Finite Group Databasebeta

← C3×#23934 #10642541

Subgroups — C3×#23934 #10642541

Order 768.

1272 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic
2C2×C3×#451C212yesmaximal
3C2×C3×#391C211yesmaximal, characteristic
4C3×#2024C212yesmaximal, complemented
5C3×#2016C212yesmaximal, complemented
6C3×#1878C214yesmaximal, complemented
7C3×#1872C212yesmaximal
8C3×#1817C211yesmaximal, characteristic
9#23934C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×Q8C2211yescharacteristic
11C22×C3×Q16C2214yes—
12C3×#533—28no—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#125—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×C4⋊C4C2211yescharacteristic
22C22×C3×C42C2211yescharacteristic
23C3×#451—28no—
24C2×C3×#103C2214yes—
25C2×C3×#103C2214yes—
26C2×C3×C22⋊C8C2212yes—
27C3×#391—21no—
28C3×#391—21no—
29C3×#391C2212yes—
30C3×#391C2212yes—
31C2×C3×#95—21no—
32C2×C3×#95—21no—
33C3×#340C2214yes—
34C3×#336C2212yes—
35C3×#336C2212yes—
36C2×#533C2×C311yescharacteristic
37C2×#451C2×C312yes—
38C2×#391C2×C311yescharacteristic
39#2024C2×C312yescomplemented
40#2016C2×C312yescomplemented
41#1878C2×C314yescomplemented
42#1872C2×C312yes—
43#1817C2×C311yescharacteristic
44C22×C3×Q8—24no—
45C22×C3×Q8—21no—
46C22×C3×Q8—21no—
47C22×C3×Q8—24no—
48C22×C3×Q8—42no—
49C22×C3×Q8C2312yes—
50C22×C3×Q8C2312yes—
51C23×C3×C4—21no—
52C23×C3×C4—21no—
53C23×C3×C4D411yescharacteristic
54C23×C3×C4D411yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C2×C3×Q16—28no—
57C2×C3×Q16—28no—
58C2×C3×Q16—44no—
59C22×C3×C8C2314yes—
60C3×C4⋊Q8—24no—
61C3×C4⋊Q8—24no—
62C3×#125—28no—
63C3×C22⋊Q8—28no—
64C3×C22⋊Q8—24no—
65C3×C22⋊Q8D418yes—
66C3×C4×Q8—28no—
67C3×C4×Q8—24no—
68C3×C4×Q8—24no—
69C3×#117—28no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—42no—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—22no—
80C2×C3×C42—24no—
81C2×C3×C42—22no—
82C2×C3×C42D411yescharacteristic
83C2×C3×C42D411yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C3×#103—28no—
87C3×#103—44no—
88C3×C22⋊C8—24no—
89C3×#95—22no—
90C3×#95—22no—
91C3×#95—22no—
92C3×#95—22no—
93C23×Q8C22×C311yescharacteristic
94C22×Q16C22×C314yes—
95#533—28no—
96C2×C4⋊Q8C22×C311yescharacteristic
97C2×C4⋊Q8C22×C311yescharacteristic
98C2×#125—21no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8C22×C312yes—
101C2×C4×Q8—21no—
102C2×C4×Q8C22×C312yes—
103C2×#117—21no—
104C22×C4⋊C4C22×C311yescharacteristic
105C22×C42C22×C311yescharacteristic
106#451—28no—
107C2×#103C22×C314yes—
108C2×#103C22×C314yes—
109C2×C22⋊C8C22×C312yes—
110#391—21no—
111#391—21no—
112#391C22×C312yes—
113#391C22×C312yes—
114C2×#95—21no—
115C2×#95—21no—
116#340C22×C314yes—
117#336C22×C312yes—
118#336C22×C312yes—
119C24×C3C2×D411yescharacteristic
120C2×C3×Q8—24no—
121C2×C3×Q8—24no—
122C2×C3×Q8—22no—
123C2×C3×Q8—22no—
124C2×C3×Q8—24no—
125C2×C3×Q8—24no—
126C2×C3×Q8—24no—
127C2×C3×Q8—24no—
128C2×C3×Q8—416no—
129C2×C3×Q8—42no—
130C2×C3×Q8—42no—
131C2×C3×Q8—42no—
132C2×C3×Q8—42no—
133C2×C3×Q8C2×D414yes—
134C2×C3×Q8#4114yes—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—24no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—41no—
147C22×C3×C4—42no—
148C22×C3×C4—42no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4—42no—
152C22×C3×C4Q1614yescomplemented
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D414yes—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2411yescharacteristic
160C3×Q16—416no—
161C2×C3×C8—24no—
162C2×C3×C8—24no—
163C2×C3×C8—24no—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—24no—
166C3×C4⋊C4—28no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—28no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4C2×D414yes—
173C3×C22⋊C4—28no—
174C3×C42—28no—
175C3×C42—22no—
176C3×C42—22no—
177C3×C42—22no—
178C3×C42—22no—
179C22×Q8—24no—
180C22×Q8—21no—
181C22×Q8—21no—
182C22×Q8—24no—
183C22×Q8—42no—
184C22×Q8C23×C312yes—
185C22×Q8C23×C312yes—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4C3×D411yescharacteristic
189C23×C4C3×D411yescharacteristic
190C23×C4C23×C311yescharacteristic
191C2×Q16—28no—
192C2×Q16—28no—
193C2×Q16—44no—
194C22×C8C23×C314yes—
195C4⋊Q8—24no—
196C4⋊Q8—24no—
197#125—28no—
198C22⋊Q8—28no—
199C22⋊Q8—24no—
200C22⋊Q8C3×D418yes—
201C4×Q8—28no—
202C4×Q8—24no—
203C4×Q8—24no—
204#117—28no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4C23×C312yes—
