Finite Group Databasebeta

← C22×C3×D32⋊C4 #13181045

Subgroups — C22×C3×D32⋊C4 #13181045

Order 1728.

2520 conjugacy classes of subgroups; only the 370 normal subgroups are shown.

normal subgroups only

This group has 2520 conjugacy classes of subgroups — too many to draw. Only its 370 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
1C23×C3×D32C211yesmaximal, characteristic
2C23×C3×#157C211yesmaximal, characteristic, complemented
3C22×C3×#624C211yesmaximal, characteristic, complemented
4C2×C3×D32⋊C4C2112yesmaximal, direct product factor
5C22×D32⋊C4C311yesmaximal, characteristic, direct product factor
6C23×C3×C3⋊D3C2211yescharacteristic
7C23×C32×D3—21no—
8C22×C3×D32—23no—
9C22×C3×D32—24no—
10C22×C3×D32C418yescomplemented
11C22×C3×D32C2216yes—
12C22×C3×#157—24no—
13C22×C3×#157C2216yescomplemented
14C22×C32×Dic3—21no—
15C2×C3×#624—23no—
16C2×C3×#624C2216yescomplemented
17C3×D32⋊C4C22116yesdirect product factor
18C24×C3×D3—61no—
19C23×D32C2×C311yescharacteristic
20C23×#157C2×C311yescharacteristic, complemented
21C22×C3×C4×D3—61no—
22C22×#624C2×C311yescharacteristic, complemented
23C2×D32⋊C4C2×C3112yesdirect product factor
24C23×C33D411yescharacteristic
25C22×C3×C3⋊D3C2×C414yes—
26C22×C3×C3⋊D3D413yes—
27C22×C3×C3⋊D3D414yes—
28C22×C3×C3⋊D3C2313yes—
29C22×C32×D3—28no—
30C22×C32×D3—26no—
31C2×C3×D32—224no—
32C2×C3×D32—26no—
33C2×C3×D32—212no—
34C2×C3×D32C2×C4124yescomplemented
35C2×C3×D32C2314yes—
36C2×C3×#157—212no—
37C2×C3×#157C2314yescomplemented
38C2×C32×Dic3—26no—
39C3×#624—26no—
40C3×#624C2314yescomplemented
41C22×C3×C22⋊C4—91nomaximal
42C24×C32—61no—
43C23×C3⋊D3C22×C311yescharacteristic
44C23×C3×D3—21no—
45C23×C3×D3—21no—
46C23×C3×D3—41no—
47C23×C3×D3—68no—
48C23×C3×D3—66no—
49C23×C3×D3—66no—
50C23×C3×D3—68no—
51C23×C3×D3—61no—
52C23×C3×D3—61no—
53C22×D32—23no—
54C22×D32—24no—
55C22×D32C3×C418yescomplemented
56C22×D32C22×C316yes—
57C22×#157—24no—
58C22×#157C22×C316yescomplemented
59C22×C32×C4—61no—
60C22×C3×Dic3—21no—
61C22×C3×Dic3—21no—
62C22×C3×Dic3—41no—
63C2×C3×C4×D3—66no—
64C2×C3×C4×D3—66no—
65C2×#624—23no—
66C2×#624C22×C316yescomplemented
67D32⋊C4C22×C3116yesdirect product factor
68C22×C33C22⋊C414yescomplemented
69C22×C33C2×D413yes—
70C2×C3×C3⋊D3C22⋊C4112yescomplemented
71C2×C3×C3⋊D3C22×C416yes—
72C2×C3×C3⋊D3C2×D413yes—
73C2×C3×C3⋊D3C2×D416yes—
74C2×C3×C3⋊D3C2411yescharacteristic
75C2×C32×D3—224no—
76C2×C32×D3—24no—
77C3×D32—224no—
78C3×D32—24no—
79C3×D32C22×C418yescomplemented
80C3×#157—24no—
81C32×Dic3—24no—
82C25×C3—91no—
83C24×D3—61no—
84C23×C3×C4—91no—
