Finite Group Databasebeta

← C2×C3×#63729 #13166710

Subgroups — C2×C3×#63729 #13166710

Order 1728.

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

normal subgroups only

This group has 792 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×#3802C211yesmaximal, characteristic
2C2×C3×C4⋊#157C211yesmaximal, characteristic, complemented
3C2×C3×#3710C211yesmaximal, characteristic, complemented
4C3×#63729C214yesmaximal, direct product factor
5C2×#63729C311yesmaximal, characteristic, direct product factor
6C22×C3×D32—21no—
7C22×C3×#157—21no—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×D12—21no—
10C2×C3×#625C412yescomplemented
11C3×#3802—21no—
12C3×#3802—21no—
13C3×#3802C412yescomplemented
14C3×#3802C2212yes—
15C3×C4⋊#157—21no—
16C3×C4⋊#157C2212yescomplemented
17C2×C32×C3⋊C8—21no—
18C3×#3710—21no—
19C3×#3710C2212yescomplemented
20C2×C3×D3×D4—61no—
21C2×#3802C2×C311yescharacteristic
22C2×C4⋊#157C2×C311yescharacteristic, complemented
23C2×C3×D3×C8—61no—
24C2×#3710C2×C311yescharacteristic, complemented
25#63729C2×C314yesdirect product factor
26C22×C3×C3⋊D3D411yescharacteristic
27C22×C32×D3—22no—
28C2×C3×D32—22no—
29C2×C3×D32—22no—
30C2×C3×D32—42no—
31C2×C3×D32—42no—
32C2×C3×#157—22no—
33C2×C3×#157—42no—
34C2×C33×C4D411yescharacteristic
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C3×C4×C3⋊D3C2×C411yescharacteristic
37C3×C4×C3⋊D3D411yescharacteristic
38C3×C4×C3⋊D3D411yescharacteristic
39C3×C4×C3⋊D3C2311yescharacteristic
40C32×D12—22no—
41C32×D12—22no—
42C3×#625—22no—
43C3×#625C2×C414yescomplemented
44C32×C3⋊C8—22no—
45C2×C3×#102—91nomaximal
46C23×C3×D3—62no—
47C22×D32—21no—
48C22×#157—21no—
49C2×C32×D4—61no—
50C2×C4×C3⋊D3C22×C311yescharacteristic
51C2×C3×#67—62no—
52C3×D3×D4—62no—
53C3×D3×D4—62no—
54C3×D3×D4—62no—
55C3×D3×D4—62no—
56C2×C3×D12—21no—
57C2×C3×D12—21no—
58C2×C3×D12—41no—
59C2×C3×C4×D3—61no—
60C2×C3×C4×D3—61no—
61C2×#625C3×C412yescomplemented
62#3802—21no—
63#3802—21no—
64#3802C3×C412yescomplemented
65#3802C22×C312yes—
66C4⋊#157—21no—
67C4⋊#157C22×C312yescomplemented
68C2×C32×C8—61no—
69C2×C3×C3⋊C8—21no—
70C2×C3×C3⋊C8—21no—
71C2×C3×C3⋊C8—41no—
72C3×D3×C8—62no—
73C3×D3×C8—62no—
74#3710—21no—
75#3710C22×C312yescomplemented
76C22×C33C22⋊C411yescharacteristic
77C2×C3×C3⋊D3C22⋊C411yescharacteristic
78C2×C3×C3⋊D3D812yes—
79C2×C3×C3⋊D3SD1612yes—
80C2×C3×C3⋊D3C2×D411yescharacteristic
81C2×C32×D3—24no—
82C2×C32×D3—44no—
83C3×D32—44no—
84C3×D32—42no—
85C3×D32—42no—
86C3×#157—42no—
87C33×C4C22⋊C411yescharacteristic
88C33×C4C2×D411yescharacteristic
89C3×C3⋊Dic3C22⋊C411yescharacteristic
