Finite Group Databasebeta

← C3×#29298 #10647905

Subgroups — C3×#29298 #10647905

Order 768.

810 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2851C211yesmaximal, characteristic
2C2×C3×#389C211yesmaximal, characteristic
3C3×#2085C212yesmaximal, complemented
4C3×#2059C212yesmaximal, complemented
5C3×#1942C212yesmaximal, complemented
6C3×#1941C212yesmaximal, complemented
7C3×#1853C211yesmaximal, characteristic
8C3×#1853C211yesmaximal, characteristic
9C3×#1852C211yesmaximal, characteristic
10C3×#1852C211yesmaximal, characteristic
11C3×C4×C22⋊C8C211yesmaximal, characteristic
12#29298C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#125C2211yescharacteristic
14C22×C3×C4⋊C4—21no—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C2×C3×#107C2212yes—
19C2×C3×#107C2212yes—
20C2×C3×#106C2212yes—
21C2×C3×#106C2212yes—
22C2×C3×C22⋊C8C2211yescharacteristic
23C2×C3×C22⋊C8C2211yescharacteristic
24C2×C3×C4×C8C2212yes—
25C3×#389—21no—
26C3×#389—21no—
27C3×#389C2212yes—
28C3×#389C2212yes—
29C3×#389C2211yescharacteristic
30C3×#389C2211yescharacteristic
31C3×#387—21no—
32C3×#387—21no—
33C3×#385—21no—
34C3×#385—21no—
35C3×#382C2211yescharacteristic
36C3×#380C2211yescharacteristic
37C3×#380C2211yescharacteristic
38C2×C3×#95—21no—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44#2851C2×C311yescharacteristic
45C2×#389C2×C311yescharacteristic
46#2085C2×C312yescomplemented
47#2059C2×C312yescomplemented
48#1942C2×C312yescomplemented
49#1941C2×C312yescomplemented
50#1853C2×C311yescharacteristic
51#1853C2×C311yescharacteristic
52#1852C2×C311yescharacteristic
53#1852C2×C311yescharacteristic
54C4×C22⋊C8C2×C311yescharacteristic
55C23×C3×C4—21no—
56C23×C3×C4—21no—
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×C3×C8C2312yes—
61C22×C3×C8C2312yes—
62C3×#125—42no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—42no—
73C2×C3×C4⋊C4—44no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C42—42no—
83C2×C3×C42D411yescharacteristic
84C2×C3×C42D411yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C3×#107—24no—
88C3×#107—42no—
89C3×#106—24no—
90C3×#106—42no—
91C3×C22⋊C8Q814yes—
92C3×C22⋊C8Q814yes—
93C3×C4×C8—42no—
94C3×#95—22no—
95C3×#95—22no—
96C3×#95—21no—
97C3×#95—21no—
98C3×#95—21no—
99C3×#95—21no—
100C3×#95—21no—
101C3×#95—21no—
102C2×#125C22×C311yescharacteristic
103C22×C4⋊C4—21no—
104C22×C4⋊C4C22×C311yescharacteristic
105C22×C4⋊C4C22×C311yescharacteristic
106C22×C42C22×C311yescharacteristic
107C2×#107C22×C312yes—
108C2×#107C22×C312yes—
109C2×#106C22×C312yes—
110C2×#106C22×C312yes—
111C2×C22⋊C8C22×C311yescharacteristic
112C2×C22⋊C8C22×C311yescharacteristic
113C2×C4×C8C22×C312yes—
114#389—21no—
115#389—21no—
116#389C22×C312yes—
117#389C22×C311yescharacteristic
118#389C22×C311yescharacteristic
119#389C22×C312yes—
120#387—21no—
121#387—21no—
122#385—21no—
123#385—21no—
124#382C22×C311yescharacteristic
125#380C22×C311yescharacteristic
126#380C22×C311yescharacteristic
127C2×#95—21no—
128#340C22×C312yes—
129#340C22×C312yes—
130#340C22×C312yes—
131#340C22×C312yes—
132#336C22×C312yes—
133C24×C3C2×D411yescharacteristic
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—24no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—41no—
148C22×C3×C4—41no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4—42no—
152C22×C3×C4—42no—
153C22×C3×C4—42no—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×Q812yes—
160C22×C3×C4C2×Q812yes—
161C22×C3×C4C2411yescharacteristic
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8C2×Q814yes—
167C2×C3×C8C2×Q814yes—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4—44no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4—44no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4#4114yes—
184C3×C22⋊C4—42no—
185C3×C22⋊C4—42no—
186C3×C42—22no—
187C3×C42—22no—
188C3×C42—41no—
189C3×C42—41no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4C3×D411yescharacteristic
193C23×C4C3×D411yescharacteristic
194C23×C4C23×C311yescharacteristic
195C22×C8C23×C312yes—
196C22×C8C23×C312yes—
197#125—42no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4C23×C312yes—
210C2×C4⋊C4C23×C311yescharacteristic
211C2×C4⋊C4C23×C311yescharacteristic
212C2×C4⋊C4C23×C311yescharacteristic
213C2×C4⋊C4C23×C311yescharacteristic
