Finite Group Databasebeta

← C3×#25702 #10644309

Subgroups — C3×#25702 #10644309

Order 768.

1102 conjugacy classes of subgroups.

The subgroup lattice has 418 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×#475C211yesmaximal, characteristic, complemented
4C3×#2547C211yesmaximal, characteristic
5C3×#2035C214yesmaximal, complemented
6C3×#1987C211yesmaximal, characteristic, complemented
7C3×#1954C211yesmaximal, characteristic
8C3×#1853C214yesmaximal, complemented
9C3×C4×#103C211yesmaximal, characteristic, complemented
10#25702C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#554—24no—
12C3×#554—24no—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#125—21no—
16C2×C3×#124—21no—
17C2×C3×C22⋊Q8—22no—
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4—21no—
21C2×C3×C4×D4C2211yescharacteristic
22C22×C3×C4⋊C4C2212yes—
23C3×#498—24no—
24C3×#475—24no—
25C2×C3×#106C2214yes—
26C2×C3×#106C2212yes—
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×C22⋊C8C2214yes—
31C2×C3×C4×C8C2211yescharacteristic
32C3×#397—22no—
33C3×#391—21no—
34C3×#380C2214yes—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#340C2212yes—
37C3×#340C2214yes—
38C2×#554C2×C311yescharacteristic
39C2×#498C2×C311yescharacteristic, complemented
40C2×#475C2×C311yescharacteristic, complemented
41#2547C2×C311yescharacteristic
42#2035C2×C314yescomplemented
43#1987C2×C311yescharacteristic, complemented
44#1954C2×C311yescharacteristic
45#1853C2×C314yescomplemented
46C4×#103C2×C311yescharacteristic, complemented
47C22×C3×Q8—21no—
48C22×C3×Q8—21no—
49C22×C3×Q8C2311yescharacteristic
50C22×C3×D4D411yescharacteristic
51C23×C3×C4—22no—
52C23×C3×C4C2312yes—
53C22×C3×C8C2312yes—
54C22×C3×C8C2312yes—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8—22no—
57C3×C4⋊Q8—22no—
58C3×C4⋊Q8D414yes—
59C3×#125—28no—
60C3×#124—44no—
61C3×C22⋊Q8—216no—
62C3×C22⋊Q8—24no—
63C3×C22⋊Q8—24no—
64C3×C4×Q8—22no—
65C3×C4×Q8D414yes—
66C3×C4×D4—28no—
67C3×C4×D4—22no—
68C3×C4×D4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—24no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4D411yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4D412yes—
84C2×C3×C42—21no—
85C2×C3×C42—21no—
86C2×C3×C42C2311yescharacteristic
87C3×#106—24no—
88C3×#106—28no—
89C3×C4⋊C8—22no—
90C3×#103—24no—
91C3×#103—24no—
92C3×C22⋊C8—28no—
93C3×C4×C8—22no—
94C3×#95—21no—
95C3×#95—24no—
96#554—24no—
97#554—24no—
98C2×C4⋊Q8C22×C311yescharacteristic
99C2×C4⋊Q8C22×C311yescharacteristic
100C2×#125—21no—
101C2×#124—21no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8C22×C312yes—
104C2×C4×Q8C22×C311yescharacteristic
105C2×C4×D4—21no—
106C2×C4×D4C22×C311yescharacteristic
107C22×C4⋊C4C22×C312yes—
108#498—24no—
109#475—24no—
110C2×#106C22×C312yes—
111C2×#106C22×C314yes—
112C2×C4⋊C8C22×C311yescharacteristic
113C2×#103C22×C312yes—
114C2×#103C22×C312yes—
115C2×C22⋊C8C22×C314yes—
116C2×C4×C8C22×C311yescharacteristic
117#397—22no—
118#391—21no—
119#380C22×C314yes—
120C4×C4⋊C4C22×C311yescharacteristic
121#340C22×C312yes—
122#340C22×C314yes—
123C24×C3C2×D412yes—
124C2×C3×Q8—24no—
125C2×C3×Q8—21no—
126C2×C3×Q8—22no—
127C2×C3×Q8—22no—
128C2×C3×Q8—42no—
129C2×C3×Q8—42no—
130C2×C3×Q8—42no—
131C2×C3×Q8—82no—
132C2×C3×Q8C2×D412yes—
133C2×C3×D4—22no—
134C2×C3×D4—22no—
135C2×C3×D4SD1614yes—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—22no—
139C22×C3×C4—28no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—24no—
143C22×C3×C4—24no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—84no—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2411yescharacteristic
151C2×C3×C8—22no—
152C2×C3×C8—22no—
153C2×C3×C8—22no—
154C2×C3×C8—24no—
155C2×C3×C8#4114yes—
156C3×C4⋊C4—28no—
157C3×C4⋊C4—28no—
158C3×C4⋊C4—24no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—42no—
168C3×C4⋊C4C2×D412yes—
169C3×C4⋊C4C2×D414yes—
170C3×C22⋊C4—28no—
171C3×C22⋊C4—24no—
172C3×C22⋊C4—48no—
173C3×C42—24no—
174C3×C42—42no—
175C3×C42C2×D412yes—
176C3×C42C2×D412yes—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8C23×C311yescharacteristic
180C22×D4C3×D411yescharacteristic
181C23×C4—22no—
182C23×C4C23×C312yes—
183C22×C8C23×C312yes—
184C22×C8C23×C312yes—
185C4⋊Q8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4⋊Q8C3×D414yes—
189#125—28no—
