Finite Group Databasebeta

← C3×#23554 #10642161

Subgroups — C3×#23554 #10642161

Order 768.

1202 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic
2C2×C3×#428C212yesmaximal
3C2×C3×#389C211yesmaximal, characteristic
4C3×#2059C214yesmaximal, complemented
5C3×#2058C212yesmaximal, complemented
6C3×#2057C212yesmaximal, complemented
7C3×#1736C212yesmaximal
8C3×#1728C211yesmaximal, characteristic
9#23554C311yesmaximal, characteristic, direct product factor, Sylow
10C22×C3×M16C2212yes—
11C3×#533—28no—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#125C2212yes—
15C2×C3×C22⋊Q8—21no—
16C2×C3×C22⋊Q8—21no—
17C2×C3×C4×Q8—21no—
18C2×C3×C4×Q8—21no—
19C2×C3×#117C2212yes—
20C22×C3×C4⋊C4—21no—
21C22×C3×C4⋊C4C2211yescharacteristic
22C22×C3×C4⋊C4C2211yescharacteristic
23C22×C3×C42C2211yescharacteristic
24C3×#428—28no—
25C2×C3×#107C2214yes—
26C2×C3×#106C2214yes—
27C3×#389—21no—
28C3×#389—21no—
29C3×#389C2212yes—
30C3×#389C2212yes—
31C2×C3×#95—21no—
32C3×#340C2214yes—
33C3×#340C2214yes—
34C3×#336C2212yes—
35C3×#336C2212yes—
36C2×#533C2×C311yescharacteristic
37C2×#428C2×C312yes—
38C2×#389C2×C311yescharacteristic
39#2059C2×C314yescomplemented
40#2058C2×C312yescomplemented
41#2057C2×C312yescomplemented
42#1736C2×C312yes—
43#1728C2×C311yescharacteristic
44C22×C3×Q8—21no—
45C22×C3×Q8—21no—
46C23×C3×C4—21no—
47C23×C3×C4—21no—
48C23×C3×C4D411yescharacteristic
49C23×C3×C4D411yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C2×C3×M16—24no—
52C2×C3×M16—24no—
53C2×C3×M16Q818yescomplemented
54C22×C3×C8C2314yes—
55C3×C4⋊Q8—24no—
56C3×C4⋊Q8—24no—
57C3×#125—24no—
58C3×#125—24no—
59C3×C22⋊Q8—28no—
60C3×C22⋊Q8—28no—
61C3×C4×Q8—28no—
62C3×C4×Q8—28no—
63C3×#117—24no—
64C3×#117D418yes—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—44no—
76C2×C3×C4⋊C4Q814yescomplemented
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—22no—
82C2×C3×C42—24no—
83C2×C3×C42—22no—
84C2×C3×C42D411yescharacteristic
85C2×C3×C42D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C3×#107—28no—
89C3×#106—28no—
90C3×#95—22no—
91C3×#95—22no—
92C22×M16C22×C312yes—
93#533—28no—
94C2×C4⋊Q8C22×C311yescharacteristic
95C2×C4⋊Q8C22×C311yescharacteristic
96C2×#125C22×C312yes—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—21no—
99C2×C4×Q8—21no—
100C2×C4×Q8—21no—
101C2×#117C22×C312yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4C22×C311yescharacteristic
104C22×C4⋊C4C22×C311yescharacteristic
105C22×C42C22×C311yescharacteristic
106#428—28no—
107C2×#107C22×C314yes—
108C2×#106C22×C314yes—
109#389—21no—
110#389—21no—
111#389C22×C312yes—
112#389C22×C312yes—
113C2×#95—21no—
114#340C22×C314yes—
115#340C22×C314yes—
116#336C22×C312yes—
117#336C22×C312yes—
118C24×C3C2×D411yescharacteristic
119C2×C3×Q8—24no—
120C2×C3×Q8—24no—
121C2×C3×Q8—42no—
122C2×C3×Q8—42no—
123C2×C3×Q8—42no—
124C2×C3×Q8—42no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—24no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4—42no—
