Finite Group Databasebeta

← C22×C3×#916 #11918600

Subgroups — C22×C3×#916 #11918600

Order 1152.

3157 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#239C211yesmaximal, characteristic, complemented
2C22×C3×#237C211yesmaximal, characteristic, complemented
3C22×C32×SD16C211yesmaximal, characteristic, complemented
4C22×C3×#216C211yesmaximal, characteristic, complemented
5C22×C3×#213C211yesmaximal, characteristic, complemented
6C2×C3×#916C2124yesmaximal, direct product factor
7C22×C3×#204C211yesmaximal, characteristic, complemented
8C22×C3×D3×C8C211yesmaximal, characteristic, complemented
9C22×C3×#135—31nomaximal
10C22×#916C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C22×C32×D4C2211yescharacteristic
13C22×C3×#67—21no—
14C23×C3×Dic3—21no—
15C2×C3×#239—26no—
16C2×C3×#239C22112yescomplemented
17C2×C3×#237—26no—
18C2×C3×#237C22112yescomplemented
19C22×C3×D12—21no—
20C22×C3×D12C2211yescharacteristic
21C22×C3×C4×D3—21no—
22C22×C3×C4×D3C2211yescharacteristic
23C22×C3×Dic6C2211yescharacteristic
24C2×C32×SD16C22112yescomplemented
25C22×C32×C8C2211yescharacteristic
26C2×C3×#216C22112yescomplemented
27C2×C3×#213C22112yescomplemented
28C22×C3×C3⋊C8C2211yescharacteristic
29C3×#916C22164yesdirect product factor
30C2×C3×#204C22112yescomplemented
31C2×C3×D3×C8C22112yescomplemented
32C22×C3×#41—31no—
33C22×C3×#41—31no—
34C22×#239C2×C311yescharacteristic, complemented
35C22×#237C2×C311yescharacteristic, complemented
36C2×C3×#135—324no—
37C22×C3×Q16—31no—
38C22×C3×SD16—21no—
39C22×C3×SD16—31no—
40C22×C3×SD16D311yescharacteristic, complemented
41C22×C3×SD16C2×C311yescharacteristic, complemented
42C22×C3×D8—31no—
43C23×C3×C8—31no—
44C22×#216C2×C311yescharacteristic, complemented
45C22×#213C2×C311yescharacteristic, complemented
46C2×#916C2×C3124yesdirect product factor
47C22×#204C2×C311yescharacteristic, complemented
48C22×D3×C8C2×C311yescharacteristic, complemented
49C24×C32—21no—
50C23×C3×D3—21no—
51C23×C3×D3D411yescharacteristic
52C2×C32×Q8—23no—
53C2×C32×Q8C2316yes—
54C2×C32×D4—23no—
55C2×C32×D4C2316yes—
56C22×C32×C4—21no—
57C22×C32×C4C2311yescharacteristic
58C2×C3×#67—212no—
59C22×C3×Dic3—21no—
60C22×C3×Dic3—212no—
61C22×C3×Dic3D411yescharacteristic
62C3×#239—224no—
63C3×#239C23116yescomplemented
64C3×#237—224no—
65C3×#237C23116yescomplemented
66C2×C3×D12—212no—
67C2×C3×D12—23no—
68C2×C3×D12C2316yes—
69C2×C3×C4×D3—212no—
70C2×C3×C4×D3D416yes—
71C2×C3×C4×D3C2316yes—
72C2×C3×Dic6—23no—
73C2×C3×Dic6C2316yes—
74C32×SD16C23116yescomplemented
75C2×C32×C8C2316yes—
76C3×#216C23116yescomplemented
77C3×#213C23116yescomplemented
78C2×C3×C3⋊C8C2316yes—
