Finite Group Databasebeta

← C3×#10593365 #13166706

Subgroups — C3×#10593365 #13166706

Order 1728.

812 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 114 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
2C2×C3×#3787C211yesmaximal, characteristic, complemented
3C3×#63900C211yesmaximal, characteristic, complemented
4C3×#63739C214yesmaximal, complemented
5#10593365C311yesmaximal, characteristic, direct product factor
6C22×C3×D32—21no—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×D12—21no—
9C2×C3×#625C2212yes—
10C2×C3×#624—21no—
11C3×#3802—22no—
12C3×#3802C414yescomplemented
13C2×C3×#622C2212yescomplemented
14C3×#3787—22no—
15C32×C4⋊Dic3—21no—
16C3×#3724C2212yescomplemented
17C2×C3×D3×D4—61no—
18C2×#3802C2×C311yescharacteristic
19C2×#3787C2×C311yescharacteristic, complemented
20C3×D3×C4⋊C4—61no—
21#63900C2×C311yescharacteristic, complemented
22#63739C2×C314yescomplemented
23C22×C3×C3⋊D3D411yescharacteristic
24C22×C32×D3—22no—
25C2×C3×D32—24no—
26C2×C3×D32—42no—
27C2×C3×D32—42no—
28C2×C33×C4D411yescharacteristic
29C2×C3×C3⋊Dic3C2311yescharacteristic
30C3×C4×C3⋊D3C2×C412yes—
31C3×C4×C3⋊D3D412yes—
32C2×C32×Dic3—22no—
33C32×D12—24no—
34C3×#625—22no—
35C3×#625C2×C414yescomplemented
36C3×#624—42no—
37C3×#622—22no—
38C2×C3×#102—91nomaximal
39C23×C3×D3—62no—
40C22×D32—21no—
41C2×C32×D4—61no—
42C2×C4×C3⋊D3C22×C311yescharacteristic
43C2×C3×#67—62no—
44C3×D3×D4—64no—
45C3×D3×D4—64no—
46C2×C3×D12—21no—
47C2×C3×D12—21no—
48C2×C3×D12—41no—
49C2×C3×C4×D3—61no—
50C2×C3×C4×D3—61no—
51C2×C3×C4×D3—62no—
52C2×#625C22×C312yes—
53C2×#624—21no—
54#3802—22no—
55#3802C3×C414yescomplemented
56C2×#622C22×C312yescomplemented
57#3787—22no—
58C32×C4⋊C4—61no—
59C3×C4⋊Dic3—21no—
60C3×C4⋊Dic3—21no—
61C3×C4⋊Dic3—41no—
62C3×Dic3⋊C4—62no—
63#3724C22×C312yescomplemented
64C22×C33C2×D411yescharacteristic
65C2×C3×C3⋊D3C22⋊C412yes—
66C2×C3×C3⋊D3D812yes—
67C2×C3×C3⋊D3SD1612yes—
68C2×C32×D3—24no—
69C2×C32×D3—42no—
70C2×C32×D3—42no—
71C3×D32—48no—
72C33×C4C22⋊C412yes—
73C3×C3⋊Dic3C22×C411yescharacteristic
74C3×C3⋊Dic3C2×D411yescharacteristic
75C32×Dic3—42no—
76C22×C3×D4—91no—
77C2×D3×D4—61no—
78C22×C3×C8—91no—
79C2×C3×C4⋊C4—91no—
80D3×C4⋊C4—61no—
81C3×#102—94no—
82C23×C32—62no—
83C22×C3⋊D3C3×D411yescharacteristic
84C22×C3×D3—22no—
85C22×C3×D3—22no—
86C22×C3×D3—42no—
87C22×C3×D3—61no—
88C22×C3×D3—61no—
89C22×C3×D3—68no—
90C22×C3×D3—124no—
91C22×C3×D3—124no—
92C2×D32—24no—
93C2×D32—42no—
94C2×D32—42no—
95C32×D4—64no—
96C2×C32×C4—21no—
97C2×C32×C4—21no—
98C2×C32×C4—41no—
99C2×C32×C4—41no—
100C2×C32×C4—62no—
101C2×C32×C4C3×D411yescharacteristic
102C2×C3⋊Dic3C23×C311yescharacteristic
103C4×C3⋊D3C2×C3×C412yes—
104C4×C3⋊D3C3×D412yes—
105C3×#67—64no—
106C3×#67—64no—
107C2×C3×Dic3—22no—
108C2×C3×Dic3—22no—
109C2×C3×Dic3—42no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C3×D12—24no—