137C22×C3×C4—42no—
138C22×C3×C4—42no—
139C22×C3×C4—42no—
140C22×C3×C4—42no—
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D414yes—
147C22×C3×C4C2×Q814yes—
148C22×C3×C4C2×Q812yes—
149C22×C3×C4#4112yes—
150C22×C3×C4C2411yescharacteristic
151C3×M16—216no—
152C2×C3×C8—24no—
153C2×C3×C8—24no—
154C2×C3×C8C2×Q818yes—
155C3×C4⋊C4—28no—
156C3×C4⋊C4—28no—
157C3×C4⋊C4—24no—
158C3×C4⋊C4—24no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—24no—
164C3×C4⋊C4—44no—
165C3×C4⋊C4—42no—
166C3×C4⋊C4—44no—
167C3×C4⋊C4C2×D414yes—
168C3×C4⋊C4C2×D414yes—
169C3×C4⋊C4C2×Q814yes—
170C3×C22⋊C4—28no—
171C3×C42—22no—
172C3×C42—22no—
173C3×C42—28no—
174C3×C42—22no—
175C3×C42—22no—
176C22×Q8—21no—
177C22×Q8—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4C3×D411yescharacteristic
181C23×C4C3×D411yescharacteristic
182C23×C4C23×C311yescharacteristic
183C2×M16—24no—
184C2×M16—24no—
185C2×M16C3×Q818yescomplemented
186C22×C8C23×C314yes—
187C4⋊Q8—24no—
188C4⋊Q8—24no—
189#125—24no—
190#125—24no—
191C22⋊Q8—28no—
192C22⋊Q8—28no—
193C4×Q8—28no—
194C4×Q8—28no—
195#117—24no—
196#117C3×D418yes—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4C3×Q814yescomplemented
209C2×C4⋊C4C23×C312yes—
210C2×C4⋊C4C23×C312yes—
211C2×C4⋊C4C23×C312yes—
212C2×C4⋊C4C23×C312yes—
213C2×C22⋊C4—22no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42C3×D411yescharacteristic
217C2×C42C3×D411yescharacteristic
218C2×C42C23×C311yescharacteristic
219C2×C42C23×C311yescharacteristic
220#107—28no—
221#106—28no—
222#95—22no—
223#95—22no—
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3#12012yes—
229C23×C3C22⋊Q812yes—
230C23×C3C4⋊Q812yes—
231C23×C3C22×D411yescharacteristic
232C3×Q8—48no—
233C3×Q8—48no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—24no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—44no—
248C2×C3×C4—44no—
249C2×C3×C4—48no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—84no—
260C2×C3×C4#12012yes—
261C2×C3×C4C22⋊Q818yes—
262C2×C3×C4C22⋊Q812yes—
263C2×C3×C4C4⋊Q818yes—
264C2×C3×C4C4⋊Q812yes—
265C2×C3×C4AGL(1,ℤ/8)12yes—
266C2×C3×C4#13712yes—
267C2×C3×C4C22×D412yes—
268C2×C3×C4C22×Q812yes—
269C2×C3×C4C22×Q811yescharacteristic
270C2×C3×C4C2×#4111yescharacteristic
271C3×C8—28no—
272C24C2×C3×D411yescharacteristic
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4C2×C3×D411yescharacteristic
296C22×C4C2×C3×D411yescharacteristic
297C22×C4C2×C3×D411yescharacteristic
298C22×C4C2×C3×D411yescharacteristic
299C22×C4C2×C3×D411yescharacteristic
300C22×C4C2×C3×D414yes—
301C22×C4C2×C3×Q814yes—
302C22×C4C2×C3×Q812yes—
303C22×C4C3×#4112yes—
304C22×C4C24×C311yescharacteristic
305M16—216no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8C2×C3×Q818yes—
309C4⋊C4—28no—
310C4⋊C4—28no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—44no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4C2×C3×D414yes—
