Finite Group Databasebeta

← C3×#25629 #10644236

Subgroups — C3×#25629 #10644236

Order 768.

1178 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#559C211yesmaximal, characteristic
2C2×C3×#492C211yesmaximal, characteristic
3C2×C3×#462C211yesmaximal, characteristic, complemented
4C3×#2568C211yesmaximal, characteristic, complemented
5C3×#2015C214yesmaximal, complemented
6C3×#1984C211yesmaximal, characteristic, complemented
7C3×#1982C211yesmaximal, characteristic
8C3×#1877C214yesmaximal, complemented
9C3×C4×#103C211yesmaximal, characteristic, complemented
10#25629C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×Q16C2212yes—
12C22×C3×SD16C2214yes—
13C3×#559—24no—
14C3×#559—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#127C2211yescharacteristic
17C2×C3×#124—21no—
18C2×C3×#124C2212yes—
19C2×C3×#123—22no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4—21no—
23C2×C3×C4×D4C2212yes—
24C3×#492—24no—
25C3×#462—24no—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#102C2214yes—
30C2×C3×C4×C8C2211yescharacteristic
31C3×#400—22no—
32C3×#391—21no—
33C3×#391C2214yes—
34C3×#382—21no—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#340C2212yes—
37C3×#336C2214yes—
38C2×#559C2×C311yescharacteristic
39C2×#492C2×C311yescharacteristic
40C2×#462C2×C311yescharacteristic, complemented
41#2568C2×C311yescharacteristic, complemented
42#2015C2×C314yescomplemented
43#1984C2×C311yescharacteristic, complemented
44#1982C2×C311yescharacteristic
45#1877C2×C314yescomplemented
46C4×#103C2×C311yescharacteristic, complemented
47C22×C3×Q8D411yescharacteristic
48C22×C3×Q8C2312yes—
49C22×C3×Q8C2311yescharacteristic
50C22×C3×D4—21no—
51C22×C3×D4—21no—
52C22×C3×D4C2312yes—
53C23×C3×C4—22no—
54C2×C3×Q16—24no—
55C2×C3×Q16—44no—
56C2×C3×SD16—28no—
57C2×C3×SD16—28no—
58C2×C3×SD16—44no—
59C22×C3×C8C2312yes—
60C22×C3×C8C2312yes—
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8—22no—
63C3×#127—22no—
64C3×#127D414yes—
65C3×#124—28no—
66C3×#124—24no—
67C3×#124—24no—
68C3×#123—216no—
69C3×C4×Q8—22no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×Q8D414yes—
73C3×C4×D4—28no—
74C3×C4×D4—24no—
75C3×C4×D4—24no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4D411yescharacteristic
80C2×C3×C4⋊C4D412yes—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C42—21no—
87C2×C3×C42—21no—
88C2×C3×C42C2312yes—
89C2×C3×C42C2311yescharacteristic
90C3×C4⋊C8—22no—
91C3×#103—24no—
92C3×#103—24no—
93C3×#102—44no—
94C3×C4×C8—22no—
95C3×#95—21no—
96C3×#95—24no—
97C3×#95—24no—
98C22×Q16C22×C312yes—
99C22×SD16C22×C314yes—
100#559—24no—
101#559—24no—
102C2×C4⋊Q8C22×C311yescharacteristic
103C2×#127C22×C311yescharacteristic
104C2×#124—21no—
105C2×#124C22×C312yes—
106C2×#123—22no—
107C2×C4×Q8C22×C311yescharacteristic
108C2×C4×Q8C22×C311yescharacteristic
109C2×C4×D4—21no—
110C2×C4×D4C22×C312yes—
111#492—24no—
112#462—24no—
113C2×C4⋊C8C22×C311yescharacteristic
114C2×#103C22×C312yes—
115C2×#103C22×C312yes—
116C2×#102C22×C314yes—
117C2×C4×C8C22×C311yescharacteristic
