Finite Group Databasebeta

← C3×#10593360 #13166700

Subgroups — C3×#10593360 #13166700

Order 1728.

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

normal subgroups only

This group has 660 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×#3798C211yesmaximal, characteristic
2C2×C3×#3788C211yesmaximal, characteristic, complemented
3C3×#63900C211yesmaximal, characteristic, complemented
4C3×#63740C212yesmaximal, complemented
5C3×#63739C212yesmaximal, complemented
6#10593360C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×C4×D3—21no—
9C2×C3×#627C2211yescharacteristic
10C2×C3×#626—21no—
11C2×C3×#625C2211yescharacteristic
12C2×C3×#624—21no—
13C3×#3798—22no—
14C3×#3798C414yescomplemented
15C2×C3×#622C2212yes—
16C3×#3788—22no—
17C32×C4⋊Dic3—21no—
18C3×#3724C2212yes—
19C2×C3×#235—61no—
20C2×#3798C2×C311yescharacteristic
21C2×#3788C2×C311yescharacteristic, complemented
22C3×D3×C4⋊C4—61no—
23#63900C2×C311yescharacteristic, complemented
24#63740C2×C312yescomplemented
25#63739C2×C312yescomplemented
26C22×C3×C3⋊D3D411yescharacteristic
27C22×C32×D3—21no—
28C2×C33×C4D411yescharacteristic
29C2×C3×C3⋊Dic3C2311yescharacteristic
30C3×C4×C3⋊D3C2×C412yes—
31C3×C4×C3⋊D3D412yes—
32C2×C32×Dic3—21no—
33C2×C32×Dic3—22no—
34C32×C4×D3—24no—
35C3×#627—21no—
36C3×#627C2×C412yescomplemented
37C3×#626—24no—
38C3×#625—21no—
39C3×#625C2×C412yescomplemented
40C3×#624—42no—
41C3×#622—22no—
42C3×#417—91nomaximal
43C22×C32×C4—61no—
44C2×C4×C3⋊D3C22×C311yescharacteristic
45C2×C3×#67—61no—
46C2×C3×#67—61no—
47C3×#235—64no—
48C3×#235—64no—
49C2×C3×D12—61no—
50C2×C3×C4×D3—21no—
51C2×C3×C4×D3—21no—
52C2×C3×C4×D3—41no—
53C2×C3×C4×D3—61no—
54C2×C3×C4×D3—61no—
55C2×C3×C4×D3—62no—
56C2×C3×Dic6—61no—
57C2×#627C22×C311yescharacteristic
58C2×#626—21no—
59C2×#625C22×C311yescharacteristic
60C2×#624—21no—
61#3798—22no—
62#3798C3×C414yescomplemented
63C2×#622C22×C312yes—
64#3788—22no—
65C32×C4⋊C4—61no—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3—21no—
68C3×C4⋊Dic3—41no—
69C3×Dic3⋊C4—62no—
70#3724C22×C312yes—
71C22×C33C2×D411yescharacteristic
72C2×C3×C3⋊D3—21no—
73C2×C3×C3⋊D3—21no—
74C2×C3×C3⋊D3C22⋊C412yes—
75C2×C32×D3—22no—
76C2×C32×D3—41no—
77C2×C32×D3—41no—
78C33×C4C22⋊C412yes—
79C3×C3⋊Dic3C22×C411yescharacteristic
80C3×C3⋊Dic3C2×D411yescharacteristic
81C32×Dic3—22no—
82C32×Dic3—42no—
83C2×C3×#41—91no—
84C2×#235—61no—
85C2×C3×M16—91no—
86C2×C3×C4⋊C4—91no—
87D3×C4⋊C4—61no—
88C3×#103—92no—
89C3×#102—92no—
90C23×C32—61no—
91C22×C3⋊D3C3×D411yescharacteristic
92C22×C3×D3—21no—
93C22×C3×D3—21no—
94C22×C3×D3—41no—
95C22×C3×D3—61no—
96C22×C3×D3—61no—
97C2×C32×C4—21no—
98C2×C32×C4—21no—
99C2×C32×C4—41no—
100C2×C32×C4—41no—
101C2×C32×C4—64no—
102C2×C32×C4—61no—
103C2×C32×C4—62no—
104C2×C32×C4C3×D411yescharacteristic
105C2×C3⋊Dic3C23×C311yescharacteristic
106C4×C3⋊D3C2×C3×C412yes—
107C4×C3⋊D3C3×D412yes—
108C3×#67—62no—
109C3×#67—64no—
110C3×#67—62no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—22no—
114C2×C3×Dic3—22no—
115C2×C3×Dic3—41no—
116C2×C3×Dic3—42no—
