Finite Group Databasebeta

← C32×#3516 #11770565

Subgroups — C32×#3516 #11770565

Order 1152.

3036 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#557C211yesmaximal, characteristic, complemented
2C32×#556C213yesmaximal, complemented
3C32×#554C213yesmaximal, complemented
4C32×#553C213yesmaximal, complemented
5C32×#552C213yesmaximal, complemented
6C32×D4×Q8C211yesmaximal, characteristic, complemented
7C32×#548C211yesmaximal, characteristic, complemented
8C32×#548C211yesmaximal, characteristic, complemented
9C32×#542C213yesmaximal, complemented
10C32×#541C213yesmaximal, complemented
11C32×#529C213yesmaximal, complemented
12C32×#529C213yesmaximal, complemented
13C32×#520C211yesmaximal, characteristic, complemented
14C32×#519C211yesmaximal, characteristic, complemented
15C2×C32×C4×Q8C211yesmaximal, characteristic, complemented
16C3×#3516C314yesmaximal, direct product factor
17C2×C32×#41C2211yescharacteristic
18C22×C32×Q8C2211yescharacteristic
19C22×C32×Q8C2211yescharacteristic
20C32×C4⋊Q8C2213yes—
21C32×C4⋊Q8C2213yes—
22C32×#126C2213yes—
23C32×#126C2216yes—
24C32×#126C2213yes—
25C32×#126C2216yes—
26C32×#125C2216yes—
27C32×#125C2213yes—
28C32×#125C2213yes—
29C32×#125C2213yes—
30C32×#124C2213yes—
31C32×#124C2213yes—
32C32×#124C2213yes—
33C32×#123C2216yes—
34C32×#123C2211yescharacteristic
35C32×#123C2213yes—
36C32×C22⋊Q8C2213yes—
37C32×C22⋊Q8C2213yes—
38C32×C22⋊Q8C2213yes—
39C32×C22⋊Q8C2216yes—
40C32×C22⋊Q8C2213yes—
41C32×C22⋊Q8C2213yes—
42C32×#121C2213yes—
43C32×#121C2213yes—
44C32×#121C2213yes—
45C32×C4×Q8C2213yes—
46C32×C4×Q8C2213yes—
47C32×C4×Q8C2213yes—
48C32×C4×Q8C2211yescharacteristic
49C32×C4×Q8C2211yescharacteristic
50C32×C4×Q8C2211yescharacteristic
51C32×C4×Q8C2211yescharacteristic
52C32×C4×Q8C2213yes—
53C32×C4×Q8C2213yes—
54C32×C4×D4C2213yes—
55C32×C4×D4C2216yes—
56C32×C4×D4C2213yes—
57C32×C4×D4C2213yes—
58C32×C4×D4C2213yes—
59C32×C4×D4C2213yes—
60C32×#117C2213yes—
61C32×#117C2216yes—
62C32×#117C2213yes—
63C32×#117C2213yes—
64C2×C32×C4⋊C4C2213yes—
65C2×C32×C4⋊C4C2213yes—
66C2×C32×C42C2213yes—
67C3×#557C2×C314yescomplemented
68C3×#556C2×C3112yescomplemented
69C3×#554C2×C3112yescomplemented
70C3×#553C2×C3112yescomplemented
71C3×#552C2×C3112yescomplemented
72C3×D4×Q8C2×C314yescomplemented
73C3×#548C2×C314yescomplemented
74C3×#548C2×C314yescomplemented
75C3×#542C2×C3112yescomplemented
76C3×#541C2×C3112yescomplemented
77C3×#529C2×C3112yescomplemented
78C3×#529C2×C3112yescomplemented