113C3×D12—24no—
114C3×D12—44no—
115C3×C4×D3—62no—
116C3×C4×D3—62no—
117C3×C4×D3—62no—
118C3×C4×D3—62no—
119C3×C4×D3—124no—
120#625—22no—
121#625C2×C3×C414yescomplemented
122#624—42no—
123#622—22no—
124C2×#102—91noSylow
125C2×C33C2×C22⋊C411yescharacteristic
126C2×C33C2×D811yescharacteristic
127C2×C33C2×SD1611yescharacteristic
128C3×C3⋊D3#10214yescomplemented
129C32×D3—44no—
130C24×C3—181no—
131C23×D3—62no—
132C2×C3×D4—61no—
133C2×C3×D4—61no—
134C2×C3×D4—94no—
135C2×C3×D4—92no—
136C2×C3×D4—121no—
137C2×C3×D4—182no—
138C22×C3×C4—91no—
139C22×C3×C4—181no—
140C2×#67—62no—
141D3×D4—64no—
142D3×D4—64no—
143C2×D12—21no—
144C2×C4×D3—61no—
145C2×C4×D3—61no—
146C2×C4×D3—62no—
147C2×C3×C8—92no—
148C2×C3×C8—182no—
149C3×C4⋊C4—61no—
150C3×C4⋊C4—61no—
151C3×C4⋊C4—92no—
152C3×C4⋊C4—121no—
153C4⋊Dic3—21no—
154Dic3⋊C4—62no—
155C22×C32—21no—
156C22×C32—21no—
157C22×C32—41no—
158C22×C32—41no—
159C22×C32—64no—
160C22×C32—122no—
161C22×C32—122no—
162C22×C32C2×C3×D411yescharacteristic
163C2×C3⋊D3C3×C22⋊C412yes—
164C2×C3⋊D3C3×D812yes—
165C2×C3⋊D3C3×SD1612yes—
166C2×C3×D3—24no—
167C2×C3×D3—24no—
168C2×C3×D3—44no—
169C2×C3×D3—42no—
170C2×C3×D3—42no—
171C2×C3×D3—42no—
172C2×C3×D3—42no—
173C2×C3×D3—62no—
174C2×C3×D3—62no—
175C2×C3×D3—62no—
176C2×C3×D3—62no—
177C2×C3×D3—62no—
178C2×C3×D3—62no—
179C2×C3×D3—82no—
180C2×C3×D3—82no—
181C2×C3×D3—1216no—
182D32—48no—
183C32×C4—22no—
184C32×C4—22no—
185C32×C4—42no—
186C32×C4—42no—
187C32×C4—122no—
188C32×C4C3×C22⋊C412yes—
189C3⋊Dic3C22×C3×C411yescharacteristic
190C3⋊Dic3C2×C3×D411yescharacteristic
191C3×Dic3—42no—
192C3×Dic3—42no—
193C3×Dic3—61no—
194C3×Dic3—61no—
195C3×Dic3—61no—
196C3×Dic3—61no—
197C3×Dic3—82no—
198C22×D4—91no—
199C22×C8—91no—
200C2×C4⋊C4—91no—
201#102—94no—
202C33C2×#10211yescharacteristic, complemented, Sylow
203C23×C3—62no—
204C23×C3—62no—
205C23×C3—91no—
206C23×C3—122no—
207C23×C3—184no—
208C23×C3—362no—
209C23×C3—362no—
210C22×D3—22no—
211C22×D3—61no—
212C22×D3—61no—
213C22×D3—68no—
214C22×D3—124no—
215C22×D3—124no—
216C3×D4—64no—
217C3×D4—64no—
218C3×D4—94no—
219C3×D4—124no—
220C3×D4—182no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—62no—
226C2×C3×C4—62no—
227C2×C3×C4—91no—
228C2×C3×C4—92no—
229C2×C3×C4—92no—
230C2×C3×C4—122no—
231C2×C3×C4—362no—
232C2×C3×C4GO+(4,𝔽2)11yescharacteristic
233#67—64no—
234#67—64no—
235C2×Dic3—22no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238D12—24no—
239C4×D3—62no—
240C4×D3—62no—
241C4×D3—62no—
242C4×D3—62no—
243C4×D3—124no—
244C3×C8—182no—
245C2×C32—21no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—41no—
252C2×C32—41no—
253C2×C32—41no—
254C2×C32—41no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32—124no—
258C2×C32C2×C3×C22⋊C411yescharacteristic
259C2×C32C2×C3×D811yescharacteristic
260C2×C32C2×C3×SD1611yescharacteristic
261C3⋊D3C3×#10214yescomplemented
262C3×D3—44no—
