Finite Group Databasebeta

← C3×C4×D32⋊C4 #13166758

Subgroups — C3×C4×D32⋊C4 #13166758

Order 1728.

944 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 162 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×C4×D32C211yesmaximal, characteristic
2C2×C3×C4×#157C211yesmaximal, characteristic, complemented
3C2×C3×D32⋊C4C212yesmaximal
4C3×C4×#624C211yesmaximal, characteristic, complemented
5C3×#63737C212yesmaximal, complemented
6C4×D32⋊C4C311yesmaximal, characteristic, direct product factor
7C22×C3×D32C2211yescharacteristic
8C22×C3×#157C2212yes—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×C4×D3—21no—
11C2×C3×#624C2211yescharacteristic
12C2×C3×#624C2212yes—
13C2×C3×D3×Dic3—21no—
14C3×C4×D32—22no—
15C3×C4×D32C414yescomplemented
16C3×C4×#157—22no—
17C3×D32⋊C4C418yesdirect product factor
18C32×C4×Dic3—21no—
19C3×Dic32—21no—
20C22×C3×C4×D3—61no—
21C2×C4×D32C2×C311yescharacteristic
22C2×C4×#157C2×C311yescharacteristic, complemented
23C2×D32⋊C4C2×C312yes—
24C3×C42×D3—61no—
25C4×#624C2×C311yescharacteristic, complemented
26#63737C2×C312yescomplemented
27C22×C3×C3⋊D3C2311yescharacteristic
28C22×C32×D3—21no—
29C2×C3×D32—21no—
30C2×C3×D32—22no—
31C2×C3×D32C2×C414yes—
32C2×C3×D32C2×C412yes—
33C2×C3×#157—22no—
34C2×C3×#157—22no—
35C2×C3×#157C2×C414yescomplemented
36C2×C33×C4D411yescharacteristic
37C2×C3×C3⋊Dic3D411yescharacteristic
38C3×C4×C3⋊D3C2×C412yes—
39C3×C4×C3⋊D3D412yes—
40C2×C32×Dic3—21no—
41C2×C32×Dic3—22no—
42C32×C4×D3—24no—
43C3×#624—21no—
44C3×#624—22no—
45C3×#624C2×C412yes—
46C3×#624C2×C414yescomplemented
47C3×D3×Dic3—24no—
48C3×C4×C22⋊C4—91nomaximal
49C23×C3×D3—61no—
50C22×D32C22×C311yescharacteristic
51C22×#157C22×C312yes—
52C22×C32×C4—61no—
53C2×C4×C3⋊D3C22×C311yescharacteristic
54C22×C3×Dic3—61no—
55C2×C3×C4×D3—21no—
56C2×C3×C4×D3—21no—
57C2×C3×C4×D3—41no—
58C2×C3×C4×D3—64no—
59C2×C3×C4×D3—64no—
60C2×C3×C4×D3—61no—
61C2×C3×C4×D3—62no—
62C2×C3×C4×D3—61no—
63C2×C3×C4×D3—61no—
64C2×C3×C4×D3—61no—
65C2×#624C22×C311yescharacteristic
66C2×#624C22×C312yes—
67C2×D3×Dic3—21no—
68C4×D32—22no—
69C4×D32C3×C414yescomplemented
70C4×#157—22no—
71D32⋊C4C3×C418yesdirect product factor
72C32×C42—61no—
73C3×C4×Dic3—21no—
74C3×C4×Dic3—21no—
75C3×C4×Dic3—41no—
76C3×C4×Dic3—62no—
77Dic32—21no—
78C22×C33C2×D411yescharacteristic
79C2×C3×C3⋊D3C22×C411yescharacteristic
80C2×C3×C3⋊D3C22×C412yes—
81C2×C3×C3⋊D3C2×D411yescharacteristic
82C2×C3×C3⋊D3#4112yes—
83C2×C32×D3—22no—
84C2×C32×D3—24no—
85C3×D32—22no—
86C3×D32—24no—
87C3×D32C4214yescomplemented
88C3×#157—24no—
89C33×C4C22⋊C412yescomplemented
90C3×C3⋊Dic3C22⋊C412yescomplemented
91C32×Dic3—22no—
92C32×Dic3—24no—
