Finite Group Databasebeta

← C3×#16688 #10635295

Subgroups — C3×#16688 #10635295

Order 768.

1150 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#398C211yesmaximal, characteristic
2C2×C3×#387C212yesmaximal
3C2×C3×#382C211yesmaximal, characteristic
4C3×#1443C212yesmaximal
5C3×#1442C212yesmaximal
6C3×#1442C212yesmaximal
7C3×#1441C212yesmaximal
8C3×#1441C211yesmaximal, characteristic
9C3×#1441C212yesmaximal
10#16688C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×C4⋊C4C2212yes—
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C3×#398—24no—
19C3×#387—28no—
20C3×#382—24no—
21C2×C3×#95C2212yes—
22C2×C3×#95C2211yescharacteristic
23C2×C3×#95C2212yes—
24C2×C3×#95C2212yes—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2211yescharacteristic
35C2×#398C2×C311yescharacteristic
36C2×#387C2×C312yes—
37C2×#382C2×C311yescharacteristic
38#1443C2×C312yes—
39#1442C2×C312yes—
40#1442C2×C312yes—
41#1441C2×C312yes—
42#1441C2×C312yes—
43#1441C2×C311yescharacteristic
44C23×C3×C4C2312yes—
45C23×C3×C4C2312yes—
46C23×C3×C4C2312yes—
47C23×C3×C4C2312yes—
48C23×C3×C4C2311yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2312yes—
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4—24no—
55C2×C3×C4⋊C4—24no—
56C2×C3×C4⋊C4—22no—
57C2×C3×C4⋊C4—24no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—24no—
61C2×C3×C4⋊C4—24no—
62C2×C3×C4⋊C4—41no—
63C2×C3×C4⋊C4—41no—
64C2×C3×C4⋊C4—41no—
65C2×C3×C4⋊C4—41no—
66C2×C3×C4⋊C4—41no—
67C2×C3×C4⋊C4—41no—
68C2×C3×C4⋊C4—41no—
69C2×C3×C4⋊C4D414yes—
70C2×C3×C42—22no—
71C2×C3×C42—22no—
72C2×C3×C42D414yes—
73C3×#95—24no—
74C3×#95—24no—
75C3×#95—24no—
76C3×#95—24no—
77C3×#95—24no—
78C3×#95—24no—
79C3×#95—22no—
80C3×#95—24no—
81C3×#95—24no—
82C3×#95—24no—
83C3×#95—24no—
84C3×#95—41no—
85C3×#95—42no—
86C3×#95—42no—
87C22×C4⋊C4C22×C312yes—
88C22×C4⋊C4C22×C312yes—
89C22×C4⋊C4C22×C311yescharacteristic
90C22×C4⋊C4C22×C311yescharacteristic
91C22×C4⋊C4C22×C311yescharacteristic
92C22×C4⋊C4C22×C311yescharacteristic
93C22×C42C22×C311yescharacteristic
94#398—24no—
95#387—28no—
96#382—24no—
97C2×#95C22×C312yes—
98C2×#95C22×C311yescharacteristic
99C2×#95C22×C312yes—
100C2×#95C22×C312yes—
101C2×#95C22×C312yes—
102C2×#95C22×C312yes—
103C2×#95C22×C312yes—
104C2×#95C22×C312yes—
105C2×#95C22×C312yes—
106C2×#95C22×C312yes—
107C2×#95C22×C312yes—
108C2×#95C22×C312yes—
109C2×#95C22×C312yes—
110C2×#95C22×C311yescharacteristic
111C24×C3C2411yescharacteristic
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D414yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4#4114yes—
169C22×C3×C4#4114yes—
170C22×C3×C4#4112yes—
171C3×C4⋊C4—216no—
172C3×C4⋊C4—44no—
173C3×C4⋊C4—44no—
174C3×C4⋊C4—44no—
175C3×C4⋊C4—44no—
176C3×C4⋊C4—48no—
177C3×C42—28no—
178C23×C4C23×C312yes—
179C23×C4C23×C312yes—
180C23×C4C23×C312yes—
181C23×C4C23×C311yescharacteristic
182C23×C4C23×C311yescharacteristic
183C23×C4C23×C312yes—
184C23×C4C23×C312yes—
185C23×C4C23×C312yes—
186C23×C4C23×C311yescharacteristic
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4C3×D414yes—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42C3×D414yes—
207#95—24no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—24no—
213#95—22no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—41no—
219#95—42no—
220#95—42no—
221C23×C3C22×D411yescharacteristic
222C23×C3C22×D411yescharacteristic
223C23×C3C2×#4111yescharacteristic
224C23×C3C2×#4112yes—
225C23×C3C2×#4112yes—
226C23×C3H(5,𝔽2)11yescharacteristic
227C23×C3H(5,𝔽2)11yescharacteristic
228C23×C32−1+411yescharacteristic
