Finite Group Databasebeta

← C3×#29664 #10648271

Subgroups — C3×#29664 #10648271

Order 768.

1132 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic
2C2×C3×#462C212yesmaximal, complemented
3C3×#2857C211yesmaximal, characteristic
4C3×#2024C212yesmaximal, complemented
5C3×#2016C212yesmaximal, complemented
6C3×#2001C214yesmaximal
7C3×#1803C212yesmaximal, complemented
8C3×#1797C211yesmaximal, characteristic
9#29664C311yesmaximal, characteristic, direct product factor, Sylow
10C22×C3×Q16C2214yes—
11C3×#533—28no—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#125—21no—
16C2×C3×C22⋊Q8—21no—
17C2×C3×C22⋊Q8C2212yes—
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8C2212yes—
21C2×C3×#117—21no—
22C22×C3×C42C2211yescharacteristic
23C3×#462—28no—
24C2×C3×C4⋊C8C2212yes—
25C2×C3×#103C2214yes—
26C2×C3×#103C2214yes—
27C2×C3×C22⋊C8C2212yes—
28C3×#391C2212yes—
29C3×#389—21no—
30C3×#389—21no—
31C3×#384C2211yescharacteristic
32C3×#380—21no—
33C3×#380—21no—
34C3×#340C2214yes—
35C3×#336C2212yes—
36C2×#533C2×C311yescharacteristic
37C2×#462C2×C312yescomplemented
38#2857C2×C311yescharacteristic
39#2024C2×C312yescomplemented
40#2016C2×C312yescomplemented
41#2001C2×C314yes—
42#1803C2×C312yescomplemented
43#1797C2×C311yescharacteristic
44C22×C3×Q8—21no—
45C22×C3×Q8C2312yes—
46C22×C3×Q8C2312yes—
47C23×C3×C4D411yescharacteristic
48C23×C3×C4D411yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C2×C3×Q16—28no—
51C2×C3×Q16—44no—
52C2×C3×Q16—44no—
53C22×C3×C8C2314yes—
54C3×C4⋊Q8—24no—
55C3×C4⋊Q8—24no—
56C3×C4⋊Q8—24no—
57C3×#125—28no—
58C3×C22⋊Q8—28no—
59C3×C22⋊Q8—24no—
60C3×C22⋊Q8—24no—
61C3×C22⋊Q8—44no—
62C3×C4×Q8—28no—
63C3×C4×Q8—24no—
64C3×C4×Q8D418yes—
65C3×#117—28no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C42—22no—
81C2×C3×C42—24no—
82C2×C3×C42D411yescharacteristic
83C2×C3×C42D411yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C3×C4⋊C8—24no—
87C3×#103—28no—
88C3×#103—44no—
89C3×C22⋊C8—42no—
90C3×#95—22no—
91C3×#95—22no—
92C22×Q16C22×C314yes—
93#533—28no—
94C2×C4⋊Q8C22×C311yescharacteristic
95C2×C4⋊Q8C22×C311yescharacteristic
96C2×C4⋊Q8C22×C311yescharacteristic
97C2×#125—21no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8C22×C312yes—
100C2×C22⋊Q8C22×C312yes—
101C2×C4×Q8—21no—
102C2×C4×Q8C22×C312yes—
103C2×#117—21no—
104C22×C42C22×C311yescharacteristic
105#462—28no—
106C2×C4⋊C8C22×C312yes—
107C2×#103C22×C314yes—
108C2×#103C22×C314yes—
109C2×C22⋊C8C22×C312yes—
110#391C22×C312yes—
111#389—21no—
112#389—21no—
113#384C22×C311yescharacteristic
114#380—21no—
115#380—21no—
116#340C22×C314yes—
117#336C22×C312yes—
118C24×C3C2×D411yescharacteristic
119C2×C3×Q8—24no—
120C2×C3×Q8—22no—
121C2×C3×Q8—24no—
122C2×C3×Q8—22no—
123C2×C3×Q8—22no—
124C2×C3×Q8—22no—
125C2×C3×Q8—24no—
126C2×C3×Q8—22no—
127C2×C3×Q8—22no—
128C2×C3×Q8—22no—
129C2×C3×Q8—42no—
130C2×C3×Q8—42no—
131C2×C3×Q8C2×D414yes—
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—21no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—41no—
145C22×C3×C4Q1614yescomplemented
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2411yescharacteristic
152C3×Q16—416no—
153C2×C3×C8—24no—
154C2×C3×C8—24no—
155C2×C3×C8—24no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4—24no—
158C3×C4⋊C4—28no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—28no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—24no—
163C3×C4⋊C4—42no—
164C3×C4⋊C4—44no—
165C3×C4⋊C4—42no—
166C3×C4⋊C4—42no—
167C3×C4⋊C4—42no—
168C3×C4⋊C4C2×D414yes—
169C3×C22⋊C4—28no—
170C3×C22⋊C4—44no—
171C3×C42—22no—
172C3×C42—22no—
173C3×C42—28no—
174C3×C42—22no—
175C3×C42C2×D414yes—
176C22×Q8—21no—
177C22×Q8C23×C312yes—
178C22×Q8C23×C312yes—
179C23×C4C3×D411yescharacteristic
180C23×C4C3×D411yescharacteristic
181C23×C4C23×C311yescharacteristic
182C2×Q16—28no—
183C2×Q16—44no—
184C2×Q16—44no—
185C22×C8C23×C314yes—
186C4⋊Q8—24no—
187C4⋊Q8—24no—
188C4⋊Q8—24no—
189#125—28no—
190C22⋊Q8—28no—
191C22⋊Q8—24no—
192C22⋊Q8—24no—
193C22⋊Q8—44no—
194C4×Q8—28no—
195C4×Q8—24no—
196C4×Q8C3×D418yes—
197#117—28no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C23×C312yes—
209C2×C4⋊C4C23×C312yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C42—22no—
213C2×C42—24no—
214C2×C42C3×D411yescharacteristic
215C2×C42C3×D411yescharacteristic
216C2×C42C23×C311yescharacteristic
