Finite Group Databasebeta

← C2×C3×#63730 #13166714

Subgroups — C2×C3×#63730 #13166714

Order 1728.

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

normal subgroups only

This group has 616 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×#3797C211yesmaximal, characteristic
2C2×C3×C4⋊#157C211yesmaximal, characteristic
3C2×C3×#3710C211yesmaximal, characteristic
4C3×#63730C214yesmaximal, direct product factor
5C2×#63730C311yesmaximal, characteristic, direct product factor
6C22×C3×#157—21no—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×Dic6—21no—
9C2×C3×#627C412yescomplemented
10C2×C3×#624—21no—
11C3×#3797—21no—
12C3×#3797—21no—
13C3×#3797C412yescomplemented
14C3×#3797C2212yes—
15C3×C4⋊#157—21no—
16C3×C4⋊#157C2212yes—
17C2×C32×C3⋊C8—21no—
18C3×#3710—21no—
19C3×#3710C2212yes—
20C2×C3×D3×Q8—61no—
21C2×#3797C2×C311yescharacteristic
22C2×C4⋊#157C2×C311yescharacteristic
23C2×C3×D3×C8—61no—
24C2×#3710C2×C311yescharacteristic
25#63730C2×C314yesdirect product factor
26C22×C3×C3⋊D3D411yescharacteristic
27C2×C3×#157—22no—
28C2×C3×#157—42no—
29C2×C33×C4D411yescharacteristic
30C2×C3×C3⋊Dic3C2×C411yescharacteristic
31C3×C4×C3⋊D3C2×C411yescharacteristic
32C3×C4×C3⋊D3D411yescharacteristic
33C3×C4×C3⋊D3D411yescharacteristic
34C3×C4×C3⋊D3C2311yescharacteristic
35C2×C32×Dic3—22no—
36C32×Dic6—22no—
37C32×Dic6—22no—
38C3×#627—22no—
39C3×#627C2×C414yescomplemented
40C3×#624—22no—
41C3×#624—22no—
42C32×C3⋊C8—22no—
43C2×C3×#103—91nomaximal
44C22×#157—21no—
45C2×C32×Q8—61no—
46C2×C4×C3⋊D3C22×C311yescharacteristic
47C3×D3×Q8—62no—
48C3×D3×Q8—62no—
49C3×D3×Q8—62no—
50C3×D3×Q8—62no—
51C2×C3×C4×D3—62no—
52C2×C3×C4×D3—61no—
53C2×C3×C4×D3—61no—
54C2×C3×Dic6—21no—
55C2×C3×Dic6—21no—
56C2×C3×Dic6—41no—
57C2×C3×Dic6—62no—
58C2×#627C3×C412yescomplemented
59C2×#624—21no—
60#3797—21no—
61#3797—21no—
62#3797C3×C412yescomplemented
63#3797C22×C312yes—
64C4⋊#157—21no—
65C4⋊#157C22×C312yes—
66C2×C32×C8—61no—
67C2×C3×C3⋊C8—21no—
68C2×C3×C3⋊C8—21no—
69C2×C3×C3⋊C8—41no—
70C3×D3×C8—62no—
71C3×D3×C8—62no—
72#3710—21no—
73#3710C22×C312yes—
74C22×C33C22⋊C411yescharacteristic
75C2×C3×C3⋊D3C22⋊C411yescharacteristic
76C2×C3×C3⋊D3SD1612yes—
77C2×C3×C3⋊D3Q1612yes—
78C2×C3×C3⋊D3C2×D411yescharacteristic
79C3×#157—42no—
80C33×C4C22⋊C411yescharacteristic
81C33×C4C2×D411yescharacteristic
82C3×C3⋊Dic3C22⋊C411yescharacteristic
83C3×C3⋊Dic3C22×C411yescharacteristic
84C32×Dic3—24no—
85C22×C3×Q8—91no—
86C2×D3×Q8—61no—
87C22×C3×C8—91no—
88C2×C3×C4⋊C4—91no—
89C2×D3×C8—61no—
90C3×#103—94no—
91C22×C3⋊D3C3×D411yescharacteristic
92C22×C3×D3—61no—
93C22×C3×D3—61no—
94C2×#157—22no—
95C2×#157—42no—
96C32×Q8—62no—
97C32×Q8—62no—
98C2×C32×C4—21no—
99C2×C32×C4—21no—
100C2×C32×C4—41no—
101C2×C32×C4—41no—
102C2×C32×C4—62no—
103C2×C32×C4C3×D411yescharacteristic
104C2×C3⋊Dic3C2×C3×C411yescharacteristic
105C4×C3⋊D3C2×C3×C411yescharacteristic
106C4×C3⋊D3C3×D411yescharacteristic
107C4×C3⋊D3C3×D411yescharacteristic
108C4×C3⋊D3C23×C311yescharacteristic
109C2×C3×Dic3—22no—
110C2×C3×Dic3—22no—
111C2×C3×Dic3—42no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C3×C4×D3—61no—
115C3×C4×D3—61no—