79C3×#204C23116yescomplemented
80C3×D3×C8C23116yescomplemented
81C22×#135—31noSylow
82C2×C3×#41—312no—
83C2×C3×#41—312no—
84C2×C3×#41—66no—
85C2×C3×#41—66no—
86C22×C3×Q8—21no—
87C22×C3×Q8—31no—
88C22×C3×Q8C2×D311yescharacteristic
89C22×C3×Q8C22×C311yescharacteristic
90C22×C3×D4—21no—
91C22×C3×D4—31no—
92C22×C3×D4—61no—
93C22×C3×D4—61no—
94C22×C3×D4C2×D311yescharacteristic
95C22×C3×D4C22×C311yescharacteristic
96C23×C3×C4—31no—
97C23×C3×C4—61no—
98C23×C3×C4—61no—
99C22×#67—21no—
100C23×Dic3—21no—
101C2×#239—26no—
102C2×#239C22×C3112yescomplemented
103C2×#237—26no—
104C2×#237C22×C3112yescomplemented
105C22×D12—21no—
106C22×D12C22×C311yescharacteristic
107C22×C4×D3—21no—
108C22×C4×D3C22×C311yescharacteristic
109C22×Dic6C22×C311yescharacteristic
110C3×#135—364no—
111C2×C3×Q16—312no—
112C2×C3×SD16—212no—
113C2×C3×SD16—312no—
114C2×C3×SD16C2×D3112yescomplemented
115C2×C3×SD16C22×C3112yescomplemented
116C2×C3×D8—312no—
117C22×C3×C8—21no—
118C22×C3×C8—31no—
119C22×C3×C8—312no—
120C22×C3×C8C2×D311yescharacteristic
121C22×C3×C8C22×C311yescharacteristic
122C2×#216C22×C3112yescomplemented
123C2×#213C22×C3112yescomplemented
124C22×C3⋊C8C22×C311yescharacteristic
125#916C22×C3164yesdirect product factor
126C2×#204C22×C3112yescomplemented
127C2×D3×C8C22×C3112yescomplemented
128C23×C32—26no—
129C23×C32—44no—
130C23×C32C2×D411yescharacteristic
131C22×C3×D3—24no—
132C22×C3×D3—26no—
133C22×C3×D3—44no—
134C22×C3×D3C2×D416yes—
135C32×Q8—26no—
136C32×Q8C2414yes—
137C32×D4—26no—
138C32×D4C2414yes—
139C2×C32×C4—26no—
140C2×C32×C4C2×D413yes—
141C2×C32×C4C2413yes—
142C3×#67—216no—
143C2×C3×Dic3—26no—
144C2×C3×Dic3—216no—
145C2×C3×Dic3C2×D416yes—
146C3×D12—216no—
147C3×D12—26no—
148C3×D12C2414yes—
149C3×C4×D3—216no—
150C3×C4×D3C2×D4112yes—
151C3×C4×D3C2414yes—
152C3×Dic6—26no—
153C3×Dic6C2414yes—
154C32×C8C2414yes—
155C3×C3⋊C8C2414yes—
156C22×#41—31no—
157C22×#41—31no—
158C2×#135—324no—
159C22×Q16—31no—
160C22×SD16—31no—
161C22×SD16C3×D311yescharacteristic, complemented
162C22×D8—31no—
163C23×C8—31no—
164C24×C3—21no—
165C24×C3—21no—
166C24×C3—31no—
167C24×C3—41no—
168C24×C3—61no—
169C23×D3—21no—
170C23×D3C3×D411yescharacteristic
171C3×#41—316no—
172C3×#41—316no—
173C3×#41—624no—
174C3×#41—624no—
175C2×C3×Q8—23no—
176C2×C3×Q8—23no—
177C2×C3×Q8—26no—
178C2×C3×Q8—36no—
179C2×C3×Q8—43no—
180C2×C3×Q8—63no—
181C2×C3×Q8C22×D316yes—
182C2×C3×Q8C23×C316yes—
183C2×C3×D4—23no—
184C2×C3×D4—23no—
185C2×C3×D4—26no—
186C2×C3×D4—36no—
187C2×C3×D4—43no—
188C2×C3×D4—612no—
189C2×C3×D4—612no—
190C2×C3×D4—63no—
191C2×C3×D4C22×D316yes—
192C2×C3×D4C23×C316yes—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—31no—
