C4×D3—61no—
160C3×D8—184no—
161C2×C3×C8—92no—
162C3×C4⋊C4—91no—
163C3×C22⋊C4—61no—
164C3×C22⋊C4—61no—
165C3×C22⋊C4—121no—
166(C2×D3)⋊C4—21no—
167Dic3⋊C4—61no—
168C22×C32—21no—
169C22×C32—21no—
170C22×C32—41no—
171C22×C32—41no—
172C22×C32—64no—
173C22×C32—121no—
174C22×C32—122no—
175C22×C32—122no—
176C22×C32—121no—
177C22×C32—121no—
178C22×C32C2×C3×D411yescharacteristic
179C2×C3⋊D3—21no—
180C2×C3⋊D3—21no—
181C2×C3⋊D3C3×D812yescomplemented
182C2×C3×D3—24no—
183C2×C3×D3—24no—
184C2×C3×D3—44no—
185C2×C3×D3—42no—
186C2×C3×D3—42no—
187C2×C3×D3—42no—
188C2×C3×D3—42no—
189C2×C3×D3—41no—
190C2×C3×D3—41no—
191C2×C3×D3—41no—
192C2×C3×D3—42no—
193C2×C3×D3—41no—
194C2×C3×D3—41no—
195C2×C3×D3—41no—
196C2×C3×D3—62no—
197C2×C3×D3—62no—
198C2×C3×D3—62no—
199C2×C3×D3—62no—
200C2×C3×D3—82no—
201C2×C3×D3—82no—
202C2×C3×D3—81no—
203C2×C3×D3—81no—
204C2×C3×D3—121no—
205C2×C3×D3—121no—
206C2×C3×D3—1216no—
207D32—48no—
208C32×C4—22no—
209C32×C4—21no—
210C32×C4—41no—
211C32×C4—42no—
212C32×C4—81no—
213C32×C4—121no—
214C3⋊Dic3C2×C3×D412yes—
215C3×Dic3—41no—
216C3×Dic3—41no—
217C3×Dic3—62no—
218C3×Dic3—62no—
219C3×Dic3—81no—
220C22×D4—91no—
221C2×D8—92no—
222#121—91no—
223#102—92no—
224C22⋊C8—91no—
225C33#44711yescharacteristic, complemented, Sylow
226C23×C3—62no—
227C23×C3—61no—
228C23×C3—62no—
229C23×C3—61no—
230C23×C3—91no—
231C23×C3—121no—
232C23×C3—122no—
233C23×C3—184no—
234C23×C3—362no—
235C23×C3—362no—
236C22×D3—22no—
237C22×D3—21no—
238C22×D3—61no—
239C22×D3—61no—
240C22×D3—68no—
241C22×D3—124no—
242C22×D3—124no—
243C3×D4—64no—
244C3×D4—64no—
245C3×D4—94no—
246C3×D4—124no—
247C3×D4—182no—
248C3×D4—182no—
249C3×D4—362no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—61no—
255C2×C3×C4—61no—
256C2×C3×C4—91no—
257C2×C3×C4—121no—
258C2×C3×C4—182no—
259C2×C3×C4GO+(4,𝔽2)11yescharacteristic
260#67—64no—
261#67—64no—
262#67—122no—
263#67—122no—
264C2×Dic3—21no—
265C2×Dic3—61no—
266C2×Dic3—61no—
267D12—24no—
268D12—122no—
269C4×D3—64no—
270C4×D3—122no—
271C3×C8—182no—
272C2×C32—21no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—21no—
278C2×C32—41no—
279C2×C32—41no—
280C2×C32—41no—
281C2×C32—41no—
282C2×C32—41no—
283C2×C32—41no—
284C2×C32—124no—
285C2×C32—241no—
286C2×C32C3×#12011yescharacteristic
287C2×C32C2×C3×D811yescharacteristic
288C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
289C3⋊D3—22no—
290C3×D3—44no—
291C3×D3—44no—
292C3×D3—64no—
293C3×D3—84no—
294C3×D3—81no—
295C3×D3—82no—
296C3×D3—81no—
297C3×D3—122no—
298C24—181no—
299C2×D4—61no—
300C2×D4—92no—
301C2×D4—91no—
302C2×D4—181no—
303C2×D4—184no—
304C22×C4—91no—
305D8—184no—
306C2×C8—92no—
307C4⋊C4—91no—
308C22⋊C4—61no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—64no—
314C22×C3—64no—
315C22×C3—92no—
316C22×C3—124no—
317C22×C3—122no—
