Finite Group Databasebeta

← C2×C3×#63739 #13166858

Subgroups — C2×C3×#63739 #13166858

Order 1728.

920 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C22×C3×#625C211yesmaximal, characteristic
2C22×C3×#622C211yesmaximal, characteristic, complemented
3C2×C3×#3724C211yesmaximal, characteristic, complemented
4C3×#63739C214yesmaximal, direct product factor
5C2×#63739C311yesmaximal, characteristic, direct product factor
6C23×C32×D3—21no—
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C22×C32×Dic3—21no—
9C2×C3×#625—21no—
10C2×C3×#625—22no—
11C2×C3×#625C414yescomplemented
12C2×C3×#625C2212yes—
13C2×C3×#622—22no—
14C2×C3×#622C2212yescomplemented
15C3×#3724—21no—
16C3×#3724C2212yescomplemented
17C22×C3×#67—61no—
18C22×#625C2×C311yescharacteristic
19C22×#622C2×C311yescharacteristic, complemented
20C2×C3×Dic3⋊C4—61no—
21C2×#3724C2×C311yescharacteristic, complemented
22#63739C2×C314yesdirect product factor
23C23×C33D411yescharacteristic
24C22×C32×D3—24no—
25C22×C32×D3—22no—
26C22×C32×D3—42no—
27C22×C32×D3—42no—
28C2×C3×C3⋊Dic3C2×C412yes—
29C2×C3×C3⋊Dic3D411yescharacteristic
30C2×C3×C3⋊Dic3D412yes—
31C2×C3×C3⋊Dic3C2311yescharacteristic
32C2×C32×Dic3—22no—
33C2×C32×Dic3—42no—
34C3×#625—24no—
35C3×#625—22no—
36C3×#625C2×C414yescomplemented
37C3×#622—22no—
38C2×C3×#102—91nomaximal
39C24×C32—61no—
40C23×C3×D3—21no—
41C23×C3×D3—21no—
42C23×C3×D3—41no—
43C22×C32×C4—61no—
44C22×C3⋊Dic3C22×C311yescharacteristic
45C2×C3×#67—64no—
46C2×C3×#67—62no—
47C2×C3×#67—62no—
48C2×C3×#67—64no—
49C22×C3×Dic3—21no—
50C22×C3×Dic3—21no—
51C22×C3×Dic3—41no—
52C22×C3×Dic3—61no—
53C22×C3×Dic3—61no—
54C2×#625—21no—
55C2×#625—22no—
56C2×#625C3×C414yescomplemented
57C2×#625C22×C312yes—
58C2×#622—22no—
59C2×#622C22×C312yescomplemented
60C3×Dic3⋊C4—62no—
61C3×Dic3⋊C4—62no—
62#3724—21no—
63#3724C22×C312yescomplemented
64C22×C33C22⋊C412yes—
65C22×C33D812yes—
66C22×C33SD1612yes—
67C22×C33C2×D411yescharacteristic
68C2×C32×D3—24no—
69C2×C32×D3—48no—
70C2×C32×D3—42no—
71C2×C32×D3—42no—
72C3×C3⋊Dic3C22⋊C412yes—
73C3×C3⋊Dic3C22×C411yescharacteristic
74C3×C3⋊Dic3C2×D411yescharacteristic
75C32×Dic3—42no—
76C22×C3×D4—91no—
77C22×#67—61no—
78C22×C3×C8—91no—
79C2×C3×C4⋊C4—91no—
80C2×Dic3⋊C4—61no—
81C3×#102—94no—
82C23×C32—21no—
83C23×C32—21no—
84C23×C32—41no—
85C23×C32—41no—
86C23×C32—62no—
87C23×C32—64no—
88C23×C32—122no—
89C23×C32—122no—
90C23×C32C3×D411yescharacteristic
91C22×C3×D3—22no—
92C22×C3×D3—24no—
93C22×C3×D3—24no—
94C22×C3×D3—22no—
95C22×C3×D3—44no—
96C22×C3×D3—42no—
97C22×C3×D3—42no—
98C22×C3×D3—42no—
99C22×C3×D3—44no—
100C22×C3×D3—44no—
101C22×C3×D3—42no—
102C22×C3×D3—42no—
103C2×C32×C4—62no—
104C2×C32×C4—122no—
105C2×C3⋊Dic3C2×C3×C412yes—
106C2×C3⋊Dic3C3×D411yescharacteristic
107C2×C3⋊Dic3C3×D412yes—
108C2×C3⋊Dic3C23×C311yescharacteristic
109C3×#67—64no—
110C3×#67—68no—
111C3×#67—64no—
112C2×C3×Dic3—22no—
113C2×C3×Dic3—22no—
114C2×C3×Dic3—42no—
115C2×C3×Dic3—42no—
116C2×C3×Dic3—44no—
117C2×C3×Dic3—42no—
118C2×C3×Dic3—61no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—61no—
126#625—24no—
127#625—22no—
128#625C2×C3×C414yescomplemented
