Finite Group Databasebeta

← C3×#29199 #10647806

Subgroups — C3×#29199 #10647806

Order 768.

706 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2647C211yesmaximal, characteristic, complemented
2C3×#2326C211yesmaximal, characteristic
3C3×#2066C212yesmaximal, complemented
4C3×#2065C212yesmaximal
5C3×#1959C211yesmaximal, characteristic
6C3×#1958C211yesmaximal, characteristic
7C3×#1865C212yesmaximal, complemented
8C3×#1864C212yesmaximal
9C3×#1844C211yesmaximal, characteristic, complemented
10C3×#1836C211yesmaximal, characteristic, complemented
11C3×#1768C211yesmaximal, characteristic, complemented
12#29199C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#124C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×#106C2212yes—
17C2×C3×#106C2212yes—
18C2×C3×C4⋊C8C2212yes—
19C2×C3×#103C2212yes—
20C2×C3×#103C2212yes—
21C2×C3×#102C2212yes—
22C2×C3×#102C2212yes—
23C2×C3×#97C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×#400—22no—
26C3×#398—22no—
27C3×#391C2211yescharacteristic
28C3×#390C2211yescharacteristic
29C3×#389C2212yes—
30C3×#389C2212yes—
31C3×#388—22no—
32C3×#385—22no—
33C3×#384C2211yescharacteristic
34C3×#384C2211yescharacteristic
35C3×#382C2211yescharacteristic
36C3×#376C2211yescharacteristic
37C3×#340C2212yes—
38C3×#340C2212yes—
39C3×#336C2212yes—
40#2647C2×C311yescharacteristic, complemented
41#2326C2×C311yescharacteristic
42#2066C2×C312yescomplemented
43#2065C2×C312yes—
44#1959C2×C311yescharacteristic
45#1958C2×C311yescharacteristic
46#1865C2×C312yescomplemented
47#1864C2×C312yes—
48#1844C2×C311yescharacteristic, complemented
49#1836C2×C311yescharacteristic, complemented
50#1768C2×C311yescharacteristic, complemented
51C22×C3×Q8C2311yescharacteristic
52C22×C3×D4C2311yescharacteristic
53C23×C3×C4—21no—
54C22×C3×C8C2312yes—
55C22×C3×C8C2312yes—
56C3×#124—22no—
57C3×#124Q814yes—
58C3×C4×Q8—42no—
59C3×C4×D4—42no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—22no—
72C2×C3×C42—21no—
73C2×C3×C42C2311yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C3×#106—42no—
77C3×#106—42no—
78C3×C4⋊C8—24no—
79C3×#103—24no—
80C3×#103—42no—
81C3×#102—24no—
82C3×#102—42no—
83C3×#97—22no—
84C3×C4×C8—22no—
85C3×#95—22no—
86C3×#95—22no—
87C3×#95—22no—
88C3×#95—22no—
89C3×#95D412yes—
90C3×#95D412yes—
91C2×#124C22×C311yescharacteristic
92C2×C4×Q8C22×C311yescharacteristic
93C2×C4×D4C22×C311yescharacteristic
94C2×#106C22×C312yes—
95C2×#106C22×C312yes—
96C2×C4⋊C8C22×C312yes—
97C2×#103C22×C312yes—
98C2×#103C22×C312yes—
99C2×#102C22×C312yes—
100C2×#102C22×C312yes—
101C2×#97C22×C311yescharacteristic
102C2×C4×C8C22×C311yescharacteristic
103#400—22no—
104#398—22no—
105#391C22×C311yescharacteristic
106#390C22×C311yescharacteristic
107#389C22×C312yes—
108#389C22×C312yes—
109#388—22no—
110#385—22no—
111#384C22×C311yescharacteristic
112#384C22×C311yescharacteristic
113#382C22×C311yescharacteristic
114#376C22×C311yescharacteristic
115#340C22×C312yes—
116#340C22×C312yes—
117#336C22×C312yes—
118C24×C3—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—22no—
121C2×C3×Q8—22no—
122C2×C3×Q8C2×Q812yes—
123C2×C3×D4—22no—
124C2×C3×D4—21no—
125C2×C3×D4—22no—
126C2×C3×D4C2×Q812yes—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—42no—
133C22×C3×C4—82no—
134C22×C3×C4C2×D412yes—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D412yes—
138C22×C3×C4C2411yescharacteristic
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8—22no—
143C2×C3×C8—22no—
144C2×C3×C8#4114yes—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—22no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—42no—
152C3×C4⋊C4—82no—
153C3×C4⋊C4—82no—
154C3×C4⋊C4—82no—
155C3×C4⋊C4—82no—
156C3×C22⋊C4—28no—
157C3×C22⋊C4—42no—
158C3×C22⋊C4—82no—
159C3×C42—22no—
160C3×C42—22no—
161C3×C42—42no—
162C3×C42C2×Q812yes—
163C3×C42#4112yes—
164C22×Q8C23×C311yescharacteristic
165C22×D4C23×C311yescharacteristic
166C23×C4—21no—
