Finite Group Databasebeta

← C3×#25837 #10644444

Subgroups — C3×#25837 #10644444

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q82C211yesmaximal, characteristic
2C2×C3×#498C211yesmaximal, characteristic
3C2×C3×#475C211yesmaximal, characteristic
4C3×#2546C211yesmaximal, characteristic
5C3×#2058C214yesmaximal
6C3×#1987C211yesmaximal, characteristic
7C3×#1984C211yesmaximal, characteristic
8C3×#1865C214yesmaximal
9C3×C4×#103C211yesmaximal, characteristic
10#25837C311yesmaximal, characteristic, direct product factor, Sylow
11C3×Q82—24no—
12C3×Q82—24no—
13C2×C3×C4⋊Q8—21no—
14C2×C3×C4⋊Q8—21no—
15C2×C3×C4⋊Q8—22no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×C4⋊Q8C2212yes—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8C2212yes—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C3×#498—24no—
24C3×#475—24no—
25C2×C3×#106C2214yes—
26C2×C3×#106C2212yes—
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×C4×C8C2211yescharacteristic
31C3×#398—22no—
32C3×#391—21no—
33C3×#389C2214yes—
34C3×C4×C4⋊C4C2211yescharacteristic
35C3×#340C2212yes—
36C3×#340C2214yes—
37C3×#336C2214yes—
38C2×Q82C2×C311yescharacteristic
39C2×#498C2×C311yescharacteristic
40C2×#475C2×C311yescharacteristic
41#2546C2×C311yescharacteristic
42#2058C2×C314yes—
43#1987C2×C311yescharacteristic
44#1984C2×C311yescharacteristic
45#1865C2×C314yes—
46C4×#103C2×C311yescharacteristic
47C22×C3×Q8—22no—
48C22×C3×Q8—21no—
49C22×C3×Q8—21no—
50C22×C3×Q8D411yescharacteristic
51C22×C3×Q8C2311yescharacteristic
52C22×C3×C8C2312yes—
53C22×C3×C8C2312yes—
54C3×C4⋊Q8—216no—
55C3×C4⋊Q8—28no—
56C3×C4⋊Q8—24no—
57C3×C4⋊Q8—24no—
58C3×C4⋊Q8—22no—
59C3×C4⋊Q8—22no—
60C3×C4⋊Q8—22no—
61C3×C4⋊Q8—44no—
62C3×C4⋊Q8D414yes—
63C3×C4×Q8—28no—
64C3×C4×Q8—24no—
65C3×C4×Q8—24no—
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C3×C4×Q8—22no—
69C3×C4×Q8D414yes—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4D411yescharacteristic
77C2×C3×C4⋊C4D412yes—
78C2×C3×C4⋊C4C2312yes—
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×#106—28no—
89C3×C4⋊C8—22no—
90C3×#103—24no—
91C3×#103Q818yescomplemented
92C3×C4×C8—22no—
93C3×#95—21no—
94C3×#95—24no—
95Q82—24no—
96Q82—24no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8C22×C311yescharacteristic
101C2×C4⋊Q8C22×C312yes—
102C2×C4⋊Q8C22×C311yescharacteristic
103C2×C4×Q8—21no—
104C2×C4×Q8C22×C312yes—
105C2×C4×Q8C22×C311yescharacteristic
106C2×C4×Q8C22×C311yescharacteristic
107#498—24no—
108#475—24no—
109C2×#106C22×C312yes—
110C2×#106C22×C314yes—
111C2×C4⋊C8C22×C311yescharacteristic
112C2×#103C22×C312yes—
113C2×#103C22×C312yes—
114C2×C4×C8C22×C311yescharacteristic
115#398—22no—
116#391—21no—
117#389C22×C314yes—
118C4×C4⋊C4C22×C311yescharacteristic
119#340C22×C312yes—
120#340C22×C314yes—
121#336C22×C314yes—
122C2×C3×Q8—28no—
123C2×C3×Q8—22no—
124C2×C3×Q8—24no—
125C2×C3×Q8—22no—
126C2×C3×Q8—21no—
127C2×C3×Q8—22no—
128C2×C3×Q8—42no—
129C2×C3×Q8—42no—
130C2×C3×Q8—42no—
131C2×C3×Q8—82no—
132C2×C3×Q8—84no—
133C2×C3×Q8SD1614yes—
134C2×C3×Q8C2×D412yes—
135C2×C3×Q8C2×Q814yes—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2411yescharacteristic
146C2×C3×C8—22no—
147C2×C3×C8—22no—
148C2×C3×C8—22no—
149C2×C3×C8—24no—
150C2×C3×C8C2×Q814yes—
151C3×C4⋊C4—28no—
152C3×C4⋊C4—28no—
153C3×C4⋊C4—24no—
154C3×C4⋊C4—24no—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—42no—
163C3×C4⋊C4—48no—
164C3×C4⋊C4C2×D414yes—
165C3×C4⋊C4C2×D412yes—
166C3×C4⋊C4C2×Q814yes—
167C3×C42—28no—
168C3×C42—24no—
169C3×C42—22no—
170C3×C42—22no—
171C3×C42—42no—
172C3×C42C2×D412yes—
173C3×C42C2×D412yes—
174C22×Q8—22no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8C3×D411yescharacteristic
178C22×Q8C23×C311yescharacteristic
179C22×C8C23×C312yes—
180C22×C8C23×C312yes—
181C4⋊Q8—216no—
182C4⋊Q8—28no—
183C4⋊Q8—24no—
184C4⋊Q8—24no—
185C4⋊Q8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—44no—
189C4⋊Q8C3×D414yes—