214C2×C4⋊C4C23×C312yes—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C42—42no—
218C2×C42C3×D411yescharacteristic
219C2×C42C3×D411yescharacteristic
220C2×C42C23×C311yescharacteristic
221C2×C42C23×C311yescharacteristic
222#107—24no—
223#107—42no—
224#106—24no—
225#106—42no—
226C22⋊C8C3×Q814yes—
227C22⋊C8C3×Q814yes—
228C4×C8—42no—
229#95—22no—
230#95—22no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—21no—
240C23×C3—21no—
241C23×C3C4⋊Q812yes—
242C23×C3#13512yes—
243C23×C3#13512yes—
244C23×C3C22×D411yescharacteristic
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—24no—
252C2×C3×C4—24no—
253C2×C3×C4—24no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—44no—
261C2×C3×C4—44no—
262C2×C3×C4—48no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—84no—
281C2×C3×C4C4⋊Q812yes—
282C2×C3×C4AGL(1,ℤ/8)12yes—
283C2×C3×C4#13712yes—
284C2×C3×C4C22×Q811yescharacteristic
285C2×C3×C4C22×Q811yescharacteristic
286C2×C3×C4C2×#4112yes—
287C2×C3×C4H(5,𝔽2)12yes—
288C3×C8—24no—
289C3×C8—24no—
290C24C2×C3×D411yescharacteristic
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4C2×C3×D411yescharacteristic
312C22×C4C2×C3×D411yescharacteristic
313C22×C4C2×C3×D411yescharacteristic
314C22×C4C2×C3×D411yescharacteristic
315C22×C4C2×C3×D411yescharacteristic
316C22×C4C2×C3×Q812yes—
317C22×C4C2×C3×Q812yes—
318C22×C4C24×C311yescharacteristic
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8C2×C3×Q814yes—
324C2×C8C2×C3×Q814yes—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—42no—
334C4⋊C4—44no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—44no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4C3×#4114yes—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C42—22no—
344C42—22no—
345C42—41no—
346C42—41no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3#49914yes—
354C22×C3#52511yescharacteristic
355C22×C3C2×C4⋊Q811yescharacteristic
356C22×C3#53411yescharacteristic
357C22×C3C2×#13511yescharacteristic
358C22×C3C2×#13511yescharacteristic
359C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
360C22×C3C2×#13711yescharacteristic
361C3×C4—22no—
362C3×C4—22no—
363C3×C4—22no—
364C3×C4—44no—
365C3×C4—41no—
366C3×C4—41no—
367C3×C4—81no—
368C3×C4—81no—
369C3×C4—81no—
370C3×C4—82no—
371C3×C4—81no—
372C3×C4#55412yes—
373C3×C4#55412yes—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23C3×C4⋊Q812yes—
379C23C3×#13512yes—
380C23C3×#13512yes—
381C23C22×C3×D411yescharacteristic
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—24no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—84no—
418C2×C4C3×C4⋊Q812yes—
419C2×C4C3×AGL(1,ℤ/8)12yes—
420C2×C4C3×#13712yes—
421C2×C4C22×C3×Q811yescharacteristic
422C2×C4C22×C3×Q811yescharacteristic
423C2×C4C2×C3×#4112yes—
424C2×C4C3×H(5,𝔽2)12yes—
425C8—24no—
426C8—24no—
427C2×C3—24no—
428C2×C3#258311yescharacteristic
429C2×C3#310511yescharacteristic
430C2×C3#310811yescharacteristic
431C2×C3C2×#49911yescharacteristic
432C2×C3#316511yescharacteristic
433C2×C3#318811yescharacteristic
434C2×C3#319011yescharacteristic
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22C3×#49914yes—
442C22C3×#52511yescharacteristic
443C22C2×C3×C4⋊Q811yescharacteristic
444C22C3×#53411yescharacteristic
445C22C2×C3×#13511yescharacteristic
446C22C2×C3×#13511yescharacteristic
447C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
448C22C2×C3×#13711yescharacteristic
449C4—22no—
450C4—22no—
451C4—22no—
452C4—41no—
453C4—41no—
454C4—44no—
455C4—82no—
456C4—81no—
457C4—81no—
458C4—81no—
459C4—81no—
460C4C3×#55412yes—
461C4C3×#55412yes—
462C3#2929811yescharacteristic, direct product factor, Sylow
463C2—24no—
464C2C3×#258311yescharacteristic
465C2C3×#310511yescharacteristic
466C2C3×#310811yescharacteristic
467C2C2×C3×#49911yescharacteristic
468C2C3×#316511yescharacteristic
469C2C3×#318811yescharacteristic
470C2C3×#319011yescharacteristic
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×#2085C2semidirect
4. C3×#2059C2semidirect
5. C3×#1942C2semidirect
6. C3×#1941C2semidirect
12. #29298C3direct (N × H)
46. #2085C2×C3semidirect
47. #2059C2×C3semidirect
48. #1942C2×C3semidirect
49. #1941C2×C3semidirect
462. C3#29298direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28512
C2×C3×#3892
C3×#20852
C3×#20592
C3×#19422
C3×#19412
C3×#18532
C3×#18522
C3×C4×C22⋊C82
#292983