Finite Group Databasebeta

← C3×#10593321 #13166715

Subgroups — C3×#10593321 #13166715

Order 1728.

582 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 94 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×#3797C211yesmaximal, characteristic
2C3×#64286C211yesmaximal, characteristic
3C3×#63818C211yesmaximal, characteristic
4C3×#63730C214yesmaximal, complemented
5#10593321C311yesmaximal, characteristic, direct product factor
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×C32×Dic6—21no—
8C2×C3×#627C412yescomplemented
9C2×C3×#624—21no—
10C3×#3797—21no—
11C3×#3797—21no—
12C3×#3797C412yescomplemented
13C3×#3797C2212yes—
14C3×C22⋊#157—21no—
15C3×C4⋊#157C2212yes—
16C3×C4×#157—21no—
17C32×#208—21no—
18C3×#3712—21no—
19C3×#3710C2212yes—
20C2×C3×D3×Q8—61no—
21C2×#3797C2×C311yescharacteristic
22#64286C2×C311yescharacteristic
23C3×D3×M16—61no—
24#63818C2×C311yescharacteristic
25#63730C2×C314yescomplemented
26C22×C3×C3⋊D3D411yescharacteristic
27C2×C3×#157—22no—
28C2×C33×C4D411yescharacteristic
29C2×C3×C3⋊Dic3C2×C411yescharacteristic
30C3×C4×C3⋊D3C2×C411yescharacteristic
31C3×C4×C3⋊D3D411yescharacteristic
32C3×C4×C3⋊D3D411yescharacteristic
33C3×C4×C3⋊D3C2311yescharacteristic
34C2×C32×Dic3—22no—
35C32×Dic6—22no—
36C32×Dic6—22no—
37C3×#627—22no—
38C3×#627C2×C414yes—
39C3×#624—22no—
40C3×#624—22no—
41C32×C3⋊C8—22no—
42C3×#419—91nomaximal
43C2×C32×Q8—61no—
44C2×C4×C3⋊D3C22×C311yescharacteristic
45C3×D3×Q8—62no—
46C3×D3×Q8—62no—
47C3×D3×Q8—62no—
48C3×D3×Q8—62no—
49C2×C3×C4×D3—62no—
50C2×C3×C4×D3—61no—
51C2×C3×C4×D3—61no—
52C2×C3×Dic6—21no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—41no—
55C2×C3×Dic6—62no—
56C2×#627C3×C412yescomplemented
57C2×#624—21no—
58#3797—21no—
59#3797—21no—
60#3797C3×C412yescomplemented
61#3797C22×C312yes—
62C22⋊#157—21no—
63C4⋊#157C22×C312yes—
64C4×#157—21no—
65C32×M16—61no—
66C3×#208—21no—
67C3×#208—21no—
68C3×#208—41no—
69C3×ΓL(1,𝔽25)—62no—
70C3×D3×C8—62no—
71#3712—21no—
72#3710C22×C312yes—
73C22×C33C22⋊C411yescharacteristic
74C2×C3×C3⋊D3—22no—
75C2×C3×C3⋊D3C22⋊C411yescharacteristic
76C2×C3×C3⋊D3C2×D411yescharacteristic
77C3×#157—42no—
78C33×C4C22⋊C411yescharacteristic
79C33×C4C2×D411yescharacteristic
80C3×C3⋊Dic3C22⋊C411yescharacteristic
81C3×C3⋊Dic3C22×C411yescharacteristic
82C32×Dic3—24no—
83C22×C3×Q8—91no—
84C2×D3×Q8—61no—
85C2×C3×M16—91no—
86C3×#117—91no—
87D3×M16—61no—
88C3×#103—94no—
89C22×C3⋊D3C3×D411yescharacteristic
90C22×C3×D3—61no—
91C22×C3×D3—61no—
92C2×#157—22no—
93C32×Q8—62no—
94C32×Q8—62no—
95C2×C32×C4—21no—
96C2×C32×C4—21no—
97C2×C32×C4—41no—
98C2×C32×C4—41no—
99C2×C32×C4—62no—
100C2×C32×C4C3×D411yescharacteristic
101C2×C3⋊Dic3C2×C3×C411yescharacteristic
102C4×C3⋊D3C2×C3×C411yescharacteristic
103C4×C3⋊D3C3×D411yescharacteristic
104C4×C3⋊D3C3×D411yescharacteristic