93C23×C3×C4—91no—
94C22×C4×D3—61no—
95C2×C3×C22⋊C4—92no—
96C2×C3×C42—91no—
97C2×C3×C42—91no—
98C42×D3—61no—
99C3×#95—92no—
100C23×C32—61no—
101C22×C3⋊D3C23×C311yescharacteristic
102C22×C3×D3—21no—
103C22×C3×D3—21no—
104C22×C3×D3—41no—
105C22×C3×D3—64no—
106C22×C3×D3—62no—
107C22×C3×D3—61no—
108C22×C3×D3—61no—
109C22×C3×D3—64no—
110C22×C3×D3—62no—
111C2×D32—21no—
112C2×D32—22no—
113C2×D32C2×C3×C414yes—
114C2×D32C2×C3×C412yes—
115C2×#157—22no—
116C2×#157—22no—
117C2×#157C2×C3×C414yescomplemented
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—41no—
121C2×C32×C4—41no—
122C2×C32×C4—64no—
123C2×C32×C4—62no—
124C2×C32×C4—61no—
125C2×C32×C4C3×D411yescharacteristic
126C2×C3⋊Dic3C3×D411yescharacteristic
127C4×C3⋊D3C2×C3×C412yes—
128C4×C3⋊D3C3×D412yes—
129C2×C3×Dic3—22no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—22no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—42no—
134C2×C3×Dic3—41no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—64no—
138C3×C4×D3—24no—
139C3×C4×D3—24no—
140C3×C4×D3—44no—
141C3×C4×D3—62no—
142C3×C4×D3—64no—
143C3×C4×D3—62no—
144C3×C4×D3—62no—
145C3×C4×D3—62no—
146C3×C4×D3—62no—
147C3×C4×D3—64no—
148C3×C4×D3—62no—
149C3×C4×D3—64no—
150#624—21no—
151#624—22no—
152#624C2×C3×C412yes—
153#624C2×C3×C414yescomplemented
154D3×Dic3—24no—
155C4×C22⋊C4—91noSylow
156C2×C33C2×C22⋊C411yescharacteristic
157C2×C33C4×D412yes—
158C3×C3⋊D3C2×C4211yescharacteristic
159C3×C3⋊D3#11711yescharacteristic
160C3×C3⋊D3C4×D412yes—
161C32×D3—24no—
162C24×C3—91no—
163C23×D3—61no—
164C22×C3×C4—61no—
165C22×C3×C4—61no—
166C22×C3×C4—91no—
167C22×C3×C4—92no—
168C22×C3×C4—94no—
169C22×C3×C4—91no—
170C22×C3×C4—92no—
171C22×C3×C4—121no—
172C22×C3×C4—182no—
173C22×C3×C4—181no—
174C22×Dic3—61no—
175C2×C4×D3—21no—
176C2×C4×D3—64no—
177C2×C4×D3—62no—
178C2×C4×D3—61no—
179C2×C4×D3—61no—
180C2×C4×D3—61no—
181C2×C4×D3—64no—
182C2×C4×D3—61no—
183C3×C22⋊C4—98no—
184C3×C42—61no—
185C3×C42—61no—
186C3×C42—121no—
187C3×C42—181no—
188C3×C42—182no—
189C4×Dic3—21no—
190C4×Dic3—62no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—41no—
194C22×C32—41no—
195C22×C32—62no—
196C22×C32—64no—
197C22×C32C2×C3×D411yescharacteristic
198C2×C3⋊D3C22×C3×C411yescharacteristic
199C2×C3⋊D3C22×C3×C412yes—
200C2×C3⋊D3C2×C3×D411yescharacteristic
201C2×C3⋊D3C3×#4112yes—
202C2×C3×D3—22no—
203C2×C3×D3—24no—
204C2×C3×D3—22no—
205C2×C3×D3—24no—
206C2×C3×D3—42no—
207C2×C3×D3—44no—
208C2×C3×D3—64no—
209C2×C3×D3—61no—
210C2×C3×D3—62no—
211C2×C3×D3—61no—
212C2×C3×D3—62no—
213C2×C3×D3—61no—
214C2×C3×D3—61no—
215C2×C3×D3—62no—
216C2×C3×D3—62no—
217C2×C3×D3—64no—
218C2×C3×D3—68no—
219D32—22no—