229C23×C32−1+411yescharacteristic
230C23×C32−1+412yes—
231C23×C32−1+412yes—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—24no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4#12114yes—
291C24C24×C311yescharacteristic
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4C2×C3×D412yes—
344C22×C4C2×C3×D414yes—
345C22×C4C2×C3×D412yes—
346C22×C4C2×C3×D412yes—
347C22×C4C2×C3×D412yes—
348C22×C4C3×#4114yes—
349C22×C4C3×#4114yes—
350C22×C4C3×#4112yes—
351C4⋊C4—216no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—48no—
357C42—28no—
358C22×C3C2×#12111yescharacteristic
359C22×C3#52512yes—
360C22×C3#53411yescharacteristic
361C22×C3#53612yes—
362C22×C3#53611yescharacteristic
363C22×C3#53711yescharacteristic
364C22×C3#53711yescharacteristic
365C22×C3#53911yescharacteristic
366C22×C3#53911yescharacteristic
367C22×C3#54112yes—
368C22×C3#54112yes—
369C22×C3#54212yes—
370C22×C3#54212yes—
371C22×C3#54712yes—
372C22×C3#54812yes—
373C22×C3#55111yescharacteristic
374C22×C3#55212yes—
375C22×C3#55212yes—
376C22×C3#55511yescharacteristic
377C22×C3#55612yes—
378C22×C3#56311yescharacteristic
379C22×C3#56311yescharacteristic
380C22×C3#56412yes—
381C3×C4—24no—
382C3×C4—44no—
383C3×C4—44no—
384C3×C4—44no—
385C3×C4—42no—
386C3×C4—42no—
387C3×C4—82no—
388C3×C4—82no—
389C3×C4—82no—
390C23C22×C3×D411yescharacteristic
391C23C22×C3×D411yescharacteristic
392C23C2×C3×#4111yescharacteristic
393C23C2×C3×#4112yes—
394C23C2×C3×#4112yes—
395C23C3×H(5,𝔽2)11yescharacteristic
396C23C3×H(5,𝔽2)11yescharacteristic
397C23C3×2−1+412yes—
398C23C3×2−1+412yes—
399C23C3×2−1+411yescharacteristic
400C23C3×2−1+411yescharacteristic
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
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—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4C3×#12114yes—
460C2×C3#241512yes—
461C2×C3#260911yescharacteristic
462C2×C3#261811yescharacteristic
463C2×C3#263712yes—
464C2×C3#271711yescharacteristic
465C2×C3#271912yes—
466C2×C3#273111yescharacteristic
467C2×C3#276712yes—
468C2×C3#276911yescharacteristic
469C2×C3#278911yescharacteristic
470C2×C3#282211yescharacteristic
471C22C2×C3×#12111yescharacteristic
472C22C3×#52512yes—
473C22C3×#53411yescharacteristic
474C22C3×#53611yescharacteristic
475C22C3×#53612yes—
476C22C3×#53711yescharacteristic
477C22C3×#53711yescharacteristic
478C22C3×#53911yescharacteristic
479C22C3×#53911yescharacteristic
480C22C3×#54112yes—
481C22C3×#54112yes—
482C22C3×#54212yes—
483C22C3×#54212yes—
484C22C3×#54712yes—
485C22C3×#54812yes—
486C22C3×#55111yescharacteristic
487C22C3×#55212yes—
488C22C3×#55212yes—
489C22C3×#55511yescharacteristic
490C22C3×#55612yes—
491C22C3×#56311yescharacteristic
492C22C3×#56311yescharacteristic
493C22C3×#56412yes—
494C4—24no—
495C4—44no—
496C4—44no—
497C4—42no—
498C4—42no—
499C4—44no—
500C4—82no—
501C4—82no—
502C4—82no—
503C3#1668811yescharacteristic, direct product factor, Sylow
504C2C3×#241512yes—
505C2C3×#260911yescharacteristic
506C2C3×#261811yescharacteristic
507C2C3×#263712yes—
508C2C3×#271711yescharacteristic
509C2C3×#271912yes—
510C2C3×#273111yescharacteristic
511C2C3×#276712yes—
512C2C3×#276911yescharacteristic
513C2C3×#278911yescharacteristic
514C2C3×#282211yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C3C24
commutator, Frattini, cosocleC24C24×C3
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
10. #16688C3direct (N × H)
503. C3#16688direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3982
C2×C3×#3872
C2×C3×#3822
C3×#14432
C3×#14422
C3×#14412
#166883