Finite Group Databasebeta

← C3×#19077 #10637684

Subgroups — C3×#19077 #10637684

Order 768.

996 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 230 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3550C211yesmaximal, characteristic, complemented
2C3×#3151C211yesmaximal, characteristic
3C3×#3138C212yesmaximal, complemented
4C3×#1657C218yesmaximal, complemented
5C3×#1526C212yesmaximal, complemented
6C3×#1525C211yesmaximal, characteristic, complemented
7#19077C311yesmaximal, characteristic, direct product factor, Sylow
8C2×C3×AGL(1,ℤ/8)—22no—
9C3×#563C2214yescomplemented
10C3×#559—21no—
11C3×Q82—21no—
12C3×#556—22no—
13C3×D4×Q8—22no—
14C3×#542—24no—
15C3×#536—22no—
16C3×#533C2212yes—
17C3×#532C2211yescharacteristic
18C2×C3×#127C2211yescharacteristic
19C3×#496C2214yes—
20C3×#493—21no—
21C3×#492—21no—
22C3×#490C2218yes—
23C3×#486—22no—
24C3×#486—22no—
25C3×#418—24no—
26C3×C4×M16C2212yes—
27C3×C4×M16C2211yescharacteristic
28C3×#330C2218yescomplemented
29C3×#329C2214yescomplemented
30#3550C2×C311yescharacteristic, complemented
31#3151C2×C311yescharacteristic
32#3138C2×C312yescomplemented
33#1657C2×C318yescomplemented
34#1526C2×C312yescomplemented
35#1525C2×C311yescharacteristic, complemented
36C2×C3×#41D412yes—
37C22×C3×Q8—22no—
38C22×C3×D4—21no—
39C22×C3×D4—22no—
40C3×AGL(1,ℤ/8)—48no—
41C2×C3×SD16—24no—
42C2×C3×D8—24no—
43C2×C3×M16—21no—
44C2×C3×M16—22no—
45C3×C4⋊Q8—24no—
46C3×C4⋊Q8—24no—
47C3×C4⋊Q8D411yescharacteristic
48C3×C4⋊Q8D412yes—
49C3×C4⋊Q8D412yes—
50C3×#127—22no—
51C3×#127—21no—
52C3×#127D411yescharacteristic
53C3×#127C2312yes—
54C3×#126—28no—
55C3×#125C2314yes—
56C3×#124—24no—
57C3×#124—24no—
58C3×#124C2312yes—
59C3×#123—24no—
60C3×#123—24no—
61C3×C22⋊Q8—24no—
62C3×C22⋊Q8—22no—
63C3×C22⋊Q8—22no—
64C3×C22⋊Q8—28no—
65C3×#121—22no—
66C3×C4×Q8—24no—
67C3×C4×Q8—22no—
68C3×C4×Q8—22no—
69C3×C4×D4—24no—
70C3×C4×D4—22no—
71C3×#117—22no—
72C3×#117D414yes—
73C2×C3×C42C2311yescharacteristic
74C3×#102—28no—
75C3×#102—28no—
76C3×#97C2314yes—
77C3×#97C2312yes—
78C3×C4×C8—22no—
79C3×C4×C8—21no—
80C2×AGL(1,ℤ/8)—22no—
81#563C22×C314yescomplemented
82#559—21no—
83Q82—21no—
84#556—22no—
85D4×Q8—22no—
86#542—24no—
87#536—22no—
88#533C22×C312yes—
89#532C22×C311yescharacteristic
90C2×#127C22×C311yescharacteristic
91#496C22×C314yes—
92#493—21no—
93#492—21no—
94#490C22×C318yes—
95#486—22no—
96#486—22no—
97#418—24no—
98C4×M16C22×C312yes—
99C4×M16C22×C311yescharacteristic
100#330C22×C318yescomplemented
101#329C22×C314yescomplemented
102C24×C3—41no—
103C3×#41—44no—
104C2×C3×Q8—24no—
105C2×C3×Q8—22no—
106C2×C3×Q8—22no—
107C2×C3×Q8—84no—
108C2×C3×Q8C2×D412yes—
109C2×C3×D4—22no—
110C2×C3×D4—22no—
111C2×C3×D4—24no—
112C2×C3×D4—22no—
113C2×C3×D4—44no—
114C2×C3×D4—48no—
115C2×C3×D4—41no—
116C2×C3×D4—42no—
117C2×C3×D4C2×D412yes—
118C22×C3×C4—24no—
119C22×C3×C4—22no—
120C22×C3×C4C2×D412yes—
121C22×C3×C4C2×D411yescharacteristic
122C3×SD16—48no—
123C3×D8—48no—
124C3×M16—24no—
125C3×M16—44no—
126C2×C3×C8—22no—
127C2×C3×C8—24no—
128C3×C4⋊C4—28no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—24no—
133C3×C4⋊C4—24no—
134C3×C4⋊C4C2×D414yes—
135C3×C4⋊C4C2×D414yes—
136C3×C22⋊C4—24no—
137C3×C22⋊C4—28no—
138C3×C22⋊C4—24no—
139C3×C22⋊C4—24no—
140C3×C42—24no—
141C3×C42—24no—
142C3×C42C2×D412yes—
143C3×C42C2×D411yescharacteristic
144C3×C42C2411yescharacteristic
145C2×#41C3×D412yes—
146C22×Q8—22no—
147C22×D4—21no—
148C22×D4—22no—
149AGL(1,ℤ/8)—48no—
150C2×SD16—24no—
151C2×D8—24no—
152C2×M16—21no—
153C2×M16—22no—
154C4⋊Q8—24no—
155C4⋊Q8—24no—
156C4⋊Q8C3×D411yescharacteristic
157C4⋊Q8C3×D412yes—
158C4⋊Q8C3×D412yes—
159#127—22no—
160#127—21no—
161#127C3×D411yescharacteristic
162#127C23×C312yes—
