Finite Group Databasebeta

← C3×#29328 #10647935

Subgroups — C3×#29328 #10647935

Order 768.

994 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2856C211yesmaximal, characteristic
2C2×C3×#389C211yesmaximal, characteristic, complemented
3C3×#2099C212yesmaximal, complemented
4C3×#2041C212yesmaximal, complemented
5C3×#1934C214yesmaximal, complemented
6C3×#1874C212yesmaximal, complemented
7C3×#1874C212yesmaximal, complemented
8C3×C4×C22⋊C8C211yesmaximal, characteristic, complemented
9#29328C311yesmaximal, characteristic, direct product factor, Sylow
10C2×C3×#127C2211yescharacteristic
11C2×C3×#121—21no—
12C2×C3×#121—21no—
13C2×C3×#121C2212yes—
14C22×C3×C4⋊C4—21no—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C2×C3×#102C2214yes—
19C2×C3×#102C2214yes—
20C2×C3×C22⋊C8C2212yes—
21C2×C3×C4×C8C2212yes—
22C3×#390—21no—
23C3×#390—21no—
24C3×#390C2212yes—
25C3×#389—21no—
26C3×#389—21no—
27C3×#389C2212yes—
28C3×#389C2212yes—
29C3×#384C2211yescharacteristic
30C2×C3×#95—21no—
31C3×#340C2214yes—
32C3×#340C2214yes—
33C3×#336C2212yes—
34#2856C2×C311yescharacteristic
35C2×#389C2×C311yescharacteristic, complemented
36#2099C2×C312yescomplemented
37#2041C2×C312yescomplemented
38#1934C2×C314yescomplemented
39#1874C2×C312yescomplemented
40#1874C2×C312yescomplemented
41C4×C22⋊C8C2×C311yescharacteristic, complemented
42C22×C3×D4—22no—
43C22×C3×D4—21no—
44C22×C3×D4—21no—
45C22×C3×D4C2312yes—
46C23×C3×C4—21no—
47C23×C3×C4—21no—
48C23×C3×C4D411yescharacteristic
49C23×C3×C4D411yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C22×C3×C8C2314yes—
52C3×#127—42no—
53C3×#121—28no—
54C3×#121—44no—
55C3×#121—44no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—24no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—42no—
64C2×C3×C4⋊C4—44no—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C42—42no—
72C2×C3×C42D411yescharacteristic
73C2×C3×C42D411yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C3×#102—28no—
77C3×#102—44no—
78C3×C22⋊C8—24no—
79C3×C4×C8—42no—
80C3×#95—22no—
81C3×#95—22no—
82C2×#127C22×C311yescharacteristic
83C2×#121—21no—
84C2×#121—21no—
85C2×#121C22×C312yes—
86C22×C4⋊C4—21no—
87C22×C4⋊C4C22×C311yescharacteristic
88C22×C4⋊C4C22×C311yescharacteristic
89C22×C42C22×C311yescharacteristic
90C2×#102C22×C314yes—
91C2×#102C22×C314yes—
92C2×C22⋊C8C22×C312yes—
93C2×C4×C8C22×C312yes—
94#390—21no—
95#390—21no—
96#390C22×C312yes—
97#389—21no—
98#389—21no—
99#389C22×C312yes—
100#389C22×C312yes—
101#384C22×C311yescharacteristic
102C2×#95—21no—
103#340C22×C314yes—
104#340C22×C314yes—
105#336C22×C312yes—
106C24×C3—22no—
107C24×C3C2×D411yescharacteristic
108C2×C3×D4—24no—
109C2×C3×D4—24no—
110C2×C3×D4—24no—
111C2×C3×D4—42no—
112C2×C3×D4—42no—
113C2×C3×D4—42no—
114C2×C3×D4—42no—
115C2×C3×D4—44no—
116C2×C3×D4—44no—
117C2×C3×D4—44no—
118C2×C3×D4—42no—
119C2×C3×D4—42no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—24no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4—42no—
131C22×C3×C4—42no—
132C22×C3×C4—42no—
133C22×C3×C4—42no—
134C22×C3×C4—42no—
135C22×C3×C4D814yes—
136C22×C3×C4SD1614yes—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4#4114yes—
143C22×C3×C4C2411yescharacteristic
144C2×C3×C8—24no—
145C2×C3×C8—24no—
146C2×C3×C8—24no—
147C3×C4⋊C4—24no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—42no—
152C3×C4⋊C4—44no—
153C3×C4⋊C4—44no—
154C3×C4⋊C4—42no—
155C3×C4⋊C4—42no—
156C3×C4⋊C4#4114yes—
157C3×C22⋊C4—44no—
158C3×C22⋊C4—44no—
159C3×C22⋊C4—44no—
160C3×C42—22no—
161C3×C42—22no—
162C3×C42—41no—
163C3×C42—41no—
164C22×D4—22no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4C23×C312yes—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4C3×D411yescharacteristic
171C23×C4C3×D411yescharacteristic
172C23×C4C23×C311yescharacteristic
173C22×C8C23×C314yes—
174#127—42no—
175#121—28no—
176#121—44no—
177#121—44no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4C23×C312yes—
188C2×C4⋊C4C23×C312yes—
189C2×C4⋊C4C23×C312yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C42—42no—
194C2×C42C3×D411yescharacteristic
195C2×C42C3×D411yescharacteristic
196C2×C42C23×C311yescharacteristic
197C2×C42C23×C311yescharacteristic
198#102—28no—
199#102—44no—
200C22⋊C8—24no—
201C4×C8—42no—
202#95—22no—
203#95—22no—
204C23×C3—21no—
205C23×C3—21no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—42no—