197C22×C3×C4—36no—
198C22×C3×C4—36no—
199C22×C3×C4—41no—
200C22×C3×C4—612no—
201C22×C3×C4—612no—
202C22×C3×C4—61no—
203C22×C3×C4C22×D311yescharacteristic
204C22×C3×C4C23×C311yescharacteristic
205C2×#67—212no—
206C22×Dic3—21no—
207C22×Dic3—212no—
208C22×Dic3C3×D411yescharacteristic
209#239—224no—
210#239C23×C3116yescomplemented
211#237—224no—
212#237C23×C3116yescomplemented
213C2×D12—212no—
214C2×D12—23no—
215C2×D12C23×C316yes—
216C2×C4×D3—212no—
217C2×C4×D3C3×D416yes—
218C2×C4×D3C23×C316yes—
219C2×Dic6—23no—
220C2×Dic6C23×C316yes—
221C3×Q16—316no—
222C3×SD16—216no—
223C3×SD16—316no—
224C3×SD16C22×D3116yescomplemented
225C3×SD16C23×C3116yescomplemented
226C3×D8—316no—
227C2×C3×C8—26no—
228C2×C3×C8—36no—
229C2×C3×C8—316no—
230C2×C3×C8C22×D316yes—
231C2×C3×C8C23×C316yes—
232#216C23×C3116yescomplemented
233#213C23×C3116yescomplemented
234C2×C3⋊C8C23×C316yes—
235#204C23×C3116yescomplemented
236D3×C8C23×C3116yescomplemented
237C22×C32—24no—
238C22×C32—412no—
239C22×C32#13514yescomplemented
240C22×C32C22×D413yes—
241C2×C3×D3—212no—
242C2×C3×D3—24no—
243C2×C3×D3—412no—
244C2×C3×D3C22×D414yes—
245C32×C4—24no—
246C32×C4C22×D413yes—
247C32×C4C2511yescharacteristic
248C3×Dic3—24no—
249C3×Dic3C22×D414yes—
250C2×#41—312no—
251C2×#41—312no—
252C2×#41—66no—
253C2×#41—66no—
254C22×Q8—31no—
255C22×Q8C2×C3×D311yescharacteristic
256C22×D4—31no—
257C22×D4—61no—
258C22×D4—61no—
259C22×D4C2×C3×D311yescharacteristic
260C23×C4—31no—
261C23×C4—61no—
262C23×C4—61no—
263#135—364no—
264C2×Q16—312no—
265C2×SD16—312no—
266C2×SD16C2×C3×D3112yescomplemented
267C2×D8—312no—
268C22×C8—31no—
269C22×C8—312no—
270C22×C8C2×C3×D311yescharacteristic
271C23×C3—26no—
272C23×C3—26no—
273C23×C3—21no—
274C23×C3—36no—
275C23×C3—46no—
276C23×C3—44no—
277C23×C3—44no—
278C23×C3—64no—
279C23×C3—66no—
280C23×C3—84no—
281C23×C3—124no—
282C23×C3D3×D411yescharacteristic
283C23×C3C2×C3×D411yescharacteristic
284C22×D3—24no—
285C22×D3—26no—
286C22×D3—44no—
287C22×D3C2×C3×D416yes—
288C3×Q8—26no—
289C3×Q8—26no—
290C3×Q8—24no—
291C3×Q8—34no—
292C3×Q8—46no—
293C3×Q8—66no—
294C3×Q8C23×D314yes—
295C3×Q8C24×C314yes—
296C3×D4—26no—
297C3×D4—26no—
298C3×D4—24no—
299C3×D4—34no—
300C3×D4—46no—
301C3×D4—616no—
302C3×D4—616no—
303C3×D4—66no—
304C3×D4C23×D314yes—
305C3×D4C24×C314yes—
306C2×C3×C4—26no—
307C2×C3×C4—26no—
308C2×C3×C4—23no—
309C2×C3×C4—23no—
310C2×C3×C4—36no—
311C2×C3×C4—312no—
312C2×C3×C4—34no—
313C2×C3×C4—46no—
314C2×C3×C4—616no—
315C2×C3×C4—616no—
316C2×C3×C4—66no—
317C2×C3×C4D3×D413yes—
318C2×C3×C4C2×C3×D413yes—
319C2×C3×C4C23×D313yes—
320C2×C3×C4C24×C313yes—