163#126—28no—
164#125C23×C314yes—
165#124—24no—
166#124—24no—
167#124C23×C312yes—
168#123—24no—
169#123—24no—
170C22⋊Q8—24no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173C22⋊Q8—28no—
174#121—22no—
175C4×Q8—24no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—24no—
179C4×D4—22no—
180#117—22no—
181#117C3×D414yes—
182C2×C42C23×C311yescharacteristic
183#102—28no—
184#102—28no—
185#97C23×C314yes—
186#97C23×C312yes—
187C4×C8—22no—
188C4×C8—21no—
189C23×C3—22no—
190C23×C3—42no—
191C23×C3—82no—
192C23×C3—84no—
193C23×C3#12011yescharacteristic
194C3×Q8—42no—
195C3×Q8—44no—
196C3×Q8—44no—
197C3×Q8—84no—
198C3×D4—42no—
199C3×D4—44no—
200C3×D4—48no—
201C3×D4—44no—
202C3×D4—82no—
203C3×D4—82no—
204C3×D4—84no—
205C2×C3×C4—22no—
206C2×C3×C4—24no—
207C2×C3×C4—24no—
208C2×C3×C4—22no—
209C2×C3×C4—24no—
210C2×C3×C4—44no—
211C2×C3×C4—84no—
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4#12012yes—
214C2×C3×C4C22×D412yes—
215C2×C3×C4C22×D411yescharacteristic
216C3×C8—24no—
217C3×C8—44no—
218C24—41no—
219#41—44no—
220C2×Q8—24no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—84no—
224C2×Q8C2×C3×D412yes—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—24no—
228C2×D4—22no—
229C2×D4—44no—
230C2×D4—48no—
231C2×D4—41no—
232C2×D4—42no—
233C2×D4C2×C3×D412yes—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4C2×C3×D412yes—
237C22×C4C2×C3×D411yescharacteristic
238SD16—48no—
239D8—48no—
240M16—24no—
241M16—44no—
242C2×C8—22no—
243C2×C8—24no—
244C4⋊C4—28no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4C2×C3×D414yes—
251C4⋊C4C2×C3×D414yes—
252C22⋊C4—24no—
253C22⋊C4—28no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C42—24no—
257C42—24no—
258C42C2×C3×D412yes—
259C42C2×C3×D411yescharacteristic
260C42C24×C311yescharacteristic
261C22×C3—21no—
262C22×C3—22no—
263C22×C3—42no—
264C22×C3—44no—
265C22×C3—82no—
266C22×C3—84no—
267C22×C3—164no—
268C22×C3C2×#12011yescharacteristic
269C3×C4—21no—
270C3×C4—22no—
271C3×C4—42no—
272C3×C4—44no—
273C3×C4—44no—
274C3×C4#45312yes—
275C3×C4#45414yes—
276C23—22no—
277C23—42no—
278C23—82no—
279C23—84no—
280C23C3×#12011yescharacteristic
281Q8—42no—
282Q8—44no—
283Q8—44no—
284Q8—84no—
285D4—42no—
286D4—44no—
287D4—48no—
288D4—44no—
289D4—82no—
290D4—82no—
291D4—84no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—44no—
298C2×C4—84no—
299C2×C4C3×#12011yescharacteristic
300C2×C4C3×#12012yes—
301C2×C4C22×C3×D412yes—
302C2×C4C22×C3×D411yescharacteristic
303C8—24no—
304C8—44no—
305C2×C3—41no—
306C2×C3—82no—
307C2×C3—162no—
308C2×C3C2×#45311yescharacteristic
309C2×C3C2×#45412yes—
310C22—21no—
311C22—22no—
312C22—42no—
313C22—44no—
314C22—82no—
315C22—84no—
316C22—164no—
317C22C2×C3×#12011yescharacteristic
318C4—21no—
319C4—22no—
320C4—42no—
321C4—44no—
322C4—44no—
323C4C3×#45312yes—
324C4C3×#45414yes—
325C3#1907711yescharacteristic, direct product factor, Sylow
326C2—41no—
327C2—82no—
328C2—162no—
329C2C2×C3×#45311yescharacteristic
330C2C2×C3×#45412yes—
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC3×#3151C2
LCSG ▷ 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×#3550C2semidirect
3. C3×#3138C2semidirect
4. C3×#1657C2semidirect
5. C3×#1526C2semidirect
6. C3×#1525C2semidirect
7. #19077C3direct (N × H)
9. C3×#563C22semidirect
28. C3×#330C22semidirect
29. C3×#329C22semidirect
30. #3550C2×C3semidirect
32. #3138C2×C3semidirect
33. #1657C2×C3semidirect
34. #1526C2×C3semidirect
35. #1525C2×C3semidirect
81. #563C22×C3semidirect
100. #330C22×C3semidirect
101. #329C22×C3semidirect
325. C3#19077direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35502
C3×#31512
C3×#31382
C3×#16572
C3×#15262
C3×#15252
#190773