Finite Group Databasebeta

← C32×#2654 #11769703

Subgroups — C32×#2654 #11769703

Order 1152.

1134 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#400C211yesmaximal, characteristic, complemented
2C32×#400C211yesmaximal, characteristic, complemented
3C32×#399C211yesmaximal, characteristic
4C32×#398C211yesmaximal, characteristic, complemented
5C32×#398C211yesmaximal, characteristic, complemented
6C32×#397C211yesmaximal, characteristic
7C32×#397C211yesmaximal, characteristic
8C32×#396C211yesmaximal, characteristic
9C32×#389C211yesmaximal, characteristic, complemented
10C32×#388C211yesmaximal, characteristic
11C32×#388C211yesmaximal, characteristic
12C32×#388C211yesmaximal, characteristic
13C32×#387C211yesmaximal, characteristic, complemented
14C32×#387C211yesmaximal, characteristic, complemented
15C32×C4×C4⋊C4C211yesmaximal, characteristic, complemented
16C3×#2654C314yesmaximal, direct product factor
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C32×C4⋊C4C2211yescharacteristic
19C2×C32×C4⋊C4C2211yescharacteristic
20C2×C32×C4⋊C4C2211yescharacteristic
21C2×C32×C4⋊C4C2211yescharacteristic
22C2×C32×C4⋊C4C2211yescharacteristic
23C2×C32×C4⋊C4C2211yescharacteristic
24C2×C32×C4⋊C4C2211yescharacteristic
25C2×C32×C4⋊C4C2211yescharacteristic
26C2×C32×C4⋊C4C2211yescharacteristic
27C2×C32×C4⋊C4C2211yescharacteristic
28C2×C32×C22⋊C4C2211yescharacteristic
29C2×C32×C22⋊C4C2211yescharacteristic
30C2×C32×C22⋊C4C2211yescharacteristic
31C2×C32×C22⋊C4C2211yescharacteristic
32C2×C32×C22⋊C4C2211yescharacteristic
33C2×C32×C22⋊C4C2211yescharacteristic
34C2×C32×C22⋊C4C2211yescharacteristic
35C2×C32×C42C2211yescharacteristic
36C2×C32×C42C2211yescharacteristic
37C2×C32×C42C2211yescharacteristic
38C32×#95C2211yescharacteristic
39C32×#95C2211yescharacteristic
40C32×#95C2211yescharacteristic
41C32×#95C2211yescharacteristic
42C32×#95C2211yescharacteristic
43C32×#95C2211yescharacteristic
44C32×#95C2211yescharacteristic
45C32×#95C2211yescharacteristic
46C32×#95C2211yescharacteristic
47C32×#95C2211yescharacteristic
48C32×#95C2211yescharacteristic
49C32×#95C2211yescharacteristic
50C32×#95C2211yescharacteristic
51C32×#95C2211yescharacteristic
52C3×#400C2×C314yescomplemented
53C3×#400C2×C314yescomplemented
54C3×#399C2×C314yes—
55C3×#398C2×C314yescomplemented
56C3×#398C2×C314yescomplemented
57C3×#397C2×C314yes—
58C3×#397C2×C314yes—
59C3×#396C2×C314yes—
60C3×#389C2×C314yescomplemented
61C3×#388C2×C314yes—
62C3×#388C2×C314yes—
63C3×#388C2×C314yes—
64C3×#387C2×C314yescomplemented
65C3×#387C2×C314yescomplemented
66C3×C4×C4⋊C4C2×C314yescomplemented
67C24×C32C2311yescharacteristic
68C22×C32×C4C2311yescharacteristic
69C22×C32×C4C2311yescharacteristic
70C22×C32×C4C2311yescharacteristic
71C22×C32×C4C2311yescharacteristic
72C22×C32×C4C2311yescharacteristic
73C22×C32×C4C2311yescharacteristic
74C22×C32×C4C2311yescharacteristic
75C22×C32×C4C2311yescharacteristic
76C22×C32×C4C2311yescharacteristic
77C22×C32×C4C2311yescharacteristic
78C22×C32×C4C2311yescharacteristic
79C22×C32×C4C2311yescharacteristic
80C22×C32×C4C2311yescharacteristic
81C22×C32×C4C2311yescharacteristic
82C32×C4⋊C4—22no—
83C32×C4⋊C4—22no—
84C32×C4⋊C4—22no—
85C32×C4⋊C4—22no—
86C32×C4⋊C4—22no—
87C32×C4⋊C4—41no—
88C32×C4⋊C4—41no—
89C32×C4⋊C4—41no—
90C32×C4⋊C4—41no—
91C32×C4⋊C4—41no—
92C32×C4⋊C4—41no—
93C32×C22⋊C4—22no—
94C32×C22⋊C4—22no—
95C32×C22⋊C4—22no—
96C32×C22⋊C4—41no—
97C32×C22⋊C4—41no—
98C32×C22⋊C4—41no—
99C32×C22⋊C4—41no—
100C32×C42—22no—
101C32×C42—41no—
102C32×C42—41no—
103#2654C3211yescharacteristic, direct product factor, Sylow
104C2×C3×C4⋊C4C22×C314yes—
105C2×C3×C4⋊C4C22×C314yes—
106C2×C3×C4⋊C4C22×C314yes—
