Finite Group Databasebeta

← #215929

Subgroups — #215929

Order 512.

798 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic
2#21916C211yesmaximal, characteristic
3#13949C211yesmaximal, characteristic
4#9913C212yesmaximal
5#9910C212yesmaximal
6#9887C212yesmaximal
7#9887C212yesmaximal
8#8717C211yesmaximal, characteristic
9#8717C211yesmaximal, characteristic
10#8716C211yesmaximal, characteristic
11#8716C211yesmaximal, characteristic
12#2851C2212yes—
13#2851C2212yes—
14#2646—21no—
15#2646—21no—
16#2571C412yescomplemented
17#2571C412yescomplemented
18#2570—22no—
19#2570—22no—
20#2552C412yescomplemented
21#2552C412yescomplemented
22#2317—21no—
23#2317—21no—
24C2×#387—21no—
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1917—22no—
29#1917—22no—
30#1817C2211yescharacteristic
31#1817C2211yescharacteristic
32#1797C2212yes—
33#1749C2212yes—
34#1442—21no—
35#1441—21no—
36#1436—21no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39#1434C2211yescharacteristic
40#1434C2211yescharacteristic
41#1434C2211yescharacteristic
42C4×#95C2211yescharacteristic
43#1294C2212yes—
44#1294C2212yes—
45#1294C2212yes—
46#1294C2212yes—
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51C2×#125C2×C412yes—
52C2×#125C2×C412yes—
53C2×#117—21no—
54C2×#117—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×C4⋊C8—22no—
64C2×C22⋊C8C2312yes—
65C2×C22⋊C8C2312yes—
66C2×#97—22no—
67#400—22no—
68#400—22no—
69#400—22no—
70#400—22no—
71#398—22no—
72#398—22no—
73#398—22no—
74#398—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389C2×C412yes—
80#389C2×C412yes—
81#389C2×C412yes—
82#389C2×C412yes—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—44no—
88#385—22no—
89#385—22no—
90#385—22no—
91#385—22no—
92#384—22no—
93#384—22no—
94#384C2×C411yescharacteristic
95#384C2×C411yescharacteristic
96#384C2×C411yescharacteristic
97#384C2×C411yescharacteristic
98#382—22no—
99#382—22no—
100#382—21no—
101#382—21no—
102#382C2×C412yes—
103#382C2×C412yes—
104#380—21no—
105#380—21no—
106#380C2312yes—
107#380C2312yes—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4D412yes—
115C4×C4⋊C4D412yes—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95D411yescharacteristic
126C2×#95D411yescharacteristic
127C2×#95C2311yescharacteristic
128C2×#95C2311yescharacteristic
129C2×#95C2311yescharacteristic
130C2×#95C2311yescharacteristic
131#336—22no—
132#336—21no—
133#336—21no—
134#336—21no—
135#336—21no—
136#336—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—22no—
150#125—82no—
151#125—82no—
152#117—42no—
153#117—42no—
154#117—42no—
155#117—42no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—81no—
181C2×C4⋊C4—81no—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42—81no—
199C2×C42—81no—
200C2×C42—81no—
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C412yes—
204C2×C42C22×C412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C4⋊C8—82no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214#97—82no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—42no—
232#95—42no—
233#95—42no—
234#95—42no—
235#95—42no—
236#95—42no—
237#95—42no—
238#95—42no—
239#95—81no—
240#95C22⋊C412yes—
241#95C22⋊C412yes—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244C24C22×D411yescharacteristic
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4AGL(1,ℤ/8)12yes—
312C22×C4#13712yes—
313C22×C4C23×C411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C42−1+412yes—
316C22×C42−1+412yes—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—81no—
336C4⋊C4—81no—
337C4⋊C4—81no—
338C4⋊C4—81no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4—84no—
348C4⋊C4—84no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—44no—
362C42—44no—
363C42—84no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23C22×C22⋊C411yescharacteristic
369C23#53611yescharacteristic
370C23#53611yescharacteristic
371C23C2×AGL(1,ℤ/8)11yescharacteristic
372C23C2×#13711yescharacteristic
373C23#57511yescharacteristic
374C23#57511yescharacteristic
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
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—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4#41712yes—
458C2×C4#41712yes—
459C2×C4#51912yes—
460C2×C4#53612yes—
461C8—82no—
462C8—82no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22#231011yescharacteristic
476C22#288311yescharacteristic
477C22#288411yescharacteristic
478C22C2×#41711yescharacteristic
479C22#289611yescharacteristic
480C22#290111yescharacteristic
481C22#290111yescharacteristic
482C22#323411yescharacteristic
483C22#323411yescharacteristic
484C22#323611yescharacteristic
485C22#323611yescharacteristic
486C4—22no—
487C4—22no—
488C4—22no—
489C4—22no—
490C4—41no—
491C4—41no—
492C4—42no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—82no—
497C4—82no—
498C4—82no—
499C4—82no—
500C4—82no—
501C4—82no—
502C4—161no—
503C4—161no—
504C2—21no—
505C2—21no—
506C2—21no—
507C2—21no—
508C2#1762411yescharacteristic
509C2#1762811yescharacteristic
510C2#1762911yescharacteristic
511C2#1763011yescharacteristic
512C2#2182811yescharacteristic
513C2#2183811yescharacteristic
514C2#2184611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ 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
16. #2571C4semidirect
17. #2571C4semidirect
20. #2552C4semidirect
21. #2552C4semidirect

Maximal subgroups

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

subgroupindex
#351812
#219162
#139492
#99132
#99102
#98872
#87172
#87162