Finite Group Databasebeta

← C3×#27681 #10646288

Subgroups — C3×#27681 #10646288

Order 768.

886 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C22⋊Q8C211yesmaximal, characteristic
2C2×C3×#384C211yesmaximal, characteristic
3C3×#2100C212yesmaximal, complemented
4C3×#2095C212yesmaximal, complemented
5C3×#1947C212yesmaximal, complemented
6C3×#1872C212yesmaximal
7C3×#1852C212yesmaximal
8C3×#1803C212yesmaximal, complemented
9C3×#1797C211yesmaximal, characteristic
10#27681C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×C22⋊Q8C2211yescharacteristic
12C2×C3×C4×Q8C2211yescharacteristic
13C22×C3×C4⋊C4—21no—
14C22×C3×C4⋊C4—21no—
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C42C2211yescharacteristic
17C2×C3×#107C2214yes—
18C2×C3×C4⋊C8C2212yes—
19C2×C3×#103C2214yes—
20C2×C3×C22⋊C8C2212yes—
21C3×#391C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×#387—21no—
24C3×#385—21no—
25C3×#384—21no—
26C3×#384—21no—
27C3×#384C2212yes—
28C3×#384C2212yes—
29C3×#380C2211yescharacteristic
30C3×C4×C4⋊C4—21no—
31C3×C4×C4⋊C4C2211yescharacteristic
32C3×C4×C22⋊C4—21no—
33C3×#340C2214yes—
34C3×#340C2214yes—
35C3×#336C2212yes—
36C4×C22⋊Q8C2×C311yescharacteristic
37C2×#384C2×C311yescharacteristic
38#2100C2×C312yescomplemented
39#2095C2×C312yescomplemented
40#1947C2×C312yescomplemented
41#1872C2×C312yes—
42#1852C2×C312yes—
43#1803C2×C312yescomplemented
44#1797C2×C311yescharacteristic
45C22×C3×Q8C2311yescharacteristic
46C23×C3×C4—22no—
47C23×C3×C4D411yescharacteristic
48C23×C3×C4D411yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C22×C3×C8C2314yes—
51C3×C22⋊Q8—22no—
52C3×C22⋊Q8Q814yes—
53C3×C4×Q8—24no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—22no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—44no—
63C2×C3×C4⋊C4—44no—
64C2×C3×C4⋊C4—44no—
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C4⋊C4C2314yes—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C22⋊C4—21no—
70C2×C3×C22⋊C4—21no—
71C2×C3×C42—24no—
72C2×C3×C42—21no—
73C2×C3×C42—21no—
74C2×C3×C42D411yescharacteristic
75C2×C3×C42D411yescharacteristic
76C2×C3×C42C2311yescharacteristic
77C2×C3×C42C2311yescharacteristic
78C3×#107—44no—
79C3×C4⋊C8—24no—
80C3×#103—44no—
81C3×C22⋊C8—24no—
82C3×#95—21no—
83C3×#95—21no—
84C3×#95—21no—
85C3×#95—21no—
86C3×#95—21no—
87C2×C22⋊Q8C22×C311yescharacteristic
88C2×C4×Q8C22×C311yescharacteristic
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4C22×C312yes—
92C22×C42C22×C311yescharacteristic
93C2×#107C22×C314yes—
94C2×C4⋊C8C22×C312yes—
95C2×#103C22×C314yes—
96C2×C22⋊C8C22×C312yes—
97#391C22×C311yescharacteristic
98#389C22×C311yescharacteristic
99#387—21no—
100#385—21no—
101#384—21no—
102#384—21no—
103#384C22×C312yes—
104#384C22×C312yes—
105#380C22×C311yescharacteristic
106C4×C4⋊C4—21no—
107C4×C4⋊C4C22×C311yescharacteristic
108C4×C22⋊C4—21no—
109#340C22×C314yes—
110#340C22×C314yes—
111#336C22×C312yes—
112C24×C3C2×D411yescharacteristic
113C2×C3×Q8—21no—
114C2×C3×Q8—22no—