79C3×#520C2×C314yescomplemented
80C3×#519C2×C314yescomplemented
81C2×C3×C4×Q8C2×C314yescomplemented
82C32×#41—24no—
83C2×C32×Q8—24no—
84C2×C32×Q8—24no—
85C2×C32×Q8C2313yes—
86C2×C32×Q8C2313yes—
87C2×C32×Q8C2313yes—
88C2×C32×Q8C2311yescharacteristic
89C2×C32×D4C2313yes—
90C2×C32×D4C2313yes—
91C2×C32×D4C2311yescharacteristic
92C2×C32×D4C2313yes—
93C22×C32×C4C2313yes—
94C22×C32×C4C2313yes—
95C22×C32×C4C2313yes—
96C22×C32×C4C2311yescharacteristic
97C22×C32×C4C2313yes—
98C22×C32×C4C2316yes—
99C32×C4⋊C4C2313yes—
100C32×C4⋊C4C2313yes—
101C32×C4⋊C4C2313yes—
102C32×C4⋊C4C2313yes—
103C32×C4⋊C4C2316yes—
104C32×C4⋊C4C2311yescharacteristic
105C32×C4⋊C4C2313yes—
106C32×C4⋊C4C2313yes—
107C32×C4⋊C4C2313yes—
108C32×C4⋊C4C2313yes—
109C32×C4⋊C4C2316yes—
110C32×C4⋊C4C2313yes—
111C32×C4⋊C4C2313yes—
112C32×C4⋊C4C2316yes—
113C32×C4⋊C4C2313yes—
114C32×C4⋊C4C2311yescharacteristic
115C32×C4⋊C4C2313yes—
116C32×C22⋊C4C2313yes—
117C32×C22⋊C4C2311yescharacteristic
118C32×C22⋊C4C2313yes—
119C32×C22⋊C4C2313yes—
120C32×C22⋊C4C2313yes—
121C32×C22⋊C4C2316yes—
122C32×C22⋊C4C2311yescharacteristic
123C32×C22⋊C4C2313yes—
124C32×C22⋊C4C2311yescharacteristic
125C32×C22⋊C4C2316yes—
126C32×C22⋊C4C2313yes—
127C32×C22⋊C4C2313yes—
128C32×C42C2313yes—
129C32×C42C2313yes—
130C32×C42C2313yes—
131C32×C42C2313yes—
132C32×C42C2316yes—
133C32×C42C2313yes—
134C32×C42C2313yes—
135#3516C3211yescharacteristic, direct product factor, Sylow
136C2×C3×#41C22×C314yes—
137C22×C3×Q8C22×C314yes—
138C22×C3×Q8C22×C314yes—
139C3×C4⋊Q8C22×C3112yes—
140C3×C4⋊Q8C22×C3112yes—
141C3×#126C22×C3124yes—
142C3×#126C22×C3112yes—
143C3×#126C22×C3112yes—
144C3×#126C22×C3124yes—
145C3×#125C22×C3112yes—
146C3×#125C22×C3112yes—
147C3×#125C22×C3124yes—
148C3×#125C22×C3112yes—
149C3×#124C22×C3112yes—
150C3×#124C22×C3112yes—
151C3×#124C22×C3112yes—
152C3×#123C22×C314yes—
153C3×#123C22×C3112yes—
154C3×#123C22×C3124yes—
155C3×C22⋊Q8C22×C3112yes—
156C3×C22⋊Q8C22×C3112yes—
157C3×C22⋊Q8C22×C3112yes—
158C3×C22⋊Q8C22×C3112yes—
159C3×C22⋊Q8C22×C3112yes—
160C3×C22⋊Q8C22×C3124yes—
161C3×#121C22×C3112yes—
162C3×#121C22×C3112yes—
163C3×#121C22×C3112yes—
164C3×C4×Q8C22×C314yes—
165C3×C4×Q8C22×C3112yes—
166C3×C4×Q8C22×C314yes—
167C3×C4×Q8C22×C3112yes—
168C3×C4×Q8C22×C3112yes—
169C3×C4×Q8C22×C314yes—
170C3×C4×Q8C22×C3112yes—
171C3×C4×Q8C22×C314yes—
172C3×C4×Q8C22×C3112yes—
173C3×C4×D4C22×C3112yes—
174C3×C4×D4C22×C3112yes—
175C3×C4×D4C22×C3112yes—
176C3×C4×D4C22×C3112yes—
177C3×C4×D4C22×C3112yes—
