Finite Group Databasebeta

← C3×#10065 #10628672

Subgroups — C3×#10065 #10628672

Order 768.

582 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#487C211yesmaximal, characteristic
2C2×C3×#462C211yesmaximal, characteristic
3C3×#2567C211yesmaximal, characteristic
4C3×#1988C211yesmaximal, characteristic
5C2×C3×#333C211yesmaximal, characteristic
6C2×C3×Q8⋊C8C211yesmaximal, characteristic
7C3×#1279C211yesmaximal, characteristic
8#10065C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×Q16—21no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C4×Q8C2211yescharacteristic
13C3×#487—22no—
14C3×#487—22no—
15C3×#462—22no—
16C3×#462—22no—
17C2×C3×C4⋊C8C2211yescharacteristic
18C2×C3×C4⋊C8C2211yescharacteristic
19C2×C3×#103—22no—
20C2×C3×#103—21no—
21C2×C3×C4×C8C2211yescharacteristic
22C3×#395—22no—
23C3×#391—21no—
24C3×#389—22no—
25C3×#384—21no—
26C3×C4×C4⋊C4C2211yescharacteristic
27C3×#340—22no—
28C3×#333—22no—
29C3×Q8⋊C8—22no—
30C2×#487C2×C311yescharacteristic
31C2×#462C2×C311yescharacteristic
32#2567C2×C311yescharacteristic
33#1988C2×C311yescharacteristic
34C2×#333C2×C311yescharacteristic
35C2×Q8⋊C8C2×C311yescharacteristic
36#1279C2×C311yescharacteristic
37C22×C3×Q8—21no—
38C22×C3×Q8D411yescharacteristic
39C22×C3×Q8D412yes—
40C2×C3×Q16—24no—
41C2×C3×Q16—42no—
42C2×C3×Q16—42no—
43C22×C3×C8—21no—
44C22×C3×C8—21no—
45C22×C3×C8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8—21no—
50C3×C4⋊Q8—21no—
51C3×C4⋊Q8—21no—
52C3×C4⋊Q8—21no—
53C3×C4⋊Q8D412yes—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×C4×Q8—21no—
57C3×C4×Q8—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4D411yescharacteristic
65C2×C3×C4⋊C4D412yes—
66C2×C3×C42—21no—
67C2×C3×C42—21no—
68C2×C3×C42C2311yescharacteristic
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×#103—24no—
74C3×#103—28no—
75C3×C4×C8—21no—
76C3×C4×C8D412yes—
77C3×#95—24no—
78C3×#95—21no—
79C22×Q16—21no—
80C2×C4⋊Q8C22×C311yescharacteristic
81C2×C4⋊Q8C22×C311yescharacteristic
82C2×C4×Q8C22×C311yescharacteristic
83#487—22no—
84#487—22no—
85#462—22no—
86#462—22no—
87C2×C4⋊C8C22×C311yescharacteristic
88C2×C4⋊C8C22×C311yescharacteristic
89C2×#103—22no—
90C2×#103—21no—
91C2×C4×C8C22×C311yescharacteristic
92#395—22no—
93#391—21no—
94#389—22no—
95#384—21no—
96C4×C4⋊C4C22×C311yescharacteristic
97#340—22no—
98#333—22no—
99Q8⋊C8—22no—
100C2×C3×Q8—21no—
101C2×C3×Q8—21no—
102C2×C3×Q8—22no—
103C2×C3×Q8—22no—
104C2×C3×Q8—21no—
105C2×C3×Q8—22no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—44no—
109C2×C3×Q8—41no—
110C2×C3×Q8—41no—
111C2×C3×Q8—41no—
112C2×C3×Q8—41no—
113C2×C3×Q8—41no—
114C2×C3×Q8SD1612yes—
115C22×C3×C4—21no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—21no—
119C22×C3×C4—41no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C3×Q16—48no—
124C2×C3×C8—22no—
125C2×C3×C8—22no—
126C2×C3×C8—42no—
127C2×C3×C8—41no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C2×C3×C8—42no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—84no—
145C3×C4⋊C4Q1614yescomplemented
146C3×C42—41no—
147C3×C42—41no—
148C3×C42—41no—
149C3×C42—41no—
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42#4111yescharacteristic
154C22×Q8—21no—
155C22×Q8C3×D411yescharacteristic
156C22×Q8C3×D412yes—
157C2×Q16—24no—
158C2×Q16—42no—
159C2×Q16—42no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8C3×D412yes—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4C3×D411yescharacteristic
182C2×C4⋊C4C3×D412yes—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42C23×C311yescharacteristic
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8C3×D412yes—
190#103—24no—
191#103—28no—
192C4×C8—21no—
193C4×C8C3×D412yes—
194#95—24no—
195#95—21no—
196C23×C3#12011yescharacteristic
197C3×Q8—22no—
198C3×Q8—22no—
199C3×Q8—44no—