90C3×C3⋊Dic3C22×C411yescharacteristic
91C22×C3×D4—91no—
92C2×D3×D4—61no—
93C22×C3×C8—91no—
94C2×C3×C4⋊C4—91no—
95C2×D3×C8—61no—
96C3×#102—94no—
97C23×C32—62no—
98C22×C3⋊D3C3×D411yescharacteristic
99C22×C3×D3—22no—
100C22×C3×D3—22no—
101C22×C3×D3—42no—
102C22×C3×D3—61no—
103C22×C3×D3—61no—
104C22×C3×D3—64no—
105C22×C3×D3—64no—
106C22×C3×D3—124no—
107C22×C3×D3—124no—
108C2×D32—22no—
109C2×D32—22no—
110C2×D32—42no—
111C2×D32—42no—
112C2×#157—22no—
113C2×#157—42no—
114C32×D4—62no—
115C32×D4—62no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—41no—
119C2×C32×C4—41no—
120C2×C32×C4C3×D411yescharacteristic
121C2×C3⋊Dic3C2×C3×C411yescharacteristic
122C4×C3⋊D3C2×C3×C411yescharacteristic
123C4×C3⋊D3C3×D411yescharacteristic
124C4×C3⋊D3C3×D411yescharacteristic
125C4×C3⋊D3C23×C311yescharacteristic
126C3×#67—64no—
127C3×#67—64no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C3×D12—22no—
131C3×D12—22no—
132C3×D12—22no—
133C3×D12—22no—
134C3×D12—42no—
135C3×D12—42no—
136C3×C4×D3—61no—
137C3×C4×D3—61no—
138C3×C4×D3—61no—
139C3×C4×D3—61no—
140C3×C4×D3—61no—
141C3×C4×D3—61no—
142C3×C4×D3—61no—
143C3×C4×D3—61no—
144#625—22no—
145#625C2×C3×C414yescomplemented
146C32×C8—62no—
147C3×C3⋊C8—22no—
148C3×C3⋊C8—22no—
149C3×C3⋊C8—42no—
150C2×#102—91noSylow
151C2×C33#10212yescomplemented
152C2×C33C2×C22⋊C411yescharacteristic
153C3×C3⋊D3#10212yescomplemented
154C3×C3⋊D3C2×D811yescharacteristic
155C3×C3⋊D3C2×SD1611yescharacteristic
156C32×D3—44no—
157C24×C3—181no—
158C23×D3—62no—
159C2×C3×D4—61no—
160C2×C3×D4—61no—
161C2×C3×D4—92no—
162C2×C3×D4—92no—
163C2×C3×D4—92no—
164C2×C3×D4—121no—
165C2×C3×D4—181no—
166C2×C3×D4—181no—
167C22×C3×C4—91no—
168C22×C3×C4—181no—
169C2×#67—62no—
170D3×D4—62no—
171D3×D4—62no—
172D3×D4—62no—
173D3×D4—62no—
174C2×D12—21no—
175C2×C4×D3—61no—
176C2×C4×D3—61no—
177C2×C3×C8—61no—
178C2×C3×C8—61no—
179C2×C3×C8—92no—
180C2×C3×C8—121no—
181C2×C3×C8—181no—
182C3×C4⋊C4—92no—
183C3×C4⋊C4—181no—
184C2×C3⋊C8—21no—
185D3×C8—62no—
186D3×C8—62no—
187C22×C32—21no—
188C22×C32—21no—
189C22×C32—41no—
190C22×C32—41no—
191C22×C32—64no—
192C22×C32—124no—
193C22×C32C3×C22⋊C411yescharacteristic
194C2×C3⋊D3C3×C22⋊C411yescharacteristic
195C2×C3⋊D3C3×D812yes—
196C2×C3⋊D3C3×SD1612yes—
197C2×C3⋊D3C2×C3×D411yescharacteristic
198C2×C3×D3—24no—
199C2×C3×D3—24no—
200C2×C3×D3—44no—
201C2×C3×D3—44no—
202C2×C3×D3—44no—
203C2×C3×D3—61no—
204C2×C3×D3—61no—
205C2×C3×D3—62no—
206C2×C3×D3—62no—
207C2×C3×D3—62no—
208C2×C3×D3—62no—
209C2×C3×D3—61no—
210C2×C3×D3—61no—
211C2×C3×D3—84no—
212C2×C3×D3—128no—
213C2×C3×D3—124no—
214C2×C3×D3—124no—
215D32—44no—
216D32—42no—
