Finite Group Databasebeta

← C3×#10249 #10628856

Subgroups — C3×#10249 #10628856

Order 768.

536 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3230C211yesmaximal, characteristic, complemented
2C3×#3077C211yesmaximal, characteristic
3C3×#2563C211yesmaximal, characteristic
4C3×#1549C211yesmaximal, characteristic, complemented
5C3×#1503C211yesmaximal, characteristic, complemented
6C3×#1478C211yesmaximal, characteristic
7C3×#1303C211yesmaximal, characteristic, complemented
8#10249C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C3×#500—21no—
13C3×#487—21no—
14C3×#484—22no—
15C3×#478—22no—
16C3×#477—22no—
17C3×#476—21no—
18C3×#474—21no—
19C3×#431C2211yescharacteristic
20C3×#431C2211yescharacteristic
21C3×#428—21no—
22C3×#423C2211yescharacteristic
23C3×#418—21no—
24C3×#393—22no—
25C3×#390—21no—
26C3×#382—21no—
27C3×#380—22no—
28C3×C4×C4⋊C4C2211yescharacteristic
29C3×#332—22no—
30C3×#328—22no—
31C3×D4⋊C8—22no—
32#3230C2×C311yescharacteristic, complemented
33#3077C2×C311yescharacteristic
34#2563C2×C311yescharacteristic
35#1549C2×C311yescharacteristic, complemented
36#1503C2×C311yescharacteristic, complemented
37#1478C2×C311yescharacteristic
38#1303C2×C311yescharacteristic, complemented
39C22×C3×Q8—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C2×C3×M16—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8D411yescharacteristic
48C3×C4⋊Q8D411yescharacteristic
49C3×#125—21no—
50C3×C22⋊Q8—21no—
51C3×C22⋊Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C3×#117—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4D412yes—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—24no—
65C2×C3×C42—21no—
66C2×C3×C42C2311yescharacteristic
67C3×#107—22no—
68C3×#107—24no—
69C3×#106—22no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×C4⋊C8D412yes—
74C3×#103—24no—
75C3×#102—22no—
76C3×#102—22no—
77C3×C22⋊C8—21no—
78C3×C22⋊C8—21no—
79C3×C4×C8—21no—
80C3×C4×C8—21no—
81C3×#95—24no—
82C3×#95D412yes—
83#533C22×C311yescharacteristic
84C2×C4⋊Q8C22×C311yescharacteristic
85C2×C4×D4C22×C311yescharacteristic
86#500—21no—
87#487—21no—
88#484—22no—
89#478—22no—
90#477—22no—
91#476—21no—
92#474—21no—
93#431C22×C311yescharacteristic
94#431C22×C311yescharacteristic
95#428—21no—
96#423C22×C311yescharacteristic
97#418—21no—
98#393—22no—
99#390—21no—
100#382—21no—
101#380—22no—
102C4×C4⋊C4C22×C311yescharacteristic
103#332—22no—
104#328—22no—
105D4⋊C8—22no—
106C24×C3—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—41no—
110C2×C3×Q8—41no—
111C2×C3×Q8—82no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4Q1612yes—
118C22×C3×C4—21no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—44no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C3×M16—41no—
128C3×M16—41no—
129C2×C3×C8—22no—
130C2×C3×C8—22no—
131C2×C3×C8—41no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—42no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—84no—
149C3×C42—41no—
150C3×C42—41no—
151C3×C42—41no—
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42#4111yescharacteristic
156C22×Q8—21no—
157C22×D4C3×D411yescharacteristic
158C23×C4—21no—
159C2×M16—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8C3×D411yescharacteristic
165C4⋊Q8C3×D411yescharacteristic
166#125—21no—
167C22⋊Q8—21no—
168C22⋊Q8—21no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×D4—21no—
172C4×D4—21no—
173C4×D4—21no—
174C4×D4—21no—
175#117—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4C3×D411yescharacteristic
179C2×C4⋊C4C3×D412yes—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—24no—
182C2×C42—21no—
183C2×C42C23×C311yescharacteristic
184#107—22no—
185#107—24no—
186#106—22no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8C3×D412yes—
190C4⋊C8C3×D412yes—
191#103—24no—
192#102—22no—
193#102—22no—
194C22⋊C8—21no—
195C22⋊C8—21no—
196C4×C8—21no—
197C4×C8—21no—
198#95—24no—
