Finite Group Databasebeta

← #204253

Subgroups — #204253

Order 512.

1188 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2592C211yesmaximal, characteristic
2#21953C211yesmaximal, characteristic, complemented
3#20535C211yesmaximal, characteristic, complemented
4C2×#1469C211yesmaximal, characteristic
5C2×#1469C211yesmaximal, characteristic
6C2×#1467C211yesmaximal, characteristic, complemented
7C2×#1461C211yesmaximal, characteristic, complemented
8#9652C214yesmaximal, complemented
9#9651C214yesmaximal, complemented
10C22×#124C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2592—22no—
17#2592—22no—
18#2592—21no—
19#2592—21no—
20#2592—22no—
21C2×#397C412yes—
22C2×#393C412yes—
23C2×#391C412yes—
24C2×#390C412yes—
25C2×#388C2214yes—
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#1469—24no—
36#1469—24no—
37#1467—24no—
38#1461—24no—
39C2×#324C2212yes—
40C2×#324C2214yes—
41C2×#324C2212yes—
42C2×#323C2212yes—
43C2×#323C2214yes—
44C2×#323C2212yes—
45C23×Q8C2×C411yescharacteristic
46C23×Q8C2×C411yescharacteristic
47C23×D4C2×C411yescharacteristic
48C2×#124—21no—
49C2×#124—21no—
50C2×#124—21no—
51C2×#124—21no—
52C2×#124—22no—
53C2×#124—22no—
54C2×#124—22no—
55C2×#124—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2×C414yes—
69C22×C22⋊C4C2×C414yes—
70C22×C22⋊C4C2312yes—
71C22×C42C2×C412yes—
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#397—24no—
99#397—22no—
100#397—22no—
101#393—24no—
102#393—22no—
103#393—22no—
104#391—22no—
105#391—22no—
106#391—24no—
107#390—22no—
108#390—22no—
109#390—24no—
110#388—24no—
111#388—24no—
112#388—24no—
113#388—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—24no—
126#324—24no—
127#324—28no—
128#323—28no—
129#323—24no—
130#323—24no—
131C25C22×C412yes—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—24no—
141C22×Q8—42no—
142C22×Q8—44no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—44no—
148C23×C4C22×C412yes—
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C412yes—
152C23×C4C22×C412yes—
153C23×C4C22×C414yes—
154C23×C4C2411yescharacteristic
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160#124—42no—
161#124—42no—
162#124—42no—
163#124—42no—
164#124—42no—
165#124—42no—
166#124—42no—
167#124—42no—
168C4×Q8—42no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×Q8—42no—
174C4×Q8—42no—
175C4×Q8—42no—
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—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C414yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—84no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—24no—
210C2×C42—24no—
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—24no—
255C24—24no—
256C24—42no—
257C24—44no—
258C24—42no—
259C24C23×C411yescharacteristic
260C2×Q8—28no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—48no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—44no—
271C2×Q8—84no—
272C2×Q8—84no—
273C2×D4—44no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—44no—
277C2×D4—84no—
278C2×D4—84no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4C2×C22⋊C414yes—
304C22×C4C2×C22⋊C414yes—
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C4×D412yes—
310C22×C4C4×D412yes—
311C22×C4C4×D412yes—
312C22×C4C4×D412yes—
313C22×C4#13114yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—24no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—44no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C22⋊C4—48no—
345C22⋊C4—48no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C22⋊C4—84no—
353C22⋊C4—84no—
354C42—42no—
355C42—42no—
356C42—44no—
357C42—44no—
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—44no—
371C23—42no—
372C23—42no—
373C23—44no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—44no—
378C23—82no—
379C23—82no—
380C23—84no—
381C23—84no—
382C23—84no—
383C23—84no—
384C23—82no—
385C23—82no—
386C23C22×C22⋊C411yescharacteristic
387C23C2×C4×D411yescharacteristic
388C23C2×C4×D411yescharacteristic
389C23C2×#13112yes—
390C23#56812yes—
391Q8—48no—
392Q8—48no—
393Q8—84no—
394Q8—84no—
395D4—84no—
396D4—84no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
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—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4#39012yes—
438C2×C4#39012yes—
439C2×C4#39012yes—
440C2×C4#39012yes—
441C2×C4#41314yes—
442C2×C4C2×#12111yescharacteristic
443C2×C4C2×#12111yescharacteristic
444C2×C4C2×#12411yescharacteristic
445C2×C4C2×#12711yescharacteristic
446C8—42no—
447C8—42no—
448C8—44no—
449C8—82no—
450C8—82no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—84no—
460C22—84no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—162no—
465C22—162no—
466C22—164no—
467C22C2×#39011yescharacteristic
468C22#288012yes—
469C22#288311yescharacteristic
470C22#288411yescharacteristic
471C22C2×#41312yes—
472C22#292911yescharacteristic
473C22#293011yescharacteristic
474C22#293312yes—
475C4—22no—
476C4—22no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—22no—
482C4—44no—
483C4—44no—
484C4—44no—
485C4—82no—
486C4—82no—
487C4#196714yes—
488C2—21no—
489C2—21no—
490C2—22no—
491C2—162no—
492C2#1742812yes—
493C2#1742912yes—
494C2#2119111yescharacteristic
495C2#2121411yescharacteristic
496C2C2×#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
6. C2×#1467C2semidirect
7. C2×#1461C2semidirect
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×#25922
#219532
#205352
C2×#14692
C2×#14672
C2×#14612
#96522
#96512