220D32—24no—
221D32C3×C4214yescomplemented
222#157—24no—
223C32×C4—22no—
224C32×C4—22no—
225C32×C4—42no—
226C32×C4—42no—
227C32×C4—62no—
228C32×C4—64no—
229C32×C4C3×C22⋊C412yescomplemented
230C3⋊Dic3C3×C22⋊C412yescomplemented
231C3×Dic3—22no—
232C3×Dic3—24no—
233C3×Dic3—22no—
234C3×Dic3—24no—
235C3×Dic3—42no—
236C3×Dic3—44no—
237C3×Dic3—62no—
238C3×Dic3—62no—
239C23×C4—91no—
240C2×C22⋊C4—92no—
241C2×C42—91no—
242C2×C42—91no—
243#95—92no—
244C33C4×C22⋊C411yescharacteristic, complemented, Sylow
245C23×C3—61no—
246C23×C3—61no—
247C23×C3—94no—
248C23×C3—92no—
249C23×C3—91no—
250C23×C3—121no—
251C23×C3—181no—
252C23×C3—182no—
253C22×D3—21no—
254C22×D3—62no—
255C22×D3—64no—
256C22×D3—61no—
257C22×D3—61no—
258C22×D3—62no—
259C22×D3—64no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—64no—
265C2×C3×C4—62no—
266C2×C3×C4—61no—
267C2×C3×C4—62no—
268C2×C3×C4—61no—
269C2×C3×C4—64no—
270C2×C3×C4—94no—
271C2×C3×C4—92no—
272C2×C3×C4—91no—
273C2×C3×C4—92no—
274C2×C3×C4—92no—
275C2×C3×C4—94no—
276C2×C3×C4—124no—
277C2×C3×C4—122no—
278C2×C3×C4—121no—
279C2×C3×C4—182no—
280C2×C3×C4—181no—
281C2×C3×C4—184no—
282C2×C3×C4—182no—
283C2×C3×C4—182no—
284C2×C3×C4GO+(4,𝔽2)11yescharacteristic
285C2×Dic3—22no—
286C2×Dic3—21no—
287C2×Dic3—61no—
288C2×Dic3—61no—
289C2×Dic3—64no—
290C4×D3—24no—
291C4×D3—62no—
292C4×D3—64no—
293C4×D3—62no—
294C4×D3—62no—
295C4×D3—62no—
296C4×D3—62no—
297C4×D3—64no—
298C4×D3—62no—
299C4×D3—64no—
300C2×C32—22no—
301C2×C32—22no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32—41no—
305C2×C32—41no—
306C2×C32—42no—
307C2×C32—42no—
308C2×C32—64no—
309C2×C32C2×C3×C22⋊C411yescharacteristic
310C2×C32C3×C4×D412yes—
311C3⋊D3C2×C3×C4211yescharacteristic
312C3⋊D3C3×#11711yescharacteristic
313C3⋊D3C3×C4×D412yes—
314C3×D3—24no—
315C3×D3—24no—
316C3×D3—44no—
317C3×D3—61no—
318C3×D3—61no—
319C3×D3—61no—
320C3×D3—61no—
321C3×D3—62no—
322C3×D3—62no—
323C24—91no—
324C22×C4—61no—
325C22×C4—91no—
326C22×C4—92no—
327C22×C4—94no—
328C22×C4—91no—
329C22×C4—92no—
330C22×C4—182no—
331C22×C4—181no—
332C22⋊C4—98no—
333C42—61no—
334C42—181no—
335C42—182no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—62no—
341C22×C3—64no—
342C22×C3—64no—
343C22×C3—62no—
344C22×C3—94no—
345C22×C3—92no—
346C22×C3—92no—
347C22×C3—91no—
348C22×C3—91no—
349C22×C3—122no—
350C22×C3—124no—
351C22×C3—184no—
352C22×C3—182no—
353C22×C3C2×GO+(4,𝔽2)11yescharacteristic
354C2×D3—22no—
355C2×D3—24no—
356C2×D3—64no—
357C2×D3—61no—
358C2×D3—61no—
359C2×D3—62no—
360C2×D3—61no—
361C2×D3—61no—
362C2×D3—62no—
363C2×D3—62no—
364C2×D3—62no—
365C2×D3—64no—
366C2×D3—68no—
367C3×C4—22no—
368C3×C4—22no—
369C3×C4—42no—
