Finite Group Databasebeta

← C3×#25820 #10644427

Subgroups — C3×#25820 #10644427

Order 768.

990 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#544C211yesmaximal, characteristic
2C2×C3×#501C211yesmaximal, characteristic, complemented
3C2×C3×#498C211yesmaximal, characteristic, complemented
4C3×#2628C211yesmaximal, characteristic
5C3×#2059C212yesmaximal, complemented
6C3×#2058C212yesmaximal, complemented
7C3×#1993C211yesmaximal, characteristic, complemented
8C3×#1991C211yesmaximal, characteristic, complemented
9C3×#1854C212yesmaximal
10C3×#1853C212yesmaximal
11C3×#1750C211yesmaximal, characteristic
12#25820C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#544—24no—
14C3×#544—24no—
15C2×C3×C4⋊Q8—21no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#125—21no—
20C2×C3×#125C2211yescharacteristic
21C2×C3×#125C2211yescharacteristic
22C2×C3×C22⋊Q8—21no—
23C2×C3×C22⋊Q8—21no—
24C2×C3×#117C2211yescharacteristic
25C2×C3×#117C2211yescharacteristic
26C22×C3×C4⋊C4C2211yescharacteristic
27C3×#501—24no—
28C3×#498—24no—
29C2×C3×#107C2212yes—
30C2×C3×#106C2212yes—
31C2×C3×#106C2212yes—
32C2×C3×#106C2212yes—
33C2×C3×C22⋊C8C2212yes—
34C2×C3×#97C2211yescharacteristic
35C2×C3×C4×C8C2211yescharacteristic
36C3×#400—21no—
37C3×#399—21no—
38C3×#395—21no—
39C3×#393—21no—
40C3×#389C2212yes—
41C3×#382C2211yescharacteristic
42C3×#380C2212yes—
43C3×#340C2212yes—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#544C2×C311yescharacteristic
49C2×#501C2×C311yescharacteristic, complemented
50C2×#498C2×C311yescharacteristic, complemented
51#2628C2×C311yescharacteristic
52#2059C2×C312yescomplemented
53#2058C2×C312yescomplemented
54#1993C2×C311yescharacteristic, complemented
55#1991C2×C311yescharacteristic, complemented
56#1854C2×C312yes—
57#1853C2×C312yes—
58#1750C2×C311yescharacteristic
59C22×C3×Q8—21no—
60C22×C3×Q8—21no—
61C22×C3×Q8—21no—
62C23×C3×C4—21no—
63C23×C3×C4C2311yescharacteristic
64C22×C3×C8C2312yes—
65C22×C3×C8C2312yes—
66C3×C4⋊Q8—28no—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8—22no—
72C3×C4⋊Q8D414yes—
73C3×#125—28no—
74C3×#125—22no—
75C3×#125—22no—
76C3×#125—22no—
77C3×#125D414yes—
78C3×C22⋊Q8—28no—
79C3×C22⋊Q8—28no—
80C3×#117—22no—
81C3×#117—22no—
82C3×#117—22no—
83C3×#117—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4D411yescharacteristic
106C2×C3×C22⋊C4D411yescharacteristic
107C2×C3×C42—21no—
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C3×#107—24no—
111C3×#106—24no—
112C3×#106—24no—
113C3×#106—24no—
114C3×C22⋊C8Q818yescomplemented
115C3×#97—22no—
116C3×C4×C8—22no—
117C3×#95—22no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—22no—
121#544—24no—
122#544—24no—
123C2×C4⋊Q8—21no—
124C2×C4⋊Q8C22×C311yescharacteristic
125C2×C4⋊Q8C22×C311yescharacteristic
126C2×C4⋊Q8C22×C311yescharacteristic
127C2×#125—21no—
128C2×#125C22×C311yescharacteristic
129C2×#125C22×C311yescharacteristic
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8—21no—
132C2×#117C22×C311yescharacteristic
133C2×#117C22×C311yescharacteristic
