Finite Group Databasebeta

← C3×#33049 #10651656

Subgroups — C3×#33049 #10651656

Order 768.

914 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×#487C211yesmaximal, characteristic, complemented
2C2×C3×C4×M16C211yesmaximal, characteristic
3C3×#2857C211yesmaximal, characteristic
4C3×#2857C211yesmaximal, characteristic
5C3×#1990C211yesmaximal, characteristic, complemented
6C3×#1956C211yesmaximal, characteristic, complemented
7C3×#1935C214yesmaximal, complemented
8C3×#1894C214yesmaximal
9C3×#1834C211yesmaximal, characteristic, complemented
10#33049C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×M16C2212yes—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C22⋊Q8—21no—
15C2×C3×C22⋊Q8C2212yes—
16C22×C3×C42C2211yescharacteristic
17C3×#487—24no—
18C2×C3×#103C2214yes—
19C2×C3×#103C2214yes—
20C3×C4×M16—24no—
21C2×C3×#97C2212yes—
22C2×C3×C4×C8C2211yescharacteristic
23C2×C3×C4×C8C2211yescharacteristic
24C3×#391—21no—
25C3×#391C2212yes—
26C3×#389—21no—
27C3×#389—21no—
28C3×#389—21no—
29C3×#389C2212yes—
30C3×#384C2211yescharacteristic
31C3×#384C2211yescharacteristic
32C3×#380—21no—
33C3×#380—21no—
34C3×#380—21no—
35C3×#380C2212yes—
36C3×#340C2214yes—
37C3×#340C2214yes—
38C2×#487C2×C311yescharacteristic, complemented
39C2×C4×M16C2×C311yescharacteristic
40#2857C2×C311yescharacteristic
41#2857C2×C311yescharacteristic
42#1990C2×C311yescharacteristic, complemented
43#1956C2×C311yescharacteristic, complemented
44#1935C2×C314yescomplemented
45#1894C2×C314yes—
46#1834C2×C311yescharacteristic, complemented
47C22×C3×Q8—21no—
48C22×C3×Q8C2312yes—
49C23×C3×C4D411yescharacteristic
50C23×C3×C4D411yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C2×C3×M16—24no—
53C2×C3×M16—24no—
54C2×C3×M16—24no—
55C22×C3×C8C2312yes—
56C22×C3×C8C2312yes—
57C3×C4⋊Q8—22no—
58C3×C4⋊Q8—22no—
59C3×C4⋊Q8—22no—
60C3×C4⋊Q8—22no—
61C3×C22⋊Q8—28no—
62C3×C22⋊Q8—44no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C42—22no—
78C2×C3×C42—22no—
79C2×C3×C42D411yescharacteristic
80C2×C3×C42D411yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C3×#103—28no—
84C3×#103—28no—
85C3×#97—24no—
86C3×C4×C8—22no—
87C3×C4×C8D414yes—
88C3×#95—22no—
89C3×#95—22no—
90C3×#95—22no—
91C3×#95—22no—
92C22×M16C22×C312yes—
93C2×C4⋊Q8C22×C311yescharacteristic
94C2×C4⋊Q8C22×C311yescharacteristic
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8C22×C312yes—
97C22×C42C22×C311yescharacteristic
98#487—24no—
99C2×#103C22×C314yes—
100C2×#103C22×C314yes—
101C4×M16—24no—
102C2×#97C22×C312yes—
103C2×C4×C8C22×C311yescharacteristic
104C2×C4×C8C22×C311yescharacteristic
105#391—21no—
106#391C22×C312yes—
107#389—21no—
108#389—21no—
109#389—21no—
110#389C22×C312yes—
111#384C22×C311yescharacteristic
112#384C22×C311yescharacteristic
113#380—21no—
114#380—21no—
115#380—21no—
116#380C22×C312yes—
117#340C22×C314yes—
118#340C22×C314yes—
119C24×C3C2×D411yescharacteristic
120C2×C3×Q8—22no—
121C2×C3×Q8—24no—
122C2×C3×Q8—24no—
123C2×C3×Q8—22no—
124C2×C3×Q8—42no—
125C2×C3×Q8—42no—
126C2×C3×Q8—42no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4#4114yes—
146C22×C3×C4C2411yescharacteristic
147C3×M16—216no—
148C2×C3×C8—22no—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C2×C3×C8C2×D414yes—
153C2×C3×C8#4114yes—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—44no—
159C3×C4⋊C4—42no—
160C3×C4⋊C4—42no—
161C3×C4⋊C4—42no—
162C3×C4⋊C4—44no—
163C3×C4⋊C4—42no—
164C3×C4⋊C4—42no—
165C3×C4⋊C4—42no—
166C3×C4⋊C4—42no—
167C3×C22⋊C4—44no—
168C3×C22⋊C4—44no—
169C3×C42—22no—
170C3×C42—22no—
171C3×C42—21no—
172C3×C42—21no—
173C3×C42C2×D412yes—
174C3×C42#4112yes—
175C22×Q8—21no—
176C22×Q8C23×C312yes—
177C23×C4C3×D411yescharacteristic
178C23×C4C3×D411yescharacteristic
179C23×C4C23×C311yescharacteristic
180C2×M16—24no—
181C2×M16—24no—
182C2×M16—24no—
183C22×C8C23×C312yes—
184C22×C8C23×C312yes—
185C4⋊Q8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C22⋊Q8—28no—
190C22⋊Q8—44no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4C23×C312yes—
201C2×C4⋊C4C23×C312yes—
202C2×C4⋊C4C23×C312yes—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42C3×D411yescharacteristic
208C2×C42C3×D411yescharacteristic