209C23×C3—42no—
210C23×C3—42no—
211C23×C3—42no—
212C23×C3—42no—
213C23×C3—42no—
214C23×C3—42no—
215C23×C3#12412yes—
216C23×C3C2×D812yes—
217C23×C3C2×SD1612yes—
218C23×C3C22×D411yescharacteristic
219C3×D4—48no—
220C3×D4—84no—
221C3×D4—84no—
222C3×D4—88no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—28no—
234C2×C3×C4—24no—
235C2×C3×C4—44no—
236C2×C3×C4—44no—
237C2×C3×C4—48no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—84no—
247C2×C3×C4#12412yes—
248C2×C3×C4C2×D812yes—
249C2×C3×C4C2×D812yes—
250C2×C3×C4C2×SD1612yes—
251C2×C3×C4C2×SD1612yes—
252C2×C3×C4C2×#4112yes—
253C2×C3×C4C2×#4112yes—
254C2×C3×C4H(5,𝔽2)12yes—
255C3×C8—44no—
256C24—22no—
257C24C2×C3×D411yescharacteristic
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—44no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—42no—
269C2×D4—42no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4C3×D814yes—
286C22×C4C3×SD1614yes—
287C22×C4C2×C3×D411yescharacteristic
288C22×C4C2×C3×D411yescharacteristic
289C22×C4C2×C3×D411yescharacteristic
290C22×C4C2×C3×D411yescharacteristic
291C22×C4C2×C3×D411yescharacteristic
292C22×C4C3×#4114yes—
293C22×C4C24×C311yescharacteristic
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4C3×#4114yes—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C42—22no—
311C42—22no—
312C42—41no—
313C42—41no—
314C22×C3—22no—
315C22×C3—22no—
316C22×C3—22no—
317C22×C3—22no—
318C22×C3—22no—
319C22×C3—22no—
320C22×C3—82no—
321C22×C3—82no—
322C22×C3—82no—
323C22×C3—82no—
324C22×C3—82no—
325C22×C3—82no—
326C22×C3—82no—
327C22×C3#48614yes—
328C22×C3#52511yescharacteristic
329C22×C3C2×#12411yescharacteristic
330C22×C3#53411yescharacteristic
331C22×C3C22×D811yescharacteristic
332C22×C3C22×SD1611yescharacteristic
333C22×C3#57511yescharacteristic
334C22×C3#57511yescharacteristic
335C3×C4—22no—
336C3×C4—22no—
337C3×C4—22no—
338C3×C4—22no—
339C3×C4—22no—
340C3×C4—44no—
341C3×C4—82no—
342C3×C4—82no—
343C3×C4#55314yes—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23C3×#12412yes—
356C23C2×C3×D812yes—
357C23C2×C3×SD1612yes—
358C23C22×C3×D411yescharacteristic
359D4—48no—
360D4—84no—
361D4—84no—
362D4—88no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—28no—
374C2×C4—24no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—48no—
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—84no—
387C2×C4C3×#12412yes—
388C2×C4C2×C3×D812yes—
389C2×C4C2×C3×D812yes—
390C2×C4C2×C3×SD1612yes—
391C2×C4C2×C3×SD1612yes—
392C2×C4C2×C3×#4112yes—
393C2×C4C2×C3×#4112yes—
394C2×C4C3×H(5,𝔽2)12yes—
395C8—44no—
396C2×C3—24no—
397C2×C3—162no—
398C2×C3#259111yescharacteristic
399C2×C3#310011yescharacteristic
400C2×C3#310311yescharacteristic
401C2×C3C2×#48611yescharacteristic
402C2×C3#314211yescharacteristic
403C2×C3#318611yescharacteristic
404C2×C3#318711yescharacteristic
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—82no—
412C22—82no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22C3×#48614yes—
419C22C3×#52511yescharacteristic
420C22C2×C3×#12411yescharacteristic
421C22C3×#53411yescharacteristic
422C22C22×C3×D811yescharacteristic
423C22C22×C3×SD1611yescharacteristic
424C22C3×#57511yescharacteristic
425C22C3×#57511yescharacteristic
426C4—22no—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—22no—
431C4—44no—
432C4—82no—
433C4—82no—
434C4C3×#55314yes—
435C3#2932811yescharacteristic, direct product factor, Sylow
436C2—24no—
437C2—162no—
438C2C3×#259111yescharacteristic
439C2C3×#310011yescharacteristic
440C2C3×#310311yescharacteristic
441C2C2×C3×#48611yescharacteristic
442C2C3×#314211yescharacteristic
443C2C3×#318611yescharacteristic
444C2C3×#318711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2856C2
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×#389C2semidirect
3. C3×#2099C2semidirect
4. C3×#2041C2semidirect
5. C3×#1934C2semidirect
6. C3×#1874C2semidirect
7. C3×#1874C2semidirect
8. C3×C4×C22⋊C8C2semidirect
9. #29328C3direct (N × H)
35. C2×#389C2×C3semidirect
36. #2099C2×C3semidirect
37. #2041C2×C3semidirect
38. #1934C2×C3semidirect
39. #1874C2×C3semidirect
40. #1874C2×C3semidirect
41. C4×C22⋊C8C2×C3semidirect
435. C3#29328direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28562
C2×C3×#3892
C3×#20992
C3×#20412
C3×#19342
C3×#18742
C3×C4×C22⋊C82
#293283