217D32—42no—
218#157—42no—
219C32×C4—21no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4—41no—
226C32×C4—41no—
227C32×C4C3×C22⋊C411yescharacteristic
228C32×C4C2×C3×D411yescharacteristic
229C3⋊Dic3C3×C22⋊C411yescharacteristic
230C3⋊Dic3C22×C3×C411yescharacteristic
231C3×Dic3—61no—
232C3×Dic3—61no—
233C3×Dic3—61no—
234C3×Dic3—61no—
235C22×D4—91no—
236C22×C8—91no—
237C2×C4⋊C4—91no—
238#102—94no—
239C33C2×#10211yescharacteristic, complemented, Sylow
240C23×C3—62no—
241C23×C3—62no—
242C23×C3—91no—
243C23×C3—122no—
244C23×C3—182no—
245C23×C3—182no—
246C23×C3—362no—
247C23×C3—362no—
248C22×D3—22no—
249C22×D3—61no—
250C22×D3—61no—
251C22×D3—64no—
252C22×D3—64no—
253C22×D3—124no—
254C22×D3—124no—
255C3×D4—62no—
256C3×D4—62no—
257C3×D4—62no—
258C3×D4—62no—
259C3×D4—94no—
260C3×D4—122no—
261C3×D4—122no—
262C3×D4—182no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—91no—
268C2×C3×C4—91no—
269C2×C3×C4—91no—
270C2×C3×C4—91no—
271C2×C3×C4—91no—
272C2×C3×C4—182no—
273C2×C3×C4—362no—
274C2×C3×C4GO+(4,𝔽2)11yescharacteristic
275#67—64no—
276#67—64no—
277C2×Dic3—61no—
278C2×Dic3—61no—
279D12—22no—
280D12—22no—
281C4×D3—61no—
282C4×D3—61no—
283C4×D3—61no—
284C4×D3—61no—
285C4×D3—61no—
286C4×D3—61no—
287C4×D3—61no—
288C4×D3—61no—
289C3×C8—62no—
290C3×C8—62no—
291C3×C8—122no—
292C3⋊C8—22no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32—22no—
296C2×C32—22no—
297C2×C32—42no—
298C2×C32—42no—
299C2×C32—41no—
300C2×C32—41no—
301C2×C32—124no—
302C2×C32C3×#10212yescomplemented
303C2×C32C2×C3×C22⋊C411yescharacteristic
304C3⋊D3C3×#10212yescomplemented
305C3⋊D3C2×C3×D811yescharacteristic
306C3⋊D3C2×C3×SD1611yescharacteristic
307C3×D3—44no—
308C3×D3—44no—
309C3×D3—62no—
310C3×D3—62no—
311C3×D3—61no—
312C3×D3—61no—
313C3×D3—61no—
314C3×D3—61no—
315C3×D3—84no—
316C24—181no—
317C2×D4—61no—
318C2×D4—92no—
319C2×D4—92no—
320C2×D4—92no—
321C2×D4—181no—
322C2×D4—181no—
323C22×C4—91no—
324C22×C4—181no—
325C2×C8—61no—
326C2×C8—92no—
327C2×C8—181no—
328C4⋊C4—92no—
329C4⋊C4—181no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—64no—
335C22×C3—64no—
336C22×C3—91no—
337C22×C3—91no—
338C22×C3—92no—
339C22×C3—92no—
340C22×C3—124no—
341C22×C3—124no—
342C22×C3—124no—
343C22×C3—244no—
344C22×C3—362no—
345C22×C3—362no—
346C22×C3—364no—
347C22×C3D32⋊C411yescharacteristic
348C2×D3—24no—
349C2×D3—44no—
350C2×D3—61no—
351C2×D3—61no—
352C2×D3—61no—
353C2×D3—61no—
354C2×D3—62no—
355C2×D3—62no—
356C2×D3—62no—
357C2×D3—62no—
358C2×D3—128no—
359C2×D3—124no—
360C2×D3—124no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—41no—
366C3×C4—41no—
367C3×C4—41no—
