Finite Group Databasebeta

← C2×C3×#3079 #10679089

Subgroups — C2×C3×#3079 #10679089

Order 768.

2358 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×C4×#41C211yesmaximal, characteristic
3C3×#3079C2116yesmaximal, direct product factor
4C2×C3×#477C212yesmaximal, complemented
5C2×C3×#475C212yesmaximal, complemented
6C2×C3×#476C212yesmaximal, complemented
7C2×C3×#474C212yesmaximal, complemented
8C2×C3×#428C212yesmaximal, complemented
9C2×C3×#423C211yesmaximal, characteristic, complemented
10C2×C3×#417C212yesmaximal
11C2×#3079C311yesmaximal, characteristic, direct product factor, Sylow
12C22×C3×#41C2211yescharacteristic
13C23×C3×Q8—21no—
14C22×C3×M16C2212yes—
15C2×C3×C4⋊Q8—21no—
16C2×C3×C4⋊Q8—21no—
17C2×C3×C4⋊Q8—24no—
18C2×C3×C4⋊Q8C2212yes—
19C2×C3×C4⋊Q8C2212yes—
20C2×C3×C4⋊Q8C2218yescomplemented
21C3×C4×#41—24no—
22C3×C4×#41C2218yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×D4—21no—
26C2×C3×C4×D4—21no—
27C2×C3×C4×D4C2211yescharacteristic
28C2×C3×C4×D4C2211yescharacteristic
29C2×C3×#117—21no—
30C2×C3×#117C2211yescharacteristic
31C22×C3×C4⋊C4—21no—
32C22×C3×C4⋊C4C2212yes—
33C22×C3×C42—21no—
34C22×C3×C42C2211yescharacteristic
35C3×#477C22116yescomplemented
36C3×#476C22116yescomplemented
37C3×#475C22116yescomplemented
38C3×#474C22116yescomplemented
39C3×#428C22116yescomplemented
40C2×C3×#107C2214yes—
41C2×C3×#106C2214yes—
42C3×#423C2218yescomplemented
43C2×C3×C4⋊C8C2212yes—
44C2×C3×C4⋊C8C2212yes—
45C3×#417C22116yes—
46C2×C3×#103C2214yes—
47C2×C3×#102C2214yes—
48C22×C4⋊Q8C2×C311yescharacteristic, complemented
49C2×C4×#41C2×C311yescharacteristic
50#3079C2×C3116yesdirect product factor
51C2×#477C2×C312yescomplemented
52C2×#475C2×C312yescomplemented
53C2×#476C2×C312yescomplemented
54C2×#474C2×C312yescomplemented
55C2×#428C2×C312yescomplemented
56C2×#423C2×C311yescharacteristic, complemented
57C2×#417C2×C312yes—
58C2×C3×#41—22no—
59C2×C3×#41—24no—
60C2×C3×#41Q818yescomplemented
61C2×C3×#41C2314yes—
62C22×C3×Q8—22no—
63C22×C3×Q8—22no—
64C22×C3×Q8—28no—
65C22×C3×Q8—48no—
66C22×C3×Q8C2311yescharacteristic
67C22×C3×D4—21no—
68C22×C3×D4C2311yescharacteristic
69C23×C3×C4—21no—
70C23×C3×C4—22no—
71C23×C3×C4—21no—
72C23×C3×C4—21no—
73C23×C3×C4D411yescharacteristic
74C23×C3×C4D411yescharacteristic
75C23×C3×C4C2311yescharacteristic
76C2×C3×M16—28no—
77C2×C3×M16C2318yes—
78C22×C3×C8C2314yes—
79C3×C4⋊Q8—24no—
80C3×C4⋊Q8—24no—
81C3×C4⋊Q8—28no—
82C3×C4⋊Q8—28no—
83C3×C4⋊Q8C2318yes—
84C3×C4⋊Q8C2318yes—
85C3×C4×Q8—22no—
86C3×C4×Q8—22no—
87C3×C4×Q8C2314yes—
88C3×C4×Q8C2314yes—
89C3×C4×D4—28no—
90C3×C4×D4—28no—
91C3×C4×D4—22no—
92C3×C4×D4—22no—
93C3×C4×D4C2314yes—
94C3×C4×D4C2314yes—
95C3×#117—22no—
96C3×#117—28no—