209C2×C42C23×C311yescharacteristic
210C2×C42C23×C311yescharacteristic
211#103—28no—
212#103—28no—
213#97—24no—
214C4×C8—22no—
215C4×C8C3×D414yes—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—21no—
224C23×C3—21no—
225C23×C3—21no—
226C23×C3#12412yes—
227C23×C3C22×D411yescharacteristic
228C3×Q8—44no—
229C3×Q8—44no—
230C3×Q8—84no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—24no—
234C2×C3×C4—24no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—44no—
242C2×C3×C4—44no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4#12112yes—
256C2×C3×C4#12112yes—
257C2×C3×C4#12412yes—
258C2×C3×C4AGL(1,ℤ/8)12yes—
259C2×C3×C4#13712yes—
260C2×C3×C4#13712yes—
261C2×C3×C4#13712yes—
262C2×C3×C4C22×D411yescharacteristic
263C2×C3×C4C2×#4112yes—
264C2×C3×C4C2×#4112yes—
265C2×C3×C4C2×#4111yescharacteristic
266C3×C8—24no—
267C3×C8—24no—
268C24C2×C3×D411yescharacteristic
269C2×Q8—22no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—22no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4C2×C3×D411yescharacteristic
290C22×C4C2×C3×D411yescharacteristic
291C22×C4C2×C3×D411yescharacteristic
292C22×C4C2×C3×D411yescharacteristic
293C22×C4C2×C3×D411yescharacteristic
294C22×C4C3×#4114yes—
295C22×C4C24×C311yescharacteristic
296M16—216no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8C2×C3×D414yes—
302C2×C8C3×#4114yes—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—22no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C42—22no—
319C42—22no—
320C42—21no—
321C42—21no—
322C42C2×C3×D412yes—
323C42C3×#4112yes—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3C2×#12111yescharacteristic
332C22×C3C2×#12411yescharacteristic
333C22×C3#53211yescharacteristic
334C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
335C22×C3C2×#13711yescharacteristic
336C22×C3C2×#13711yescharacteristic
337C22×C3C2×#13711yescharacteristic
338C3×C4—22no—
339C3×C4—41no—
340C3×C4—41no—
341C3×C4—82no—
342C3×C4—82no—
343C3×C4—82no—
344C3×C4—82no—
345C3×C4#46814yes—
346C3×C4#47014yes—
347C3×C4#52512yes—
348C3×C4#52912yes—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23C3×#12412yes—
356C23C22×C3×D411yescharacteristic
357Q8—44no—
358Q8—44no—
359Q8—84no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4C3×#12112yes—
385C2×C4C3×#12112yes—
386C2×C4C3×#12412yes—
387C2×C4C3×AGL(1,ℤ/8)12yes—
388C2×C4C3×#13712yes—
389C2×C4C3×#13712yes—
390C2×C4C3×#13712yes—
391C2×C4C22×C3×D411yescharacteristic
392C2×C4C2×C3×#4112yes—
393C2×C4C2×C3×#4112yes—
394C2×C4C2×C3×#4111yescharacteristic
395C8—24no—
396C8—24no—
397C2×C3—42no—
398C2×C3#285011yescharacteristic
399C2×C3C2×#46811yescharacteristic
400C2×C3C2×#47011yescharacteristic
401C2×C3#313911yescharacteristic
402C2×C3#314111yescharacteristic
403C2×C3#318211yescharacteristic
404C2×C3#318511yescharacteristic
405C22—22no—
406C22—22no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22C2×C3×#12111yescharacteristic
413C22C2×C3×#12411yescharacteristic
414C22C3×#53211yescharacteristic
415C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
416C22C2×C3×#13711yescharacteristic
417C22C2×C3×#13711yescharacteristic
418C22C2×C3×#13711yescharacteristic
419C4—22no—
420C4—41no—
421C4—41no—
422C4—82no—
423C4—82no—
424C4—82no—
425C4—82no—
426C4C3×#46814yes—
427C4C3×#47014yes—
428C4C3×#52512yes—
429C4C3×#52912yes—
430C3#3304911yescharacteristic, direct product factor, Sylow
431C2—42no—
432C2C3×#285011yescharacteristic
433C2C2×C3×#46811yescharacteristic
434C2C2×C3×#47011yescharacteristic
435C2C3×#313911yescharacteristic
436C2C3×#314111yescharacteristic
437C2C3×#318211yescharacteristic
438C2C3×#318511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
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×#487C2semidirect
5. C3×#1990C2semidirect
6. C3×#1956C2semidirect
7. C3×#1935C2semidirect
9. C3×#1834C2semidirect
10. #33049C3direct (N × H)
38. C2×#487C2×C3semidirect
42. #1990C2×C3semidirect
43. #1956C2×C3semidirect
44. #1935C2×C3semidirect
46. #1834C2×C3semidirect
430. C3#33049direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4872
C2×C3×C4×M162
C3×#28572
C3×#19902
C3×#19562
C3×#19352
C3×#18942
C3×#18342
#330493