Finite Group Databasebeta

← #162438

Subgroups — #162438

Order 512.

905 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35198C211yesmaximal, characteristic
2#22734C211yesmaximal, characteristic, complemented
3#22018C211yesmaximal, characteristic, complemented
4#7732C212yesmaximal, complemented
5#7719C212yesmaximal, complemented
6#7672C212yesmaximal, complemented
7#7659C212yesmaximal, complemented
8#7587C211yesmaximal, characteristic, complemented
9#7574C211yesmaximal, characteristic, complemented
10#7569C211yesmaximal, characteristic, complemented
11#7560C211yesmaximal, characteristic, complemented
12C2×#542—21no—
13C2×C8×Q8—21no—
14#2860C2212yes—
15#2858C2212yes—
16#2662—22no—
17#2551C411yescharacteristic
18#2548C411yescharacteristic
19#2541C411yescharacteristic
20#2538C411yescharacteristic
21#2491—21no—
22#2461—22no—
23#2452C412yes—
24#2445C412yes—
25#2426—21no—
26#2324—21no—
27C4×#127C2211yescharacteristic
28#2293C2211yescharacteristic
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—21no—
32#1939—21no—
33#1939—22no—
34#1916—22no—
35#1839C2212yes—
36#1831C2212yes—
37#1831C2212yes—
38#1831C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2212yes—
42#1744C2211yescharacteristic
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2211yescharacteristic
46#1270C2211yescharacteristic
47#1270C2212yes—
48#1270C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51#1270C2211yescharacteristic
52#1270C2212yes—
53#542—44no—
54#542—44no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C412yes—
57C2×#127C2×C411yescharacteristic
58C2×#127C2×C412yes—
59C2×#126—22no—
60C2×#124—21no—
61C2×#124—22no—
62C2×#124—22no—
63C2×#123—22no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—22no—
66C2×#121—21no—
67C2×#121—21no—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C412yes—
75C2×#117—21no—
76C8×Q8—44no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—22no—
83C2×C4×C8—21no—
84C2×C4×C8—22no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87#396—22no—
88#396—22no—
89#395—22no—
90#394—22no—
91#394—22no—
92#394—22no—
93#393—22no—
94#392—22no—
95#390—21no—
96#390—21no—
97#390—21no—
98#390—22no—
99#390—22no—
100#390C2×C412yes—
101#390C2×C412yes—
102#390C2×C412yes—
103#390C2×C412yes—
104#390C2×C411yescharacteristic
105#390C2312yes—
106#390C2311yescharacteristic
107#389—21no—
108#389C2×C412yes—
109#389C2×C412yes—
110#389C2×C411yescharacteristic
111#389C2311yescharacteristic
112#388—22no—
113#388—22no—
114#388—22no—
115#388—21no—
116#388—21no—
117#386—22no—
118#385—22no—
119#384C2×C411yescharacteristic
120#380—21no—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D411yescharacteristic
128C4×C22⋊C4—21no—
129C43C2311yescharacteristic
130C43C2312yes—
131#336—22no—
132#336—22no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336—21no—
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143C22×Q8—22no—
144C22×Q8C22⋊C411yescharacteristic
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4C22×C411yescharacteristic
150C22×D4C22×C412yes—
151C23×C4—21no—
152C23×C4—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—22no—
159C4⋊Q8—44no—
160C4⋊Q8—82no—
161#127—24no—
162#127—82no—
163#126—44no—
164#126—44no—
165#124—44no—
166#124—44no—
167#124—44no—
168#124—82no—
169#124—82no—
170#123—48no—
171C22⋊Q8—44no—
172C22⋊Q8—48no—
173#121—44no—
174#121—44no—
175C4×Q8—24no—
176C4×Q8—42no—
177C4×Q8D814yes—
178C4×Q8SD1614yes—
179C4×D4—24no—
180C4×D4—42no—
181C4×D4—42no—
182C4×D4—81no—
183C4×D4—81no—
184C4×D4—82no—
185#117—42no—
186#117—42no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C42—21no—
221C2×C42C22×C412yes—
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C412yes—
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D412yes—