321#67—216no—
322C2×Dic3—26no—
323C2×Dic3—216no—
324C2×Dic3C2×C3×D416yes—
325D12—216no—
326D12—26no—
327D12C24×C314yes—
328C4×D3—216no—
329C4×D3C2×C3×D4112yes—
330C4×D3C24×C314yes—
331Dic6—26no—
332Dic6C24×C314yes—
333C3×C8—24no—
334C3×C8—34no—
335C3×C8C23×D314yes—
336C3×C8C24×C314yes—
337C3⋊C8C24×C314yes—
338C2×C32—44no—
339C2×C32C2×#13516yescomplemented
340C2×C32C23×D411yescharacteristic
341C3×D3—24no—
342C3×D3—44no—
343C24—21no—
344C24—31no—
345C24—61no—
346#41—316no—
347#41—316no—
348#41—624no—
349#41—624no—
350C2×Q8—23no—
351C2×Q8—36no—
352C2×Q8—63no—
353C2×Q8C22×C3×D316yes—
354C2×D4—23no—
355C2×D4—36no—
356C2×D4—612no—
357C2×D4—612no—
358C2×D4—63no—
359C2×D4C22×C3×D316yes—
360C22×C4—21no—
361C22×C4—31no—
362C22×C4—36no—
363C22×C4—36no—
364C22×C4—612no—
365C22×C4—612no—
366C22×C4—61no—
367C22×C4C22×C3×D311yescharacteristic
368Q16—316no—
369SD16—316no—
370SD16C22×C3×D3116yescomplemented
371D8—316no—
372C2×C8—36no—
373C2×C8—316no—
374C2×C8C22×C3×D316yes—
375C22×C3—24no—
376C22×C3—24no—
377C22×C3—24no—
378C22×C3—23no—
379C22×C3—34no—
380C22×C3—44no—
381C22×C3—412no—
382C22×C3—412no—
383C22×C3—612no—
384C22×C3—64no—
385C22×C3—812no—
386C22×C3—1212no—
387C22×C3#91614yesdirect product factor
388C22×C3C3×#13514yescomplemented
389C22×C3C2×D3×D413yes—
390C22×C3C22×C3×D413yes—
391C2×D3—212no—
392C2×D3—24no—
393C2×D3—412no—
394C2×D3C22×C3×D414yes—
395C3×C4—24no—
396C3×C4—24no—
397C3×C4—23no—
398C3×C4—21no—
399C3×C4—34no—
400C3×C4—44no—
401C3×C4—64no—
402C3×C4C2×D3×D413yes—
403C3×C4C22×C3×D413yes—
404C3×C4C24×D311yescharacteristic
405C3×C4C25×C311yescharacteristic
406Dic3—24no—
407Dic3C22×C3×D414yes—
408C32C22×#13511yescharacteristic, complemented, Sylow
409C23—26no—
410C23—36no—
411C23—44no—
412C23—64no—
413C23—66no—
414C23—124no—
415C23C3×D3×D411yescharacteristic
416Q8—26no—
417Q8—34no—
418Q8—66no—
419Q8C23×C3×D314yes—
420D4—26no—
421D4—34no—
422D4—616no—
423D4—616no—
424D4—66no—
425D4C23×C3×D314yes—
426C2×C4—26no—
427C2×C4—36no—
428C2×C4—312no—
429C2×C4—34no—
430C2×C4—616no—
431C2×C4—616no—
432C2×C4—66no—
433C2×C4C3×D3×D413yes—
434C2×C4C23×C3×D313yes—
435C8—34no—
436C8C23×C3×D314yes—
437C2×C3—26no—
438C2×C3—21no—
439C2×C3—44no—
440C2×C3—44no—
441C2×C3—64no—
442C2×C3—84no—
443C2×C3—124no—
444C2×C3C2×#91616yesdirect product factor
445C2×C3C2×C3×#13516yescomplemented
446C2×C3C22×D3×D411yescharacteristic
447C2×C3C23×C3×D411yescharacteristic
448D3—24no—
449D3—44no—
450C22—24no—
451C22—34no—
452C22—412no—
453C22—612no—
454C22—64no—
455C22—1212no—
456C22C3×#91614yesdirect product factor
457C22C2×C3×D3×D413yes—
458C4—24no—
459C4—34no—
