Finite Group Databasebeta

← C22×C3⋊D3×C22⋊A4 #72252709

Subgroups — C22×C3⋊D3×C22⋊A4 #72252709

Order 3456.

59824 conjugacy classes of subgroups; only the 328 normal subgroups are shown.

normal subgroups only

This group has 59824 conjugacy classes of subgroups — too many to draw. Only its 328 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×C32×C22⋊A4C211yesmaximal, characteristic, complemented
2C2×C3⋊D3×C22⋊A4C216yesmaximal, direct product factor
3C22×D3×C22⋊A4—34nomaximal
4C26×C3⋊D3C311yesmaximal, characteristic, complemented
5C2×C32×C22⋊A4C2213yescomplemented
6C3⋊D3×C22⋊A4C2214yesdirect product factor
7C22×A4×C3⋊D3—45nomaximal
8C26×C32C2×C311yescharacteristic, complemented
9C25×C3⋊D3—35no—
10C25×C3⋊D3—35no—
11C25×C3⋊D3—330no—
12C25×C3⋊D3C2×C316yescomplemented
13C22×C3×C22⋊A4—24no—
14C22×C3×C22⋊A4D314yescomplemented
15C2×D3×C22⋊A4—324no—
16C32×C22⋊A4C2311yescharacteristic, complemented
17C22×C32×A4—45no—
18C2×A4×C3⋊D3—430no—
19C26×D3—34no—
20C23×C22⋊A4—91no—
21C25×C32—315no—
22C25×C32—35no—
23C25×C32C22×C313yescomplemented
24C24×C3⋊D3—3180no—
25C24×C3⋊D3—3240no—
26C24×C3⋊D3—3180no—
27C24×C3⋊D3—330no—
28C24×C3⋊D3—330no—
29C24×C3⋊D3—340no—
30C24×C3⋊D3—310no—
31C24×C3⋊D3—35no—
32C24×C3⋊D3—330no—
33C24×C3⋊D3—330no—
34C24×C3⋊D3—330no—
35C24×C3⋊D3—360no—
36C24×C3⋊D3A415yescomplemented
37C24×C3⋊D3C22×C314yescomplemented
38C2×C3×C22⋊A4—212no—
39C2×C3×C22⋊A4C2×D3112yescomplemented
40D3×C22⋊A4—316no—
41C22×D3×A4—1220no—
42C22×C3×C3⋊D3—161no—
43C2×C32×A4—415no—
44A4×C3⋊D3—420no—
45C26×C3C3×D314yescomplemented
46C25×D3—324no—
47C25×D3—920no—
48C25×D3—9120no—
49C25×D3—920no—
50C22×C22⋊A4—24no—
51C22×C22⋊A4—96no—
52C22×C22⋊A4C3⋊D311yescharacteristic, direct product factor
53C24×C32—390no—
54C24×C32—360no—
55C24×C32—310no—
56C24×C32—310no—
57C24×C32—315no—
58C24×C32—315no—
59C24×C32—315no—
60C24×C32C2×A415yescomplemented
61C24×C32C23×C311yescharacteristic, complemented
62C23×C3⋊D3—390no—
63C23×C3⋊D3—3360no—
64C23×C3⋊D3—3480no—
65C23×C3⋊D3—3360no—
66C23×C3⋊D3—3480no—
67C23×C3⋊D3—3360no—
68C23×C3⋊D3—3360no—
69C23×C3⋊D3—3180no—
70C23×C3⋊D3—3360no—
71C23×C3⋊D3—3180no—
72C23×C3⋊D3—360no—
73C23×C3⋊D3—330no—
74C23×C3⋊D3—3180no—
75C23×C3⋊D3—360no—
76C23×C3⋊D3—330no—
77C23×C3⋊D3—360no—
78C23×C3⋊D3—320no—
79C23×C3⋊D3—35no—
80C23×C3⋊D3—35no—
81C23×C3⋊D3—330no—
82C23×C3⋊D3—320no—
83C23×C3⋊D3C2×A4130yescomplemented
84C3×C22⋊A4—24no—
85C3×C22⋊A4C22×D314yescomplemented
86C22×C3×A4—420no—
87C22×C3×A4—820no—
88C2×D3×A4—12120no—
89C27—91noSylow
90C22×C33—161no—
