Finite Group Databasebeta

← C3×#10275 #10628882

Subgroups — C3×#10275 #10628882

Order 768.

562 conjugacy classes of subgroups.

The subgroup lattice has 382 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
3C2×C3×#474C211yesmaximal, characteristic
4C3×#1954C211yesmaximal, characteristic
5C2×C3×#333C211yesmaximal, characteristic, complemented
6C2×C3×#332C211yesmaximal, characteristic, complemented
7C3×#1277C211yesmaximal, characteristic, complemented
8#10275C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C3×#487—22no—
13C3×#487—22no—
14C3×#476—22no—
15C3×#476—22no—
16C3×#474—22no—
17C3×#474—22no—
18C2×C3×#107—21no—
19C2×C3×#106—21no—
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#103—22no—
24C2×C3×#102—21no—
25C2×C3×#102—21no—
26C2×C3×C22⋊C8—22no—
27C2×C3×C4×C8C2211yescharacteristic
28C3×#333—22no—
29C3×#332—22no—
30C2×#487C2×C311yescharacteristic, complemented
31C2×#476C2×C311yescharacteristic
32C2×#474C2×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—
41C22×C3×C8—21no—
42C22×C3×C8—21no—
43C22×C3×C8—21no—
44C22×C3×C8D412yes—
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×C4⋊Q8D412yes—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C22⋊C4—21no—
63C2×C3×C42C2311yescharacteristic
64C3×#107—24no—
65C3×#106—24no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8D412yes—
71C3×C4⋊C8D412yes—
72C3×#103—28no—
73C3×#102—24no—
74C3×#102—24no—
75C3×C22⋊C8—44no—
76C3×C4×C8—21no—
77C3×C4×C8—21no—
78C2×C4⋊Q8C22×C311yescharacteristic
79C2×C4⋊Q8C22×C311yescharacteristic
80C2×C4×D4C22×C311yescharacteristic
81#487—22no—
82#487—22no—
83#476—22no—
84#476—22no—
85#474—22no—
86#474—22no—
87C2×#107—21no—
88C2×#106—21no—
89C2×C4⋊C8C22×C311yescharacteristic
90C2×C4⋊C8C22×C311yescharacteristic
91C2×C4⋊C8C22×C311yescharacteristic
92C2×#103—22no—
93C2×#102—21no—
94C2×#102—21no—
95C2×C22⋊C8—22no—
96C2×C4×C8C22×C311yescharacteristic
97#333—22no—
98#332—22no—
99C24×C3—21no—
100C2×C3×Q8—22no—
101C2×C3×Q8—22no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—42no—
105C2×C3×Q8—42no—
106C2×C3×Q8—82no—
107C2×C3×D4—21no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4Q1612yes—
113C22×C3×C4—21no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4—42no—
119C22×C3×C4—42no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4#4111yescharacteristic
123C2×C3×C8—22no—
124C2×C3×C8—22no—
125C2×C3×C8—22no—
126C2×C3×C8—22no—
127C2×C3×C8—22no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C2×C3×C8—41no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C2×C3×C8—42no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C22×Q8—21no—
151C22×Q8—21no—
152C22×D4C3×D411yescharacteristic
153C23×C4—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8C3×D412yes—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8C3×D412yes—
165C4⋊Q8C3×D412yes—
166C4×D4—21no—
167C4×D4—21no—
168C4×D4—21no—
169C4×D4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4C3×D411yescharacteristic
175C2×C22⋊C4—21no—
176C2×C42C23×C311yescharacteristic
177#107—24no—
178#106—24no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8C3×D412yes—
184C4⋊C8C3×D412yes—
185#103—28no—
186#102—24no—
187#102—24no—
188C22⋊C8—44no—
189C4×C8—21no—
190C4×C8—21no—
191C23×C3—22no—
192C23×C3—41no—
193C23×C3—42no—
194C23×C3—41no—
195C23×C3—42no—
196C23×C3#12111yescharacteristic
197C3×Q8—44no—
198C3×Q8—82no—
199C3×Q8—82no—
200C3×Q8—81no—
201C3×Q8—81no—
202C3×D4—22no—
203C3×D4—22no—
204C3×D4—41no—
205C3×D4—41no—
206C2×C3×C4—22no—
207C2×C3×C4—41no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—82no—