134C22×C4⋊C4C22×C311yescharacteristic
135#501—24no—
136#498—24no—
137C2×#107C22×C312yes—
138C2×#106C22×C312yes—
139C2×#106C22×C312yes—
140C2×#106C22×C312yes—
141C2×C22⋊C8C22×C312yes—
142C2×#97C22×C311yescharacteristic
143C2×C4×C8C22×C311yescharacteristic
144#400—21no—
145#399—21no—
146#395—21no—
147#393—21no—
148#389C22×C312yes—
149#382C22×C311yescharacteristic
150#380C22×C312yes—
151#340C22×C312yes—
152#340C22×C312yes—
153#340C22×C312yes—
154#340C22×C312yes—
155#336C22×C312yes—
156C24×C3C2×D411yescharacteristic
157C2×C3×Q8—24no—
158C2×C3×Q8—24no—
159C2×C3×Q8—42no—
160C2×C3×Q8—82no—
161C2×C3×Q8—82no—
162C2×C3×Q8—82no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—24no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—82no—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×Q814yes—
180C22×C3×C4C2411yescharacteristic
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C2×C3×C8C2×Q814yes—
186C2×C3×C8C2×Q814yes—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—42no—
204C3×C4⋊C4—82no—
205C3×C4⋊C4C2×D412yes—
206C3×C4⋊C4C2×D412yes—
207C3×C4⋊C4C2×D412yes—
208C3×C4⋊C4C2×D412yes—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—22no—
211C3×C22⋊C4—22no—
212C3×C22⋊C4—82no—
213C3×C42—24no—
214C3×C42—21no—
215C3×C42—21no—
216C3×C42C2×D412yes—
217C3×C42C2×D412yes—
218C22×Q8—21no—
219C22×Q8—21no—
220C22×Q8—21no—
221C23×C4—21no—
222C23×C4C23×C311yescharacteristic
223C22×C8C23×C312yes—
224C22×C8C23×C312yes—
225C4⋊Q8—28no—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8C3×D414yes—
232#125—28no—
233#125—22no—
234#125—22no—
235#125—22no—
236#125C3×D414yes—
237C22⋊Q8—28no—
238C22⋊Q8—28no—
239#117—22no—
240#117—22no—
241#117—22no—
242#117—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4C3×D411yescharacteristic
255C2×C4⋊C4C3×D411yescharacteristic
256C2×C4⋊C4C23×C312yes—
257C2×C4⋊C4C23×C312yes—
258C2×C4⋊C4C23×C311yescharacteristic
259C2×C4⋊C4C23×C311yescharacteristic
260C2×C4⋊C4C23×C311yescharacteristic
261C2×C4⋊C4C23×C311yescharacteristic
262C2×C22⋊C4—21no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4C3×D411yescharacteristic
265C2×C22⋊C4C3×D411yescharacteristic
266C2×C42—21no—
267C2×C42C23×C311yescharacteristic
268C2×C42C23×C311yescharacteristic
269#107—24no—
270#106—24no—
271#106—24no—
272#106—24no—
273C22⋊C8C3×Q818yescomplemented
274#97—22no—
275C4×C8—22no—
276#95—22no—
277#95—21no—
278#95—21no—
279#95—22no—
280C23×C3—21no—
281C23×C3—21no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3C4⋊Q812yes—
285C23×C3C22×D411yescharacteristic
286C3×Q8—84no—
287C3×Q8—84no—
288C3×Q8—84no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—21no—
296C2×C3×C4—22no—
297C2×C3×C4—21no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—84no—
314C2×C3×C4#12012yes—
315C2×C3×C4#12012yes—
316C2×C3×C4C4⋊Q812yes—
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×Q812yes—
320C2×C3×C42−1+412yes—
321C3×C8—24no—
322C3×C8—24no—
323C24C2×C3×D411yescharacteristic
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—42no—