200C3×Q8—41no—
201C3×Q8—41no—
202C3×Q8—42no—
203C3×Q8—42no—
204C3×Q8—82no—
205C3×Q8—81no—
206C3×Q8—81no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—24no—
210C2×C3×C4—41no—
211C2×C3×C4—42no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—44no—
219C2×C3×C4—81no—
220C2×C3×C4—81no—
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12711yescharacteristic
227C2×C3×C4C2×SD1611yescharacteristic
228C2×C3×C4C2×Q1612yes—
229C2×C3×C4#13511yescharacteristic
230C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
231C2×C3×C4#13711yescharacteristic
232C2×C3×C4#13712yes—
233C3×C8—42no—
234C3×C8—42no—
235C3×C8—81no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—44no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8C3×SD1612yes—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4C2×C3×D411yescharacteristic
257C22×C4C2×C3×D411yescharacteristic
258C22×C4C2×C3×D411yescharacteristic
259Q16—48no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—42no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—42no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—21no—
270C4⋊C4—22no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—84no—
281C4⋊C4C3×Q1614yescomplemented
282C42—41no—
283C42—41no—
284C42—41no—
285C42—41no—
286C42C2×C3×D411yescharacteristic
287C42C2×C3×D411yescharacteristic
288C42C2×C3×D411yescharacteristic
289C42C3×#4111yescharacteristic
290C22×C3#39211yescharacteristic
291C22×C3#44811yescharacteristic
292C22×C3#44911yescharacteristic
293C22×C3#45112yes—
294C22×C3#45512yes—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—44no—
298C3×C4—42no—
299C3×C4—82no—
300C3×C4—81no—
301C3×C4—82no—
302C3×C4#45011yescharacteristic
303C3×C4#45211yescharacteristic
304C3×C4#46212yes—
305C3×C4#46511yescharacteristic
306C3×C4#46811yescharacteristic
307C3×C4#49411yescharacteristic
308C3×C4#49711yescharacteristic
309C23C3×#12011yescharacteristic
310Q8—22no—
311Q8—22no—
312Q8—44no—
313Q8—41no—
314Q8—41no—
315Q8—42no—
316Q8—42no—
317Q8—82no—
318Q8—81no—
319Q8—81no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—44no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4C3×#12011yescharacteristic
335C2×C4C3×#12011yescharacteristic
336C2×C4C3×#12111yescharacteristic
337C2×C4C3×#12111yescharacteristic
338C2×C4C3×#12111yescharacteristic
339C2×C4C3×#12711yescharacteristic
340C2×C4C2×C3×SD1611yescharacteristic
341C2×C4C2×C3×Q1612yes—
342C2×C4C3×#13511yescharacteristic
343C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
344C2×C4C3×#13711yescharacteristic
345C2×C4C3×#13712yes—
346C8—42no—
347C8—42no—
348C8—81no—
349C2×C3#164412yes—
350C2×C3#164512yes—
351C2×C3#200011yescharacteristic
352C2×C3#200411yescharacteristic
353C2×C3#201611yescharacteristic
354C22C3×#39211yescharacteristic
355C22C3×#44811yescharacteristic
356C22C3×#44911yescharacteristic
357C22C3×#45112yes—
358C22C3×#45512yes—
359C4—21no—
360C4—21no—
361C4—42no—
362C4—44no—
363C4—82no—
364C4—82no—
365C4—81no—
366C4C3×#45011yescharacteristic
367C4C3×#45211yescharacteristic
368C4C3×#46212yes—
369C4C3×#46511yescharacteristic
370C4C3×#46811yescharacteristic
371C4C3×#49411yescharacteristic
372C4C3×#49711yescharacteristic
373C3#1006511yescharacteristic, direct product factor, Sylow
374C2C3×#164412yes—
375C2C3×#164512yes—
376C2C3×#200011yescharacteristic
377C2C3×#200411yescharacteristic
378C2C3×#201611yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C3#120
commutator, Frattini, cosocleC2×C42C23×C3
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
8. #10065C3direct (N × H)
145. C3×C4⋊C4Q16semidirect
281. C4⋊C4C3×Q16semidirect
373. C3#10065direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4872
C2×C3×#4622
C3×#25672
C3×#19882
C2×C3×#3332
C2×C3×Q8⋊C82
C3×#12792
#100653