318C22×C3—122no—
319C22×C3—121no—
320C22×C3—121no—
321C22×C3—121no—
322C22×C3—122no—
323C22×C3—122no—
324C22×C3—122no—
325C22×C3—121no—
326C22×C3—121no—
327C22×C3—121no—
328C22×C3—181no—
329C22×C3—181no—
330C22×C3—242no—
331C22×C3—242no—
332C22×C3—241no—
333C22×C3—241no—
334C22×C3—368no—
335C22×C3C2×GO+(4,𝔽2)11yescharacteristic
336C2×D3—24no—
337C2×D3—42no—
338C2×D3—42no—
339C2×D3—41no—
340C2×D3—41no—
341C2×D3—41no—
342C2×D3—62no—
343C2×D3—62no—
344C2×D3—62no—
345C2×D3—62no—
346C2×D3—121no—
347C2×D3—121no—
348C2×D3—1216no—
349C3×C4—22no—
350C3×C4—21no—
351C3×C4—41no—
352C3×C4—42no—
353C3×C4—81no—
354C3×C4—92no—
355C3×C4—121no—
356C3×C4—121no—
357C3×C4—241no—
358Dic3—41no—
359Dic3—62no—
360Dic3—62no—
361C32—21no—
362C32—21no—
363C32—41no—
364C32—41no—
365C32C3×#44711yescharacteristic, complemented
366C23—62no—
367C23—61no—
368C23—91no—
369C23—184no—
370C23—362no—
371C23—362no—
372D4—64no—
373D4—94no—
374D4—182no—
375D4—182no—
376D4—362no—
377C2×C4—61no—
378C2×C4—91no—
379C2×C4—182no—
380C2×C4C3×GO+(4,𝔽2)11yescharacteristic
381C8—182no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—41no—
389C2×C3—41no—
390C2×C3—41no—
391C2×C3—41no—
392C2×C3—41no—
393C2×C3—41no—
394C2×C3—124no—
395C2×C3—124no—
396C2×C3—182no—
397C2×C3—244no—
398C2×C3—241no—
399C2×C3—242no—
400C2×C3—241no—
401C2×C3#6422411yescharacteristic
402C2×C3#6422511yescharacteristic
403C2×C3#6424111yescharacteristic
404D3—44no—
405D3—64no—
406D3—81no—
407D3—122no—
408C22—64no—
409C22—92no—
410C22—122no—
411C22—122no—
412C22—121no—
413C22—121no—
414C22—121no—
415C22—181no—
416C22—181no—
417C22—368no—
418C22C2×C3×GO+(4,𝔽2)11yescharacteristic
419C4—21no—
420C4—92no—
421C4—121no—
422C3—21no—
423C3—21no—
424C3—41no—
425C3—41no—
426C3#1059342511yescharacteristic, direct product factor
427C2—124no—
428C2—182no—
429C2—241no—
430C2C3×#6422411yescharacteristic
431C2C3×#6422511yescharacteristic
432C2C3×#6424111yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#447
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3802C2semidirect
2. C2×C3×#3774C2semidirect
3. C3×#63912C2semidirect
4. C3×#63778C2semidirect
5. C3×#63739C2semidirect
6. #10593425C3direct (N × H)
12. C2×C3×#625C22semidirect
14. C2×C3×#622C22semidirect
17. C3×#3724C22semidirect
19. C2×#3802C2×C3semidirect
20. C2×#3774C2×C3semidirect
22. #63912C2×C3semidirect
23. #63778C2×C3semidirect
24. #63739C2×C3semidirect
39. C3×#625D4semidirect
58. C2×#625C22×C3semidirect
60. C2×#622C22×C3semidirect
67. #3724C22×C3semidirect
71. C2×C3×C3⋊D3D8semidirect
133. #625C3×D4semidirect
181. C2×C3⋊D3C3×D8semidirect
225. C33#447semidirect
365. C32C3×#447semidirect
426. C3#10593425direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38022
C2×C3×#37742
C3×#639122
C3×#637782
C3×#637392
#105934253
C3×#4479