Finite Group Databasebeta

← C3×#33060 #10651667

Subgroups — C3×#33060 #10651667

Order 768.

822 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#486C211yesmaximal, characteristic, complemented
2C3×#3079C211yesmaximal, characteristic, complemented
3C3×#3038C211yesmaximal, characteristic
4C3×C4×#131C211yesmaximal, characteristic, complemented
5C3×#1997C211yesmaximal, characteristic, complemented
6C3×#1965C211yesmaximal, characteristic, complemented
7C3×#1931C214yesmaximal, complemented
8C3×#1907C214yesmaximal, complemented
9C3×#1837C211yesmaximal, characteristic, complemented
10#33060C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×AGL(1,ℤ/8)C2212yes—
12C2×C3×#131C2212yes—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#127C2211yescharacteristic
15C3×C4×#41C2211yescharacteristic
16C3×#486—21no—
17C3×#486—21no—
18C3×#486—22no—
19C3×#477—21no—
20C3×#476—21no—
21C3×#475—21no—
22C3×#474—21no—
23C3×#460—21no—
24C3×#460—21no—
25C3×#459—21no—
26C3×#459—21no—
27C3×#428C2212yes—
28C3×#423C2211yescharacteristic
29C3×#423C2211yescharacteristic
30C3×#418C2212yes—
31C3×#417C2212yes—
32C2×C3×#102C2214yes—
33C2×C3×#100C2214yes—
34C3×#405—22no—
35C3×#405C2212yes—
36C3×C4×M16—21no—
37C3×C4×M16C2211yescharacteristic
38C2×C3×C4×C8—21no—
39C2×C3×C4×C8C2211yescharacteristic
40C3×#343C2214yes—
41C3×#341C2214yes—
42C2×#486C2×C311yescharacteristic, complemented
43#3079C2×C311yescharacteristic, complemented
44#3038C2×C311yescharacteristic
45C4×#131C2×C311yescharacteristic, complemented
46#1997C2×C311yescharacteristic, complemented
47#1965C2×C311yescharacteristic, complemented
48#1931C2×C314yescomplemented
49#1907C2×C314yescomplemented
50#1837C2×C311yescharacteristic, complemented
51C2×C3×#41C2311yescharacteristic
52C22×C3×Q8—21no—
53C22×C3×D4—21no—
54C22×C3×D4—21no—
55C22×C3×D4C2312yes—
56C3×AGL(1,ℤ/8)—28no—
57C2×C3×SD16—22no—
58C2×C3×D8—22no—
59C3×#131—24no—
60C3×#131—24no—
61C2×C3×M16—22no—
62C2×C3×M16C2312yes—
63C2×C3×M16C2312yes—
64C2×C3×M16C2312yes—
65C22×C3×C8—22no—
66C22×C3×C8C2312yes—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8—21no—
69C3×C4⋊Q8—21no—
70C3×C4⋊Q8—21no—
71C3×#127—21no—
72C3×#127—21no—
73C3×#127—21no—
74C3×#127—21no—
75C3×C4×Q8D411yescharacteristic
76C3×C4×Q8D411yescharacteristic
77C3×C4×D4—21no—
78C3×C4×D4—21no—
79C3×C4×D4D411yescharacteristic
80C3×C4×D4D411yescharacteristic
81C3×#117—21no—
82C3×#117C2311yescharacteristic
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C42—21no—
86C2×C3×C42C2311yescharacteristic
87C3×#107—22no—
88C3×#106—22no—
89C3×C4⋊C8—21no—
90C3×C4⋊C8—21no—
91C3×C4⋊C8—21no—
92C3×C4⋊C8—21no—
93C3×#103—22no—
94C3×#102—22no—
95C3×#102—24no—
96C3×#102—24no—
97C3×#102—22no—
98C3×#102—22no—
99C3×#100—28no—
100C3×#97—22no—
101C3×#97—21no—
102C3×C4×C8—22no—
103C3×C4×C8—21no—
104C3×C4×C8D411yescharacteristic
105C3×C4×C8D411yescharacteristic
106C3×C4×C8D412yes—
107C2×AGL(1,ℤ/8)C22×C312yes—
108C2×#131C22×C312yes—
109C2×C4⋊Q8C22×C311yescharacteristic
110C2×#127C22×C311yescharacteristic
111C4×#41C22×C311yescharacteristic
112#486—21no—
113#486—21no—
114#486—22no—
115#477—21no—
116#476—21no—
117#475—21no—
118#474—21no—
119#460—21no—
120#460—21no—
121#459—21no—
122#459—21no—
123#428C22×C312yes—
124#423C22×C311yescharacteristic
125#423C22×C311yescharacteristic
126#418C22×C312yes—
127#417C22×C312yes—
128C2×#102C22×C314yes—