213C2×C4⋊C4C23×C312yes—
214C2×C22⋊C4—22no—
215C2×C42—22no—
216C2×C42—24no—
217C2×C42C3×D411yescharacteristic
218C2×C42C3×D411yescharacteristic
219C2×C42C23×C311yescharacteristic
220C2×C42C23×C311yescharacteristic
221#103—28no—
222#103—44no—
223C22⋊C8—24no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—21no—
231C23×C3—21no—
232C23×C3#12012yes—
233C23×C3C2×Q1612yes—
234C23×C3#13712yes—
235C23×C3C22×D411yescharacteristic
236C3×Q8—28no—
237C3×Q8—28no—
238C3×Q8—48no—
239C3×Q8—48no—
240C3×Q8—44no—
241C3×Q8—44no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—28no—
251C2×C3×C4—28no—
252C2×C3×C4—21no—
253C2×C3×C4—24no—
254C2×C3×C4—24no—
255C2×C3×C4—24no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—44no—
259C2×C3×C4—48no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—84no—
270C2×C3×C4#12012yes—
271C2×C3×C4C2×Q1612yes—
272C2×C3×C4C2×Q1612yes—
273C2×C3×C4#13712yes—
274C2×C3×C4C22×D412yes—
275C2×C3×C4C2×#4112yes—
276C2×C3×C4H(5,𝔽2)12yes—
277C3×C8—44no—
278C24C2×C3×D411yescharacteristic
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—416no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8C2×C3×D414yes—
293C2×Q8C3×#4114yes—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4C3×Q1614yescomplemented
312C22×C4C2×C3×D411yescharacteristic
313C22×C4C2×C3×D411yescharacteristic
314C22×C4C2×C3×D414yes—
315C22×C4C2×C3×D411yescharacteristic
316C22×C4C2×C3×D411yescharacteristic
317C22×C4C2×C3×D411yescharacteristic
318C22×C4C24×C311yescharacteristic
319Q16—416no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—28no—
326C4⋊C4—24no—
327C4⋊C4—28no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—42no—
331C4⋊C4C2×C3×D414yes—
332C22⋊C4—28no—
333C42—28no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3#45114yes—
345C22×C3C2×#12011yescharacteristic
346C22×C3#52511yescharacteristic
347C22×C3#53511yescharacteristic
348C22×C3C22×Q1611yescharacteristic
349C22×C3C2×#13711yescharacteristic
350C22×C3C2×#13711yescharacteristic
351C22×C3#57511yescharacteristic
352C3×C4—22no—
353C3×C4—22no—
354C3×C4—22no—
355C3×C4—22no—
356C3×C4—44no—
357C3×C4—44no—
358C3×C4—41no—
359C3×C4—44no—
360C3×C4—82no—
361C3×C4#54614yes—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23C3×#12012yes—
367C23C2×C3×Q1612yes—
368C23C3×#13712yes—
369C23C22×C3×D411yescharacteristic
370Q8—28no—
371Q8—28no—
372Q8—48no—
373Q8—48no—
374Q8—44no—
375Q8—44no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—28no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—28no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—44no—
393C2×C4—48no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—84no—
404C2×C4C3×#12012yes—
405C2×C4C2×C3×Q1612yes—
406C2×C4C2×C3×Q1612yes—
407C2×C4C3×#13712yes—
408C2×C4C22×C3×D412yes—
409C2×C4C2×C3×#4112yes—
410C2×C4C3×H(5,𝔽2)12yes—
411C8—44no—
412C2×C3—24no—
413C2×C3#240411yescharacteristic
414C2×C3C2×#45111yescharacteristic
415C2×C3#300711yescharacteristic
416C2×C3#310411yescharacteristic
417C2×C3#311311yescharacteristic
418C2×C3#320911yescharacteristic
419C2×C3#321511yescharacteristic
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22C3×#45114yes—
427C22C2×C3×#12011yescharacteristic
428C22C3×#52511yescharacteristic
429C22C3×#53511yescharacteristic
430C22C22×C3×Q1611yescharacteristic
431C22C2×C3×#13711yescharacteristic
432C22C2×C3×#13711yescharacteristic
433C22C3×#57511yescharacteristic
434C4—22no—
435C4—22no—
436C4—22no—
437C4—22no—
438C4—44no—
439C4—44no—
440C4—41no—
441C4—44no—
442C4—82no—
443C4C3×#54614yes—
444C3#2393411yescharacteristic, direct product factor, Sylow
445C2—24no—
446C2C3×#240411yescharacteristic
447C2C2×C3×#45111yescharacteristic
448C2C3×#300711yescharacteristic
449C2C3×#310411yescharacteristic
450C2C3×#311311yescharacteristic
451C2C3×#320911yescharacteristic
452C2C3×#321511yescharacteristic
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×#2024C2semidirect
5. C3×#2016C2semidirect
6. C3×#1878C2semidirect
9. #23934C3direct (N × H)
39. #2024C2×C3semidirect
40. #2016C2×C3semidirect
41. #1878C2×C3semidirect
152. C22×C3×C4Q16semidirect
311. C22×C4C3×Q16semidirect
444. C3#23934direct (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×#4512
C2×C3×#3912
C3×#20242
C3×#20162
C3×#18782
C3×#18722
C3×#18172
#239343