118#400—22no—
119#391—21no—
120#391C22×C314yes—
121#382—21no—
122C4×C4⋊C4C22×C311yescharacteristic
123#340C22×C312yes—
124#336C22×C314yes—
125C24×C3—22no—
126C2×C3×Q8—22no—
127C2×C3×Q8—22no—
128C2×C3×Q8—22no—
129C2×C3×Q8—21no—
130C2×C3×Q8—22no—
131C2×C3×Q8—22no—
132C2×C3×Q8—22no—
133C2×C3×Q8—24no—
134C2×C3×Q8—22no—
135C2×C3×Q8—24no—
136C2×C3×Q8C2×D412yes—
137C2×C3×D4—24no—
138C2×C3×D4—24no—
139C2×C3×D4—22no—
140C2×C3×D4—24no—
141C2×C3×D4—24no—
142C2×C3×D4—42no—
143C2×C3×D4—42no—
144C2×C3×D4—82no—
145C2×C3×D4C2×D414yes—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—28no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—84no—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2411yescharacteristic
157C3×Q16—48no—
158C3×SD16—416no—
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C2×C3×C8—24no—
163C2×C3×C8#4114yes—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—24no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4C2×D412yes—
173C3×C22⋊C4—28no—
174C3×C22⋊C4—28no—
175C3×C22⋊C4—28no—
176C3×C42—24no—
177C3×C42—22no—
178C3×C42—22no—
179C3×C42—42no—
180C3×C42C2×D412yes—
181C3×C42C2×D412yes—
182C22×Q8C3×D411yescharacteristic
183C22×Q8C23×C312yes—
184C22×Q8C23×C311yescharacteristic
185C22×D4—21no—
186C22×D4—21no—
187C22×D4C23×C312yes—
188C23×C4—22no—
189C2×Q16—24no—
190C2×Q16—44no—
191C2×SD16—28no—
192C2×SD16—28no—
193C2×SD16—44no—
194C22×C8C23×C312yes—
195C22×C8C23×C312yes—
196C4⋊Q8—22no—
197C4⋊Q8—22no—
198#127—22no—
199#127C3×D414yes—
200#124—28no—
201#124—24no—
202#124—24no—
203#123—216no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8C3×D414yes—
208C4×D4—28no—
209C4×D4—24no—
210C4×D4—24no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C3×D411yescharacteristic
215C2×C4⋊C4C3×D412yes—
216C2×C4⋊C4C23×C312yes—
217C2×C4⋊C4C23×C311yescharacteristic
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42C23×C312yes—
224C2×C42C23×C311yescharacteristic
225C4⋊C8—22no—
226#103—24no—
227#103—24no—
228#102—44no—
229C4×C8—22no—
230#95—21no—
231#95—24no—
232#95—24no—
233C23×C3—24no—
234C23×C3—44no—
235C23×C3—42no—
236C23×C3—42no—
237C23×C3—44no—
238C23×C3C22×D411yescharacteristic
239C3×Q8—22no—
240C3×Q8—22no—
241C3×Q8—42no—
242C3×Q8—44no—
243C3×Q8—42no—
244C3×Q8—42no—
245C3×Q8—44no—
246C3×D4—28no—
247C3×D4—42no—
248C3×D4—42no—
249C3×D4—48no—
250C3×D4—84no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—24no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—44no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—88no—
267C2×C3×C4#12012yes—
268C2×C3×C4#12012yes—
269C2×C3×C4#13512yes—
270C2×C3×C4#13712yes—
271C2×C3×C4C22×D411yescharacteristic
272C2×C3×C4C22×D411yescharacteristic
273C2×C3×C4C2×#4112yes—
274C2×C3×C4H(5,𝔽2)12yes—
275C3×C8—24no—
276C3×C8—42no—
277C24—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—21no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—24no—
286C2×Q8—22no—
287C2×Q8—24no—
288C2×Q8C2×C3×D412yes—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—82no—
