Finite Group Databasebeta

← C3×#33003 #10651610

Subgroups — C3×#33003 #10651610

Order 768.

1376 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#480C211yesmaximal, characteristic
3C2×C3×#448C211yesmaximal, characteristic
4C3×#2854C211yesmaximal, characteristic
5C3×#1880C214yesmaximal, complemented
6C3×#1863C214yesmaximal, complemented
7C3×#1823C211yesmaximal, characteristic, complemented
8C3×#1817C211yesmaximal, characteristic, complemented
9C3×#1787C211yesmaximal, characteristic, complemented
10#33003C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×C3×SD16C2212yes—
13C2×C3×#123—21no—
14C2×C3×#121C2211yescharacteristic
15C2×C3×C4×Q8—24no—
16C2×C3×C4×Q8—24no—
17C2×C3×C4×Q8—22no—
18C2×C3×C4×Q8C2214yes—
19C22×C3×C4⋊C4—21no—
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C42—21no—
22C22×C3×C42C2211yescharacteristic
23C3×#480—24no—
24C3×#448—24no—
25C2×C3×#107C2212yes—
26C2×C3×#103C2214yes—
27C2×C3×#103C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×C22⋊C8C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31C3×#390C2212yes—
32C3×#389C2212yes—
33C3×#388—22no—
34C3×#386—21no—
35C3×#380C2211yescharacteristic
36C3×#340C2214yes—
37C3×#336C2214yes—
38C22×C4×Q8C2×C311yescharacteristic, complemented
39C2×#480C2×C311yescharacteristic
40C2×#448C2×C311yescharacteristic
41#2854C2×C311yescharacteristic
42#1880C2×C314yescomplemented
43#1863C2×C314yescomplemented
44#1823C2×C311yescharacteristic, complemented
45#1817C2×C311yescharacteristic, complemented
46#1787C2×C311yescharacteristic, complemented
47C22×C3×Q8—28no—
48C22×C3×Q8—22no—
49C22×C3×Q8—21no—
50C22×C3×Q8D414yes—
51C22×C3×Q8C2312yes—
52C22×C3×D4—21no—
53C22×C3×D4C2311yescharacteristic
54C23×C3×C4—21no—
55C23×C3×C4—21no—
56C23×C3×C4D411yescharacteristic
57C23×C3×C4D411yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C2×C3×SD16—24no—
60C2×C3×SD16—28no—
61C22×C3×C8C2312yes—
62C22×C3×C8C2312yes—
63C3×#123—44no—
64C3×#121—22no—
65C3×#121—22no—
66C3×C4×Q8—216no—
67C3×C4×Q8—28no—
68C3×C4×Q8—28no—
69C2×C3×C4⋊C4—28no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4D414yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C42—28no—
83C2×C3×C42—24no—
84C2×C3×C42—22no—
85C2×C3×C42—22no—
86C2×C3×C42D412yes—
87C2×C3×C42C2312yes—
88C3×#107—24no—
89C3×#103—24no—
90C3×#103—28no—
91C3×#102—24no—
92C3×C22⋊C8—22no—
93C3×C22⋊C8—22no—
94C3×#95—21no—
95C3×#95—22no—
96C23×Q8C22×C311yescharacteristic
97C22×SD16C22×C312yes—
98C2×#123—21no—
99C2×#121C22×C311yescharacteristic
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×Q8—22no—
103C2×C4×Q8C22×C314yes—
104C22×C4⋊C4—21no—
105C22×C4⋊C4C22×C311yescharacteristic
106C22×C42—21no—
107C22×C42C22×C311yescharacteristic
108#480—24no—
109#448—24no—
110C2×#107C22×C312yes—
111C2×#103C22×C312yes—
112C2×#103C22×C314yes—
113C2×#102C22×C312yes—
114C2×C22⋊C8C22×C311yescharacteristic
115C2×C22⋊C8C22×C311yescharacteristic
116#390C22×C312yes—
117#389C22×C312yes—
118#388—22no—
119#386—21no—
120#380C22×C311yescharacteristic
121#340C22×C314yes—
122#336C22×C314yes—
123C24×C3—21no—
