Finite Group Databasebeta

← #493884

Subgroups — #493884

Order 512.

1910 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2311C211yesmaximal, characteristic
2#20584C212yesmaximal
3#12859C214yesmaximal, complemented
4#12856C214yesmaximal, complemented
5#12583C214yesmaximal, complemented
6C23×#41C2211yescharacteristic
7C22×#117C411yescharacteristic, complemented
8C23×C4⋊C4C411yescharacteristic, complemented
9C2×#410—22no—
10C2×#410C2214yes—
11#2311—22no—
12#2311—22no—
13#2311C412yescomplemented
14#2311C412yescomplemented
15#2311C2212yes—
16#2311C2212yes—
17C2×#391C412yescomplemented
18C2×#390C2212yes—
19C2×#386—22no—
20#2280—22no—
21C2×#380—22no—
22#1782—22no—
23#1782C2214yes—
24#1781—22no—
25#1781C2214yes—
26#1393C22116yescomplemented
27C22×#41—28no—
28C22×#41—42no—
29C22×#41—42no—
30C22×#41—44no—
31C22×#41C2×C411yescharacteristic
32C22×#41C2×C411yescharacteristic
33C22×#41D412yes—
34C22×#41D412yes—
35C22×#41C2311yescharacteristic
36C22×#41C2311yescharacteristic
37C23×Q8C2×C411yescharacteristic
38C23×D4D412yes—
39C23×D4C2311yescharacteristic
40C24×C4—21no—
41C24×C4C2×C411yescharacteristic
42C24×C4D412yes—
43C2×#117—21no—
44C2×#117—21no—
45C2×#117—24no—
46C2×#117—44no—
47C2×#117C2×C412yes—
48C2×#117C2×C412yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—42no—
54C22×C4⋊C4—42no—
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—24no—
60C22×C42C2×C412yes—
61#410—216no—
62#410—48no—
63C2×#99—24no—
64C2×#99—24no—
65C2×#99—24no—
66C2×#99C2318yes—
67#391—24no—
68#391C2×C414yes—
69#391C2×C414yes—
70#390—24no—
71#390C2×C414yes—
72#390C2314yes—
73#386—24no—
74#386—24no—
75#386—24no—
76#386—24no—
77#381—24no—
78#380—24no—
79#380—24no—
80#380—24no—
81#380—24no—
82#380—42no—
83#380—42no—
84#379—24no—
85C4×C22⋊C4—42no—
86C4×C22⋊C4—42no—
87C2×#95—24no—
88C25C22⋊C411yescharacteristic
89C25C2×D412yes—
90C2×#41—24no—
91C2×#41—28no—
92C2×#41—24no—
93C2×#41—28no—
94C2×#41—24no—
95C2×#41—28no—
96C2×#41—42no—
97C2×#41—42no—
98C2×#41—42no—
99C2×#41—42no—
100C2×#41—42no—
101C2×#41—42no—
102C2×#41—42no—
103C2×#41—42no—
104C2×#41—42no—
105C2×#41—42no—
106C2×#41—42no—
107C2×#41—42no—
108C2×#41—44no—
109C2×#41—432no—
110C2×#41—432no—
111C2×#41#4118yes—
112C22×Q8—24no—
113C22×Q8—41no—
114C22×Q8—41no—
115C22×Q8—42no—
116C22×Q8C22⋊C412yes—
117C22×Q8C22×C411yescharacteristic
118C22×Q8C22×C411yescharacteristic
119C22×D4—28no—
120C22×D4—24no—
121C22×D4—42no—
122C22×D4—42no—
123C22×D4—42no—
124C22×D4—41no—
125C22×D4—41no—
126C22×D4—44no—
127C22×D4C22⋊C412yes—
128C22×D4C22⋊C412yes—
129C22×D4C22×C411yescharacteristic
130C22×D4C2×D412yes—
131C22×D4C2×D412yes—
132C22×D4C2×D412yes—
133C22×D4C2411yescharacteristic
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—28no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—44no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—84no—
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C412yes—
153C23×C4C22⋊C412yes—
154C23×C4C22⋊C412yes—
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162#117—44no—
163#117—44no—
164#117—44no—
165#117—44no—
166#117—48no—
167#117—48no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4C22×C414yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—88no—
193C2×C42—22no—
194C2×C42—44no—
195C2×C42—44no—
196C2×C42C22×C414yes—
197#99—216no—
198#99—48no—
199#99—48no—
200#99—48no—
201#95—28no—
202#95—28no—
203#95—44no—
204#95—44no—
205C24—22no—
206C24—22no—
207C24—21no—
208C24—21no—
209C24—24no—
210C24—22no—
211C24—22no—
212C24—42no—
213C24—42no—
214C24—42no—
215C24—44no—
216C24—44no—
217C24C2×C22⋊C411yescharacteristic
218C24C2×C22⋊C412yes—
219C24C2×C22⋊C411yescharacteristic
220C24C2×C22⋊C411yescharacteristic