190#124—44no—
191C22⋊Q8—216no—
192C22⋊Q8—24no—
193C22⋊Q8—24no—
194C4×Q8—22no—
195C4×Q8C3×D414yes—
196C4×D4—28no—
197C4×D4—22no—
198C4×D4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C3×D411yescharacteristic
207C2×C4⋊C4C23×C312yes—
208C2×C4⋊C4C23×C312yes—
209C2×C4⋊C4C23×C311yescharacteristic
210C2×C4⋊C4C23×C312yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4C3×D412yes—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42C23×C311yescharacteristic
217#106—24no—
218#106—28no—
219C4⋊C8—22no—
220#103—24no—
221#103—24no—
222C22⋊C8—28no—
223C4×C8—22no—
224#95—21no—
225#95—24no—
226C23×C3—22no—
227C23×C3—24no—
228C23×C3—22no—
229C23×C3—24no—
230C23×C3C2×SD1614yes—
231C23×C3C22×D411yescharacteristic
232C3×Q8—22no—
233C3×Q8—22no—
234C3×Q8—42no—
235C3×Q8—84no—
236C3×Q8—84no—
237C3×D4—28no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—44no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—44no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—88no—
257C2×C3×C4#12012yes—
258C2×C3×C4#12012yes—
259C2×C3×C4C2×SD1612yes—
260C2×C3×C4#13712yes—
261C2×C3×C4C22×D411yescharacteristic
262C2×C3×C4C22×D411yescharacteristic
263C2×C3×C4C2×#4112yes—
264C2×C3×C42−1+412yes—
265C3×C8—24no—
266C3×C8—42no—
267C24C2×C3×D412yes—
268C2×Q8—24no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—82no—
276C2×Q8C2×C3×D412yes—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4C3×SD1614yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—28no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—84no—
291C22×C4C2×C3×D412yes—
292C22×C4C2×C3×D411yescharacteristic
293C22×C4C2×C3×D411yescharacteristic
294C22×C4C24×C311yescharacteristic
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—24no—
299C2×C8C3×#4114yes—
300C4⋊C4—28no—
301C4⋊C4—28no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—42no—
312C4⋊C4C2×C3×D412yes—
313C4⋊C4C2×C3×D414yes—
314C22⋊C4—28no—
315C22⋊C4—24no—
316C22⋊C4—48no—
317C42—24no—
318C42—42no—
319C42C2×C3×D412yes—
320C42C2×C3×D412yes—
321C22×C3—28no—
322C22×C3—24no—
323C22×C3—44no—
324C22×C3—42no—
325C22×C3—42no—
326C22×C3C2×#12011yescharacteristic
327C22×C3#54712yes—
328C22×C3C22×SD1611yescharacteristic
329C22×C3C2×#13711yescharacteristic
330C22×C3#57712yes—
331C3×C4—22no—
332C3×C4—42no—
333C3×C4—42no—
334C3×C4—42no—
335C3×C4—44no—
336C3×C4—82no—
337C3×C4—82no—
338C3×C4#44814yes—
339C3×C4#53211yescharacteristic
340C3×C4#53611yescharacteristic
341C3×C4#54812yes—
342C23—22no—
343C23—24no—
344C23—22no—
345C23—24no—
346C23C2×C3×SD1614yes—
347C23C22×C3×D411yescharacteristic
348Q8—22no—
349Q8—22no—
350Q8—42no—
351Q8—84no—
352Q8—84no—
353D4—28no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—88no—
373C2×C4C3×#12012yes—
374C2×C4C3×#12012yes—
375C2×C4C2×C3×SD1612yes—
376C2×C4C3×#13712yes—
377C2×C4C22×C3×D411yescharacteristic
378C2×C4C22×C3×D411yescharacteristic
379C2×C4C2×C3×#4112yes—
380C2×C4C3×2−1+412yes—
381C8—24no—
382C8—42no—
383C2×C3—44no—
384C2×C3#243211yescharacteristic
385C2×C3C2×#44811yescharacteristic
386C2×C3#300911yescharacteristic
387C2×C3#333512yes—
388C2×C3#335612yes—
389C22—24no—
390C22—28no—
391C22—42no—
392C22—42no—
393C22—44no—
394C22C2×C3×#12011yescharacteristic
395C22C3×#54712yes—
396C22C22×C3×SD1611yescharacteristic
397C22C2×C3×#13711yescharacteristic
398C22C3×#57712yes—
399C4—22no—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—44no—
404C4—82no—
405C4—82no—
406C4C3×#44814yes—
407C4C3×#53211yescharacteristic
408C4C3×#53611yescharacteristic
409C4C3×#54812yes—
410C3#2570211yescharacteristic, direct product factor, Sylow
411C2—44no—
412C2C3×#243211yescharacteristic
413C2C2×C3×#44811yescharacteristic
414C2C3×#300911yescharacteristic
415C2C3×#333512yes—
416C2C3×#335612yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#498C2semidirect
3. C2×C3×#475C2semidirect
5. C3×#2035C2semidirect
6. C3×#1987C2semidirect
8. C3×#1853C2semidirect
9. C3×C4×#103C2semidirect
10. #25702C3direct (N × H)
39. C2×#498C2×C3semidirect
40. C2×#475C2×C3semidirect
42. #2035C2×C3semidirect
43. #1987C2×C3semidirect
45. #1853C2×C3semidirect
46. C4×#103C2×C3semidirect
410. C3#25702direct (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×#4752
C3×#25472
C3×#20352
C3×#19872
C3×#19542
C3×#18532
C3×C4×#1032
#257023