368C3×C4—41no—
369C3×C4—91no—
370C3×C4—91no—
371C3×C4—362no—
372C3×C4D32⋊C411yescharacteristic
373C3×C4C2×GO+(4,𝔽2)11yescharacteristic
374Dic3—61no—
375Dic3—61no—
376Dic3—61no—
377Dic3—61no—
378C32—21no—
379C32—21no—
380C32—41no—
381C32—41no—
382C32C2×C3×#10211yescharacteristic, complemented
383C23—62no—
384C23—91no—
385C23—182no—
386C23—182no—
387C23—362no—
388C23—362no—
389D4—62no—
390D4—62no—
391D4—94no—
392D4—182no—
393C2×C4—91no—
394C2×C4—91no—
395C2×C4—91no—
396C2×C4—91no—
397C2×C4—91no—
398C2×C4—182no—
399C2×C4—362no—
400C2×C4C3×GO+(4,𝔽2)11yescharacteristic
401C8—62no—
402C2×C3—22no—
403C2×C3—22no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—42no—
407C2×C3—42no—
408C2×C3—41no—
409C2×C3—41no—
410C2×C3—91no—
411C2×C3—91no—
412C2×C3—92no—
413C2×C3—124no—
414C2×C3—124no—
415C2×C3—244no—
416C2×C3#6372912yesdirect product factor
417C2×C3C2×D32⋊C411yescharacteristic
418D3—44no—
419D3—62no—
420D3—61no—
421D3—61no—
422D3—62no—
423D3—61no—
424D3—61no—
425C22—64no—
426C22—92no—
427C22—92no—
428C22—91no—
429C22—91no—
430C22—124no—
431C22—362no—
432C22—364no—
433C22—362no—
434C22C3×D32⋊C411yescharacteristic
435C4—91no—
436C4—91no—
437C4—362no—
438C4C3×D32⋊C411yescharacteristic
439C4C2×C3×GO+(4,𝔽2)11yescharacteristic
440C3—21no—
441C3—21no—
442C3—41no—
443C3—41no—
444C3C2×#6372911yescharacteristic, direct product factor
445C2—91no—
446C2—92no—
447C2—91no—
448C2—124no—
449C2C3×#6372912yesdirect product factor
450C2C2×C3×D32⋊C411yescharacteristic
subgroupG/H
centerC22×C3D32⋊C4
commutatorC3⋊Dic3C22×C3×C4
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C22⋊C4
prime radicalC2×C3×#3802C2
cosocleC4×C3⋊D3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C2×C3×#102
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×C4⋊#157C2semidirect
3. C2×C3×#3710C2semidirect
4. C3×#63729C2direct (N × H)
5. C2×#63729C3direct (N × H)
10. C2×C3×#625C4semidirect
13. C3×#3802C4semidirect
16. C3×C4⋊#157C22semidirect
19. C3×#3710C22semidirect
22. C2×C4⋊#157C2×C3semidirect
24. C2×#3710C2×C3semidirect
25. #63729C2×C3direct (N × H)
43. C3×#625C2×C4semidirect
61. C2×#625C3×C4semidirect
64. #3802C3×C4semidirect
67. C4⋊#157C22×C3semidirect
75. #3710C22×C3semidirect
145. #625C2×C3×C4semidirect
151. C2×C33#102semidirect
153. C3×C3⋊D3#102semidirect
239. C33C2×#102semidirect
302. C2×C32C3×#102semidirect
304. C3⋊D3C3×#102semidirect
382. C32C2×C3×#102semidirect
416. C2×C3#63729direct (N × H)
444. C3C2×#63729direct (N × H)
449. C2C3×#63729direct (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×C4⋊#1572
C2×C3×#37102
C3×#637292
C2×#637293
C2×C3×#1029