Finite Group Databasebeta

← C3×#13332 #10631939

Subgroups — C3×#13332 #10631939

Order 768.

566 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#510C211yesmaximal, characteristic, complemented
2C2×C3×#509C211yesmaximal, characteristic
3C2×C3×#495C211yesmaximal, characteristic, complemented
4C2×C3×#362C211yesmaximal, characteristic, complemented
5C2×C3×#361C211yesmaximal, characteristic, complemented
6C3×#1974C211yesmaximal, characteristic, complemented
7C3×#1376C211yesmaximal, characteristic, complemented
8#13332C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×#135C2211yescharacteristic
10C22×C3×Q16C2211yescharacteristic
11C22×C3×SD16—21no—
12C22×C3×D8C2211yescharacteristic
13C2×C3×#124—21no—
14C3×#510—22no—
15C3×#510—22no—
16C3×#509—22no—
17C3×#509—22no—
18C2×C3×Q32—21no—
19C2×C3×SD32—21no—
20C2×C3×SD32—21no—
21C2×C3×D16—21no—
22C2×C3×M32C2211yescharacteristic
23C2×C3×M32C2211yescharacteristic
24C3×#495—22no—
25C3×#495—22no—
26C2×C3×#108C2211yescharacteristic
27C2×C3×#104—21no—
28C3×#408—21no—
29C2×C3×C4×C8C2211yescharacteristic
30C3×#362—22no—
31C3×#361—22no—
32C2×#510C2×C311yescharacteristic, complemented
33C2×#509C2×C311yescharacteristic
34C2×#495C2×C311yescharacteristic, complemented
35C2×#362C2×C311yescharacteristic, complemented
36C2×#361C2×C311yescharacteristic, complemented
37#1974C2×C311yescharacteristic, complemented
38#1376C2×C311yescharacteristic, complemented
39C2×C3×#41—21no—
40C22×C3×Q8—21no—
41C22×C3×D4—21no—
42C3×#135—21no—
43C3×#135—21no—
44C3×#135—21no—
45C3×#135—21no—
46C2×C3×Q16—21no—
47C2×C3×Q16—21no—
48C2×C3×Q16—21no—
49C2×C3×Q16—21no—
50C2×C3×Q16—21no—
51C2×C3×Q16D411yescharacteristic
52C2×C3×Q16D412yes—
53C2×C3×SD16—24no—
54C2×C3×SD16—21no—
55C2×C3×SD16—42no—
56C2×C3×SD16—42no—
57C2×C3×D8—21no—
58C2×C3×D8—21no—
59C2×C3×D8—21no—
60C2×C3×D8—21no—
61C2×C3×D8—21no—
62C2×C3×D8D411yescharacteristic
63C2×C3×D8D412yes—
64C2×C3×M16—21no—
65C22×C3×C8D411yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C3×#124—24no—
68C3×#124—42no—
69C2×C3×C22⋊C4—42no—
70C2×C3×C42D411yescharacteristic
71C3×Q32—24no—
72C3×SD32—24no—
73C3×SD32—24no—
74C3×D16—24no—
75C3×M32—21no—
76C3×M32—21no—
77C3×M32D412yes—
78C3×M32D412yes—
79C2×C3×C16—21no—
80C2×C3×C16—21no—
81C3×#108—21no—
82C3×#108—21no—
83C3×#104—42no—
84C3×C22⋊C8—42no—
85C3×C4×C8—21no—
86C3×C4×C8—21no—
87C2×#135C22×C311yescharacteristic
88C22×Q16C22×C311yescharacteristic
89C22×SD16—21no—
90C22×D8C22×C311yescharacteristic
91C2×#124—21no—
92#510—22no—
93#510—22no—
94#509—22no—
95#509—22no—
96C2×Q32—21no—
97C2×SD32—21no—
98C2×SD32—21no—
99C2×D16—21no—
100C2×M32C22×C311yescharacteristic
101C2×M32C22×C311yescharacteristic
102#495—22no—
103#495—22no—
104C2×#108C22×C311yescharacteristic
105C2×#104—21no—
106#408—21no—
107C2×C4×C8C22×C311yescharacteristic
108#362—22no—
109#361—22no—
110C24×C3—41no—
111C3×#41—41no—
112C3×#41—41no—
113C3×#41—41no—
114C3×#41—41no—
115C2×C3×Q8—22no—
116C2×C3×Q8—21no—
117C2×C3×Q8—41no—