105C4×C3⋊D3C23×C311yescharacteristic
106C2×C3×Dic3—22no—
107C2×C3×Dic3—22no—
108C2×C3×Dic3—42no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C3×C4×D3—61no—
112C3×C4×D3—61no—
113C3×C4×D3—61no—
114C3×C4×D3—61no—
115C3×C4×D3—61no—
116C3×C4×D3—61no—
117C3×C4×D3—61no—
118C3×C4×D3—61no—
119C3×C4×D3—64no—
120C3×C4×D3—64no—
121C3×Dic6—22no—
122C3×Dic6—22no—
123C3×Dic6—22no—
124C3×Dic6—22no—
125C3×Dic6—42no—
126C3×Dic6—42no—
127C3×Dic6—64no—
128C3×Dic6—64no—
129#627—22no—
130#627C2×C3×C414yes—
131#624—22no—
132#624—22no—
133C32×C8—62no—
134C3×C3⋊C8—22no—
135C3×C3⋊C8—22no—
136C3×C3⋊C8—42no—
137#419—91noSylow
138C2×C33—21no—
139C2×C33C2×C22⋊C411yescharacteristic
140C3×C3⋊D3—21no—
141C3×C3⋊D3#13712yes—
142C2×C3×Q8—61no—
143C2×C3×Q8—61no—
144C2×C3×Q8—92no—
145C2×C3×Q8—92no—
146C2×C3×Q8—92no—
147C2×C3×Q8—121no—
148C2×C3×Q8—181no—
149C2×C3×Q8—181no—
150C22×C3×C4—91no—
151C22×C3×C4—181no—
152D3×Q8—62no—
153D3×Q8—62no—
154D3×Q8—62no—
155D3×Q8—62no—
156C2×C4×D3—61no—
157C2×C4×D3—61no—
158C2×C4×D3—62no—
159C2×Dic6—21no—
160C2×Dic6—62no—
161C3×M16—61no—
162C3×M16—61no—
163C3×M16—121no—
164C3×M16—181no—
165C2×C3×C8—92no—
166C3×C4⋊C4—92no—
167C3×C22⋊C4—181no—
168C3×C42—181no—
169#208—21no—
170ΓL(1,𝔽25)—62no—
171D3×C8—62no—
172C22×C32—21no—
173C22×C32—21no—
174C22×C32—41no—
175C22×C32—41no—
176C22×C32C3×C22⋊C411yescharacteristic
177C2×C3⋊D3—22no—
178C2×C3⋊D3C3×C22⋊C411yescharacteristic
179C2×C3⋊D3C2×C3×D411yescharacteristic
180C2×C3×D3—61no—
181C2×C3×D3—61no—
182C2×C3×D3—61no—
183C2×C3×D3—61no—
184C2×C3×D3—64no—
185C2×C3×D3—122no—
186#157—42no—
187C32×C4—21no—
188C32×C4—21no—
189C32×C4—21no—
190C32×C4—21no—
191C32×C4—41no—
192C32×C4—41no—
193C32×C4—41no—
194C32×C4—41no—
195C32×C4—64no—
196C32×C4C3×C22⋊C411yescharacteristic
197C32×C4C2×C3×D411yescharacteristic
198C3⋊Dic3C3×C22⋊C411yescharacteristic
199C3⋊Dic3C22×C3×C411yescharacteristic
200C3×Dic3—24no—
201C3×Dic3—24no—
202C3×Dic3—44no—
203C3×Dic3—61no—
204C3×Dic3—61no—
205C3×Dic3—61no—
206C3×Dic3—61no—
207C22×Q8—91no—
208C2×M16—91no—
209#117—91no—
210#103—94no—
211C33#41911yescharacteristic, complemented, Sylow
212C23×C3—91no—
213C22×D3—61no—
214C22×D3—61no—
215C3×Q8—62no—
216C3×Q8—62no—
217C3×Q8—62no—
218C3×Q8—62no—
219C3×Q8—94no—
220C3×Q8—122no—
221C3×Q8—122no—
222C3×Q8—182no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—62no—
228C2×C3×C4—62no—
229C2×C3×C4—91no—
230C2×C3×C4—91no—
231C2×C3×C4—91no—
232C2×C3×C4—91no—
233C2×C3×C4—91no—
234C2×C3×C4—122no—
235C2×C3×C4—182no—
236C2×C3×C4—182no—
237C2×C3×C4—182no—
238C2×C3×C4GO+(4,𝔽2)11yescharacteristic
239C2×Dic3—22no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242C4×D3—61no—
243C4×D3—61no—
244C4×D3—61no—
245C4×D3—61no—
246C4×D3—61no—
247C4×D3—61no—
248C4×D3—61no—
249C4×D3—61no—
250C4×D3—64no—
251C4×D3—64no—
