Finite Group Databasebeta

← C32×#2704 #11769753

Subgroups — C32×#2704 #11769753

Order 1152.

1416 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×#125C211yesmaximal, characteristic, complemented
2C2×C32×C22⋊Q8C211yesmaximal, characteristic
3C32×#398C211yesmaximal, characteristic, complemented
4C32×#398C211yesmaximal, characteristic, complemented
5C32×#398C211yesmaximal, characteristic, complemented
6C32×#396C211yesmaximal, characteristic
7C32×#396C211yesmaximal, characteristic
8C32×#395C211yesmaximal, characteristic, complemented
9C32×#391C211yesmaximal, characteristic, complemented
10C32×#389C211yesmaximal, characteristic, complemented
11C32×#389C211yesmaximal, characteristic, complemented
12C32×#388C211yesmaximal, characteristic
13C32×#388C211yesmaximal, characteristic
14C32×#385C211yesmaximal, characteristic
15C32×#380C211yesmaximal, characteristic
16C3×#2704C314yesmaximal, direct product factor
17C22×C32×Q8C2211yescharacteristic
18C23×C32×C4C2211yescharacteristic
19C32×#125—24no—
20C32×C22⋊Q8—24no—
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×C4⋊C4C2211yescharacteristic
29C2×C32×C4⋊C4C2211yescharacteristic
30C2×C32×C4⋊C4C2211yescharacteristic
31C2×C32×C4⋊C4C2211yescharacteristic
32C2×C32×C4⋊C4C2211yescharacteristic
33C2×C32×C4⋊C4C2211yescharacteristic
34C2×C32×C4⋊C4C2211yescharacteristic
35C2×C32×C4⋊C4C2211yescharacteristic
36C2×C32×C22⋊C4C2211yescharacteristic
37C2×C32×C22⋊C4C2211yescharacteristic
38C2×C32×C22⋊C4C2211yescharacteristic
39C2×C32×C22⋊C4C2211yescharacteristic
40C2×C32×C22⋊C4C2211yescharacteristic
41C2×C32×C22⋊C4C2211yescharacteristic
42C2×C32×C42C2211yescharacteristic
43C2×C32×C42C2211yescharacteristic
44C32×#95C2211yescharacteristic
45C32×#95C2211yescharacteristic
46C32×#95C2211yescharacteristic
47C32×#95C2211yescharacteristic
48C32×#95C2211yescharacteristic
49C32×#95C2211yescharacteristic
50C32×#95C2211yescharacteristic
51C32×#95C2211yescharacteristic
52C32×#95C2211yescharacteristic
53C32×#95C2211yescharacteristic
54C2×C3×#125C2×C314yescomplemented
55C2×C3×C22⋊Q8C2×C314yes—
56C3×#398C2×C314yescomplemented
57C3×#398C2×C314yescomplemented
58C3×#398C2×C314yescomplemented
59C3×#396C2×C314yes—
60C3×#396C2×C314yes—
61C3×#395C2×C314yescomplemented
62C3×#391C2×C314yescomplemented
63C3×#389C2×C314yescomplemented
64C3×#389C2×C314yescomplemented
65C3×#388C2×C314yes—
66C3×#388C2×C314yes—
67C3×#385C2×C314yes—
68C3×#380C2×C314yes—
69C24×C32C2311yescharacteristic
70C2×C32×Q8—22no—
71C2×C32×Q8—22no—
72C2×C32×Q8—41no—
73C22×C32×C4—22no—
74C22×C32×C4—22no—
75C22×C32×C4—41no—
76C22×C32×C4C2311yescharacteristic
77C22×C32×C4C2311yescharacteristic
