Finite Group Databasebeta

← C3×#25838 #10644445

Subgroups — C3×#25838 #10644445

Order 768.

1266 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic
2C2×C3×#500C211yesmaximal, characteristic, complemented
3C2×C3×#474C211yesmaximal, characteristic
4C3×#2546C211yesmaximal, characteristic, complemented
5C3×#2057C214yesmaximal, complemented
6C3×#1988C211yesmaximal, characteristic, complemented
7C3×#1984C211yesmaximal, characteristic, complemented
8C3×#1862C214yesmaximal
9C3×C4×#102C211yesmaximal, characteristic
10#25838C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8—21no—
12C3×D4×Q8—24no—
13C3×D4×Q8—24no—
14C2×C3×C4⋊Q8—21no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4⋊Q8C2212yes—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8—22no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2212yes—
22C2×C3×C4×D4C2211yescharacteristic
23C3×#500—24no—
24C3×#474—24no—
25C2×C3×#107C2214yes—
26C2×C3×#107C2212yes—
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×D4×Q8C2×C311yescharacteristic
39C2×#500C2×C311yescharacteristic, complemented
40C2×#474C2×C311yescharacteristic
41#2546C2×C311yescharacteristic, complemented
42#2057C2×C314yescomplemented
43#1988C2×C311yescharacteristic, complemented
44#1984C2×C311yescharacteristic, complemented
45#1862C2×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⋊Q8—44no—
65C3×C4⋊Q8D414yes—
66C3×C22⋊Q8—28no—
67C3×C22⋊Q8—216no—
68C3×C4×Q8—22no—
69C3×C4×Q8—22no—
70C3×C4×D4—24no—
71C3×C4×D4—24no—
72C3×C4×D4—22no—
73C3×C4×D4D414yes—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—24no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4D411yescharacteristic
79C2×C3×C4⋊C4D412yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C42—21no—
87C2×C3×C42C2312yes—
88C2×C3×C42C2311yescharacteristic
89C3×#107—24no—
90C3×#107—28no—
91C3×C4⋊C8—22no—
92C3×#102—24no—
93C3×#102Q818yescomplemented
94C3×C4×C8—22no—
95C3×#95—21no—
96C3×#95—24no—
97C23×Q8—21no—
98D4×Q8—24no—
99D4×Q8—24no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C22×C312yes—
102C2×C4⋊Q8C22×C311yescharacteristic
103C2×C4⋊Q8C22×C311yescharacteristic
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—22no—
106C2×C4×Q8C22×C311yescharacteristic
107C2×C4×D4C22×C312yes—
108C2×C4×D4C22×C311yescharacteristic
109#500—24no—
110#474—24no—
111C2×#107C22×C312yes—
112C2×#107C22×C314yes—
113C2×C4⋊C8C22×C311yescharacteristic
114C2×#102C22×C312yes—
115C2×#102C22×C312yes—
116C2×C4×C8C22×C311yescharacteristic
117#398—22no—
118#391—21no—
119#389C22×C314yes—
120C4×C4⋊C4C22×C311yescharacteristic
121#340C22×C312yes—
122#340C22×C314yes—
123#336C22×C314yes—
124C24×C3—21no—
125C2×C3×Q8—28no—
126C2×C3×Q8—22no—
127C2×C3×Q8—24no—
128C2×C3×Q8—22no—
129C2×C3×Q8—44no—
130C2×C3×Q8—432no—
131C2×C3×Q8—42no—
132C2×C3×Q8—44no—
133C2×C3×Q8—42no—
134C2×C3×Q8—42no—
135C2×C3×Q8—42no—
136C2×C3×Q8—42no—
137C2×C3×Q8D814yes—
138C2×C3×D4—21no—
139C2×C3×D4—22no—
140C2×C3×D4C2×D412yes—
141C2×C3×D4C2×Q814yes—
