Finite Group Databasebeta

← C3×#25830 #10644437

Subgroups — C3×#25830 #10644437

Order 768.

1054 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3503C211yesmaximal, characteristic
2C3×#3165C211yesmaximal, characteristic, complemented
3C3×#3079C211yesmaximal, characteristic
4C3×#2868C211yesmaximal, characteristic, complemented
5C3×#2060C214yesmaximal, complemented
6C3×#1997C211yesmaximal, characteristic, complemented
7C3×#1994C211yesmaximal, characteristic, complemented
8C3×#1859C214yesmaximal
9C3×C4×#104C211yesmaximal, characteristic
10#25830C311yesmaximal, characteristic, direct product factor, Sylow
11C3×Q82—21no—
12C3×#557—21no—
13C3×#554—21no—
14C3×#554—22no—
15C3×D4×Q8—22no—
16C3×D4×Q8—21no—
17C3×#544—22no—
18C3×#544C2212yes—
19C3×#533—21no—
20C3×#533C2211yescharacteristic
21C2×C3×C4⋊Q8—21no—
22C2×C3×C4⋊Q8—22no—
23C2×C3×C4⋊Q8C2211yescharacteristic
24C2×C3×C4⋊Q8C2211yescharacteristic
25C3×#520C2212yes—
26C3×C4×#41C2211yescharacteristic
27C3×#501—22no—
28C3×#500—21no—
29C3×#498—21no—
30C3×#477—21no—
31C3×#476—21no—
32C3×#475—21no—
33C3×#474—21no—
34C3×#428C2214yes—
35C3×#428C2212yes—
36C3×#423C2211yescharacteristic
37C2×C3×#104C2212yes—
38C3×#417C2212yes—
39C3×C4×M16C2211yescharacteristic
40C3×#384—22no—
41C3×#384C2214yes—
42C3×C43C2211yescharacteristic
43C3×#343C2214yes—
44C3×#339C2214yes—
45C3×#339C2212yes—
46#3503C2×C311yescharacteristic
47#3165C2×C311yescharacteristic, complemented
48#3079C2×C311yescharacteristic
49#2868C2×C311yescharacteristic, complemented
50#2060C2×C314yescomplemented
51#1997C2×C311yescharacteristic, complemented
52#1994C2×C311yescharacteristic, complemented
53#1859C2×C314yes—
54C4×#104C2×C311yescharacteristic
55C2×C3×#41C2311yescharacteristic
56C22×C3×Q8—22no—
57C22×C3×Q8—21no—
58C22×C3×Q8D412yes—
59C2×C3×M16C2312yes—
60C2×C3×M16C2312yes—
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8—24no—
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8—21no—
69C3×C4⋊Q8—21no—
70C3×C4⋊Q8—21no—
71C3×C4⋊Q8—21no—
72C3×C4⋊Q8—21no—
73C3×C4⋊Q8—21no—
74C3×C4⋊Q8—48no—
75C3×C4⋊Q8D411yescharacteristic
76C3×C4⋊Q8D411yescharacteristic
77C3×#125—24no—
78C3×#125—22no—
79C3×#125—22no—
80C3×#125D412yes—
81C3×C22⋊Q8—22no—
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8—24no—
84C3×C22⋊Q8—24no—
85C3×C22⋊Q8—21no—
86C3×C22⋊Q8—21no—
87C3×C22⋊Q8—24no—
88C3×C4×Q8—22no—
89C3×C4×Q8—22no—
90C3×C4×Q8—22no—
91C3×C4×Q8—21no—
92C3×C4×Q8—21no—
93C3×C4×Q8D411yescharacteristic
94C3×C4×Q8D411yescharacteristic
95C3×C4×D4—22no—
96C3×C4×D4—21no—
97C3×C4×D4—21no—
98C3×C4×D4—24no—
99C3×C4×D4D411yescharacteristic
100C3×C4×D4D411yescharacteristic
101C3×#117—22no—
102C3×#117—21no—
103C3×#117C2311yescharacteristic
104C3×#117C2312yes—
105C2×C3×C4⋊C4—22no—
106C2×C3×C4⋊C4—22no—
107C2×C3×C4⋊C4—21no—
108C2×C3×C4⋊C4—28no—
109C2×C3×C4⋊C4D412yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C4⋊C4C2312yes—
112C2×C3×C42—21no—
113C2×C3×C42—21no—
114C2×C3×C42—21no—
115C2×C3×C42C2312yes—
116C2×C3×C42C2311yescharacteristic
117C3×#107—22no—
118C3×#107—24no—
119C3×#106—22no—
120C3×#106—24no—
121C3×C4⋊C8—21no—
122C3×C4⋊C8—21no—
123C3×#104Q818yescomplemented
124C3×#103—22no—
125C3×#102—22no—
126C3×#97—21no—
127C3×C4×C8—21no—
128Q82—21no—
129#557—21no—
130#554—21no—
131#554—22no—
132D4×Q8—22no—
133D4×Q8—21no—
134#544—22no—
135#544C22×C312yes—
136#533—21no—
137#533C22×C311yescharacteristic
138C2×C4⋊Q8—21no—
139C2×C4⋊Q8—22no—
140C2×C4⋊Q8C22×C311yescharacteristic
141C2×C4⋊Q8C22×C311yescharacteristic
142#520C22×C312yes—
143C4×#41C22×C311yescharacteristic