107C2×C3×C4⋊C4C22×C314yes—
108C2×C3×C4⋊C4C22×C314yes—
109C2×C3×C4⋊C4C22×C314yes—
110C2×C3×C4⋊C4C22×C314yes—
111C2×C3×C4⋊C4C22×C314yes—
112C2×C3×C4⋊C4C22×C314yes—
113C2×C3×C4⋊C4C22×C314yes—
114C2×C3×C4⋊C4C22×C314yes—
115C2×C3×C22⋊C4C22×C314yes—
116C2×C3×C22⋊C4C22×C314yes—
117C2×C3×C22⋊C4C22×C314yes—
118C2×C3×C22⋊C4C22×C314yes—
119C2×C3×C22⋊C4C22×C314yes—
120C2×C3×C22⋊C4C22×C314yes—
121C2×C3×C22⋊C4C22×C314yes—
122C2×C3×C42C22×C314yes—
123C2×C3×C42C22×C314yes—
124C2×C3×C42C22×C314yes—
125C3×#95C22×C314yes—
126C3×#95C22×C314yes—
127C3×#95C22×C314yes—
128C3×#95C22×C314yes—
129C3×#95C22×C314yes—
130C3×#95C22×C314yes—
131C3×#95C22×C314yes—
132C3×#95C22×C314yes—
133C3×#95C22×C314yes—
134C3×#95C22×C314yes—
135C3×#95C22×C314yes—
136C3×#95C22×C314yes—
137C3×#95C22×C314yes—
138C3×#95C22×C314yes—
139C23×C32—21no—
140C23×C32—21no—
141C23×C32—21no—
142C23×C32—21no—
143C23×C32—21no—
144C23×C32—21no—
145C23×C32—21no—
146C23×C32C2411yescharacteristic
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—21no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4#4112yes—
187C2×C32×C4#4112yes—
188C2×C32×C4#4112yes—
189#400C2×C3211yescharacteristic, complemented
190#400C2×C3211yescharacteristic, complemented
191#399C2×C3211yescharacteristic
192#398C2×C3211yescharacteristic, complemented
193#398C2×C3211yescharacteristic, complemented
194#397C2×C3211yescharacteristic
195#397C2×C3211yescharacteristic
196#396C2×C3211yescharacteristic
197#389C2×C3211yescharacteristic, complemented
198#388C2×C3211yescharacteristic
199#388C2×C3211yescharacteristic
200#388C2×C3211yescharacteristic
201#387C2×C3211yescharacteristic, complemented
202#387C2×C3211yescharacteristic, complemented
203C4×C4⋊C4C2×C3211yescharacteristic, complemented
204C24×C3C23×C314yes—
205C22×C3×C4C23×C314yes—
206C22×C3×C4C23×C314yes—
207C22×C3×C4C23×C314yes—
208C22×C3×C4C23×C314yes—
209C22×C3×C4C23×C314yes—
210C22×C3×C4C23×C314yes—
211C22×C3×C4C23×C314yes—
212C22×C3×C4C23×C314yes—
213C22×C3×C4C23×C314yes—
214C22×C3×C4C23×C314yes—
215C22×C3×C4C23×C314yes—
216C22×C3×C4C23×C314yes—
217C22×C3×C4C23×C314yes—
218C22×C3×C4C23×C314yes—
219C3×C4⋊C4—28no—
220C3×C4⋊C4—28no—
221C3×C4⋊C4—28no—
222C3×C4⋊C4—28no—
223C3×C4⋊C4—28no—
224C3×C4⋊C4—44no—
225C3×C4⋊C4—44no—
226C3×C4⋊C4—44no—
227C3×C4⋊C4—44no—
228C3×C4⋊C4—44no—
229C3×C4⋊C4—44no—
230C3×C22⋊C4—28no—
231C3×C22⋊C4—28no—
232C3×C22⋊C4—28no—
233C3×C22⋊C4—44no—
234C3×C22⋊C4—44no—
235C3×C22⋊C4—44no—
236C3×C22⋊C4—44no—
237C3×C42—28no—
238C3×C42—44no—
239C3×C42—44no—
240C22×C32—41no—
241C22×C32—41no—
242C22×C32—41no—
243C22×C32—41no—
244C22×C32—41no—
245C22×C32—41no—
246C22×C32—41no—
247C22×C32C2×#4111yescharacteristic
248C22×C32C2×#4111yescharacteristic
249C22×C32C2×#4111yescharacteristic
250C22×C32H(5,𝔽2)11yescharacteristic
251C22×C32H(5,𝔽2)11yescharacteristic
252C22×C322−1+411yescharacteristic
253C22×C322−1+411yescharacteristic
254C32×C4—22no—
255C32×C4—22no—
256C32×C4—22no—
257C32×C4—41no—
258C32×C4—41no—
259C32×C4—41no—
260C32×C4—41no—
261C32×C4—41no—
262C32×C4—41no—
263C32×C4—41no—
264C32×C4—41no—
265C32×C4—41no—
266C32×C4—41no—
267C32×C4—41no—
268C2×C4⋊C4C22×C3211yescharacteristic
269C2×C4⋊C4C22×C3211yescharacteristic
270C2×C4⋊C4C22×C3211yescharacteristic
271C2×C4⋊C4C22×C3211yescharacteristic
272C2×C4⋊C4C22×C3211yescharacteristic
273C2×C4⋊C4C22×C3211yescharacteristic
274C2×C4⋊C4C22×C3211yescharacteristic
275C2×C4⋊C4C22×C3211yescharacteristic
276C2×C4⋊C4C22×C3211yescharacteristic
277C2×C4⋊C4C22×C3211yescharacteristic
278C2×C4⋊C4C22×C3211yescharacteristic
279C2×C22⋊C4C22×C3211yescharacteristic
280C2×C22⋊C4C22×C3211yescharacteristic
281C2×C22⋊C4C22×C3211yescharacteristic
282C2×C22⋊C4C22×C3211yescharacteristic
