Finite Group Databasebeta

← C3×#29828 #10648435

Subgroups — C3×#29828 #10648435

Order 768.

842 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
1C2×C3×#476C211yesmaximal, characteristic
2C2×C3×#461C211yesmaximal, characteristic
3C3×#2862C211yesmaximal, characteristic, complemented
4C3×#2293C211yesmaximal, characteristic
5C3×#2066C214yesmaximal, complemented
6C3×#2055C214yesmaximal
7C3×#1987C211yesmaximal, characteristic, complemented
8C3×#1981C211yesmaximal, characteristic, complemented
9C3×#1842C211yesmaximal, characteristic, complemented
10#29828C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×SD16C2212yes—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×C3×#117—21no—
16C3×#476—24no—
17C3×#461—24no—
18C2×C3×#106C2212yes—
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C4⋊C8C2214yes—
22C2×C3×#103C2212yes—
23C2×C3×#103C2214yes—
24C2×C3×#102C2212yes—
25C3×#391—22no—
26C3×#391C2214yes—
27C3×#390C2212yes—
28C3×#389C2212yes—
29C3×#388—22no—
30C3×#382—21no—
31C3×#382—21no—
32C3×C4×C4⋊C4C2211yescharacteristic
33C3×C4×C22⋊C4—21no—
34C3×C43C2211yescharacteristic
35C3×#340C2214yes—
36C2×#476C2×C311yescharacteristic
37C2×#461C2×C311yescharacteristic
38#2862C2×C311yescharacteristic, complemented
39#2293C2×C311yescharacteristic
40#2066C2×C314yescomplemented
41#2055C2×C314yes—
42#1987C2×C311yescharacteristic, complemented
43#1981C2×C311yescharacteristic, complemented
44#1842C2×C311yescharacteristic, complemented
45C22×C3×Q8—21no—
46C22×C3×Q8C2312yes—
47C22×C3×D4C2311yescharacteristic
48C23×C3×C4—21no—
49C2×C3×SD16—24no—
50C2×C3×SD16—44no—
51C22×C3×C8C2312yes—
52C22×C3×C8C2312yes—
53C3×C4⋊Q8—22no—
54C3×C4⋊Q8—22no—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8—22no—
57C3×C4×D4—22no—
58C3×C4×D4D414yes—
59C3×#117—44no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C42—21no—
71C2×C3×C42D411yescharacteristic
72C2×C3×C42D412yes—
73C2×C3×C42D411yescharacteristic
74C2×C3×C42C2312yes—
75C2×C3×C42C2311yescharacteristic
76C3×#106—24no—
77C3×C4⋊C8—22no—
78C3×C4⋊C8—22no—
79C3×C4⋊C8—44no—
80C3×#103—24no—
81C3×#103—44no—
82C3×#102—24no—
83C3×#95—22no—
84C3×#95—21no—
85C3×#95—24no—
86C3×#95—24no—
87C22×SD16C22×C312yes—
88C2×C4⋊Q8C22×C311yescharacteristic
89C2×C4⋊Q8C22×C311yescharacteristic
90C2×C4×D4C22×C311yescharacteristic
91C2×#117—21no—
92#476—24no—
93#461—24no—
94C2×#106C22×C312yes—
95C2×C4⋊C8C22×C311yescharacteristic
96C2×C4⋊C8C22×C311yescharacteristic
97C2×C4⋊C8C22×C314yes—
98C2×#103C22×C312yes—
99C2×#103C22×C314yes—
100C2×#102C22×C312yes—
101#391—22no—
102#391C22×C314yes—
103#390C22×C312yes—
104#389C22×C312yes—
105#388—22no—
106#382—21no—
107#382—21no—
108C4×C4⋊C4C22×C311yescharacteristic
109C4×C22⋊C4—21no—
110C43C22×C311yescharacteristic
111#340C22×C314yes—
112C24×C3—21no—
113C2×C3×Q8—22no—
114C2×C3×Q8—22no—
115C2×C3×Q8—24no—
116C2×C3×Q8—24no—
117C2×C3×Q8—44no—
118C2×C3×Q8—42no—
119C2×C3×D4—21no—
120C2×C3×D4—22no—
121C2×C3×D4—22no—
122C2×C3×D4C2×D412yes—
123C22×C3×C4—21no—
124C22×C3×C4—24no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—42no—
129C22×C3×C4—42no—
130C22×C3×C4C2×D412yes—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2411yescharacteristic
135C3×SD16—48no—
136C2×C3×C8—22no—
137C2×C3×C8—22no—
138C2×C3×C8—24no—
139C2×C3×C8—24no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—42no—
146C3×C4⋊C4—42no—
147C3×C4⋊C4—42no—
148C3×C4⋊C4—44no—
149C3×C4⋊C4C2×D412yes—
150C3×C22⋊C4—24no—
151C3×C22⋊C4—44no—
152C3×C22⋊C4—44no—
153C3×C42—22no—
154C3×C42—22no—
155C3×C42—22no—
156C3×C42—42no—
157C3×C42—42no—
158C3×C42SD1614yescomplemented
159C3×C42C2×D412yes—
160C3×C42#4112yes—
161C22×Q8—21no—
162C22×Q8C23×C312yes—
163C22×D4C23×C311yescharacteristic
164C23×C4—21no—
165C2×SD16—24no—
166C2×SD16—44no—
167C22×C8C23×C312yes—
168C22×C8C23×C312yes—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4×D4—22no—
174C4×D4C3×D414yes—
175#117—44no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4C23×C312yes—
181C2×C4⋊C4C23×C312yes—
182C2×C4⋊C4C23×C311yescharacteristic
183C2×C22⋊C4—21no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C42—21no—
