Finite Group Databasebeta

← C3×#10250 #10628857

Subgroups — C3×#10250 #10628857

Order 768.

604 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3208C211yesmaximal, characteristic
2C3×#3078C211yesmaximal, characteristic, complemented
3C3×#2548C211yesmaximal, characteristic
4C3×#1545C211yesmaximal, characteristic
5C3×#1503C211yesmaximal, characteristic, complemented
6C3×#1479C211yesmaximal, characteristic, complemented
7C3×#1303C211yesmaximal, characteristic, complemented
8#10250C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#127C2211yescharacteristic
11C2×C3×#124—21no—
12C2×C3×C4×Q8C2211yescharacteristic
13C3×#492—21no—
14C3×#486—21no—
15C3×#479—22no—
16C3×#477—21no—
17C3×#475—21no—
18C3×#460—22no—
19C3×#450—22no—
20C3×#431C2211yescharacteristic
21C3×#431C2211yescharacteristic
22C3×#428—21no—
23C3×#423C2211yescharacteristic
24C3×#419—21no—
25C3×#394—22no—
26C3×#390—22no—
27C3×#390—21no—
28C3×C4×C4⋊C4C2211yescharacteristic
29C3×#331—22no—
30C3×#328—22no—
31C3×Q8⋊C8—22no—
32#3208C2×C311yescharacteristic
33#3078C2×C311yescharacteristic, complemented
34#2548C2×C311yescharacteristic
35#1545C2×C311yescharacteristic
36#1503C2×C311yescharacteristic, complemented
37#1479C2×C311yescharacteristic, complemented
38#1303C2×C311yescharacteristic, complemented
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C22×C3×D4D412yes—
43C2×C3×SD16—24no—
44C2×C3×M16—21no—
45C3×C4⋊Q8D411yescharacteristic
46C3×C4⋊Q8D411yescharacteristic
47C3×#127—21no—
48C3×#127—21no—
49C3×#127—21no—
50C3×#127—21no—
51C3×#125—21no—
52C3×#124—42no—
53C3×#124—42no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C3×C4×Q8—21no—
57C3×C4×Q8—21no—
58C3×C4×Q8—21no—
59C3×C4×Q8—21no—
60C3×C4×Q8—21no—
61C3×C4×Q8—21no—
62C3×#117—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4D411yescharacteristic
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4—24no—
68C2×C3×C42—21no—
69C2×C3×C42—21no—
70C2×C3×C42C2311yescharacteristic
71C3×#107—22no—
72C3×#106—22no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8D412yes—
76C3×C4⋊C8D412yes—
77C3×#103—22no—
78C3×#103—22no—
79C3×#102—24no—
80C3×C22⋊C8—21no—
81C3×C22⋊C8—21no—
82C3×C4×C8—21no—
83C3×C4×C8—21no—
84C3×#95D412yes—
85#533C22×C311yescharacteristic
86C2×#127C22×C311yescharacteristic
87C2×#124—21no—
88C2×C4×Q8C22×C311yescharacteristic
89#492—21no—
90#486—21no—
91#479—22no—
92#477—21no—
93#475—21no—
94#460—22no—
95#450—22no—
96#431C22×C311yescharacteristic
97#431C22×C311yescharacteristic
98#428—21no—
99#423C22×C311yescharacteristic
100#419—21no—
101#394—22no—
102#390—22no—
103#390—21no—
104C4×C4⋊C4C22×C311yescharacteristic
105#331—22no—
106#328—22no—
107Q8⋊C8—22no—
108C24×C3—22no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C2×C3×Q8SD1612yescomplemented
117C2×C3×D4—22no—
118C2×C3×D4—22no—
119C2×C3×D4—44no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—82no—
129C22×C3×C4—21no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C3×SD16—84no—
136C3×M16—41no—
137C3×M16—41no—
138C2×C3×C8—22no—
139C2×C3×C8—22no—
140C2×C3×C8—41no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—42no—
148C3×C4⋊C4—41no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C22⋊C4—48no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—84no—
155C3×C42—41no—
156C3×C42—41no—
157C3×C42—41no—
158C3×C42—41no—
159C3×C42—41no—
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42C2×D411yescharacteristic
163C3×C42#4111yescharacteristic
164C22×Q8C3×D411yescharacteristic
165C22×D4—21no—
166C22×D4—21no—
167C22×D4C3×D412yes—
168C2×SD16—24no—
169C2×M16—21no—
170C4⋊Q8C3×D411yescharacteristic
171C4⋊Q8C3×D411yescharacteristic
172#127—21no—
173#127—21no—
174#127—21no—
175#127—21no—
176#125—21no—
177#124—42no—
178#124—42no—
179C22⋊Q8—21no—
180C22⋊Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187#117—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4C3×D411yescharacteristic
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42C23×C311yescharacteristic
196#107—22no—
197#106—22no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8C3×D412yes—
201C4⋊C8C3×D412yes—
202#103—22no—
203#103—22no—
