Finite Group Databasebeta

← C2×C3×#64221 #13181012

Subgroups — C2×C3×#64221 #13181012

Order 1728.

1144 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1144 conjugacy classes of subgroups — too many to draw. Only its 194 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×#3797C211yesmaximal, characteristic, complemented
3C2×C3×C4×#157C211yesmaximal, characteristic, complemented
4C2×C3×D32⋊C4C214yesmaximal, complemented
5C3×#64221C218yesmaximal, direct product factor
6C2×#64221C311yesmaximal, characteristic, direct product factor
7C22×C3×D32C2212yes—
8C22×C3×#157C2212yes—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×D12—21no—
11C2×C32×Dic6—21no—
12C2×C3×#627—21no—
13C2×C3×#625—21no—
14C2×C3×#624C2212yes—
15C3×#3802—22no—
16C3×#3802C2214yes—
17C3×#3797—22no—
18C3×#3797C2214yescomplemented
19C3×C4×#157C2214yescomplemented
20C3×D32⋊C4C22116yescomplemented
21C2×C3×D3×Q8—61no—
22C2×C3×D3×D4—61no—
23C2×#3802C2×C311yescharacteristic
24C2×#3797C2×C311yescharacteristic, complemented
25C2×C4×#157C2×C311yescharacteristic, complemented
26C2×D32⋊C4C2×C314yescomplemented
27#64221C2×C318yesdirect product factor
28C22×C3×C3⋊D3C2311yescharacteristic
29C22×C32×D3—22no—
30C2×C3×D32—28no—
31C2×C3×D32—22no—
32C2×C3×D32C2314yes—
33C2×C3×#157—24no—
34C2×C3×#157C2314yes—
35C2×C33×C4D411yescharacteristic
36C2×C3×C3⋊Dic3D411yescharacteristic
37C3×C4×C3⋊D3D412yes—
38C3×C4×C3⋊D3C2312yes—
39C2×C32×Dic3—22no—
40C32×D12—24no—
41C32×Dic6—24no—
42C3×#627—24no—
43C3×#625—24no—
44C3×#624—22no—
45C3×#624C2314yes—
46C2×C3×#124—91nomaximal
47C23×C3×D3—62no—
48C22×D32C22×C312yes—
49C22×#157C22×C312yes—
50C2×C32×Q8—61no—
51C2×C32×D4—61no—
52C2×C4×C3⋊D3C22×C311yescharacteristic
53C2×C3×#67—62no—
54C3×D3×Q8—64no—
55C3×D3×Q8—64no—
56C3×D3×D4—64no—
57C3×D3×D4—64no—
58C2×C3×D12—21no—
59C2×C3×D12—21no—
60C2×C3×D12—41no—
61C2×C3×C4×D3—62no—
62C2×C3×C4×D3—61no—
63C2×C3×C4×D3—61no—
64C2×C3×Dic6—21no—
65C2×C3×Dic6—21no—
66C2×C3×Dic6—41no—
67C2×C3×Dic6—62no—
68C2×#627—21no—
69C2×#625—21no—
70C2×#624C22×C312yes—
71#3802—22no—
72#3802C22×C314yes—
73#3797—22no—
74#3797C22×C314yescomplemented
75C4×#157C22×C314yescomplemented
76D32⋊C4C22×C3116yescomplemented
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3C2×D411yescharacteristic
79C2×C3×C3⋊D3#4114yes—
80C2×C3×C3⋊D3C2411yescharacteristic
81C2×C32×D3—24no—
82C2×C32×D3—44no—
83C3×D32—28no—
84C3×D32—44no—
85C3×#157—24no—
86C33×C4C2×D412yes—
87C3×C3⋊Dic3C2×D412yes—
88C32×Dic3—24no—
89C22×C3×Q8—91no—
90C22×C3×D4—91no—
91C2×D3×Q8—61no—
92C2×D3×D4—61no—
93C3×#124—98no—
94C2×C3×C22⋊C4—94no—
95C2×C3×C42—91no—
96C23×C32—62no—
97C22×C3⋊D3C23×C311yescharacteristic
98C22×C3×D3—22no—
99C22×C3×D3—22no—
100C22×C3×D3—42no—
101C22×C3×D3—64no—
102C22×C3×D3—61no—
103C22×C3×D3—61no—
104C22×C3×D3—64no—
105C22×C3×D3—128no—
106C2×D32—22no—
107C2×D32—28no—
108C2×D32C23×C314yes—
109C2×#157—24no—
110C2×#157C23×C314yes—
111C32×Q8—64no—
112C32×D4—64no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—41no—
116C2×C32×C4—41no—
117C2×C32×C4—62no—
118C2×C32×C4C3×D411yescharacteristic
119C2×C3⋊Dic3C3×D411yescharacteristic
120C4×C3⋊D3C3×D412yes—