187C2×C42C3×D411yescharacteristic
188C2×C42C3×D412yes—
189C2×C42C3×D411yescharacteristic
190C2×C42C23×C312yes—
191C2×C42C23×C311yescharacteristic
192#106—24no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—44no—
196#103—24no—
197#103—44no—
198#102—24no—
199#95—22no—
200#95—21no—
201#95—24no—
202#95—24no—
203C23×C3—22no—
204C23×C3—42no—
205C23×C3—41no—
206C23×C3—41no—
207C23×C3—42no—
208C23×C3C22×D411yescharacteristic
209C3×Q8—42no—
210C3×Q8—42no—
211C3×Q8—44no—
212C3×Q8—84no—
213C3×D4—22no—
214C3×D4—22no—
215C3×D4—42no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—48no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—44no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—44no—
233C2×C3×C4#12112yes—
234C2×C3×C4#12112yes—
235C2×C3×C4C2×SD1614yes—
236C2×C3×C4C2×SD1612yes—
237C2×C3×C4#13712yes—
238C2×C3×C4C22×D411yescharacteristic
239C2×C3×C4C2×#4111yescharacteristic
240C2×C3×C4H(5,𝔽2)12yes—
241C2×C3×C42−1+412yes—
242C3×C8—42no—
243C3×C8—42no—
244C24—21no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—44no—
250C2×Q8—42no—
251C2×D4—21no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4C2×C3×D412yes—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4C2×C3×D412yes—
263C22×C4C2×C3×D411yescharacteristic
264C22×C4C2×C3×D412yes—
265C22×C4C2×C3×D411yescharacteristic
266C22×C4C24×C311yescharacteristic
267SD16—48no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—24no—
271C2×C8—24no—
272C4⋊C4—21no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4C2×C3×D412yes—
282C22⋊C4—24no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—42no—
289C42—42no—
290C42C3×SD1614yescomplemented
291C42C2×C3×D412yes—
292C42C3×#4112yes—
293C22×C3—42no—
294C22×C3—42no—
295C22×C3—42no—
296C22×C3—82no—
297C22×C3—82no—
298C22×C3C2×#12111yescharacteristic
299C22×C3#53712yes—
300C22×C3C22×SD1611yescharacteristic
301C22×C3C2×#13711yescharacteristic
302C22×C3#57512yes—
303C3×C4—24no—
304C3×C4—22no—
305C3×C4—42no—
306C3×C4—42no—
307C3×C4—82no—
308C3×C4—82no—
309C3×C4—82no—
310C3×C4#46114yes—
311C3×C4#53511yescharacteristic
312C3×C4#53611yescharacteristic
313C3×C4#54212yes—
314C23—22no—
315C23—41no—
316C23—41no—
317C23—42no—
318C23—42no—
319C23C22×C3×D411yescharacteristic
320Q8—42no—
321Q8—42no—
322Q8—44no—
323Q8—84no—
324D4—22no—
325D4—22no—
326D4—42no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—48no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4C3×#12112yes—
345C2×C4C3×#12112yes—
346C2×C4C2×C3×SD1614yes—
347C2×C4C2×C3×SD1612yes—
348C2×C4C3×#13712yes—
349C2×C4C22×C3×D411yescharacteristic
350C2×C4C2×C3×#4111yescharacteristic
351C2×C4C3×H(5,𝔽2)12yes—
352C2×C4C3×2−1+412yes—
353C8—42no—
354C8—42no—
355C2×C3—82no—
356C2×C3#262011yescharacteristic
357C2×C3C2×#46111yescharacteristic
358C2×C3#304011yescharacteristic
359C2×C3#324612yes—
360C2×C3#325212yes—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—82no—
365C22—82no—
366C22C2×C3×#12111yescharacteristic
367C22C3×#53712yes—
368C22C22×C3×SD1611yescharacteristic
369C22C2×C3×#13711yescharacteristic
370C22C3×#57512yes—
371C4—24no—
372C4—22no—
373C4—42no—
374C4—42no—
375C4—82no—
376C4—82no—
377C4—82no—
378C4C3×#46114yes—
379C4C3×#53511yescharacteristic
380C4C3×#53611yescharacteristic
381C4C3×#54212yes—
382C3#2982811yescharacteristic, direct product factor, Sylow
383C2—82no—
384C2C3×#262011yescharacteristic
385C2C2×C3×#46111yescharacteristic
386C2C3×#304011yescharacteristic
387C2C3×#324612yes—
388C2C3×#325212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C43 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2862C2semidirect
5. C3×#2066C2semidirect
7. C3×#1987C2semidirect
8. C3×#1981C2semidirect
9. C3×#1842C2semidirect
10. #29828C3direct (N × H)
38. #2862C2×C3semidirect
40. #2066C2×C3semidirect
42. #1987C2×C3semidirect
43. #1981C2×C3semidirect
44. #1842C2×C3semidirect
158. C3×C42SD16semidirect
290. C42C3×SD16semidirect
382. C3#29828direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4762
C2×C3×#4612
C3×#28622
C3×#22932
C3×#20662
C3×#20552
C3×#19872
C3×#19812
C3×#18422
#298283