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—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4C3×D411yescharacteristic
204C2×C4⋊C4C3×D412yes—
205C2×C4⋊C4C23×C312yes—
206C2×C4⋊C4C23×C312yes—
207C2×C4⋊C4C23×C311yescharacteristic
208C2×C4⋊C4C23×C312yes—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42C23×C312yes—
213C2×C42C23×C311yescharacteristic
214#106—24no—
215#106—28no—
216C4⋊C8—22no—
217#103—24no—
218#103C3×Q818yescomplemented
219C4×C8—22no—
220#95—21no—
221#95—24no—
222C23×C3C22×D411yescharacteristic
223C3×Q8—22no—
224C3×Q8—24no—
225C3×Q8—28no—
226C3×Q8—84no—
227C3×Q8—88no—
228C3×Q8—84no—
229C3×Q8C4⋊Q814yescomplemented
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—24no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—24no—
237C2×C3×C4—22no—
238C2×C3×C4—24no—
239C2×C3×C4—24no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—44no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4#12012yes—
248C2×C3×C4#12012yes—
249C2×C3×C4C2×SD1614yes—
250C2×C3×C4C2×SD1612yes—
251C2×C3×C4#13712yes—
252C2×C3×C4C22×D411yescharacteristic
253C2×C3×C4C22×D411yescharacteristic
254C2×C3×C4C22×Q812yes—
255C2×C3×C42−1+412yes—
256C3×C8—24no—
257C3×C8—42no—
258C2×Q8—28no—
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—21no—
263C2×Q8—22no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—82no—
268C2×Q8—84no—
269C2×Q8C3×SD1614yes—
270C2×Q8C2×C3×D412yes—
271C2×Q8C2×C3×Q814yes—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4C2×C3×D412yes—
278C22×C4C2×C3×D411yescharacteristic
279C22×C4C2×C3×D412yes—
280C22×C4C2×C3×D411yescharacteristic
281C22×C4C24×C311yescharacteristic
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—24no—
286C2×C8C2×C3×Q814yes—
287C4⋊C4—28no—
288C4⋊C4—28no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—42no—
299C4⋊C4—48no—
300C4⋊C4C2×C3×D414yes—
301C4⋊C4C2×C3×D412yes—
302C4⋊C4C2×C3×Q814yes—
303C42—28no—
304C42—24no—
305C42—22no—
306C42—22no—
307C42—42no—
308C42C2×C3×D412yes—
309C42C2×C3×D412yes—
310C22×C3C2×#12011yescharacteristic
311C22×C3D4×Q812yes—
312C22×C3C22×SD1611yescharacteristic
313C22×C3C2×#13711yescharacteristic
314C22×C3#57712yes—
315C3×C4—24no—
316C3×C4—24no—
317C3×C4—22no—
318C3×C4—44no—
319C3×C4—42no—
320C3×C4—44no—
321C3×C4—42no—
322C3×C4—82no—
323C3×C4#44814yes—
324C3×C4C2×C4⋊Q811yescharacteristic
325C3×C4#53611yescharacteristic
326C3×C4D4×Q812yes—
327C23C22×C3×D411yescharacteristic
328Q8—28no—
329Q8—22no—
330Q8—24no—
331Q8—84no—
332Q8—88no—
333Q8—84no—
334Q8C3×C4⋊Q814yescomplemented
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4C3×#12012yes—
353C2×C4C3×#12012yes—
354C2×C4C2×C3×SD1614yes—
355C2×C4C2×C3×SD1612yes—
356C2×C4C3×#13712yes—
357C2×C4C22×C3×D411yescharacteristic
358C2×C4C22×C3×D411yescharacteristic
359C2×C4C22×C3×Q812yes—
360C2×C4C3×2−1+412yes—
361C8—24no—
362C8—42no—
363C2×C3#243411yescharacteristic
364C2×C3C2×#44811yescharacteristic
365C2×C3#300911yescharacteristic
366C2×C3Q8×SD1612yes—
367C2×C3#338712yes—
368C22C2×C3×#12011yescharacteristic
369C22C3×D4×Q812yes—
370C22C22×C3×SD1611yescharacteristic
371C22C2×C3×#13711yescharacteristic
372C22C3×#57712yes—
373C4—24no—
374C4—24no—
375C4—22no—
376C4—42no—
377C4—44no—
378C4—42no—
379C4—44no—
380C4—82no—
381C4C3×#44814yes—
382C4C2×C3×C4⋊Q811yescharacteristic
383C4C3×#53611yescharacteristic
384C4C3×D4×Q812yes—
385C3#2583711yescharacteristic, direct product factor, Sylow
386C2C3×#243411yescharacteristic
387C2C2×C3×#44811yescharacteristic
388C2C3×#300911yescharacteristic
389C2C3×Q8×SD1612yes—
390C2C3×#338712yes—
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. #25837C3direct (N × H)
91. C3×#103Q8semidirect
218. #103C3×Q8semidirect
229. C3×Q8C4⋊Q8semidirect
334. Q8C3×C4⋊Q8semidirect
385. C3#25837direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q822
C2×C3×#4982
C2×C3×#4752
C3×#25462
C3×#20582
C3×#19872
C3×#19842
C3×#18652
C3×C4×#1032
#258373