129#622—22no—
130C2×#102—91noSylow
131C2×C33#10214yescomplemented
132C2×C33C2×C22⋊C411yescharacteristic
133C2×C33C2×D811yescharacteristic
134C2×C33C2×SD1611yescharacteristic
135C32×D3—44no—
136C24×C3—61no—
137C24×C3—61no—
138C24×C3—121no—
139C23×D3—21no—
140C2×C3×D4—92no—
141C2×C3×D4—94no—
142C2×C3×D4—181no—
143C2×C3×D4—182no—
144C22×C3×C4—61no—
145C22×C3×C4—61no—
146C22×C3×C4—91no—
147C22×C3×C4—121no—
148C2×#67—64no—
149C2×#67—62no—
150C2×#67—62no—
151C2×#67—64no—
152C22×Dic3—21no—
153C22×Dic3—61no—
154C22×Dic3—61no—
155C2×C3×C8—92no—
156C2×C3×C8—182no—
157C3×C4⋊C4—92no—
158C3×C4⋊C4—181no—
159Dic3⋊C4—62no—
160Dic3⋊C4—62no—
161C22×C32—21no—
162C22×C32—21no—
163C22×C32—22no—
164C22×C32—22no—
165C22×C32—22no—
166C22×C32—22no—
167C22×C32—22no—
168C22×C32—22no—
169C22×C32—42no—
170C22×C32—42no—
171C22×C32—42no—
172C22×C32—42no—
173C22×C32—42no—
174C22×C32—42no—
175C22×C32—41no—
176C22×C32—41no—
177C22×C32—64no—
178C22×C32—122no—
179C22×C32—122no—
180C22×C32—128no—
181C22×C32C3×C22⋊C412yes—
182C22×C32C3×D812yes—
183C22×C32C3×SD1612yes—
184C22×C32C2×C3×D411yescharacteristic
185C2×C3×D3—24no—
186C2×C3×D3—24no—
187C2×C3×D3—44no—
188C2×C3×D3—42no—
189C2×C3×D3—42no—
190C2×C3×D3—48no—
191C2×C3×D3—416no—
192C2×C3×D3—44no—
193C2×C3×D3—44no—
194C2×C3×D3—48no—
195C2×C3×D3—42no—
196C2×C3×D3—42no—
197C32×C4—122no—
198C3⋊Dic3C3×C22⋊C412yes—
199C3⋊Dic3C22×C3×C411yescharacteristic
200C3⋊Dic3C2×C3×D411yescharacteristic
201C3×Dic3—42no—
202C3×Dic3—44no—
203C3×Dic3—42no—
204C3×Dic3—62no—
205C3×Dic3—62no—
206C3×Dic3—61no—
207C3×Dic3—61no—
208C3×Dic3—61no—
209C3×Dic3—61no—
210C22×D4—91no—
211C22×C8—91no—
212C2×C4⋊C4—91no—
213#102—94no—
214C33C2×#10211yescharacteristic, complemented, Sylow
215C23×C3—21no—
216C23×C3—21no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—62no—
220C23×C3—64no—
221C23×C3—62no—
222C23×C3—64no—
223C23×C3—122no—
224C23×C3—124no—
225C23×C3—122no—
226C23×C3—122no—
227C23×C3—124no—
228C23×C3—124no—
229C23×C3—122no—
230C23×C3—122no—
231C23×C3GO+(4,𝔽2)11yescharacteristic
232C22×D3—22no—
233C22×D3—24no—
234C22×D3—42no—
235C22×D3—42no—
236C3×D4—94no—
237C3×D4—182no—
238C3×D4—184no—
239C2×C3×C4—62no—
240C2×C3×C4—62no—
241C2×C3×C4—91no—
242C2×C3×C4—91no—
243C2×C3×C4—92no—
244C2×C3×C4—92no—
245C2×C3×C4—122no—
246C2×C3×C4—122no—
247C2×C3×C4—124no—
248C2×C3×C4—122no—
249#67—64no—
250#67—68no—
251#67—64no—
252C2×Dic3—22no—
253C2×Dic3—42no—
254C2×Dic3—62no—
255C2×Dic3—62no—
256C2×Dic3—61no—
257C2×Dic3—62no—
258C2×Dic3—61no—
259C2×Dic3—62no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C3×C8—182no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—24no—
270C2×C32—24no—
271C2×C32—44no—
272C2×C32—44no—
273C2×C32—41no—
274C2×C32—41no—
275C2×C32—41no—
276C2×C32—41no—
277C2×C32—41no—
278C2×C32—41no—
279C2×C32—124no—
280C2×C32C3×#10214yescomplemented
281C2×C32C2×C3×C22⋊C411yescharacteristic
282C2×C32C2×C3×D811yescharacteristic
283C2×C32C2×C3×SD1611yescharacteristic
284C3×D3—44no—
285C3×D3—48no—
286C3×D3—44no—
287C24—61no—
288C2×D4—92no—
289C2×D4—94no—
290C2×D4—181no—
291C2×D4—182no—
292C22×C4—61no—
293C22×C4—91no—
294C2×C8—92no—