78C22×C32×C4C2311yescharacteristic
79C22×C32×C4C2311yescharacteristic
80C22×C32×C4C2311yescharacteristic
81C22×C32×C4C2311yescharacteristic
82C22×C32×C4C2311yescharacteristic
83C22×C32×C4C2311yescharacteristic
84C22×C32×C4C2311yescharacteristic
85C22×C32×C4C2311yescharacteristic
86C22×C32×C4C2311yescharacteristic
87C22×C32×C4C2311yescharacteristic
88C22×C32×C4C2311yescharacteristic
89C22×C32×C4C2311yescharacteristic
90C32×C4⋊C4—22no—
91C32×C4⋊C4—22no—
92C32×C4⋊C4—22no—
93C32×C4⋊C4—22no—
94C32×C4⋊C4—22no—
95C32×C4⋊C4—22no—
96C32×C4⋊C4—22no—
97C32×C4⋊C4—22no—
98C32×C4⋊C4—41no—
99C32×C4⋊C4—41no—
100C32×C4⋊C4—41no—
101C32×C4⋊C4—41no—
102C32×C4⋊C4—41no—
103C32×C4⋊C4—41no—
104C32×C4⋊C4D414yes—
105C32×C22⋊C4—22no—
106C32×C22⋊C4—22no—
107C32×C22⋊C4—41no—
108C32×C22⋊C4—41no—
109C32×C22⋊C4—41no—
110C32×C22⋊C4Q814yescomplemented
111C32×C42—22no—
112C32×C42—41no—
113#2704C3211yescharacteristic, direct product factor, Sylow
114C22×C3×Q8C22×C314yes—
115C23×C3×C4C22×C314yes—
116C3×#125—216no—
117C3×C22⋊Q8—216no—
118C2×C3×C4⋊C4C22×C314yes—
119C2×C3×C4⋊C4C22×C314yes—
120C2×C3×C4⋊C4C22×C314yes—
121C2×C3×C4⋊C4C22×C314yes—
122C2×C3×C4⋊C4C22×C314yes—
123C2×C3×C4⋊C4C22×C314yes—
124C2×C3×C4⋊C4C22×C314yes—
125C2×C3×C4⋊C4C22×C314yes—
126C2×C3×C4⋊C4C22×C314yes—
127C2×C3×C4⋊C4C22×C314yes—
128C2×C3×C4⋊C4C22×C314yes—
129C2×C3×C4⋊C4C22×C314yes—
130C2×C3×C4⋊C4C22×C314yes—
131C2×C3×C4⋊C4C22×C314yes—
132C2×C3×C4⋊C4C22×C314yes—
133C2×C3×C22⋊C4C22×C314yes—
134C2×C3×C22⋊C4C22×C314yes—
135C2×C3×C22⋊C4C22×C314yes—
136C2×C3×C22⋊C4C22×C314yes—
137C2×C3×C22⋊C4C22×C314yes—
138C2×C3×C22⋊C4C22×C314yes—
139C2×C3×C42C22×C314yes—
140C2×C3×C42C22×C314yes—
141C3×#95C22×C314yes—
142C3×#95C22×C314yes—
143C3×#95C22×C314yes—
144C3×#95C22×C314yes—
145C3×#95C22×C314yes—
146C3×#95C22×C314yes—
147C3×#95C22×C314yes—
148C3×#95C22×C314yes—
149C3×#95C22×C314yes—
150C3×#95C22×C314yes—
151C23×C32—21no—
152C23×C32—21no—
153C23×C32—21no—
154C23×C32—21no—
155C23×C32—21no—
156C23×C32—21no—
157C23×C32C2×Q812yes—
158C23×C32C2411yescharacteristic
159C32×Q8—44no—
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—21no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—21no—
191C2×C32×C4—21no—
192C2×C32×C4—21no—
193C2×C32×C4—21no—
194C2×C32×C4—21no—
195C2×C32×C4—21no—
196C2×C32×C4—44no—
197C2×C32×C4C2×D412yes—
198C2×C32×C4C2×D412yes—
199C2×C32×C4C2×D412yes—
200C2×C32×C4C2×Q812yes—
201C2×C32×C4C2×Q812yes—
202C2×C32×C4#4112yes—
203C2×#125C2×C3211yescharacteristic, complemented
204C2×C22⋊Q8C2×C3211yescharacteristic
205#398C2×C3211yescharacteristic, complemented
206#398C2×C3211yescharacteristic, complemented