142C22×C3×C4—28no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—24no—
147C22×C3×C4—22no—
148C22×C3×C4—44no—
149C22×C3×C4—44no—
150C22×C3×C4—42no—
151C22×C3×C4—42no—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2411yescharacteristic
157C2×C3×C8—22no—
158C2×C3×C8—22no—
159C2×C3×C8—22no—
160C2×C3×C8—24no—
161C2×C3×C8C2×Q814yescomplemented
162C3×C4⋊C4—28no—
163C3×C4⋊C4—24no—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—48no—
173C3×C4⋊C4C2×D414yes—
174C3×C4⋊C4C2×D412yes—
175C3×C4⋊C4C2×Q814yescomplemented
176C3×C22⋊C4—28no—
177C3×C22⋊C4—24no—
178C3×C42—22no—
179C3×C42—22no—
180C3×C42—42no—
181C3×C42C2×D412yes—
182C3×C42C2×D412yes—
183C22×Q8—28no—
184C22×Q8—22no—
185C22×Q8—21no—
186C22×Q8—21no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8C3×D411yescharacteristic
190C22×D4C23×C311yescharacteristic
191C23×C4—22no—
192C23×C4—21no—
193C22×C8C23×C312yes—
194C22×C8C23×C312yes—
195C4⋊Q8—24no—
196C4⋊Q8—24no—
197C4⋊Q8—22no—
198C4⋊Q8—22no—
199C4⋊Q8—22no—
200C4⋊Q8—44no—
201C4⋊Q8C3×D414yes—
202C22⋊Q8—28no—
203C22⋊Q8—216no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×D4—24no—
207C4×D4—24no—
208C4×D4—22no—
209C4×D4C3×D414yes—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C3×D411yescharacteristic
215C2×C4⋊C4C3×D412yes—
216C2×C4⋊C4C23×C312yes—
217C2×C4⋊C4C23×C312yes—
218C2×C4⋊C4C23×C311yescharacteristic
219C2×C4⋊C4C23×C312yes—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C42—21no—
223C2×C42C23×C312yes—
224C2×C42C23×C311yescharacteristic
225#107—24no—
226#107—28no—
227C4⋊C8—22no—
228#102—24no—
229#102C3×Q818yescomplemented
230C4×C8—22no—
231#95—21no—
232#95—24no—
233C23×C3—22no—
234C23×C3—24no—
235C23×C3—41no—
236C23×C3—41no—
237C23×C3—42no—
238C23×C3C22×D411yescharacteristic
239C3×Q8—28no—
240C3×Q8—416no—
241C3×Q8—48no—
242C3×Q8—84no—
243C3×D4—22no—
244C3×D4—24no—
245C3×D4C4⋊Q814yescomplemented
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—24no—
251C2×C3×C4—22no—
252C2×C3×C4—24no—
253C2×C3×C4—24no—
254C2×C3×C4—416no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—42no—
259C2×C3×C4—41no—
260C2×C3×C4—42no—
261C2×C3×C4—48no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4#12012yes—
266C2×C3×C4#12012yes—
267C2×C3×C4C2×D814yes—
268C2×C3×C4C2×D812yes—
269C2×C3×C4AGL(1,ℤ/8)12yes—
270C2×C3×C4C22×D411yescharacteristic
271C2×C3×C4C22×D411yescharacteristic
272C2×C3×C4C22×Q812yes—
273C2×C3×C42−1+412yes—
274C3×C8—24no—
275C3×C8—42no—
276C24—21no—
277C2×Q8—28no—
278C2×Q8—22no—
279C2×Q8—24no—
280C2×Q8—22no—
281C2×Q8—44no—
282C2×Q8—432no—
283C2×Q8—42no—
284C2×Q8—44no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8C3×D814yes—
290C2×D4—21no—
291C2×D4—22no—
292C2×D4C2×C3×D412yes—
293C2×D4C2×C3×Q814yes—
294C22×C4—28no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4C2×C3×D412yes—
305C22×C4C2×C3×D411yescharacteristic
306C22×C4C2×C3×D412yes—
307C22×C4C2×C3×D411yescharacteristic
308C22×C4C24×C311yescharacteristic
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—24no—
