Finite Group Databasebeta

← C3×#10593498 #13166864

Subgroups — C3×#10593498 #13166864

Order 1728.

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

normal subgroups only

This group has 644 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
2C22×C3×#622C211yesmaximal, characteristic, complemented
3C3×#63957C211yesmaximal, characteristic, complemented
4C3×#63740C212yesmaximal, complemented
5C3×#63739C212yesmaximal, complemented
6#10593498C311yesmaximal, characteristic, direct product factor
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×C32×#67—21no—
9C2×C3×#627C2211yescharacteristic
10C2×C3×#625C2211yescharacteristic
11C3×#3805—22no—
12C3×#3805C414yescomplemented
13C2×C3×D3×Dic3—21no—
14C2×C3×#622—22no—
15C2×C3×#622C2212yes—
16C32×#217—21no—
17C3×#3724C2212yes—
18C3×Dic32—21no—
19C2×C3×#237—61no—
20C2×#3805C2×C311yescharacteristic
21C22×#622C2×C311yescharacteristic, complemented
22C3×#877—61no—
23#63957C2×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—
35C3×#627—21no—
36C3×#627C2×C412yes—
37C3×#625—21no—
38C3×#625C2×C412yes—
39C3×D3×Dic3—24no—
40C3×#622—22no—
41C3×#416—91nomaximal
42C2×C32×D4—61no—
43C22×C3⋊Dic3C22×C311yescharacteristic
44C2×C3×#67—21no—
45C2×C3×#67—21no—
46C2×C3×#67—41no—
47C2×C3×#67—61no—
48C22×C3×Dic3—61no—
49C22×C3×Dic3—61no—
50C22×C3×Dic3—61no—
51C3×#237—64no—
52C3×#237—64no—
53C2×C3×C4×D3—61no—
54C2×C3×Dic6—61no—
55C2×#627C22×C311yescharacteristic
56C2×#625C22×C311yescharacteristic
57#3805—22no—
58#3805C3×C414yescomplemented
59C2×D3×Dic3—21no—
60C2×#622—22no—
61C2×#622C22×C312yes—
62C32×C22⋊C4—61no—
63C3×#217—21no—
64C3×#217—21no—
65C3×#217—41no—
66C3×Dic3⋊C4—62no—
67C3×C4×Dic3—62no—
68#3724C22×C312yes—
69Dic32—21no—
70C22×C33—22no—
71C22×C33C22⋊C412yes—
72C22×C33C2×D411yescharacteristic
73C2×C32×D3—22no—
74C2×C32×D3—42no—
75C3×C3⋊Dic3C22⋊C412yes—
76C3×C3⋊Dic3C22×C411yescharacteristic
77C3×C3⋊Dic3C2×D411yescharacteristic
78C32×Dic3—22no—
79C32×Dic3—42no—
80C2×C3×#41—91no—
81C2×#237—61no—
82C22×C3×C8—91no—
83C3×#117—91no—
84#877—61no—
85C3×#103—92no—
86C3×#102—92no—
87C23×C32—21no—
88C23×C32—21no—
89C23×C32—41no—
90C23×C32—41no—
91C23×C32—61no—
92C23×C32C3×D411yescharacteristic
93C22×C3×D3—21no—
94C22×C3×D3—21no—
95C22×C3×D3—41no—
96C32×D4—64no—
97C2×C32×C4—61no—
98C2×C32×C4—62no—
99C2×C3⋊Dic3C2×C3×C412yes—
100C2×C3⋊Dic3C3×D411yescharacteristic
101C2×C3⋊Dic3C3×D412yes—
102C2×C3⋊Dic3C23×C311yescharacteristic
103C3×#67—24no—
104C3×#67—24no—
105C3×#67—44no—
106C3×#67—62no—
107C3×#67—62no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—21no—
110C2×C3×Dic3—22no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—41no—