252Dic6—22no—
253Dic6—22no—
254Dic6—64no—
255Dic6—64no—
256C3×C8—62no—
257C3×C8—62no—
258C3×C8—122no—
259C3⋊C8—22no—
260C2×C32—22no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—42no—
268C2×C32—81no—
269C2×C32C2×C3×C22⋊C411yescharacteristic
270C3⋊D3—21no—
271C3⋊D3C3×#13712yes—
272C3×D3—62no—
273C3×D3—62no—
274C3×D3—62no—
275C3×D3—121no—
276C2×Q8—61no—
277C2×Q8—92no—
278C2×Q8—92no—
279C2×Q8—92no—
280C2×Q8—181no—
281C2×Q8—181no—
282C22×C4—91no—
283C22×C4—181no—
284M16—61no—
285M16—181no—
286C2×C8—92no—
287C4⋊C4—92no—
288C22⋊C4—181no—
289C42—181no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—91no—
295C22×C3—91no—
296C22×C3—182no—
297C22×C3D32⋊C411yescharacteristic
298C2×D3—61no—
299C2×D3—61no—
300C2×D3—61no—
301C2×D3—61no—
302C2×D3—64no—
303C2×D3—122no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—41no—
309C3×C4—41no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4—64no—
313C3×C4—64no—
314C3×C4—91no—
315C3×C4—91no—
316C3×C4—124no—
317C3×C4—362no—
318C3×C4D32⋊C411yescharacteristic
319C3×C4C2×GO+(4,𝔽2)11yescharacteristic
320Dic3—24no—
321Dic3—61no—
322Dic3—61no—
323Dic3—61no—
324Dic3—61no—
325C32—21no—
326C32—21no—
327C32—41no—
328C32—41no—
329C32C3×#41911yescharacteristic, complemented
330C23—91no—
331Q8—62no—
332Q8—62no—
333Q8—94no—
334Q8—182no—
335C2×C4—62no—
336C2×C4—91no—
337C2×C4—91no—
338C2×C4—91no—
339C2×C4—91no—
340C2×C4—91no—
341C2×C4—182no—
342C2×C4—182no—
343C2×C4—182no—
344C2×C4C3×GO+(4,𝔽2)11yescharacteristic
345C8—62no—
346C2×C3—22no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—41no—
351C2×C3—42no—
352C2×C3—41no—
353C2×C3—41no—
354C2×C3—81no—
355C2×C3—92no—
356C2×C3—181no—
357C2×C3C2×D32⋊C411yescharacteristic
358D3—62no—
359D3—62no—
360D3—62no—
361D3—121no—
362C22—91no—
363C22—91no—
364C22—182no—
365C22C3×D32⋊C411yescharacteristic
366C4—64no—
367C4—91no—
368C4—91no—
369C4—362no—
370C4C3×D32⋊C411yescharacteristic
371C4C2×C3×GO+(4,𝔽2)11yescharacteristic
372C3—21no—
373C3—21no—
374C3—41no—
375C3—41no—
376C3#1059332111yescharacteristic, direct product factor
377C2—21no—
378C2—92no—
379C2—181no—
380C2C2×C3×D32⋊C411yescharacteristic
subgroupG/H
centerC2×C3C2×D32⋊C4
commutatorC3⋊Dic3C22×C3×C4
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33C2×C22⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC4×C3⋊D3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#419
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×#63730C2semidirect
5. #10593321C3direct (N × H)
8. C2×C3×#627C4semidirect
12. C3×#3797C4semidirect
25. #63730C2×C3semidirect
56. C2×#627C3×C4semidirect
60. #3797C3×C4semidirect
211. C33#419semidirect
329. C32C3×#419semidirect
376. C3#10593321direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37972
C3×#642862
C3×#638182
C3×#637302
#105933213
C3×#4199