283C2×C22⋊C4C22×C3211yescharacteristic
284C2×C22⋊C4C22×C3211yescharacteristic
285C2×C22⋊C4C22×C3211yescharacteristic
286C2×C42C22×C3211yescharacteristic
287C2×C42C22×C3211yescharacteristic
288C2×C42C22×C3211yescharacteristic
289#95C22×C3211yescharacteristic
290#95C22×C3211yescharacteristic
291#95C22×C3211yescharacteristic
292#95C22×C3211yescharacteristic
293#95C22×C3211yescharacteristic
294#95C22×C3211yescharacteristic
295#95C22×C3211yescharacteristic
296#95C22×C3211yescharacteristic
297#95C22×C3211yescharacteristic
298#95C22×C3211yescharacteristic
299#95C22×C3211yescharacteristic
300#95C22×C3211yescharacteristic
301#95C22×C3211yescharacteristic
302#95C22×C3211yescharacteristic
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—24no—
306C23×C3—24no—
307C23×C3—24no—
308C23×C3—24no—
309C23×C3—24no—
310C23×C3C24×C314yes—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4C3×#4118yes—
351C2×C3×C4C3×#4118yes—
352C2×C3×C4C3×#4118yes—
353C2×C32—81no—
354C2×C32#52911yescharacteristic
355C2×C32#53911yescharacteristic
356C2×C32#53911yescharacteristic
357C2×C32#54011yescharacteristic
358C2×C32#54111yescharacteristic
359C2×C32#54211yescharacteristic
360C2×C32#54211yescharacteristic
361C24C23×C3211yescharacteristic
362C22×C4C23×C3211yescharacteristic
363C22×C4C23×C3211yescharacteristic
364C22×C4C23×C3211yescharacteristic
365C22×C4C23×C3211yescharacteristic
366C22×C4C23×C3211yescharacteristic
367C22×C4C23×C3211yescharacteristic
368C22×C4C23×C3211yescharacteristic
369C22×C4C23×C3211yescharacteristic
370C22×C4C23×C3211yescharacteristic
371C22×C4C23×C3211yescharacteristic
372C22×C4C23×C3211yescharacteristic
373C22×C4C23×C3211yescharacteristic
374C22×C4C23×C3211yescharacteristic
375C22×C4C23×C3211yescharacteristic
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C42—22no—
395C42—41no—
396C42—41no—
397C22×C3—44no—
398C22×C3—44no—
399C22×C3—44no—
400C22×C3—44no—
401C22×C3—44no—
402C22×C3—44no—
403C22×C3—44no—
404C22×C3C2×C3×#4114yes—
405C22×C3C2×C3×#4114yes—
406C22×C3C2×C3×#4114yes—
407C22×C3C3×H(5,𝔽2)14yes—
408C22×C3C3×H(5,𝔽2)14yes—
409C22×C3C3×2−1+414yes—
410C22×C3C3×2−1+414yes—
411C3×C4—28no—
412C3×C4—28no—
413C3×C4—28no—
414C3×C4—44no—
415C3×C4—44no—
416C3×C4—44no—
417C3×C4—44no—
418C3×C4—44no—
419C3×C4—44no—
420C3×C4—44no—
421C3×C4—44no—
422C3×C4—44no—
423C3×C4—44no—
424C3×C4—44no—
425C32#265411yescharacteristic, direct product factor, Sylow
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23C24×C3211yescharacteristic
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4C32×#4112yes—
474C2×C4C32×#4112yes—
475C2×C4C32×#4112yes—
476C2×C3—84no—
477C2×C3C3×#52914yes—
478C2×C3C3×#53914yes—
479C2×C3C3×#53914yes—
480C2×C3C3×#54014yes—
481C2×C3C3×#54114yes—
482C2×C3C3×#54214yes—
483C2×C3C3×#54214yes—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22C2×C32×#4111yescharacteristic
492C22C2×C32×#4111yescharacteristic
493C22C2×C32×#4111yescharacteristic
494C22C32×H(5,𝔽2)11yescharacteristic
495C22C32×H(5,𝔽2)11yescharacteristic
496C22C32×2−1+411yescharacteristic
497C22C32×2−1+411yescharacteristic
498C4—22no—
499C4—22no—
500C4—22no—
501C4—41no—
502C4—41no—
503C4—41no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4—41no—
512C3C3×#265414yesdirect product factor
513C2—81no—
514C2C32×#52911yescharacteristic
515C2C32×#53911yescharacteristic
516C2C32×#53911yescharacteristic
517C2C32×#54011yescharacteristic
518C2C32×#54111yescharacteristic
519C2C32×#54211yescharacteristic
520C2C32×#54211yescharacteristic
subgroupG/H
center, socleC23×C32C24
commutator, Frattini, cosocleC23C24×C32
prime radicalC24×C32C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C32 ◁ G