178C3×C4×D4C22×C3124yes—
179C3×#117C22×C3112yes—
180C3×#117C22×C3112yes—
181C3×#117C22×C3112yes—
182C3×#117C22×C3124yes—
183C2×C3×C4⋊C4C22×C3112yes—
184C2×C3×C4⋊C4C22×C3112yes—
185C2×C3×C42C22×C3112yes—
186C23×C32C2411yescharacteristic
187C23×C32C2413yes—
188C23×C32C2411yescharacteristic
189C32×Q8—26no—
190C32×Q8—26no—
191C32×Q8—43no—
192C32×Q8#4114yes—
193C32×D4—26no—
194C32×D4—22no—
195C32×D4—26no—
196C32×D4—43no—
197C2×C32×C4—22no—
198C2×C32×C4—26no—
199C2×C32×C4—26no—
200C2×C32×C4—26no—
201C2×C32×C4—46no—
202C2×C32×C4—43no—
203C2×C32×C4C2411yescharacteristic
204C2×C32×C4C2411yescharacteristic
205C2×C32×C4C2413yes—
206C2×C32×C4C2411yescharacteristic
207C2×C32×C4C2413yes—
208C2×C32×C4C2411yescharacteristic
209C2×C32×C4C2413yes—
210C2×C32×C4C2413yes—
211C2×C32×C4C2413yes—
212C2×C32×C4C2413yes—
213C2×C32×C4C2411yescharacteristic
214C2×C32×C4C2413yes—
215#557C2×C3211yescharacteristic, complemented
216#556C2×C3213yescomplemented
217#554C2×C3213yescomplemented
218#553C2×C3213yescomplemented
219#552C2×C3213yescomplemented
220D4×Q8C2×C3211yescharacteristic, complemented
221#548C2×C3211yescharacteristic, complemented
222#548C2×C3211yescharacteristic, complemented
223#542C2×C3213yescomplemented
224#541C2×C3213yescomplemented
225#529C2×C3213yescomplemented
226#529C2×C3213yescomplemented
227#520C2×C3211yescharacteristic, complemented
228#519C2×C3211yescharacteristic, complemented
229C2×C4×Q8C2×C3211yescharacteristic, complemented
230C3×#41—216no—
231C2×C3×Q8—216no—
232C2×C3×Q8—216no—
233C2×C3×Q8C23×C3112yes—
234C2×C3×Q8C23×C3112yes—
235C2×C3×Q8C23×C3112yes—
236C2×C3×Q8C23×C314yes—
237C2×C3×D4C23×C3112yes—
238C2×C3×D4C23×C314yes—
239C2×C3×D4C23×C3112yes—
240C2×C3×D4C23×C3112yes—
241C22×C3×C4C23×C314yes—
242C22×C3×C4C23×C3112yes—
243C22×C3×C4C23×C3112yes—
244C22×C3×C4C23×C3112yes—
245C22×C3×C4C23×C3112yes—
246C22×C3×C4C23×C3124yes—
247C3×C4⋊C4C23×C3112yes—
248C3×C4⋊C4C23×C3112yes—
249C3×C4⋊C4C23×C3112yes—
250C3×C4⋊C4C23×C3112yes—
251C3×C4⋊C4C23×C3124yes—
252C3×C4⋊C4C23×C314yes—
253C3×C4⋊C4C23×C3112yes—
254C3×C4⋊C4C23×C3112yes—
255C3×C4⋊C4C23×C3112yes—
256C3×C4⋊C4C23×C3112yes—
257C3×C4⋊C4C23×C3124yes—
258C3×C4⋊C4C23×C3112yes—
259C3×C4⋊C4C23×C3112yes—
260C3×C4⋊C4C23×C3124yes—
261C3×C4⋊C4C23×C3112yes—
262C3×C4⋊C4C23×C314yes—
263C3×C4⋊C4C23×C3112yes—
264C3×C22⋊C4C23×C3112yes—
265C3×C22⋊C4C23×C314yes—
266C3×C22⋊C4C23×C3112yes—
267C3×C22⋊C4C23×C3112yes—
268C3×C22⋊C4C23×C3112yes—
269C3×C22⋊C4C23×C3124yes—
270C3×C22⋊C4C23×C314yes—
