Finite Group Databasebeta

← C32×#2416 #11769465

Subgroups — C32×#2416 #11769465

Order 1152.

2364 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#121C211yesmaximal, characteristic, complemented
2C2×C32×#121C212yesmaximal, complemented
3C2×C32×#121C211yesmaximal, characteristic, complemented
4C2×C32×C4×D4C211yesmaximal, characteristic, complemented
5C22×C32×C4⋊C4C211yesmaximal, characteristic, complemented
6C32×#396C212yesmaximal, complemented
7C32×#394C212yesmaximal, complemented
8C32×#390C211yesmaximal, characteristic, complemented
9C32×#389C211yesmaximal, characteristic, complemented
10C32×#386C212yesmaximal, complemented
11C32×#380C211yesmaximal, characteristic, complemented
12C3×#2416C314yesmaximal, direct product factor
13C22×C32×D4C2212yes—
14C22×C32×D4C2211yescharacteristic
15C22×C32×D4C2212yes—
16C22×C32×D4C2211yescharacteristic
17C23×C32×C4C2211yescharacteristic
18C23×C32×C4C2212yes—
19C23×C32×C4C2211yescharacteristic
20C32×#121—28no—
21C32×#121—24no—
22C32×#121—24no—
23C32×C4×D4—24no—
24C2×C32×C4⋊C4—24no—
25C2×C32×C4⋊C4C2211yescharacteristic
26C2×C32×C4⋊C4C2211yescharacteristic
27C2×C32×C4⋊C4C2212yes—
28C2×C32×C4⋊C4C2212yes—
29C2×C32×C4⋊C4C2211yescharacteristic
30C2×C32×C4⋊C4C2211yescharacteristic
31C2×C32×C22⋊C4C2212yes—
32C2×C32×C22⋊C4C2211yescharacteristic
33C2×C32×C22⋊C4C2212yes—
34C2×C32×C22⋊C4C2212yes—
35C2×C32×C22⋊C4C2211yescharacteristic
36C2×C32×C22⋊C4C2212yes—
37C2×C32×C22⋊C4C2212yes—
38C2×C32×C42C2211yescharacteristic
39C32×#95C2212yes—
40C32×#95C2212yes—
41C2×C3×#121C2×C318yescomplemented
42C2×C3×#121C2×C314yescomplemented
43C2×C3×#121C2×C314yescomplemented
44C2×C3×C4×D4C2×C314yescomplemented
45C22×C3×C4⋊C4C2×C314yescomplemented
46C3×#396C2×C318yescomplemented
47C3×#394C2×C318yescomplemented
48C3×#390C2×C314yescomplemented
49C3×#389C2×C314yescomplemented
50C3×#386C2×C318yescomplemented
51C3×#380C2×C314yescomplemented
52C24×C32C2312yes—
53C24×C32C2311yescharacteristic
54C24×C32C2311yescharacteristic
55C2×C32×D4—22no—
56C2×C32×D4—22no—
57C2×C32×D4—24no—
58C2×C32×D4—24no—
59C2×C32×D4—24no—
60C2×C32×D4—24no—
61C2×C32×D4—24no—
62C2×C32×D4—22no—
63C2×C32×D4—22no—
64C2×C32×D4—22no—
65C2×C32×D4—42no—
66C2×C32×D4D414yescomplemented
67C22×C32×C4—22no—
68C22×C32×C4—22no—
69C22×C32×C4—24no—
70C22×C32×C4—24no—
71C22×C32×C4—22no—
72C22×C32×C4—22no—
73C22×C32×C4—22no—
74C22×C32×C4—42no—
75C22×C32×C4D414yescomplemented
76C22×C32×C4C2312yes—
77C22×C32×C4C2311yescharacteristic
78C22×C32×C4C2312yes—
79C22×C32×C4C2311yescharacteristic
80C22×C32×C4C2312yes—
81C22×C32×C4C2311yescharacteristic
82C22×C32×C4C2311yescharacteristic
83C22×C32×C4C2311yescharacteristic
84C32×C4⋊C4—24no—
85C32×C4⋊C4—22no—
86C32×C4⋊C4—22no—
87C32×C4⋊C4—24no—
88C32×C4⋊C4—22no—
89C32×C4⋊C4D414yescomplemented