212C2×C3×C4—82no—
213C2×C3×C4—82no—
214C2×C3×C4—81no—
215C2×C3×C4—81no—
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12711yescharacteristic
222C2×C3×C4C2×Q1611yescharacteristic
223C2×C3×C4#13512yes—
224C2×C3×C4#13511yescharacteristic
225C2×C3×C4AGL(1,ℤ/8)12yes—
226C2×C3×C4#13711yescharacteristic
227C2×C3×C4#13711yescharacteristic
228C3×C8—42no—
229C3×C8—42no—
230C3×C8—42no—
231C3×C8—81no—
232C24—21no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—82no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4C3×Q1612yes—
246C22×C4—21no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4C2×C3×D411yescharacteristic
254C22×C4C2×C3×D411yescharacteristic
255C22×C4C3×#4111yescharacteristic
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—42no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—22no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C42C2×C3×D411yescharacteristic
280C42C2×C3×D411yescharacteristic
281C42C2×C3×D411yescharacteristic
282C42C2×C3×D411yescharacteristic
283C22×C3—44no—
284C22×C3—42no—
285C22×C3—82no—
286C22×C3—81no—
287C22×C3—81no—
288C22×C3#39211yescharacteristic
289C22×C3#46312yes—
290C22×C3#46711yescharacteristic
291C22×C3#47011yescharacteristic
292C22×C3#47112yes—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—81no—
296C3×C4—82no—
297C3×C4—82no—
298C3×C4#44912yes—
299C3×C4#45111yescharacteristic
300C3×C4#45111yescharacteristic
301C3×C4#45211yescharacteristic
302C3×C4#45211yescharacteristic
303C3×C4#49511yescharacteristic
304C3×C4#49611yescharacteristic
305C23—22no—
306C23—41no—
307C23—42no—
308C23—41no—
309C23—42no—
310C23C3×#12111yescharacteristic
311Q8—44no—
312Q8—82no—
313Q8—82no—
314Q8—81no—
315Q8—81no—
316D4—22no—
317D4—22no—
318D4—41no—
319D4—41no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4C3×#12011yescharacteristic
331C2×C4C3×#12011yescharacteristic
332C2×C4C3×#12011yescharacteristic
333C2×C4C3×#12111yescharacteristic
334C2×C4C3×#12111yescharacteristic
335C2×C4C3×#12711yescharacteristic
336C2×C4C2×C3×Q1611yescharacteristic
337C2×C4C3×#13512yes—
338C2×C4C3×#13511yescharacteristic
339C2×C4C3×AGL(1,ℤ/8)12yes—
340C2×C4C3×#13711yescharacteristic
341C2×C4C3×#13711yescharacteristic
342C8—42no—
343C8—42no—
344C8—42no—
345C8—81no—
346C2×C3—82no—
347C2×C3#168312yes—
348C2×C3#170312yes—
349C2×C3#200111yescharacteristic
350C2×C3#201211yescharacteristic
351C2×C3#201911yescharacteristic
352C22—42no—
353C22—44no—
354C22—81no—
355C22—81no—
356C22—82no—
357C22C3×#39211yescharacteristic
358C22C3×#46312yes—
359C22C3×#46711yescharacteristic
360C22C3×#47011yescharacteristic
361C22C3×#47112yes—
362C4—21no—
363C4—21no—
364C4—81no—
365C4—82no—
366C4—82no—
367C4C3×#44912yes—
368C4C3×#45111yescharacteristic
369C4C3×#45111yescharacteristic
370C4C3×#45211yescharacteristic
371C4C3×#45211yescharacteristic
372C4C3×#49511yescharacteristic
373C4C3×#49611yescharacteristic
374C3#1027511yescharacteristic, direct product factor, Sylow
375C2—82no—
376C2C3×#168312yes—
377C2C3×#170312yes—
378C2C3×#200111yescharacteristic
379C2C3×#201211yescharacteristic
380C2C3×#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×#487C2semidirect
5. C2×C3×#333C2semidirect
6. C2×C3×#332C2semidirect
7. C3×#1277C2semidirect
8. #10275C3direct (N × H)
30. C2×#487C2×C3semidirect
34. C2×#333C2×C3semidirect
35. C2×#332C2×C3semidirect
36. #1277C2×C3semidirect
374. C3#10275direct (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
C2×C3×#4742
C3×#19542
C2×C3×#3332
C2×C3×#3322
C3×#12772
#102753