Finite Group Databasebeta

← C3×#15091 #10633698

Subgroups — C3×#15091 #10633698

Order 768.

2322 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×C3×#386C211yesmaximal, characteristic
3C2×C3×#386C212yesmaximal
4C3×#1442C214yesmaximal, complemented
5C3×#1439C214yesmaximal, complemented
6C3×C22⋊#95C211yesmaximal, characteristic, complemented
7C3×C22⋊#95C212yesmaximal, complemented
8#15091C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×D4C2211yescharacteristic
10C24×C3×C4C2211yescharacteristic
11C24×C3×C4C2212yes—
12C22×C3×C4⋊C4—28no—
13C22×C3×C4⋊C4—24no—
14C22×C3×C4⋊C4C2214yes—
15C22×C3×C22⋊C4C2212yes—
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2212yes—
18C22×C3×C22⋊C4C2214yes—
19C3×#386—24no—
20C3×#386—28no—
21C2×C3×#95C2214yes—
22C2×C3×#95C2212yes—
23C2×C3×#95C2214yes—
24C2×C3×#95C2218yes—
25C23×C4⋊C4C2×C311yescharacteristic, complemented
26C2×#386C2×C311yescharacteristic
27C2×#386C2×C312yes—
28#1442C2×C314yescomplemented
29#1439C2×C314yescomplemented
30C22⋊#95C2×C311yescharacteristic, complemented
31C22⋊#95C2×C312yescomplemented
32C25×C3C2311yescharacteristic
33C25×C3C2311yescharacteristic
34C22×C3×D4—22no—
35C22×C3×D4—24no—
36C22×C3×D4—24no—
37C22×C3×D4—22no—
38C22×C3×D4Q814yes—
39C23×C3×C4—24no—
40C23×C3×C4—28no—
41C23×C3×C4—24no—
42C23×C3×C4—24no—
43C23×C3×C4—48no—
44C23×C3×C4D414yes—
45C23×C3×C4D418yes—
46C23×C3×C4C2314yes—
47C23×C3×C4C2312yes—
48C23×C3×C4C2314yes—
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2312yes—
51C2×C3×C4⋊C4—232no—
52C2×C3×C4⋊C4—216no—
53C2×C3×C4⋊C4—28no—
54C2×C3×C22⋊C4—22no—
55C2×C3×C22⋊C4—24no—
56C2×C3×C22⋊C4—24no—
57C2×C3×C22⋊C4—24no—
58C2×C3×C22⋊C4—24no—
59C2×C3×C22⋊C4—28no—
60C2×C3×C22⋊C4—24no—
61C2×C3×C22⋊C4—28no—
62C2×C3×C22⋊C4—48no—
63C3×#95—28no—
64C3×#95—24no—
65C3×#95—28no—
66C3×#95—48no—
67C23×D4C22×C311yescharacteristic
68C24×C4C22×C311yescharacteristic
69C24×C4C22×C312yes—
70C22×C4⋊C4—28no—
71C22×C4⋊C4—24no—
72C22×C4⋊C4C22×C314yes—
73C22×C22⋊C4C22×C312yes—
74C22×C22⋊C4C22×C311yescharacteristic
75C22×C22⋊C4C22×C312yes—
76C22×C22⋊C4C22×C314yes—
77#386—24no—
78#386—28no—
79C2×#95C22×C314yes—
80C2×#95C22×C314yes—
81C2×#95C22×C312yes—
82C2×#95C22×C318yes—
83C24×C3—24no—
84C24×C3—24no—
85C24×C3—21no—
86C24×C3—22no—
87C24×C3—24no—
88C24×C3—22no—
89C24×C3—21no—
90C24×C3—24no—
91C24×C3C2×D412yes—
92C24×C3C2×D414yes—
93C24×C3C2×Q812yes—
94C24×C3C2×Q812yes—
95C24×C3#4114yes—
96C24×C3#4112yes—
97C24×C3C2411yescharacteristic
98C2×C3×D4—216no—
99C2×C3×D4—28no—
100C2×C3×D4—44no—
101C2×C3×D4—48no—
102C2×C3×D4—44no—
103C22×C3×C4—232no—
104C22×C3×C4—28no—
105C22×C3×C4—24no—
106C22×C3×C4—24no—
107C22×C3×C4—216no—
108C22×C3×C4—216no—
109C22×C3×C4—21no—
110C22×C3×C4—22no—
111C22×C3×C4—24no—
112C22×C3×C4—24no—
113C22×C3×C4—22no—
114C22×C3×C4—21no—
115C22×C3×C4—28no—
116C22×C3×C4—28no—
117C22×C3×C4—22no—
118C22×C3×C4—24no—
119C22×C3×C4—28no—
120C22×C3×C4—24no—
121C22×C3×C4—24no—