113C2×C3×Dic3—42no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—62no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—64no—
123C3×C4×D3—64no—
124C3×Dic6—62no—
125C3×Dic6—62no—
126#627—21no—
127#627C2×C3×C412yes—
128#625—21no—
129#625C2×C3×C412yes—
130D3×Dic3—24no—
131#622—22no—
132#416—91noSylow
133C2×C33—22no—
134C2×C33C2×C22⋊C411yescharacteristic
135C2×C33#13512yes—
136C32×D3—42no—
137C3×#41—94no—
138C3×#41—182no—
139C2×C3×Q8—91no—
140C2×C3×D4—61no—
141C2×C3×D4—61no—
142C2×C3×D4—91no—
143C2×C3×D4—121no—
144C22×C3×C4—91no—
145C22×C3×C4—181no—
146C2×#67—21no—
147C2×#67—61no—
148C22×Dic3—61no—
149C22×Dic3—61no—
150C22×Dic3—61no—
151#237—64no—
152#237—64no—
153C2×C4×D3—61no—
154C2×Dic6—61no—
155C2×C3×C8—92no—
156C2×C3×C8—182no—
157C3×C4⋊C4—92no—
158C3×C22⋊C4—61no—
159C3×C22⋊C4—61no—
160C3×C22⋊C4—121no—
161C3×C42—181no—
162#217—21no—
163Dic3⋊C4—62no—
164C4×Dic3—62no—
165C22×C32—21no—
166C22×C32—21no—
167C22×C32—22no—
168C22×C32—22no—
169C22×C32—22no—
170C22×C32—41no—
171C22×C32—41no—
172C22×C32—42no—
173C22×C32—42no—
174C22×C32—42no—
175C22×C32—42no—
176C22×C32—62no—
177C22×C32—82no—
178C22×C32—82no—
179C22×C32—122no—
180C22×C32C3×C22⋊C412yes—
181C22×C32C2×C3×D411yescharacteristic
182C2×C3×D3—22no—
183C2×C3×D3—22no—
184C2×C3×D3—42no—
185C2×C3×D3—42no—
186C2×C3×D3—42no—
187C2×C3×D3—82no—
188C32×C4—62no—
189C32×C4—122no—
190C3⋊Dic3C3×C22⋊C412yes—
191C3⋊Dic3C22×C3×C411yescharacteristic
192C3⋊Dic3C2×C3×D411yescharacteristic
193C3×Dic3—22no—
194C3×Dic3—22no—
195C3×Dic3—42no—
196C3×Dic3—42no—
197C3×Dic3—42no—
198C3×Dic3—62no—
199C3×Dic3—62no—
200C3×Dic3—61no—
201C3×Dic3—61no—
202C3×Dic3—61no—
203C3×Dic3—61no—
204C3×Dic3—82no—
205C2×#41—91no—
206C22×C8—91no—
207#117—91no—
208#103—92no—
209#102—92no—
210C33#41611yescharacteristic, complemented, Sylow
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3—61no—
216C23×C3—61no—
217C23×C3—121no—
218C23×C3GO+(4,𝔽2)11yescharacteristic
219C22×D3—21no—
220C3×Q8—92no—
221C3×Q8—181no—
222C3×D4—64no—
223C3×D4—64no—
224C3×D4—92no—
225C3×D4—124no—
226C3×D4—181no—
227C2×C3×C4—61no—
228C2×C3×C4—61no—
229C2×C3×C4—62no—
230C2×C3×C4—62no—
231C2×C3×C4—91no—
232C2×C3×C4—91no—
233C2×C3×C4—92no—
234C2×C3×C4—92no—
235C2×C3×C4—121no—
236C2×C3×C4—122no—
237C2×C3×C4—184no—
238#67—24no—
239#67—62no—
240#67—62no—
241C2×Dic3—21no—
242C2×Dic3—22no—
243C2×Dic3—62no—
244C2×Dic3—62no—
245C2×Dic3—61no—
246C2×Dic3—61no—
247C2×Dic3—62no—
248C2×Dic3—62no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—64no—
252C4×D3—64no—
253Dic6—62no—
254Dic6—62no—
255C3×C8—182no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—22no—
259C2×C32—22no—
260C2×C32—22no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—42no—