221C24C22×D412yes—
222#41—416no—
223#41—416no—
224#41—416no—
225#41—416no—
226#41—416no—
227#41—416no—
228#41—416no—
229#41—416no—
230C2×Q8—22no—
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—42no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—48no—
239C2×Q8—48no—
240C2×Q8#11714yes—
241C2×D4—24no—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—22no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—48no—
261C2×D4—48no—
262C2×D4—416no—
263C2×D4—416no—
264C2×D4#12318yes—
265C2×D4C2×#4114yes—
266C22×C4—216no—
267C22×C4—216no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—48no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—48no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—48no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—48no—
318C22×C4—84no—
319C22×C4—84no—
320C22×C4—88no—
321C22×C4—88no—
322C22×C4#9912yes—
323C22×C4#9912yes—
324C22×C4#9914yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4#11714yes—
331C22×C4#12318yes—
332C22×C4C23×C411yescharacteristic
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—88no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—48no—
343C22⋊C4—48no—
344C22⋊C4—88no—
345C22⋊C4—88no—
346C22⋊C4—88no—
347C22⋊C4—88no—
348C42—44no—
349C42—44no—
350C42—88no—
351C23—21no—
352C23—22no—
353C23—22no—
354C23—21no—
355C23—24no—
356C23—24no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—22no—
361C23—21no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—44no—
367C23—44no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—42no—
373C23—48no—
374C23—48no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—416no—
381C23—42no—
382C23—42no—
383C23—44no—
384C23—44no—
385C23—82no—
386C23—82no—
387C23—88no—
388C23#38112yes—
389C23C2×#9911yescharacteristic
390C23C2×#9911yescharacteristic
391C23C2×#9912yes—
392C23C22×C22⋊C411yescharacteristic
393C23C2×#12314yes—
394Q8—44no—
395Q8—48no—
396Q8—44no—
397D4—48no—
398D4—44no—
399D4—48no—
400D4—48no—
401D4—44no—
402D4—48no—
403D4—48no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—28no—
407C2×C4—28no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—48no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—88no—
439C2×C4—84no—
440C2×C4—88no—
441C2×C4—88no—
442C2×C4—84no—
443C2×C4—88no—
444C2×C4—88no—
445C2×C4—88no—
446C2×C4#38114yes—
447C2×C4#38112yes—
448C2×C4C2×#9912yes—
449C2×C4C2×#9912yes—
450C2×C4C2×#9912yes—
451C2×C4C2×#9912yes—
452C2×C4C2×#11712yes—
453C22—24no—
454C22—24no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—22no—
462C22—22no—
463C22—24no—
464C22—44no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—48no—
470C22—48no—
471C22—44no—
472C22—41no—
473C22—41no—
474C22—42no—
475C22—42no—
476C22—41no—
477C22—41no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—82no—
482C22—88no—
483C22—88no—
484C22—82no—
485C22#212812yes—
486C22#212912yes—
487C22C2×#38111yescharacteristic
488C22C22×#9911yescharacteristic
489C22C22×#9911yescharacteristic
490C4—22no—
491C4—22no—
492C4—22no—
493C4—24no—
494C4—42no—
495C4—84no—
496C4—84no—
497C4—168no—
498C4#181914yes—
499C2—21no—
500C2—21no—
501C2—22no—
502C2—22no—
503C2—44no—
504C2—42no—
505C2—42no—
506C2—82no—
507C2C2×#181911yescharacteristic
508C2C2×#212811yescharacteristic
509C2C2×#212911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #12859C2semidirect
4. #12856C2semidirect
5. #12583C2semidirect
7. C22×#117C4semidirect
8. C23×C4⋊C4C4semidirect
13. #2311C4semidirect
14. #2311C4semidirect
17. C2×#391C4semidirect
26. #1393C22semidirect

Maximal subgroups

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

subgroupindex
C2×#23112
#205842
#128592
#128562
#125832