370C3×C4—42no—
371C3×C4—62no—
372C3×C4—64no—
373C3×C4—64no—
374C3×C4—62no—
375C3×C4—92no—
376C3×C4—122no—
377C3×C4—124no—
378C3×C4—184no—
379C3×C4D32⋊C412yesdirect product factor
380Dic3—22no—
381Dic3—24no—
382Dic3—62no—
383Dic3—62no—
384C32—21no—
385C32—21no—
386C32—41no—
387C32—41no—
388C32C3×C4×C22⋊C411yescharacteristic, complemented
389C23—61no—
390C23—94no—
391C23—92no—
392C23—91no—
393C23—181no—
394C23—182no—
395C2×C4—62no—
396C2×C4—61no—
397C2×C4—64no—
398C2×C4—94no—
399C2×C4—92no—
400C2×C4—91no—
401C2×C4—92no—
402C2×C4—92no—
403C2×C4—94no—
404C2×C4—181no—
405C2×C4—182no—
406C2×C4—184no—
407C2×C4—182no—
408C2×C4—182no—
409C2×C4C3×GO+(4,𝔽2)11yescharacteristic
410C2×C3—22no—
411C2×C3—22no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—42no—
415C2×C3—42no—
416C2×C3—41no—
417C2×C3—41no—
418C2×C3—64no—
419C2×C3—64no—
420C2×C3—92no—
421C2×C3—91no—
422C2×C3—91no—
423C2×C3—124no—
424C2×C3C4×GO+(4,𝔽2)12yes—
425C2×C3C2×D32⋊C411yescharacteristic
426D3—24no—
427D3—61no—
428D3—61no—
429D3—62no—
430D3—61no—
431D3—61no—
432D3—62no—
433C22—64no—
434C22—62no—
435C22—94no—
436C22—92no—
437C22—92no—
438C22—91no—
439C22—91no—
440C22—184no—
441C22—182no—
442C22C2×C3×GO+(4,𝔽2)11yescharacteristic
443C4—64no—
444C4—62no—
445C4—92no—
446C4—184no—
447C4C3×D32⋊C412yesdirect product factor
448C3—21no—
449C3—21no—
450C3—41no—
451C3—41no—
452C3C4×D32⋊C411yescharacteristic, direct product factor
453C2—64no—
454C2—92no—
455C2—91no—
456C2—91no—
457C2C3×C4×GO+(4,𝔽2)12yes—
458C2C2×C3×D32⋊C411yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4GO+(4,𝔽2)
commutatorC3⋊D3C2×C3×C42
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×D32C22
cosocleC22×C3⋊D3C23×C3
hypocenterC32C3×C4×C22⋊C4
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×#157C2semidirect
4. C3×C4×#624C2semidirect
5. C3×#63737C2semidirect
6. C4×D32⋊C4C3direct (N × H)
15. C3×C4×D32C4semidirect
17. C3×D32⋊C4C4direct (N × H)
22. C2×C4×#157C2×C3semidirect
25. C4×#624C2×C3semidirect
26. #63737C2×C3semidirect
35. C2×C3×#157C2×C4semidirect
46. C3×#624C2×C4semidirect
69. C4×D32C3×C4semidirect
71. D32⋊C4C3×C4direct (N × H)
87. C3×D32C42semidirect
89. C33×C4C22⋊C4semidirect
90. C3×C3⋊Dic3C22⋊C4semidirect
117. C2×#157C2×C3×C4semidirect
153. #624C2×C3×C4semidirect
221. D32C3×C42semidirect
229. C32×C4C3×C22⋊C4semidirect
230. C3⋊Dic3C3×C22⋊C4semidirect
244. C33C4×C22⋊C4semidirect
379. C3×C4D32⋊C4direct (N × H)
388. C32C3×C4×C22⋊C4semidirect
447. C4C3×D32⋊C4direct (N × H)
452. C3C4×D32⋊C4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×D322
C2×C3×C4×#1572
C2×C3×D32⋊C42
C3×C4×#6242
C3×#637372
C4×D32⋊C43
C3×C4×C22⋊C49