460C4—64no—
461C4C2×C3×D3×D413yes—
462C4C24×C3×D311yescharacteristic
463C3—21no—
464C3C22×#91611yescharacteristic, direct product factor
465C3C22×C3×#13511yescharacteristic, complemented
466C2—44no—
467C2—64no—
468C2—124no—
469C2C2×C3×#91616yesdirect product factor
470C2C22×C3×D3×D411yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC3×C4C25×C3
FrattiniC4C24×C3×D3
FittingC22×C32×SD16C2
socleC23×C32C2×D4
hypercenterC22×C3×SD16D3
hypocenterC3C22×C3×#135
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C22×C3×SD16

Complements

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

NH ≅ G/Ntype
1. C22×C3×#239C2semidirect
2. C22×C3×#237C2semidirect
3. C22×C32×SD16C2semidirect
4. C22×C3×#216C2semidirect
5. C22×C3×#213C2semidirect
6. C2×C3×#916C2direct (N × H)
7. C22×C3×#204C2semidirect
8. C22×C3×D3×C8C2semidirect
10. C22×#916C3direct (N × H)
16. C2×C3×#239C22semidirect
18. C2×C3×#237C22semidirect
24. C2×C32×SD16C22semidirect
26. C2×C3×#216C22semidirect
27. C2×C3×#213C22semidirect
29. C3×#916C22direct (N × H)
30. C2×C3×#204C22semidirect
31. C2×C3×D3×C8C22semidirect
34. C22×#239C2×C3semidirect
35. C22×#237C2×C3semidirect
40. C22×C3×SD16D3semidirect
41. C22×C3×SD16C2×C3semidirect
44. C22×#216C2×C3semidirect
45. C22×#213C2×C3semidirect
46. C2×#916C2×C3direct (N × H)
47. C22×#204C2×C3semidirect
48. C22×D3×C8C2×C3semidirect
63. C3×#239C23semidirect
65. C3×#237C23semidirect
74. C32×SD16C23semidirect
76. C3×#216C23semidirect
77. C3×#213C23semidirect
79. C3×#204C23semidirect
80. C3×D3×C8C23semidirect
102. C2×#239C22×C3semidirect
104. C2×#237C22×C3semidirect
114. C2×C3×SD16C2×D3semidirect
115. C2×C3×SD16C22×C3semidirect
122. C2×#216C22×C3semidirect
123. C2×#213C22×C3semidirect
125. #916C22×C3direct (N × H)
126. C2×#204C22×C3semidirect
127. C2×D3×C8C22×C3semidirect
161. C22×SD16C3×D3semidirect
210. #239C23×C3semidirect
212. #237C23×C3semidirect
224. C3×SD16C22×D3semidirect
225. C3×SD16C23×C3semidirect
232. #216C23×C3semidirect
233. #213C23×C3semidirect
235. #204C23×C3semidirect
236. D3×C8C23×C3semidirect
239. C22×C32#135semidirect
266. C2×SD16C2×C3×D3semidirect
339. C2×C32C2×#135semidirect
370. SD16C22×C3×D3semidirect
387. C22×C3#916direct (N × H)
388. C22×C3C3×#135semidirect
408. C32C22×#135semidirect
444. C2×C3C2×#916direct (N × H)
445. C2×C3C2×C3×#135semidirect
456. C22C3×#916direct (N × H)
464. C3C22×#916direct (N × H)
465. C3C22×C3×#135semidirect
469. C2C2×C3×#916direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#2392
C22×C3×#2372
C22×C32×SD162
C22×C3×#2162
C22×C3×#2132
C2×C3×#9162
C22×C3×#2042
C22×C3×D3×C82
C22×C3×#1353
C22×#9163