204#102—24no—
205C22⋊C8—21no—
206C22⋊C8—21no—
207C4×C8—21no—
208C4×C8—21no—
209#95C3×D412yes—
210C23×C3—44no—
211C23×C3—44no—
212C23×C3—42no—
213C23×C3—42no—
214C23×C3—42no—
215C23×C3#12011yescharacteristic
216C3×Q8—22no—
217C3×Q8—22no—
218C3×Q8—41no—
219C3×Q8—41no—
220C3×Q8—42no—
221C3×Q8—81no—
222C3×D4—44no—
223C3×D4—82no—
224C3×D4—82no—
225C3×D4—82no—
226C3×D4—81no—
227C3×D4—81no—
228C3×D4—81no—
229C3×D4—81no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—42no—
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12711yescharacteristic
248C2×C3×C4C2×SD1611yescharacteristic
249C2×C3×C4#13511yescharacteristic
250C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
251C2×C3×C4#13711yescharacteristic
252C3×C8—42no—
253C3×C8—42no—
254C3×C8—81no—
255C24—22no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8C3×SD1612yescomplemented
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—44no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—82no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4C2×C3×D411yescharacteristic
280C22×C4C2×C3×D411yescharacteristic
281C22×C4C2×C3×D411yescharacteristic
282SD16—84no—
283M16—41no—
284M16—41no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—41no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C22⋊C4—48no—
300C22⋊C4—41no—
301C22⋊C4—84no—
302C42—41no—
303C42—41no—
304C42—41no—
305C42—41no—
306C42—41no—
307C42C2×C3×D411yescharacteristic
308C42C2×C3×D411yescharacteristic
309C42C2×C3×D411yescharacteristic
310C42C3×#4111yescharacteristic
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—84no—
314C22×C3—84no—
315C22×C3—82no—
316C22×C3—82no—
317C22×C3—82no—
318C22×C3#39211yescharacteristic
319C22×C3#45011yescharacteristic
320C22×C3#45211yescharacteristic
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—42no—
325C3×C4—82no—
326C3×C4—81no—
327C3×C4—81no—
328C3×C4—81no—
329C3×C4#44811yescharacteristic
330C3×C4#44911yescharacteristic
331C3×C4#45512yes—
332C3×C4#46511yescharacteristic
333C3×C4#46811yescharacteristic
334C23—44no—
335C23—44no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23C3×#12011yescharacteristic
340Q8—22no—
341Q8—22no—
342Q8—41no—
343Q8—41no—
344Q8—42no—
345Q8—81no—
346D4—44no—
347D4—82no—
348D4—82no—
349D4—82no—
350D4—81no—
351D4—81no—
352D4—81no—
353D4—81no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4C3×#12011yescharacteristic
367C2×C4C3×#12011yescharacteristic
368C2×C4C3×#12111yescharacteristic
369C2×C4C3×#12111yescharacteristic
370C2×C4C3×#12111yescharacteristic
371C2×C4C3×#12711yescharacteristic
372C2×C4C2×C3×SD1611yescharacteristic
373C2×C4C3×#13511yescharacteristic
374C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
375C2×C4C3×#13711yescharacteristic
376C8—42no—
377C8—42no—
378C8—81no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—162no—
382C2×C3#200011yescharacteristic
383C2×C3#200411yescharacteristic
384C2×C3#200811yescharacteristic
385C22—21no—
386C22—21no—
387C22—84no—
388C22—84no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22C3×#39211yescharacteristic
393C22C3×#45011yescharacteristic
394C22C3×#45211yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—82no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4C3×#44811yescharacteristic
404C4C3×#44911yescharacteristic
405C4C3×#45512yes—
406C4C3×#46511yescharacteristic
407C4C3×#46811yescharacteristic
408C3#1025011yescharacteristic, direct product factor, Sylow
409C2—21no—
410C2—21no—
411C2—162no—
412C2C3×#200011yescharacteristic
413C2C3×#200411yescharacteristic
414C2C3×#200811yescharacteristic
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC2×C3×#127C22
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
2. C3×#3078C2semidirect
5. C3×#1503C2semidirect
6. C3×#1479C2semidirect
7. C3×#1303C2semidirect
8. #10250C3direct (N × H)
33. #3078C2×C3semidirect
36. #1503C2×C3semidirect
37. #1479C2×C3semidirect
38. #1303C2×C3semidirect
116. C2×C3×Q8SD16semidirect
263. C2×Q8C3×SD16semidirect
408. C3#10250direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#32082
C3×#30782
C3×#25482
C3×#15452
C3×#15032
C3×#14792
C3×#13032
#102503