Finite Group Databasebeta

← C3×#29299 #10647906

Subgroups — C3×#29299 #10647906

Order 768.

782 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2857C211yesmaximal, characteristic
2C3×#2296C211yesmaximal, characteristic
3C3×#2084C211yesmaximal, characteristic, complemented
4C3×#2083C211yesmaximal, characteristic, complemented
5C3×#2058C211yesmaximal, characteristic, complemented
6C3×#2057C211yesmaximal, characteristic, complemented
7C3×#1942C212yesmaximal, complemented
8C3×#1941C212yesmaximal, complemented
9C3×#1854C212yesmaximal
10C3×#1854C212yesmaximal
11C3×C4×C22⋊C8C211yesmaximal, characteristic
12#29299C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C22⋊Q8—21no—
15C2×C3×#117C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C2×C3×#107C2212yes—
19C2×C3×#107C2212yes—
20C2×C3×#106C2212yes—
21C2×C3×#106C2212yes—
22C2×C3×C22⋊C8C2212yes—
23C2×C3×C4×C8C2212yes—
24C3×#391—21no—
25C3×#389—21no—
26C3×#389C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C3×#389C2211yescharacteristic
30C3×#389C2212yes—
31C3×#385—21no—
32C3×#385—21no—
33C3×#384C2211yescharacteristic
34C3×#380—21no—
35C3×#380C2212yes—
36C3×C4×C4⋊C4—21no—
37C3×C4×C4⋊C4—21no—
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#336C2211yescharacteristic
43C3×#336C2211yescharacteristic
44#2857C2×C311yescharacteristic
45#2296C2×C311yescharacteristic
46#2084C2×C311yescharacteristic, complemented
47#2083C2×C311yescharacteristic, complemented
48#2058C2×C311yescharacteristic, complemented
49#2057C2×C311yescharacteristic, complemented
50#1942C2×C312yescomplemented
51#1941C2×C312yescomplemented
52#1854C2×C312yes—
53#1854C2×C312yes—
54C4×C22⋊C8C2×C311yescharacteristic
55C22×C3×Q8—21no—
56C23×C3×C4D411yescharacteristic
57C23×C3×C4D411yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C22×C3×C8C2312yes—
60C22×C3×C8C2312yes—
61C3×C4⋊Q8—42no—
62C3×C22⋊Q8—44no—
63C3×#117—22no—
64C3×#117—22no—
65C3×#117—42no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C42—21no—
84C2×C3×C42—21no—
85C2×C3×C42—42no—
86C2×C3×C42D411yescharacteristic
87C2×C3×C42D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C3×#107—24no—
91C3×#107—42no—
92C3×#106—24no—
93C3×#106—42no—
94C3×C22⋊C8Q818yescomplemented
95C3×C4×C8—42no—
96C3×#95—21no—
97C3×#95—21no—
98C3×#95—21no—
99C3×#95—21no—
100C3×#95—22no—
101C3×#95—22no—
102C2×C4⋊Q8C22×C311yescharacteristic
103C2×C22⋊Q8—21no—
104C2×#117C22×C311yescharacteristic
105C2×#117C22×C311yescharacteristic
106C22×C42C22×C311yescharacteristic
107C2×#107C22×C312yes—
108C2×#107C22×C312yes—
109C2×#106C22×C312yes—
110C2×#106C22×C312yes—
111C2×C22⋊C8C22×C312yes—
112C2×C4×C8C22×C312yes—
113#391—21no—
114#389—21no—
115#389C22×C311yescharacteristic
116#389C22×C311yescharacteristic
117#389C22×C312yes—
118#389C22×C311yescharacteristic
119#389C22×C311yescharacteristic
120#385—21no—
121#385—21no—
122#384C22×C311yescharacteristic
123#380—21no—
124#380C22×C312yes—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127#340C22×C312yes—
128#340C22×C312yes—
129#340C22×C312yes—
130#340C22×C312yes—
131#336C22×C311yescharacteristic
132#336C22×C311yescharacteristic
133C24×C3C2×D411yescharacteristic
134C2×C3×Q8—42no—
135C2×C3×Q8—42no—
136C2×C3×Q8—42no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—41no—
151C22×C3×C4—41no—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×Q814yes—
158C22×C3×C4C2411yescharacteristic
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8C2×Q814yes—
164C2×C3×C8C2×Q814yes—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—42no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4—42no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4—42no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—44no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4#4112yes—
182C3×C4⋊C4#4112yes—
183C3×C22⋊C4—24no—
184C3×C22⋊C4—42no—
185C3×C22⋊C4—44no—
186C3×C42—24no—
187C3×C42—22no—
188C3×C42—22no—
189C3×C42—41no—
190C3×C42—41no—
191C3×C42—42no—
192C22×Q8—21no—
193C23×C4C3×D411yescharacteristic
194C23×C4C3×D411yescharacteristic
195C23×C4C23×C311yescharacteristic
196C22×C8C23×C312yes—
197C22×C8C23×C312yes—
198C4⋊Q8—42no—
199C22⋊Q8—44no—
200#117—22no—
201#117—22no—
202#117—42no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C23×C311yescharacteristic
212C2×C4⋊C4C23×C311yescharacteristic
213C2×C4⋊C4C23×C312yes—
214C2×C4⋊C4C23×C312yes—
