Finite Group Databasebeta

← C3×#29688 #10648295

Subgroups — C3×#29688 #10648295

Order 768.

1068 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic
2C2×C3×#475C212yesmaximal, complemented
3C3×#2857C211yesmaximal, characteristic
4C3×#2058C212yesmaximal, complemented
5C3×#2043C212yesmaximal, complemented
6C3×#2035C214yesmaximal
7C3×#1803C212yesmaximal, complemented
8C3×#1797C211yesmaximal, characteristic
9#29688C311yesmaximal, characteristic, direct product factor, Sylow
10C3×#533—28no—
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#125—21no—
15C2×C3×C22⋊Q8—21no—
16C2×C3×C22⋊Q8—21no—
17C2×C3×C22⋊Q8C2212yes—
18C2×C3×C4×Q8—21no—
19C2×C3×C4×Q8C2212yes—
20C2×C3×#117—21no—
21C22×C3×C42C2211yescharacteristic
22C3×#475—28no—
23C2×C3×#106C2214yes—
24C2×C3×C4⋊C8C2212yes—
25C2×C3×#103C2214yes—
26C2×C3×#103C2214yes—
27C2×C3×C22⋊C8C2212yes—
28C3×#391—21no—
29C3×#389—21no—
30C3×#389C2212yes—
31C3×#384C2211yescharacteristic
32C3×#380—21no—
33C3×#380C2212yes—
34C3×#340C2214yes—
35C3×#336C2212yes—
36C2×#533C2×C311yescharacteristic
37C2×#475C2×C312yescomplemented
38#2857C2×C311yescharacteristic
39#2058C2×C312yescomplemented
40#2043C2×C312yescomplemented
41#2035C2×C314yes—
42#1803C2×C312yescomplemented
43#1797C2×C311yescharacteristic
44C22×C3×Q8—21no—
45C22×C3×Q8—21no—
46C22×C3×Q8C2312yes—
47C23×C3×C4D411yescharacteristic
48C23×C3×C4D411yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C22×C3×C8C2314yes—
51C3×C4⋊Q8—24no—
52C3×C4⋊Q8—24no—
53C3×C4⋊Q8—24no—
54C3×#125—28no—
55C3×C22⋊Q8—28no—
56C3×C22⋊Q8—24no—
57C3×C22⋊Q8—24no—
58C3×C22⋊Q8—44no—
59C3×C4×Q8—28no—
60C3×C4×Q8—24no—
61C3×C4×Q8D418yes—
62C3×#117—28no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
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—21no—
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—24no—
79C2×C3×C42D411yescharacteristic
80C2×C3×C42D411yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C3×#106—28no—
84C3×C4⋊C8—24no—
85C3×#103—28no—
86C3×#103—44no—
87C3×C22⋊C8—42no—
88C3×#95—22no—
89C3×#95—22no—
90#533—28no—
91C2×C4⋊Q8C22×C311yescharacteristic
92C2×C4⋊Q8C22×C311yescharacteristic
93C2×C4⋊Q8C22×C311yescharacteristic
94C2×#125—21no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8C22×C312yes—
98C2×C4×Q8—21no—
99C2×C4×Q8C22×C312yes—
100C2×#117—21no—
101C22×C42C22×C311yescharacteristic
102#475—28no—
103C2×#106C22×C314yes—
104C2×C4⋊C8C22×C312yes—
105C2×#103C22×C314yes—
106C2×#103C22×C314yes—
107C2×C22⋊C8C22×C312yes—
108#391—21no—
109#389—21no—
110#389C22×C312yes—
111#384C22×C311yescharacteristic
112#380—21no—
113#380C22×C312yes—
114#340C22×C314yes—
115#336C22×C312yes—
116C24×C3C2×D411yescharacteristic
117C2×C3×Q8—24no—
118C2×C3×Q8—22no—
119C2×C3×Q8—22no—
120C2×C3×Q8—22no—
121C2×C3×Q8—22no—
122C2×C3×Q8—22no—
123C2×C3×Q8—42no—
124C2×C3×Q8—42no—
125C2×C3×Q8—42no—
126C2×C3×Q8—42no—
127C2×C3×Q8—42no—
128C2×C3×Q8C2×D414yes—
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—21no—
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—41no—
142C22×C3×C4SD1614yes—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2411yescharacteristic
149C2×C3×C8—24no—
150C2×C3×C8—24no—
151C2×C3×C8—24no—
152C3×C4⋊C4—24no—
153C3×C4⋊C4—24no—
154C3×C4⋊C4—28no—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—28no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—24no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—42no—
162C3×C4⋊C4—42no—
163C3×C4⋊C4—42no—
164C3×C4⋊C4—44no—
165C3×C4⋊C4C2×D414yes—
166C3×C22⋊C4—28no—
167C3×C22⋊C4—44no—
168C3×C42—22no—
169C3×C42—22no—
170C3×C42—28no—
171C3×C42—22no—
172C3×C42C2×D414yes—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8C23×C312yes—
176C23×C4C3×D411yescharacteristic
177C23×C4C3×D411yescharacteristic
178C23×C4C23×C311yescharacteristic
179C22×C8C23×C314yes—
180C4⋊Q8—24no—
181C4⋊Q8—24no—
182C4⋊Q8—24no—
183#125—28no—
184C22⋊Q8—28no—
185C22⋊Q8—24no—
186C22⋊Q8—24no—
187C22⋊Q8—44no—
188C4×Q8—28no—
189C4×Q8—24no—
190C4×Q8C3×D418yes—
191#117—28no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C23×C312yes—
202C2×C4⋊C4C23×C312yes—
203C2×C4⋊C4C23×C312yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C42—22no—
207C2×C42—24no—
208C2×C42C3×D411yescharacteristic
