Finite Group Databasebeta

← C3×#10245 #10628852

Subgroups — C3×#10245 #10628852

Order 768.

558 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3227C211yesmaximal, characteristic
2C3×#3037C211yesmaximal, characteristic
3C3×#2567C211yesmaximal, characteristic, complemented
4C3×#1549C211yesmaximal, characteristic, complemented
5C3×#1502C211yesmaximal, characteristic
6C3×#1479C211yesmaximal, characteristic, complemented
7C3×#1303C211yesmaximal, characteristic, complemented
8#10245C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×#137—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C4×Q8C2211yescharacteristic
13C3×#498—21no—
14C3×#486—21no—
15C3×#481—22no—
16C3×#476—22no—
17C3×#464—22no—
18C3×#462—21no—
19C3×#460—21no—
20C3×#431C2211yescharacteristic
21C3×#431C2211yescharacteristic
22C3×#423C2211yescharacteristic
23C3×#417—21no—
24C3×#395—22no—
25C3×#391—21no—
26C3×#389—22no—
27C3×#384—21no—
28C3×C4×C4⋊C4C2211yescharacteristic
29C3×#332—22no—
30C3×#327—22no—
31C3×Q8⋊C8—22no—
32#3227C2×C311yescharacteristic
33#3037C2×C311yescharacteristic
34#2567C2×C311yescharacteristic, complemented
35#1549C2×C311yescharacteristic, complemented
36#1502C2×C311yescharacteristic
37#1479C2×C311yescharacteristic, complemented
38#1303C2×C311yescharacteristic, complemented
39C2×C3×#41—21no—
40C22×C3×Q8—21no—
41C22×C3×Q8D411yescharacteristic
42C3×#137—42no—
43C3×#137—42no—
44C2×C3×Q16—22no—
45C2×C3×SD16—22no—
46C2×C3×M16—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8—21no—
50C3×C4⋊Q8—21no—
51C3×C4⋊Q8D411yescharacteristic
52C3×#127D411yescharacteristic
53C3×#124—21no—
54C3×#121—21no—
55C3×#121—21no—
56C3×C4×Q8—21no—
57C3×C4×Q8—21no—
58C3×C4×Q8—21no—
59C3×C4×Q8—21no—
60C3×C4×D4—21no—
61C3×C4×D4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4D411yescharacteristic
67C2×C3×C4⋊C4D412yes—
68C2×C3×C42—21no—
69C2×C3×C42—21no—
70C2×C3×C42C2311yescharacteristic
71C3×#106—24no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8D412yes—
75C3×C4⋊C8D412yes—
76C3×#103—22no—
77C3×#102—22no—
78C3×#102—24no—
79C3×C22⋊C8—21no—
80C3×C22⋊C8—21no—
81C3×C4×C8—21no—
82C3×C4×C8—21no—
83C3×#95—24no—
84C3×#95D412yes—
85C2×#137—21no—
86#532C22×C311yescharacteristic
87C2×C4⋊Q8C22×C311yescharacteristic
88C2×C4×Q8C22×C311yescharacteristic
89#498—21no—
90#486—21no—
91#481—22no—
92#476—22no—
93#464—22no—
94#462—21no—
95#460—21no—
96#431C22×C311yescharacteristic
97#431C22×C311yescharacteristic
98#423C22×C311yescharacteristic
99#417—21no—
100#395—22no—
101#391—21no—
102#389—22no—
103#384—21no—
104C4×C4⋊C4C22×C311yescharacteristic
105#332—22no—
106#327—22no—
107Q8⋊C8—22no—
108C3×#41—41no—
109C3×#41—41no—
110C3×#41—41no—
111C3×#41—41no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—41no—
119C2×C3×Q8—41no—
120C2×C3×Q8—82no—
121C2×C3×Q8SD1612yes—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—41no—
126C22×C3×C4—21no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4—41no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C3×Q16—44no—
135C3×SD16—44no—
136C3×M16—41no—
137C3×M16—41no—
138C2×C3×C8—22no—
139C2×C3×C8—22no—
140C2×C3×C8—41no—
141C3×C4⋊C4—24no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—84no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C42—41no—
156C3×C42—41no—
157C3×C42—41no—
158C3×C42—41no—
159C3×C42C2×D411yescharacteristic
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42#4111yescharacteristic
163C2×#41—21no—
164C22×Q8—21no—
165C22×Q8C3×D411yescharacteristic
166#137—42no—
167#137—42no—
168C2×Q16—22no—
169C2×SD16—22no—
170C2×M16—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8C3×D411yescharacteristic
176#127C3×D411yescharacteristic
177#124—21no—
178#121—21no—
179#121—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×D4—21no—
185C4×D4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4C3×D411yescharacteristic
191C2×C4⋊C4C3×D412yes—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42C23×C311yescharacteristic
195#106—24no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8C3×D412yes—
199C4⋊C8C3×D412yes—
200#103—22no—
201#102—22no—
202#102—24no—
203C22⋊C8—21no—