207#398C2×C3211yescharacteristic, complemented
208#396C2×C3211yescharacteristic
209#396C2×C3211yescharacteristic
210#395C2×C3211yescharacteristic, complemented
211#391C2×C3211yescharacteristic, complemented
212#389C2×C3211yescharacteristic, complemented
213#389C2×C3211yescharacteristic, complemented
214#388C2×C3211yescharacteristic
215#388C2×C3211yescharacteristic
216#385C2×C3211yescharacteristic
217#380C2×C3211yescharacteristic
218C24×C3C23×C314yes—
219C2×C3×Q8—28no—
220C2×C3×Q8—28no—
221C2×C3×Q8—44no—
222C22×C3×C4—28no—
223C22×C3×C4—28no—
224C22×C3×C4—44no—
225C22×C3×C4C23×C314yes—
226C22×C3×C4C23×C314yes—
227C22×C3×C4C23×C314yes—
228C22×C3×C4C23×C314yes—
229C22×C3×C4C23×C314yes—
230C22×C3×C4C23×C314yes—
231C22×C3×C4C23×C314yes—
232C22×C3×C4C23×C314yes—
233C22×C3×C4C23×C314yes—
234C22×C3×C4C23×C314yes—
235C22×C3×C4C23×C314yes—
236C22×C3×C4C23×C314yes—
237C22×C3×C4C23×C314yes—
238C22×C3×C4C23×C314yes—
239C3×C4⋊C4—28no—
240C3×C4⋊C4—28no—
241C3×C4⋊C4—28no—
242C3×C4⋊C4—28no—
243C3×C4⋊C4—28no—
244C3×C4⋊C4—28no—
245C3×C4⋊C4—28no—
246C3×C4⋊C4—28no—
247C3×C4⋊C4—44no—
248C3×C4⋊C4—44no—
249C3×C4⋊C4—44no—
250C3×C4⋊C4—44no—
251C3×C4⋊C4—44no—
252C3×C4⋊C4—44no—
253C3×C4⋊C4C3×D4116yes—
254C3×C22⋊C4—28no—
255C3×C22⋊C4—28no—
256C3×C22⋊C4—44no—
257C3×C22⋊C4—44no—
258C3×C22⋊C4—44no—
259C3×C22⋊C4C3×Q8116yescomplemented
260C3×C42—28no—
261C3×C42—44no—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32—41no—
266C22×C32—41no—
267C22×C32—41no—
268C22×C32—41no—
269C22×C32C22×D411yescharacteristic
270C22×C32C22×Q811yescharacteristic
271C22×C32C2×#4111yescharacteristic
272C22×C32H(5,𝔽2)11yescharacteristic
273C22×C32H(5,𝔽2)11yescharacteristic
274C22×C322−1+411yescharacteristic
275C22×C322−1+411yescharacteristic
276C32×C4—22no—
277C32×C4—22no—
278C32×C4—22no—
279C32×C4—22no—
280C32×C4—22no—
281C32×C4—22no—
282C32×C4—41no—
283C32×C4—41no—
284C32×C4—41no—
285C32×C4—41no—
286C32×C4—41no—
287C32×C4—41no—
288C32×C4—41no—
289C32×C4—41no—
290C22×Q8C22×C3211yescharacteristic
291C23×C4C22×C3211yescharacteristic
292#125—24no—
293C22⋊Q8—24no—
294C2×C4⋊C4C22×C3211yescharacteristic
295C2×C4⋊C4C22×C3211yescharacteristic
296C2×C4⋊C4C22×C3211yescharacteristic
297C2×C4⋊C4C22×C3211yescharacteristic
298C2×C4⋊C4C22×C3211yescharacteristic
299C2×C4⋊C4C22×C3211yescharacteristic
300C2×C4⋊C4C22×C3211yescharacteristic
301C2×C4⋊C4C22×C3211yescharacteristic
302C2×C4⋊C4C22×C3211yescharacteristic
303C2×C4⋊C4C22×C3211yescharacteristic
304C2×C4⋊C4C22×C3211yescharacteristic
305C2×C4⋊C4C22×C3211yescharacteristic
306C2×C4⋊C4C22×C3211yescharacteristic
307C2×C4⋊C4C22×C3211yescharacteristic
308C2×C4⋊C4C22×C3211yescharacteristic