124C24×C3C2×D411yescharacteristic
125C2×C3×Q8—216no—
126C2×C3×Q8—28no—
127C2×C3×Q8—216no—
128C2×C3×Q8—24no—
129C2×C3×Q8—22no—
130C2×C3×Q8—24no—
131C2×C3×Q8C2×D414yes—
132C2×C3×Q8#4118yes—
133C2×C3×D4—21no—
134C2×C3×D4—21no—
135C2×C3×D4—24no—
136C2×C3×D4—42no—
137C2×C3×D4—42no—
138C2×C3×D4—42no—
139C22×C3×C4—28no—
140C22×C3×C4—22no—
141C22×C3×C4—24no—
142C22×C3×C4—22no—
143C22×C3×C4—24no—
144C22×C3×C4—24no—
145C22×C3×C4—22no—
146C22×C3×C4—24no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—42no—
153C22×C3×C4—42no—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2411yescharacteristic
160C3×SD16—48no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8—24no—
164C2×C3×C8—24no—
165C3×C4⋊C4—216no—
166C3×C4⋊C4—24no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—44no—
172C3×C4⋊C4C2×D414yes—
173C3×C22⋊C4—44no—
174C3×C22⋊C4—42no—
175C3×C22⋊C4—42no—
176C3×C22⋊C4—44no—
177C3×C42—216no—
178C3×C42—24no—
179C3×C42—22no—
180C3×C42—22no—
181C22×Q8—28no—
182C22×Q8—22no—
183C22×Q8—21no—
184C22×Q8C3×D414yes—
185C22×Q8C23×C312yes—
186C22×D4—21no—
187C22×D4C23×C311yescharacteristic
188C23×C4—21no—
189C23×C4—21no—
190C23×C4C3×D411yescharacteristic
191C23×C4C3×D411yescharacteristic
192C23×C4C23×C311yescharacteristic
193C2×SD16—24no—
194C2×SD16—28no—
195C22×C8C23×C312yes—
196C22×C8C23×C312yes—
197#123—44no—
198#121—22no—
199#121—22no—
200C4×Q8—216no—
201C4×Q8—28no—
202C4×Q8—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4C3×D414yes—
209C2×C4⋊C4C23×C311yescharacteristic
210C2×C4⋊C4C23×C312yes—
211C2×C4⋊C4C23×C312yes—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C42—28no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42C3×D412yes—
221C2×C42C23×C312yes—
222#107—24no—
223#103—24no—
224#103—28no—
225#102—24no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228#95—21no—
229#95—22no—
230C23×C3—22no—
231C23×C3—22no—
232C23×C3—41no—
233C23×C3—42no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—42no—
237C23×C3#12012yes—
238C23×C3#13512yes—
239C23×C3AGL(1,ℤ/8)12yes—
240C23×C3C22×D411yescharacteristic
241C3×Q8—216no—
242C3×Q8—24no—
243C3×Q8—28no—
244C3×Q8#12318yescomplemented
245C3×D4—42no—
246C3×D4—42no—
247C3×D4—84no—
248C2×C3×C4—28no—
249C2×C3×C4—216no—
250C2×C3×C4—22no—
251C2×C3×C4—24no—
252C2×C3×C4—28no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—22no—
258C2×C3×C4—24no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—44no—
264C2×C3×C4—48no—
265C2×C3×C4—41no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4#12012yes—
272C2×C3×C4#13714yes—
273C2×C3×C4C22×D411yescharacteristic
274C2×C3×C4C22×D411yescharacteristic
275C2×C3×C4C2×#4114yes—
276C3×C8—42no—
277C3×C8—42no—
278C24—21no—
279C24C2×C3×D411yescharacteristic
280C2×Q8—216no—
281C2×Q8—28no—
282C2×Q8—216no—
283C2×Q8—24no—
284C2×Q8—22no—
285C2×Q8—24no—
286C2×Q8C2×C3×D414yes—
287C2×Q8C3×#4118yes—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—24no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C22×C4—28no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4C2×C3×D412yes—
310C22×C4C2×C3×D411yescharacteristic