116C3×C4×D3—61no—
117C3×C4×D3—61no—
118C3×C4×D3—61no—
119C3×C4×D3—61no—
120C3×C4×D3—61no—
121C3×C4×D3—61no—
122C3×C4×D3—64no—
123C3×C4×D3—64no—
124C3×Dic6—22no—
125C3×Dic6—22no—
126C3×Dic6—22no—
127C3×Dic6—22no—
128C3×Dic6—42no—
129C3×Dic6—42no—
130C3×Dic6—64no—
131C3×Dic6—64no—
132#627—22no—
133#627C2×C3×C414yescomplemented
134#624—22no—
135#624—22no—
136C32×C8—62no—
137C3×C3⋊C8—22no—
138C3×C3⋊C8—22no—
139C3×C3⋊C8—42no—
140C2×#103—91noSylow
141C2×C33#10312yescomplemented
142C2×C33C2×C22⋊C411yescharacteristic
143C3×C3⋊D3#10312yescomplemented
144C3×C3⋊D3C2×SD1611yescharacteristic
145C3×C3⋊D3C2×Q1611yescharacteristic
146C2×C3×Q8—61no—
147C2×C3×Q8—61no—
148C2×C3×Q8—92no—
149C2×C3×Q8—92no—
150C2×C3×Q8—92no—
151C2×C3×Q8—121no—
152C2×C3×Q8—181no—
153C2×C3×Q8—181no—
154C22×C3×C4—91no—
155C22×C3×C4—181no—
156C22×C3×C4—181no—
157D3×Q8—62no—
158D3×Q8—62no—
159D3×Q8—62no—
160D3×Q8—62no—
161C2×C4×D3—61no—
162C2×C4×D3—61no—
163C2×C4×D3—62no—
164C2×Dic6—21no—
165C2×Dic6—62no—
166C2×C3×C8—61no—
167C2×C3×C8—61no—
168C2×C3×C8—92no—
169C2×C3×C8—121no—
170C2×C3×C8—181no—
171C3×C4⋊C4—92no—
172C3×C4⋊C4—181no—
173C2×C3⋊C8—21no—
174D3×C8—62no—
175D3×C8—62no—
176C22×C32—21no—
177C22×C32—21no—
178C22×C32—41no—
179C22×C32—41no—
180C22×C32C3×C22⋊C411yescharacteristic
181C2×C3⋊D3C3×C22⋊C411yescharacteristic
182C2×C3⋊D3C3×SD1612yes—
183C2×C3⋊D3C3×Q1612yes—
184C2×C3⋊D3C2×C3×D411yescharacteristic
185C2×C3×D3—61no—
186C2×C3×D3—61no—
187C2×C3×D3—62no—
188C2×C3×D3—62no—
189C2×C3×D3—62no—
190C2×C3×D3—62no—
191C2×C3×D3—61no—
192C2×C3×D3—61no—
193#157—42no—
194C32×C4—21no—
195C32×C4—21no—
196C32×C4—21no—
197C32×C4—21no—
198C32×C4—41no—
199C32×C4—41no—
200C32×C4—41no—
201C32×C4—41no—
202C32×C4—64no—
203C32×C4C3×C22⋊C411yescharacteristic
204C32×C4C2×C3×D411yescharacteristic
205C3⋊Dic3C3×C22⋊C411yescharacteristic
206C3⋊Dic3C22×C3×C411yescharacteristic
207C3×Dic3—24no—
208C3×Dic3—24no—
209C3×Dic3—44no—
210C3×Dic3—61no—
211C3×Dic3—61no—
212C3×Dic3—61no—
213C3×Dic3—61no—
214C22×Q8—91no—
215C22×C8—91no—
216C2×C4⋊C4—91no—
217#103—94no—
218C33C2×#10311yescharacteristic, complemented, Sylow
219C23×C3—91no—
220C22×D3—61no—
221C22×D3—61no—
222C3×Q8—62no—
223C3×Q8—62no—
224C3×Q8—62no—
225C3×Q8—62no—
226C3×Q8—94no—
227C3×Q8—122no—
228C3×Q8—122no—
229C3×Q8—182no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—62no—
235C2×C3×C4—62no—
236C2×C3×C4—91no—
237C2×C3×C4—91no—
238C2×C3×C4—91no—
239C2×C3×C4—91no—
240C2×C3×C4—91no—
241C2×C3×C4—122no—
242C2×C3×C4—182no—
243C2×C3×C4—182no—
244C2×C3×C4—182no—
245C2×C3×C4—362no—
246C2×C3×C4GO+(4,𝔽2)11yescharacteristic
247C2×Dic3—22no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C4×D3—61no—
251C4×D3—61no—
252C4×D3—61no—
253C4×D3—61no—
254C4×D3—61no—
255C4×D3—61no—
256C4×D3—61no—
257C4×D3—61no—
258C4×D3—64no—
259C4×D3—64no—
260Dic6—22no—
261Dic6—22no—
262Dic6—64no—
263Dic6—64no—
264C3×C8—62no—
265C3×C8—62no—
266C3×C8—122no—
267C3⋊C8—22no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32—22no—
271C2×C32—22no—
272C2×C32—42no—
273C2×C32—42no—
274C2×C32—41no—
275C2×C32—41no—
276C2×C32C3×#10312yescomplemented