233C2×C42C2411yescharacteristic
234C4⋊C8—42no—
235C4⋊C8—42no—
236C4⋊C8—44no—
237C4⋊C8—42no—
238C4⋊C8—44no—
239C4⋊C8—82no—
240C4×C8—42no—
241C4×C8—42no—
242C4×C8—44no—
243C4×C8—81no—
244#95—21no—
245#95—21no—
246#95—22no—
247#95—22no—
248#95—21no—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251#95C22⋊C412yes—
252C24—21no—
253C24—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—44no—
257C2×Q8—82no—
258C2×Q8—82no—
259C2×Q8#10214yes—
260C2×D4—22no—
261C2×D4—24no—
262C2×D4—42no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—44no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—44no—
271C2×D4—42no—
272C2×D4—81no—
273C2×D4—81no—
274C2×D4—82no—
275C2×D4—82no—
276C2×D4—82no—
277C2×D4—82no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C23×C411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C2×C8—22no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—41no—
316C2×C8—41no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—42no—
322C4⋊C4—41no—
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—81no—
332C4⋊C4—82no—
333C4⋊C4#10214yes—
334C4⋊C4#10218yes—
335C22⋊C4—44no—
336C22⋊C4—41no—
337C22⋊C4—42no—
338C22⋊C4—41no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—41no—
342C22⋊C4—42no—
343C22⋊C4—41no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—81no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C42—24no—
356C42—21no—
357C42—22no—
358C42—21no—
359C42—22no—
360C42—22no—
361C42—41no—
362C42—42no—
363C42—41no—
364C42—42no—
365C42—42no—
366C42—41no—
367C42—42no—
368C42#13114yes—
369C42C2×D812yes—
370C42C2×D814yes—
371C42C2×SD1612yes—
372C42C2×SD1614yes—
373C23—41no—
374C23—42no—
375C23—41no—
376C23—42no—
377C23—42no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23—41no—
382C23—42no—
383C23—42no—
384C23—41no—
385C23—41no—
386C23—42no—
387C23C22×C22⋊C411yescharacteristic
388Q8—28no—
389Q8—164no—
390D4—82no—
391D4—84no—
392D4—84no—
393D4—84no—
394D4—164no—
395D4—164no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—84no—
432C2×C4—164no—
433C2×C4#40812yes—
434C2×C4#40812yes—
435C2×C4C2×#10214yes—
436C2×C4C2×#10212yes—
437C2×C4C2×#10214yes—
438C2×C4C2×#10212yes—
439C2×C4#42112yes—
440C2×C4C2×#13112yes—
441C2×C4C22×D811yescharacteristic
442C2×C4C22×SD1611yescharacteristic
443C2×C4#57611yescharacteristic
444C2×C4#57711yescharacteristic
445C8—42no—
446C8—81no—
447C8—82no—
448C8—81no—
449C8—81no—
450C8—82no—
451C22—81no—
452C22—82no—
453C22—81no—
454C22—82no—
455C22—82no—
456C22—81no—
457C22—82no—
458C22—81no—
459C22—81no—
460C22—82no—
461C22—81no—
462C22—82no—
463C22—81no—
464C22—82no—
465C22C2×#40811yescharacteristic
466C22#288411yescharacteristic
467C22#288811yescharacteristic
468C22C22×#10211yescharacteristic
469C22#289711yescharacteristic
470C22C2×#42111yescharacteristic
471C22#290111yescharacteristic
472C4—24no—
473C4—22no—
474C4—24no—
475C4—22no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—22no—
481C4—41no—
482C4—42no—
483C4—42no—
484C4—81no—
485C4—82no—
486C4—81no—
487C4—81no—
488C4#148014yes—
489C2—161no—
490C2—162no—
491C2#1739011yescharacteristic
492C2C2×#148011yescharacteristic
493C2#1748311yescharacteristic
494C2#1760211yescharacteristic
495C2#1761811yescharacteristic
496C2#1768111yescharacteristic
497C2#1772011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2541C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22734C2semidirect
3. #22018C2semidirect
4. #7732C2semidirect
5. #7719C2semidirect
6. #7672C2semidirect
7. #7659C2semidirect
8. #7587C2semidirect
9. #7574C2semidirect
10. #7569C2semidirect
11. #7560C2semidirect

Maximal subgroups

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

subgroupindex
#351982
#227342
#220182
#77322
#77192
#76722
#76592
#75872
#75742
#75692
#75602