Finite Group Databasebeta

← C3×#10000 #10628607

Subgroups — C3×#10000 #10628607

Order 768.

632 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3101C211yesmaximal, characteristic, complemented
2C3×#3077C211yesmaximal, characteristic, complemented
3C3×#2535C211yesmaximal, characteristic
4C3×#1518C211yesmaximal, characteristic, complemented
5C3×#1486C211yesmaximal, characteristic, complemented
6C3×#1478C211yesmaximal, characteristic, complemented
7C3×#1286C211yesmaximal, characteristic, complemented
8#10000C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#121—21no—
11C2×C3×C4×D4C2211yescharacteristic
12C2×C3×C4×D4C2211yescharacteristic
13C3×#483—21no—
14C3×#478—22no—
15C3×#476—21no—
16C3×#476—21no—
17C3×#474—21no—
18C3×#461—21no—
19C3×#450—21no—
20C3×C4×SD16—22no—
21C3×#431C2211yescharacteristic
22C3×#431C2211yescharacteristic
23C3×#428—21no—
24C3×#423C2211yescharacteristic
25C3×#418—21no—
26C3×#397—21no—
27C3×#392—21no—
28C3×#386—22no—
29C3×#382C2211yescharacteristic
30C3×#380—21no—
31C3×#328—22no—
32C3×D4⋊C8—22no—
33C3×D4⋊C8—22no—
34#3101C2×C311yescharacteristic, complemented
35#3077C2×C311yescharacteristic, complemented
36#2535C2×C311yescharacteristic
37#1518C2×C311yescharacteristic, complemented
38#1486C2×C311yescharacteristic, complemented
39#1478C2×C311yescharacteristic, complemented
40#1286C2×C311yescharacteristic, complemented
41C22×C3×D4—22no—
42C22×C3×D4D411yescharacteristic
43C22×C3×D4D411yescharacteristic
44C23×C3×C4—21no—
45C23×C3×C4—21no—
46C2×C3×SD16—22no—
47C2×C3×M16—21no—
48C3×C4⋊Q8D411yescharacteristic, complemented
49C3×C4⋊Q8D411yescharacteristic, complemented
50C3×#125—21no—
51C3×C22⋊Q8—21no—
52C3×C22⋊Q8—21no—
53C3×#121—42no—
54C3×#121—42no—
55C3×C4×Q8—21no—
56C3×C4×Q8—21no—
57C3×C4×D4—21no—
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60C3×C4×D4—21no—
61C3×C4×D4—21no—
62C3×C4×D4—21no—
63C3×C4×D4—21no—
64C3×C4×D4—21no—
65C3×#117—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4D411yescharacteristic
68C2×C3×C4⋊C4D411yescharacteristic
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C42C2311yescharacteristic
74C3×#107—22no—
75C3×#106—22no—
76C3×#106—22no—
77C3×C4⋊C8—21no—
78C3×C4⋊C8—21no—
79C3×C4⋊C8—21no—
80C3×C4⋊C8D412yescomplemented
81C3×#103—22no—
82C3×#102—22no—
83C3×#102—22no—
84C3×#102—22no—
85C3×C22⋊C8—21no—
86C3×C22⋊C8—21no—
87C3×C4×C8—21no—
88C3×C4×C8D412yescomplemented
89C3×#95—22no—
90C3×#95—21no—
91C3×#95D412yes—
92#533C22×C311yescharacteristic
93C2×#121—21no—
94C2×C4×D4C22×C311yescharacteristic
95C2×C4×D4C22×C311yescharacteristic
96#483—21no—
97#478—22no—
98#476—21no—
99#476—21no—
100#474—21no—
101#461—21no—
102#450—21no—
103C4×SD16—22no—
104#431C22×C311yescharacteristic
105#431C22×C311yescharacteristic
106#428—21no—
107#423C22×C311yescharacteristic
108#418—21no—
109#397—21no—
110#392—21no—
111#386—22no—
112#382C22×C311yescharacteristic
113#380—21no—
114#328—22no—
115D4⋊C8—22no—
116D4⋊C8—22no—
117C24×C3—21no—
118C24×C3—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—42no—
129C2×C3×D4—44no—
130C2×C3×D4—44no—
131C2×C3×D4—82no—
132C2×C3×D4SD1612yescomplemented
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4—41no—
142C22×C3×C4—41no—
143C22×C3×C4—41no—
144C22×C3×C4—82no—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C3×SD16—44no—
149C3×M16—41no—
150C3×M16—41no—
151C2×C3×C8—22no—
152C2×C3×C8—22no—
153C2×C3×C8—41no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—41no—
163C3×C4⋊C4—41no—
164C3×C4⋊C4—41no—
165C3×C4⋊C4—41no—
166C3×C22⋊C4—44no—
167C3×C22⋊C4—41no—
168C3×C22⋊C4—41no—
169C3×C22⋊C4—41no—
170C3×C22⋊C4—41no—
171C3×C22⋊C4—41no—
172C3×C22⋊C4—82no—
173C3×C42—41no—
174C3×C42C2×D411yescharacteristic
175C3×C42C2×D411yescharacteristic
176C3×C42C2×D411yescharacteristic
177C3×C42#4111yescharacteristic
178C22×D4—22no—
179C22×D4C3×D411yescharacteristic
180C22×D4C3×D411yescharacteristic
181C23×C4—21no—
182C23×C4—21no—
183C2×SD16—22no—
184C2×M16—21no—
185C4⋊Q8C3×D411yescharacteristic, complemented
186C4⋊Q8C3×D411yescharacteristic, complemented
187#125—21no—
188C22⋊Q8—21no—
189C22⋊Q8—21no—
190#121—42no—
191#121—42no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—21no—
202#117—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4C3×D411yescharacteristic
205C2×C4⋊C4C3×D411yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C42C23×C311yescharacteristic
211#107—22no—
212#106—22no—
213#106—22no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8C3×D412yescomplemented
218#103—22no—
219#102—22no—
220#102—22no—
221#102—22no—
222C22⋊C8—21no—
223C22⋊C8—21no—
224C4×C8—21no—
225C4×C8C3×D412yescomplemented
226#95—22no—
227#95—21no—