118C2×C3×Q8—41no—
119C2×C3×Q8—41no—
120C2×C3×Q8—41no—
121C2×C3×Q8—41no—
122C2×C3×D4—22no—
123C2×C3×D4—21no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C22×C3×C4—21no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4C2×D411yescharacteristic
134C3×Q16—21no—
135C3×Q16—22no—
136C3×Q16—21no—
137C3×Q16—22no—
138C3×Q16—41no—
139C3×Q16—41no—
140C3×SD16—48no—
141C3×SD16—41no—
142C3×SD16—41no—
143C3×D8—21no—
144C3×D8—22no—
145C3×D8—21no—
146C3×D8—22no—
147C3×D8—41no—
148C3×D8—41no—
149C3×M16—41no—
150C3×M16—41no—
151C2×C3×C8—21no—
152C2×C3×C8—21no—
153C2×C3×C8—41no—
154C2×C3×C8D812yes—
155C2×C3×C8C2×D411yescharacteristic
156C2×C3×C8C2×D411yescharacteristic
157C2×C3×C8C2×D411yescharacteristic
158C2×C3×C8C2×D411yescharacteristic
159C2×C3×C8C2×D411yescharacteristic
160C2×C3×C8#4111yescharacteristic
161C3×C22⋊C4—48no—
162C3×C42—21no—
163C3×C42D812yes—
164C3×C16—22no—
165C3×C16—22no—
166C2×#41—21no—
167C22×Q8—21no—
168C22×D4—21no—
169#135—21no—
170#135—21no—
171#135—21no—
172#135—21no—
173C2×Q16—21no—
174C2×Q16—21no—
175C2×Q16—21no—
176C2×Q16—21no—
177C2×Q16—21no—
178C2×Q16C3×D412yes—
179C2×Q16C3×D411yescharacteristic
180C2×SD16—24no—
181C2×SD16—21no—
182C2×SD16—42no—
183C2×SD16—42no—
184C2×D8—21no—
185C2×D8—21no—
186C2×D8—21no—
187C2×D8—21no—
188C2×D8—21no—
189C2×D8C3×D412yes—
190C2×D8C3×D411yescharacteristic
191C2×M16—21no—
192C22×C8C3×D411yescharacteristic
193C22×C8C23×C311yescharacteristic
194#124—24no—
195#124—42no—
196C2×C22⋊C4—42no—
197C2×C42C3×D411yescharacteristic
198Q32—24no—
199SD32—24no—
200SD32—24no—
201D16—24no—
202M32—21no—
203M32—21no—
204M32C3×D412yes—
205M32C3×D412yes—
206C2×C16—21no—
207C2×C16—21no—
208#108—21no—
209#108—21no—
210#104—42no—
211C22⋊C8—42no—
212C4×C8—21no—
213C4×C8—21no—
214C23×C3—42no—
215C23×C3—41no—
216C23×C3—84no—
217C23×C3—81no—
218C23×C3—81no—
219C23×C3AGL(1,ℤ/8)11yescharacteristic
220C3×Q8—41no—
221C3×Q8—42no—
222C3×Q8—42no—
223C3×Q8—41no—
224C3×Q8—81no—
225C3×Q8—81no—
226C3×D4—42no—
227C3×D4—41no—
228C3×D4—42no—
229C3×D4—41no—
230C3×D4—81no—
231C3×D4—81no—
232C3×D4—81no—
233C3×D4—81no—
234C2×C3×C4—22no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—41no—
238C2×C3×C4—81no—
239C2×C3×C4—81no—
240C2×C3×C4—81no—
241C2×C3×C4—81no—
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12711yescharacteristic
245C2×C3×C4C2×D811yescharacteristic
246C2×C3×C4C2×D811yescharacteristic
247C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
248C3×C8—22no—
249C3×C8—81no—
250C3×C8#12011yescharacteristic
251C3×C8#12011yescharacteristic
252C3×C8#12111yescharacteristic
253C3×C8#12111yescharacteristic
254C24—41no—
255#41—41no—
256#41—41no—
257#41—41no—
258#41—41no—
259C2×Q8—22no—
260C2×Q8—21no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×D4—22no—
267C2×D4—21no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4C2×C3×D411yescharacteristic
278Q16—21no—
279Q16—22no—
280Q16—21no—
281Q16—22no—
282Q16—41no—
283Q16—41no—
284SD16—48no—