117C2×C3×Dic3—61no—
118C2×C3×Dic3—61no—
119C3×D12—64no—
120C3×C4×D3—24no—
121C3×C4×D3—24no—
122C3×C4×D3—44no—
123C3×C4×D3—62no—
124C3×C4×D3—62no—
125C3×C4×D3—62no—
126C3×C4×D3—62no—
127C3×C4×D3—124no—
128C3×Dic6—62no—
129C3×Dic6—62no—
130#627—21no—
131#627C2×C3×C412yescomplemented
132#626—24no—
133#625—21no—
134#625C2×C3×C412yescomplemented
135#624—42no—
136#622—22no—
137#417—91noSylow
138C2×C33C2×C22⋊C411yescharacteristic
139C2×C33AGL(1,ℤ/8)11yescharacteristic
140C2×C33#13711yescharacteristic
141C3×C3⋊D3—22no—
142C32×D3—42no—
143C3×#41—94no—
144C3×#41—182no—
145C2×C3×Q8—91no—
146C2×C3×D4—91no—
147C2×C3×D4—181no—
148C22×C3×C4—61no—
149C22×C3×C4—61no—
150C22×C3×C4—91no—
151C22×C3×C4—121no—
152C22×C3×C4—181no—
153C2×#67—61no—
154C2×#67—61no—
155#235—64no—
156#235—64no—
157C2×D12—61no—
158C2×C4×D3—21no—
159C2×C4×D3—61no—
160C2×C4×D3—61no—
161C2×C4×D3—62no—
162C2×Dic6—61no—
163C3×M16—182no—
164C2×C3×C8—92no—
165C3×C4⋊C4—61no—
166C3×C4⋊C4—61no—
167C3×C4⋊C4—92no—
168C3×C4⋊C4—121no—
169C4⋊Dic3—21no—
170Dic3⋊C4—62no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32—41no—
174C22×C32—41no—
175C22×C32—62no—
176C22×C32—121no—
177C22×C32—121no—
178C22×C32C2×C3×D411yescharacteristic
179C2×C3⋊D3—21no—
180C2×C3⋊D3—21no—
181C2×C3⋊D3C3×C22⋊C412yes—
182C2×C3×D3—22no—
183C2×C3×D3—22no—
184C2×C3×D3—42no—
185C2×C3×D3—41no—
186C2×C3×D3—41no—
187C2×C3×D3—42no—
188C2×C3×D3—42no—
189C2×C3×D3—41no—
190C2×C3×D3—41no—
191C2×C3×D3—62no—
192C2×C3×D3—62no—
193C2×C3×D3—62no—
194C2×C3×D3—62no—
195C2×C3×D3—121no—
196C2×C3×D3—121no—
197C32×C4—22no—
198C32×C4—22no—
199C32×C4—42no—
200C32×C4—42no—
201C32×C4—62no—
202C32×C4—122no—
203C32×C4C3×C22⋊C412yes—
204C3⋊Dic3C22×C3×C411yescharacteristic
205C3⋊Dic3C2×C3×D411yescharacteristic
206C3×Dic3—22no—
207C3×Dic3—22no—
208C3×Dic3—42no—
209C3×Dic3—42no—
210C3×Dic3—42no—
211C3×Dic3—61no—
212C3×Dic3—61no—
213C3×Dic3—61no—
214C3×Dic3—61no—
215C3×Dic3—82no—
216C2×#41—91no—
217C2×M16—91no—
218C2×C4⋊C4—91no—
219#103—92no—
220#102—92no—
221C33#41711yescharacteristic, complemented, Sylow
222C23×C3—61no—
223C23×C3—61no—
224C23×C3—91no—
225C23×C3—121no—
226C22×D3—21no—
227C22×D3—61no—
228C22×D3—61no—
229C3×Q8—92no—
230C3×Q8—181no—
231C3×D4—92no—
232C3×D4—184no—
233C3×D4—181no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—64no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—64no—
242C2×C3×C4—62no—
243C2×C3×C4—62no—
244C2×C3×C4—91no—
245C2×C3×C4—92no—
246C2×C3×C4—92no—
247C2×C3×C4—124no—
248C2×C3×C4—121no—
249C2×C3×C4—122no—
250C2×C3×C4—362no—
251C2×C3×C4GO+(4,𝔽2)11yescharacteristic
252#67—62no—
253#67—64no—
254#67—62no—
255C2×Dic3—21no—
256C2×Dic3—22no—
257C2×Dic3—61no—
258C2×Dic3—61no—
259D12—64no—
260C4×D3—24no—
261C4×D3—62no—
262C4×D3—62no—
263C4×D3—62no—
264C4×D3—62no—
265C4×D3—124no—
266Dic6—62no—
267Dic6—62no—
268C3×C8—182no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32—21no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—41no—
276C2×C32—41no—
277C2×C32—41no—
278C2×C32—41no—
279C2×C32—41no—
