Finite Group Databasebeta

← C3×#10269 #10628876

Subgroups — C3×#10269 #10628876

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#486C211yesmaximal, characteristic
2C2×C3×#477C211yesmaximal, characteristic, complemented
3C2×C3×#460C211yesmaximal, characteristic
4C3×#1984C211yesmaximal, characteristic, complemented
5C2×C3×#333C211yesmaximal, characteristic, complemented
6C2×C3×#331C211yesmaximal, characteristic
7C3×#1277C211yesmaximal, characteristic, complemented
8#10269C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×SD16—21no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×#127C2211yescharacteristic
12C2×C3×C4×Q8C2211yescharacteristic
13C3×#486—22no—
14C3×#486—22no—
15C3×#477—22no—
16C3×#477—22no—
17C3×#460—22no—
18C3×#460—22no—
19C2×C3×#107—21no—
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#103—21no—
24C2×C3×#102—22no—
25C2×C3×#102—21no—
26C2×C3×C4×C8C2211yescharacteristic
27C3×#336—22no—
28C3×#333—22no—
29C3×#331—22no—
30C2×#486C2×C311yescharacteristic
31C2×#477C2×C311yescharacteristic, complemented
32C2×#460C2×C311yescharacteristic
33#1984C2×C311yescharacteristic, complemented
34C2×#333C2×C311yescharacteristic, complemented
35C2×#331C2×C311yescharacteristic
36#1277C2×C311yescharacteristic, complemented
37C22×C3×Q8—21no—
38C22×C3×Q8D411yescharacteristic
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C2×C3×SD16—24no—
43C2×C3×SD16—42no—
44C2×C3×SD16—42no—
45C22×C3×C8—21no—
46C22×C3×C8—21no—
47C22×C3×C8—21no—
48C22×C3×C8D412yes—
49C3×C4⋊Q8—21no—
50C3×C4⋊Q8—21no—
51C3×C4⋊Q8—21no—
52C3×C4⋊Q8D412yes—
53C3×#127—21no—
54C3×#127—21no—
55C3×#127—21no—
56C3×#127D412yes—
57C3×C4×Q8—21no—
58C3×C4×Q8—21no—
59C3×C4×Q8—21no—
60C3×C4×Q8—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4D411yescharacteristic
65C2×C3×C42—21no—
66C2×C3×C42C2311yescharacteristic
67C3×#107—24no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×C4⋊C8D412yes—
74C3×#103—24no—
75C3×#102—24no—
76C3×#102—28no—
77C3×C4×C8—21no—
78C3×C4×C8—21no—
79C22×SD16—21no—
80C2×C4⋊Q8C22×C311yescharacteristic
81C2×#127C22×C311yescharacteristic
82C2×C4×Q8C22×C311yescharacteristic
83#486—22no—
84#486—22no—
85#477—22no—
86#477—22no—
87#460—22no—
88#460—22no—
89C2×#107—21no—
90C2×C4⋊C8C22×C311yescharacteristic
91C2×C4⋊C8C22×C311yescharacteristic
92C2×C4⋊C8C22×C311yescharacteristic
93C2×#103—21no—
94C2×#102—22no—
95C2×#102—21no—
96C2×C4×C8C22×C311yescharacteristic
97#336—22no—
98#333—22no—
99#331—22no—
100C24×C3—41no—
101C2×C3×Q8—21no—
102C2×C3×Q8—21no—
103C2×C3×Q8—21no—
104C2×C3×Q8—22no—
105C2×C3×Q8—21no—
106C2×C3×Q8—21no—
107C2×C3×Q8—41no—
108C2×C3×Q8—82no—
109C2×C3×Q8SD1612yes—
110C2×C3×D4—22no—
111C2×C3×D4—22no—
112C2×C3×D4—22no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—42no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—42no—
122C2×C3×D4—82no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—41no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C3×SD16—48no—
130C2×C3×C8—22no—
131C2×C3×C8—22no—
132C2×C3×C8—22no—
133C2×C3×C8—22no—
134C2×C3×C8—22no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C2×C3×C8—41no—
138C2×C3×C8—41no—
139C2×C3×C8—41no—
140C2×C3×C8—42no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C42—41no—
150C3×C42—41no—
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C22×Q8—21no—
156C22×Q8C3×D411yescharacteristic
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C2×SD16—24no—
161C2×SD16—42no—
162C2×SD16—42no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8C3×D412yes—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8C3×D412yes—
171#127—21no—
172#127—21no—
173#127—21no—
174#127C3×D412yes—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C3×D411yescharacteristic
183C2×C42—21no—
184C2×C42C23×C311yescharacteristic
185#107—24no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8C3×D412yes—
191C4⋊C8C3×D412yes—
192#103—24no—
193#102—24no—
194#102—28no—
195C4×C8—21no—
196C4×C8—21no—
197C23×C3—42no—
198C23×C3—82no—
199C23×C3—81no—
200C23×C3—81no—
201C23×C3—82no—
202C23×C3#12111yescharacteristic
203C3×Q8—22no—
204C3×Q8—22no—
205C3×Q8—41no—
