Finite Group Databasebeta

← C3×#10593523 #13166889

Subgroups — C3×#10593523 #13166889

Order 1728.

854 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 854 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C22×C3×#625C211yesmaximal, characteristic
2C2×C3×#3775C212yesmaximal, complemented
3C3×#63982C211yesmaximal, characteristic, complemented
4C3×C22⋊#622C211yesmaximal, characteristic, complemented
5C3×#63739C212yesmaximal, complemented
6#10593523C311yesmaximal, characteristic, direct product factor
7C23×C32×D3—21no—
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C2×C3×#627C2211yescharacteristic, complemented
10C2×C3×#625—21no—
11C2×C3×#625—24no—
12C2×C3×#625C2212yes—
13C2×C3×#622C2212yescomplemented
14C3×#3775—24no—
15C32×#217—21no—
16C3×#3724C2211yescharacteristic, complemented
17C3×Dic3⋊Dic3—21no—
18C22×C3×#67—61no—
19C22×#625C2×C311yescharacteristic
20C2×#3775C2×C312yescomplemented
21C3×#878—61no—
22#63982C2×C311yescharacteristic, complemented
23C22⋊#622C2×C311yescharacteristic, complemented
24#63739C2×C312yescomplemented
25C23×C33D411yescharacteristic
26C22×C32×D3—24no—
27C22×C32×D3—22no—
28C22×C32×D3—42no—
29C22×C32×D3—42no—
30C2×C3×C3⋊Dic3—22no—
31C2×C3×C3⋊Dic3D411yescharacteristic
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C2×C32×Dic3—21no—
34C2×C32×Dic3—21no—
35C3×#627—22no—
36C3×#625—24no—
37C3×#625—22no—
38C3×#625D414yes—
39C3×#622—22no—
40C3×#450—91nomaximal
41C24×C32—61no—
42C23×C3×D3—21no—
43C23×C3×D3—21no—
44C23×C3×D3—41no—
45C22×C3⋊Dic3C22×C311yescharacteristic
46C2×C3×#67—62no—
47C2×C3×#67—62no—
48C2×C3×#67—68no—
49C22×C3×Dic3—61no—
50C22×C3×Dic3—61no—
51C2×C3×Dic6—61no—
52C2×#627C22×C311yescharacteristic, complemented
53C2×#625—21no—
54C2×#625—24no—
55C2×#625C22×C312yes—
56C2×#622C22×C312yescomplemented
57#3775—24no—
58C32×C22⋊C4—61no—
59C3×#217—21no—
60C3×#217—21no—
61C3×#217—41no—
62C3×C4⋊Dic3—61no—
63C3×Dic3⋊C4—61no—
64C3×Dic3⋊C4—61no—
65#3724C22×C311yescharacteristic, complemented
66Dic3⋊Dic3—21no—
67C22×C33—21no—
68C22×C33—21no—
69C22×C33SD1612yescomplemented
70C22×C33C2×D411yescharacteristic
71C2×C32×D3—24no—
72C2×C32×D3—48no—
73C2×C32×D3—42no—
74C2×C32×D3—42no—
75C3×C3⋊Dic3—21no—
76C3×C3⋊Dic3C2×D412yes—
77C32×Dic3—41no—
78C32×Dic3—41no—
79C22×C3×D4—91no—
80C22×#67—61no—
81C2×C3×SD16—92no—
82C3×C22⋊Q8—91no—
83#878—61no—
84C3×#102—92no—
85C3×C22⋊C8—91no—
86C23×C32—21no—
87C23×C32—21no—
88C23×C32—41no—
89C23×C32—41no—
90C23×C32—64no—
91C23×C32—62no—
92C23×C32—122no—
93C23×C32—122no—
94C23×C32C3×D411yescharacteristic
95C22×C3×D3—24no—
96C22×C3×D3—24no—
97C22×C3×D3—22no—
98C22×C3×D3—22no—
99C22×C3×D3—42no—
100C22×C3×D3—44no—
101C22×C3×D3—42no—
102C22×C3×D3—42no—
103C22×C3×D3—42no—
104C22×C3×D3—44no—
105C22×C3×D3—44no—
106C22×C3×D3—42no—
107C2×C32×C4—61no—
108C2×C32×C4—61no—
109C2×C3⋊Dic3—22no—
110C2×C3⋊Dic3C3×D411yescharacteristic
111C2×C3⋊Dic3C23×C311yescharacteristic
112C3×#67—64no—
113C3×#67—64no—
114C3×#67—68no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—21no—
117C2×C3×Dic3—21no—
118C2×C3×Dic3—21no—
119C2×C3×Dic3—41no—
