Finite Group Databasebeta

← C3×#10266 #10628873

Subgroups — C3×#10266 #10628873

Order 768.

710 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#492C211yesmaximal, characteristic, complemented
2C2×C3×#474C211yesmaximal, characteristic, complemented
3C2×C3×#459C211yesmaximal, characteristic
4C3×#1954C211yesmaximal, characteristic, complemented
5C2×C3×#332C211yesmaximal, characteristic, complemented
6C2×C3×#331C211yesmaximal, characteristic, complemented
7C3×#1277C211yesmaximal, characteristic, complemented
8#10266C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×SD16—22no—
10C22×C3×D8—21no—
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×#127C2211yescharacteristic
13C2×C3×C4×D4C2211yescharacteristic
14C3×#492—22no—
15C3×#492—22no—
16C3×#474—22no—
17C3×#474—22no—
18C3×#459—22no—
19C3×#459—22no—
20C2×C3×#107—21no—
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#102—21no—
25C2×C3×#102—21no—
26C2×C3×C22⋊C8—22no—
27C2×C3×C4×C8C2211yescharacteristic
28C3×#332—22no—
29C3×#331—22no—
30C2×#492C2×C311yescharacteristic, complemented
31C2×#474C2×C311yescharacteristic, complemented
32C2×#459C2×C311yescharacteristic
33#1954C2×C311yescharacteristic, complemented
34C2×#332C2×C311yescharacteristic, complemented
35C2×#331C2×C311yescharacteristic, complemented
36#1277C2×C311yescharacteristic, complemented
37C22×C3×Q8—21no—
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×C3×SD16—28no—
44C2×C3×SD16—44no—
45C2×C3×SD16—44no—
46C2×C3×D8—24no—
47C2×C3×D8—42no—
48C2×C3×D8—42no—
49C22×C3×C8—21no—
50C22×C3×C8—21no—
51C22×C3×C8—21no—
52C22×C3×C8D412yes—
53C3×C4⋊Q8—21no—
54C3×C4⋊Q8—21no—
55C3×C4⋊Q8—21no—
56C3×C4⋊Q8D412yes—
57C3×#127—21no—
58C3×#127—21no—
59C3×#127—21no—
60C3×#127D412yes—
61C3×C4×D4—21no—
62C3×C4×D4—21no—
63C3×C4×D4—21no—
64C3×C4×D4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4D411yescharacteristic
68C2×C3×C22⋊C4—21no—
69C2×C3×C42C2311yescharacteristic
70C3×#107—24no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8D412yes—
76C3×C4⋊C8D412yes—
77C3×#102—24no—
78C3×#102—24no—
79C3×C22⋊C8—44no—
80C3×C4×C8—21no—
81C3×C4×C8—21no—
82C22×SD16—22no—
83C22×D8—21no—
84C2×C4⋊Q8C22×C311yescharacteristic
85C2×#127C22×C311yescharacteristic
86C2×C4×D4C22×C311yescharacteristic
87#492—22no—
88#492—22no—
89#474—22no—
90#474—22no—
91#459—22no—
92#459—22no—
93C2×#107—21no—
94C2×C4⋊C8C22×C311yescharacteristic
95C2×C4⋊C8C22×C311yescharacteristic
96C2×C4⋊C8C22×C311yescharacteristic
97C2×#102—21no—
98C2×#102—21no—
99C2×C22⋊C8—22no—
100C2×C4×C8C22×C311yescharacteristic
101#332—22no—
102#331—22no—
103C24×C3—21no—
104C24×C3—41no—
105C2×C3×Q8—22no—
106C2×C3×Q8—41no—
107C2×C3×Q8—42no—
108C2×C3×Q8—42no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—22no—
112C2×C3×D4—22no—
113C2×C3×D4—22no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—42no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—42no—
125C2×C3×D4—41no—
126C2×C3×D4—82no—
127C2×C3×D4SD1612yescomplemented
128C22×C3×C4—21no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—42no—
133C22×C3×C4—42no—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C3×SD16—416no—
138C3×D8—48no—
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8—22no—
143C2×C3×C8—41no—
144C2×C3×C8—41no—
145C2×C3×C8—41no—
146C2×C3×C8—41no—
147C2×C3×C8—41no—
148C2×C3×C8—42no—
149C2×C3×C8D814yescomplemented
150C3×C4⋊C4—21no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4—41no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C42C2×D411yescharacteristic
159C3×C42C2×D411yescharacteristic
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C22×Q8—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4C3×D411yescharacteristic
167C23×C4—21no—
168C2×SD16—28no—
169C2×SD16—44no—
170C2×SD16—44no—
171C2×D8—24no—
172C2×D8—42no—
173C2×D8—42no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8C3×D412yes—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8C3×D412yes—
182#127—21no—
183#127—21no—
184#127—21no—
185#127C3×D412yes—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C3×D411yescharacteristic
193C2×C22⋊C4—21no—
194C2×C42C23×C311yescharacteristic
195#107—24no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8C3×D412yes—
201C4⋊C8C3×D412yes—
202#102—24no—
203#102—24no—
204C22⋊C8—44no—
205C4×C8—21no—
206C4×C8—21no—
207C23×C3—22no—
208C23×C3—41no—
209C23×C3—42no—
210C23×C3—41no—
211C23×C3—42no—
212C23×C3—42no—
213C23×C3—82no—
214C23×C3—81no—
215C23×C3—81no—
216C23×C3—82no—
217C23×C3#12111yescharacteristic
218C3×Q8—44no—
219C3×Q8—81no—
220C3×Q8—81no—
