Finite Group Databasebeta

← #189746

Subgroups — #189746

Order 512.

1198 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2353C211yesmaximal, characteristic
2#21788C212yesmaximal
3#8704C212yesmaximal, complemented
4#8702C212yesmaximal, complemented
5#8700C212yesmaximal, complemented
6#8698C212yesmaximal, complemented
7#7916C212yesmaximal, complemented
8#7910C211yesmaximal, characteristic, complemented
9#7906C211yesmaximal, characteristic, complemented
10C22×C4×D4C2211yescharacteristic
11C2×#432C2212yes—
12C2×#431C2212yes—
13#2353—22no—
14#2353—22no—
15#2353—22no—
16#2353—22no—
17C2×#388C412yes—
18C2×#387C412yes—
19C2×#386C412yes—
20C2×#383C2211yescharacteristic
21C2×#381C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×C4×C22⋊C4C412yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25C2×#376C2211yescharacteristic
26C22×#95C2211yescharacteristic
27#1955—22no—
28#1955—22no—
29#1954—22no—
30#1954—22no—
31#1779C2214yes—
32#1779C2214yes—
33#1280C2212yes—
34#1280C2212yes—
35#1280C2214yes—
36#1280C2212yes—
37#1280C2212yes—
38#1280C2214yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2×C412yes—
41C24×C4C2311yescharacteristic
42C24×C4C2311yescharacteristic
43C2×C4×D4—22no—
44C2×C4×D4—24no—
45C2×C4×D4—22no—
46C2×C4×D4D412yes—
47C2×C4×D4D412yes—
48C2×C4×D4D412yes—
49C2×C4×D4D412yes—
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C412yes—
52C22×C22⋊C4C2×C412yes—
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C412yes—
56C22×C22⋊C4C2×C412yes—
57C22×C42C2×C412yes—
58C22×C42C2×C412yes—
59C22×C42C2311yescharacteristic
60#432—24no—
61#432—24no—
62#431—24no—
63#431—24no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8—24no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8C2314yes—
71C2×C22⋊C8C2314yes—
72C2×#97—22no—
73C2×C4×C8—22no—
74#388—24no—
75#388—24no—
76#387—24no—
77#387—24no—
78#386—24no—
79#386—24no—
80#383—22no—
81#383—22no—
82#381—22no—
83#381—22no—
84#380—22no—
85#380—22no—
86C4×C22⋊C4—22no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92#376—22no—
93#376—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95C2×C412yes—
97C2×#95C2×C412yes—
98C2×#95C2×C412yes—
99C2×#95C2×C412yes—
100C2×#95C2×C412yes—
101C2×#95C2311yescharacteristic
102C2×#95C2311yescharacteristic
103C2×#95C2311yescharacteristic
104C2×#95C2311yescharacteristic
105C25C22×C411yescharacteristic
106C25C22×C411yescharacteristic
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—42no—
110C22×D4—42no—
111C22×D4C22⋊C412yes—
112C22×D4C22⋊C412yes—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—24no—
122C23×C4—42no—
123C23×C4—44no—
124C23×C4—82no—
125C23×C4—82no—
126C23×C4—82no—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C412yes—
129C23×C4C22×C412yes—
130C23×C4C22×C412yes—
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C412yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2411yescharacteristic
140C22×C8—24no—
141C22×C8—24no—
142C4×D4—24no—
143C4×D4—24no—
144C4×D4—24no—
145C4×D4—24no—
146C4×D4—24no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22⋊C412yes—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—44no—
167C2×C22⋊C4—82no—
168C2×C22⋊C4C22⋊C411yescharacteristic
169C2×C22⋊C4C22⋊C411yescharacteristic
170C2×C22⋊C4C22⋊C411yescharacteristic
171C2×C22⋊C4C22⋊C411yescharacteristic
172C2×C22⋊C4C22⋊C412yes—
173C2×C22⋊C4C22⋊C412yes—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—24no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—42no—
180C2×C42—82no—
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42#4112yes—
