Finite Group Databasebeta

← C3×#10060 #10628667

Subgroups — C3×#10060 #10628667

Order 768.

638 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×#487C211yesmaximal, characteristic, complemented
2C2×C3×#476C211yesmaximal, characteristic
3C3×#2542C211yesmaximal, characteristic
4C3×#1987C211yesmaximal, characteristic, complemented
5C2×C3×#332C211yesmaximal, characteristic, complemented
6C2×C3×D4⋊C8C211yesmaximal, characteristic
7C3×#1279C211yesmaximal, characteristic, complemented
8#10060C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C22⋊Q8—22no—
12C2×C3×C4×D4C2211yescharacteristic
13C3×#487—22no—
14C3×#487—22no—
15C3×#476—22no—
16C3×#476—22no—
17C2×C3×#106—21no—
18C2×C3×C4⋊C8C2211yescharacteristic
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×#103—22no—
21C2×C3×#102—21no—
22C2×C3×C4×C8C2211yescharacteristic
23C3×#399—22no—
24C3×#390—21no—
25C3×#384—21no—
26C3×C4×C4⋊C4C2211yescharacteristic
27C3×#340—22no—
28C3×#332—22no—
29C3×D4⋊C8—22no—
30C2×#487C2×C311yescharacteristic, complemented
31C2×#476C2×C311yescharacteristic
32#2542C2×C311yescharacteristic
33#1987C2×C311yescharacteristic, complemented
34C2×#332C2×C311yescharacteristic, complemented
35C2×D4⋊C8C2×C311yescharacteristic
36#1279C2×C311yescharacteristic, complemented
37C22×C3×Q8—21no—
38C22×C3×Q8—21no—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4—21no—
41C22×C3×C8—21no—
42C22×C3×C8—21no—
43C22×C3×C8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8—21no—
50C3×C4⋊Q8—21no—
51C3×C4⋊Q8D412yes—
52C3×C22⋊Q8—44no—
53C3×C22⋊Q8—44no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—24no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C4⋊C4D412yes—
64C2×C3×C4⋊C4D412yes—
65C2×C3×C22⋊C4—24no—
66C2×C3×C22⋊C4—21no—
67C2×C3×C42—21no—
68C2×C3×C42C2311yescharacteristic
69C3×#106—24no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8D412yes—
74C3×#103—28no—
75C3×#102—24no—
76C3×C4×C8—21no—
77C3×C4×C8D412yes—
78C3×#95—21no—
79C2×C4⋊Q8C22×C311yescharacteristic
80C2×C4⋊Q8C22×C311yescharacteristic
81C2×C22⋊Q8—22no—
82C2×C4×D4C22×C311yescharacteristic
83#487—22no—
84#487—22no—
85#476—22no—
86#476—22no—
87C2×#106—21no—
88C2×C4⋊C8C22×C311yescharacteristic
89C2×C4⋊C8C22×C311yescharacteristic
90C2×#103—22no—
91C2×#102—21no—
92C2×C4×C8C22×C311yescharacteristic
93#399—22no—
94#390—21no—
95#384—21no—
96C4×C4⋊C4C22×C311yescharacteristic
97#340—22no—
98#332—22no—
99D4⋊C8—22no—
100C24×C3—21no—
101C2×C3×Q8—22no—
102C2×C3×Q8—44no—
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C2×C3×Q8—42no—
107C2×C3×Q8—42no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4Q1612yes—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—21no—
117C22×C3×C4—41no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—42no—
121C22×C3×C4—42no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C2×C3×C8—22no—
126C2×C3×C8—22no—
127C2×C3×C8—42no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C2×C3×C8—41no—
131C2×C3×C8—42no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—48no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4SD1614yes—
145C3×C22⋊C4—48no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C42—41no—
149C3×C42—41no—
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42#4111yescharacteristic
154C22×Q8—21no—
155C22×Q8—21no—
156C22×D4C3×D411yescharacteristic
157C23×C4—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8C3×D412yes—
169C22⋊Q8—44no—
170C22⋊Q8—44no—
171C4×D4—21no—
172C4×D4—21no—
173C4×D4—21no—
174C4×D4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4C3×D411yescharacteristic
180C2×C4⋊C4C3×D412yes—
181C2×C4⋊C4C3×D412yes—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—21no—
184C2×C42—21no—
185C2×C42C23×C311yescharacteristic
186#106—24no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8C3×D412yes—
191#103—28no—
192#102—24no—
193C4×C8—21no—
194C4×C8C3×D412yes—
195#95—21no—
196C23×C3—22no—
197C23×C3—41no—
198C23×C3—42no—
199C23×C3—41no—
200C23×C3—42no—
201C23×C3#12011yescharacteristic
