Finite Group Databasebeta

← #162431

Subgroups — #162431

Order 512.

999 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35171C211yesmaximal, characteristic
2#22696C211yesmaximal, characteristic, complemented
3#22008C211yesmaximal, characteristic, complemented
4#7724C212yesmaximal, complemented
5#7714C212yesmaximal, complemented
6#7664C212yesmaximal, complemented
7#7654C212yesmaximal, complemented
8#7587C211yesmaximal, characteristic, complemented
9#7574C211yesmaximal, characteristic, complemented
10#7569C211yesmaximal, characteristic, complemented
11#7560C211yesmaximal, characteristic, complemented
12C2×#540—21no—
13C2×C8×D4—21no—
14#2856C2212yes—
15#2854C2212yes—
16#2661—22no—
17#2552C411yescharacteristic
18#2541C411yescharacteristic
19#2541C411yescharacteristic
20#2537C411yescharacteristic
21#2490—21no—
22#2464—22no—
23#2428—21no—
24#2428C412yes—
25#2422C412yes—
26#2324—21no—
27C4×#127C2211yescharacteristic
28#2293C2211yescharacteristic
29C42×D4C2211yescharacteristic
30#1954—21no—
31#1939—21no—
32#1939—21no—
33#1923—22no—
34#1850—22no—
35#1831C2211yescharacteristic
36#1817C2212yes—
37#1797C2212yes—
38#1797C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1744C2211yescharacteristic
42C2×#336C2212yes—
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2211yescharacteristic
46#1270C2212yes—
47#1270C2212yes—
48#1270C2211yescharacteristic
49#1270C2212yes—
50#1270C2212yes—
51#1270C2211yescharacteristic
52#1270C2212yes—
53C23×C8—22no—
54#540—44no—
55#540—44no—
56C2×#127—21no—
57C2×#127C2×C411yescharacteristic
58C2×#127C2×C412yes—
59C2×#125—21no—
60C2×#123—22no—
61C2×#123—22no—
62C2×#121—21no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121—22no—
68C2×#121C2×C412yes—
69C2×#121C2×C412yes—
70C2×#121C2×C412yes—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C412yes—
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4D411yescharacteristic
77C2×C4×D4D411yescharacteristic
78C2×#117—21no—
79C22×C42C2312yes—
80C8×D4—44no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92#398—22no—
93#397—22no—
94#394—22no—
95#394—22no—
96#394—22no—
97#392—22no—
98#392—22no—
99#392—22no—
100#390—21no—
101#390—21no—
102#390—21no—
103#390—22no—
104#390—22no—
105#390C2×C412yes—
106#390C2×C411yescharacteristic
107#390C2312yes—
108#390C2311yescharacteristic
109#389—21no—
110#389C2×C412yes—
111#389C2×C411yescharacteristic
112#389C2311yescharacteristic
113#388—22no—
114#388—22no—
115#388—21no—
116#388—21no—
117#386—22no—
118#386—22no—
119#385—22no—
120#384C2×C411yescharacteristic
121#384C2×C412yes—
122#380—21no—
123#380C2×C412yes—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C22⋊C4—21no—
129C4×C22⋊C4C2×C412yes—
130C4×C22⋊C4D412yes—
131C4×C22⋊C4D412yes—
132C43C2311yescharacteristic
133#336—22no—
134#336—21no—
135#336—22no—
136#336—21no—
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4C22⋊C411yescharacteristic
151C22×D4C22×C411yescharacteristic
152C22×D4C22×C412yes—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4C22×C412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—44no—
165C22×C8—44no—
166C22×C8—82no—
167#127—24no—
168#127—44no—
169#127—82no—
170#125—44no—
171#123—44no—
172#123—44no—
173#123—48no—
174#121—44no—
175#121—44no—
176#121—44no—
177#121—44no—
178#121—44no—
179#121—48no—
180#121—82no—
181#121—82no—
182#121—82no—
183#121—82no—
184#121—82no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—42no—
188C4×D4—42no—
189C4×D4—42no—
190C4×D4—81no—
191C4×D4—81no—
192C4×D4—82no—
193C4×D4D814yes—
194C4×D4SD1614yes—
195#117—42no—
196#117—42no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4C22⋊C412yes—
226C2×C22⋊C4C22⋊C412yes—
227C2×C22⋊C4C22⋊C412yes—
228C2×C42—21no—
229C2×C42—82no—
230C2×C42C22×C411yescharacteristic
231C2×C42C22×C412yes—
232C2×C42C22×C411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D411yescharacteristic
239C2×C42C2411yescharacteristic