199#95C3×D412yes—
200C23×C3—22no—
201C23×C3—42no—
202C23×C3—42no—
203C23×C3—41no—
204C23×C3—41no—
205C23×C3#12011yescharacteristic
206C3×Q8—42no—
207C3×Q8—81no—
208C3×Q8—82no—
209C3×Q8—82no—
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—44no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
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—82no—
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12711yescharacteristic
233C2×C3×C4C2×Q1611yescharacteristic
234C2×C3×C4#13511yescharacteristic
235C2×C3×C4#13711yescharacteristic
236C2×C3×C4#13711yescharacteristic
237C3×C8—42no—
238C3×C8—42no—
239C3×C8—81no—
240C24—21no—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—82no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4C3×Q1612yes—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—44no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4C2×C3×D411yescharacteristic
259C22×C4C2×C3×D411yescharacteristic
260C22×C4C2×C3×D411yescharacteristic
261M16—41no—
262M16—41no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—41no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—22no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—22no—
273C4⋊C4—21no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—84no—
283C42—41no—
284C42—41no—
285C42—41no—
286C42C2×C3×D411yescharacteristic
287C42C2×C3×D411yescharacteristic
288C42C2×C3×D411yescharacteristic
289C42C3×#4111yescharacteristic
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—44no—
293C22×C3—42no—
294C22×C3—82no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3#39211yescharacteristic
298C22×C3#45111yescharacteristic
299C22×C3#45211yescharacteristic
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—82no—
304C3×C4—81no—
305C3×C4—81no—
306C3×C4—82no—
307C3×C4—81no—
308C3×C4#45111yescharacteristic
309C3×C4#45211yescharacteristic
310C3×C4#45412yes—
311C3×C4#46711yescharacteristic
312C3×C4#47011yescharacteristic
313C23—22no—
314C23—42no—
315C23—42no—
316C23—41no—
317C23—41no—
318C23C3×#12011yescharacteristic
319Q8—42no—
320Q8—81no—
321Q8—82no—
322Q8—82no—
323D4—22no—
324D4—22no—
325D4—41no—
326D4—41no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4C3×#12011yescharacteristic
341C2×C4C3×#12011yescharacteristic
342C2×C4C3×#12111yescharacteristic
343C2×C4C3×#12111yescharacteristic
344C2×C4C3×#12111yescharacteristic
345C2×C4C3×#12711yescharacteristic
346C2×C4C2×C3×Q1611yescharacteristic
347C2×C4C3×#13511yescharacteristic
348C2×C4C3×#13711yescharacteristic
349C2×C4C3×#13711yescharacteristic
350C8—42no—
351C8—42no—
352C8—81no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—82no—
356C2×C3#200111yescharacteristic
357C2×C3#200311yescharacteristic
358C2×C3#200811yescharacteristic
359C22—21no—
360C22—21no—
361C22—44no—
362C22—42no—
363C22—82no—
364C22—81no—
365C22—81no—
366C22C3×#39211yescharacteristic
367C22C3×#45111yescharacteristic
368C22C3×#45211yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—82no—
373C4—81no—
374C4—81no—
375C4—82no—
376C4—81no—
377C4C3×#45111yescharacteristic
378C4C3×#45211yescharacteristic
379C4C3×#45412yes—
380C4C3×#46711yescharacteristic
381C4C3×#47011yescharacteristic
382C3#1024911yescharacteristic, direct product factor, Sylow
383C2—21no—
384C2—21no—
385C2—82no—
386C2C3×#200111yescharacteristic
387C2C3×#200311yescharacteristic
388C2C3×#200811yescharacteristic
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC22×C3×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#3230C2semidirect
4. C3×#1549C2semidirect
5. C3×#1503C2semidirect
7. C3×#1303C2semidirect
8. #10249C3direct (N × H)
32. #3230C2×C3semidirect
35. #1549C2×C3semidirect
36. #1503C2×C3semidirect
38. #1303C2×C3semidirect
382. C3#10249direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#32302
C3×#30772
C3×#25632
C3×#15492
C3×#15032
C3×#14782
C3×#13032
#102493