85C23×C3×C4—91no—
86C22×C4×D3—61no—
87C2×C3×C22⋊C4—912no—
88C23×C32—21no—
89C23×C32—21no—
90C23×C32—41no—
91C23×C32—41no—
92C23×C32—66no—
93C23×C32—68no—
94C23×C32C3×D411yescharacteristic
95C22×C3⋊D3C2×C3×C414yes—
96C22×C3⋊D3C3×D413yes—
97C22×C3⋊D3C3×D414yes—
98C22×C3⋊D3C23×C313yes—
99C22×C3×D3—26no—
100C22×C3×D3—28no—
101C22×C3×D3—28no—
102C22×C3×D3—26no—
103C22×C3×D3—48no—
104C22×C3×D3—46no—
105C22×C3×D3—624no—
106C22×C3×D3—64no—
107C22×C3×D3—64no—
108C22×C3×D3—63no—
109C22×C3×D3—64no—
110C22×C3×D3—63no—
111C22×C3×D3—64no—
112C22×C3×D3—64no—
113C22×C3×D3—63no—
114C22×C3×D3—63no—
115C22×C3×D3—648no—
116C22×C3×D3—612no—
117C22×C3×D3—624no—
118C2×D32—224no—
119C2×D32—26no—
120C2×D32—212no—
121C2×D32C2×C3×C4124yescomplemented
122C2×D32C23×C314yes—
123C2×#157—212no—
124C2×#157C23×C314yescomplemented
125C2×C32×C4—66no—
126C2×C3×Dic3—26no—
127C2×C3×Dic3—26no—
128C2×C3×Dic3—46no—
129C3×C4×D3—64no—
130C3×C4×D3—612no—
131#624—26no—
132#624C23×C314yescomplemented
133C22×C22⋊C4—91noSylow
134C2×C33C2×C22⋊C416yescomplemented
135C2×C33C22×D411yescharacteristic
136C3×C3⋊D3C2×C22⋊C416yescomplemented
137C3×C3⋊D3C23×C411yescharacteristic
138C3×C3⋊D3C22×D411yescharacteristic
139C32×D3—28no—
140C24×C3—61no—
141C24×C3—61no—
142C24×C3—98no—
143C24×C3—96no—
144C24×C3—91no—
145C24×C3—121no—
146C24×C3—183no—
147C24×C3—184no—
148C23×D3—21no—
149C23×D3—68no—
150C23×D3—66no—
151C23×D3—61no—
152C23×D3—61no—
153C23×D3—66no—
154C23×D3—68no—
155C22×C3×C4—61no—
156C22×C3×C4—61no—
157C22×C3×C4—96no—
158C22×C3×C4—96no—
159C22×C3×C4—121no—
160C22×C3×C4—183no—
161C22×C3×C4—184no—
162C22×Dic3—21no—
163C2×C4×D3—66no—
164C2×C4×D3—66no—
165C3×C22⋊C4—916no—
166C22×C32—23no—
167C22×C32—23no—
168C22×C32—24no—
169C22×C32—24no—
170C22×C32—44no—
171C22×C32—44no—
172C22×C32—43no—
173C22×C32—43no—
174C22×C32—64no—
175C22×C32—624no—
176C22×C32C3×C22⋊C414yescomplemented
177C22×C32C2×C3×D413yes—
178C2×C3⋊D3C3×C22⋊C4112yescomplemented
179C2×C3⋊D3C22×C3×C416yes—
180C2×C3⋊D3C2×C3×D413yes—
181C2×C3⋊D3C2×C3×D416yes—
182C2×C3⋊D3C24×C311yescharacteristic
183C2×C3×D3—24no—
184C2×C3×D3—224no—
185C2×C3×D3—224no—
186C2×C3×D3—24no—
187C2×C3×D3—424no—
188C2×C3×D3—44no—
189C2×C3×D3—68no—
190C2×C3×D3—612no—
191C2×C3×D3—612no—
192C2×C3×D3—66no—
193C2×C3×D3—66no—
194C2×C3×D3—63no—
195C2×C3×D3—66no—
196C2×C3×D3—63no—
197C2×C3×D3—66no—
198C2×C3×D3—61no—
199C2×C3×D3—61no—
200C2×C3×D3—648no—
201C2×C3×D3—68no—
202D32—224no—
203D32—24no—
204D32C22×C3×C418yescomplemented
205#157—24no—
206C32×C4—64no—
207C3×Dic3—24no—
208C3×Dic3—24no—
209C3×Dic3—44no—