285SD16—41no—
286SD16—41no—
287D8—21no—
288D8—22no—
289D8—21no—
290D8—22no—
291D8—41no—
292D8—41no—
293M16—41no—
294M16—41no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—41no—
298C2×C8C3×D812yes—
299C2×C8C2×C3×D411yescharacteristic
300C2×C8C2×C3×D411yescharacteristic
301C2×C8C2×C3×D411yescharacteristic
302C2×C8C2×C3×D411yescharacteristic
303C2×C8C2×C3×D411yescharacteristic
304C2×C8C3×#4111yescharacteristic
305C22⋊C4—48no—
306C42—21no—
307C42C3×D812yes—
308C16—22no—
309C16—22no—
310C22×C3—22no—
311C22×C3—84no—
312C22×C3—82no—
313C22×C3—81no—
314C22×C3—82no—
315C22×C3—162no—
316C22×C3—162no—
317C22×C3#44711yescharacteristic
318C22×C3#45911yescharacteristic
319C22×C3#49611yescharacteristic
320C3×C4—42no—
321C3×C4—81no—
322C3×C4—82no—
323C3×C4#39211yescharacteristic
324C3×C4#44711yescharacteristic
325C3×C4#45911yescharacteristic
326C3×C4#49311yescharacteristic
327C23—42no—
328C23—41no—
329C23—84no—
330C23—81no—
331C23—81no—
332C23C3×AGL(1,ℤ/8)11yescharacteristic
333Q8—41no—
334Q8—42no—
335Q8—42no—
336Q8—41no—
337Q8—81no—
338Q8—81no—
339D4—42no—
340D4—41no—
341D4—42no—
342D4—41no—
343D4—81no—
344D4—81no—
345D4—81no—
346D4—81no—
347C2×C4—22no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4C3×#12011yescharacteristic
356C2×C4C3×#12111yescharacteristic
357C2×C4C3×#12711yescharacteristic
358C2×C4C2×C3×D811yescharacteristic
359C2×C4C2×C3×D811yescharacteristic
360C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
361C8—22no—
362C8—81no—
363C8C3×#12011yescharacteristic
364C8C3×#12011yescharacteristic
365C8C3×#12111yescharacteristic
366C8C3×#12111yescharacteristic
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—161no—
370C2×C3—162no—
371C2×C3#201111yescharacteristic
372C2×C3#221212yes—
373C22—22no—
374C22—84no—
375C22—81no—
376C22—82no—
377C22—82no—
378C22—162no—
379C22—162no—
380C22C3×#44711yescharacteristic
381C22C3×#45911yescharacteristic
382C22C3×#49611yescharacteristic
383C4—42no—
384C4—81no—
385C4—82no—
386C4C3×#39211yescharacteristic
387C4C3×#44711yescharacteristic
388C4C3×#45911yescharacteristic
389C4C3×#49311yescharacteristic
390C3#1333211yescharacteristic, direct product factor, Sylow
391C2—21no—
392C2—21no—
393C2—162no—
394C2—161no—
395C2C3×#201111yescharacteristic
396C2C3×#221212yes—
subgroupG/H
center, socleC22×C3#459
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC2×C3×#509C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4×C8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#510C2semidirect
3. C2×C3×#495C2semidirect
4. C2×C3×#362C2semidirect
5. C2×C3×#361C2semidirect
6. C3×#1974C2semidirect
7. C3×#1376C2semidirect
8. #13332C3direct (N × H)
32. C2×#510C2×C3semidirect
34. C2×#495C2×C3semidirect
35. C2×#362C2×C3semidirect
36. C2×#361C2×C3semidirect
37. #1974C2×C3semidirect
38. #1376C2×C3semidirect
390. C3#13332direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5102
C2×C3×#5092
C2×C3×#4952
C2×C3×#3622
C2×C3×#3612
C3×#19742
C3×#13762
#133323