311C22×C4C2×C3×D412yes—
312C22×C4C2×C3×D412yes—
313C22×C4C2×C3×D412yes—
314C22×C4C24×C311yescharacteristic
315SD16—48no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—24no—
319C2×C8—24no—
320C4⋊C4—216no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—24no—
326C4⋊C4—44no—
327C4⋊C4C2×C3×D414yes—
328C22⋊C4—44no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—44no—
332C42—216no—
333C42—24no—
334C42—22no—
335C42—22no—
336C22×C3—22no—
337C22×C3—24no—
338C22×C3—24no—
339C22×C3—22no—
340C22×C3—81no—
341C22×C3—82no—
342C22×C3—82no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3#44914yes—
346C22×C3C2×#12011yescharacteristic
347C22×C3#52512yes—
348C22×C3C2×#13511yescharacteristic
349C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
350C22×C3C2×#13712yes—
351C3×C4—28no—
352C3×C4—22no—
353C3×C4—24no—
354C3×C4—42no—
355C3×C4—44no—
356C3×C4—82no—
357C3×C4—81no—
358C3×C4C2×#12312yes—
359C3×C4#52512yes—
360C23—22no—
361C23—22no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—41no—
366C23—42no—
367C23C3×#12012yes—
368C23C3×#13512yes—
369C23C3×AGL(1,ℤ/8)12yes—
370C23C22×C3×D411yescharacteristic
371Q8—216no—
372Q8—24no—
373Q8—28no—
374Q8C3×#12318yescomplemented
375D4—42no—
376D4—42no—
377D4—84no—
378C2×C4—216no—
379C2×C4—28no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—28no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—48no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4C3×#12012yes—
402C2×C4C3×#13714yes—
403C2×C4C22×C3×D411yescharacteristic
404C2×C4C22×C3×D411yescharacteristic
405C2×C4C2×C3×#4114yes—
406C8—42no—
407C8—42no—
408C2×C3—24no—
409C2×C3—161no—
410C2×C3#284911yescharacteristic
411C2×C3C2×#44911yescharacteristic
412C2×C3#300411yescharacteristic
413C2×C3#310812yes—
414C2×C3#311212yes—
415C22—22no—
416C22—22no—
417C22—24no—
418C22—24no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—82no—
423C22—82no—
424C22C3×#44914yes—
425C22C2×C3×#12011yescharacteristic
426C22C3×#52512yes—
427C22C2×C3×#13511yescharacteristic
428C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
429C22C2×C3×#13712yes—
430C4—24no—
431C4—28no—
432C4—22no—
433C4—42no—
434C4—44no—
435C4—81no—
436C4—82no—
437C4C2×C3×#12312yes—
438C4C3×#52512yes—
439C3#3300311yescharacteristic, direct product factor, Sylow
440C2—24no—
441C2—161no—
442C2C3×#284911yescharacteristic
443C2C2×C3×#44911yescharacteristic
444C2C3×#300411yescharacteristic
445C2C3×#310812yes—
446C2C3×#311212yes—
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. C22×C3×C4×Q8C2semidirect
5. C3×#1880C2semidirect
6. C3×#1863C2semidirect
7. C3×#1823C2semidirect
8. C3×#1817C2semidirect
9. C3×#1787C2semidirect
10. #33003C3direct (N × H)
38. C22×C4×Q8C2×C3semidirect
42. #1880C2×C3semidirect
43. #1863C2×C3semidirect
44. #1823C2×C3semidirect
45. #1817C2×C3semidirect
46. #1787C2×C3semidirect
244. C3×Q8#123semidirect
374. Q8C3×#123semidirect
439. C3#33003direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×Q82
C2×C3×#4802
C2×C3×#4482
C3×#28542
C3×#18802
C3×#18632
C3×#18232
C3×#18172
C3×#17872
#330033