Finite Group Databasebeta

← #204252

Subgroups — #204252

Order 512.

1210 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2562C211yesmaximal, characteristic
2#21953C211yesmaximal, characteristic, complemented
3#20535C211yesmaximal, characteristic, complemented
4C2×#1470C211yesmaximal, characteristic, complemented
5C2×#1470C211yesmaximal, characteristic, complemented
6C2×#1466C211yesmaximal, characteristic
7C2×#1460C211yesmaximal, characteristic
8#9652C214yesmaximal, complemented
9#9651C214yesmaximal, complemented
10C22×#125C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2562—22no—
17#2562—22no—
18#2562—21no—
19#2562—21no—
20#2562—22no—
21C2×#400C412yes—
22C2×#394C412yes—
23C2×#388C2214yes—
24C2×#380C412yes—
25C2×#380C412yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27#1940—22no—
28#1939—22no—
29#1797C2214yes—
30#1771—22no—
31#1770—21no—
32C8×C4⋊C4—21no—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35#1470—24no—
36#1470—24no—
37#1466—24no—
38#1460—24no—
39C2×#324C2212yes—
40C2×#324C2214yes—
41C2×#324C2212yes—
42C2×#323C2212yes—
43C2×#323C2214yes—
44C2×#323C2212yes—
45C23×D4C2×C411yescharacteristic
46C24×C4C2×C412yes—
47C2×#125—21no—
48C2×#125—21no—
49C2×#125—21no—
50C2×#125—21no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4C2×C414yes—
70C22×C22⋊C4C2×C414yes—
71C22×C22⋊C4C2312yes—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74#432—22no—
75#432—22no—
76#432—22no—
77#432D414yes—
78#431—22no—
79#431—22no—
80#431—22no—
81#431D414yes—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—24no—
87C2×C22⋊C8C2312yes—
88C2×C22⋊C8C2312yes—
89C2×C22⋊C8C2311yescharacteristic
90C2×C22⋊C8C2311yescharacteristic
91C2×C22⋊C8C2312yes—
92C2×#97—21no—
93C2×#97—21no—
94C2×#97—22no—
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97C2×C4×C8—22no—
98#400—24no—
99#400—22no—
100#400—22no—
101#394—24no—
102#394—22no—
103#394—22no—
104#388—24no—
105#388—24no—
106#388—24no—
107#388—24no—
108#380—22no—
109#380—22no—
110#380—24no—
111#380—22no—
112#380—22no—
113#380—24no—
114C4×C4⋊C4D412yes—
115C4×C4⋊C4D412yes—
116C4×C4⋊C4D411yescharacteristic
117C4×C4⋊C4D411yescharacteristic
118C4×C4⋊C4D412yes—
119C2×#95C2×C414yes—
120C2×#95C2×C414yes—
121C2×#95C2×C412yes—
122#336—24no—
123#336—22no—
124#336—22no—
125#324—28no—
126#324—24no—
127#324—24no—
128#323—24no—
129#323—24no—
130#323—28no—
131C25C22×C412yes—
132C22×D4—24no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—42no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—44no—
143C23×C4—44no—
144C23×C4C22×C412yes—
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C412yes—
148C23×C4C22×C412yes—
149C23×C4C22×C414yes—
150C23×C4C2411yescharacteristic
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156#125—42no—
157#125—42no—
158#125—42no—
159#125—42no—
160#125—42no—
161#125—42no—
162#125—42no—
163#125—42no—
164C4×D4—42no—
165C4×D4—42no—
166C4×D4—42no—
167C4×D4—42no—
168C4×D4—42no—
169C4×D4—42no—
170C4×D4—42no—
171C4×D4—42no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C414yes—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—84no—
211C2×C42—82no—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D414yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42#4112yes—
222C2×C42#4112yes—
223C4⋊C8—24no—
224C4⋊C8—24no—
225C4⋊C8—42no—
226C4⋊C8—42no—
227C4⋊C8—84no—
228C22⋊C8—21no—
229C22⋊C8—21no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234C22⋊C8—22no—
235C22⋊C8C2×D412yes—
236C22⋊C8C2×D412yes—
237C22⋊C8C2×D414yes—
238#97—24no—
239#97—42no—
240#97—82no—
241C4×C8—24no—
242C4×C8—42no—
243C4×C8—82no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95C22⋊C414yes—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—42no—
258C24—42no—
259C24—42no—
260C24—42no—
261C24#13114yes—
262C24C23×C411yescharacteristic
263C2×D4—28no—
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—48no—
267C2×D4—44no—
268C2×D4—42no—
269C2×D4—42no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—88no—
301C22×C4—88no—
302C22×C4C2×C22⋊C414yes—
303C22×C4C2×C22⋊C414yes—
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C4×D412yes—
309C22×C4C4×D412yes—
310C22×C4C4×D412yes—
311C22×C4C4×D412yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—24no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C22⋊C4—48no—
349C22⋊C4—48no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—84no—
355C22⋊C4—84no—
356C42—42no—
357C42—42no—
358C42—44no—
359C42#12112yes—
360C42#12112yes—
361C42#12112yes—
362C42#12112yes—
363C42#12412yes—
364C42#12412yes—
365C42#12712yes—
366C42#12712yes—
367C23—21no—
368C23—21no—
369C23—22no—
370C23—24no—
371C23—24no—
372C23—24no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—82no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23C22×C22⋊C411yescharacteristic
394C23C2×C4×D411yescharacteristic
395C23C2×C4×D411yescharacteristic
396C23C2×#13112yes—
397C23#56812yes—
398D4—48no—
399D4—48no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—88no—
434C2×C4—84no—
435C2×C4—84no—
436C2×C4—88no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4#39012yes—
442C2×C4#39012yes—
443C2×C4#39012yes—
444C2×C4#39012yes—
445C2×C4#41314yes—
446C2×C4C2×#12111yescharacteristic
447C2×C4C2×#12111yescharacteristic
448C2×C4C2×#12411yescharacteristic
449C2×C4C2×#12711yescharacteristic
450C8—42no—
451C8—42no—
452C8—44no—
453C8—82no—
454C8—82no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—44no—
464C22—44no—
465C22—44no—
466C22—84no—
467C22—84no—
468C22—84no—
469C22—84no—
470C22—82no—
471C22—82no—
472C22—82no—
473C22—82no—
474C22C2×#39011yescharacteristic
475C22#288012yes—
476C22#288311yescharacteristic
477C22#288411yescharacteristic
478C22C2×#41312yes—
479C22#292911yescharacteristic
480C22#293011yescharacteristic
481C22#293312yes—
482C4—22no—
483C4—22no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—22no—
489C4—44no—
490C4—44no—
491C4—82no—
492C4—82no—
493C4—82no—
494C4#196714yes—
495C2—21no—
496C2—21no—
497C2—22no—
498C2—84no—
499C2#1742312yes—
500C2#1742412yes—
501C2#2119111yescharacteristic
502C2#2121411yescharacteristic
503C2C2×#196711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21953C2semidirect
3. #20535C2semidirect
4. C2×#1470C2semidirect
5. C2×#1470C2semidirect
8. #9652C2semidirect
9. #9651C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25622
#219532
#205352
C2×#14702
C2×#14662
C2×#14602
#96522
#96512