97C3×#117C2314yes—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4—28no—
101C2×C3×C4⋊C4—24no—
102C2×C3×C4⋊C4—22no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4C2314yes—
106C2×C3×C4⋊C4C2318yes—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C42—21no—
113C2×C3×C42—21no—
114C2×C3×C42—21no—
115C2×C3×C42—28no—
116C2×C3×C42D411yescharacteristic
117C2×C3×C42D411yescharacteristic
118C2×C3×C42D414yes—
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2311yescharacteristic
121C2×C3×C42C2314yes—
122C3×#107C23116yes—
123C3×#106C23116yes—
124C3×C4⋊C8C2318yes—
125C3×C4⋊C8C2318yes—
126C3×#103C23116yes—
127C3×#102C23116yes—
128C22×#41C22×C311yescharacteristic
129C23×Q8—21no—
130C22×M16C22×C312yes—
131C2×C4⋊Q8—21no—
132C2×C4⋊Q8—24no—
133C2×C4⋊Q8—21no—
134C2×C4⋊Q8C22×C312yes—
135C2×C4⋊Q8C22×C312yes—
136C2×C4⋊Q8C22×C318yescomplemented
137C4×#41—24no—
138C4×#41C22×C318yes—
139C2×C4×Q8C22×C311yescharacteristic
140C2×C4×Q8C22×C311yescharacteristic
141C2×C4×D4—21no—
142C2×C4×D4—21no—
143C2×C4×D4C22×C311yescharacteristic
144C2×C4×D4C22×C311yescharacteristic
145C2×#117—21no—
146C2×#117C22×C311yescharacteristic
147C22×C4⋊C4—21no—
148C22×C4⋊C4C22×C312yes—
149C22×C42—21no—
150C22×C42C22×C311yescharacteristic
151#477C22×C3116yescomplemented
152#476C22×C3116yescomplemented
153#475C22×C3116yescomplemented
154#474C22×C3116yescomplemented
155#428C22×C3116yescomplemented
156C2×#107C22×C314yes—
157C2×#106C22×C314yes—
158#423C22×C318yescomplemented
159C2×C4⋊C8C22×C312yes—
160C2×C4⋊C8C22×C312yes—
161#417C22×C3116yes—
162C2×#103C22×C314yes—
163C2×#102C22×C314yes—
164C24×C3—21no—
165C24×C3C2×D411yescharacteristic
166C3×#41—216no—
167C3×#41—28no—
168C3×#41C2×Q8116yescomplemented
169C2×C3×Q8—21no—
170C2×C3×Q8—28no—
171C2×C3×Q8—28no—
172C2×C3×Q8—22no—
173C2×C3×Q8—48no—
174C2×C3×Q8—432no—
175C2×C3×Q8—48no—
176C2×C3×Q8C2×Q814yescomplemented
177C2×C3×Q8C2412yes—
178C2×C3×D4—28no—
179C2×C3×D4—24no—
180C2×C3×D4—21no—
181C2×C3×D4—22no—
182C2×C3×D4C2×Q814yescomplemented
183C2×C3×D4C2412yes—
184C22×C3×C4—21no—
185C22×C3×C4—28no—
186C22×C3×C4—24no—
187C22×C3×C4—24no—
188C22×C3×C4—24no—
189C22×C3×C4—28no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—24no—
193C22×C3×C4—24no—
194C22×C3×C4—24no—
195C22×C3×C4—48no—
196C22×C3×C4—44no—
197C22×C3×C4—44no—
198C22×C3×C4C2×D414yes—
199C22×C3×C4C2×D414yes—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×Q814yes—
207C22×C3×C4#4114yes—
208C22×C3×C4C2412yes—
209C22×C3×C4C2411yescharacteristic
210C3×M16—216no—
211C2×C3×C8—28no—
212C2×C3×C8C2418yes—
213C3×C4⋊C4—28no—
214C3×C4⋊C4—28no—
215C3×C4⋊C4—28no—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—24no—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4C2418yes—