280C2×C32—41no—
281C2×C32—122no—
282C2×C32C2×C3×C22⋊C411yescharacteristic
283C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
284C2×C32C3×#13711yescharacteristic
285C3⋊D3—22no—
286C3×D3—42no—
287C3×D3—44no—
288C3×D3—42no—
289C3×D3—64no—
290C3×D3—122no—
291#41—94no—
292#41—182no—
293C2×Q8—91no—
294C2×D4—91no—
295C2×D4—181no—
296C22×C4—61no—
297C22×C4—91no—
298C22×C4—181no—
299M16—182no—
300C2×C8—92no—
301C4⋊C4—61no—
302C4⋊C4—92no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—62no—
308C22×C3—62no—
309C22×C3—92no—
310C22×C3—122no—
311C22×C3—121no—
312C22×C3—121no—
313C22×C3—122no—
314C22×C3—122no—
315C22×C3—121no—
316C22×C3—121no—
317C22×C3—181no—
318C22×C3—181no—
319C22×C3C2×GO+(4,𝔽2)11yescharacteristic
320C2×D3—22no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—62no—
324C2×D3—62no—
325C2×D3—62no—
326C2×D3—62no—
327C2×D3—121no—
328C2×D3—121no—
329C3×C4—22no—
330C3×C4—22no—
331C3×C4—42no—
332C3×C4—42no—
333C3×C4—62no—
334C3×C4—62no—
335C3×C4—91no—
336C3×C4—91no—
337C3×C4—122no—
338C3×C4—122no—
339C3×C4—122no—
340C3×C4—242no—
341C3×C4D32⋊C412yes—
342Dic3—22no—
343Dic3—42no—
344Dic3—61no—
345Dic3—61no—
346Dic3—61no—
347Dic3—61no—
348C32—21no—
349C32—21no—
350C32—41no—
351C32—41no—
352C32C3×#41711yescharacteristic, complemented
353C23—61no—
354C23—91no—
355Q8—92no—
356Q8—181no—
357D4—92no—
358D4—184no—
359D4—181no—
360C2×C4—61no—
361C2×C4—64no—
362C2×C4—62no—
363C2×C4—91no—
364C2×C4—92no—
365C2×C4—92no—
366C2×C4—362no—
367C2×C4C3×GO+(4,𝔽2)11yescharacteristic
368C8—182no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—41no—
378C2×C3—41no—
379C2×C3—41no—
380C2×C3—41no—
381C2×C3—122no—
382C2×C3—124no—
383C2×C3—122no—
384C2×C3—182no—
385C2×C3#6422411yescharacteristic
386C2×C3#6422611yescharacteristic
387C2×C3C2×D32⋊C411yescharacteristic
388D3—42no—
389D3—64no—
390D3—122no—
391C22—62no—
392C22—92no—
393C22—121no—
394C22—121no—
395C22—181no—
396C22—181no—
397C22C2×C3×GO+(4,𝔽2)11yescharacteristic
398C4—62no—
399C4—91no—
400C4—91no—
401C4—122no—
402C4C3×D32⋊C412yes—
403C3—21no—
404C3—21no—
405C3—41no—
406C3—41no—
407C3#1059336011yescharacteristic, direct product factor
408C2—122no—
409C2—182no—
410C2C3×#6422411yescharacteristic
411C2C3×#6422611yescharacteristic
412C2C2×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×#3798C2
cosocleC2×C3⋊Dic3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#417
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×#3788C2semidirect
3. C3×#63900C2semidirect
4. C3×#63740C2semidirect
5. C3×#63739C2semidirect
6. #10593360C3direct (N × H)
14. C3×#3798C4semidirect
21. C2×#3788C2×C3semidirect
23. #63900C2×C3semidirect
24. #63740C2×C3semidirect
25. #63739C2×C3semidirect
36. C3×#627C2×C4semidirect
39. C3×#625C2×C4semidirect
62. #3798C3×C4semidirect
131. #627C2×C3×C4semidirect
134. #625C2×C3×C4semidirect
221. C33#417semidirect
352. C32C3×#417semidirect
407. C3#10593360direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37982
C2×C3×#37882
C3×#639002
C3×#637402
C3×#637392
#105933603
C3×#4179