121C4×C3⋊D3C23×C312yes—
122C3×#67—68no—
123C2×C3×Dic3—22no—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—42no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C3×D12—24no—
129C3×D12—24no—
130C3×D12—44no—
131C3×C4×D3—64no—
132C3×C4×D3—62no—
133C3×C4×D3—62no—
134C3×C4×D3—62no—
135C3×C4×D3—62no—
136C3×C4×D3—64no—
137C3×Dic6—24no—
138C3×Dic6—24no—
139C3×Dic6—44no—
140C3×Dic6—68no—
141#627—24no—
142#625—24no—
143#624—22no—
144#624C23×C314yes—
145C2×#124—91noSylow
146C2×C33#12412yescomplemented
147C2×C33C22×D411yescharacteristic
148C3×C3⋊D3#12412yescomplemented
149C3×C3⋊D3C2×#4112yes—
150C32×D3—44no—
151C24×C3—92no—
152C23×D3—62no—
153C2×C3×Q8—61no—
154C2×C3×Q8—61no—
155C2×C3×Q8—94no—
156C2×C3×Q8—121no—
157C2×C3×Q8—181no—
158C2×C3×Q8—182no—
159C2×C3×D4—61no—
160C2×C3×D4—61no—
161C2×C3×D4—94no—
162C2×C3×D4—121no—
163C2×C3×D4—182no—
164C2×C3×D4—181no—
165C22×C3×C4—92no—
166C22×C3×C4—91no—
167C22×C3×C4—92no—
168C2×#67—62no—
169D3×Q8—64no—
170D3×Q8—64no—
171D3×D4—64no—
172D3×D4—64no—
173C2×D12—21no—
174C2×C4×D3—62no—
175C2×C4×D3—61no—
176C2×C4×D3—61no—
177C2×Dic6—21no—
178C2×Dic6—62no—
179C3×C22⋊C4—916no—
180C3×C42—94no—
181C22×C32—21no—
182C22×C32—21no—
183C22×C32—41no—
184C22×C32—41no—
185C22×C32—64no—
186C22×C32—124no—
187C22×C32C2×C3×D411yescharacteristic
188C2×C3⋊D3C2×C3×D411yescharacteristic
189C2×C3⋊D3C3×#4114yes—
190C2×C3⋊D3C24×C311yescharacteristic
191C2×C3×D3—24no—
192C2×C3×D3—24no—
193C2×C3×D3—44no—
194C2×C3×D3—44no—
195C2×C3×D3—44no—
196C2×C3×D3—61no—
197C2×C3×D3—61no—
198C2×C3×D3—64no—
199C2×C3×D3—64no—
200C2×C3×D3—61no—
201C2×C3×D3—61no—
202C2×C3×D3—84no—
203C2×C3×D3—128no—
204C2×C3×D3—128no—
205D32—28no—
206D32—44no—
207#157—24no—
208C32×C4—22no—
209C32×C4—22no—
210C32×C4—42no—
211C32×C4—42no—
212C32×C4—64no—
213C32×C4C2×C3×D412yes—
214C3⋊Dic3C2×C3×D412yes—
215C3×Dic3—24no—
216C3×Dic3—24no—
217C3×Dic3—44no—
218C3×Dic3—62no—
219C3×Dic3—62no—
220C22×Q8—91no—
221C22×D4—91no—
222#124—98no—
223C2×C22⋊C4—94no—
224C2×C42—91no—
225C33C2×#12411yescharacteristic, complemented, Sylow
226C23×C3—62no—
227C23×C3—62no—
228C23×C3—94no—
229C23×C3—91no—
230C23×C3—122no—
231C23×C3—182no—
232C23×C3—188no—
233C22×D3—22no—
234C22×D3—64no—
235C22×D3—61no—
236C22×D3—61no—
237C22×D3—64no—
238C22×D3—128no—
239C3×Q8—64no—
240C3×Q8—64no—
241C3×Q8—124no—
242C3×Q8—184no—
243C3×D4—64no—
244C3×D4—64no—
245C3×D4—124no—
246C3×D4—184no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—62no—
252C2×C3×C4—62no—
253C2×C3×C4—94no—
254C2×C3×C4—91no—
255C2×C3×C4—92no—
256C2×C3×C4—92no—
257C2×C3×C4—94no—
258C2×C3×C4—122no—
259C2×C3×C4—182no—
260C2×C3×C4—184no—
261C2×C3×C4GO+(4,𝔽2)11yescharacteristic
262#67—68no—
263C2×Dic3—22no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266D12—24no—
267C4×D3—64no—
268C4×D3—62no—
269C4×D3—62no—
270C4×D3—62no—
271C4×D3—62no—
272C4×D3—64no—
273Dic6—24no—
274Dic6—68no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—22no—
278C2×C32—22no—
279C2×C32—42no—
280C2×C32—42no—
281C2×C32—41no—
282C2×C32—41no—
283C2×C32—124no—
284C2×C32C3×#12412yescomplemented
285C2×C32C22×C3×D411yescharacteristic