120C2×C3×Dic3—41no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—64no—
126C2×C3×Dic3—122no—
127C3×Dic6—122no—
128#627—22no—
129#625—24no—
130#625—22no—
131#625C3×D414yes—
132#622—22no—
133#450—91noSylow
134C2×C33—22no—
135C2×C33#12011yescharacteristic
136C2×C33C2×SD1611yescharacteristic
137C2×C33AGL(1,ℤ/8)11yescharacteristic
138C32×D3—44no—
139C24×C3—61no—
140C24×C3—61no—
141C24×C3—121no—
142C23×D3—21no—
143C2×C3×Q8—91no—
144C2×C3×D4—92no—
145C2×C3×D4—181no—
146C2×C3×D4—184no—
147C22×C3×C4—91no—
148C2×#67—62no—
149C2×#67—62no—
150C2×#67—68no—
151C22×Dic3—61no—
152C22×Dic3—61no—
153C2×Dic6—61no—
154C3×SD16—184no—
155C2×C3×C8—92no—
156C3×C4⋊C4—91no—
157C3×C4⋊C4—181no—
158C3×C22⋊C4—61no—
159C3×C22⋊C4—61no—
160C3×C22⋊C4—121no—
161#217—21no—
162C4⋊Dic3—61no—
163Dic3⋊C4—61no—
164Dic3⋊C4—61no—
165C22×C32—22no—
166C22×C32—22no—
167C22×C32—21no—
168C22×C32—21no—
169C22×C32—21no—
170C22×C32—21no—
171C22×C32—22no—
172C22×C32—22no—
173C22×C32—42no—
174C22×C32—41no—
175C22×C32—41no—
176C22×C32—42no—
177C22×C32—41no—
178C22×C32—41no—
179C22×C32—42no—
180C22×C32—42no—
181C22×C32—64no—
182C22×C32—81no—
183C22×C32—81no—
184C22×C32—128no—
185C22×C32—122no—
186C22×C32—122no—
187C22×C32C3×SD1612yescomplemented
188C22×C32C2×C3×D411yescharacteristic
189C2×C3×D3—24no—
190C2×C3×D3—24no—
191C2×C3×D3—44no—
192C2×C3×D3—48no—
193C2×C3×D3—48no—
194C2×C3×D3—42no—
195C2×C3×D3—42no—
196C2×C3×D3—44no—
197C2×C3×D3—44no—
198C2×C3×D3—416no—
199C2×C3×D3—42no—
200C2×C3×D3—42no—
201C32×C4—121no—
202C32×C4—121no—
203C3⋊Dic3—21no—
204C3⋊Dic3C2×C3×D412yes—
205C3×Dic3—41no—
206C3×Dic3—41no—
207C3×Dic3—41no—
208C3×Dic3—41no—
209C3×Dic3—62no—
210C3×Dic3—62no—
211C3×Dic3—62no—
212C3×Dic3—81no—
213C3×Dic3—81no—
214C3×Dic3—121no—
215C22×D4—91no—
216C2×SD16—92no—
217C22⋊Q8—91no—
218#102—92no—
219C22⋊C8—91no—
220C33#45011yescharacteristic, complemented, Sylow
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—64no—
226C23×C3—62no—
227C23×C3—64no—
228C23×C3—62no—
229C23×C3—124no—
230C23×C3—122no—
231C23×C3—122no—
232C23×C3—122no—
233C23×C3—122no—
234C23×C3—124no—
235C23×C3—124no—
236C23×C3—122no—
237C23×C3GO+(4,𝔽2)11yescharacteristic
238C22×D3—24no—
239C22×D3—22no—
240C22×D3—42no—
241C22×D3—42no—
242C3×Q8—182no—
243C3×D4—94no—
244C3×D4—182no—
245C3×D4—184no—
246C2×C3×C4—61no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4—61no—
250C2×C3×C4—91no—
251C2×C3×C4—91no—
252C2×C3×C4—121no—
253C2×C3×C4—121no—
254C2×C3×C4—182no—
255#67—64no—
256#67—64no—
257#67—68no—
258C2×Dic3—21no—
259C2×Dic3—21no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264C2×Dic3—64no—
265C2×Dic3—122no—
266Dic6—122no—
267C3×C8—182no—
268C2×C32—24no—
269C2×C32—22no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32—21no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—41no—
277C2×C32—41no—
278C2×C32—41no—
279C2×C32—41no—
280C2×C32—41no—
281C2×C32—41no—
282C2×C32—42no—
283C2×C32—44no—
284C2×C32—82no—
285C2×C32—124no—
286C2×C32C3×#12011yescharacteristic
287C2×C32C2×C3×SD1611yescharacteristic
288C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
289C3×D3—44no—
