Finite Group Databasebeta

← C3×#25835 #10644442

Subgroups — C3×#25835 #10644442

Order 768.

1082 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
1C2×C3×#554C211yesmaximal, characteristic
2C2×C3×#498C211yesmaximal, characteristic, complemented
3C2×C3×#474C211yesmaximal, characteristic
4C3×#2546C211yesmaximal, characteristic, complemented
5C3×#2059C214yesmaximal, complemented
6C3×#1987C211yesmaximal, characteristic, complemented
7C3×#1984C211yesmaximal, characteristic, complemented
8C3×#1864C214yesmaximal
9C3×C4×#102C211yesmaximal, characteristic
10#25835C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#554—24no—
12C3×#554—24no—
13C2×C3×C4⋊Q8—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125C2212yes—
17C2×C3×C22⋊Q8—21no—
18C2×C3×C22⋊Q8—22no—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4C2212yes—
21C2×C3×C4×D4C2211yescharacteristic
22C22×C3×C4⋊C4—21no—
23C3×#498—24no—
24C3×#474—24no—
25C2×C3×#107C2212yes—
26C2×C3×#106C2214yes—
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×C4×C8C2211yescharacteristic
31C3×#398—22no—
32C3×#391—21no—
33C3×#389C2214yes—
34C3×C4×C4⋊C4C2211yescharacteristic
35C3×#340C2212yes—
36C3×#340C2214yes—
37C3×#336C2214yes—
38C2×#554C2×C311yescharacteristic
39C2×#498C2×C311yescharacteristic, complemented
40C2×#474C2×C311yescharacteristic
41#2546C2×C311yescharacteristic, complemented
42#2059C2×C314yescomplemented
43#1987C2×C311yescharacteristic, complemented
44#1984C2×C311yescharacteristic, complemented
45#1864C2×C314yes—
46C4×#102C2×C311yescharacteristic
47C22×C3×Q8—21no—
48C22×C3×Q8—21no—
49C22×C3×Q8D411yescharacteristic
50C22×C3×D4C2311yescharacteristic
51C23×C3×C4—21no—
52C23×C3×C4—22no—
53C22×C3×C8C2312yes—
54C22×C3×C8C2312yes—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8—22no—
57C3×C4⋊Q8—22no—
58C3×C4⋊Q8—44no—
59C3×C4⋊Q8D414yes—
60C3×#125—24no—
61C3×#125—24no—
62C3×C22⋊Q8—28no—
63C3×C22⋊Q8—216no—
64C3×C4×Q8—22no—
65C3×C4×Q8—22no—
66C3×C4×D4—24no—
67C3×C4×D4—24no—
68C3×C4×D4—22no—
69C3×C4×D4D414yes—
70C2×C3×C4⋊C4—28no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
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×C22⋊C4—22no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C42—21no—
85C2×C3×C42C2312yes—
86C2×C3×C42C2311yescharacteristic
87C3×#107—24no—
88C3×#106—28no—
89C3×C4⋊C8—22no—
90C3×#102—24no—
91C3×#102Q818yescomplemented
92C3×C4×C8—22no—
93C3×#95—21no—
94C3×#95—24no—
95#554—24no—
96#554—24no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C22×C311yescharacteristic
99C2×C4⋊Q8C22×C311yescharacteristic
100C2×#125C22×C312yes—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—22no—
103C2×C4×Q8C22×C311yescharacteristic
104C2×C4×D4C22×C312yes—
105C2×C4×D4C22×C311yescharacteristic
106C22×C4⋊C4—21no—
107#498—24no—
108#474—24no—
109C2×#107C22×C312yes—
110C2×#106C22×C314yes—
111C2×C4⋊C8C22×C311yescharacteristic
112C2×#102C22×C312yes—
113C2×#102C22×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—
122C24×C3—21no—
123C2×C3×Q8—22no—
124C2×C3×Q8—24no—
125C2×C3×Q8—22no—
126C2×C3×Q8—22no—
127C2×C3×Q8—42no—
128C2×C3×Q8—42no—
129C2×C3×Q8—42no—
130C2×C3×Q8—82no—
131C2×C3×D4—21no—
132C2×C3×D4—22no—
133C2×C3×D4C2×D412yes—
134C2×C3×D4C2×Q814yes—
135C22×C3×C4—22no—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4—28no—
139C22×C3×C4—24no—
140C22×C3×C4—22no—
141C22×C3×C4—42no—
142C22×C3×C4—42no—
143C22×C3×C4—84no—
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2411yescharacteristic
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C2×C3×C8—24no—
153C2×C3×C8C2×Q814yes—
154C3×C4⋊C4—28no—
155C3×C4⋊C4—28no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—24no—
158C3×C4⋊C4—24no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—48no—
165C3×C4⋊C4—42no—
166C3×C4⋊C4C2×D414yes—
167C3×C4⋊C4C2×D412yes—
168C3×C4⋊C4C2×Q814yes—
169C3×C22⋊C4—28no—
170C3×C22⋊C4—24no—
171C3×C42—22no—
172C3×C42—22no—
173C3×C42—42no—
174C3×C42C2×D412yes—
175C3×C42C2×D412yes—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8C3×D411yescharacteristic
179C22×D4C23×C311yescharacteristic
180C23×C4—21no—
181C23×C4—22no—
182C22×C8C23×C312yes—
183C22×C8C23×C312yes—
184C4⋊Q8—22no—
185C4⋊Q8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—44no—
188C4⋊Q8C3×D414yes—
189#125—24no—
190#125—24no—
191C22⋊Q8—28no—
192C22⋊Q8—216no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×D4—24no—
196C4×D4—24no—
197C4×D4—22no—