217C2×C42C23×C311yescharacteristic
218C4⋊C8—24no—
219#103—28no—
220#103—44no—
221C22⋊C8—42no—
222#95—22no—
223#95—22no—
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3C2×Q1612yes—
231C23×C3C22×D411yescharacteristic
232C3×Q8—24no—
233C3×Q8—24no—
234C3×Q8—48no—
235C3×Q8—42no—
236C3×Q8—42no—
237C3×Q8—44no—
238C3×Q8—44no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—28no—
246C2×C3×C4—21no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—44no—
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—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4#12012yes—
265C2×C3×C4#12012yes—
266C2×C3×C4C2×Q1612yes—
267C2×C3×C4C2×Q1612yes—
268C2×C3×C4#13712yes—
269C2×C3×C4#13712yes—
270C2×C3×C4C22×D412yes—
271C2×C3×C4H(5,𝔽2)11yescharacteristic
272C2×C3×C4H(5,𝔽2)12yes—
273C2×C3×C42−1+411yescharacteristic
274C3×C8—44no—
275C24C2×C3×D411yescharacteristic
276C2×Q8—24no—
277C2×Q8—22no—
278C2×Q8—24no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—24no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8C2×C3×D414yes—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—41no—
302C22×C4C3×Q1614yescomplemented
303C22×C4C2×C3×D411yescharacteristic
304C22×C4C2×C3×D411yescharacteristic
305C22×C4C2×C3×D411yescharacteristic
306C22×C4C2×C3×D411yescharacteristic
307C22×C4C2×C3×D411yescharacteristic
308C22×C4C24×C311yescharacteristic
309Q16—416no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—28no—
316C4⋊C4—24no—
317C4⋊C4—28no—
318C4⋊C4—22no—
319C4⋊C4—24no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4C2×C3×D414yes—
326C22⋊C4—28no—
327C22⋊C4—44no—
328C42—22no—
329C42—22no—
330C42—28no—
331C42—22no—
332C42C2×C3×D414yes—
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—22no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3C2×#12011yescharacteristic
341C22×C3#53511yescharacteristic
342C22×C3#53711yescharacteristic
343C22×C3C22×Q1611yescharacteristic
344C22×C3C2×#13711yescharacteristic
345C22×C3C2×#13711yescharacteristic
346C22×C3#57511yescharacteristic
347C3×C4—22no—
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—41no—
351C3×C4—44no—
352C3×C4—44no—
353C3×C4—82no—
354C3×C4—82no—
355C3×C4#45114yes—
356C3×C4#53512yes—
357C3×C4#53712yes—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23C2×C3×Q1612yes—
365C23C22×C3×D411yescharacteristic
366Q8—24no—
367Q8—24no—
368Q8—48no—
369Q8—42no—
370Q8—42no—
371Q8—44no—
372Q8—44no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—28no—
380C2×C4—21no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4C3×#12012yes—
399C2×C4C3×#12012yes—
400C2×C4C2×C3×Q1612yes—
401C2×C4C2×C3×Q1612yes—
402C2×C4C3×#13712yes—
403C2×C4C3×#13712yes—
404C2×C4C22×C3×D412yes—
405C2×C4C3×H(5,𝔽2)11yescharacteristic
406C2×C4C3×H(5,𝔽2)12yes—
407C2×C4C3×2−1+411yescharacteristic
408C8—44no—
409C2×C3—42no—
410C2×C3#261511yescharacteristic
411C2×C3C2×#45111yescharacteristic
412C2×C3#300711yescharacteristic
413C2×C3#320911yescharacteristic
414C2×C3#321611yescharacteristic
415C2×C3#324911yescharacteristic
416C2×C3#325511yescharacteristic
417C22—22no—
418C22—22no—
419C22—22no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22C2×C3×#12011yescharacteristic
425C22C3×#53511yescharacteristic
426C22C3×#53711yescharacteristic
427C22C22×C3×Q1611yescharacteristic
428C22C2×C3×#13711yescharacteristic
429C22C2×C3×#13711yescharacteristic
430C22C3×#57511yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—44no—
435C4—41no—
436C4—44no—
437C4—82no—
438C4—82no—
439C4C3×#45114yes—
440C4C3×#53512yes—
441C4C3×#53712yes—
442C3#2966411yescharacteristic, direct product factor, Sylow
443C2—42no—
444C2C3×#261511yescharacteristic
445C2C2×C3×#45111yescharacteristic
446C2C3×#300711yescharacteristic
447C2C3×#320911yescharacteristic
448C2C3×#321611yescharacteristic
449C2C3×#324911yescharacteristic
450C2C3×#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×#462C2semidirect
4. C3×#2024C2semidirect
5. C3×#2016C2semidirect
7. C3×#1803C2semidirect
9. #29664C3direct (N × H)
37. C2×#462C2×C3semidirect
39. #2024C2×C3semidirect
40. #2016C2×C3semidirect
42. #1803C2×C3semidirect
145. C22×C3×C4Q16semidirect
302. C22×C4C3×Q16semidirect
442. C3#29664direct (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×#4622
C3×#28572
C3×#20242
C3×#20162
C3×#20012
C3×#18032
C3×#17972
#296643