Finite Group Databasebeta

← C3×#10593496 #13166862

Subgroups — C3×#10593496 #13166862

Order 1728.

688 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3805C211yesmaximal, characteristic
2C2×C3×#3792C211yesmaximal, characteristic, complemented
3C2×C3×#3724C211yesmaximal, characteristic, complemented
4C3×#63740C212yesmaximal, complemented
5C3×#63739C212yesmaximal, complemented
6#10593496C311yesmaximal, characteristic, direct product factor
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×C32×#67—21no—
9C22×C32×Dic3—21no—
10C2×C3×#627C2211yescharacteristic
11C2×C3×#625C2211yescharacteristic
12C3×#3805—22no—
13C3×#3805C414yescomplemented
14C2×C3×D3×Dic3—21no—
15C3×#3792—22no—
16C2×C3×#622C2212yes—
17C3×#3724—21no—
18C3×#3724C2212yes—
19C2×C3×#237—61no—
20C2×#3805C2×C311yescharacteristic
21C2×#3792C2×C311yescharacteristic, complemented
22C2×C3×Dic3⋊C4—61no—
23C2×#3724C2×C311yescharacteristic, complemented
24#63740C2×C312yescomplemented
25#63739C2×C312yescomplemented
26C23×C33D411yescharacteristic
27C22×C32×D3—21no—
28C2×C3×C3⋊Dic3C2×C412yes—
29C2×C3×C3⋊Dic3D411yescharacteristic
30C2×C3×C3⋊Dic3D412yes—
31C2×C3×C3⋊Dic3C2311yescharacteristic
32C32×#67—24no—
33C2×C32×Dic3—21no—
34C2×C32×Dic3—22no—
35C2×C32×Dic3—42no—
36C3×#627—21no—
37C3×#627C2×C412yescomplemented
38C3×#625—21no—
39C3×#625C2×C412yescomplemented
40C3×D3×Dic3—24no—
41C3×#622—22no—
42C3×#417—91nomaximal
43C2×C32×D4—61no—
44C22×C32×C4—61no—
45C22×C3⋊Dic3C22×C311yescharacteristic
46C2×C3×#67—21no—
47C2×C3×#67—21no—
48C2×C3×#67—41no—
49C2×C3×#67—61no—
50C22×C3×Dic3—21no—
51C22×C3×Dic3—21no—
52C22×C3×Dic3—41no—
53C22×C3×Dic3—61no—
54C22×C3×Dic3—61no—
55C22×C3×Dic3—61no—
56C3×#237—64no—
57C3×#237—64no—
58C2×C3×C4×D3—61no—
59C2×C3×Dic6—61no—
60C2×#627C22×C311yescharacteristic
61C2×#625C22×C311yescharacteristic
62#3805—22no—
63#3805C3×C414yescomplemented
64C2×D3×Dic3—21no—
65#3792—22no—
66C2×#622C22×C312yes—
67C3×Dic3⋊C4—62no—
68C3×Dic3⋊C4—62no—
69#3724—21no—
70#3724C22×C312yes—
71C22×C33—21no—
72C22×C33—21no—
73C22×C33C22⋊C412yes—
74C22×C33C2×D411yescharacteristic
75C2×C32×D3—22no—
76C2×C32×D3—41no—
77C2×C32×D3—41no—
78C3×C3⋊Dic3C22⋊C412yes—
79C3×C3⋊Dic3C22×C411yescharacteristic
80C3×C3⋊Dic3C2×D411yescharacteristic
81C32×Dic3—22no—
82C32×Dic3—42no—
83C2×C3×#41—91no—
84C2×#237—61no—
85C2×C3×M16—91no—
86C2×C3×C4⋊C4—91no—
87C2×Dic3⋊C4—61no—
88C3×#103—92no—
89C3×#102—92no—
90C23×C32—21no—
91C23×C32—21no—
92C23×C32—41no—
93C23×C32—41no—
94C23×C32—61no—
95C23×C32C3×D411yescharacteristic
96C22×C3×D3—21no—
97C22×C3×D3—21no—
98C22×C3×D3—41no—
99C32×D4—64no—
100C2×C32×C4—61no—
101C2×C32×C4—62no—
102C2×C32×C4—122no—
103C2×C3⋊Dic3C2×C3×C412yes—
104C2×C3⋊Dic3C3×D411yescharacteristic
105C2×C3⋊Dic3C3×D412yes—
106C2×C3⋊Dic3C23×C311yescharacteristic
107C3×#67—24no—
108C3×#67—24no—
109C3×#67—44no—
110C3×#67—62no—
111C3×#67—62no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—22no—
115C2×C3×Dic3—22no—
116C2×C3×Dic3—42no—
117C2×C3×Dic3—41no—
118C2×C3×Dic3—42no—
119C2×C3×Dic3—42no—
