Finite Group Databasebeta

← C3×#25834 #10644441

Subgroups — C3×#25834 #10644441

Order 768.

1246 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic
2C2×C3×#498C211yesmaximal, characteristic, complemented
3C2×C3×#476C211yesmaximal, characteristic
4C3×#2567C211yesmaximal, characteristic, complemented
5C3×#2058C214yesmaximal, complemented
6C3×#1988C211yesmaximal, characteristic, complemented
7C3×#1984C211yesmaximal, characteristic, complemented
8C3×#1864C214yesmaximal
9C3×C4×#102C211yesmaximal, characteristic
10#25834C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8—21no—
12C3×D4×Q8—24no—
13C3×D4×Q8—24no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2212yes—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×C22⋊Q8—21no—
18C2×C3×C22⋊Q8—22no—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4C2212yes—
21C2×C3×C4×D4C2211yescharacteristic
22C3×#498—24no—
23C3×#476—24no—
24C2×C3×#106C2214yes—
25C2×C3×#106C2212yes—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×C4×C8C2211yescharacteristic
30C3×#395—22no—
31C3×#391—21no—
32C3×#389C2214yes—
33C3×#384—21no—
34C3×C4×C4⋊C4C2211yescharacteristic
35C3×#340C2212yes—
36C3×#340C2214yes—
37C3×#336C2214yes—
38C2×D4×Q8C2×C311yescharacteristic
39C2×#498C2×C311yescharacteristic, complemented
40C2×#476C2×C311yescharacteristic
41#2567C2×C311yescharacteristic, complemented
42#2058C2×C314yescomplemented
43#1988C2×C311yescharacteristic, complemented
44#1984C2×C311yescharacteristic, complemented
45#1864C2×C314yes—
46C4×#102C2×C311yescharacteristic
47C22×C3×Q8—28no—
48C22×C3×Q8—22no—
49C22×C3×Q8—21no—
50C22×C3×Q8—21no—
51C22×C3×Q8—44no—
52C22×C3×Q8—44no—
53C22×C3×Q8D411yescharacteristic
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4—22no—
56C23×C3×C4—21no—
57C22×C3×C8C2312yes—
58C22×C3×C8C2312yes—
59C3×C4⋊Q8—24no—
60C3×C4⋊Q8—24no—
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8D414yes—
65C3×C22⋊Q8—28no—
66C3×C22⋊Q8—216no—
67C3×C4×Q8—22no—
68C3×C4×Q8—22no—
69C3×C4×D4—24no—
70C3×C4×D4—24no—
71C3×C4×D4—22no—
72C3×C4×D4D414yes—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—24no—
77C2×C3×C4⋊C4D411yescharacteristic
78C2×C3×C4⋊C4D412yes—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C42—21no—
86C2×C3×C42C2312yes—
87C2×C3×C42C2311yescharacteristic
88C3×#106—24no—
89C3×#106—28no—
90C3×C4⋊C8—22no—
91C3×#102—24no—
92C3×#102Q818yescomplemented
93C3×C4×C8—22no—
94C3×#95—21no—
95C3×#95—24no—
96C23×Q8—21no—
97D4×Q8—24no—
98D4×Q8—24no—
99C2×C4⋊Q8C22×C311yescharacteristic
100C2×C4⋊Q8C22×C312yes—
101C2×C4⋊Q8C22×C311yescharacteristic
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—22no—
104C2×C4×Q8C22×C311yescharacteristic
105C2×C4×D4C22×C312yes—
106C2×C4×D4C22×C311yescharacteristic
107#498—24no—
108#476—24no—
109C2×#106C22×C312yes—
110C2×#106C22×C314yes—
111C2×C4⋊C8C22×C311yescharacteristic
112C2×#102C22×C312yes—
113C2×#102C22×C312yes—
114C2×C4×C8C22×C311yescharacteristic
115#395—22no—
116#391—21no—
117#389C22×C314yes—
118#384—21no—
119C4×C4⋊C4C22×C311yescharacteristic
120#340C22×C312yes—
121#340C22×C314yes—
122#336C22×C314yes—
123C24×C3—21no—