322C4⋊C4C2×C3×D414yes—
323C4⋊C4C2×C3×Q814yes—
324C22⋊C4—28no—
325C42—22no—
326C42—22no—
327C42—28no—
328C42—22no—
329C42—22no—
330C22×C3—22no—
331C22×C3—22no—
332C22×C3—22no—
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—22no—
336C22×C3#39514yes—
337C22×C3C2×#12011yescharacteristic
338C22×C3C2×C22⋊Q811yescharacteristic
339C22×C3C2×C4⋊Q811yescharacteristic
340C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
341C22×C3C2×#13711yescharacteristic
342C22×C3#57511yescharacteristic
343C22×C3#57511yescharacteristic
344C3×C4—22no—
345C3×C4—22no—
346C3×C4—44no—
347C3×C4—41no—
348C3×C4—44no—
349C3×C4—41no—
350C3×C4—44no—
351C3×C4—82no—
352C3×C4#39514yes—
353C3×C4C2×C22⋊Q812yes—
354C3×C4C2×C4⋊Q812yes—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23C3×#12012yes—
360C23C3×C22⋊Q812yes—
361C23C3×C4⋊Q812yes—
362C23C22×C3×D411yescharacteristic
363Q8—48no—
364Q8—48no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—48no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—84no—
391C2×C4C3×#12012yes—
392C2×C4C3×C22⋊Q812yes—
393C2×C4C3×C22⋊Q818yes—
394C2×C4C3×C4⋊Q812yes—
395C2×C4C3×C4⋊Q818yes—
396C2×C4C3×AGL(1,ℤ/8)12yes—
397C2×C4C3×#13712yes—
398C2×C4C22×C3×D412yes—
399C2×C4C22×C3×Q812yes—
400C2×C4C22×C3×Q811yescharacteristic
401C2×C4C2×C3×#4111yescharacteristic
402C8—28no—
403C2×C3—24no—
404C2×C3C2×#39511yescharacteristic
405C2×C3#300611yescharacteristic
406C2×C3#300711yescharacteristic
407C2×C3#307711yescharacteristic
408C2×C3#307811yescharacteristic
409C2×C3#316511yescharacteristic
410C2×C3#316611yescharacteristic
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22C3×#39514yes—
418C22C2×C3×#12011yescharacteristic
419C22C2×C3×C22⋊Q811yescharacteristic
420C22C2×C3×C4⋊Q811yescharacteristic
421C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
422C22C2×C3×#13711yescharacteristic
423C22C3×#57511yescharacteristic
424C22C3×#57511yescharacteristic
425C4—22no—
426C4—22no—
427C4—41no—
428C4—44no—
429C4—41no—
430C4—44no—
431C4—44no—
432C4—82no—
433C4C3×#39514yes—
434C4C2×C3×C22⋊Q812yes—
435C4C2×C3×C4⋊Q812yes—
436C3#2355411yescharacteristic, direct product factor, Sylow
437C2—24no—
438C2C2×C3×#39511yescharacteristic
439C2C3×#300611yescharacteristic
440C2C3×#300711yescharacteristic
441C2C3×#307711yescharacteristic
442C2C3×#307811yescharacteristic
443C2C3×#316511yescharacteristic
444C2C3×#316611yescharacteristic
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
4. C3×#2059C2semidirect
5. C3×#2058C2semidirect
6. C3×#2057C2semidirect
9. #23554C3direct (N × H)
39. #2059C2×C3semidirect
40. #2058C2×C3semidirect
41. #2057C2×C3semidirect
53. C2×C3×M16Q8semidirect
76. C2×C3×C4⋊C4Q8semidirect
185. C2×M16C3×Q8semidirect
208. C2×C4⋊C4C3×Q8semidirect
436. C3#23554direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×C3×#4282
C2×C3×#3892
C3×#20592
C3×#20582
C3×#20572
C3×#17362
C3×#17282
#235543