290C3×D3—48no—
291C3×D3—44no—
292C24—61no—
293C2×Q8—91no—
294C2×D4—92no—
295C2×D4—181no—
296C2×D4—184no—
297C22×C4—91no—
298SD16—184no—
299C2×C8—92no—
300C4⋊C4—91no—
301C4⋊C4—181no—
302C22⋊C4—61no—
303C22×C3—22no—
304C22×C3—22no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—22no—
310C22×C3—22no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—42no—
314C22×C3—42no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—42no—
318C22×C3—42no—
319C22×C3—64no—
320C22×C3—64no—
321C22×C3—81no—
322C22×C3—81no—
323C22×C3—124no—
324C22×C3—128no—
325C22×C3—128no—
326C22×C3—122no—
327C22×C3—122no—
328C22×C3—1216no—
329C22×C3—124no—
330C22×C3—124no—
331C22×C3—122no—
332C22×C3—122no—
333C22×C3#377512yescomplemented
334C22×C3C2×GO+(4,𝔽2)11yescharacteristic
335C2×D3—24no—
336C2×D3—48no—
337C2×D3—42no—
338C2×D3—42no—
339C3×C4—92no—
340C3×C4—121no—
341C3×C4—121no—
342C3×C4—121no—
343C3×C4—121no—
344C3×C4—181no—
345C3×C4—241no—
346C3×C4—241no—
347Dic3—41no—
348Dic3—41no—
349Dic3—62no—
350Dic3—62no—
351Dic3—62no—
352Dic3—121no—
353C32—21no—
354C32—21no—
355C32—41no—
356C32—41no—
357C32C3×#45011yescharacteristic, complemented
358C23—64no—
359C23—62no—
360C23—122no—
361C23—122no—
362C23C3×GO+(4,𝔽2)11yescharacteristic
363Q8—182no—
364D4—94no—
365D4—182no—
366D4—184no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—91no—
370C2×C4—91no—
371C2×C4—182no—
372C8—182no—
373C2×C3—24no—
374C2×C3—22no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—42no—
382C2×C3—44no—
383C2×C3—41no—
384C2×C3—41no—
385C2×C3—41no—
386C2×C3—41no—
387C2×C3—41no—
388C2×C3—41no—
389C2×C3—82no—
390C2×C3—124no—
391C2×C3—128no—
392C2×C3—124no—
393C2×C3#6423611yescharacteristic
394C2×C3C2×#377511yescharacteristic
395C2×C3#6424111yescharacteristic
396D3—44no—
397C22—21no—
398C22—21no—
399C22—64no—
400C22—128no—
401C22—122no—
402C22—122no—
403C22C3×#377512yescomplemented
404C22C2×C3×GO+(4,𝔽2)11yescharacteristic
405C4—92no—
406C4—121no—
407C4—121no—
408C4—181no—
409C3—21no—
410C3—21no—
411C3—41no—
412C3—41no—
413C3#1059352311yescharacteristic, direct product factor
414C2—22no—
415C2—124no—
416C2C3×#6423611yescharacteristic
417C2C2×C3×#377511yescharacteristic
418C2C3×#6424111yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC22×C3×#625C2
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#450
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3775C2semidirect
3. C3×#63982C2semidirect
4. C3×C22⋊#622C2semidirect
5. C3×#63739C2semidirect
6. #10593523C3direct (N × H)
9. C2×C3×#627C22semidirect
13. C2×C3×#622C22semidirect
16. C3×#3724C22semidirect
20. C2×#3775C2×C3semidirect
22. #63982C2×C3semidirect
23. C22⋊#622C2×C3semidirect
24. #63739C2×C3semidirect
52. C2×#627C22×C3semidirect
56. C2×#622C22×C3semidirect
65. #3724C22×C3semidirect
69. C22×C33SD16semidirect
187. C22×C32C3×SD16semidirect
220. C33#450semidirect
333. C22×C3#3775semidirect
357. C32C3×#450semidirect
403. C22C3×#3775semidirect
413. C3#10593523direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6252
C2×C3×#37752
C3×#639822
C3×C22⋊#6222
C3×#637392
#105935233
C3×#4509