Finite Group Databasebeta

← C3×#33044 #10651651

Subgroups — C3×#33044 #10651651

Order 768.

948 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#488C211yesmaximal, characteristic, complemented
2C22×C3×C4×C8C211yesmaximal, characteristic, complemented
3C3×#2851C211yesmaximal, characteristic, complemented
4C3×#2850C211yesmaximal, characteristic
5C3×#1991C211yesmaximal, characteristic, complemented
6C3×#1958C211yesmaximal, characteristic, complemented
7C3×#1934C212yesmaximal, complemented
8C3×#1933C212yesmaximal, complemented
9C3×#1893C212yesmaximal
10C3×#1892C212yesmaximal, complemented
11C3×#1835C211yesmaximal, characteristic, complemented
12#33044C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×C8C2212yes—
14C2×C3×#125C2211yescharacteristic
15C2×C3×#124C2211yescharacteristic
16C2×C3×C22⋊Q8C2211yescharacteristic
17C2×C3×#121C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C3×#488—24no—
20C2×C3×#103C2212yes—
21C2×C3×#103C2212yes—
22C2×C3×#102C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×C4×C8—24no—
25C2×C3×C4×C8C2211yescharacteristic
26C2×C3×C4×C8C2212yes—
27C2×C3×C4×C8C2211yescharacteristic
28C3×#391C2211yescharacteristic
29C3×#390C2211yescharacteristic
30C3×#389C2212yes—
31C3×#388—22no—
32C3×#387—22no—
33C3×#386—22no—
34C3×#385—22no—
35C3×#382C2211yescharacteristic
36C3×#382C2211yescharacteristic
37C3×#380C2212yes—
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C2×#488C2×C311yescharacteristic, complemented
43C22×C4×C8C2×C311yescharacteristic, complemented
44#2851C2×C311yescharacteristic, complemented
45#2850C2×C311yescharacteristic
46#1991C2×C311yescharacteristic, complemented
47#1958C2×C311yescharacteristic, complemented
48#1934C2×C312yescomplemented
49#1933C2×C312yescomplemented
50#1893C2×C312yes—
51#1892C2×C312yescomplemented
52#1835C2×C311yescharacteristic, complemented
53C22×C3×Q8C2311yescharacteristic
54C22×C3×D4—21no—
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4D412yes—
57C23×C3×C4C2311yescharacteristic
58C22×C3×C8—24no—
59C22×C3×C8—24no—
60C22×C3×C8—24no—
61C22×C3×C8C2312yes—
62C22×C3×C8C2312yes—
63C3×#125—22no—
64C3×#125—22no—
65C3×#124—22no—
66C3×#124—22no—
67C3×C22⋊Q8—24no—
68C3×#121—24no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C42—22no—
82C2×C3×C42—22no—
83C2×C3×C42D412yes—
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C3×#103—24no—
87C3×#103—24no—
88C3×#102—24no—
89C3×#102—24no—
90C3×C4×C8—24no—
91C3×C4×C8—22no—
92C3×C4×C8D414yescomplemented
93C3×#95—22no—
94C3×#95—22no—
95C3×#95—22no—
96C3×#95—22no—
97C3×#95—22no—
98C23×C8C22×C312yes—
99C2×#125C22×C311yescharacteristic
100C2×#124C22×C311yescharacteristic
101C2×C22⋊Q8C22×C311yescharacteristic
102C2×#121C22×C311yescharacteristic
103C22×C42C22×C311yescharacteristic
104#488—24no—
105C2×#103C22×C312yes—
106C2×#103C22×C312yes—
107C2×#102C22×C312yes—
108C2×#102C22×C312yes—
109C2×C4×C8—24no—
110C2×C4×C8C22×C312yes—
111C2×C4×C8C22×C311yescharacteristic
112C2×C4×C8C22×C311yescharacteristic