129C2×#100C22×C314yes—
130#405—22no—
131#405C22×C312yes—
132C4×M16—21no—
133C4×M16C22×C311yescharacteristic
134C2×C4×C8—21no—
135C2×C4×C8C22×C311yescharacteristic
136#343C22×C314yes—
137#341C22×C314yes—
138C24×C3—22no—
139C3×#41—22no—
140C3×#41#4114yes—
141C2×C3×Q8—41no—
142C2×C3×Q8—41no—
143C2×C3×Q8—82no—
144C2×C3×Q8C2×D411yescharacteristic
145C2×C3×D4—21no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—24no—
149C2×C3×D4—24no—
150C2×C3×D4—41no—
151C2×C3×D4—41no—
152C2×C3×D4—41no—
153C2×C3×D4—41no—
154C2×C3×D4—82no—
155C2×C3×D4—82no—
156C2×C3×D4C2×D411yescharacteristic
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4#4111yescharacteristic
162C22×C3×C4#4111yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C3×SD16—44no—
165C3×D8—44no—
166C3×M16—28no—
167C3×M16—22no—
168C3×M16—24no—
169C3×M16—24no—
170C3×M16#4114yes—
171C2×C3×C8—28no—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8C2×D412yes—
178C2×C3×C8C2×D412yes—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—41no—
184C3×C4⋊C4—41no—
185C3×C4⋊C4C2×D411yescharacteristic
186C3×C4⋊C4C2×D411yescharacteristic
187C3×C22⋊C4—21no—
188C3×C22⋊C4—21no—
189C3×C22⋊C4—21no—
190C3×C42—21no—
191C3×C42—21no—
192C3×C42—21no—
193C3×C42C2×D411yescharacteristic
194C3×C42C2×D411yescharacteristic
195C3×C42C2×D411yescharacteristic
196C3×C42C2×D411yescharacteristic
197C2×#41C23×C311yescharacteristic
198C22×Q8—21no—
199C22×D4—21no—
200C22×D4—21no—
201C22×D4C23×C312yes—
202AGL(1,ℤ/8)—28no—
203C2×SD16—22no—
204C2×D8—22no—
205#131—24no—
206#131—24no—
207C2×M16—22no—
208C2×M16C23×C312yes—
209C2×M16C23×C312yes—
210C2×M16C23×C312yes—
211C22×C8—22no—
212C22×C8C23×C312yes—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217#127—21no—
218#127—21no—
219#127—21no—
220#127—21no—
221C4×Q8C3×D411yescharacteristic
222C4×Q8C3×D411yescharacteristic
223C4×D4—21no—
224C4×D4—21no—
225C4×D4C3×D411yescharacteristic
226C4×D4C3×D411yescharacteristic
227#117—21no—
228#117C23×C311yescharacteristic
229C2×C4⋊C4—21no—
230C2×C4⋊C4C23×C312yes—
231C2×C42—21no—
232C2×C42C23×C311yescharacteristic
233#107—22no—
234#106—22no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239#103—22no—
240#102—22no—
241#102—24no—
242#102—24no—
243#102—22no—
244#102—22no—
245#100—28no—
246#97—22no—
247#97—21no—
248C4×C8—22no—
249C4×C8—21no—
250C4×C8C3×D411yescharacteristic
251C4×C8C3×D411yescharacteristic
252C4×C8C3×D412yes—
253C23×C3—21no—
254C23×C3—42no—
255C23×C3—44no—
256C23×C3—82no—
257C23×C3—82no—
258C23×C3C2×#4111yescharacteristic
259C3×Q8—21no—
260C3×Q8—82no—
261C3×Q8—82no—
262C3×Q8#12412yes—
263C3×D4—24no—
264C3×D4—21no—
265C3×D4—44no—
266C3×D4—42no—
267C3×D4—42no—
268C3×D4—82no—
269C3×D4—82no—
270C3×D4—82no—
271C3×D4—82no—
272C3×D4#12412yes—
273C2×C3×C4—21no—
274C2×C3×C4—24no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—82no—
283C2×C3×C4#12111yescharacteristic
284C2×C3×C4#12111yescharacteristic
285C2×C3×C4#12111yescharacteristic
286C2×C3×C4#12111yescharacteristic
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C4C2×#4112yes—
290C2×C3×C4C2×#4112yes—
291C3×C8—22no—
292C3×C8—24no—
293C3×C8—22no—
294C3×C8—42no—
295C3×C8—42no—
296C24—22no—
297#41—22no—
298#41C3×#4114yes—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—82no—
302C2×Q8C2×C3×D411yescharacteristic
303C2×D4—21no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—82no—