115C2×C3×Q8—22no—
116C2×C3×Q8C2×Q812yes—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—24no—
129C22×C3×C4—44no—
130C22×C3×C4—44no—
131C22×C3×C4—44no—
132C22×C3×C4Q1614yes—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×Q812yes—
139C22×C3×C4#4112yes—
140C22×C3×C4C2411yescharacteristic
141C2×C3×C8—24no—
142C2×C3×C8—24no—
143C2×C3×C8—24no—
144C3×C4⋊C4—24no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—24no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—44no—
151C3×C4⋊C4—44no—
152C3×C4⋊C4—44no—
153C3×C4⋊C4—44no—
154C3×C4⋊C4—42no—
155C3×C4⋊C4—44no—
156C3×C4⋊C4—42no—
157C3×C4⋊C4C2×Q812yes—
158C3×C22⋊C4—24no—
159C3×C22⋊C4—42no—
160C3×C42—22no—
161C3×C42—22no—
162C3×C42—22no—
163C3×C42—42no—
164C3×C42—42no—
165C3×C42#4114yes—
166C22×Q8C23×C311yescharacteristic
167C23×C4—22no—
168C23×C4C3×D411yescharacteristic
169C23×C4C3×D411yescharacteristic
170C23×C4C23×C311yescharacteristic
171C22×C8C23×C314yes—
172C22⋊Q8—22no—
173C22⋊Q8C3×Q814yes—
174C4×Q8—24no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4C23×C314yes—
187C2×C4⋊C4C23×C311yescharacteristic
188C2×C4⋊C4C23×C311yescharacteristic
189C2×C4⋊C4C23×C311yescharacteristic
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C42—24no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42C3×D411yescharacteristic
196C2×C42C3×D411yescharacteristic
197C2×C42C23×C311yescharacteristic
198C2×C42C23×C311yescharacteristic
199#107—44no—
200C4⋊C8—24no—
201#103—44no—
202C22⋊C8—24no—
203#95—21no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—21no—
208C23×C3—21no—
209C23×C3—21no—
210C23×C3—21no—
211C23×C3—21no—
212C23×C3C22⋊Q812yes—
213C23×C3C2×Q1612yes—
214C23×C3AGL(1,ℤ/8)12yes—
215C23×C3C22×D411yescharacteristic
216C3×Q8—24no—
217C3×Q8—42no—
218C2×C3×C4—28no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—24no—
228C2×C3×C4—24no—
229C2×C3×C4—28no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—44no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—44no—
243C2×C3×C4—44no—
244C2×C3×C4—41no—
245C2×C3×C4—88no—
246C2×C3×C4C22⋊Q812yes—
247C2×C3×C4#12312yes—
248C2×C3×C4#12312yes—
249C2×C3×C4C2×Q1612yes—
250C2×C3×C4C2×Q1612yes—
251C2×C3×C4AGL(1,ℤ/8)12yes—
252C2×C3×C4#13712yes—
253C2×C3×C4C22×Q811yescharacteristic
254C2×C3×C4C2×#4111yescharacteristic
255C2×C3×C4C2×#4112yes—
256C2×C3×C4H(5,𝔽2)12yes—
257C3×C8—44no—
258C24C2×C3×D411yescharacteristic
259C2×Q8—21no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8C2×C3×Q812yes—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—24no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4C3×Q1614yes—
279C22×C4C2×C3×D411yescharacteristic
280C22×C4C2×C3×D411yescharacteristic
281C22×C4C2×C3×D411yescharacteristic
282C22×C4C2×C3×D411yescharacteristic
283C22×C4C2×C3×D411yescharacteristic
284C22×C4C2×C3×Q812yes—
285C22×C4C3×#4112yes—
286C22×C4C24×C311yescharacteristic
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—24no—
290C4⋊C4—24no—
291C4⋊C4—21no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4C2×C3×Q812yes—