91C2×C3×C3⋊D3—166no—
92C32×A4—45no—
93C25×C3—360no—
94C25×C3—320no—
95C25×C3C2×C3×D3112yescomplemented
96C24×D3—320no—
97C24×D3—316no—
98C24×D3—9120no—
99C24×D3—9120no—
100C24×D3—940no—
101C24×D3—9160no—
102C24×D3—9720no—
103C24×D3—9960no—
104C24×D3—9720no—
105C24×D3—9120no—
106C24×D3—9120no—
107C24×D3—920no—
108C24×D3—9240no—
109C24×D3—9120no—
110C2×C22⋊A4—212no—
111C2×C22⋊A4—94no—
112C2×C22⋊A4C2×C3⋊D313yesdirect product factor
113C23×A4—365no—
114C23×C32—315no—
115C23×C32—390no—
116C23×C32—390no—
117C23×C32—3120no—
118C23×C32—390no—
119C23×C32—330no—
120C23×C32—315no—
121C23×C32—35no—
122C23×C32—35no—
123C23×C32C22×A4115yescomplemented
124C22×C3⋊D3—3180no—
125C22×C3⋊D3—3120no—
126C22×C3⋊D3—3720no—
127C22×C3⋊D3—3720no—
128C22×C3⋊D3—3360no—
129C22×C3⋊D3—3120no—
130C22×C3⋊D3—3180no—
131C22×C3⋊D3—330no—
132C22×C3⋊D3—3180no—
133C22×C3⋊D3—3240no—
134C22×C3⋊D3—340no—
135C22×C3⋊D3—3360no—
136C22×C3⋊D3—340no—
137C22×C3⋊D3—320no—
138C22×C3⋊D3—35no—
139C22×C3⋊D3—330no—
140C22×C3⋊D3—330no—
141C22×C3⋊D3—330no—
142C22×C3⋊D3—360no—
143C22×C3⋊D3C22×A4120yescomplemented
144C22×C3⋊D3C22⋊A411yescharacteristic, direct product factor
145C22×C3×D3—484no—
146C2×C3×A4—460no—
147C2×C3×A4—860no—
148D3×A4—1280no—
149C26—96no—
150C26—275no—
151C26—2730no—
152C26—275no—
153C26C3×C3⋊D311yescharacteristic, complemented
154C2×C33—163no—
155C3×C3⋊D3—164no—
156C24×C3—360no—
157C24×C3—3360no—
158C24×C3—3240no—
159C24×C3—340no—
160C24×C3—340no—
161C24×C3—360no—
162C24×C3—360no—
163C24×C3D3×A4120yescomplemented
164C24×C3C22×C3×D314yescomplemented
165C23×D3—3120no—
166C23×D3—920no—
167C23×D3—980no—
168C23×D3—9240no—
169C23×D3—9240no—
170C23×D3—9720no—
171C23×D3—9360no—
172C23×D3—91920no—
173C23×D3—91440no—
174C23×D3—91920no—
175C23×D3—91440no—
176C23×D3—91440no—
177C23×D3—91440no—
178C23×D3—9720no—
179C23×D3—91440no—
180C23×D3—9720no—
181C23×D3—9240no—
182C23×D3—9120no—
183C23×D3—9120no—
184C23×D3—9120no—
185C23×D3—980no—
186C23×D3—920no—
187C22⋊A4—24no—
188C22⋊A4C22×C3⋊D311yescharacteristic, direct product factor
189C22×A4—45no—
190C22×A4—820no—
191C22×A4—3630no—
192C22×C32—315no—
193C22×C32—310no—
194C22×C32—360no—
195C22×C32—390no—
196C22×C32—310no—
197C22×C32—315no—
198C22×C32—315no—
199C22×C32—164no—
200C22×C32—324no—
201C22×C32C23×A415yescomplemented
202C22×C32C2×C22⋊A411yescharacteristic, complemented
203C2×C3⋊D3—3360no—
204C2×C3⋊D3—360no—
205C2×C3⋊D3—3120no—
206C2×C3⋊D3—330no—
207C2×C3⋊D3—360no—
208C2×C3⋊D3—320no—
209C2×C3⋊D3—320no—
210C2×C3⋊D3C2×C22⋊A416yesdirect product factor