222C3×C4⋊C4C2412yes—
223C3×C4⋊C4C2412yes—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—24no—
227C3×C42—24no—
228C3×C42—24no—
229C3×C42—24no—
230C3×C42C2×D414yes—
231C3×C42C2×D414yes—
232C3×C42C2×D412yes—
233C3×C42C2×D412yes—
234C3×C42C2412yes—
235C3×C42C2412yes—
236C2×#41—22no—
237C2×#41—24no—
238C2×#41C3×Q818yescomplemented
239C2×#41C23×C314yes—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—28no—
243C22×Q8—48no—
244C22×Q8C23×C311yescharacteristic
245C22×D4—21no—
246C22×D4C23×C311yescharacteristic
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4C3×D411yescharacteristic
252C23×C4C3×D411yescharacteristic
253C23×C4C23×C311yescharacteristic
254C2×M16—28no—
255C2×M16C23×C318yes—
256C22×C8C23×C314yes—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—28no—
260C4⋊Q8—28no—
261C4⋊Q8C23×C318yes—
262C4⋊Q8C23×C318yes—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8C23×C314yes—
266C4×Q8C23×C314yes—
267C4×D4—28no—
268C4×D4—28no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4C23×C314yes—
272C4×D4C23×C314yes—
273#117—22no—
274#117—28no—
275#117C23×C314yes—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4C23×C314yes—
284C2×C4⋊C4C23×C318yes—
285C2×C4⋊C4C23×C311yescharacteristic
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—28no—
294C2×C42C3×D411yescharacteristic
295C2×C42C3×D414yes—
296C2×C42C3×D411yescharacteristic
297C2×C42C23×C311yescharacteristic
298C2×C42C23×C311yescharacteristic
299C2×C42C23×C314yes—
300#107C23×C3116yes—
301#106C23×C3116yes—
302C4⋊C8C23×C318yes—
303C4⋊C8C23×C318yes—
304#103C23×C3116yes—
305#102C23×C3116yes—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—24no—
309C23×C3—22no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3C22⋊Q814yes—
313C23×C3C22×D412yes—
314C23×C3C22×D411yescharacteristic
315C3×Q8—24no—
316C3×Q8—22no—
317C3×Q8—416no—
318C3×Q8—416no—
319C3×Q8C22×Q814yescomplemented
320C3×D4—216no—
321C3×D4—24no—
322C3×D4—22no—
323C3×D4C22×Q814yescomplemented
324C2×C3×C4—24no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—216no—
329C2×C3×C4—28no—
330C2×C3×C4—28no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—28no—
334C2×C3×C4—416no—
335C2×C3×C4—42no—
336C2×C3×C4—48no—
337C2×C3×C4—42no—
338C2×C3×C4—48no—
339C2×C3×C4—48no—
340C2×C3×C4C22⋊Q814yes—
341C2×C3×C4C22⋊Q818yes—
342C2×C3×C4AGL(1,ℤ/8)14yes—
343C2×C3×C4#13714yes—
344C2×C3×C4C22×D412yes—
345C2×C3×C4C22×D412yes—
346C2×C3×C4C22×D412yes—
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C22×D412yes—
349C2×C3×C4C22×D412yes—
350C2×C3×C4C22×Q812yes—
351C2×C3×C4C22×Q814yes—
352C2×C3×C4C2×#4112yes—
353C2×C3×C4C2×#4114yes—