313C2×D4—82no—
314C2×D4C2×C3×D411yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4C3×#4111yescharacteristic
320C22×C4C3×#4111yescharacteristic
321C22×C4C24×C311yescharacteristic
322SD16—44no—
323D8—44no—
324M16—28no—
325M16—22no—
326M16—24no—
327M16—24no—
328M16C3×#4114yes—
329C2×C8—28no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8C2×C3×D412yes—
336C2×C8C2×C3×D412yes—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4C2×C3×D411yescharacteristic
344C4⋊C4C2×C3×D411yescharacteristic
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42C2×C3×D411yescharacteristic
352C42C2×C3×D411yescharacteristic
353C42C2×C3×D411yescharacteristic
354C42C2×C3×D411yescharacteristic
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—22no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—82no—
361C22×C3—82no—
362C22×C3—82no—
363C22×C3—164no—
364C22×C3C2×#12111yescharacteristic
365C22×C3#52912yes—
366C3×C4—22no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—41no—
370C3×C4—41no—
371C3×C4—82no—
372C3×C4#47112yes—
373C3×C4#47212yes—
374C3×C4#52512yes—
375C3×C4C2×#12411yescharacteristic
376C3×C4#53211yescharacteristic
377C23—21no—
378C23—42no—
379C23—44no—
380C23—82no—
381C23—82no—
382C23C2×C3×#4111yescharacteristic
383Q8—21no—
384Q8—82no—
385Q8—82no—
386Q8C3×#12412yes—
387D4—24no—
388D4—21no—
389D4—44no—
390D4—42no—
391D4—42no—
392D4—82no—
393D4—82no—
394D4—82no—
395D4—82no—
396D4C3×#12412yes—
397C2×C4—21no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—82no—
407C2×C4C3×#12111yescharacteristic
408C2×C4C3×#12111yescharacteristic
409C2×C4C3×#12111yescharacteristic
410C2×C4C3×#12111yescharacteristic
411C2×C4C22×C3×D411yescharacteristic
412C2×C4C22×C3×D411yescharacteristic
413C2×C4C2×C3×#4112yes—
414C2×C4C2×C3×#4112yes—
415C8—22no—
416C8—24no—
417C8—22no—
418C8—42no—
419C8—42no—
420C2×C3—22no—
421C2×C3—42no—
422C2×C3—162no—
423C2×C3#285011yescharacteristic
424C2×C3C2×#47111yescharacteristic
425C2×C3C2×#47211yescharacteristic
426C22—21no—
427C22—21no—
428C22—22no—
429C22—41no—
430C22—41no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22—164no—
435C22C2×C3×#12111yescharacteristic
436C22C3×#52912yes—
437C4—21no—
438C4—21no—
439C4—22no—
440C4—41no—
441C4—41no—
442C4—82no—
443C4C3×#47112yes—
444C4C3×#47212yes—
445C4C3×#52512yes—
446C4C2×C3×#12411yescharacteristic
447C4C3×#53211yescharacteristic
448C3#3306011yescharacteristic, direct product factor, Sylow
449C2—22no—
450C2—42no—
451C2—162no—
452C2C3×#285011yescharacteristic
453C2C2×C3×#47111yescharacteristic
454C2C2×C3×#47211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#121
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3038C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#486C2semidirect
2. C3×#3079C2semidirect
4. C3×C4×#131C2semidirect
5. C3×#1997C2semidirect
6. C3×#1965C2semidirect
7. C3×#1931C2semidirect
8. C3×#1907C2semidirect
9. C3×#1837C2semidirect
10. #33060C3direct (N × H)
42. C2×#486C2×C3semidirect
43. #3079C2×C3semidirect
45. C4×#131C2×C3semidirect
46. #1997C2×C3semidirect
47. #1965C2×C3semidirect
48. #1931C2×C3semidirect
49. #1907C2×C3semidirect
50. #1837C2×C3semidirect
448. C3#33060direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4862
C3×#30792
C3×#30382
C3×C4×#1312
C3×#19972
C3×#19652
C3×#19312
C3×#19072
C3×#18372
#330603