198C4×D4C3×D414yes—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
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×C4⋊C4C23×C312yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C42—21no—
214C2×C42C23×C312yes—
215C2×C42C23×C311yescharacteristic
216#107—24no—
217#106—28no—
218C4⋊C8—22no—
219#102—24no—
220#102C3×Q818yescomplemented
221C4×C8—22no—
222#95—21no—
223#95—24no—
224C23×C3—22no—
225C23×C3—24no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—42no—
229C23×C3C22×D411yescharacteristic
230C3×Q8—44no—
231C3×Q8—84no—
232C3×Q8—84no—
233C3×D4—22no—
234C3×D4—24no—
235C3×D4C4⋊Q814yescomplemented
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—24no—
245C2×C3×C4—41no—
246C2×C3×C4—42no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—44no—
250C2×C3×C4—48no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—88no—
255C2×C3×C4#12012yes—
256C2×C3×C4#12012yes—
257C2×C3×C4#13512yes—
258C2×C3×C4AGL(1,ℤ/8)12yes—
259C2×C3×C4C22×D411yescharacteristic
260C2×C3×C4C22×D411yescharacteristic
261C2×C3×C4C22×Q812yes—
262C2×C3×C42−1+412yes—
263C3×C8—24no—
264C3×C8—42no—
265C24—21no—
266C2×Q8—22no—
267C2×Q8—24no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—82no—
274C2×D4—21no—
275C2×D4—22no—
276C2×D4C2×C3×D412yes—
277C2×D4C2×C3×Q814yes—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—28no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—84no—
287C22×C4C2×C3×D412yes—
288C22×C4C2×C3×D412yes—
289C22×C4C2×C3×D411yescharacteristic
290C22×C4C2×C3×D411yescharacteristic
291C22×C4C24×C311yescharacteristic
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—24no—
296C2×C8C2×C3×Q814yes—
297C4⋊C4—28no—
298C4⋊C4—28no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—48no—
308C4⋊C4—42no—
309C4⋊C4C2×C3×D414yes—
310C4⋊C4C2×C3×D412yes—
311C4⋊C4C2×C3×Q814yes—
312C22⋊C4—28no—
313C22⋊C4—24no—
314C42—22no—
315C42—22no—
316C42—42no—
317C42C2×C3×D412yes—
318C42C2×C3×D412yes—
319C22×C3—24no—
320C22×C3—44no—
321C22×C3—44no—
322C22×C3—42no—
323C22×C3—42no—
324C22×C3C2×#12011yescharacteristic
325C22×C3D4×Q812yes—
326C22×C3C2×#13511yescharacteristic
327C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
328C22×C3#57712yes—
329C3×C4—22no—
330C3×C4—44no—
331C3×C4—42no—
332C3×C4—42no—
333C3×C4—42no—
334C3×C4—44no—
335C3×C4—82no—
336C3×C4#44914yes—
337C3×C4C2×C4⋊Q811yescharacteristic
338C3×C4#53611yescharacteristic
339C3×C4D4×Q812yes—
340C23—22no—
341C23—24no—
342C23—41no—
343C23—41no—
344C23—42no—
345C23C22×C3×D411yescharacteristic
346Q8—44no—
347Q8—84no—
348Q8—84no—
349D4—22no—
350D4—24no—
351D4C3×C4⋊Q814yescomplemented
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—48no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—88no—
371C2×C4C3×#12012yes—
372C2×C4C3×#12012yes—
373C2×C4C3×#13512yes—
374C2×C4C3×AGL(1,ℤ/8)12yes—
375C2×C4C22×C3×D411yescharacteristic
376C2×C4C22×C3×D411yescharacteristic
377C2×C4C22×C3×Q812yes—
378C2×C4C3×2−1+412yes—
379C8—24no—
380C8—42no—
381C2×C3—44no—
382C2×C3#243411yescharacteristic
383C2×C3C2×#44911yescharacteristic
384C2×C3#300911yescharacteristic
385C2×C3#338112yes—
386C2×C3#338412yes—
387C22—24no—
388C22—44no—
389C22—42no—
390C22—42no—
391C22—44no—
392C22C2×C3×#12011yescharacteristic
393C22C3×D4×Q812yes—
394C22C2×C3×#13511yescharacteristic
395C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
396C22C3×#57712yes—
397C4—22no—
398C4—44no—
399C4—42no—
400C4—42no—
401C4—42no—
402C4—44no—
403C4—82no—
404C4C3×#44914yes—
405C4C2×C3×C4⋊Q811yescharacteristic
406C4C3×#53611yescharacteristic
407C4C3×D4×Q812yes—
408C3#2583511yescharacteristic, direct product factor, Sylow
409C2—44no—
410C2C3×#243411yescharacteristic
411C2C2×C3×#44911yescharacteristic
412C2C3×#300911yescharacteristic
413C2C3×#338112yes—
414C2C3×#338412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
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
2. C2×C3×#498C2semidirect
4. C3×#2546C2semidirect
5. C3×#2059C2semidirect
6. C3×#1987C2semidirect
7. C3×#1984C2semidirect
10. #25835C3direct (N × H)
39. C2×#498C2×C3semidirect
41. #2546C2×C3semidirect
42. #2059C2×C3semidirect
43. #1987C2×C3semidirect
44. #1984C2×C3semidirect
91. C3×#102Q8semidirect
220. #102C3×Q8semidirect
235. C3×D4C4⋊Q8semidirect
351. D4C3×C4⋊Q8semidirect
408. C3#25835direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5542
C2×C3×#4982
C2×C3×#4742
C3×#25462
C3×#20592
C3×#19872
C3×#19842
C3×#18642
C3×C4×#1022
#258353