Finite Group Databasebeta

← C3×#29160 #10647767

Subgroups — C3×#29160 #10647767

Order 768.

828 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2857C211yesmaximal, characteristic
2C3×C4×C22⋊Q8C211yesmaximal, characteristic
3C3×#2100C211yesmaximal, characteristic, complemented
4C3×#2095C211yesmaximal, characteristic, complemented
5C3×#2057C211yesmaximal, characteristic, complemented
6C3×#2024C211yesmaximal, characteristic, complemented
7C3×#1921C212yesmaximal, complemented
8C3×#1892C212yesmaximal
9C3×#1863C212yesmaximal, complemented
10C3×#1808C212yesmaximal
11C2×C3×#336C211yesmaximal, characteristic
12#29160C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×C8C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C22⋊Q8—21no—
16C2×C3×C22⋊Q8C2211yescharacteristic
17C2×C3×C4×Q8C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C2×C3×#107C2212yes—
20C2×C3×#107C2212yes—
21C2×C3×#103C2212yes—
22C2×C3×#103C2212yes—
23C3×#391—21no—
24C3×#391C2211yescharacteristic
25C3×#389—21no—
26C3×#389C2211yescharacteristic
27C3×#389C2212yes—
28C3×#387—21no—
29C3×#385—21no—
30C3×#384C2211yescharacteristic
31C3×#380—21no—
32C3×#380C2211yescharacteristic
33C3×#380C2212yes—
34C3×C4×C4⋊C4—21no—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×C4×C22⋊C4—21no—
37C3×#340C2212yes—
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#336C2211yescharacteristic
42C3×#336C2211yescharacteristic
43C3×#336C2212yes—
44#2857C2×C311yescharacteristic
45C4×C22⋊Q8C2×C311yescharacteristic
46#2100C2×C311yescharacteristic, complemented
47#2095C2×C311yescharacteristic, complemented
48#2057C2×C311yescharacteristic, complemented
49#2024C2×C311yescharacteristic, complemented
50#1921C2×C312yescomplemented
51#1892C2×C312yes—
52#1863C2×C312yescomplemented
53#1808C2×C312yes—
54C2×#336C2×C311yescharacteristic
55C22×C3×Q8—21no—
56C22×C3×Q8C2311yescharacteristic
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×C3×C8—24no—
61C22×C3×C8—24no—
62C22×C3×C8—42no—
63C22×C3×C8C2312yes—
64C22×C3×C8C2312yes—
65C3×C4⋊Q8—42no—
66C3×C22⋊Q8—22no—
67C3×C22⋊Q8—44no—
68C3×C22⋊Q8Q814yescomplemented
69C3×C4×Q8—42no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C42—21no—
86C2×C3×C42—21no—
87C2×C3×C42—42no—
88C2×C3×C42D411yescharacteristic
89C2×C3×C42D411yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C3×#107—24no—
93C3×#107—42no—
94C3×#103—24no—
95C3×#103—42no—
96C3×#95—21no—
97C3×#95—21no—
98C3×#95—21no—
99C3×#95—21no—
100C3×#95—21no—
101C3×#95—22no—
102C3×#95—22no—
103C23×C8C22×C312yes—
104C2×C4⋊Q8C22×C311yescharacteristic
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8C22×C311yescharacteristic
107C2×C4×Q8C22×C311yescharacteristic
108C22×C42C22×C311yescharacteristic
109C2×#107C22×C312yes—
110C2×#107C22×C312yes—
111C2×#103C22×C312yes—
112C2×#103C22×C312yes—
113#391—21no—
114#391C22×C311yescharacteristic
115#389—21no—
116#389C22×C311yescharacteristic
117#389C22×C312yes—
118#387—21no—
119#385—21no—
120#384C22×C311yescharacteristic
121#380—21no—
122#380C22×C311yescharacteristic
123#380C22×C312yes—
124C4×C4⋊C4—21no—
125C4×C4⋊C4C22×C311yescharacteristic
126C4×C22⋊C4—21no—
127#340C22×C312yes—
128#340C22×C312yes—
129#340C22×C312yes—
130#340C22×C312yes—
131#336C22×C311yescharacteristic
132#336C22×C311yescharacteristic
133#336C22×C312yes—
134C24×C3C2×D411yescharacteristic
135C2×C3×Q8—21no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—42no—
139C2×C3×Q8—42no—
140C2×C3×Q8—42no—
141C2×C3×Q8C2×Q812yes—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—41no—
153C22×C3×C4—41no—
154C22×C3×C4—41no—
155C22×C3×C4Q1614yes—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×Q812yes—
162C22×C3×C4#4112yes—
163C22×C3×C4C2411yescharacteristic
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—48no—
170C2×C3×C8#4114yes—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4—44no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—42no—
185C3×C4⋊C4C2×Q812yes—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—42no—
188C3×C22⋊C4—44no—
189C3×C42—22no—
190C3×C42—22no—
191C3×C42—41no—
192C3×C42—41no—
193C3×C42—42no—
194C3×C42—42no—
195C22×Q8—21no—
196C22×Q8C23×C311yescharacteristic
197C23×C4C3×D411yescharacteristic
198C23×C4C3×D411yescharacteristic
199C23×C4C23×C311yescharacteristic
200C22×C8—24no—
201C22×C8—24no—
202C22×C8—42no—
203C22×C8C23×C312yes—
204C22×C8C23×C312yes—
205C4⋊Q8—42no—
206C22⋊Q8—22no—
207C22⋊Q8—44no—
208C22⋊Q8C3×Q814yescomplemented
209C4×Q8—42no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C23×C311yescharacteristic
218C2×C4⋊C4C23×C312yes—