215C2×C4⋊C4C23×C311yescharacteristic
216C2×C4⋊C4C23×C311yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—42no—
223C2×C42C3×D411yescharacteristic
224C2×C42C3×D411yescharacteristic
225C2×C42C23×C311yescharacteristic
226C2×C42C23×C311yescharacteristic
227#107—24no—
228#107—42no—
229#106—24no—
230#106—42no—
231C22⋊C8C3×Q818yescomplemented
232C4×C8—42no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—22no—
238#95—22no—
239C23×C3—21no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3C4⋊Q812yes—
246C23×C3C22×D411yescharacteristic
247C3×Q8—84no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—24no—
259C2×C3×C4—24no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—44no—
263C2×C3×C4—44no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4C4⋊Q812yes—
281C2×C3×C4#13512yes—
282C2×C3×C4#13512yes—
283C2×C3×C4AGL(1,ℤ/8)12yes—
284C2×C3×C4#13712yes—
285C2×C3×C4C22×Q812yes—
286C2×C3×C4C2×#4111yescharacteristic
287C2×C3×C4C2×#4111yescharacteristic
288C2×C3×C4H(5,𝔽2)12yes—
289C3×C8—24no—
290C3×C8—24no—
291C24C2×C3×D411yescharacteristic
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4C2×C3×D411yescharacteristic
311C22×C4C2×C3×D411yescharacteristic
312C22×C4C2×C3×D411yescharacteristic
313C22×C4C2×C3×D411yescharacteristic
314C22×C4C2×C3×D411yescharacteristic
315C22×C4C2×C3×Q814yes—
316C22×C4C24×C311yescharacteristic
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8C2×C3×Q814yes—
322C2×C8C2×C3×Q814yes—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—44no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4C3×#4112yes—
340C4⋊C4C3×#4112yes—
341C22⋊C4—24no—
342C22⋊C4—42no—
343C22⋊C4—44no—
344C42—24no—
345C42—22no—
346C42—22no—
347C42—41no—
348C42—41no—
349C42—42no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3#52511yescharacteristic
358C22×C3C2×C4⋊Q811yescharacteristic
359C22×C3#53411yescharacteristic
360C22×C3C2×#13511yescharacteristic
361C22×C3C2×#13511yescharacteristic
362C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
363C22×C3C2×#13711yescharacteristic
364C3×C4—22no—
365C3×C4—22no—
366C3×C4—22no—
367C3×C4—22no—
368C3×C4—22no—
369C3×C4—42no—
370C3×C4—42no—
371C3×C4—82no—
372C3×C4—81no—
373C3×C4—82no—
374C3×C4—81no—
375C3×C4#55412yes—
376C3×C4#55412yes—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23C3×C4⋊Q812yes—
384C23C22×C3×D411yescharacteristic
385Q8—84no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4C3×C4⋊Q812yes—
419C2×C4C3×#13512yes—
420C2×C4C3×#13512yes—
421C2×C4C3×AGL(1,ℤ/8)12yes—
422C2×C4C3×#13712yes—
423C2×C4C22×C3×Q812yes—
424C2×C4C2×C3×#4111yescharacteristic
425C2×C4C2×C3×#4111yescharacteristic
426C2×C4C3×H(5,𝔽2)12yes—
427C8—24no—
428C8—24no—
429C2×C3—42no—
430C2×C3#258311yescharacteristic
431C2×C3#310611yescharacteristic
432C2×C3#310711yescharacteristic
433C2×C3#316011yescharacteristic
434C2×C3#316511yescharacteristic
435C2×C3#318811yescharacteristic
436C2×C3#319011yescharacteristic
437C22—22no—
438C22—22no—
439C22—22no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22C3×#52511yescharacteristic
445C22C2×C3×C4⋊Q811yescharacteristic
446C22C3×#53411yescharacteristic
447C22C2×C3×#13511yescharacteristic
448C22C2×C3×#13511yescharacteristic
449C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
450C22C2×C3×#13711yescharacteristic
451C4—22no—
452C4—22no—
453C4—22no—
454C4—22no—
455C4—22no—
456C4—42no—
457C4—42no—
458C4—82no—
459C4—81no—
460C4—81no—
461C4—82no—
462C4C3×#55412yes—
463C4C3×#55412yes—
464C3#2929911yescharacteristic, direct product factor, Sylow
465C2—42no—
466C2C3×#258311yescharacteristic
467C2C3×#310611yescharacteristic
468C2C3×#310711yescharacteristic
469C2C3×#316011yescharacteristic
470C2C3×#316511yescharacteristic
471C2C3×#318811yescharacteristic
472C2C3×#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×#2084C2semidirect
4. C3×#2083C2semidirect
5. C3×#2058C2semidirect
6. C3×#2057C2semidirect
7. C3×#1942C2semidirect
8. C3×#1941C2semidirect
12. #29299C3direct (N × H)
46. #2084C2×C3semidirect
47. #2083C2×C3semidirect
48. #2058C2×C3semidirect
49. #2057C2×C3semidirect
50. #1942C2×C3semidirect
51. #1941C2×C3semidirect
94. C3×C22⋊C8Q8semidirect
231. C22⋊C8C3×Q8semidirect
464. C3#29299direct (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×#22962
C3×#20842
C3×#20832
C3×#20582
C3×#20572
C3×#19422
C3×#19412
C3×#18542
C3×C4×C22⋊C82
#292993