297C2×D4C2×C3×D414yes—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—28no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—84no—
304C22×C4C2×C3×D412yes—
305C22×C4C2×C3×D412yes—
306C22×C4C2×C3×D411yescharacteristic
307C22×C4C2×C3×D411yescharacteristic
308C22×C4C24×C311yescharacteristic
309Q16—48no—
310SD16—416no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—24no—
315C2×C8C3×#4114yes—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—24no—
319C4⋊C4—21no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4C2×C3×D412yes—
325C22⋊C4—28no—
326C22⋊C4—28no—
327C22⋊C4—28no—
328C42—24no—
329C42—22no—
330C42—22no—
331C42—42no—
332C42C2×C3×D412yes—
333C42C2×C3×D412yes—
334C22×C3—44no—
335C22×C3—44no—
336C22×C3—44no—
337C22×C3—84no—
338C22×C3—84no—
339C22×C3C2×#12011yescharacteristic
340C22×C3#55012yes—
341C22×C3C2×#13511yescharacteristic
342C22×C3C2×#13711yescharacteristic
343C22×C3#57712yes—
344C3×C4—22no—
345C3×C4—42no—
346C3×C4—42no—
347C3×C4—42no—
348C3×C4—42no—
349C3×C4—82no—
350C3×C4—82no—
351C3×C4#45214yes—
352C3×C4#53211yescharacteristic
353C3×C4#53511yescharacteristic
354C3×C4#54612yes—
355C23—24no—
356C23—42no—
357C23—42no—
358C23—44no—
359C23—44no—
360C23C22×C3×D411yescharacteristic
361Q8—22no—
362Q8—22no—
363Q8—44no—
364Q8—42no—
365Q8—42no—
366Q8—42no—
367Q8—44no—
368D4—28no—
369D4—48no—
370D4—42no—
371D4—42no—
372D4—84no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—88no—
389C2×C4C3×#12012yes—
390C2×C4C3×#12012yes—
391C2×C4C3×#13512yes—
392C2×C4C3×#13712yes—
393C2×C4C22×C3×D411yescharacteristic
394C2×C4C22×C3×D411yescharacteristic
395C2×C4C2×C3×#4112yes—
396C2×C4C3×H(5,𝔽2)12yes—
397C8—24no—
398C8—42no—
399C2×C3—84no—
400C2×C3#243111yescharacteristic
401C2×C3C2×#45211yescharacteristic
402C2×C3#300911yescharacteristic
403C2×C3#339412yes—
404C2×C3#340012yes—
405C22—44no—
406C22—44no—
407C22—44no—
408C22—84no—
409C22—84no—
410C22C2×C3×#12011yescharacteristic
411C22C3×#55012yes—
412C22C2×C3×#13511yescharacteristic
413C22C2×C3×#13711yescharacteristic
414C22C3×#57712yes—
415C4—22no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—82no—
421C4—82no—
422C4C3×#45214yes—
423C4C3×#53211yescharacteristic
424C4C3×#53511yescharacteristic
425C4C3×#54612yes—
426C3#2562911yescharacteristic, direct product factor, Sylow
427C2—84no—
428C2C3×#243111yescharacteristic
429C2C2×C3×#45211yescharacteristic
430C2C3×#300911yescharacteristic
431C2C3×#339412yes—
432C2C3×#340012yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×C3×#462C2semidirect
4. C3×#2568C2semidirect
5. C3×#2015C2semidirect
6. C3×#1984C2semidirect
8. C3×#1877C2semidirect
9. C3×C4×#103C2semidirect
10. #25629C3direct (N × H)
40. C2×#462C2×C3semidirect
41. #2568C2×C3semidirect
42. #2015C2×C3semidirect
43. #1984C2×C3semidirect
45. #1877C2×C3semidirect
46. C4×#103C2×C3semidirect
426. C3#25629direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5592
C2×C3×#4922
C2×C3×#4622
C3×#25682
C3×#20152
C3×#19842
C3×#19822
C3×#18772
C3×C4×#1032
#256293