295C2×C8—182no—
296C4⋊C4—92no—
297C4⋊C4—181no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—22no—
301C22×C3—22no—
302C22×C3—22no—
303C22×C3—22no—
304C22×C3—22no—
305C22×C3—22no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—42no—
309C22×C3—42no—
310C22×C3—42no—
311C22×C3—42no—
312C22×C3—42no—
313C22×C3—42no—
314C22×C3—64no—
315C22×C3—64no—
316C22×C3—124no—
317C22×C3—122no—
318C22×C3—122no—
319C22×C3—128no—
320C22×C3—124no—
321C22×C3—124no—
322C22×C3—1216no—
323C22×C3—122no—
324C22×C3—122no—
325C22×C3—128no—
326C22×C3D32⋊C412yes—
327C22×C3#377412yes—
328C22×C3#377512yes—
329C22×C3C2×GO+(4,𝔽2)11yescharacteristic
330C2×D3—24no—
331C2×D3—42no—
332C2×D3—42no—
333C2×D3—48no—
334C3×C4—91no—
335C3×C4—91no—
336C3×C4—92no—
337C3×C4—122no—
338C3×C4—124no—
339C3×C4—122no—
340Dic3—42no—
341Dic3—62no—
342Dic3—61no—
343Dic3—61no—
344Dic3—62no—
345Dic3—61no—
346Dic3—61no—
347C32—21no—
348C32—21no—
349C32—41no—
350C32—41no—
351C32C2×C3×#10211yescharacteristic, complemented
352C23—62no—
353C23—64no—
354C23—122no—
355C23—122no—
356C23C3×GO+(4,𝔽2)11yescharacteristic
357D4—94no—
358D4—182no—
359D4—184no—
360C2×C4—62no—
361C2×C4—91no—
362C2×C4—91no—
363C2×C4—92no—
364C2×C4—92no—
365C2×C4—122no—
366C8—182no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—24no—
372C2×C3—24no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—41no—
378C2×C3—41no—
379C2×C3—44no—
380C2×C3—44no—
381C2×C3—41no—
382C2×C3—41no—
383C2×C3—124no—
384C2×C3—128no—
385C2×C3—124no—
386C2×C3#6373914yesdirect product factor
387C2×C3C2×D32⋊C411yescharacteristic
388C2×C3C2×#377411yescharacteristic
389C2×C3C2×#377511yescharacteristic
390D3—44no—
391C22—64no—
392C22—122no—
393C22—122no—
394C22—128no—
395C22C3×D32⋊C412yes—
396C22C3×#377412yes—
397C22C3×#377512yes—
398C22C2×C3×GO+(4,𝔽2)11yescharacteristic
399C4—91no—
400C4—91no—
401C4—92no—
402C4—122no—
403C3—21no—
404C3—21no—
405C3—41no—
406C3—41no—
407C3C2×#6373911yescharacteristic, direct product factor
408C2—124no—
409C2C3×#6373914yesdirect product factor
410C2C2×C3×D32⋊C411yescharacteristic
411C2C2×C3×#377411yescharacteristic
412C2C2×C3×#377511yescharacteristic
subgroupG/H
center, hypercenterC23×C3GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
Fitting, socleC23×C33D4
prime radicalC22×C3×#625C2
cosocleC2×C3⋊Dic3C23×C3
hypocenterC32C2×C3×#102
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C22×C3×#622C2semidirect
3. C2×C3×#3724C2semidirect
4. C3×#63739C2direct (N × H)
5. C2×#63739C3direct (N × H)
11. C2×C3×#625C4semidirect
14. C2×C3×#622C22semidirect
16. C3×#3724C22semidirect
19. C22×#622C2×C3semidirect
21. C2×#3724C2×C3semidirect
22. #63739C2×C3direct (N × H)
36. C3×#625C2×C4semidirect
56. C2×#625C3×C4semidirect
59. C2×#622C22×C3semidirect
63. #3724C22×C3semidirect
128. #625C2×C3×C4semidirect
131. C2×C33#102semidirect
214. C33C2×#102semidirect
280. C2×C32C3×#102semidirect
351. C32C2×C3×#102semidirect
386. C2×C3#63739direct (N × H)
407. C3C2×#63739direct (N × H)
409. C2C3×#63739direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6252
C22×C3×#6222
C2×C3×#37242
C3×#637392
C2×#637393
C2×C3×#1029