313C2×C8C2×C3×Q814yescomplemented
314C4⋊C4—28no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—21no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—42no—
324C4⋊C4—48no—
325C4⋊C4C2×C3×D414yes—
326C4⋊C4C2×C3×D412yes—
327C4⋊C4C2×C3×Q814yescomplemented
328C22⋊C4—28no—
329C22⋊C4—24no—
330C42—22no—
331C42—22no—
332C42—42no—
333C42C2×C3×D412yes—
334C42C2×C3×D412yes—
335C22×C3—24no—
336C22×C3—44no—
337C22×C3—44no—
338C22×C3—42no—
339C22×C3—42no—
340C22×C3C2×#12011yescharacteristic
341C22×C3D4×Q812yes—
342C22×C3C22×D811yescharacteristic
343C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
344C22×C3#57812yes—
345C3×C4—24no—
346C3×C4—22no—
347C3×C4—44no—
348C3×C4—44no—
349C3×C4—42no—
350C3×C4—42no—
351C3×C4—82no—
352C3×C4#44714yes—
353C3×C4C2×C4⋊Q811yescharacteristic
354C3×C4#53611yescharacteristic
355C3×C4D4×Q812yes—
356C23—22no—
357C23—24no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23C22×C3×D411yescharacteristic
362Q8—28no—
363Q8—416no—
364Q8—48no—
365Q8—84no—
366D4—22no—
367D4—24no—
368D4C3×C4⋊Q814yescomplemented
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—416no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—48no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4C3×#12012yes—
389C2×C4C3×#12012yes—
390C2×C4C2×C3×D814yes—
391C2×C4C2×C3×D812yes—
392C2×C4C3×AGL(1,ℤ/8)12yes—
393C2×C4C22×C3×D411yescharacteristic
394C2×C4C22×C3×D411yescharacteristic
395C2×C4C22×C3×Q812yes—
396C2×C4C3×2−1+412yes—
397C8—24no—
398C8—42no—
399C2×C3—44no—
400C2×C3#243411yescharacteristic
401C2×C3C2×#44711yescharacteristic
402C2×C3#301111yescharacteristic
403C2×C3Q8×D812yes—
404C2×C3#338612yes—
405C22—24no—
406C22—44no—
407C22—42no—
408C22—42no—
409C22—44no—
410C22C2×C3×#12011yescharacteristic
411C22C3×D4×Q812yes—
412C22C22×C3×D811yescharacteristic
413C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
414C22C3×#57812yes—
415C4—24no—
416C4—22no—
417C4—44no—
418C4—42no—
419C4—42no—
420C4—44no—
421C4—82no—
422C4C3×#44714yes—
423C4C2×C3×C4⋊Q811yescharacteristic
424C4C3×#53611yescharacteristic
425C4C3×D4×Q812yes—
426C3#2583811yescharacteristic, direct product factor, Sylow
427C2—44no—
428C2C3×#243411yescharacteristic
429C2C2×C3×#44711yescharacteristic
430C2C3×#301111yescharacteristic
431C2C3×Q8×D812yes—
432C2C3×#338612yes—
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×#500C2semidirect
4. C3×#2546C2semidirect
5. C3×#2057C2semidirect
6. C3×#1988C2semidirect
7. C3×#1984C2semidirect
10. #25838C3direct (N × H)
39. C2×#500C2×C3semidirect
41. #2546C2×C3semidirect
42. #2057C2×C3semidirect
43. #1988C2×C3semidirect
44. #1984C2×C3semidirect
93. C3×#102Q8semidirect
161. C2×C3×C8C2×Q8semidirect
175. C3×C4⋊C4C2×Q8semidirect
229. #102C3×Q8semidirect
245. C3×D4C4⋊Q8semidirect
313. C2×C8C2×C3×Q8semidirect
327. C4⋊C4C2×C3×Q8semidirect
368. D4C3×C4⋊Q8semidirect
426. C3#25838direct (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×#5002
C2×C3×#4742
C3×#25462
C3×#20572
C3×#19882
C3×#19842
C3×#18622
C3×C4×#1022
#258383