124C2×C3×Q8—28no—
125C2×C3×Q8—22no—
126C2×C3×Q8—24no—
127C2×C3×Q8—22no—
128C2×C3×Q8—44no—
129C2×C3×Q8—432no—
130C2×C3×Q8—42no—
131C2×C3×Q8—44no—
132C2×C3×Q8—42no—
133C2×C3×Q8—42no—
134C2×C3×Q8—82no—
135C2×C3×Q8SD1614yes—
136C2×C3×D4—21no—
137C2×C3×D4—22no—
138C2×C3×D4C2×D412yes—
139C2×C3×D4C2×Q814yes—
140C22×C3×C4—28no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—24no—
145C22×C3×C4—22no—
146C22×C3×C4—44no—
147C22×C3×C4—44no—
148C22×C3×C4—42no—
149C22×C3×C4—42no—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2411yescharacteristic
155C2×C3×C8—22no—
156C2×C3×C8—22no—
157C2×C3×C8—22no—
158C2×C3×C8—24no—
159C2×C3×C8C2×Q814yescomplemented
160C3×C4⋊C4—28no—
161C3×C4⋊C4—24no—
162C3×C4⋊C4—24no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—42no—
170C3×C4⋊C4—84no—
171C3×C4⋊C4C2×D414yes—
172C3×C4⋊C4C2×D412yes—
173C3×C4⋊C4C2×Q814yescomplemented
174C3×C22⋊C4—28no—
175C3×C22⋊C4—24no—
176C3×C42—22no—
177C3×C42—22no—
178C3×C42—42no—
179C3×C42C2×D412yes—
180C3×C42C2×D412yes—
181C22×Q8—28no—
182C22×Q8—22no—
183C22×Q8—21no—
184C22×Q8—21no—
185C22×Q8—44no—
186C22×Q8—44no—
187C22×Q8C3×D411yescharacteristic
188C22×D4C23×C311yescharacteristic
189C23×C4—22no—
190C23×C4—21no—
191C22×C8C23×C312yes—
192C22×C8C23×C312yes—
193C4⋊Q8—24no—
194C4⋊Q8—24no—
195C4⋊Q8—22no—
196C4⋊Q8—22no—
197C4⋊Q8—22no—
198C4⋊Q8C3×D414yes—
199C22⋊Q8—28no—
200C22⋊Q8—216no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4—22no—
206C4×D4C3×D414yes—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4C3×D411yescharacteristic
212C2×C4⋊C4C3×D412yes—
213C2×C4⋊C4C23×C312yes—
214C2×C4⋊C4C23×C312yes—
215C2×C4⋊C4C23×C311yescharacteristic
216C2×C4⋊C4C23×C312yes—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C42—21no—
220C2×C42C23×C312yes—
221C2×C42C23×C311yescharacteristic
222#106—24no—
223#106—28no—
224C4⋊C8—22no—
225#102—24no—
226#102C3×Q818yescomplemented
227C4×C8—22no—
228#95—21no—
229#95—24no—
230C23×C3—22no—
231C23×C3—24no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—42no—
235C23×C3C22×D411yescharacteristic
236C3×Q8—28no—
237C3×Q8—416no—
238C3×Q8—48no—
239C3×Q8—84no—
240C3×D4—22no—
241C3×D4—24no—
242C3×D4C4⋊Q814yes—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—24no—
248C2×C3×C4—22no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—416no—
252C2×C3×C4—42no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—41no—
257C2×C3×C4—42no—
258C2×C3×C4—48no—
259C2×C3×C4—42no—
260C2×C3×C4—44no—
261C2×C3×C4#12012yes—
262C2×C3×C4#12012yes—
263C2×C3×C4C2×SD1614yes—
264C2×C3×C4C2×SD1612yes—
265C2×C3×C4AGL(1,ℤ/8)12yes—
266C2×C3×C4C22×D411yescharacteristic
267C2×C3×C4C22×D411yescharacteristic
268C2×C3×C4C22×Q812yes—
269C2×C3×C42−1+412yes—
270C3×C8—24no—
271C3×C8—42no—
272C24—21no—
273C2×Q8—28no—
274C2×Q8—22no—
275C2×Q8—24no—
276C2×Q8—22no—
277C2×Q8—44no—
278C2×Q8—432no—
279C2×Q8—42no—
280C2×Q8—44no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—82no—
284C2×Q8C3×SD1614yes—
285C2×D4—21no—
286C2×D4—22no—
287C2×D4C2×C3×D412yes—
288C2×D4C2×C3×Q814yes—
289C22×C4—28no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4C2×C3×D412yes—