327C2×Q8—82no—
328C2×Q8—82no—
329C2×Q8—82no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—82no—
341C22×C4C2×C3×D411yescharacteristic
342C22×C4C2×C3×D411yescharacteristic
343C22×C4C2×C3×D411yescharacteristic
344C22×C4C2×C3×D411yescharacteristic
345C22×C4C2×C3×D411yescharacteristic
346C22×C4C2×C3×Q814yes—
347C22×C4C24×C311yescharacteristic
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8C2×C3×Q814yes—
353C2×C8C2×C3×Q814yes—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—42no—
371C4⋊C4—82no—
372C4⋊C4C2×C3×D412yes—
373C4⋊C4C2×C3×D412yes—
374C4⋊C4C2×C3×D412yes—
375C4⋊C4C2×C3×D412yes—
376C22⋊C4—24no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—82no—
380C42—24no—
381C42—21no—
382C42—21no—
383C42C2×C3×D412yes—
384C42C2×C3×D412yes—
385C22×C3—24no—
386C22×C3—22no—
387C22×C3—42no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3C2×#12011yescharacteristic
391C22×C3C2×C4⋊Q811yescharacteristic
392C22×C3#53611yescharacteristic
393C22×C3#57712yes—
394C22×C3#57712yes—
395C3×C4—22no—
396C3×C4—22no—
397C3×C4—22no—
398C3×C4—41no—
399C3×C4—42no—
400C3×C4—42no—
401C3×C4—42no—
402C3×C4—41no—
403C3×C4—42no—
404C3×C4—82no—
405C3×C4—82no—
406C3×C4D4×Q812yes—
407C3×C4D4×Q812yes—
408C23—22no—
409C23—22no—
410C23—21no—
411C23—21no—
412C23C3×C4⋊Q812yes—
413C23C22×C3×D411yescharacteristic
414Q8—84no—
415Q8—84no—
416Q8—84no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—84no—
442C2×C4C3×#12012yes—
443C2×C4C3×#12012yes—
444C2×C4C3×C4⋊Q812yes—
445C2×C4C22×C3×D411yescharacteristic
446C2×C4C22×C3×D411yescharacteristic
447C2×C4C22×C3×Q812yes—
448C2×C4C3×2−1+412yes—
449C8—24no—
450C8—24no—
451C2×C3—42no—
452C2×C3#243411yescharacteristic
453C2×C3#300911yescharacteristic
454C2×C3#301011yescharacteristic
455C2×C3#316312yes—
456C2×C3#324412yes—
457C22—24no—
458C22—22no—
459C22—42no—
460C22—41no—
461C22—41no—
462C22C2×C3×#12011yescharacteristic
463C22C2×C3×C4⋊Q811yescharacteristic
464C22C3×#53611yescharacteristic
465C22C3×#57712yes—
466C22C3×#57712yes—
467C4—22no—
468C4—22no—
469C4—22no—
470C4—41no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—41no—
475C4—42no—
476C4—82no—
477C4—82no—
478C4C3×D4×Q812yes—
479C4C3×D4×Q812yes—
480C3#2582011yescharacteristic, direct product factor, Sylow
481C2—42no—
482C2C3×#243411yescharacteristic
483C2C3×#300911yescharacteristic
484C2C3×#301011yescharacteristic
485C2C3×#316312yes—
486C2C3×#324412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#501C2semidirect
3. C2×C3×#498C2semidirect
5. C3×#2059C2semidirect
6. C3×#2058C2semidirect
7. C3×#1993C2semidirect
8. C3×#1991C2semidirect
12. #25820C3direct (N × H)
49. C2×#501C2×C3semidirect
50. C2×#498C2×C3semidirect
52. #2059C2×C3semidirect
53. #2058C2×C3semidirect
54. #1993C2×C3semidirect
55. #1991C2×C3semidirect
114. C3×C22⋊C8Q8semidirect
273. C22⋊C8C3×Q8semidirect
480. C3#25820direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5442
C2×C3×#5012
C2×C3×#4982
C3×#26282
C3×#20592
C3×#20582
C3×#19932
C3×#19912
C3×#18542
C3×#18532
C3×#17502
#258203