228#95C3×D412yes—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—41no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—41no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3#12011yescharacteristic
242C3×Q8—42no—
243C3×Q8—81no—
244C3×D4—22no—
245C3×D4—22no—
246C3×D4—22no—
247C3×D4—22no—
248C3×D4—41no—
249C3×D4—41no—
250C3×D4—41no—
251C3×D4—41no—
252C3×D4—88no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—42no—
264C2×C3×C4—41no—
265C2×C3×C4—82no—
266C2×C3×C4—82no—
267C2×C3×C4—81no—
268C2×C3×C4—81no—
269C2×C3×C4—81no—
270C2×C3×C4—81no—
271C2×C3×C4#12011yescharacteristic
272C2×C3×C4#12011yescharacteristic
273C2×C3×C4#12111yescharacteristic
274C2×C3×C4#12111yescharacteristic
275C2×C3×C4#12111yescharacteristic
276C2×C3×C4#12711yescharacteristic
277C2×C3×C4C2×SD1611yescharacteristic
278C2×C3×C4#13511yescharacteristic
279C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
280C2×C3×C4#13711yescharacteristic
281C3×C8—42no—
282C3×C8—42no—
283C3×C8—81no—
284C24—21no—
285C24—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—42no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—82no—
299C2×D4C3×SD1612yescomplemented
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—82no—
312C22×C4C2×C3×D411yescharacteristic
313C22×C4C2×C3×D411yescharacteristic
314C22×C4C2×C3×D411yescharacteristic
315SD16—44no—
316M16—41no—
317M16—41no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—41no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C22⋊C4—44no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—82no—
340C42—41no—
341C42C2×C3×D411yescharacteristic
342C42C2×C3×D411yescharacteristic
343C42C2×C3×D411yescharacteristic
344C42C3×#4111yescharacteristic
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—42no—
348C22×C3—44no—
349C22×C3—42no—
350C22×C3—42no—
351C22×C3—42no—
352C22×C3—81no—
353C22×C3—82no—
354C22×C3—81no—
355C22×C3—81no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3#39211yescharacteristic
360C22×C3#45011yescharacteristic
361C22×C3#45211yescharacteristic
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—82no—
367C3×C4—81no—
368C3×C4—81no—
369C3×C4—81no—
370C3×C4—81no—
371C3×C4#44811yescharacteristic
372C3×C4#44911yescharacteristic
373C3×C4#46511yescharacteristic
374C3×C4#46811yescharacteristic
375C23—22no—
376C23—22no—
377C23—41no—
378C23—42no—
379C23—42no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23C3×#12011yescharacteristic
388Q8—42no—
389Q8—81no—
390D4—22no—
391D4—22no—
392D4—22no—
393D4—22no—
394D4—41no—
395D4—41no—
396D4—41no—
397D4—41no—
398D4—88no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4C3×#12011yescharacteristic
418C2×C4C3×#12011yescharacteristic
419C2×C4C3×#12111yescharacteristic
420C2×C4C3×#12111yescharacteristic
421C2×C4C3×#12111yescharacteristic
422C2×C4C3×#12711yescharacteristic
423C2×C4C2×C3×SD1611yescharacteristic
424C2×C4C3×#13511yescharacteristic
425C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
426C2×C4C3×#13711yescharacteristic
427C8—42no—
428C8—42no—
429C8—81no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—82no—
433C2×C3—82no—
434C2×C3#200011yescharacteristic
435C2×C3#200811yescharacteristic
436C2×C3#201011yescharacteristic
437C22—21no—
438C22—21no—
439C22—42no—
440C22—44no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—81no—
445C22—82no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22C3×#39211yescharacteristic
452C22C3×#45011yescharacteristic
453C22C3×#45211yescharacteristic
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—82no—
459C4—81no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4C3×#44811yescharacteristic
464C4C3×#44911yescharacteristic
465C4C3×#46511yescharacteristic
466C4C3×#46811yescharacteristic
467C3#1000011yescharacteristic, direct product factor, Sylow
468C2—21no—
469C2—21no—
470C2—82no—
471C2—82no—
472C2C3×#200011yescharacteristic
473C2C3×#200811yescharacteristic
474C2C3×#201011yescharacteristic
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC3×#2535C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#3101C2semidirect
2. C3×#3077C2semidirect
4. C3×#1518C2semidirect
5. C3×#1486C2semidirect
6. C3×#1478C2semidirect
7. C3×#1286C2semidirect
8. #10000C3direct (N × H)
34. #3101C2×C3semidirect
35. #3077C2×C3semidirect
37. #1518C2×C3semidirect
38. #1486C2×C3semidirect
39. #1478C2×C3semidirect
40. #1286C2×C3semidirect
48. C3×C4⋊Q8D4semidirect
49. C3×C4⋊Q8D4semidirect
80. C3×C4⋊C8D4semidirect
88. C3×C4×C8D4semidirect
132. C2×C3×D4SD16semidirect
185. C4⋊Q8C3×D4semidirect
186. C4⋊Q8C3×D4semidirect
217. C4⋊C8C3×D4semidirect
225. C4×C8C3×D4semidirect
299. C2×D4C3×SD16semidirect
467. C3#10000direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#31012
C3×#30772
C3×#25352
C3×#15182
C3×#14862
C3×#14782
C3×#12862
#100003