167C22×C8C23×C312yes—
168C22×C8C23×C312yes—
169#124—22no—
170#124C3×Q814yes—
171C4×Q8—42no—
172C4×D4—42no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4C23×C311yescharacteristic
179C2×C4⋊C4C23×C311yescharacteristic
180C2×C4⋊C4C23×C312yes—
181C2×C4⋊C4C23×C312yes—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—22no—
185C2×C42—21no—
186C2×C42C23×C311yescharacteristic
187C2×C42C23×C311yescharacteristic
188C2×C42C23×C311yescharacteristic
189#106—42no—
190#106—42no—
191C4⋊C8—24no—
192#103—24no—
193#103—42no—
194#102—24no—
195#102—42no—
196#97—22no—
197C4×C8—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95C3×D412yes—
203#95C3×D412yes—
204C23×C3—22no—
205C23×C3—41no—
206C23×C3—41no—
207C23×C3—42no—
208C23×C3—42no—
209C23×C3C22×D411yescharacteristic
210C3×Q8—24no—
211C3×Q8—42no—
212C3×D4—24no—
213C3×D4—42no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—84no—
237C2×C3×C4C22⋊Q814yes—
238C2×C3×C4C22×Q811yescharacteristic
239C2×C3×C4C2×#4111yescharacteristic
240C2×C3×C4C2×#4112yes—
241C2×C3×C4H(5,𝔽2)12yes—
242C3×C8—24no—
243C3×C8—42no—
244C24—21no—
245C2×Q8—21no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8C2×C3×Q812yes—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—22no—
252C2×D4C2×C3×Q812yes—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—82no—
260C22×C4C2×C3×D412yes—
261C22×C4C2×C3×D411yescharacteristic
262C22×C4C2×C3×D411yescharacteristic
263C22×C4C2×C3×D412yes—
264C22×C4C24×C311yescharacteristic
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8C3×#4114yes—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—42no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C22⋊C4—28no—
283C22⋊C4—42no—
284C22⋊C4—82no—
285C42—22no—
286C42—22no—
287C42—42no—
288C42C2×C3×Q812yes—
289C42C3×#4112yes—
290C22×C3—42no—
291C22×C3—44no—
292C22×C3—82no—
293C22×C3—81no—
294C22×C3—81no—
295C22×C3C2×C22⋊Q811yescharacteristic
296C22×C3#53211yescharacteristic
297C22×C3#53511yescharacteristic
298C22×C3#57712yes—
299C22×C3#57712yes—
300C3×C4—24no—
301C3×C4—42no—
302C3×C4—42no—
303C3×C4—41no—
304C3×C4—41no—
305C3×C4—82no—
306C3×C4—82no—
307C3×C4—81no—
308C3×C4—81no—
309C3×C4#55312yes—
310C3×C4#55412yes—
311C23—22no—
312C23—42no—
313C23—42no—
314C23—41no—
315C23—41no—
316C23C22×C3×D411yescharacteristic
317Q8—24no—
318Q8—42no—
319D4—24no—
320D4—42no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—84no—
344C2×C4C3×C22⋊Q814yes—
345C2×C4C22×C3×Q811yescharacteristic
346C2×C4C2×C3×#4111yescharacteristic
347C2×C4C2×C3×#4112yes—
348C2×C4C3×H(5,𝔽2)12yes—
349C8—24no—
350C8—42no—
351C2×C3—82no—
352C2×C3#256611yescharacteristic
353C2×C3#308412yes—
354C2×C3#317312yes—
355C2×C3#322111yescharacteristic
356C2×C3#322211yescharacteristic
357C22—42no—
358C22—44no—
359C22—82no—
360C22—81no—
361C22—81no—
362C22C2×C3×C22⋊Q811yescharacteristic
363C22C3×#53211yescharacteristic
364C22C3×#53511yescharacteristic
365C22C3×#57712yes—
366C22C3×#57712yes—
367C4—24no—
368C4—42no—
369C4—41no—
370C4—41no—
371C4—42no—
372C4—82no—
373C4—82no—
374C4—81no—
375C4—81no—
376C4C3×#55312yes—
377C4C3×#55412yes—
378C3#2919911yescharacteristic, direct product factor, Sylow
379C2—82no—
380C2C3×#256611yescharacteristic
381C2C3×#308412yes—
382C2C3×#317312yes—
383C2C3×#322111yescharacteristic
384C2C3×#322211yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×#376 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2647C2semidirect
3. C3×#2066C2semidirect
7. C3×#1865C2semidirect
9. C3×#1844C2semidirect
10. C3×#1836C2semidirect
11. C3×#1768C2semidirect
12. #29199C3direct (N × H)
40. #2647C2×C3semidirect
42. #2066C2×C3semidirect
46. #1865C2×C3semidirect
48. #1844C2×C3semidirect
49. #1836C2×C3semidirect
50. #1768C2×C3semidirect
378. C3#29199direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#26472
C3×#23262
C3×#20662
C3×#20652
C3×#19592
C3×#19582
C3×#18652
C3×#18642
C3×#18442
C3×#18362
C3×#17682
#291993