309C2×C22⋊C4C22×C3211yescharacteristic
310C2×C22⋊C4C22×C3211yescharacteristic
311C2×C22⋊C4C22×C3211yescharacteristic
312C2×C22⋊C4C22×C3211yescharacteristic
313C2×C22⋊C4C22×C3211yescharacteristic
314C2×C22⋊C4C22×C3211yescharacteristic
315C2×C42C22×C3211yescharacteristic
316C2×C42C22×C3211yescharacteristic
317#95C22×C3211yescharacteristic
318#95C22×C3211yescharacteristic
319#95C22×C3211yescharacteristic
320#95C22×C3211yescharacteristic
321#95C22×C3211yescharacteristic
322#95C22×C3211yescharacteristic
323#95C22×C3211yescharacteristic
324#95C22×C3211yescharacteristic
325#95C22×C3211yescharacteristic
326#95C22×C3211yescharacteristic
327C23×C3—24no—
328C23×C3—24no—
329C23×C3—24no—
330C23×C3—24no—
331C23×C3—24no—
332C23×C3—24no—
333C23×C3C2×C3×Q818yes—
334C23×C3C24×C314yes—
335C3×Q8—416no—
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×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—24no—
359C2×C3×C4—24no—
360C2×C3×C4—24no—
361C2×C3×C4—24no—
362C2×C3×C4—24no—
363C2×C3×C4—24no—
364C2×C3×C4—24no—
365C2×C3×C4—24no—
366C2×C3×C4—24no—
367C2×C3×C4—24no—
368C2×C3×C4—24no—
369C2×C3×C4—24no—
370C2×C3×C4—24no—
371C2×C3×C4—24no—
372C2×C3×C4—416no—
373C2×C3×C4C2×C3×D418yes—
374C2×C3×C4C2×C3×D418yes—
375C2×C3×C4C2×C3×D418yes—
376C2×C3×C4C2×C3×Q818yes—
377C2×C3×C4C2×C3×Q818yes—
378C2×C3×C4C3×#4118yes—
379C2×C32—42no—
380C2×C32#53711yescharacteristic
381C2×C32#54111yescharacteristic
382C2×C32#54411yescharacteristic
383C2×C32#54611yescharacteristic
384C2×C32D4×Q811yescharacteristic
385C2×C32#55411yescharacteristic
386C2×C32#56211yescharacteristic
387C24C23×C3211yescharacteristic
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—41no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—41no—
394C22×C4C23×C3211yescharacteristic
395C22×C4C23×C3211yescharacteristic
396C22×C4C23×C3211yescharacteristic
397C22×C4C23×C3211yescharacteristic
398C22×C4C23×C3211yescharacteristic
399C22×C4C23×C3211yescharacteristic
400C22×C4C23×C3211yescharacteristic
401C22×C4C23×C3211yescharacteristic
402C22×C4C23×C3211yescharacteristic
403C22×C4C23×C3211yescharacteristic
404C22×C4C23×C3211yescharacteristic
405C22×C4C23×C3211yescharacteristic
406C22×C4C23×C3211yescharacteristic
407C22×C4C23×C3211yescharacteristic
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4C32×D414yes—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4C32×Q814yescomplemented
429C42—22no—
430C42—41no—
431C22×C3—28no—
432C22×C3—28no—
433C22×C3—28no—
434C22×C3—44no—
435C22×C3—44no—
436C22×C3—44no—
437C22×C3—44no—
438C22×C3C22×C3×D414yes—
439C22×C3C22×C3×Q814yes—
440C22×C3C2×C3×#4114yes—
441C22×C3C3×H(5,𝔽2)14yes—