219C2×C4⋊C4C23×C311yescharacteristic
220C2×C4⋊C4C23×C312yes—
221C2×C4⋊C4C23×C311yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—42no—
228C2×C42C3×D411yescharacteristic
229C2×C42C3×D411yescharacteristic
230C2×C42C23×C311yescharacteristic
231C2×C42C23×C311yescharacteristic
232#107—24no—
233#107—42no—
234#103—24no—
235#103—42no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—22no—
242#95—22no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3—21no—
246C23×C3—21no—
247C23×C3C22⋊Q812yes—
248C23×C3C2×Q1612yes—
249C23×C3AGL(1,ℤ/8)12yes—
250C23×C3C22×D411yescharacteristic
251C3×Q8—24no—
252C3×Q8—42no—
253C3×Q8—84no—
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—21no—
261C2×C3×C4—21no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—44no—
270C2×C3×C4—44no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—42no—
287C2×C3×C4C22⋊Q812yes—
288C2×C3×C4C2×Q1612yes—
289C2×C3×C4C2×Q1612yes—
290C2×C3×C4#13512yes—
291C2×C3×C4C22×Q811yescharacteristic
292C2×C3×C4C2×#4111yescharacteristic
293C2×C3×C4C2×#4112yes—
294C2×C3×C4H(5,𝔽2)12yes—
295C3×C8—24no—
296C3×C8—42no—
297C24C2×C3×D411yescharacteristic
298C2×Q8—21no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8C2×C3×Q812yes—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4C3×Q1614yes—
319C22×C4C2×C3×D411yescharacteristic
320C22×C4C2×C3×D411yescharacteristic
321C22×C4C2×C3×D411yescharacteristic
322C22×C4C2×C3×D411yescharacteristic
323C22×C4C2×C3×D411yescharacteristic
324C22×C4C2×C3×Q812yes—
325C22×C4C3×#4112yes—
326C22×C4C24×C311yescharacteristic
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—48no—
333C2×C8C3×#4114yes—
334C4⋊C4—24no—
335C4⋊C4—21no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4C2×C3×Q812yes—
349C22⋊C4—24no—
350C22⋊C4—44no—
351C22⋊C4—42no—
352C42—22no—
353C42—22no—
354C42—41no—
355C42—41no—
356C42—42no—
357C42—42no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3—22no—
364C22×C3#47714yes—
365C22×C3C2×C22⋊Q811yescharacteristic
366C22×C3#53211yescharacteristic
367C22×C3#53411yescharacteristic
368C22×C3C22×Q1611yescharacteristic
369C22×C3C2×#13511yescharacteristic
370C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
371C22×C3#57511yescharacteristic
372C3×C4—22no—
373C3×C4—22no—
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—42no—
377C3×C4—42no—
378C3×C4—41no—
379C3×C4—82no—
380C3×C4—81no—
381C3×C4—81no—
382C3×C4—82no—
383C3×C4#55112yes—
384C3×C4#55412yes—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23C3×C22⋊Q812yes—
390C23C2×C3×Q1612yes—
391C23C3×AGL(1,ℤ/8)12yes—
392C23C22×C3×D411yescharacteristic
393Q8—24no—
394Q8—42no—
395Q8—84no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4C3×C22⋊Q812yes—
430C2×C4C2×C3×Q1612yes—
431C2×C4C2×C3×Q1612yes—
432C2×C4C3×#13512yes—
433C2×C4C22×C3×Q811yescharacteristic
434C2×C4C2×C3×#4111yescharacteristic
435C2×C4C2×C3×#4112yes—
436C2×C4C3×H(5,𝔽2)12yes—
437C8—24no—
438C8—42no—
439C2×C3—24no—
440C2×C3#256411yescharacteristic
441C2×C3C2×#47711yescharacteristic
442C2×C3#307611yescharacteristic
443C2×C3#317011yescharacteristic
444C2×C3#318011yescharacteristic
445C2×C3#318811yescharacteristic
446C2×C3#318911yescharacteristic
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22C3×#47714yes—
454C22C2×C3×C22⋊Q811yescharacteristic
455C22C3×#53211yescharacteristic
456C22C3×#53411yescharacteristic
457C22C22×C3×Q1611yescharacteristic
458C22C2×C3×#13511yescharacteristic
459C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
460C22C3×#57511yescharacteristic
461C4—22no—
462C4—22no—
463C4—22no—
464C4—22no—
465C4—41no—
466C4—42no—
467C4—42no—
468C4—82no—
469C4—81no—
470C4—82no—
471C4—81no—
472C4C3×#55112yes—
473C4C3×#55412yes—
474C3#2916011yescharacteristic, direct product factor, Sylow
475C2—24no—
476C2C3×#256411yescharacteristic
477C2C2×C3×#47711yescharacteristic
478C2C3×#307611yescharacteristic
479C2C3×#317011yescharacteristic
480C2C3×#318011yescharacteristic
481C2C3×#318811yescharacteristic
482C2C3×#318911yescharacteristic
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×#2057C2semidirect
6. C3×#2024C2semidirect
7. C3×#1921C2semidirect
9. C3×#1863C2semidirect
12. #29160C3direct (N × H)
46. #2100C2×C3semidirect
47. #2095C2×C3semidirect
48. #2057C2×C3semidirect
49. #2024C2×C3semidirect
50. #1921C2×C3semidirect
52. #1863C2×C3semidirect
68. C3×C22⋊Q8Q8semidirect
208. C22⋊Q8C3×Q8semidirect
474. C3#29160direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28572
C3×C4×C22⋊Q82
C3×#21002
C3×#20952
C3×#20572
C3×#20242
C3×#19212
C3×#18922
C3×#18632
C3×#18082
C2×C3×#3362
#291603