113#391C22×C311yescharacteristic
114#390C22×C311yescharacteristic
115#389C22×C312yes—
116#388—22no—
117#387—22no—
118#386—22no—
119#385—22no—
120#382C22×C311yescharacteristic
121#382C22×C311yescharacteristic
122#380C22×C312yes—
123#340C22×C312yes—
124#340C22×C312yes—
125#340C22×C312yes—
126#340C22×C312yes—
127C24×C3—21no—
128C24×C3C2×D411yescharacteristic
129C2×C3×Q8—21no—
130C2×C3×Q8—22no—
131C2×C3×Q8—22no—
132C2×C3×Q8—21no—
133C2×C3×D4—21no—
134C2×C3×D4—22no—
135C2×C3×D4—22no—
136C2×C3×D4—21no—
137C2×C3×D4—42no—
138C2×C3×D4—44no—
139C22×C3×C4—24no—
140C22×C3×C4—24no—
141C22×C3×C4—24no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D412yes—
153C22×C3×C4#4114yes—
154C22×C3×C4C2411yescharacteristic
155C2×C3×C8—216no—
156C2×C3×C8—22no—
157C2×C3×C8—22no—
158C2×C3×C8—22no—
159C2×C3×C8—22no—
160C2×C3×C8C2×D414yes—
161C2×C3×C8#4114yes—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—44no—
169C3×C4⋊C4—42no—
170C3×C4⋊C4—44no—
171C3×C22⋊C4—42no—
172C3×C22⋊C4—44no—
173C3×C22⋊C4—44no—
174C3×C22⋊C4—42no—
175C3×C22⋊C4—44no—
176C3×C42—24no—
177C3×C42—21no—
178C3×C42—21no—
179C3×C42C2×D412yes—
180C3×C42#4112yes—
181C22×Q8C23×C311yescharacteristic
182C22×D4—21no—
183C22×D4C23×C311yescharacteristic
184C23×C4C3×D412yes—
185C23×C4C23×C311yescharacteristic
186C22×C8—24no—
187C22×C8—24no—
188C22×C8—24no—
189C22×C8C23×C312yes—
190C22×C8C23×C312yes—
191#125—22no—
192#125—22no—
193#124—22no—
194#124—22no—
195C22⋊Q8—24no—
196#121—24no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4C23×C311yescharacteristic
201C2×C4⋊C4C23×C311yescharacteristic
202C2×C4⋊C4C23×C312yes—
203C2×C4⋊C4C23×C312yes—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42C3×D412yes—
212C2×C42C23×C311yescharacteristic
213C2×C42C23×C311yescharacteristic
214#103—24no—
215#103—24no—
216#102—24no—
217#102—24no—
218C4×C8—24no—
219C4×C8—22no—
220C4×C8C3×D414yescomplemented
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3—22no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3#12412yes—
235C23×C3#13514yes—
236C23×C3C22×D411yescharacteristic
237C3×Q8—42no—
238C3×Q8—42no—
239C3×D4—42no—
240C3×D4—42no—
241C3×D4—84no—
242C2×C3×C4—216no—
243C2×C3×C4—21no—
244C2×C3×C4—22no—
245C2×C3×C4—21no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—41no—
252C2×C3×C4—42no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4#12114yes—
265C2×C3×C4#12412yes—
266C2×C3×C4#13514yes—
267C2×C3×C4C22×D411yescharacteristic
268C2×C3×C4C2×#4112yes—
269C2×C3×C4C2×#4112yes—
270C2×C3×C4C2×#4111yescharacteristic
271C3×C8—24no—
272C3×C8—24no—
273C24—21no—
274C24C2×C3×D411yescharacteristic
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—21no—
279C2×D4—21no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—42no—
284C2×D4—44no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4C2×C3×D412yes—
297C22×C4C2×C3×D411yescharacteristic
298C22×C4C2×C3×D412yes—
299C22×C4C3×#4114yes—
300C22×C4C24×C311yescharacteristic
301C2×C8—216no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8C2×C3×D414yes—
