Finite Group Databasebeta

← C3×#29687 #10648294

Subgroups — C3×#29687 #10648294

Order 768.

1348 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C2×C3×#474C212yesmaximal, complemented
3C3×#2857C211yesmaximal, characteristic, complemented
4C3×#2057C212yesmaximal, complemented
5C3×#2041C212yesmaximal, complemented
6C3×#2033C214yesmaximal, complemented
7C3×#1802C212yesmaximal, complemented
8C3×#1797C211yesmaximal, characteristic, complemented
9#29687C311yesmaximal, characteristic, direct product factor, Sylow
10C22×C3×#41—21no—
11C3×#532—28no—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#127C2211yescharacteristic
15C2×C3×#124—21no—
16C2×C3×C22⋊Q8—21no—
17C2×C3×#121—21no—
18C2×C3×#121C2212yes—
19C2×C3×C4×D4—21no—
20C2×C3×C4×D4C2212yes—
21C22×C3×C42C2211yescharacteristic
22C3×#474—28no—
23C2×C3×#107C2214yes—
24C2×C3×C4⋊C8C2212yes—
25C2×C3×#102C2214yes—
26C2×C3×#102C2214yes—
27C2×C3×C22⋊C8C2212yes—
28C3×#391—21no—
29C3×#389—21no—
30C3×#389C2212yes—
31C3×#384C2211yescharacteristic
32C3×#380—21no—
33C3×#380C2212yes—
34C3×#340C2214yes—
35C3×#336C2212yes—
36C2×#532C2×C311yescharacteristic
37C2×#474C2×C312yescomplemented
38#2857C2×C311yescharacteristic, complemented
39#2057C2×C312yescomplemented
40#2041C2×C312yescomplemented
41#2033C2×C314yescomplemented
42#1802C2×C312yescomplemented
43#1797C2×C311yescharacteristic, complemented
44C2×C3×#41—28no—
45C2×C3×#41—44no—
46C2×C3×#41—44no—
47C22×C3×Q8—21no—
48C22×C3×Q8—21no—
49C22×C3×D4—22no—
50C22×C3×D4—21no—
51C22×C3×D4—21no—
52C22×C3×D4C2312yes—
53C23×C3×C4—22no—
54C23×C3×C4D411yescharacteristic
55C23×C3×C4D411yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C22×C3×C8C2314yes—
58C3×C4⋊Q8—24no—
59C3×C4⋊Q8—24no—
60C3×#127—24no—
61C3×#124—28no—
62C3×C22⋊Q8—44no—
63C3×#121—28no—
64C3×#121—24no—
65C3×#121—24no—
66C3×C4×D4—28no—
67C3×C4×D4—24no—
68C3×C4×D4D418yes—
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⋊C4C2312yes—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C42—24no—
81C2×C3×C42D411yescharacteristic
82C2×C3×C42D411yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C3×#107—28no—
86C3×C4⋊C8—24no—
87C3×#102—28no—
88C3×#102—44no—
89C3×C22⋊C8—42no—
90C3×#95—22no—
91C3×#95—22no—
92C22×#41—21no—
93#532—28no—
94C2×C4⋊Q8C22×C311yescharacteristic
95C2×C4⋊Q8C22×C311yescharacteristic
96C2×#127C22×C311yescharacteristic
97C2×#124—21no—
98C2×C22⋊Q8—21no—
99C2×#121—21no—
100C2×#121C22×C312yes—
101C2×C4×D4—21no—
102C2×C4×D4C22×C312yes—
103C22×C42C22×C311yescharacteristic
104#474—28no—
105C2×#107C22×C314yes—
106C2×C4⋊C8C22×C312yes—
107C2×#102C22×C314yes—
108C2×#102C22×C314yes—
109C2×C22⋊C8C22×C312yes—
110#391—21no—
111#389—21no—
112#389C22×C312yes—
113#384C22×C311yescharacteristic
114#380—21no—
115#380C22×C312yes—
116#340C22×C314yes—
117#336C22×C312yes—
118C24×C3—22no—
119C24×C3C2×D411yescharacteristic
120C3×#41—432no—
121C2×C3×Q8—24no—
122C2×C3×Q8—42no—
123C2×C3×Q8—42no—
124C2×C3×Q8—42no—
125C2×C3×Q8—42no—
126C2×C3×Q8—42no—
127C2×C3×D4—24no—
128C2×C3×D4—28no—
129C2×C3×D4—24no—
130C2×C3×D4—22no—
131C2×C3×D4—22no—
132C2×C3×D4—22no—
133C2×C3×D4—22no—
134C2×C3×D4—22no—
135C2×C3×D4—42no—
136C2×C3×D4—42no—
137C2×C3×D4—44no—
138C2×C3×D4—44no—