90C32×C22⋊C4—22no—
91C32×C22⋊C4—24no—
92C32×C22⋊C4—22no—
93C32×C22⋊C4—24no—
94C32×C22⋊C4—24no—
95C32×C22⋊C4—24no—
96C32×C22⋊C4—42no—
97C32×C42—22no—
98#2416C3211yescharacteristic, direct product factor, Sylow
99C22×C3×D4C22×C318yes—
100C22×C3×D4C22×C314yes—
101C22×C3×D4C22×C318yes—
102C22×C3×D4C22×C314yes—
103C23×C3×C4C22×C314yes—
104C23×C3×C4C22×C318yes—
105C23×C3×C4C22×C314yes—
106C3×#121—232no—
107C3×#121—216no—
108C3×#121—216no—
109C3×C4×D4—216no—
110C2×C3×C4⋊C4—216no—
111C2×C3×C4⋊C4C22×C314yes—
112C2×C3×C4⋊C4C22×C314yes—
113C2×C3×C4⋊C4C22×C318yes—
114C2×C3×C4⋊C4C22×C318yes—
115C2×C3×C4⋊C4C22×C314yes—
116C2×C3×C4⋊C4C22×C314yes—
117C2×C3×C22⋊C4C22×C318yes—
118C2×C3×C22⋊C4C22×C314yes—
119C2×C3×C22⋊C4C22×C318yes—
120C2×C3×C22⋊C4C22×C318yes—
121C2×C3×C22⋊C4C22×C314yes—
122C2×C3×C22⋊C4C22×C318yes—
123C2×C3×C22⋊C4C22×C318yes—
124C2×C3×C42C22×C314yes—
125C3×#95C22×C318yes—
126C3×#95C22×C318yes—
127C23×C32—21no—
128C23×C32—21no—
129C23×C32—21no—
130C23×C32—21no—
131C23×C32—21no—
132C23×C32—21no—
133C23×C32—21no—
134C23×C32—21no—
135C23×C32—21no—
136C23×C32—21no—
137C23×C32—22no—
138C23×C32—22no—
139C23×C32—22no—
140C23×C32—22no—
141C23×C32—22no—
142C23×C32—22no—
143C23×C32—22no—
144C23×C32C2×D412yes—
145C23×C32C2×D412yes—
146C23×C32C2×D412yes—
147C23×C32#4112yes—
148C23×C32C2411yescharacteristic
149C32×D4—28no—
150C32×D4—48no—
151C32×D4—48no—
152C32×D4—44no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—22no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—22no—
159C2×C32×C4—28no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—22no—
163C2×C32×C4—21no—
164C2×C32×C4—22no—
165C2×C32×C4—22no—
166C2×C32×C4—22no—
167C2×C32×C4—21no—
168C2×C32×C4—22no—
169C2×C32×C4—21no—
170C2×C32×C4—22no—
171C2×C32×C4—21no—
172C2×C32×C4—48no—
173C2×C32×C4—44no—
174C2×C32×C4C2×D412yes—
175C2×C32×C4C2×D412yes—
176C2×C32×C4C2×D414yes—
177C2×C32×C4C2×D412yes—
178C2×C32×C4C2×D412yes—
179C2×C32×C4#4112yes—
180C2×#121C2×C3212yescomplemented
181C2×#121C2×C3211yescharacteristic, complemented
182C2×#121C2×C3211yescharacteristic, complemented
183C2×C4×D4C2×C3211yescharacteristic, complemented
184C22×C4⋊C4C2×C3211yescharacteristic, complemented
185#396C2×C3212yescomplemented
186#394C2×C3212yescomplemented
187#390C2×C3211yescharacteristic, complemented
188#389C2×C3211yescharacteristic, complemented
189#386C2×C3212yescomplemented
190#380C2×C3211yescharacteristic, complemented
191C24×C3C23×C318yes—
192C24×C3C23×C314yes—
193C24×C3C23×C314yes—
194C2×C3×D4—28no—
195C2×C3×D4—28no—
196C2×C3×D4—216no—
197C2×C3×D4—216no—
198C2×C3×D4—216no—
199C2×C3×D4—216no—
200C2×C3×D4—216no—
201C2×C3×D4—28no—
202C2×C3×D4—28no—
203C2×C3×D4—28no—
204C2×C3×D4—48no—
205C2×C3×D4C3×D4116yescomplemented
206C22×C3×C4—28no—