211C2×C3×D3—4824no—
212C3×A4—420no—
213C3×A4—820no—
214C25—315no—
215C25—35no—
216C25—95no—
217C25—94no—
218C25—2730no—
219C25—2730no—
220C25—2760no—
221C25—2730no—
222C25—2710no—
223C25—2740no—
224C25—27180no—
225C25—2730no—
226C25—27180no—
227C25—27240no—
228C25—2730no—
229C25—275no—
230C25C2×C3×C3⋊D313yescomplemented
231C33—161noSylow
232C23×C3—360no—
233C23×C3—3360no—
234C23×C3—3360no—
235C23×C3—3480no—
236C23×C3—3120no—
237C23×C3—3360no—
238C23×C3—360no—
239C23×C3—320no—
240C23×C3—320no—
241C23×C3—1441no—
242C23×C3C2×D3×A4160yescomplemented
243C22×D3—34no—
244C22×D3—380no—
245C22×D3—9120no—
246C22×D3—9120no—
247C22×D3—9160no—
248C22×D3—91440no—
249C22×D3—9960no—
250C22×D3—9160no—
251C22×D3—9720no—
252C22×D3—92880no—
253C22×D3—9480no—
254C22×D3—92880no—
255C22×D3—9480no—
256C22×D3—91440no—
257C22×D3—9720no—
258C22×D3—9720no—
259C22×D3—9120no—
260C22×D3—980no—
261C22×D3—920no—
262C22×D3—9240no—
263C22×D3—9120no—
264C2×A4—415no—
265C2×A4—860no—
266C2×A4—3620no—
267C2×C32—315no—
268C2×C32—35no—
269C2×C32—1612no—
270C2×C32—3212no—
271C2×C32C22×C22⋊A413yescomplemented
272C3⋊D3—320no—
273C3⋊D3C22×C22⋊A414yesdirect product factor
274C3×D3—4816no—
275C24—315no—
276C24—390no—
277C24—360no—
278C24—310no—
279C24—310no—
280C24—315no—
281C24—315no—
282C24—930no—
283C24—2720no—
284C24—275no—
285C24—2760no—
286C24—2730no—
287C24—2720no—
288C24—275no—
289C24—2790no—
290C24—2760no—
291C24—27180no—
292C24—27360no—
293C24—27180no—
294C24—27360no—
295C24—27180no—
296C24—27360no—
297C24—27360no—
298C24—27480no—
299C24—27480no—
300C24—27360no—
301C24—2760no—
302C24—2730no—
303C24—2730no—
304C24A4×C3⋊D315yescomplemented
305C24C22×C3×C3⋊D311yescharacteristic, complemented
306C22×C3—360no—
307C22×C3—340no—
308C22×C3—3360no—
309C22×C3—340no—
310C22×C3—3240no—
311C22×C3—360no—
312C22×C3—360no—
313C22×C3—161no—
314C22×C3—324no—
315C22×C3—1446no—
316C22×C3C22×D3×A4120yescomplemented
317C22×C3D3×C22⋊A414yescomplemented
318C2×D3—324no—
319C2×D3—980no—
320C2×D3—9240no—
321C2×D3—9480no—
322C2×D3—91440no—
323C2×D3—9240no—
324C2×D3—9120no—
325C2×D3—980no—
326A4—45no—
327A4—820no—
328C32—164no—
329C32—324no—
330C32C23×C22⋊A411yescharacteristic, complemented
331C23—315no—
332C23—390no—
333C23—390no—
334C23—3120no—
335C23—330no—
336C23—390no—
337C23—35no—
338C23—315no—
339C23—35no—
340C23—91no—
341C23—920no—
342C23—2730no—
343C23—2730no—
344C23—2760no—
345C23—2730no—
346C23—27180no—
347C23—2740no—
348C23—27360no—
349C23—2740no—
350C23—27240no—
351C23—27120no—
352C23—27360no—