263C3×D3—44no—
264C3×D3—64no—
265C3×D3—64no—
266C3×D3—84no—
267C24—181no—
268C2×D4—61no—
269C2×D4—94no—
270C2×D4—92no—
271C2×D4—182no—
272C22×C4—91no—
273C22×C4—181no—
274C2×C8—92no—
275C2×C8—182no—
276C4⋊C4—61no—
277C4⋊C4—92no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—64no—
283C22×C3—64no—
284C22×C3—92no—
285C22×C3—92no—
286C22×C3—92no—
287C22×C3—124no—
288C22×C3—122no—
289C22×C3—122no—
290C22×C3—122no—
291C22×C3—122no—
292C22×C3—242no—
293C22×C3—242no—
294C22×C3—368no—
295C22×C3C2×GO+(4,𝔽2)11yescharacteristic
296C2×D3—24no—
297C2×D3—42no—
298C2×D3—42no—
299C2×D3—62no—
300C2×D3—62no—
301C2×D3—62no—
302C2×D3—62no—
303C2×D3—62no—
304C2×D3—62no—
305C2×D3—1216no—
306C3×C4—22no—
307C3×C4—22no—
308C3×C4—42no—
309C3×C4—42no—
310C3×C4—91no—
311C3×C4—91no—
312C3×C4—122no—
313C3×C4—122no—
314C3×C4—242no—
315C3×C4D32⋊C412yes—
316Dic3—42no—
317Dic3—61no—
318Dic3—61no—
319Dic3—61no—
320Dic3—61no—
321C32—21no—
322C32—21no—
323C32—41no—
324C32—41no—
325C32C2×C3×#10211yescharacteristic, complemented
326C23—62no—
327C23—91no—
328C23—184no—
329C23—362no—
330C23—362no—
331D4—64no—
332D4—94no—
333D4—182no—
334C2×C4—62no—
335C2×C4—91no—
336C2×C4—92no—
337C2×C4—92no—
338C2×C4—362no—
339C2×C4C3×GO+(4,𝔽2)11yescharacteristic
340C8—182no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—41no—
350C2×C3—41no—
351C2×C3—41no—
352C2×C3—41no—
353C2×C3—94no—
354C2×C3—124no—
355C2×C3—124no—
356C2×C3—244no—
357C2×C3#6422511yescharacteristic
358C2×C3#6422711yescharacteristic
359C2×C3C2×D32⋊C411yescharacteristic
360D3—44no—
361D3—64no—
362D3—64no—
363C22—64no—
364C22—92no—
365C22—92no—
366C22—92no—
367C22—122no—
368C22—122no—
369C22—368no—
370C22C2×C3×GO+(4,𝔽2)11yescharacteristic
371C4—91no—
372C4—91no—
373C4—122no—
374C4C3×D32⋊C412yes—
375C3—21no—
376C3—21no—
377C3—41no—
378C3—41no—
379C3#1059336511yescharacteristic, direct product factor
380C2—94no—
381C2—124no—
382C2C3×#6422511yescharacteristic
383C2C3×#6422711yescharacteristic
384C2C2×C3×D32⋊C411yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#3802C2
cosocleC2×C3⋊Dic3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C2×C3×#102
LCSG ▷ 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
2. C2×C3×#3787C2semidirect
3. C3×#63900C2semidirect
4. C3×#63739C2semidirect
5. #10593365C3direct (N × H)
12. C3×#3802C4semidirect
13. C2×C3×#622C22semidirect
16. C3×#3724C22semidirect
19. C2×#3787C2×C3semidirect
21. #63900C2×C3semidirect
22. #63739C2×C3semidirect
35. C3×#625C2×C4semidirect
55. #3802C3×C4semidirect
56. C2×#622C22×C3semidirect
63. #3724C22×C3semidirect
121. #625C2×C3×C4semidirect
128. C3×C3⋊D3#102semidirect
202. C33C2×#102semidirect
261. C3⋊D3C3×#102semidirect
325. C32C2×C3×#102semidirect
379. C3#10593365direct (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×#37872
C3×#639002
C3×#637392
#105933653
C2×C3×#1029