120C2×C3×Dic3—44no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—64no—
130C3×C4×D3—64no—
131C3×Dic6—62no—
132C3×Dic6—62no—
133#627—21no—
134#627C2×C3×C412yescomplemented
135#625—21no—
136#625C2×C3×C412yescomplemented
137D3×Dic3—24no—
138#622—22no—
139#417—91noSylow
140C2×C33—22no—
141C2×C33C2×C22⋊C411yescharacteristic
142C2×C33AGL(1,ℤ/8)11yescharacteristic
143C2×C33#13711yescharacteristic
144C32×D3—42no—
145C3×#41—94no—
146C3×#41—182no—
147C2×C3×Q8—91no—
148C2×C3×D4—61no—
149C2×C3×D4—61no—
150C2×C3×D4—91no—
151C2×C3×D4—121no—
152C22×C3×C4—61no—
153C22×C3×C4—61no—
154C22×C3×C4—91no—
155C22×C3×C4—121no—
156C22×C3×C4—181no—
157C2×#67—21no—
158C2×#67—61no—
159C22×Dic3—21no—
160C22×Dic3—61no—
161C22×Dic3—61no—
162C22×Dic3—61no—
163#237—64no—
164#237—64no—
165C2×C4×D3—61no—
166C2×Dic6—61no—
167C3×M16—182no—
168C2×C3×C8—92no—
169C3×C4⋊C4—92no—
170C3×C4⋊C4—181no—
171Dic3⋊C4—62no—
172Dic3⋊C4—62no—
173C22×C32—22no—
174C22×C32—22no—
175C22×C32—21no—
176C22×C32—21no—
177C22×C32—21no—
178C22×C32—21no—
179C22×C32—22no—
180C22×C32—22no—
181C22×C32—42no—
182C22×C32—41no—
183C22×C32—41no—
184C22×C32—42no—
185C22×C32—41no—
186C22×C32—41no—
187C22×C32—42no—
188C22×C32—42no—
189C22×C32—62no—
190C22×C32—81no—
191C22×C32—81no—
192C22×C32—121no—
193C22×C32—121no—
194C22×C32C3×C22⋊C412yes—
195C22×C32C2×C3×D411yescharacteristic
196C2×C3×D3—22no—
197C2×C3×D3—22no—
198C2×C3×D3—42no—
199C2×C3×D3—41no—
200C2×C3×D3—41no—
201C2×C3×D3—41no—
202C2×C3×D3—41no—
203C2×C3×D3—81no—
204C2×C3×D3—81no—
205C32×C4—62no—
206C32×C4—122no—
207C3⋊Dic3C3×C22⋊C412yes—
208C3⋊Dic3C22×C3×C411yescharacteristic
209C3⋊Dic3C2×C3×D411yescharacteristic
210C3×Dic3—22no—
211C3×Dic3—22no—
212C3×Dic3—42no—
213C3×Dic3—42no—
214C3×Dic3—44no—
215C3×Dic3—42no—
216C3×Dic3—62no—
217C3×Dic3—62no—
218C3×Dic3—61no—
219C3×Dic3—61no—
220C3×Dic3—61no—
221C3×Dic3—61no—
222C2×#41—91no—
223C2×M16—91no—
224C2×C4⋊C4—91no—
225#103—92no—
226#102—92no—
227C33#41711yescharacteristic, complemented, Sylow
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—61no—
233C23×C3—61no—
234C23×C3—121no—
235C23×C3GO+(4,𝔽2)11yescharacteristic
236C22×D3—21no—
237C3×Q8—92no—
238C3×Q8—181no—
239C3×D4—64no—
240C3×D4—64no—
241C3×D4—92no—
242C3×D4—124no—
243C3×D4—181no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4—62no—
247C2×C3×C4—62no—
248C2×C3×C4—91no—
249C2×C3×C4—91no—
250C2×C3×C4—92no—
251C2×C3×C4—92no—
252C2×C3×C4—122no—
253C2×C3×C4—121no—
254C2×C3×C4—122no—
255C2×C3×C4—122no—
256C2×C3×C4—124no—
257C2×C3×C4—184no—
258#67—24no—
259#67—62no—
260#67—62no—
261C2×Dic3—21no—
262C2×Dic3—22no—
263C2×Dic3—42no—
264C2×Dic3—62no—
265C2×Dic3—62no—
266C2×Dic3—61no—
267C2×Dic3—61no—
268C2×Dic3—62no—
269C2×Dic3—62no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272C2×Dic3—64no—
273C4×D3—64no—
274Dic6—62no—
275Dic6—62no—
276C3×C8—182no—
277C2×C32—24no—
278C2×C32—22no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—41no—
286C2×C32—41no—
287C2×C32—41no—
288C2×C32—41no—
289C2×C32—41no—
290C2×C32—41no—
291C2×C32—42no—
292C2×C32—44no—
293C2×C32—82no—
294C2×C32—122no—