204C22⋊C8—21no—
205C4×C8—21no—
206C4×C8—21no—
207#95—24no—
208#95C3×D412yes—
209C23×C3—41no—
210C23×C3#12011yescharacteristic
211C3×Q8—22no—
212C3×Q8—22no—
213C3×Q8—41no—
214C3×Q8—41no—
215C3×Q8—41no—
216C3×Q8—41no—
217C3×Q8—82no—
218C3×Q8—82no—
219C3×D4—41no—
220C3×D4—41no—
221C3×D4—42no—
222C3×D4—81no—
223C3×D4—81no—
224C3×D4—81no—
225C2×C3×C4—22no—
226C2×C3×C4—24no—
227C2×C3×C4—22no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4—44no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—81no—
239C2×C3×C4—81no—
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12711yescharacteristic
246C2×C3×C4C2×SD1611yescharacteristic
247C2×C3×C4#13511yescharacteristic
248C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
249C2×C3×C4#13711yescharacteristic
250C3×C8—42no—
251C3×C8—42no—
252C3×C8—81no—
253#41—41no—
254#41—41no—
255#41—41no—
256#41—41no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—82no—
266C2×Q8C3×SD1612yes—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—41no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4C2×C3×D411yescharacteristic
277C22×C4C2×C3×D411yescharacteristic
278C22×C4C2×C3×D411yescharacteristic
279Q16—44no—
280SD16—44no—
281M16—41no—
282M16—41no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—41no—
286C4⋊C4—24no—
287C4⋊C4—21no—
288C4⋊C4—22no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4—21no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—84no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C42—41no—
301C42—41no—
302C42—41no—
303C42—41no—
304C42C2×C3×D411yescharacteristic
305C42C2×C3×D411yescharacteristic
306C42C2×C3×D411yescharacteristic
307C42C3×#4111yescharacteristic
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—81no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3#39211yescharacteristic
314C22×C3#44811yescharacteristic
315C22×C3#44911yescharacteristic
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—44no—
320C3×C4—42no—
321C3×C4—81no—
322C3×C4—81no—
323C3×C4—82no—
324C3×C4#45011yescharacteristic
325C3×C4#45211yescharacteristic
326C3×C4#45612yes—
327C3×C4#46511yescharacteristic
328C3×C4#46811yescharacteristic
329C23—41no—
330C23C3×#12011yescharacteristic
331Q8—22no—
332Q8—22no—
333Q8—41no—
334Q8—41no—
335Q8—41no—
336Q8—41no—
337Q8—82no—
338Q8—82no—
339D4—41no—
340D4—41no—
341D4—42no—
342D4—81no—
343D4—81no—
344D4—81no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4C3×#12011yescharacteristic
361C2×C4C3×#12011yescharacteristic
362C2×C4C3×#12111yescharacteristic
363C2×C4C3×#12111yescharacteristic
364C2×C4C3×#12111yescharacteristic
365C2×C4C3×#12711yescharacteristic
366C2×C4C2×C3×SD1611yescharacteristic
367C2×C4C3×#13511yescharacteristic
368C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
369C2×C4C3×#13711yescharacteristic
370C8—42no—
371C8—42no—
372C8—81no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—161no—
376C2×C3#200011yescharacteristic
377C2×C3#200511yescharacteristic
378C2×C3#200811yescharacteristic
379C22—21no—
380C22—21no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22C3×#39211yescharacteristic
385C22C3×#44811yescharacteristic
386C22C3×#44911yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—44no—
391C4—42no—
392C4—81no—
393C4—82no—
394C4—81no—
395C4C3×#45011yescharacteristic
396C4C3×#45211yescharacteristic
397C4C3×#45612yes—
398C4C3×#46511yescharacteristic
399C4C3×#46811yescharacteristic
400C3#1024511yescharacteristic, direct product factor, Sylow
401C2—21no—
402C2—21no—
403C2—161no—
404C2C3×#200011yescharacteristic
405C2C3×#200511yescharacteristic
406C2C3×#200811yescharacteristic
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC3×#532C22
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
3. C3×#2567C2semidirect
4. C3×#1549C2semidirect
6. C3×#1479C2semidirect
7. C3×#1303C2semidirect
8. #10245C3direct (N × H)
34. #2567C2×C3semidirect
35. #1549C2×C3semidirect
37. #1479C2×C3semidirect
38. #1303C2×C3semidirect
400. C3#10245direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#32272
C3×#30372
C3×#25672
C3×#15492
C3×#15022
C3×#14792
C3×#13032
#102453