Finite Group Databasebeta

← C3×#10270 #10628877

Subgroups — C3×#10270 #10628877

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×#487C211yesmaximal, characteristic, complemented
2C2×C3×#474C211yesmaximal, characteristic
3C2×C3×#461C211yesmaximal, characteristic
4C3×#1954C211yesmaximal, characteristic
5C2×C3×#332C211yesmaximal, characteristic, complemented
6C2×C3×#332C211yesmaximal, characteristic, complemented
7C3×#1277C211yesmaximal, characteristic, complemented
8#10270C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×SD16—21no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C4×D4C2211yescharacteristic
13C3×#487—22no—
14C3×#487—22no—
15C3×#474—22no—
16C3×#474—22no—
17C3×#461—22no—
18C3×#461—22no—
19C2×C3×#107—21no—
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#103—22no—
24C2×C3×#103—21no—
25C2×C3×#102—21no—
26C2×C3×C22⋊C8—22no—
27C2×C3×C4×C8C2211yescharacteristic
28C3×#332—22no—
29C3×#332—22no—
30C2×#487C2×C311yescharacteristic, complemented
31C2×#474C2×C311yescharacteristic
32C2×#461C2×C311yescharacteristic
33#1954C2×C311yescharacteristic
34C2×#332C2×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×#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×#103—28no—
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#487—22no—
85#487—22no—
86#474—22no—
87#474—22no—
88#461—22no—
89#461—22no—
90C2×#107—21no—
91C2×C4⋊C8C22×C311yescharacteristic
92C2×C4⋊C8C22×C311yescharacteristic
93C2×C4⋊C8C22×C311yescharacteristic
94C2×#103—22no—
95C2×#103—21no—
96C2×#102—21no—
97C2×C22⋊C8—22no—
98C2×C4×C8C22×C311yescharacteristic
99#332—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—42no—
108C2×C3×Q8—41no—
109C2×C3×Q8—41no—
110C2×C3×Q8—42no—
111C2×C3×Q8—41no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4SD1612yes—
118C22×C3×C4—21no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—42no—
124C22×C3×C4—42no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C3×SD16—48no—
129C2×C3×C8—22no—
130C2×C3×C8—22no—
131C2×C3×C8—22no—
132C2×C3×C8—22no—
133C2×C3×C8—22no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C2×C3×C8—41no—
138C2×C3×C8—41no—
139C2×C3×C8—42no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156C22×Q8—21no—
157C22×Q8—21no—
158C22×D4C3×D411yescharacteristic
159C23×C4—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⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8C3×D412yes—
174C4⋊Q8C3×D412yes—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C3×D411yescharacteristic
184C2×C22⋊C4—21no—
185C2×C42C23×C311yescharacteristic
186#107—24no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8C3×D412yes—
192C4⋊C8C3×D412yes—
193#103—24no—
194#103—28no—
195#102—24no—
196C22⋊C8—44no—
197C4×C8—21no—
198C4×C8—21no—
199C23×C3—22no—
200C23×C3—41no—
201C23×C3—42no—
202C23×C3—41no—
203C23×C3—42no—
204C23×C3#12111yescharacteristic
205C3×Q8—44no—
206C3×Q8—42no—
207C3×Q8—42no—
208C3×Q8—81no—
209C3×Q8—81no—
210C3×Q8—81no—
211C3×Q8—81no—
212C3×D4—22no—
213C3×D4—22no—
214C3×D4—41no—
215C3×D4—41no—
216C2×C3×C4—22no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—82no—
222C2×C3×C4—81no—
223C2×C3×C4—81no—
224C2×C3×C4—82no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12711yescharacteristic
233C2×C3×C4C2×SD1611yescharacteristic
234C2×C3×C4#13511yescharacteristic
235C2×C3×C4#13512yes—
236C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
237C2×C3×C4AGL(1,ℤ/8)12yes—
238C2×C3×C4#13711yescharacteristic
239C3×C8—42no—
240C3×C8—42no—
241C3×C8—42no—
242C3×C8—81no—
243C24—21no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—42no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—41no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4C3×SD1612yes—
260C22×C4—21no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4C2×C3×D411yescharacteristic
268C22×C4C2×C3×D411yescharacteristic
269C22×C4C3×#4111yescharacteristic
270SD16—48no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—42no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C42C2×C3×D411yescharacteristic
295C42C2×C3×D411yescharacteristic
296C42C2×C3×D411yescharacteristic
297C42C2×C3×D411yescharacteristic
298C22×C3—44no—
299C22×C3—42no—
300C22×C3—82no—
301C22×C3—81no—
302C22×C3—81no—
303C22×C3#39211yescharacteristic
304C22×C3#46312yes—
305C22×C3#46511yescharacteristic
306C22×C3#46811yescharacteristic
307C22×C3#47312yes—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—82no—
311C3×C4—81no—
312C3×C4—82no—
313C3×C4#44811yescharacteristic
314C3×C4#44911yescharacteristic
315C3×C4#44912yes—
316C3×C4#45011yescharacteristic
317C3×C4#45211yescharacteristic
318C3×C4#49511yescharacteristic
319C3×C4#49611yescharacteristic
320C23—22no—
321C23—41no—
322C23—42no—
323C23—41no—
324C23—42no—
325C23C3×#12111yescharacteristic
326Q8—44no—
327Q8—42no—
328Q8—42no—
329Q8—81no—
330Q8—81no—
331Q8—81no—
332Q8—81no—
333D4—22no—
334D4—22no—
335D4—41no—
336D4—41no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—82no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—81no—
347C2×C4—81no—
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×AGL(1,ℤ/8)12yes—
359C2×C4C3×#13711yescharacteristic
360C8—42no—
361C8—42no—
362C8—42no—
363C8—81no—
364C2×C3—82no—
365C2×C3#167912yes—
366C2×C3#169912yes—
367C2×C3#200011yescharacteristic
368C2×C3#201211yescharacteristic
369C2×C3#202011yescharacteristic
370C22—42no—
371C22—44no—
372C22—81no—
373C22—81no—
374C22—82no—
375C22C3×#39211yescharacteristic
376C22C3×#46312yes—
377C22C3×#46511yescharacteristic
378C22C3×#46811yescharacteristic
379C22C3×#47312yes—
380C4—21no—
381C4—21no—
382C4—82no—
383C4—81no—
384C4—82no—
385C4C3×#44811yescharacteristic
386C4C3×#44911yescharacteristic
387C4C3×#44912yes—
388C4C3×#45011yescharacteristic
389C4C3×#45211yescharacteristic
390C4C3×#49511yescharacteristic
391C4C3×#49611yescharacteristic
392C3#1027011yescharacteristic, direct product factor, Sylow
393C2—82no—
394C2C3×#167912yes—
395C2C3×#169912yes—
396C2C3×#200011yescharacteristic
397C2C3×#201211yescharacteristic
398C2C3×#202011yescharacteristic
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×#487C2semidirect
5. C2×C3×#332C2semidirect
6. C2×C3×#332C2semidirect
7. C3×#1277C2semidirect
8. #10270C3direct (N × H)
30. C2×#487C2×C3semidirect
34. C2×#332C2×C3semidirect
35. C2×#332C2×C3semidirect
36. #1277C2×C3semidirect
392. C3#10270direct (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×#4742
C2×C3×#4612
C3×#19542
C2×C3×#3322
C3×#12772
#102703