144#501—22no—
145#500—21no—
146#498—21no—
147#477—21no—
148#476—21no—
149#475—21no—
150#474—21no—
151#428C22×C312yes—
152#428C22×C314yes—
153#423C22×C311yescharacteristic
154C2×#104C22×C312yes—
155#417C22×C312yes—
156C4×M16C22×C311yescharacteristic
157#384—22no—
158#384C22×C314yes—
159C43C22×C311yescharacteristic
160#343C22×C314yes—
161#339C22×C312yes—
162#339C22×C314yes—
163C3×#41—22no—
164C3×#41C2×Q814yes—
165C2×C3×Q8—24no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—44no—
171C2×C3×Q8—44no—
172C2×C3×Q8—41no—
173C2×C3×Q8—41no—
174C2×C3×Q8—84no—
175C2×C3×Q8C2×D411yescharacteristic
176C2×C3×D4—21no—
177C2×C3×D4C2×D411yescharacteristic
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—24no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D414yes—
187C22×C3×C4C2411yescharacteristic
188C3×M16—22no—
189C3×M16—24no—
190C3×M16C2×Q814yescomplemented
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—416no—
210C3×C4⋊C4—41no—
211C3×C4⋊C4—41no—
212C3×C4⋊C4C2×D412yes—
213C3×C4⋊C4C2×D411yescharacteristic
214C3×C4⋊C4C2×D412yes—
215C3×C4⋊C4C2×D411yescharacteristic
216C3×C22⋊C4—24no—
217C3×C22⋊C4—22no—
218C3×C22⋊C4—21no—
219C3×C22⋊C4—21no—
220C3×C22⋊C4—21no—
221C3×C42—24no—
222C3×C42—24no—
223C3×C42—24no—
224C3×C42—21no—
225C3×C42—21no—
226C3×C42—21no—
227C3×C42—22no—
228C3×C42—22no—
229C3×C42C2×D411yescharacteristic
230C3×C42C2×D411yescharacteristic
231C3×C42C2×D411yescharacteristic
232C3×C42C2×D411yescharacteristic
233C3×C42C2×Q814yescomplemented
234C2×#41C23×C311yescharacteristic
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8C3×D412yes—
238C2×M16C23×C312yes—
239C2×M16C23×C312yes—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242C4⋊Q8—24no—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—48no—
254C4⋊Q8C3×D411yescharacteristic
255C4⋊Q8C3×D411yescharacteristic
256#125—24no—
257#125—22no—
258#125—22no—
259#125C3×D412yes—
260C22⋊Q8—22no—
261C22⋊Q8—22no—
262C22⋊Q8—24no—
263C22⋊Q8—24no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C22⋊Q8—24no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8C3×D411yescharacteristic
273C4×Q8C3×D411yescharacteristic
274C4×D4—22no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—24no—
278C4×D4C3×D411yescharacteristic
279C4×D4C3×D411yescharacteristic
280#117—22no—
281#117—21no—
282#117C23×C311yescharacteristic
283#117C23×C312yes—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4C3×D412yes—
289C2×C4⋊C4C23×C312yes—
290C2×C4⋊C4C23×C312yes—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42C23×C312yes—
295C2×C42C23×C311yescharacteristic
296#107—22no—
297#107—24no—
298#106—22no—
299#106—24no—
300C4⋊C8—21no—
301C4⋊C8—21no—
302#104C3×Q818yescomplemented
303#103—22no—
304#102—22no—
305#97—21no—
306C4×C8—21no—
307C23×C3—21no—
308C23×C3C22×D411yescharacteristic
309C3×Q8—21no—
310C3×Q8—44no—
311C3×Q8—42no—
312C3×Q8—42no—
313C3×Q8—82no—
314C3×Q8—82no—
315C3×Q8—84no—
316C3×Q8—82no—
317C3×Q8C4⋊Q812yescomplemented
318C3×D4—24no—
319C3×D4—21no—
320C3×D4C4⋊Q812yescomplemented
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—21no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—44no—
336C2×C3×C4—44no—
337C2×C3×C4—42no—
338C2×C3×C4—84no—
339C2×C3×C4#12011yescharacteristic
340C2×C3×C4#12011yescharacteristic
341C2×C3×C4#12011yescharacteristic
342C2×C3×C4#12011yescharacteristic
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C22×Q812yes—
346C2×C3×C42−1+412yes—
347C3×C8—24no—
348C3×C8—42no—
349#41—22no—
350#41C2×C3×Q814yes—
351C2×Q8—24no—
352C2×Q8—21no—
353C2×Q8—22no—
354C2×Q8—21no—