Complements

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

NH ≅ G/Ntype
1. C32×#400C2semidirect
2. C32×#400C2semidirect
4. C32×#398C2semidirect
5. C32×#398C2semidirect
9. C32×#389C2semidirect
13. C32×#387C2semidirect
14. C32×#387C2semidirect
15. C32×C4×C4⋊C4C2semidirect
16. C3×#2654C3direct (N × H)
52. C3×#400C2×C3semidirect
53. C3×#400C2×C3semidirect
55. C3×#398C2×C3semidirect
56. C3×#398C2×C3semidirect
60. C3×#389C2×C3semidirect
64. C3×#387C2×C3semidirect
65. C3×#387C2×C3semidirect
66. C3×C4×C4⋊C4C2×C3semidirect
103. #2654C32direct (N × H)
189. #400C2×C32semidirect
190. #400C2×C32semidirect
192. #398C2×C32semidirect
193. #398C2×C32semidirect
197. #389C2×C32semidirect
201. #387C2×C32semidirect
202. #387C2×C32semidirect
203. C4×C4⋊C4C2×C32semidirect
425. C32#2654direct (N × H)
512. C3C3×#2654direct (N × H)

Maximal subgroups

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

subgroupindex
C32×#4002
C32×#3992
C32×#3982
C32×#3972
C32×#3962
C32×#3892
C32×#3882
C32×#3872
C32×C4×C4⋊C42
C3×#26543