353C23—27720no—
354C23—27120no—
355C23—27720no—
356C23—27180no—
357C23—2730no—
358C23—27180no—
359C23—2720no—
360C23—275no—
361C23C2×A4×C3⋊D3115yescomplemented
362C2×C3—320no—
363C2×C3—360no—
364C2×C3—163no—
365C2×C3—3212no—
366C2×C3—1444no—
367C2×C3C2×D3×C22⋊A4112yescomplemented
368D3—316no—
369D3—980no—
370C22—315no—
371C22—310no—
372C22—390no—
373C22—310no—
374C22—360no—
375C22—315no—
376C22—315no—
377C22—96no—
378C22—2720no—
379C22—2760no—
380C22—2720no—
381C22—27120no—
382C22—2760no—
383C22—27360no—
384C22—2730no—
385C22C22×A4×C3⋊D315yescomplemented
386C22C3⋊D3×C22⋊A411yescharacteristic, direct product factor
387C3—161no—
388C3—324no—
389C3C22×D3×C22⋊A414yescomplemented
390C2—35no—
391C2—315no—
392C2—94no—
393C2—2720no—
394C2C2×C3⋊D3×C22⋊A413yesdirect product factor
subgroupG/H
center, hypercenterC22C3⋊D3×C22⋊A4
commutator, cosocle, hypocenterC24×C32C23×C3
FrattiniC1G
Fitting, socleC26×C32C2×C3
LCSG ▷ C24×C32
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C32×C22⋊A4C2semidirect
2. C2×C3⋊D3×C22⋊A4C2direct (N × H)
4. C26×C3⋊D3C3semidirect
5. C2×C32×C22⋊A4C22semidirect
6. C3⋊D3×C22⋊A4C22direct (N × H)
8. C26×C32C2×C3semidirect
12. C25×C3⋊D3C2×C3semidirect
14. C22×C3×C22⋊A4D3semidirect
16. C32×C22⋊A4C23semidirect
23. C25×C32C22×C3semidirect
36. C24×C3⋊D3A4semidirect
37. C24×C3⋊D3C22×C3semidirect
39. C2×C3×C22⋊A4C2×D3semidirect
45. C26×C3C3×D3semidirect
52. C22×C22⋊A4C3⋊D3direct (N × H)
60. C24×C32C2×A4semidirect
61. C24×C32C23×C3semidirect
83. C23×C3⋊D3C2×A4semidirect
85. C3×C22⋊A4C22×D3semidirect
95. C25×C3C2×C3×D3semidirect
112. C2×C22⋊A4C2×C3⋊D3direct (N × H)
123. C23×C32C22×A4semidirect
143. C22×C3⋊D3C22×A4semidirect
144. C22×C3⋊D3C22⋊A4direct (N × H)
153. C26C3×C3⋊D3semidirect
163. C24×C3D3×A4semidirect
164. C24×C3C22×C3×D3semidirect
188. C22⋊A4C22×C3⋊D3direct (N × H)
201. C22×C32C23×A4semidirect
202. C22×C32C2×C22⋊A4semidirect
210. C2×C3⋊D3C2×C22⋊A4direct (N × H)
230. C25C2×C3×C3⋊D3semidirect
242. C23×C3C2×D3×A4semidirect
271. C2×C32C22×C22⋊A4semidirect
273. C3⋊D3C22×C22⋊A4direct (N × H)
304. C24A4×C3⋊D3semidirect
305. C24C22×C3×C3⋊D3semidirect
316. C22×C3C22×D3×A4semidirect
317. C22×C3D3×C22⋊A4semidirect
330. C32C23×C22⋊A4semidirect
361. C23C2×A4×C3⋊D3semidirect
367. C2×C3C2×D3×C22⋊A4semidirect
385. C22C22×A4×C3⋊D3semidirect
386. C22C3⋊D3×C22⋊A4direct (N × H)
389. C3C22×D3×C22⋊A4semidirect
394. C2C2×C3⋊D3×C22⋊A4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×C22⋊A42
C2×C3⋊D3×C22⋊A42
C22×D3×C22⋊A43
C26×C3⋊D33
C22×A4×C3⋊D34