304C22⋊C4—24no—
305C22⋊C4—42no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—42no—
310C42—42no—
311C42C3×#4114yes—
312C22×C3—22no—
313C22×C3—22no—
314C22×C3—22no—
315C22×C3—22no—
316C22×C3—22no—
317C22×C3—22no—
318C22×C3#47714yes—
319C22×C3C2×C22⋊Q811yescharacteristic
320C22×C3C2×#12311yescharacteristic
321C22×C3#53511yescharacteristic
322C22×C3C22×Q1611yescharacteristic
323C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
324C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
325C22×C3C2×#13711yescharacteristic
326C3×C4—22no—
327C3×C4—22no—
328C3×C4—22no—
329C3×C4—22no—
330C3×C4—42no—
331C3×C4—42no—
332C3×C4—81no—
333C3×C4—81no—
334C3×C4—84no—
335C3×C4#48414yes—
336C3×C4#55312yes—
337C3×C4#55412yes—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23C3×C22⋊Q812yes—
343C23C2×C3×Q1612yes—
344C23C3×AGL(1,ℤ/8)12yes—
345C23C22×C3×D411yescharacteristic
346Q8—24no—
347Q8—42no—
348C2×C4—28no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—28no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—44no—
374C2×C4—41no—
375C2×C4—88no—
376C2×C4C3×C22⋊Q812yes—
377C2×C4C3×#12312yes—
378C2×C4C3×#12312yes—
379C2×C4C2×C3×Q1612yes—
380C2×C4C2×C3×Q1612yes—
381C2×C4C3×AGL(1,ℤ/8)12yes—
382C2×C4C3×#13712yes—
383C2×C4C22×C3×Q811yescharacteristic
384C2×C4C2×C3×#4111yescharacteristic
385C2×C4C2×C3×#4112yes—
386C2×C4C3×H(5,𝔽2)12yes—
387C8—44no—
388C2×C3—24no—
389C2×C3#248711yescharacteristic
390C2×C3C2×#47711yescharacteristic
391C2×C3#307911yescharacteristic
392C2×C3C2×#48411yescharacteristic
393C2×C3#309211yescharacteristic
394C2×C3#320911yescharacteristic
395C2×C3#321711yescharacteristic
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22C3×#47714yes—
403C22C2×C3×C22⋊Q811yescharacteristic
404C22C2×C3×#12311yescharacteristic
405C22C3×#53511yescharacteristic
406C22C22×C3×Q1611yescharacteristic
407C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
408C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
409C22C2×C3×#13711yescharacteristic
410C4—22no—
411C4—22no—
412C4—22no—
413C4—22no—
414C4—42no—
415C4—42no—
416C4—84no—
417C4—81no—
418C4—81no—
419C4C3×#48414yes—
420C4C3×#55312yes—
421C4C3×#55412yes—
422C3#2768111yescharacteristic, direct product factor, Sylow
423C2—24no—
424C2C3×#248711yescharacteristic
425C2C2×C3×#47711yescharacteristic
426C2C3×#307911yescharacteristic
427C2C2×C3×#48411yescharacteristic
428C2C3×#309211yescharacteristic
429C2C3×#320911yescharacteristic
430C2C3×#321711yescharacteristic
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
3. C3×#2100C2semidirect
4. C3×#2095C2semidirect
5. C3×#1947C2semidirect
8. C3×#1803C2semidirect
10. #27681C3direct (N × H)
38. #2100C2×C3semidirect
39. #2095C2×C3semidirect
40. #1947C2×C3semidirect
43. #1803C2×C3semidirect
422. C3#27681direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×C22⋊Q82
C2×C3×#3842
C3×#21002
C3×#20952
C3×#19472
C3×#18722
C3×#18522
C3×#18032
C3×#17972
#276813