184C2×C42#4114yes—
185C2×C42#4112yes—
186C4⋊C8—44no—
187C4⋊C8—44no—
188C22⋊C8—24no—
189C22⋊C8—24no—
190C22⋊C8—24no—
191C22⋊C8—24no—
192C22⋊C8—84no—
193C22⋊C8—84no—
194C22⋊C8—84no—
195C22⋊C8—84no—
196#97—44no—
197C4×C8—44no—
198#95—24no—
199#95—24no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—24no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209C24—21no—
210C24—21no—
211C24—22no—
212C24—22no—
213C24—21no—
214C24—21no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24—22no—
219C24—42no—
220C24—42no—
221C24—41no—
222C24—41no—
223C24C2×C22⋊C412yes—
224C24C2×C22⋊C411yescharacteristic
225C24C2×C22⋊C411yescharacteristic
226C24C23×C411yescharacteristic
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—48no—
234C2×D4—48no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—84no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—84no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—88no—
290C22×C4—88no—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4#11712yes—
298C22×C4#11712yes—
299C22×C4#11712yes—
300C22×C4#11712yes—
301C22×C4#13114yes—
302C22×C4C22×D412yes—
303C22×C4C2×#4112yes—
304C22×C4C2×#4111yescharacteristic
305C22×C4C2×#4111yescharacteristic
306C2×C8—44no—
307C2×C8—44no—
308C2×C8—44no—
309C2×C8—44no—
310C2×C8—44no—
311C2×C8—44no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—84no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—48no—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C22⋊C4—84no—
325C42—44no—
326C42—44no—
327C42—82no—
328C42—82no—
329C42#12314yes—
330C42#12314yes—
331C42#12314yes—
332C42#12314yes—
333C23—22no—
334C23—21no—
335C23—21no—
336C23—24no—
337C23—24no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—24no—
347C23—24no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—44no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—42no—
367C23—82no—
368C23—82no—
369C23—82no—
370C23—82no—
371C23#41312yes—
372C23#41312yes—
373C23C22×C22⋊C411yescharacteristic
374C23C2×#11711yescharacteristic
375C23C2×#11711yescharacteristic
376C23C2×#13112yes—
377C23#56812yes—
378D4—48no—
379D4—48no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—88no—
426C2×C4—88no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4#38112yes—
434C2×C4#38112yes—
435C2×C4#38114yes—
436C2×C4C2×#12312yes—
437C2×C4C2×#12312yes—
438C8—84no—
439C8—84no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—24no—
449C22—24no—
450C22—22no—
451C22—22no—
452C22—44no—
453C22—44no—
454C22—48no—
455C22—42no—
456C22—42no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—44no—
462C22—44no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—82no—
467C22—82no—
468C22—84no—
469C22C2×#38111yescharacteristic
470C22#288012yes—
471C22#288312yes—
472C22C2×#41311yescharacteristic
473C22C2×#41311yescharacteristic
474C22#292211yescharacteristic
475C22#292311yescharacteristic
476C22#292512yes—
477C4—22no—
478C4—22no—
479C4—22no—
480C4—22no—
481C4—24no—
482C4—44no—
483C4—41no—
484C4—41no—
485C4—42no—
486C4—82no—
487C4—82no—
488C4—82no—
489C2—21no—
490C2—21no—
491C2—22no—
492C2—42no—
493C2—42no—
494C2—82no—
495C2#1742212yes—
496C2#1742712yes—
497C2#2068411yescharacteristic
498C2#2069011yescharacteristic
499C2#2069411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8704C2semidirect
4. #8702C2semidirect
5. #8700C2semidirect
6. #8698C2semidirect
7. #7916C2semidirect
8. #7910C2semidirect
9. #7906C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23532
#217882
#87042
#87022
#87002
#86982
#79162
#79102
#79062