271C3×C22⋊C4C23×C3112yes—
272C3×C22⋊C4C23×C314yes—
273C3×C22⋊C4C23×C3124yes—
274C3×C22⋊C4C23×C3112yes—
275C3×C22⋊C4C23×C3112yes—
276C3×C42C23×C3112yes—
277C3×C42C23×C3112yes—
278C3×C42C23×C3112yes—
279C3×C42C23×C3112yes—
280C3×C42C23×C3124yes—
281C3×C42C23×C3112yes—
282C3×C42C23×C3112yes—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—23no—
287C22×C32—21no—
288C22×C32—23no—
289C22×C32—23no—
290C22×C32—21no—
291C22×C322−1+412yes—
292C22×C32C2511yescharacteristic
293C32×C4—21no—
294C32×C4—23no—
295C32×C4—23no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—23no—
299C32×C4—21no—
300C32×C4—23no—
301C32×C4—23no—
302C32×C4—23no—
303C32×C4—21no—
304C32×C4C2×#4116yes—
305C2×#41C22×C3211yescharacteristic
306C22×Q8C22×C3211yescharacteristic
307C22×Q8C22×C3211yescharacteristic
308C4⋊Q8C22×C3213yes—
309C4⋊Q8C22×C3213yes—
310#126C22×C3216yes—
311#126C22×C3213yes—
312#126C22×C3213yes—
313#126C22×C3216yes—
314#125C22×C3213yes—
315#125C22×C3213yes—
316#125C22×C3216yes—
317#125C22×C3213yes—
318#124C22×C3213yes—
319#124C22×C3213yes—
320#124C22×C3213yes—
321#123C22×C3211yescharacteristic
322#123C22×C3213yes—
323#123C22×C3216yes—
324C22⋊Q8C22×C3213yes—
325C22⋊Q8C22×C3213yes—
326C22⋊Q8C22×C3213yes—
327C22⋊Q8C22×C3213yes—
328C22⋊Q8C22×C3213yes—
329C22⋊Q8C22×C3216yes—
330#121C22×C3213yes—
331#121C22×C3213yes—
332#121C22×C3213yes—
333C4×Q8C22×C3211yescharacteristic
334C4×Q8C22×C3213yes—
335C4×Q8C22×C3211yescharacteristic
336C4×Q8C22×C3213yes—
337C4×Q8C22×C3213yes—
338C4×Q8C22×C3211yescharacteristic
339C4×Q8C22×C3213yes—
340C4×Q8C22×C3211yescharacteristic
341C4×Q8C22×C3213yes—
342C4×D4C22×C3213yes—
343C4×D4C22×C3213yes—
344C4×D4C22×C3213yes—
345C4×D4C22×C3213yes—
346C4×D4C22×C3213yes—
347C4×D4C22×C3216yes—
348#117C22×C3213yes—
349#117C22×C3213yes—
350#117C22×C3213yes—
351#117C22×C3216yes—
352C2×C4⋊C4C22×C3213yes—
353C2×C4⋊C4C22×C3213yes—
354C2×C42C22×C3213yes—
355C23×C3C24×C314yes—
356C23×C3C24×C3112yes—
357C23×C3C24×C314yes—
358C3×Q8—224no—
359C3×Q8—224no—
360C3×Q8—412no—
361C3×Q8C3×#41116yes—
362C3×D4—224no—
363C3×D4—28no—
364C3×D4—224no—
365C3×D4—412no—
366C2×C3×C4—28no—
367C2×C3×C4—224no—
368C2×C3×C4—224no—
369C2×C3×C4—224no—
370C2×C3×C4—424no—
371C2×C3×C4—412no—
372C2×C3×C4C24×C314yes—
373C2×C3×C4C24×C314yes—
374C2×C3×C4C24×C3112yes—
375C2×C3×C4C24×C314yes—
376C2×C3×C4C24×C3112yes—
377C2×C3×C4C24×C314yes—
378C2×C3×C4C24×C3112yes—
379C2×C3×C4C24×C3112yes—
380C2×C3×C4C24×C3112yes—
381C2×C3×C4C24×C3112yes—