139C2×C3×D4—42no—
140C2×C3×D4—42no—
141C2×C3×D4C2×D414yes—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—28no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—41no—
155C22×C3×C4—44no—
156C22×C3×C4—44no—
157C22×C3×C4D814yes—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C2×C3×C8—24no—
165C2×C3×C8—24no—
166C2×C3×C8—24no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—24no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—42no—
173C3×C4⋊C4—42no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4—44no—
176C3×C4⋊C4C2×D414yes—
177C3×C22⋊C4—28no—
178C3×C22⋊C4—28no—
179C3×C22⋊C4—44no—
180C3×C42—22no—
181C3×C42—22no—
182C3×C42—22no—
183C3×C42C2×D414yes—
184C2×#41—28no—
185C2×#41—44no—
186C2×#41—44no—
187C22×Q8—21no—
188C22×Q8—21no—
189C22×D4—22no—
190C22×D4—21no—
191C22×D4—21no—
192C22×D4C23×C312yes—
193C23×C4—22no—
194C23×C4C3×D411yescharacteristic
195C23×C4C3×D411yescharacteristic
196C23×C4C23×C311yescharacteristic
197C22×C8C23×C314yes—
198C4⋊Q8—24no—
199C4⋊Q8—24no—
200#127—24no—
201#124—28no—
202C22⋊Q8—44no—
203#121—28no—
204#121—24no—
205#121—24no—
206C4×D4—28no—
207C4×D4—24no—
208C4×D4C3×D418yes—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C23×C312yes—
215C2×C4⋊C4C23×C312yes—
216C2×C4⋊C4C23×C312yes—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C42—24no—
221C2×C42C3×D411yescharacteristic
222C2×C42C3×D411yescharacteristic
223C2×C42C23×C311yescharacteristic
224C2×C42C23×C311yescharacteristic
225#107—28no—
226C4⋊C8—24no—
227#102—28no—
228#102—44no—
229C22⋊C8—42no—
230#95—22no—
231#95—22no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—21no—
238C23×C3—24no—
239C23×C3—42no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3C2×D812yes—
246C23×C3C22×D411yescharacteristic
247C3×Q8—48no—
248C3×Q8—84no—
249C3×D4—24no—
250C3×D4—24no—
251C3×D4—48no—
252C3×D4—48no—
253C3×D4—416no—
254C3×D4—42no—
255C3×D4—42no—
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—28no—
263C2×C3×C4—21no—
264C2×C3×C4—24no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—44no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—416no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4#12012yes—
280C2×C3×C4#12012yes—
281C2×C3×C4C2×D812yes—
282C2×C3×C4C2×D812yes—
283C2×C3×C4AGL(1,ℤ/8)12yes—
284C2×C3×C4#13712yes—
285C2×C3×C4C22×D412yes—
286C2×C3×C4H(5,𝔽2)11yescharacteristic
287C2×C3×C42−1+411yescharacteristic
288C2×C3×C42−1+412yes—
289C3×C8—44no—
290C24—22no—
291C24C2×C3×D411yescharacteristic
292#41—432no—
293C2×Q8—24no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×D4—24no—
300C2×D4—28no—
301C2×D4—24no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4C2×C3×D414yes—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—28no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—41no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4C3×D814yes—
330C22×C4C2×C3×D411yescharacteristic
331C22×C4C2×C3×D411yescharacteristic
332C22×C4C2×C3×D411yescharacteristic
333C22×C4C2×C3×D411yescharacteristic
334C22×C4C2×C3×D411yescharacteristic
335C22×C4C24×C311yescharacteristic
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—44no—
348C4⋊C4C2×C3×D414yes—
349C22⋊C4—28no—
350C22⋊C4—28no—
