Finite Group Databasebeta

← C3×#10271 #10628878

Subgroups — C3×#10271 #10628878

Order 768.

594 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#498C211yesmaximal, characteristic, complemented
2C2×C3×#476C211yesmaximal, characteristic
3C2×C3×#461C211yesmaximal, characteristic
4C3×#1954C211yesmaximal, characteristic
5C2×C3×#333C211yesmaximal, characteristic, complemented
6C2×C3×#332C211yesmaximal, characteristic, complemented
7C3×#1277C211yesmaximal, characteristic, complemented
8#10271C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×SD16—21no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C4×D4C2211yescharacteristic
13C3×#498—22no—
14C3×#498—22no—
15C3×#476—22no—
16C3×#476—22no—
17C3×#461—22no—
18C3×#461—22no—
19C2×C3×#106—22no—
20C2×C3×#106—21no—
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103—21no—
25C2×C3×#102—21no—
26C2×C3×C22⋊C8—22no—
27C2×C3×C4×C8C2211yescharacteristic
28C3×#333—22no—
29C3×#332—22no—
30C2×#498C2×C311yescharacteristic, complemented
31C2×#476C2×C311yescharacteristic
32C2×#461C2×C311yescharacteristic
33#1954C2×C311yescharacteristic
34C2×#333C2×C311yescharacteristic, complemented
35C2×#332C2×C311yescharacteristic, complemented
36#1277C2×C311yescharacteristic, complemented
37C22×C3×Q8—21no—
38C22×C3×Q8—21no—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4—21no—
41C2×C3×SD16—24no—
42C2×C3×SD16—42no—
43C2×C3×SD16—42no—
44C22×C3×C8—21no—
45C22×C3×C8—21no—
46C22×C3×C8—21no—
47C22×C3×C8D412yes—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8—21no—
50C3×C4⋊Q8—21no—
51C3×C4⋊Q8—21no—
52C3×C4⋊Q8—21no—
53C3×C4⋊Q8—21no—
54C3×C4⋊Q8D412yes—
55C3×C4⋊Q8D412yes—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4D411yescharacteristic
65C2×C3×C22⋊C4—21no—
66C2×C3×C42C2311yescharacteristic
67C3×#106—24no—
68C3×#106—28no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8D412yes—
74C3×C4⋊C8D412yes—
75C3×#103—24no—
76C3×#102—24no—
77C3×C22⋊C8—44no—
78C3×C4×C8—21no—
79C3×C4×C8—21no—
80C22×SD16—21no—
81C2×C4⋊Q8C22×C311yescharacteristic
82C2×C4⋊Q8C22×C311yescharacteristic
83C2×C4×D4C22×C311yescharacteristic
84#498—22no—
85#498—22no—
86#476—22no—
87#476—22no—
88#461—22no—
89#461—22no—
90C2×#106—22no—
91C2×#106—21no—
92C2×C4⋊C8C22×C311yescharacteristic
93C2×C4⋊C8C22×C311yescharacteristic
94C2×C4⋊C8C22×C311yescharacteristic
95C2×#103—21no—
96C2×#102—21no—
97C2×C22⋊C8—22no—
98C2×C4×C8C22×C311yescharacteristic
99#333—22no—
100#332—22no—
101C24×C3—21no—
102C2×C3×Q8—22no—
103C2×C3×Q8—22no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C2×C3×Q8—41no—
110C2×C3×Q8—82no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4SD1612yes—
117C22×C3×C4—21no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—42no—
123C22×C3×C4—42no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4#4111yescharacteristic
127C3×SD16—48no—
128C2×C3×C8—22no—
129C2×C3×C8—22no—
130C2×C3×C8—22no—
131C2×C3×C8—22no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C2×C3×C8—42no—
138C2×C3×C8SD1614yescomplemented
139C3×C4⋊C4—22no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C22×Q8—21no—
156C22×Q8—21no—
157C22×D4C3×D411yescharacteristic
158C23×C4—21no—
159C2×SD16—24no—
160C2×SD16—42no—
161C2×SD16—42no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8C3×D412yes—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8C3×D412yes—
173C4⋊Q8C3×D412yes—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C3×D411yescharacteristic
183C2×C22⋊C4—21no—
184C2×C42C23×C311yescharacteristic
185#106—24no—
186#106—28no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8C3×D412yes—
192C4⋊C8C3×D412yes—
193#103—24no—
194#102—24no—
195C22⋊C8—44no—
196C4×C8—21no—
197C4×C8—21no—
198C23×C3—22no—
199C23×C3—41no—
200C23×C3—42no—
201C23×C3—41no—
202C23×C3—42no—
203C23×C3#12111yescharacteristic
204C3×Q8—42no—
205C3×Q8—42no—
206C3×Q8—82no—
207C3×Q8—82no—
208C3×Q8—81no—
209C3×Q8—81no—
210C3×D4—22no—
211C3×D4—22no—
212C3×D4—41no—