207C22×C3×C4—28no—
208C22×C3×C4—216no—
209C22×C3×C4—216no—
210C22×C3×C4—28no—
211C22×C3×C4—28no—
212C22×C3×C4—28no—
213C22×C3×C4—48no—
214C22×C3×C4C3×D4116yescomplemented
215C22×C3×C4C23×C314yes—
216C22×C3×C4C23×C314yes—
217C22×C3×C4C23×C318yes—
218C22×C3×C4C23×C318yes—
219C22×C3×C4C23×C318yes—
220C22×C3×C4C23×C314yes—
221C22×C3×C4C23×C314yes—
222C22×C3×C4C23×C314yes—
223C3×C4⋊C4—216no—
224C3×C4⋊C4—28no—
225C3×C4⋊C4—28no—
226C3×C4⋊C4—216no—
227C3×C4⋊C4—28no—
228C3×C4⋊C4C3×D4116yescomplemented
229C3×C22⋊C4—28no—
230C3×C22⋊C4—216no—
231C3×C22⋊C4—28no—
232C3×C22⋊C4—216no—
233C3×C22⋊C4—216no—
234C3×C22⋊C4—216no—
235C3×C22⋊C4—48no—
236C3×C42—28no—
237C22×C32—22no—
238C22×C32—22no—
239C22×C32—22no—
240C22×C32—22no—
241C22×C32—22no—
242C22×C32—22no—
243C22×C32—22no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—41no—
247C22×C32—41no—
248C22×C32—41no—
249C22×C32—41no—
250C22×C32—42no—
251C22×C32—42no—
252C22×C32—42no—
253C22×C32—42no—
254C22×C32—42no—
255C22×C32—42no—
256C22×C32—42no—
257C22×C32#12114yescomplemented
258C22×C32C22×D411yescharacteristic
259C22×C32C22×D411yescharacteristic
260C22×C32C22×D411yescharacteristic
261C22×C32C2×#4111yescharacteristic
262C22×C32C2×#4111yescharacteristic
263C22×C32H(5,𝔽2)11yescharacteristic
264C22×C322−1+411yescharacteristic
265C32×C4—22no—
266C32×C4—22no—
267C32×C4—24no—
268C32×C4—41no—
269C32×C4—42no—
270C32×C4—41no—
271C32×C4—42no—
272C32×C4#12114yescomplemented
273C22×D4C22×C3212yes—
274C22×D4C22×C3211yescharacteristic
275C22×D4C22×C3212yes—
276C22×D4C22×C3211yescharacteristic
277C23×C4C22×C3211yescharacteristic
278C23×C4C22×C3212yes—
279C23×C4C22×C3211yescharacteristic
280#121—28no—
281#121—24no—
282#121—24no—
283C4×D4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4C22×C3211yescharacteristic
286C2×C4⋊C4C22×C3212yes—
287C2×C4⋊C4C22×C3211yescharacteristic
288C2×C4⋊C4C22×C3211yescharacteristic
289C2×C4⋊C4C22×C3212yes—
290C2×C4⋊C4C22×C3211yescharacteristic
291C2×C22⋊C4C22×C3211yescharacteristic
292C2×C22⋊C4C22×C3211yescharacteristic
293C2×C22⋊C4C22×C3212yes—
294C2×C22⋊C4C22×C3212yes—
295C2×C22⋊C4C22×C3212yes—
296C2×C22⋊C4C22×C3212yes—
297C2×C22⋊C4C22×C3212yes—
298C2×C42C22×C3211yescharacteristic
299#95C22×C3212yes—
300#95C22×C3212yes—
301C23×C3—24no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—24no—
306C23×C3—24no—
307C23×C3—24no—
308C23×C3—24no—
309C23×C3—24no—
310C23×C3—24no—
311C23×C3—28no—
312C23×C3—28no—
313C23×C3—28no—
314C23×C3—28no—
315C23×C3—28no—
316C23×C3—28no—
317C23×C3—28no—
318C23×C3C2×C3×D418yes—
319C23×C3C2×C3×D418yes—
320C23×C3C2×C3×D418yes—
321C23×C3C3×#4118yes—
322C23×C3C24×C314yes—
323C3×D4—232no—
324C3×D4—432no—
325C3×D4—432no—
326C3×D4—416no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—28no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—28no—