210C25—91no—
211C23×C4—91no—
212C23×C4—91no—
213C2×C22⋊C4—912no—
214C33C22×C22⋊C411yescharacteristic, complemented, Sylow
215C23×C3—21no—
216C23×C3—21no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—66no—
220C23×C3—68no—
221C23×C3—66no—
222C23×C3—68no—
223C23×C3—94no—
224C23×C3—924no—
225C23×C3—93no—
226C23×C3—93no—
227C23×C3—94no—
228C23×C3—94no—
229C23×C3—126no—
230C23×C3—128no—
231C23×C3—186no—
232C23×C3—1812no—
233C23×C3—1824no—
234C23×C3GO+(4,𝔽2)11yescharacteristic
235C22×D3—26no—
236C22×D3—28no—
237C22×D3—624no—
238C22×D3—64no—
239C22×D3—64no—
240C22×D3—64no—
241C22×D3—63no—
242C22×D3—64no—
243C22×D3—63no—
244C22×D3—64no—
245C22×D3—63no—
246C22×D3—63no—
247C22×D3—648no—
248C22×D3—612no—
249C22×D3—624no—
250C2×C3×C4—66no—
251C2×C3×C4—66no—
252C2×C3×C4—94no—
253C2×C3×C4—94no—
254C2×C3×C4—126no—
255C2×C3×C4—186no—
256C2×C3×C4—1812no—
257C2×Dic3—26no—
258C4×D3—64no—
259C4×D3—612no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—26no—
263C2×C32—26no—
264C2×C32—46no—
265C2×C32—46no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—68no—
269C2×C32C2×C3×C22⋊C416yescomplemented
270C2×C32C22×C3×D411yescharacteristic
271C3⋊D3C2×C3×C22⋊C416yescomplemented
272C3⋊D3C23×C3×C411yescharacteristic
273C3⋊D3C22×C3×D411yescharacteristic
274C3×D3—28no—
275C3×D3—28no—
276C3×D3—48no—
277C3×D3—66no—
278C3×D3—66no—
279C3×D3—61no—
280C3×D3—61no—
281C3×D3—61no—
282C3×D3—61no—
283C24—61no—
284C24—96no—
285C24—98no—
286C24—91no—
287C24—183no—
288C24—184no—
289C22×C4—61no—
290C22×C4—96no—
291C22×C4—96no—
292C22×C4—183no—
293C22×C4—184no—
294C22⋊C4—916no—
295C22×C3—23no—
296C22×C3—23no—
297C22×C3—24no—
298C22×C3—24no—
299C22×C3—43no—
300C22×C3—43no—
301C22×C3—44no—
302C22×C3—44no—
303C22×C3—64no—
304C22×C3—624no—
305C22×C3—64no—
306C22×C3—624no—
307C22×C3—98no—
308C22×C3—91no—
309C22×C3—93no—
310C22×C3—96no—
311C22×C3—96no—
312C22×C3—912no—
313C22×C3—124no—
314C22×C3—1224no—
315C22×C3—184no—
316C22×C3—1824no—
317C22×C3D32⋊C414yesdirect product factor
318C22×C3C2×GO+(4,𝔽2)13yes—
319C2×D3—24no—
320C2×D3—224no—
321C2×D3—68no—
322C2×D3—63no—
323C2×D3—61no—
324C2×D3—612no—
325C2×D3—66no—
326C2×D3—63no—
327C2×D3—66no—
328C2×D3—61no—
329C2×D3—612no—
330C2×D3—66no—
331C2×D3—66no—
332C2×D3—648no—
333C2×D3—68no—
334C3×C4—64no—
335C3×C4—64no—
336C3×C4—124no—
337C3×C4—184no—
338Dic3—24no—
339C32—21no—
340C32—21no—
341C32—41no—
342C32—41no—
343C32C22×C3×C22⋊C411yescharacteristic, complemented
344C23—66no—
345C23—68no—
346C23—94no—
347C23—924no—
348C23—93no—
349C23—93no—
350C23—94no—
351C23—94no—