286C3⋊D3C3×#12412yescomplemented
287C3⋊D3C2×C3×#4112yes—
288C3×D3—44no—
289C3×D3—44no—
290C3×D3—62no—
291C3×D3—62no—
292C3×D3—62no—
293C3×D3—62no—
294C3×D3—84no—
295C24—92no—
296C2×Q8—61no—
297C2×Q8—94no—
298C2×Q8—181no—
299C2×Q8—182no—
300C2×D4—61no—
301C2×D4—94no—
302C2×D4—182no—
303C2×D4—181no—
304C22×C4—92no—
305C22×C4—91no—
306C22×C4—92no—
307C22⋊C4—916no—
308C42—94no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—64no—
314C22×C3—64no—
315C22×C3—91no—
316C22×C3—91no—
317C22×C3—94no—
318C22×C3—124no—
319C22×C3—124no—
320C22×C3—124no—
321C22×C3—188no—
322C22×C3—244no—
323C22×C3—364no—
324C22×C3C2×GO+(4,𝔽2)11yescharacteristic
325C2×D3—24no—
326C2×D3—44no—
327C2×D3—61no—
328C2×D3—61no—
329C2×D3—61no—
330C2×D3—61no—
331C2×D3—64no—
332C2×D3—64no—
333C2×D3—128no—
334C2×D3—128no—
335C3×C4—22no—
336C3×C4—22no—
337C3×C4—42no—
338C3×C4—42no—
339C3×C4—64no—
340C3×C4—64no—
341C3×C4—92no—
342C3×C4—124no—
343C3×C4—184no—
344C3×C4C2×GO+(4,𝔽2)12yes—
345Dic3—24no—
346Dic3—62no—
347Dic3—62no—
348C32—21no—
349C32—21no—
350C32—41no—
351C32—41no—
352C32C2×C3×#12411yescharacteristic, complemented
353C23—62no—
354C23—94no—
355C23—91no—
356C23—182no—
357C23—188no—
358Q8—64no—
359Q8—184no—
360D4—64no—
361D4—184no—
362C2×C4—62no—
363C2×C4—94no—
364C2×C4—91no—
365C2×C4—92no—
366C2×C4—92no—
367C2×C4—94no—
368C2×C4—182no—
369C2×C4—184no—
370C2×C4C3×GO+(4,𝔽2)11yescharacteristic
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—22no—
374C2×C3—22no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—42no—
378C2×C3—42no—
379C2×C3—92no—
380C2×C3—92no—
381C2×C3—124no—
382C2×C3—124no—
383C2×C3—244no—
384C2×C3#6422112yesdirect product factor
385C2×C3C22×GO+(4,𝔽2)11yescharacteristic
386D3—44no—
387D3—62no—
388D3—62no—
389D3—62no—
390D3—62no—
391C22—64no—
392C22—91no—
393C22—91no—
394C22—94no—
395C22—124no—
396C22—188no—
397C22—364no—
398C22C2×C3×GO+(4,𝔽2)11yescharacteristic
399C4—64no—
400C4—92no—
401C4—184no—
402C4C2×C3×GO+(4,𝔽2)12yes—
403C3—21no—
404C3—21no—
405C3—41no—
406C3—41no—
407C3C2×#6422111yescharacteristic, direct product factor
408C2—92no—
409C2—92no—
410C2—124no—
411C2C3×#6422112yesdirect product factor
412C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#3802C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C2×C3×#124
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3797C2semidirect
3. C2×C3×C4×#157C2semidirect
4. C2×C3×D32⋊C4C2semidirect
5. C3×#64221C2direct (N × H)
6. C2×#64221C3direct (N × H)
18. C3×#3797C22semidirect
19. C3×C4×#157C22semidirect
20. C3×D32⋊C4C22semidirect
24. C2×#3797C2×C3semidirect
25. C2×C4×#157C2×C3semidirect
26. C2×D32⋊C4C2×C3semidirect
27. #64221C2×C3direct (N × H)
74. #3797C22×C3semidirect
75. C4×#157C22×C3semidirect
76. D32⋊C4C22×C3semidirect
146. C2×C33#124semidirect
148. C3×C3⋊D3#124semidirect
225. C33C2×#124semidirect
284. C2×C32C3×#124semidirect
286. C3⋊D3C3×#124semidirect
352. C32C2×C3×#124semidirect
384. C2×C3#64221direct (N × H)
407. C3C2×#64221direct (N × H)
411. C2C3×#64221direct (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×#37972
C2×C3×C4×#1572
C2×C3×D32⋊C42
C3×#642212
C2×#642213
C2×C3×#1249