382C2×C3×C4C24×C314yes—
383C2×C3×C4C24×C3112yes—
384C2×C32—22no—
385C2×C32—41no—
386C2×C32—43no—
387C2×C32C22×#4111yescharacteristic
388C2×C32C2×2−1+411yescharacteristic
389C2×C32#58511yescharacteristic
390#41—24no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8C23×C3213yes—
394C2×Q8C23×C3213yes—
395C2×Q8C23×C3213yes—
396C2×Q8C23×C3211yescharacteristic
397C2×D4C23×C3213yes—
398C2×D4C23×C3211yescharacteristic
399C2×D4C23×C3213yes—
400C2×D4C23×C3213yes—
401C22×C4C23×C3211yescharacteristic
402C22×C4C23×C3213yes—
403C22×C4C23×C3213yes—
404C22×C4C23×C3213yes—
405C22×C4C23×C3213yes—
406C22×C4C23×C3216yes—
407C4⋊C4C23×C3213yes—
408C4⋊C4C23×C3213yes—
409C4⋊C4C23×C3213yes—
410C4⋊C4C23×C3213yes—
411C4⋊C4C23×C3216yes—
412C4⋊C4C23×C3211yescharacteristic
413C4⋊C4C23×C3213yes—
414C4⋊C4C23×C3213yes—
415C4⋊C4C23×C3213yes—
416C4⋊C4C23×C3213yes—
417C4⋊C4C23×C3216yes—
418C4⋊C4C23×C3213yes—
419C4⋊C4C23×C3213yes—
420C4⋊C4C23×C3216yes—
421C4⋊C4C23×C3213yes—
422C4⋊C4C23×C3211yescharacteristic
423C4⋊C4C23×C3213yes—
424C22⋊C4C23×C3213yes—
425C22⋊C4C23×C3211yescharacteristic
426C22⋊C4C23×C3213yes—
427C22⋊C4C23×C3213yes—
428C22⋊C4C23×C3213yes—
429C22⋊C4C23×C3216yes—
430C22⋊C4C23×C3211yescharacteristic
431C22⋊C4C23×C3213yes—
432C22⋊C4C23×C3211yescharacteristic
433C22⋊C4C23×C3216yes—
434C22⋊C4C23×C3213yes—
435C22⋊C4C23×C3213yes—
436C42C23×C3213yes—
437C42C23×C3213yes—
438C42C23×C3213yes—
439C42C23×C3213yes—
440C42C23×C3216yes—
441C42C23×C3213yes—
442C42C23×C3213yes—
443C22×C3—24no—
444C22×C3—24no—
445C22×C3—24no—
446C22×C3—212no—
447C22×C3—24no—
448C22×C3—212no—
449C22×C3—212no—
450C22×C3—24no—
451C22×C3C3×2−1+418yes—
452C22×C3C25×C314yes—
453C3×C4—24no—
454C3×C4—212no—
455C3×C4—212no—
456C3×C4—24no—
457C3×C4—24no—
458C3×C4—212no—
459C3×C4—24no—
460C3×C4—212no—
461C3×C4—212no—
462C3×C4—212no—
463C3×C4—24no—
464C3×C4C2×C3×#41124yes—
465C32#351611yescharacteristic, direct product factor, Sylow
466C23C24×C3211yescharacteristic
467C23C24×C3213yes—
468C23C24×C3211yescharacteristic
469Q8—26no—
470Q8—26no—
471Q8—43no—
472Q8C32×#4114yes—
473D4—26no—
474D4—22no—
475D4—26no—
476D4—43no—
477C2×C4—22no—
478C2×C4—26no—
479C2×C4—26no—
480C2×C4—26no—
481C2×C4—46no—
482C2×C4—43no—
483C2×C4C24×C3211yescharacteristic
484C2×C4C24×C3211yescharacteristic
485C2×C4C24×C3213yes—
486C2×C4C24×C3211yescharacteristic
487C2×C4C24×C3213yes—
488C2×C4C24×C3211yescharacteristic
489C2×C4C24×C3213yes—
490C2×C4C24×C3213yes—
491C2×C4C24×C3213yes—