202C3×Q8—44no—
203C3×Q8—82no—
204C3×Q8—82no—
205C3×Q8—81no—
206C3×Q8—81no—
207C3×D4—22no—
208C3×D4—22no—
209C3×D4—41no—
210C3×D4—41no—
211C2×C3×C4—22no—
212C2×C3×C4—24no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—44no—
219C2×C3×C4—42no—
220C2×C3×C4—82no—
221C2×C3×C4—82no—
222C2×C3×C4—81no—
223C2×C3×C4—81no—
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12711yescharacteristic
230C2×C3×C4C2×SD1612yes—
231C2×C3×C4C2×Q1611yescharacteristic
232C2×C3×C4#13511yescharacteristic
233C2×C3×C4AGL(1,ℤ/8)12yes—
234C2×C3×C4#13711yescharacteristic
235C2×C3×C4#13711yescharacteristic
236C3×C8—42no—
237C3×C8—42no—
238C3×C8—81no—
239C24—21no—
240C2×Q8—22no—
241C2×Q8—44no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4C3×Q1612yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4C2×C3×D411yescharacteristic
262C22×C4C2×C3×D411yescharacteristic
263C22×C4C2×C3×D411yescharacteristic
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—42no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—42no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—21no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—48no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4C3×SD1614yes—
284C22⋊C4—48no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C42—41no—
288C42—41no—
289C42C2×C3×D411yescharacteristic
290C42C2×C3×D411yescharacteristic
291C42C2×C3×D411yescharacteristic
292C42C3×#4111yescharacteristic
293C22×C3—44no—
294C22×C3—42no—
295C22×C3—82no—
296C22×C3—81no—
297C22×C3—81no—
298C22×C3#39211yescharacteristic
299C22×C3#45012yes—
300C22×C3#45111yescharacteristic
301C22×C3#45211yescharacteristic
302C22×C3#45412yes—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—44no—
306C3×C4—82no—
307C3×C4—81no—
308C3×C4—82no—
309C3×C4#45111yescharacteristic
310C3×C4#45211yescharacteristic
311C3×C4#46012yes—
312C3×C4#46711yescharacteristic
313C3×C4#47011yescharacteristic
314C3×C4#49211yescharacteristic
315C3×C4#49611yescharacteristic
316C23—22no—
317C23—41no—
318C23—42no—
319C23—41no—
320C23—42no—
321C23C3×#12011yescharacteristic
322Q8—44no—
323Q8—82no—
324Q8—82no—
325Q8—81no—
326Q8—81no—
327D4—22no—
328D4—22no—
329D4—41no—
330D4—41no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4C3×#12011yescharacteristic
345C2×C4C3×#12011yescharacteristic
346C2×C4C3×#12111yescharacteristic
347C2×C4C3×#12111yescharacteristic
348C2×C4C3×#12111yescharacteristic
349C2×C4C3×#12711yescharacteristic
350C2×C4C2×C3×SD1612yes—
351C2×C4C2×C3×Q1611yescharacteristic
352C2×C4C3×#13511yescharacteristic
353C2×C4C3×AGL(1,ℤ/8)12yes—
354C2×C4C3×#13711yescharacteristic
355C2×C4C3×#13711yescharacteristic
356C8—42no—
357C8—42no—
358C8—81no—
359C2×C3—82no—
360C2×C3#163512yes—
361C2×C3#163712yes—
362C2×C3#200111yescharacteristic
363C2×C3#200311yescharacteristic
364C2×C3#201411yescharacteristic
365C22—42no—
366C22—44no—
367C22—81no—
368C22—81no—
369C22—82no—
370C22C3×#39211yescharacteristic
371C22C3×#45012yes—
372C22C3×#45111yescharacteristic
373C22C3×#45211yescharacteristic
374C22C3×#45412yes—
375C4—21no—
376C4—21no—
377C4—44no—
378C4—82no—
379C4—81no—
380C4—82no—
381C4C3×#45111yescharacteristic
382C4C3×#45211yescharacteristic
383C4C3×#46012yes—
384C4C3×#46711yescharacteristic
385C4C3×#47011yescharacteristic
386C4C3×#49211yescharacteristic
387C4C3×#49611yescharacteristic
388C3#1006011yescharacteristic, direct product factor, Sylow
389C2—82no—
390C2C3×#163512yes—
391C2C3×#163712yes—
392C2C3×#200111yescharacteristic
393C2C3×#200311yescharacteristic
394C2C3×#201411yescharacteristic
subgroupG/H
center, socleC23×C3#120
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×#487C2semidirect
4. C3×#1987C2semidirect
5. C2×C3×#332C2semidirect
7. C3×#1279C2semidirect
8. #10060C3direct (N × H)
30. C2×#487C2×C3semidirect
33. #1987C2×C3semidirect
34. C2×#332C2×C3semidirect
36. #1279C2×C3semidirect
388. C3#10060direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4872
C2×C3×#4762
C3×#25422
C3×#19872
C2×C3×#3322
C2×C3×D4⋊C82
C3×#12792
#100603