354C2×C3×C4C2511yescharacteristic
355C3×C8—28no—
356C24—21no—
357C24C2×C3×D411yescharacteristic
358#41—216no—
359#41—28no—
360#41C2×C3×Q8116yescomplemented
361C2×Q8—21no—
362C2×Q8—28no—
363C2×Q8—28no—
364C2×Q8—22no—
365C2×Q8—48no—
366C2×Q8—432no—
367C2×Q8—48no—
368C2×Q8C2×C3×Q814yescomplemented
369C2×Q8C24×C312yes—
370C2×D4—28no—
371C2×D4—24no—
372C2×D4—21no—
373C2×D4—22no—
374C2×D4C2×C3×Q814yescomplemented
375C2×D4C24×C312yes—
376C22×C4—21no—
377C22×C4—28no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—28no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—48no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4C2×C3×D411yescharacteristic
391C22×C4C2×C3×D411yescharacteristic
392C22×C4C2×C3×D414yes—
393C22×C4C2×C3×D414yes—
394C22×C4C2×C3×D411yescharacteristic
395C22×C4C2×C3×D412yes—
396C22×C4C2×C3×D411yescharacteristic
397C22×C4C2×C3×D411yescharacteristic
398C22×C4C2×C3×Q814yes—
399C22×C4C3×#4114yes—
400C22×C4C24×C311yescharacteristic
401C22×C4C24×C312yes—
402M16—216no—
403C2×C8—28no—
404C2×C8C24×C318yes—
405C4⋊C4—28no—
406C4⋊C4—28no—
407C4⋊C4—28no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4C24×C318yes—
414C4⋊C4C24×C312yes—
415C4⋊C4C24×C312yes—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C42—24no—
420C42—24no—
421C42—24no—
422C42C2×C3×D414yes—
423C42C2×C3×D414yes—
424C42C2×C3×D412yes—
425C42C2×C3×D412yes—
426C42C24×C312yes—
427C42C24×C312yes—
428C22×C3—28no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—24no—
432C22×C3—48no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3C2×C22⋊Q814yes—
436C22×C3C2×C22⋊Q812yes—
437C22×C3C2×AGL(1,ℤ/8)12yes—
438C22×C3C2×#13712yes—
439C22×C3C23×D411yescharacteristic
440C3×C4—24no—
441C3×C4—22no—
442C3×C4—22no—
443C3×C4—42no—
444C3×C4—48no—
445C3×C4—42no—
446C3×C4C2×C22⋊Q814yes—
447C3×C4C2×C22⋊Q812yes—
448C3×C4C2×AGL(1,ℤ/8)14yes—
449C3×C4C2×#13714yes—
450C3×C4C23×Q811yescharacteristic
451C3×C4C22×#4111yescharacteristic
452C23—22no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—42no—
457C23—42no—
458C23C3×C22⋊Q814yes—
459C23C22×C3×D412yes—
460C23C22×C3×D411yescharacteristic
461Q8—22no—
462Q8—24no—
463Q8—416no—
464Q8—416no—
465Q8C22×C3×Q814yescomplemented
466D4—216no—
467D4—22no—
468D4—24no—
469D4C22×C3×Q814yescomplemented
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—28no—
475C2×C4—28no—
476C2×C4—216no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—28no—
480C2×C4—416no—
481C2×C4—48no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—48no—
485C2×C4—48no—
486C2×C4C3×C22⋊Q814yes—
487C2×C4C3×C22⋊Q818yes—
488C2×C4C3×AGL(1,ℤ/8)14yes—
489C2×C4C3×#13714yes—
490C2×C4C22×C3×D412yes—
491C2×C4C22×C3×D412yes—
492C2×C4C22×C3×D412yes—
