Finite Group Databasebeta

← C3×#25647 #10644254

Subgroups — C3×#25647 #10644254

Order 768.

1022 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic
2C2×C3×#487C211yesmaximal, characteristic
3C2×C3×#475C211yesmaximal, characteristic
4C3×#2546C211yesmaximal, characteristic
5C3×#2043C214yesmaximal
6C3×#1984C211yesmaximal, characteristic
7C3×#1981C211yesmaximal, characteristic
8C3×#1865C214yesmaximal
9C3×C4×#103C211yesmaximal, characteristic
10#25647C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#557—24no—
12C3×#557—24no—
13C2×C3×C4⋊Q8—21no—
14C2×C3×C4⋊Q8C2212yes—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#125—21no—
18C2×C3×#125—22no—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8C2212yes—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C3×#487—24no—
24C3×#475—24no—
25C2×C3×#106C2212yes—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103C2214yes—
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#103C2214yes—
31C2×C3×C4×C8C2211yescharacteristic
32C3×#398—22no—
33C3×#391—21no—
34C3×#389C2214yes—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#340C2212yes—
37C3×#336C2214yes—
38C2×#557C2×C311yescharacteristic
39C2×#487C2×C311yescharacteristic
40C2×#475C2×C311yescharacteristic
41#2546C2×C311yescharacteristic
42#2043C2×C314yes—
43#1984C2×C311yescharacteristic
44#1981C2×C311yescharacteristic
45#1865C2×C314yes—
46C4×#103C2×C311yescharacteristic
47C22×C3×Q8—21no—
48C22×C3×Q8D411yescharacteristic
49C22×C3×Q8C2312yes—
50C22×C3×Q8C2311yescharacteristic
51C22×C3×C8C2312yes—
52C22×C3×C8C2312yes—
53C3×C4⋊Q8—24no—
54C3×C4⋊Q8—24no—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8—22no—
57C3×C4⋊Q8—22no—
58C3×C4⋊Q8—44no—
59C3×C4⋊Q8D414yes—
60C3×#125—216no—
61C3×#125—28no—
62C3×C4×Q8—28no—
63C3×C4×Q8—24no—
64C3×C4×Q8—24no—
65C3×C4×Q8—22no—
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C3×C4×Q8D414yes—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4D411yescharacteristic
78C2×C3×C4⋊C4D412yes—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C42—22no—
83C2×C3×C42—21no—
84C2×C3×C42—21no—
85C2×C3×C42C2312yes—
86C2×C3×C42C2311yescharacteristic
87C3×#106—24no—
88C3×C4⋊C8—22no—
89C3×#103—24no—
90C3×#103—28no—
91C3×#103—24no—
92C3×#103—28no—
93C3×C4×C8—22no—
94C3×#95—21no—
95C3×#95—24no—
96#557—24no—
97#557—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C22×C312yes—
100C2×C4⋊Q8C22×C311yescharacteristic
101C2×C4⋊Q8C22×C311yescharacteristic
102C2×#125—21no—
103C2×#125—22no—
104C2×C4×Q8—21no—
105C2×C4×Q8C22×C312yes—
106C2×C4×Q8C22×C311yescharacteristic
107C2×C4×Q8C22×C311yescharacteristic
108#487—24no—
109#475—24no—
110C2×#106C22×C312yes—
111C2×C4⋊C8C22×C311yescharacteristic
112C2×#103C22×C312yes—
113C2×#103C22×C314yes—
114C2×#103C22×C314yes—
115C2×#103C22×C312yes—
116C2×C4×C8C22×C311yescharacteristic
117#398—22no—
118#391—21no—
119#389C22×C314yes—
120C4×C4⋊C4C22×C311yescharacteristic
121#340C22×C312yes—
122#336C22×C314yes—
123C2×C3×Q8—22no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8—21no—
127C2×C3×Q8—22no—
128C2×C3×Q8—24no—
129C2×C3×Q8—22no—
130C2×C3×Q8—24no—
131C2×C3×Q8—42no—
132C2×C3×Q8—42no—
133C2×C3×Q8—42no—
134C2×C3×Q8SD1614yes—
135C2×C3×Q8C2×D414yes—
136C2×C3×Q8C2×D412yes—
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2411yescharacteristic
147C2×C3×C8—22no—
148C2×C3×C8—22no—
149C2×C3×C8—24no—
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C3×C4⋊C4—28no—
153C3×C4⋊C4—28no—
154C3×C4⋊C4—28no—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4—24no—
158C3×C4⋊C4—24no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—42no—
165C3×C4⋊C4—48no—
166C3×C4⋊C4C2×D412yes—
167C3×C4⋊C4#4114yes—
168C3×C42—28no—
169C3×C42—24no—
170C3×C42—22no—
171C3×C42—22no—
172C3×C42—42no—
173C3×C42C2×D412yes—
174C3×C42C2×D412yes—
175C22×Q8—21no—
176C22×Q8C3×D411yescharacteristic
177C22×Q8C23×C312yes—
178C22×Q8C23×C311yescharacteristic
179C22×C8C23×C312yes—