277C2×C32C2×C3×C22⋊C411yescharacteristic
278C3⋊D3C3×#10312yescomplemented
279C3⋊D3C2×C3×SD1611yescharacteristic
280C3⋊D3C2×C3×Q1611yescharacteristic
281C3×D3—62no—
282C3×D3—62no—
283C3×D3—61no—
284C3×D3—61no—
285C3×D3—61no—
286C3×D3—61no—
287C2×Q8—61no—
288C2×Q8—92no—
289C2×Q8—92no—
290C2×Q8—92no—
291C2×Q8—181no—
292C2×Q8—181no—
293C22×C4—91no—
294C22×C4—181no—
295C22×C4—181no—
296C2×C8—61no—
297C2×C8—92no—
298C2×C8—181no—
299C4⋊C4—92no—
300C4⋊C4—181no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—91no—
306C22×C3—91no—
307C22×C3—92no—
308C22×C3—92no—
309C22×C3D32⋊C411yescharacteristic
310C2×D3—61no—
311C2×D3—61no—
312C2×D3—61no—
313C2×D3—61no—
314C2×D3—62no—
315C2×D3—62no—
316C2×D3—62no—
317C2×D3—62no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—41no—
323C3×C4—41no—
324C3×C4—41no—
325C3×C4—41no—
326C3×C4—64no—
327C3×C4—64no—
328C3×C4—91no—
329C3×C4—91no—
330C3×C4—124no—
331C3×C4—362no—
332C3×C4D32⋊C411yescharacteristic
333C3×C4C2×GO+(4,𝔽2)11yescharacteristic
334Dic3—24no—
335Dic3—61no—
336Dic3—61no—
337Dic3—61no—
338Dic3—61no—
339C32—21no—
340C32—21no—
341C32—41no—
342C32—41no—
343C32C2×C3×#10311yescharacteristic, complemented
344C23—91no—
345Q8—62no—
346Q8—62no—
347Q8—94no—
348Q8—182no—
349C2×C4—62no—
350C2×C4—91no—
351C2×C4—91no—
352C2×C4—91no—
353C2×C4—91no—
354C2×C4—91no—
355C2×C4—182no—
356C2×C4—182no—
357C2×C4—182no—
358C2×C4—362no—
359C2×C4C3×GO+(4,𝔽2)11yescharacteristic
360C8—62no—
361C2×C3—22no—
362C2×C3—22no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—42no—
366C2×C3—42no—
367C2×C3—41no—
368C2×C3—41no—
369C2×C3—91no—
370C2×C3—91no—
371C2×C3—92no—
372C2×C3#6373012yesdirect product factor
373C2×C3C2×D32⋊C411yescharacteristic
374D3—62no—
375D3—62no—
376D3—61no—
377D3—61no—
378D3—61no—
379D3—61no—
380C22—92no—
381C22—92no—
382C22—91no—
383C22—91no—
384C22C3×D32⋊C411yescharacteristic
385C4—64no—
386C4—91no—
387C4—91no—
388C4—362no—
389C4C3×D32⋊C411yescharacteristic
390C4C2×C3×GO+(4,𝔽2)11yescharacteristic
391C3—21no—
392C3—21no—
393C3—41no—
394C3—41no—
395C3C2×#6373011yescharacteristic, direct product factor
396C2—91no—
397C2—92no—
398C2—91no—
399C2C3×#6373012yesdirect product factor
400C2C2×C3×D32⋊C411yescharacteristic
subgroupG/H
centerC22×C3D32⋊C4
commutatorC3⋊Dic3C22×C3×C4
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C22⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC4×C3⋊D3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C2×C3×#103
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
4. C3×#63730C2direct (N × H)
5. C2×#63730C3direct (N × H)
9. C2×C3×#627C4semidirect
13. C3×#3797C4semidirect
25. #63730C2×C3direct (N × H)
39. C3×#627C2×C4semidirect
58. C2×#627C3×C4semidirect
62. #3797C3×C4semidirect
133. #627C2×C3×C4semidirect
141. C2×C33#103semidirect
143. C3×C3⋊D3#103semidirect
218. C33C2×#103semidirect
276. C2×C32C3×#103semidirect
278. C3⋊D3C3×#103semidirect
343. C32C2×C3×#103semidirect
372. C2×C3#63730direct (N × H)
395. C3C2×#63730direct (N × H)
399. C2C3×#63730direct (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
C2×C3×C4⋊#1572
C2×C3×#37102
C3×#637302
C2×#637303
C2×C3×#1039