209C2×C42C3×D411yescharacteristic
210C2×C42C23×C311yescharacteristic
211C2×C42C23×C311yescharacteristic
212#106—28no—
213C4⋊C8—24no—
214#103—28no—
215#103—44no—
216C22⋊C8—42no—
217#95—22no—
218#95—22no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—21no—
224C23×C3—21no—
225C23×C3C2×SD1612yes—
226C23×C3C22×D411yescharacteristic
227C3×Q8—24no—
228C3×Q8—24no—
229C3×Q8—48no—
230C3×Q8—42no—
231C3×Q8—42no—
232C3×Q8—84no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—28no—
240C2×C3×C4—21no—
241C2×C3×C4—24no—
242C2×C3×C4—24no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—44no—
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—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4#12012yes—
259C2×C3×C4#12012yes—
260C2×C3×C4C2×SD1612yes—
261C2×C3×C4C2×SD1612yes—
262C2×C3×C4#13712yes—
263C2×C3×C4#13712yes—
264C2×C3×C4C22×D412yes—
265C2×C3×C4H(5,𝔽2)11yescharacteristic
266C2×C3×C42−1+411yescharacteristic
267C2×C3×C42−1+412yes—
268C3×C8—44no—
269C24C2×C3×D411yescharacteristic
270C2×Q8—24no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8C2×C3×D414yes—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4C3×SD1614yes—
296C22×C4C2×C3×D411yescharacteristic
297C22×C4C2×C3×D411yescharacteristic
298C22×C4C2×C3×D411yescharacteristic
299C22×C4C2×C3×D411yescharacteristic
300C22×C4C2×C3×D411yescharacteristic
301C22×C4C24×C311yescharacteristic
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—28no—
308C4⋊C4—24no—
309C4⋊C4—28no—
310C4⋊C4—22no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4C2×C3×D414yes—
319C22⋊C4—28no—
320C22⋊C4—44no—
321C42—22no—
322C42—22no—
323C42—28no—
324C42—22no—
325C42C2×C3×D414yes—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3C2×#12011yescharacteristic
334C22×C3#53611yescharacteristic
335C22×C3#53711yescharacteristic
336C22×C3C22×SD1611yescharacteristic
337C22×C3C2×#13711yescharacteristic
338C22×C3C2×#13711yescharacteristic
339C22×C3#57511yescharacteristic
340C3×C4—22no—
341C3×C4—22no—
342C3×C4—22no—
343C3×C4—41no—
344C3×C4—44no—
345C3×C4—44no—
346C3×C4—82no—
347C3×C4—82no—
348C3×C4#44814yes—
349C3×C4#53612yes—
350C3×C4#53712yes—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23C2×C3×SD1612yes—
358C23C22×C3×D411yescharacteristic
359Q8—24no—
360Q8—24no—
361Q8—48no—
362Q8—42no—
363Q8—42no—
364Q8—84no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—28no—
372C2×C4—21no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
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×C4C3×#12012yes—
391C2×C4C3×#12012yes—
392C2×C4C2×C3×SD1612yes—
393C2×C4C2×C3×SD1612yes—
394C2×C4C3×#13712yes—
395C2×C4C3×#13712yes—
396C2×C4C22×C3×D412yes—
397C2×C4C3×H(5,𝔽2)11yescharacteristic
398C2×C4C3×2−1+411yescharacteristic
399C2×C4C3×2−1+412yes—
400C8—44no—
401C2×C3—42no—
402C2×C3#261611yescharacteristic
403C2×C3C2×#44811yescharacteristic
404C2×C3#300711yescharacteristic
405C2×C3#322911yescharacteristic
406C2×C3#323711yescharacteristic
407C2×C3#324611yescharacteristic
408C2×C3#325511yescharacteristic
409C22—22no—
410C22—22no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22C2×C3×#12011yescharacteristic
417C22C3×#53611yescharacteristic
418C22C3×#53711yescharacteristic
419C22C22×C3×SD1611yescharacteristic
420C22C2×C3×#13711yescharacteristic
421C22C2×C3×#13711yescharacteristic
422C22C3×#57511yescharacteristic
423C4—22no—
424C4—22no—
425C4—22no—
426C4—41no—
427C4—44no—
428C4—44no—
429C4—82no—
430C4—82no—
431C4C3×#44814yes—
432C4C3×#53612yes—
433C4C3×#53712yes—
434C3#2968811yescharacteristic, direct product factor, Sylow
435C2—42no—
436C2C3×#261611yescharacteristic
437C2C2×C3×#44811yescharacteristic
438C2C3×#300711yescharacteristic
439C2C3×#322911yescharacteristic
440C2C3×#323711yescharacteristic
441C2C3×#324611yescharacteristic
442C2C3×#325511yescharacteristic
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
2. C2×C3×#475C2semidirect
4. C3×#2058C2semidirect
5. C3×#2043C2semidirect
7. C3×#1803C2semidirect
9. #29688C3direct (N × H)
37. C2×#475C2×C3semidirect
39. #2058C2×C3semidirect
40. #2043C2×C3semidirect
42. #1803C2×C3semidirect
434. C3#29688direct (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×#4752
C3×#28572
C3×#20582
C3×#20432
C3×#20352
C3×#18032
C3×#17972
#296883