442C22×C3C3×H(5,𝔽2)14yes—
443C22×C3C3×2−1+414yes—
444C22×C3C3×2−1+414yes—
445C3×C4—28no—
446C3×C4—28no—
447C3×C4—28no—
448C3×C4—28no—
449C3×C4—28no—
450C3×C4—28no—
451C3×C4—44no—
452C3×C4—44no—
453C3×C4—44no—
454C3×C4—44no—
455C3×C4—44no—
456C3×C4—44no—
457C3×C4—44no—
458C3×C4—44no—
459C32#270411yescharacteristic, direct product factor, Sylow
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23C2×C32×Q812yes—
467C23C24×C3211yescharacteristic
468Q8—44no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—44no—
506C2×C4C2×C32×D412yes—
507C2×C4C2×C32×D412yes—
508C2×C4C2×C32×D412yes—
509C2×C4C2×C32×Q812yes—
510C2×C4C2×C32×Q812yes—
511C2×C4C32×#4112yes—
512C2×C3—48no—
513C2×C3C3×#53714yes—
514C2×C3C3×#54114yes—
515C2×C3C3×#54414yes—
516C2×C3C3×#54614yes—
517C2×C3C3×D4×Q814yes—
518C2×C3C3×#55414yes—
519C2×C3C3×#56214yes—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22C22×C32×D411yescharacteristic
528C22C22×C32×Q811yescharacteristic
529C22C2×C32×#4111yescharacteristic
530C22C32×H(5,𝔽2)11yescharacteristic
531C22C32×H(5,𝔽2)11yescharacteristic
532C22C32×2−1+411yescharacteristic
533C22C32×2−1+411yescharacteristic
534C4—22no—
535C4—22no—
536C4—22no—
537C4—22no—
538C4—22no—
539C4—22no—
540C4—41no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—41no—
547C4—41no—
548C3C3×#270414yesdirect product factor
549C2—42no—
550C2C32×#53711yescharacteristic
551C2C32×#54111yescharacteristic
552C2C32×#54411yescharacteristic
553C2C32×#54611yescharacteristic
554C2C32×D4×Q811yescharacteristic
555C2C32×#55411yescharacteristic
556C2C32×#56211yescharacteristic
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. C2×C32×#125C2semidirect
3. C32×#398C2semidirect
4. C32×#398C2semidirect
5. C32×#398C2semidirect
8. C32×#395C2semidirect
9. C32×#391C2semidirect
10. C32×#389C2semidirect
11. C32×#389C2semidirect
16. C3×#2704C3direct (N × H)
54. C2×C3×#125C2×C3semidirect
56. C3×#398C2×C3semidirect
57. C3×#398C2×C3semidirect
58. C3×#398C2×C3semidirect
61. C3×#395C2×C3semidirect
62. C3×#391C2×C3semidirect
63. C3×#389C2×C3semidirect
64. C3×#389C2×C3semidirect
110. C32×C22⋊C4Q8semidirect
113. #2704C32direct (N × H)
203. C2×#125C2×C32semidirect
205. #398C2×C32semidirect
206. #398C2×C32semidirect
207. #398C2×C32semidirect
210. #395C2×C32semidirect
211. #391C2×C32semidirect
212. #389C2×C32semidirect
213. #389C2×C32semidirect
259. C3×C22⋊C4C3×Q8semidirect
428. C22⋊C4C32×Q8semidirect
459. C32#2704direct (N × H)
548. C3C3×#2704direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#1252
C2×C32×C22⋊Q82
C32×#3982
C32×#3962
C32×#3952
C32×#3912
C32×#3892
C32×#3882
C32×#3852
C32×#3802
C3×#27043