355C2×Q8—22no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—84no—
361C2×Q8C2×C3×D411yescharacteristic
362C2×D4—21no—
363C2×D4C2×C3×D411yescharacteristic
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4C2×C3×D411yescharacteristic
371C22×C4C2×C3×D411yescharacteristic
372C22×C4C2×C3×D414yes—
373C22×C4C24×C311yescharacteristic
374M16—22no—
375M16—24no—
376M16C2×C3×Q814yescomplemented
377C2×C8—22no—
378C2×C8—22no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—416no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4C2×C3×D412yes—
399C4⋊C4C2×C3×D411yescharacteristic
400C4⋊C4C2×C3×D412yes—
401C4⋊C4C2×C3×D411yescharacteristic
402C22⋊C4—24no—
403C22⋊C4—22no—
404C22⋊C4—21no—
405C22⋊C4—21no—
406C22⋊C4—21no—
407C42—24no—
408C42—24no—
409C42—24no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—22no—
414C42—22no—
415C42C2×C3×D411yescharacteristic
416C42C2×C3×D411yescharacteristic
417C42C2×C3×D411yescharacteristic
418C42C2×C3×D411yescharacteristic
419C42C2×C3×Q814yescomplemented
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—22no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3C2×#12011yescharacteristic
426C22×C3D4×Q812yes—
427C3×C4—28no—
428C3×C4—22no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—41no—
432C3×C4—44no—
433C3×C4—42no—
434C3×C4—41no—
435C3×C4—42no—
436C3×C4—82no—
437C3×C4#45312yes—
438C3×C4#45612yes—
439C3×C4C2×C4⋊Q811yescharacteristic
440C3×C4#53611yescharacteristic
441C3×C4D4×Q812yes—
442C23—21no—
443C23C22×C3×D411yescharacteristic
444Q8—21no—
445Q8—44no—
446Q8—42no—
447Q8—42no—
448Q8—82no—
449Q8—82no—
450Q8—84no—
451Q8—82no—
452Q8C3×C4⋊Q812yescomplemented
453D4—24no—
454D4—21no—
455D4C3×C4⋊Q812yescomplemented
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—21no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—84no—
474C2×C4C3×#12011yescharacteristic
475C2×C4C3×#12011yescharacteristic
476C2×C4C3×#12011yescharacteristic
477C2×C4C3×#12011yescharacteristic
478C2×C4C22×C3×D411yescharacteristic
479C2×C4C22×C3×D411yescharacteristic
480C2×C4C22×C3×Q812yes—
481C2×C4C3×2−1+412yes—
482C8—24no—
483C8—42no—
484C2×C3—22no—
485C2×C3—42no—
486C2×C3#243411yescharacteristic
487C2×C3C2×#45311yescharacteristic
488C2×C3C2×#45611yescharacteristic
489C22—21no—
490C22—21no—
491C22—22no—
492C22—41no—
493C22—41no—
494C22C2×C3×#12011yescharacteristic
495C22C3×D4×Q812yes—
496C4—28no—
497C4—22no—
498C4—21no—
499C4—21no—
500C4—41no—
501C4—44no—
502C4—42no—
503C4—41no—
504C4—42no—
505C4—82no—
506C4C3×#45312yes—
507C4C3×#45612yes—
508C4C2×C3×C4⋊Q811yescharacteristic
509C4C3×#53611yescharacteristic
510C4C3×D4×Q812yes—
511C3#2583011yescharacteristic, direct product factor, Sylow
512C2—22no—
513C2—42no—
514C2C3×#243411yescharacteristic
515C2C2×C3×#45311yescharacteristic
516C2C2×C3×#45611yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3165C2semidirect
4. C3×#2868C2semidirect
5. C3×#2060C2semidirect
6. C3×#1997C2semidirect
7. C3×#1994C2semidirect
10. #25830C3direct (N × H)
47. #3165C2×C3semidirect
49. #2868C2×C3semidirect
50. #2060C2×C3semidirect
51. #1997C2×C3semidirect
52. #1994C2×C3semidirect
123. C3×#104Q8semidirect
190. C3×M16C2×Q8semidirect
233. C3×C42C2×Q8semidirect
302. #104C3×Q8semidirect
317. C3×Q8C4⋊Q8semidirect
320. C3×D4C4⋊Q8semidirect
376. M16C2×C3×Q8semidirect
419. C42C2×C3×Q8semidirect
452. Q8C3×C4⋊Q8semidirect
455. D4C3×C4⋊Q8semidirect
511. C3#25830direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35032
C3×#31652
C3×#30792
C3×#28682
C3×#20602
C3×#19972
C3×#19942
C3×#18592
C3×C4×#1042
#258303