295C2×C32C2×C3×C22⋊C411yescharacteristic
296C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
297C2×C32C3×#13711yescharacteristic
298C3×D3—42no—
299C3×D3—42no—
300C3×D3—82no—
301#41—94no—
302#41—182no—
303C2×Q8—91no—
304C2×D4—61no—
305C2×D4—91no—
306C22×C4—61no—
307C22×C4—91no—
308C22×C4—181no—
309M16—182no—
310C2×C8—92no—
311C4⋊C4—92no—
312C4⋊C4—181no—
313C22×C3—22no—
314C22×C3—22no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—22no—
320C22×C3—22no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—42no—
328C22×C3—42no—
329C22×C3—62no—
330C22×C3—62no—
331C22×C3—81no—
332C22×C3—81no—
333C22×C3—122no—
334C22×C3—121no—
335C22×C3—121no—
336C22×C3—121no—
337C22×C3—121no—
338C22×C3—241no—
339C22×C3—241no—
340C22×C3D32⋊C412yes—
341C22×C3C2×GO+(4,𝔽2)11yescharacteristic
342C2×D3—22no—
343C2×D3—41no—
344C2×D3—41no—
345C3×C4—62no—
346C3×C4—62no—
347C3×C4—91no—
348C3×C4—92no—
349C3×C4—91no—
350C3×C4—122no—
351C3×C4—122no—
352C3×C4—124no—
353C3×C4—122no—
354Dic3—22no—
355Dic3—42no—
356Dic3—61no—
357Dic3—61no—
358Dic3—62no—
359Dic3—62no—
360Dic3—61no—
361Dic3—61no—
362C32—21no—
363C32—21no—
364C32—41no—
365C32—41no—
366C32C3×#41711yescharacteristic, complemented
367C23—61no—
368C23C3×GO+(4,𝔽2)11yescharacteristic
369Q8—92no—
370Q8—181no—
371D4—64no—
372D4—92no—
373D4—181no—
374C2×C4—61no—
375C2×C4—62no—
376C2×C4—91no—
377C2×C4—91no—
378C2×C4—92no—
379C2×C4—92no—
380C2×C4—122no—
381C2×C4—184no—
382C8—182no—
383C2×C3—24no—
384C2×C3—22no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—42no—
392C2×C3—44no—
393C2×C3—41no—
394C2×C3—41no—
395C2×C3—41no—
396C2×C3—41no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—82no—
400C2×C3—122no—
401C2×C3—122no—
402C2×C3—242no—
403C2×C3C2×D32⋊C411yescharacteristic
404C2×C3#6423611yescharacteristic
405C2×C3#6423911yescharacteristic
406D3—42no—
407C22—21no—
408C22—21no—
409C22—62no—
410C22—121no—
411C22—121no—
412C22C3×D32⋊C412yes—
413C22C2×C3×GO+(4,𝔽2)11yescharacteristic
414C4—62no—
415C4—91no—
416C4—92no—
417C4—91no—
418C4—122no—
419C3—21no—
420C3—21no—
421C3—41no—
422C3—41no—
423C3#1059349611yescharacteristic, direct product factor
424C2—22no—
425C2—122no—
426C2C2×C3×D32⋊C411yescharacteristic
427C2C3×#6423611yescharacteristic
428C2C3×#6423911yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC2×C3×#3805C2
cosocleC2×C3⋊Dic3C23×C3
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#417
LCSG ▷ 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×#3792C2semidirect
3. C2×C3×#3724C2semidirect
4. C3×#63740C2semidirect
5. C3×#63739C2semidirect
6. #10593496C3direct (N × H)
13. C3×#3805C4semidirect
21. C2×#3792C2×C3semidirect
23. C2×#3724C2×C3semidirect
24. #63740C2×C3semidirect
25. #63739C2×C3semidirect
37. C3×#627C2×C4semidirect
39. C3×#625C2×C4semidirect
63. #3805C3×C4semidirect
134. #627C2×C3×C4semidirect
136. #625C2×C3×C4semidirect
227. C33#417semidirect
366. C32C3×#417semidirect
423. C3#10593496direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38052
C2×C3×#37922
C2×C3×#37242
C3×#637402
C3×#637392
#105934963
C3×#4179