351C22⋊C4—44no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42C2×C3×D414yes—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3—44no—
364C22×C3—44no—
365C22×C3—44no—
366C22×C3—82no—
367C22×C3—82no—
368C22×C3—82no—
369C22×C3—82no—
370C22×C3C2×#12011yescharacteristic
371C22×C3#53611yescharacteristic
372C22×C3#53711yescharacteristic
373C22×C3C22×D811yescharacteristic
374C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
375C22×C3C2×#13711yescharacteristic
376C22×C3#57511yescharacteristic
377C3×C4—22no—
378C3×C4—22no—
379C3×C4—22no—
380C3×C4—41no—
381C3×C4—44no—
382C3×C4—82no—
383C3×C4—82no—
384C3×C4#44714yes—
385C3×C4#53612yes—
386C3×C4#53712yes—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—24no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23C2×C3×D812yes—
401C23C22×C3×D411yescharacteristic
402Q8—48no—
403Q8—84no—
404D4—24no—
405D4—24no—
406D4—48no—
407D4—48no—
408D4—416no—
409D4—42no—
410D4—42no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—28no—
418C2×C4—21no—
419C2×C4—24no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—416no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4C3×#12012yes—
435C2×C4C3×#12012yes—
436C2×C4C2×C3×D812yes—
437C2×C4C2×C3×D812yes—
438C2×C4C3×AGL(1,ℤ/8)12yes—
439C2×C4C3×#13712yes—
440C2×C4C22×C3×D412yes—
441C2×C4C3×H(5,𝔽2)11yescharacteristic
442C2×C4C3×2−1+411yescharacteristic
443C2×C4C3×2−1+412yes—
444C8—44no—
445C2×C3—42no—
446C2×C3—84no—
447C2×C3#261611yescharacteristic
448C2×C3C2×#44711yescharacteristic
449C2×C3#300711yescharacteristic
450C2×C3#322611yescharacteristic
451C2×C3#323511yescharacteristic
452C2×C3#324511yescharacteristic
453C2×C3#325411yescharacteristic
454C22—22no—
455C22—22no—
456C22—22no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—44no—
462C22—44no—
463C22—44no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22C2×C3×#12011yescharacteristic
469C22C3×#53611yescharacteristic
470C22C3×#53711yescharacteristic
471C22C22×C3×D811yescharacteristic
472C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
473C22C2×C3×#13711yescharacteristic
474C22C3×#57511yescharacteristic
475C4—22no—
476C4—22no—
477C4—22no—
478C4—41no—
479C4—44no—
480C4—82no—
481C4—82no—
482C4C3×#44714yes—
483C4C3×#53612yes—
484C4C3×#53712yes—
485C3#2968711yescharacteristic, direct product factor, Sylow
486C2—42no—
487C2—84no—
488C2C3×#261611yescharacteristic
489C2C2×C3×#44711yescharacteristic
490C2C3×#300711yescharacteristic
491C2C3×#322611yescharacteristic
492C2C3×#323511yescharacteristic
493C2C3×#324511yescharacteristic
494C2C3×#325411yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#532C2
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×#474C2semidirect
3. C3×#2857C2semidirect
4. C3×#2057C2semidirect
5. C3×#2041C2semidirect
6. C3×#2033C2semidirect
7. C3×#1802C2semidirect
8. C3×#1797C2semidirect
9. #29687C3direct (N × H)
37. C2×#474C2×C3semidirect
38. #2857C2×C3semidirect
39. #2057C2×C3semidirect
40. #2041C2×C3semidirect
41. #2033C2×C3semidirect
42. #1802C2×C3semidirect
43. #1797C2×C3semidirect
485. C3#29687direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×C3×#4742
C3×#28572
C3×#20572
C3×#20412
C3×#20332
C3×#18022
C3×#17972
#296873