240C4⋊C8—42no—
241C4⋊C8—42no—
242C4⋊C8—42no—
243C4⋊C8—82no—
244C4⋊C8—82no—
245C22⋊C8—44no—
246C22⋊C8—82no—
247C22⋊C8—82no—
248C4×C8—42no—
249C4×C8—42no—
250C4×C8—81no—
251#95—21no—
252#95—21no—
253#95—22no—
254#95—22no—
255#95—21no—
256#95C22⋊C412yes—
257#95C22⋊C412yes—
258#95C22⋊C412yes—
259C24—21no—
260C24—22no—
261C24C2×C22⋊C412yes—
262C2×D4—22no—
263C2×D4—24no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—42no—
267C2×D4—41no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—44no—
273C2×D4—44no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—44no—
278C2×D4—42no—
279C2×D4—81no—
280C2×D4—81no—
281C2×D4—82no—
282C2×D4—82no—
283C2×D4—82no—
284C2×D4—82no—
285C2×D4—82no—
286C2×D4—82no—
287C2×D4—82no—
288C2×D4—82no—
289C2×D4—82no—
290C2×D4—82no—
291C2×D4#10214yes—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×D814yes—
319C22×C4C2×SD1614yes—
320C22×C4C23×C411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C2×C8—22no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—41no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—88no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—81no—
355C4⋊C4—82no—
356C4⋊C4#10214yes—
357C22⋊C4—24no—
358C22⋊C4—42no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—41no—
362C22⋊C4—42no—
363C22⋊C4—41no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—41no—
367C22⋊C4—42no—
368C22⋊C4—41no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—44no—
372C22⋊C4—81no—
373C22⋊C4—82no—
374C22⋊C4—82no—
375C22⋊C4—82no—
376C22⋊C4—82no—
377C22⋊C4#10218yes—
378C42—21no—
379C42—21no—
380C42—22no—
381C42—41no—
382C42—42no—
383C42—42no—
384C42—41no—
385C42—42no—
386C42—41no—
387C42—42no—
388C42#13114yes—
389C42C2×D812yes—
390C42C2×SD1612yes—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—41no—
398C23—42no—
399C23—42no—
400C23—41no—
401C23—42no—
402C23—41no—
403C23—41no—
404C23—42no—
405C23—42no—
406C23—41no—
407C23—42no—
408C23—41no—
409C23—41no—
410C23—42no—
411C23C2×#10214yes—
412C23C22×C22⋊C411yescharacteristic
413D4—28no—
414D4—82no—
415D4—84no—
416D4—84no—
417D4—84no—
418D4—164no—
419D4—164no—
420D4—164no—
421D4—164no—
422D4—164no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—84no—
461C2×C4—84no—
462C2×C4—164no—
463C2×C4#40812yes—
464C2×C4#40812yes—
465C2×C4C2×#10212yes—
466C2×C4C2×#10214yes—
467C2×C4C2×#10212yes—
468C2×C4#42112yes—
469C2×C4C2×#13112yes—
470C2×C4C22×D811yescharacteristic
471C2×C4C22×SD1611yescharacteristic
472C2×C4#57611yescharacteristic
473C2×C4#57711yescharacteristic
474C8—42no—
475C8—81no—
476C8—82no—
477C8—81no—
478C8—81no—
479C8—82no—
480C22—24no—
481C22—24no—
482C22—24no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—82no—
488C22—81no—
489C22—81no—
490C22—82no—
491C22—82no—
492C22—81no—
493C22—82no—
494C22—81no—
495C22—81no—
496C22—82no—
497C22—81no—
498C22—82no—
499C22—81no—
500C22—82no—
501C22C2×#40811yescharacteristic
502C22#288311yescharacteristic
503C22#288811yescharacteristic
504C22C22×#10211yescharacteristic
505C22#289711yescharacteristic
506C22C2×#42111yescharacteristic
507C22#290111yescharacteristic
508C4—22no—
509C4—22no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—24no—
515C4—22no—
516C4—41no—
517C4—42no—
518C4—42no—
519C4—81no—
520C4—82no—
521C4—81no—
522C4—81no—
523C4#148014yes—
524C2—44no—
525C2—161no—
526C2—162no—
527C2#1738911yescharacteristic
528C2C2×#148011yescharacteristic
529C2#1748311yescharacteristic
530C2#1759911yescharacteristic
531C2#1761311yescharacteristic
532C2#1768411yescharacteristic
533C2#1772011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#35171C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22696C2semidirect
3. #22008C2semidirect
4. #7724C2semidirect
5. #7714C2semidirect
6. #7664C2semidirect
7. #7654C2semidirect
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
#351712
#226962
#220082
#77242
#77142
#76642
#76542
#75872
#75742
#75692
#75602