307C2×C8C3×#4114yes—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—44no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C22⋊C4—42no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—42no—
321C22⋊C4—44no—
322C42—24no—
323C42—21no—
324C42—21no—
325C42C2×C3×D412yes—
326C42C3×#4112yes—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—24no—
330C22×C3—24no—
331C22×C3—81no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3—82no—
335C22×C3—82no—
336C22×C3#48814yescomplemented
337C22×C3C2×#12111yescharacteristic
338C22×C3C2×#12411yescharacteristic
339C22×C3#53211yescharacteristic
340C22×C3C2×#13512yes—
341C22×C3C2×#13512yes—
342C3×C4—22no—
343C3×C4—24no—
344C3×C4—24no—
345C3×C4—81no—
346C3×C4—81no—
347C3×C4—81no—
348C3×C4—82no—
349C3×C4—82no—
350C3×C4#52512yes—
351C3×C4#52912yes—
352C23—22no—
353C23—21no—
354C23—21no—
355C23—42no—
356C23—42no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23C3×#12412yes—
361C23C3×#13514yes—
362C23C22×C3×D411yescharacteristic
363Q8—42no—
364Q8—42no—
365D4—42no—
366D4—42no—
367D4—84no—
368C2×C4—216no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4C3×#12114yes—
391C2×C4C3×#12412yes—
392C2×C4C3×#13514yes—
393C2×C4C22×C3×D411yescharacteristic
394C2×C4C2×C3×#4112yes—
395C2×C4C2×C3×#4112yes—
396C2×C4C2×C3×#4111yescharacteristic
397C8—24no—
398C8—24no—
399C2×C3—24no—
400C2×C3—161no—
401C2×C3#285011yescharacteristic
402C2×C3#305012yes—
403C2×C3C2×#48811yescharacteristic
404C2×C3#313111yescharacteristic
405C2×C3#316812yes—
406C22—22no—
407C22—24no—
408C22—24no—
409C22—22no—
410C22—81no—
411C22—81no—
412C22—82no—
413C22—82no—
414C22—81no—
415C22C3×#48814yescomplemented
416C22C2×C3×#12111yescharacteristic
417C22C2×C3×#12411yescharacteristic
418C22C3×#53211yescharacteristic
419C22C2×C3×#13512yes—
420C22C2×C3×#13512yes—
421C4—24no—
422C4—24no—
423C4—22no—
424C4—81no—
425C4—81no—
426C4—82no—
427C4—82no—
428C4—81no—
429C4C3×#52512yes—
430C4C3×#52912yes—
431C3#3304411yescharacteristic, direct product factor, Sylow
432C2—24no—
433C2—161no—
434C2C3×#285011yescharacteristic
435C2C3×#305012yes—
436C2C2×C3×#48811yescharacteristic
437C2C3×#313111yescharacteristic
438C2C3×#316812yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#488C2semidirect
2. C22×C3×C4×C8C2semidirect
3. C3×#2851C2semidirect
5. C3×#1991C2semidirect
6. C3×#1958C2semidirect
7. C3×#1934C2semidirect
8. C3×#1933C2semidirect
10. C3×#1892C2semidirect
11. C3×#1835C2semidirect
12. #33044C3direct (N × H)
42. C2×#488C2×C3semidirect
43. C22×C4×C8C2×C3semidirect
44. #2851C2×C3semidirect
46. #1991C2×C3semidirect
47. #1958C2×C3semidirect
48. #1934C2×C3semidirect
49. #1933C2×C3semidirect
51. #1892C2×C3semidirect
52. #1835C2×C3semidirect
92. C3×C4×C8D4semidirect
220. C4×C8C3×D4semidirect
336. C22×C3#488semidirect
415. C22C3×#488semidirect
431. C3#33044direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4882
C22×C3×C4×C82
C3×#28512
C3×#28502
C3×#19912
C3×#19582
C3×#19342
C3×#19332
C3×#18932
C3×#18922
C3×#18352
#330443