352C23—186no—
353C23—1812no—
354C23—1824no—
355C23C3×GO+(4,𝔽2)11yescharacteristic
356C2×C4—66no—
357C2×C4—94no—
358C2×C4—94no—
359C2×C4—186no—
360C2×C4—1812no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—26no—
364C2×C3—26no—
365C2×C3—41no—
366C2×C3—41no—
367C2×C3—46no—
368C2×C3—46no—
369C2×C3—68no—
370C2×C3—68no—
371C2×C3—91no—
372C2×C3—91no—
373C2×C3—96no—
374C2×C3—128no—
375C2×C3C2×D32⋊C416yesdirect product factor
376C2×C3C22×GO+(4,𝔽2)11yescharacteristic
377D3—28no—
378D3—66no—
379D3—61no—
380D3—61no—
381D3—66no—
382D3—61no—
383D3—61no—
384C22—64no—
385C22—624no—
386C22—98no—
387C22—91no—
388C22—93no—
389C22—96no—
390C22—96no—
391C22—912no—
392C22—184no—
393C22—1824no—
394C22C3×D32⋊C414yesdirect product factor
395C22C2×C3×GO+(4,𝔽2)13yes—
396C4—64no—
397C4—184no—
398C3—21no—
399C3—21no—
400C3—41no—
401C3—41no—
402C3C22×D32⋊C411yescharacteristic, direct product factor
403C2—68no—
404C2—91no—
405C2—91no—
406C2—96no—
407C2C2×C3×D32⋊C416yesdirect product factor
408C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, hypercenterC23×C3GO+(4,𝔽2)
commutatorC3⋊D3C23×C3×C4
FrattiniC2C22×C3×GO+(4,𝔽2)
Fitting, socleC23×C33D4
prime radicalC23×C3×D32C2
cosocleC2×C3⋊D3C24×C3
hypocenterC32C22×C3×C22⋊C4
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C23×C3×#157C2semidirect
3. C22×C3×#624C2semidirect
4. C2×C3×D32⋊C4C2direct (N × H)
5. C22×D32⋊C4C3direct (N × H)
10. C22×C3×D32C4semidirect
13. C22×C3×#157C22semidirect
16. C2×C3×#624C22semidirect
17. C3×D32⋊C4C22direct (N × H)
20. C23×#157C2×C3semidirect
22. C22×#624C2×C3semidirect
23. C2×D32⋊C4C2×C3direct (N × H)
34. C2×C3×D32C2×C4semidirect
37. C2×C3×#157C23semidirect
40. C3×#624C23semidirect
55. C22×D32C3×C4semidirect
58. C22×#157C22×C3semidirect
66. C2×#624C22×C3semidirect
67. D32⋊C4C22×C3direct (N × H)
68. C22×C33C22⋊C4semidirect
70. C2×C3×C3⋊D3C22⋊C4semidirect
79. C3×D32C22×C4semidirect
121. C2×D32C2×C3×C4semidirect
124. C2×#157C23×C3semidirect
132. #624C23×C3semidirect
134. C2×C33C2×C22⋊C4semidirect
136. C3×C3⋊D3C2×C22⋊C4semidirect
176. C22×C32C3×C22⋊C4semidirect
178. C2×C3⋊D3C3×C22⋊C4semidirect
204. D32C22×C3×C4semidirect
214. C33C22×C22⋊C4semidirect
269. C2×C32C2×C3×C22⋊C4semidirect
271. C3⋊D3C2×C3×C22⋊C4semidirect
317. C22×C3D32⋊C4direct (N × H)
343. C32C22×C3×C22⋊C4semidirect
375. C2×C3C2×D32⋊C4direct (N × H)
394. C22C3×D32⋊C4direct (N × H)
402. C3C22×D32⋊C4direct (N × H)
407. C2C2×C3×D32⋊C4direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×D322
C23×C3×#1572
C22×C3×#6242
C2×C3×D32⋊C42
C22×D32⋊C43
C22×C3×C22⋊C49