221C3×D4—22no—
222C3×D4—22no—
223C3×D4—44no—
224C3×D4—41no—
225C3×D4—42no—
226C3×D4—41no—
227C3×D4—42no—
228C3×D4—82no—
229C3×D4—82no—
230C3×D4—81no—
231C3×D4—81no—
232C3×D4—81no—
233C3×D4—81no—
234C2×C3×C4—22no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—82no—
239C2×C3×C4—82no—
240C2×C3×C4—81no—
241C2×C3×C4—81no—
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12711yescharacteristic
248C2×C3×C4C2×D812yes—
249C2×C3×C4C2×SD1611yescharacteristic
250C2×C3×C4#13511yescharacteristic
251C2×C3×C4AGL(1,ℤ/8)12yes—
252C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
253C2×C3×C4#13711yescharacteristic
254C3×C8—24no—
255C3×C8—42no—
256C3×C8—42no—
257C3×C8—81no—
258C24—21no—
259C24—41no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—42no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—42no—
280C2×D4—41no—
281C2×D4—82no—
282C2×D4C3×SD1612yescomplemented
283C22×C4—21no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4C2×C3×D411yescharacteristic
290C22×C4C2×C3×D411yescharacteristic
291C22×C4C3×#4111yescharacteristic
292SD16—416no—
293D8—48no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8C3×D814yescomplemented
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C42C2×C3×D411yescharacteristic
314C42C2×C3×D411yescharacteristic
315C42C2×C3×D411yescharacteristic
316C42C2×C3×D411yescharacteristic
317C22×C3—44no—
318C22×C3—42no—
319C22×C3—82no—
320C22×C3—81no—
321C22×C3—81no—
322C22×C3—82no—
323C22×C3—82no—
324C22×C3—82no—
325C22×C3—162no—
326C22×C3—162no—
327C22×C3#39211yescharacteristic
328C22×C3#45912yes—
329C22×C3#46511yescharacteristic
330C22×C3#46811yescharacteristic
331C22×C3#47112yes—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—81no—
335C3×C4—82no—
336C3×C4#44712yes—
337C3×C4#44811yescharacteristic
338C3×C4#44911yescharacteristic
339C3×C4#45011yescharacteristic
340C3×C4#45211yescharacteristic
341C3×C4#49311yescharacteristic
342C3×C4#49611yescharacteristic
343C23—22no—
344C23—41no—
345C23—42no—
346C23—41no—
347C23—42no—
348C23—42no—
349C23—82no—
350C23—81no—
351C23—81no—
352C23—82no—
353C23C3×#12111yescharacteristic
354Q8—44no—
355Q8—81no—
356Q8—81no—
357D4—22no—
358D4—22no—
359D4—44no—
360D4—41no—
361D4—42no—
362D4—41no—
363D4—42no—
364D4—82no—
365D4—82no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—81no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4C3×#12011yescharacteristic
379C2×C4C3×#12011yescharacteristic
380C2×C4C3×#12011yescharacteristic
381C2×C4C3×#12111yescharacteristic
382C2×C4C3×#12111yescharacteristic
383C2×C4C3×#12711yescharacteristic
384C2×C4C2×C3×D812yes—
385C2×C4C2×C3×SD1611yescharacteristic
386C2×C4C3×#13511yescharacteristic
387C2×C4C3×AGL(1,ℤ/8)12yes—
388C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
389C2×C4C3×#13711yescharacteristic
390C8—24no—
391C8—42no—
392C8—42no—
393C8—81no—
394C2×C3—82no—
395C2×C3—162no—
396C2×C3#166512yes—
397C2×C3#168512yes—
398C2×C3#200011yescharacteristic
399C2×C3#201111yescharacteristic
400C2×C3#201911yescharacteristic
401C22—42no—
402C22—44no—
403C22—81no—
404C22—81no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—162no—
410C22—162no—
411C22C3×#39211yescharacteristic
412C22C3×#45912yes—
413C22C3×#46511yescharacteristic
414C22C3×#46811yescharacteristic
415C22C3×#47112yes—
416C4—21no—
417C4—21no—
418C4—81no—
419C4—82no—
420C4C3×#44712yes—
421C4C3×#44811yescharacteristic
422C4C3×#44911yescharacteristic
423C4C3×#45011yescharacteristic
424C4C3×#45211yescharacteristic
425C4C3×#49311yescharacteristic
426C4C3×#49611yescharacteristic
427C3#1026611yescharacteristic, direct product factor, Sylow
428C2—82no—
429C2—162no—
430C2C3×#166512yes—
431C2C3×#168512yes—
432C2C3×#200011yescharacteristic
433C2C3×#201111yescharacteristic
434C2C3×#201911yescharacteristic
subgroupG/H
center, socleC23×C3#121
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC2×C3×#459C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#492C2semidirect
2. C2×C3×#474C2semidirect
4. C3×#1954C2semidirect
5. C2×C3×#332C2semidirect
6. C2×C3×#331C2semidirect
7. C3×#1277C2semidirect
8. #10266C3direct (N × H)
30. C2×#492C2×C3semidirect
31. C2×#474C2×C3semidirect
33. #1954C2×C3semidirect
34. C2×#332C2×C3semidirect
35. C2×#331C2×C3semidirect
36. #1277C2×C3semidirect
127. C2×C3×D4SD16semidirect
149. C2×C3×C8D8semidirect
282. C2×D4C3×SD16semidirect
304. C2×C8C3×D8semidirect
427. C3#10266direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4922
C2×C3×#4742
C2×C3×#4592
C3×#19542
C2×C3×#3322
C2×C3×#3312
C3×#12772
#102663