333C2×C3×C4—232no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—28no—
337C2×C3×C4—24no—
338C2×C3×C4—28no—
339C2×C3×C4—28no—
340C2×C3×C4—28no—
341C2×C3×C4—24no—
342C2×C3×C4—28no—
343C2×C3×C4—24no—
344C2×C3×C4—28no—
345C2×C3×C4—24no—
346C2×C3×C4—432no—
347C2×C3×C4—416no—
348C2×C3×C4C2×C3×D418yes—
349C2×C3×C4C2×C3×D418yes—
350C2×C3×C4C2×C3×D4116yes—
351C2×C3×C4C2×C3×D418yes—
352C2×C3×C4C2×C3×D418yes—
353C2×C3×C4C3×#4118yes—
354C2×C32—24no—
355C2×C32—42no—
356C2×C32—82no—
357C2×C32C2×#12111yescharacteristic
358C2×C32C2×#12111yescharacteristic
359C2×C32#53711yescharacteristic
360C2×C32D4211yescharacteristic
361C2×C32#54711yescharacteristic
362C2×C32#54811yescharacteristic
363C2×C32#55311yescharacteristic
364C24C23×C3212yes—
365C24C23×C3211yescharacteristic
366C24C23×C3211yescharacteristic
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—42no—
378C2×D4C32×D414yescomplemented
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—42no—
387C22×C4C32×D414yescomplemented
388C22×C4C23×C3211yescharacteristic
389C22×C4C23×C3211yescharacteristic
390C22×C4C23×C3212yes—
391C22×C4C23×C3212yes—
392C22×C4C23×C3212yes—
393C22×C4C23×C3211yescharacteristic
394C22×C4C23×C3211yescharacteristic
395C22×C4C23×C3211yescharacteristic
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4C32×D414yescomplemented
402C22⋊C4—22no—
403C22⋊C4—24no—
404C22⋊C4—22no—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C22⋊C4—24no—
408C22⋊C4—42no—
409C42—22no—
410C22×C3—28no—
411C22×C3—28no—
412C22×C3—28no—
413C22×C3—28no—
414C22×C3—28no—
415C22×C3—28no—
416C22×C3—28no—
417C22×C3—28no—
418C22×C3—28no—
419C22×C3—44no—
420C22×C3—44no—
421C22×C3—44no—
422C22×C3—44no—
423C22×C3—48no—
424C22×C3—48no—
425C22×C3—48no—
426C22×C3—48no—
427C22×C3—48no—
428C22×C3—48no—
429C22×C3—48no—
430C22×C3C3×#121116yescomplemented
431C22×C3C22×C3×D414yes—
432C22×C3C22×C3×D414yes—
433C22×C3C22×C3×D414yes—
434C22×C3C2×C3×#4114yes—
435C22×C3C2×C3×#4114yes—
436C22×C3C3×H(5,𝔽2)14yes—
437C22×C3C3×2−1+414yes—
438C3×C4—28no—
439C3×C4—28no—
440C3×C4—216no—
441C3×C4—44no—
442C3×C4—48no—
443C3×C4—44no—
444C3×C4—48no—
445C3×C4C3×#121116yescomplemented
446C32#241611yescharacteristic, direct product factor, Sylow
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23C2×C32×D412yes—
465C23C2×C32×D412yes—
466C23C2×C32×D412yes—
467C23C32×#4112yes—
468C23C24×C3211yescharacteristic
469D4—28no—
470D4—48no—
471D4—48no—
472D4—44no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—28no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—48no—
493C2×C4—44no—
494C2×C4C2×C32×D412yes—
495C2×C4C2×C32×D412yes—
496C2×C4C2×C32×D414yes—
497C2×C4C2×C32×D412yes—
498C2×C4C2×C32×D412yes—
499C2×C4C32×#4112yes—