122C22×C3×C4—24no—
123C22×C3×C4—24no—
124C22×C3×C4—24no—
125C22×C3×C4—28no—
126C22×C3×C4—416no—
127C22×C3×C4—416no—
128C22×C3×C4—416no—
129C22×C3×C4C2×D418yes—
130C22×C3×C4C2×D414yes—
131C22×C3×C4C2×Q812yes—
132C22×C3×C4#4118yes—
133C3×C4⋊C4—232no—
134C3×C22⋊C4—416no—
135C3×C22⋊C4—48no—
136C3×C22⋊C4—44no—
137C3×C22⋊C4—48no—
138C25C23×C311yescharacteristic
139C25C23×C311yescharacteristic
140C22×D4—22no—
141C22×D4—24no—
142C22×D4—22no—
143C22×D4—24no—
144C22×D4C3×Q814yes—
145C23×C4—24no—
146C23×C4—28no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—48no—
150C23×C4C3×D414yes—
151C23×C4C3×D418yes—
152C23×C4C23×C314yes—
153C23×C4C23×C312yes—
154C23×C4C23×C314yes—
155C23×C4C23×C311yescharacteristic
156C23×C4C23×C312yes—
157C2×C4⋊C4—232no—
158C2×C4⋊C4—216no—
159C2×C4⋊C4—28no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—48no—
169#95—28no—
170#95—24no—
171#95—28no—
172#95—48no—
173C23×C3—24no—
174C23×C3—28no—
175C23×C3—28no—
176C23×C3—24no—
177C23×C3—216no—
178C23×C3—28no—
179C23×C3—28no—
180C23×C3—24no—
181C23×C3—22no—
182C23×C3—28no—
183C23×C3—41no—
184C23×C3—42no—
185C23×C3—44no—
186C23×C3—42no—
187C23×C3—41no—
188C23×C3—44no—
189C23×C3—42no—
190C23×C3—48no—
191C23×C3—44no—
192C23×C3#12014yes—
193C23×C3C22⋊Q818yes—
194C23×C3C22⋊Q814yes—
195C23×C3#12418yes—
196C23×C3#12414yes—
197C23×C3C22×D411yescharacteristic
198C23×C3C22×D412yes—
199C23×C3C22×Q811yescharacteristic
200C23×C3C2×#4112yes—
201C23×C3C2×#4111yescharacteristic
202C23×C3C2×#4114yes—
203C23×C3H(5,𝔽2)14yes—
204C3×D4—416no—
205C2×C3×C4—232no—
206C2×C3×C4—24no—
207C2×C3×C4—24no—
208C2×C3×C4—216no—
209C2×C3×C4—28no—
210C2×C3×C4—24no—
211C2×C3×C4—22no—
212C2×C3×C4—28no—
213C2×C3×C4—28no—
214C2×C3×C4—432no—
215C2×C3×C4—41no—
216C2×C3×C4—42no—
217C2×C3×C4—41no—
218C2×C3×C4—48no—
219C2×C3×C4—48no—
220C2×C3×C4—42no—
221C2×C3×C4—44no—
222C2×C3×C4—44no—
223C2×C3×C4—42no—
224C2×C3×C4—44no—
225C2×C3×C4—42no—
226C2×C3×C4—48no—
227C2×C3×C4—48no—
228C2×C3×C4—48no—
229C2×C3×C4#12318yes—
230C24—24no—
231C24—24no—
232C24—21no—
233C24—22no—
234C24—24no—
235C24—21no—
236C24—22no—
237C24—24no—
238C24C2×C3×D412yes—
239C24C2×C3×D414yes—
240C24C2×C3×Q812yes—
241C24C2×C3×Q812yes—
242C24C3×#4112yes—
243C24C3×#4114yes—
244C24C24×C311yescharacteristic
245C2×D4—28no—
246C2×D4—216no—
247C2×D4—44no—
248C2×D4—44no—
249C2×D4—48no—
250C22×C4—232no—
251C22×C4—28no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—216no—
255C22×C4—216no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—28no—
263C22×C4—28no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—28no—
273C22×C4—416no—
274C22×C4—416no—
275C22×C4—416no—
276C22×C4C2×C3×D418yes—
277C22×C4C2×C3×D414yes—
278C22×C4C2×C3×Q812yes—
279C22×C4C3×#4118yes—
280C4⋊C4—232no—
281C22⋊C4—416no—
282C22⋊C4—44no—
283C22⋊C4—48no—
284C22⋊C4—48no—
285C22×C3—216no—
286C22×C3—216no—
287C22×C3—24no—
288C22×C3—24no—
289C22×C3—28no—
290C22×C3—28no—
291C22×C3—48no—