180C22×C8C23×C312yes—
181C4⋊Q8—24no—
182C4⋊Q8—24no—
183C4⋊Q8—22no—
184C4⋊Q8—22no—
185C4⋊Q8—22no—
186C4⋊Q8—44no—
187C4⋊Q8C3×D414yes—
188#125—216no—
189#125—28no—
190C4×Q8—28no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8C3×D414yes—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C3×D411yescharacteristic
206C2×C4⋊C4C3×D412yes—
207C2×C4⋊C4C23×C312yes—
208C2×C4⋊C4C23×C311yescharacteristic
209C2×C4⋊C4C23×C312yes—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42C23×C312yes—
214C2×C42C23×C311yescharacteristic
215#106—24no—
216C4⋊C8—22no—
217#103—24no—
218#103—28no—
219#103—28no—
220#103—24no—
221C4×C8—22no—
222#95—21no—
223#95—24no—
224C23×C3C22×D411yescharacteristic
225C3×Q8—28no—
226C3×Q8—22no—
227C3×Q8—22no—
228C3×Q8—28no—
229C3×Q8—42no—
230C3×Q8—42no—
231C3×Q8—42no—
232C3×Q8—84no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—24no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—42no—
243C2×C3×C4—44no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4#12012yes—
251C2×C3×C4#12012yes—
252C2×C3×C4C2×SD1614yes—
253C2×C3×C4C2×SD1612yes—
254C2×C3×C4#13712yes—
255C2×C3×C4C22×D411yescharacteristic
256C2×C3×C4C22×D411yescharacteristic
257C2×C3×C4C2×#4112yes—
258C2×C3×C42−1+412yes—
259C3×C8—42no—
260C3×C8—42no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—21no—
265C2×Q8—22no—
266C2×Q8—24no—
267C2×Q8—22no—
268C2×Q8—24no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8C3×SD1614yes—
273C2×Q8C2×C3×D414yes—
274C2×Q8C2×C3×D412yes—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4C2×C3×D412yes—
281C22×C4C2×C3×D411yescharacteristic
282C22×C4C2×C3×D412yes—
283C22×C4C2×C3×D411yescharacteristic
284C22×C4C24×C311yescharacteristic
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—24no—
288C2×C8—22no—
289C2×C8—22no—
290C4⋊C4—28no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—21no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—42no—
303C4⋊C4—48no—
304C4⋊C4C2×C3×D412yes—
305C4⋊C4C3×#4114yes—
306C42—28no—
307C42—24no—
308C42—22no—
309C42—22no—
310C42—42no—
311C42C2×C3×D412yes—
312C42C2×C3×D412yes—
313C22×C3C2×#12011yescharacteristic
314C22×C3#54812yes—
315C22×C3C22×SD1611yescharacteristic
316C22×C3C2×#13711yescharacteristic
317C22×C3#57812yes—
318C3×C4—24no—
319C3×C4—22no—
320C3×C4—44no—
321C3×C4—42no—
322C3×C4—44no—
323C3×C4—42no—
324C3×C4—42no—
325C3×C4—82no—
326C3×C4#44814yes—
327C3×C4#53211yescharacteristic
328C3×C4#53611yescharacteristic
329C3×C4#54712yes—
330C23C22×C3×D411yescharacteristic
331Q8—28no—
332Q8—22no—
333Q8—22no—
334Q8—28no—
335Q8—42no—
336Q8—42no—
337Q8—42no—
338Q8—84no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4C3×#12012yes—
357C2×C4C3×#12012yes—
358C2×C4C2×C3×SD1614yes—
359C2×C4C2×C3×SD1612yes—
360C2×C4C3×#13712yes—
361C2×C4C22×C3×D411yescharacteristic
362C2×C4C22×C3×D411yescharacteristic
363C2×C4C2×C3×#4112yes—
364C2×C4C3×2−1+412yes—
365C8—42no—
366C8—42no—
367C2×C3#243211yescharacteristic
368C2×C3C2×#44811yescharacteristic
369C2×C3#301111yescharacteristic
370C2×C3#335912yes—
371C2×C3#337412yes—
372C22C2×C3×#12011yescharacteristic
373C22C3×#54812yes—
374C22C22×C3×SD1611yescharacteristic
375C22C2×C3×#13711yescharacteristic
376C22C3×#57812yes—
377C4—24no—
378C4—22no—
379C4—42no—
380C4—44no—
381C4—42no—
382C4—42no—
383C4—44no—
384C4—82no—
385C4C3×#44814yes—
386C4C3×#53211yescharacteristic
387C4C3×#53611yescharacteristic
388C4C3×#54712yes—
389C3#2564711yescharacteristic, direct product factor, Sylow
390C2C3×#243211yescharacteristic
391C2C2×C3×#44811yescharacteristic
392C2C3×#301111yescharacteristic
393C2C3×#335912yes—
394C2C3×#337412yes—
subgroupG/H
center, socle, prime radicalC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
10. #25647C3direct (N × H)
389. C3#25647direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5572
C2×C3×#4872
C2×C3×#4752
C3×#25462
C3×#20432
C3×#19842
C3×#19812
C3×#18652
C3×C4×#1032
#256473