264C2×C32—42no—
265C2×C32—42no—
266C2×C32—42no—
267C2×C32—82no—
268C2×C32—82no—
269C2×C32—122no—
270C2×C32C2×C3×C22⋊C411yescharacteristic
271C2×C32C3×#13512yes—
272C3×D3—42no—
273C3×D3—42no—
274C3×D3—82no—
275#41—94no—
276#41—182no—
277C2×Q8—91no—
278C2×D4—61no—
279C2×D4—91no—
280C22×C4—91no—
281C22×C4—181no—
282C2×C8—92no—
283C2×C8—182no—
284C4⋊C4—92no—
285C22⋊C4—61no—
286C42—181no—
287C22×C3—22no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—22no—
291C22×C3—22no—
292C22×C3—42no—
293C22×C3—42no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—42no—
297C22×C3—42no—
298C22×C3—62no—
299C22×C3—62no—
300C22×C3—82no—
301C22×C3—82no—
302C22×C3—122no—
303C22×C3—122no—
304C22×C3—122no—
305C22×C3—242no—
306C22×C3D32⋊C412yes—
307C22×C3C2×GO+(4,𝔽2)11yescharacteristic
308C2×D3—22no—
309C2×D3—42no—
310C3×C4—62no—
311C3×C4—62no—
312C3×C4—91no—
313C3×C4—92no—
314C3×C4—91no—
315C3×C4—122no—
316C3×C4—122no—
317C3×C4—122no—
318C3×C4—242no—
319Dic3—22no—
320Dic3—42no—
321Dic3—61no—
322Dic3—61no—
323Dic3—62no—
324Dic3—62no—
325Dic3—61no—
326Dic3—61no—
327C32—21no—
328C32—21no—
329C32—41no—
330C32—41no—
331C32C3×#41611yescharacteristic, complemented
332C23—61no—
333C23C3×GO+(4,𝔽2)11yescharacteristic
334Q8—92no—
335Q8—181no—
336D4—64no—
337D4—92no—
338D4—181no—
339C2×C4—61no—
340C2×C4—62no—
341C2×C4—91no—
342C2×C4—91no—
343C2×C4—92no—
344C2×C4—92no—
345C2×C4—184no—
346C8—182no—
347C2×C3—22no—
348C2×C3—22no—
349C2×C3—22no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—42no—
353C2×C3—42no—
354C2×C3—42no—
355C2×C3—42no—
356C2×C3—41no—
357C2×C3—41no—
358C2×C3—82no—
359C2×C3—82no—
360C2×C3—122no—
361C2×C3—122no—
362C2×C3—242no—
363C2×C3C2×D32⋊C411yescharacteristic
364C2×C3#6423712yes—
365D3—42no—
366C22—22no—
367C22—62no—
368C22—122no—
369C22C3×D32⋊C412yes—
370C22C2×C3×GO+(4,𝔽2)11yescharacteristic
371C4—62no—
372C4—91no—
373C4—92no—
374C4—91no—
375C4—122no—
376C3—21no—
377C3—21no—
378C3—41no—
379C3—41no—
380C3#1059349811yescharacteristic, direct product factor
381C2—22no—
382C2—122no—
383C2C2×C3×D32⋊C411yescharacteristic
384C2C3×#6423712yes—
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×#416
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. C22×C3×#622C2semidirect
3. C3×#63957C2semidirect
4. C3×#63740C2semidirect
5. C3×#63739C2semidirect
6. #10593498C3direct (N × H)
12. C3×#3805C4semidirect
21. C22×#622C2×C3semidirect
23. #63957C2×C3semidirect
24. #63740C2×C3semidirect
25. #63739C2×C3semidirect
58. #3805C3×C4semidirect
210. C33#416semidirect
331. C32C3×#416semidirect
380. C3#10593498direct (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
C22×C3×#6222
C3×#639572
C3×#637402
C3×#637392
#105934983
C3×#4169