292C22×C3—42no—
293C22×C3—44no—
294C22×C3—84no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3—82no—
298C22×C3#39318yes—
299C22×C3C2×#12011yescharacteristic
300C22×C3C2×C22⋊Q812yes—
301C22×C3C2×C22⋊Q811yescharacteristic
302C22×C3C2×#12312yes—
303C22×C3#52514yes—
304C22×C3C2×#12412yes—
305C22×C3C2×#12411yescharacteristic
306C22×C3#53414yes—
307C22×C3#53512yes—
308C22×C3#55318yes—
309C22×C3#55314yes—
310C22×C3#55414yes—
311C3×C4—28no—
312C3×C4—48no—
313C3×C4—42no—
314C3×C4—84no—
315C3×C4—82no—
316C23—24no—
317C23—28no—
318C23—28no—
319C23—24no—
320C23—28no—
321C23—216no—
322C23—28no—
323C23—22no—
324C23—24no—
325C23—28no—
326C23—41no—
327C23—42no—
328C23—44no—
329C23—41no—
330C23—42no—
331C23—44no—
332C23—42no—
333C23—44no—
334C23—48no—
335C23C3×#12014yes—
336C23C3×C22⋊Q814yes—
337C23C3×C22⋊Q818yes—
338C23C3×#12414yes—
339C23C3×#12418yes—
340C23C22×C3×D411yescharacteristic
341C23C22×C3×D412yes—
342C23C22×C3×Q811yescharacteristic
343C23C2×C3×#4111yescharacteristic
344C23C2×C3×#4112yes—
345C23C2×C3×#4114yes—
346C23C3×H(5,𝔽2)14yes—
347D4—416no—
348C2×C4—232no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—216no—
352C2×C4—28no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—28no—
356C2×C4—28no—
357C2×C4—432no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—48no—
362C2×C4—48no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—48no—
370C2×C4—48no—
371C2×C4—48no—
372C2×C4C3×#12318yes—
373C2×C3—28no—
374C2×C3—82no—
375C2×C3C2×#39311yescharacteristic
376C2×C3#248712yes—
377C2×C3#250412yes—
378C2×C3#257914yes—
379C2×C3#259114yes—
380C2×C3#261311yescharacteristic
381C2×C3#284911yescharacteristic
382C22—28no—
383C22—28no—
384C22—216no—
385C22—216no—
386C22—24no—
387C22—24no—
388C22—42no—
389C22—44no—
390C22—48no—
391C22—82no—
392C22—84no—
393C22—81no—
394C22—81no—
395C22C3×#39318yes—
396C22C2×C3×#12011yescharacteristic
397C22C2×C3×C22⋊Q811yescharacteristic
398C22C2×C3×C22⋊Q812yes—
399C22C2×C3×#12312yes—
400C22C3×#52514yes—
401C22C2×C3×#12411yescharacteristic
402C22C2×C3×#12412yes—
403C22C3×#53414yes—
404C22C3×#53512yes—
405C22C3×#55314yes—
406C22C3×#55318yes—
407C22C3×#55414yes—
408C4—28no—
409C4—42no—
410C4—48no—
411C4—82no—
412C4—84no—
413C3#1509111yescharacteristic, direct product factor, Sylow
414C2—28no—
415C2—82no—
416C2C2×C3×#39311yescharacteristic
417C2C3×#248712yes—
418C2C3×#250412yes—
419C2C3×#257914yes—
420C2C3×#259114yes—
421C2C3×#261311yescharacteristic
422C2C3×#284911yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×C3×C4⋊C4C2semidirect
4. C3×#1442C2semidirect
5. C3×#1439C2semidirect
6. C3×C22⋊#95C2semidirect
7. C3×C22⋊#95C2semidirect
8. #15091C3direct (N × H)
25. C23×C4⋊C4C2×C3semidirect
28. #1442C2×C3semidirect
29. #1439C2×C3semidirect
30. C22⋊#95C2×C3semidirect
31. C22⋊#95C2×C3semidirect
413. C3#15091direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4⋊C42
C2×C3×#3862
C3×#14422
C3×#14392
C3×C22⋊#952
#150913