492C2×C4C24×C3213yes—
493C2×C4C24×C3211yescharacteristic
494C2×C4C24×C3213yes—
495C2×C3—28no—
496C2×C3—44no—
497C2×C3—412no—
498C2×C3C22×C3×#4114yes—
499C2×C3C2×C3×2−1+414yes—
500C2×C3C3×#58514yes—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—23no—
505C22—21no—
506C22—23no—
507C22—23no—
508C22—21no—
509C22C32×2−1+412yes—
510C22C25×C3211yescharacteristic
511C4—21no—
512C4—23no—
513C4—23no—
514C4—21no—
515C4—21no—
516C4—23no—
517C4—21no—
518C4—23no—
519C4—23no—
520C4—23no—
521C4—21no—
522C4C2×C32×#4116yes—
523C3C3×#351614yesdirect product factor
524C2—22no—
525C2—41no—
526C2—43no—
527C2C22×C32×#4111yescharacteristic
528C2C2×C32×2−1+411yescharacteristic
529C2C32×#58511yescharacteristic
subgroupG/H
center, socleC22×C32C25
commutator, Frattini, cosocleC22C25×C32
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C32 ◁ G

Complements

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

NH ≅ G/Ntype
1. C32×#557C2semidirect
2. C32×#556C2semidirect
3. C32×#554C2semidirect
4. C32×#553C2semidirect
5. C32×#552C2semidirect
6. C32×D4×Q8C2semidirect
7. C32×#548C2semidirect
8. C32×#548C2semidirect
9. C32×#542C2semidirect
10. C32×#541C2semidirect
11. C32×#529C2semidirect
12. C32×#529C2semidirect
13. C32×#520C2semidirect
14. C32×#519C2semidirect
15. C2×C32×C4×Q8C2semidirect
16. C3×#3516C3direct (N × H)
67. C3×#557C2×C3semidirect
68. C3×#556C2×C3semidirect
69. C3×#554C2×C3semidirect
70. C3×#553C2×C3semidirect
71. C3×#552C2×C3semidirect
72. C3×D4×Q8C2×C3semidirect
73. C3×#548C2×C3semidirect
74. C3×#548C2×C3semidirect
75. C3×#542C2×C3semidirect
76. C3×#541C2×C3semidirect
77. C3×#529C2×C3semidirect
78. C3×#529C2×C3semidirect
79. C3×#520C2×C3semidirect
80. C3×#519C2×C3semidirect
81. C2×C3×C4×Q8C2×C3semidirect
135. #3516C32direct (N × H)
215. #557C2×C32semidirect
216. #556C2×C32semidirect
217. #554C2×C32semidirect
218. #553C2×C32semidirect
219. #552C2×C32semidirect
220. D4×Q8C2×C32semidirect
221. #548C2×C32semidirect
222. #548C2×C32semidirect
223. #542C2×C32semidirect
224. #541C2×C32semidirect
225. #529C2×C32semidirect
226. #529C2×C32semidirect
227. #520C2×C32semidirect
228. #519C2×C32semidirect
229. C2×C4×Q8C2×C32semidirect
465. C32#3516direct (N × H)
523. C3C3×#3516direct (N × H)

Maximal subgroups

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

subgroupindex
C32×#5572
C32×#5562
C32×#5542
C32×#5532
C32×#5522
C32×D4×Q82
C32×#5482
C32×#5422
C32×#5412
C32×#5292
C32×#5202
C32×#5192
C2×C32×C4×Q82
C3×#35163