500C2×C3—216no—
501C2×C3—48no—
502C2×C3—88no—
503C2×C3C2×C3×#12114yes—
504C2×C3C2×C3×#12114yes—
505C2×C3C3×#53714yes—
506C2×C3C3×D4214yes—
507C2×C3C3×#54714yes—
508C2×C3C3×#54814yes—
509C2×C3C3×#55314yes—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22C32×#12114yescomplemented
531C22C22×C32×D411yescharacteristic
532C22C22×C32×D411yescharacteristic
533C22C22×C32×D411yescharacteristic
534C22C2×C32×#4111yescharacteristic
535C22C2×C32×#4111yescharacteristic
536C22C32×H(5,𝔽2)11yescharacteristic
537C22C32×2−1+411yescharacteristic
538C4—22no—
539C4—22no—
540C4—24no—
541C4—41no—
542C4—42no—
543C4—41no—
544C4—42no—
545C4C32×#12114yescomplemented
546C3C3×#241614yesdirect product factor
547C2—24no—
548C2—42no—
549C2—82no—
550C2C2×C32×#12111yescharacteristic
551C2C2×C32×#12111yescharacteristic
552C2C32×#53711yescharacteristic
553C2C32×D4211yescharacteristic
554C2C32×#54711yescharacteristic
555C2C32×#54811yescharacteristic
556C2C32×#55311yescharacteristic
subgroupG/H
center, socleC23×C32C24
commutator, Frattini, cosocleC23C24×C32
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C32 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C32×#121C2semidirect
2. C2×C32×#121C2semidirect
3. C2×C32×#121C2semidirect
4. C2×C32×C4×D4C2semidirect
5. C22×C32×C4⋊C4C2semidirect
6. C32×#396C2semidirect
7. C32×#394C2semidirect
8. C32×#390C2semidirect
9. C32×#389C2semidirect
10. C32×#386C2semidirect
11. C32×#380C2semidirect
12. C3×#2416C3direct (N × H)
41. C2×C3×#121C2×C3semidirect
42. C2×C3×#121C2×C3semidirect
43. C2×C3×#121C2×C3semidirect
44. C2×C3×C4×D4C2×C3semidirect
45. C22×C3×C4⋊C4C2×C3semidirect
46. C3×#396C2×C3semidirect
47. C3×#394C2×C3semidirect
48. C3×#390C2×C3semidirect
49. C3×#389C2×C3semidirect
50. C3×#386C2×C3semidirect
51. C3×#380C2×C3semidirect
66. C2×C32×D4D4semidirect
75. C22×C32×C4D4semidirect
89. C32×C4⋊C4D4semidirect
98. #2416C32direct (N × H)
180. C2×#121C2×C32semidirect
181. C2×#121C2×C32semidirect
182. C2×#121C2×C32semidirect
183. C2×C4×D4C2×C32semidirect
184. C22×C4⋊C4C2×C32semidirect
185. #396C2×C32semidirect
186. #394C2×C32semidirect
187. #390C2×C32semidirect
188. #389C2×C32semidirect
189. #386C2×C32semidirect
190. #380C2×C32semidirect
205. C2×C3×D4C3×D4semidirect
214. C22×C3×C4C3×D4semidirect
228. C3×C4⋊C4C3×D4semidirect
257. C22×C32#121semidirect
272. C32×C4#121semidirect
378. C2×D4C32×D4semidirect
387. C22×C4C32×D4semidirect
401. C4⋊C4C32×D4semidirect
430. C22×C3C3×#121semidirect
445. C3×C4C3×#121semidirect
446. C32#2416direct (N × H)
530. C22C32×#121semidirect
545. C4C32×#121semidirect
546. C3C3×#2416direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#1212
C2×C32×C4×D42
C22×C32×C4⋊C42
C32×#3962
C32×#3942
C32×#3902
C32×#3892
C32×#3862
C32×#3802
C3×#24163