493C2×C4C22×C3×D412yes—
494C2×C4C22×C3×D412yes—
495C2×C4C22×C3×D411yescharacteristic
496C2×C4C22×C3×Q812yes—
497C2×C4C22×C3×Q814yes—
498C2×C4C2×C3×#4112yes—
499C2×C4C2×C3×#4114yes—
500C2×C4C25×C311yescharacteristic
501C8—28no—
502C2×C3—24no—
503C2×C3—44no—
504C2×C3#307914yesdirect product factor
505C2×C3C22×C22⋊Q811yescharacteristic
506C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
507C2×C3C22×#13711yescharacteristic
508C22—22no—
509C22—22no—
510C22—28no—
511C22—24no—
512C22—42no—
513C22—42no—
514C22—48no—
515C22C2×C3×C22⋊Q814yes—
516C22C2×C3×C22⋊Q812yes—
517C22C2×C3×AGL(1,ℤ/8)12yes—
518C22C2×C3×#13712yes—
519C22C23×C3×D411yescharacteristic
520C4—24no—
521C4—22no—
522C4—22no—
523C4—48no—
524C4—42no—
525C4—42no—
526C4C2×C3×C22⋊Q814yes—
527C4C2×C3×C22⋊Q812yes—
528C4C2×C3×AGL(1,ℤ/8)14yes—
529C4C2×C3×#13714yes—
530C4C23×C3×Q811yescharacteristic
531C4C22×C3×#4111yescharacteristic
532C3C2×#307911yescharacteristic, direct product factor, Sylow
533C2—24no—
534C2—44no—
535C2C3×#307914yesdirect product factor
536C2C22×C3×C22⋊Q811yescharacteristic
537C2C22×C3×AGL(1,ℤ/8)11yescharacteristic
538C2C22×C3×#13711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC22×C3×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4⋊Q8C2semidirect
3. C3×#3079C2direct (N × H)
4. C2×C3×#477C2semidirect
5. C2×C3×#475C2semidirect
6. C2×C3×#476C2semidirect
7. C2×C3×#474C2semidirect
8. C2×C3×#428C2semidirect
9. C2×C3×#423C2semidirect
11. C2×#3079C3direct (N × H)
20. C2×C3×C4⋊Q8C22semidirect
35. C3×#477C22semidirect
36. C3×#476C22semidirect
37. C3×#475C22semidirect
38. C3×#474C22semidirect
39. C3×#428C22semidirect
42. C3×#423C22semidirect
48. C22×C4⋊Q8C2×C3semidirect
50. #3079C2×C3direct (N × H)
51. C2×#477C2×C3semidirect
52. C2×#475C2×C3semidirect
53. C2×#476C2×C3semidirect
54. C2×#474C2×C3semidirect
55. C2×#428C2×C3semidirect
56. C2×#423C2×C3semidirect
60. C2×C3×#41Q8semidirect
136. C2×C4⋊Q8C22×C3semidirect
151. #477C22×C3semidirect
152. #476C22×C3semidirect
153. #475C22×C3semidirect
154. #474C22×C3semidirect
155. #428C22×C3semidirect
158. #423C22×C3semidirect
168. C3×#41C2×Q8semidirect
176. C2×C3×Q8C2×Q8semidirect
182. C2×C3×D4C2×Q8semidirect
238. C2×#41C3×Q8semidirect
319. C3×Q8C22×Q8semidirect
323. C3×D4C22×Q8semidirect
360. #41C2×C3×Q8semidirect
368. C2×Q8C2×C3×Q8semidirect
374. C2×D4C2×C3×Q8semidirect
465. Q8C22×C3×Q8semidirect
469. D4C22×C3×Q8semidirect
504. C2×C3#3079direct (N × H)
532. C3C2×#3079direct (N × H)
535. C2C3×#3079direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Q82
C2×C3×C4×#412
C3×#30792
C2×C3×#4772
C2×C3×#4752
C2×C3×#4762
C2×C3×#4742
C2×C3×#4282
C2×C3×#4232
C2×C3×#4172
C2×#30793