300C22×C4C2×C3×D411yescharacteristic
301C22×C4C2×C3×D412yes—
302C22×C4C2×C3×D411yescharacteristic
303C22×C4C24×C311yescharacteristic
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—24no—
308C2×C8C2×C3×Q814yescomplemented
309C4⋊C4—28no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—21no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—42no—
319C4⋊C4—84no—
320C4⋊C4C2×C3×D414yes—
321C4⋊C4C2×C3×D412yes—
322C4⋊C4C2×C3×Q814yescomplemented
323C22⋊C4—28no—
324C22⋊C4—24no—
325C42—22no—
326C42—22no—
327C42—42no—
328C42C2×C3×D412yes—
329C42C2×C3×D412yes—
330C22×C3—24no—
331C22×C3—44no—
332C22×C3—44no—
333C22×C3—42no—
334C22×C3—42no—
335C22×C3C2×#12011yescharacteristic
336C22×C3D4×Q812yes—
337C22×C3C22×SD1611yescharacteristic
338C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
339C22×C3#57712yes—
340C3×C4—24no—
341C3×C4—22no—
342C3×C4—44no—
343C3×C4—44no—
344C3×C4—42no—
345C3×C4—42no—
346C3×C4—82no—
347C3×C4#45014yes—
348C3×C4C2×C4⋊Q811yescharacteristic
349C3×C4#53611yescharacteristic
350C3×C4D4×Q812yes—
351C23—22no—
352C23—24no—
353C23—41no—
354C23—41no—
355C23—42no—
356C23C22×C3×D411yescharacteristic
357Q8—28no—
358Q8—416no—
359Q8—48no—
360Q8—84no—
361D4—22no—
362D4—24no—
363D4C3×C4⋊Q814yes—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—416no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—48no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4C3×#12012yes—
383C2×C4C3×#12012yes—
384C2×C4C2×C3×SD1614yes—
385C2×C4C2×C3×SD1612yes—
386C2×C4C3×AGL(1,ℤ/8)12yes—
387C2×C4C22×C3×D411yescharacteristic
388C2×C4C22×C3×D411yescharacteristic
389C2×C4C22×C3×Q812yes—
390C2×C4C3×2−1+412yes—
391C8—24no—
392C8—42no—
393C2×C3—44no—
394C2×C3#243411yescharacteristic
395C2×C3C2×#45011yescharacteristic
396C2×C3#301011yescharacteristic
397C2×C3Q8×SD1612yes—
398C2×C3#338412yes—
399C22—24no—
400C22—44no—
401C22—42no—
402C22—42no—
403C22—44no—
404C22C2×C3×#12011yescharacteristic
405C22C3×D4×Q812yes—
406C22C22×C3×SD1611yescharacteristic
407C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
408C22C3×#57712yes—
409C4—24no—
410C4—22no—
411C4—44no—
412C4—44no—
413C4—42no—
414C4—42no—
415C4—82no—
416C4C3×#45014yes—
417C4C2×C3×C4⋊Q811yescharacteristic
418C4C3×#53611yescharacteristic
419C4C3×D4×Q812yes—
420C3#2583411yescharacteristic, direct product factor, Sylow
421C2—44no—
422C2C3×#243411yescharacteristic
423C2C2×C3×#45011yescharacteristic
424C2C3×#301011yescharacteristic
425C2C3×Q8×SD1612yes—
426C2C3×#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×#2567C2semidirect
5. C3×#2058C2semidirect
6. C3×#1988C2semidirect
7. C3×#1984C2semidirect
10. #25834C3direct (N × H)
39. C2×#498C2×C3semidirect
41. #2567C2×C3semidirect
42. #2058C2×C3semidirect
43. #1988C2×C3semidirect
44. #1984C2×C3semidirect
92. C3×#102Q8semidirect
159. C2×C3×C8C2×Q8semidirect
173. C3×C4⋊C4C2×Q8semidirect
226. #102C3×Q8semidirect
308. C2×C8C2×C3×Q8semidirect
322. C4⋊C4C2×C3×Q8semidirect
420. C3#25834direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×Q82
C2×C3×#4982
C2×C3×#4762
C3×#25672
C3×#20582
C3×#19882
C3×#19842
C3×#18642
C3×C4×#1022
#258343