213C3×D4—41no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—81no—
220C2×C3×C4—81no—
221C2×C3×C4—82no—
222C2×C3×C4—82no—
223C2×C3×C4—81no—
224C2×C3×C4—81no—
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12711yescharacteristic
231C2×C3×C4C2×SD1611yescharacteristic
232C2×C3×C4C2×SD1612yes—
233C2×C3×C4#13511yescharacteristic
234C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
235C2×C3×C4#13711yescharacteristic
236C2×C3×C4#13712yes—
237C3×C8—24no—
238C3×C8—42no—
239C3×C8—42no—
240C3×C8—81no—
241C24—21no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—82no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4C3×SD1612yes—
257C22×C4—21no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4C2×C3×D411yescharacteristic
265C22×C4C2×C3×D411yescharacteristic
266C22×C4C3×#4111yescharacteristic
267SD16—48no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8C3×SD1614yescomplemented
279C4⋊C4—22no—
280C4⋊C4—21no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—21no—
284C4⋊C4—22no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C42C2×C3×D411yescharacteristic
292C42C2×C3×D411yescharacteristic
293C42C2×C3×D411yescharacteristic
294C42C2×C3×D411yescharacteristic
295C22×C3—44no—
296C22×C3—42no—
297C22×C3—82no—
298C22×C3—81no—
299C22×C3—81no—
300C22×C3#39211yescharacteristic
301C22×C3#46112yes—
302C22×C3#46511yescharacteristic
303C22×C3#46811yescharacteristic
304C22×C3#47112yes—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—82no—
308C3×C4—81no—
309C3×C4—82no—
310C3×C4#44811yescharacteristic
311C3×C4#44812yes—
312C3×C4#44911yescharacteristic
313C3×C4#45011yescharacteristic
314C3×C4#45211yescharacteristic
315C3×C4#49211yescharacteristic
316C3×C4#49711yescharacteristic
317C23—22no—
318C23—41no—
319C23—42no—
320C23—41no—
321C23—42no—
322C23C3×#12111yescharacteristic
323Q8—42no—
324Q8—42no—
325Q8—82no—
326Q8—82no—
327Q8—81no—
328Q8—81no—
329D4—22no—
330D4—22no—
331D4—41no—
332D4—41no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4C3×#12011yescharacteristic
345C2×C4C3×#12011yescharacteristic
346C2×C4C3×#12011yescharacteristic
347C2×C4C3×#12111yescharacteristic
348C2×C4C3×#12111yescharacteristic
349C2×C4C3×#12711yescharacteristic
350C2×C4C2×C3×SD1611yescharacteristic
351C2×C4C2×C3×SD1612yes—
352C2×C4C3×#13511yescharacteristic
353C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
354C2×C4C3×#13711yescharacteristic
355C2×C4C3×#13712yes—
356C8—24no—
357C8—42no—
358C8—42no—
359C8—81no—
360C2×C3—82no—
361C2×C3#167112yes—
362C2×C3#169112yes—
363C2×C3#200011yescharacteristic
364C2×C3#201311yescharacteristic
365C2×C3#201911yescharacteristic
366C22—42no—
367C22—44no—
368C22—81no—
369C22—81no—
370C22—82no—
371C22C3×#39211yescharacteristic
372C22C3×#46112yes—
373C22C3×#46511yescharacteristic
374C22C3×#46811yescharacteristic
375C22C3×#47112yes—
376C4—21no—
377C4—21no—
378C4—82no—
379C4—81no—
380C4—82no—
381C4C3×#44811yescharacteristic
382C4C3×#44812yes—
383C4C3×#44911yescharacteristic
384C4C3×#45011yescharacteristic
385C4C3×#45211yescharacteristic
386C4C3×#49211yescharacteristic
387C4C3×#49711yescharacteristic
388C3#1027111yescharacteristic, direct product factor, Sylow
389C2—82no—
390C2C3×#167112yes—
391C2C3×#169112yes—
392C2C3×#200011yescharacteristic
393C2C3×#201311yescharacteristic
394C2C3×#201911yescharacteristic
subgroupG/H
center, socleC23×C3#121
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC22×C3×D4D4
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×#498C2semidirect
5. C2×C3×#333C2semidirect
6. C2×C3×#332C2semidirect
7. C3×#1277C2semidirect
8. #10271C3direct (N × H)
30. C2×#498C2×C3semidirect
34. C2×#333C2×C3semidirect
35. C2×#332C2×C3semidirect
36. #1277C2×C3semidirect
138. C2×C3×C8SD16semidirect
278. C2×C8C3×SD16semidirect
388. C3#10271direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4982
C2×C3×#4762
C2×C3×#4612
C3×#19542
C2×C3×#3332
C2×C3×#3322
C3×#12772
#102713