206C3×Q8—41no—
207C3×Q8—82no—
208C3×Q8—82no—
209C3×D4—42no—
210C3×D4—42no—
211C3×D4—44no—
212C3×D4—82no—
213C3×D4—82no—
214C3×D4—81no—
215C3×D4—81no—
216C3×D4—81no—
217C3×D4—81no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—82no—
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12711yescharacteristic
232C2×C3×C4C2×SD1611yescharacteristic
233C2×C3×C4#13511yescharacteristic
234C2×C3×C4#13512yes—
235C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
236C2×C3×C4#13711yescharacteristic
237C2×C3×C4#13712yes—
238C3×C8—42no—
239C3×C8—42no—
240C3×C8—42no—
241C3×C8—81no—
242C24—41no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—22no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—41no—
250C2×Q8—82no—
251C2×Q8C3×SD1612yes—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—42no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—42no—
264C2×D4—82no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4C2×C3×D411yescharacteristic
269C22×C4C2×C3×D411yescharacteristic
270C22×C4C3×#4111yescharacteristic
271SD16—48no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C4⋊C4—22no—
284C4⋊C4—21no—
285C4⋊C4—22no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C42—41no—
292C42—41no—
293C42C2×C3×D411yescharacteristic
294C42C2×C3×D411yescharacteristic
295C42C2×C3×D411yescharacteristic
296C42C2×C3×D411yescharacteristic
297C22×C3—82no—
298C22×C3—82no—
299C22×C3—82no—
300C22×C3—162no—
301C22×C3—162no—
302C22×C3#39211yescharacteristic
303C22×C3#46412yes—
304C22×C3#46511yescharacteristic
305C22×C3#46811yescharacteristic
306C22×C3#47212yes—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—42no—
310C3×C4—81no—
311C3×C4—82no—
312C3×C4#44811yescharacteristic
313C3×C4#44911yescharacteristic
314C3×C4#45011yescharacteristic
315C3×C4#45211yescharacteristic
316C3×C4#45212yes—
317C3×C4#49511yescharacteristic
318C3×C4#49711yescharacteristic
319C23—42no—
320C23—82no—
321C23—81no—
322C23—81no—
323C23—82no—
324C23C3×#12111yescharacteristic
325Q8—22no—
326Q8—22no—
327Q8—41no—
328Q8—41no—
329Q8—82no—
330Q8—82no—
331D4—42no—
332D4—42no—
333D4—44no—
334D4—82no—
335D4—82no—
336D4—81no—
337D4—81no—
338D4—81no—
339D4—81no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—82no—
348C2×C4C3×#12011yescharacteristic
349C2×C4C3×#12011yescharacteristic
350C2×C4C3×#12011yescharacteristic
351C2×C4C3×#12111yescharacteristic
352C2×C4C3×#12111yescharacteristic
353C2×C4C3×#12711yescharacteristic
354C2×C4C2×C3×SD1611yescharacteristic
355C2×C4C3×#13511yescharacteristic
356C2×C4C3×#13512yes—
357C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
358C2×C4C3×#13711yescharacteristic
359C2×C4C3×#13712yes—
360C8—42no—
361C8—42no—
362C8—42no—
363C8—81no—
364C2×C3—162no—
365C2×C3#167812yes—
366C2×C3#169812yes—
367C2×C3#200011yescharacteristic
368C2×C3#201511yescharacteristic
369C2×C3#201811yescharacteristic
370C22—82no—
371C22—82no—
372C22—82no—
373C22—162no—
374C22—162no—
375C22C3×#39211yescharacteristic
376C22C3×#46412yes—
377C22C3×#46511yescharacteristic
378C22C3×#46811yescharacteristic
379C22C3×#47212yes—
380C4—21no—
381C4—21no—
382C4—42no—
383C4—81no—
384C4—82no—
385C4C3×#44811yescharacteristic
386C4C3×#44911yescharacteristic
387C4C3×#45011yescharacteristic
388C4C3×#45211yescharacteristic
389C4C3×#45212yes—
390C4C3×#49511yescharacteristic
391C4C3×#49711yescharacteristic
392C3#1026911yescharacteristic, direct product factor, Sylow
393C2—162no—
394C2C3×#167812yes—
395C2C3×#169812yes—
396C2C3×#200011yescharacteristic
397C2C3×#201511yescharacteristic
398C2C3×#201811yescharacteristic
subgroupG/H
center, socleC23×C3#121
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC2×C3×#127C22
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
2. C2×C3×#477C2semidirect
4. C3×#1984C2semidirect
5. C2×C3×#333C2semidirect
7. C3×#1277C2semidirect
8. #10269C3direct (N × H)
31. C2×#477C2×C3semidirect
33. #1984C2×C3semidirect
34. C2×#333C2×C3semidirect
36. #1277C2×C3semidirect
392. C3#10269direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4862
C2×C3×#4772
C2×C3×#4602
C3×#19842
C2×C3×#3332
C2×C3×#3312
C3×#12772
#102693