Finite Group Databasebeta

← #200543

Subgroups — #200543

Order 512.

1125 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60250C211yesmaximal, characteristic
2#35281C211yesmaximal, characteristic
3#21879C211yesmaximal, characteristic
4#17742C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic
6#9402C214yesmaximal, complemented
7#9394C214yesmaximal, complemented
8#8097C211yesmaximal, characteristic
9#8092C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#2986—22no—
13#2985—22no—
14#2984—22no—
15#2983C2212yes—
16#2982C2212yes—
17#2962—22no—
18#2960—22no—
19#2924—22no—
20#2920C2211yescharacteristic
21#2920C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24#2868—21no—
25#2862—21no—
26#2660C414yes—
27#2567—22no—
28#2546C2214yes—
29#2502C414yes—
30#2434—22no—
31#2317—22no—
32C4×#117C2211yescharacteristic
33#1997—21no—
34#1966—21no—
35#1901—22no—
36#1801C2214yes—
37#1556—22no—
38#1549—22no—
39#1546—22no—
40#1546C2212yes—
41#1546C2212yes—
42#1546C2214yes—
43C2×#332—22no—
44#1303C2212yes—
45#1303C2212yes—
46#1303C2214yes—
47C23×Q8—21no—
48C22×M16—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8C2×C412yes—
69C2×C4⋊Q8D411yescharacteristic
70C2×C4⋊Q8D411yescharacteristic
71C2×C4⋊Q8C2312yes—
72C2×#125C2×C412yes—
73C2×C22⋊Q8C2×C412yes—
74C2×C22⋊Q8D412yes—
75C2×C4×Q8D412yes—
76C2×C4×Q8C2312yes—
77C2×#117—21no—
78C2×#117D412yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2×C412yes—
81C22×C42C2311yescharacteristic
82#446—24no—
83#446—28no—
84#446—24no—
85#446—24no—
86C8×Q8—24no—
87C8×Q8—24no—
88C8×Q8—24no—
89#433—28no—
90#432—21no—
91#432—24no—
92#432—21no—
93#432—21no—
94#432D412yes—
95#431—21no—
96#431—21no—
97#431—21no—
98#431—24no—
99#431—21no—
100#431—21no—
101#431D412yes—
102#431C2311yescharacteristic
103#431C2311yescharacteristic
104#431C2312yes—
105#424—24no—
106#423—21no—
107#423—21no—
108#423—21no—
109#423D412yes—
110#423C2312yes—
111#423C2312yes—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8D412yes—
115C2×#101—24no—
116#407—21no—
117#407—21no—
118C2×C22⋊C8—21no—
119#405—24no—
120C4×M16—21no—
121C4×M16—21no—
122C2×C4×C8—21no—
123#398C2×C4116yes—
124#395—28no—
125#391—24no—
126#391—24no—
127#389—28no—
128#385—28no—
129#384—24no—
130#384—24no—
131#382—24no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2312yes—
134C4×C22⋊C4—21no—
135C4×C22⋊C4C2×C412yes—
136#376C2×C412yes—
137C43—21no—
138#343—24no—
139#332—24no—
140#332—24no—
141#332—28no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—41no—
145C22×Q8—41no—
146C22×Q8—41no—
147C22×Q8—41no—
148C22×Q8—82no—
149C22×Q8—82no—
150C22×Q8C22⋊C412yes—
151C22×Q8C2×D412yes—
152C23×C4—22no—
153C23×C4C22×C411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—41no—
160C2×M16—82no—
161C22×C8—22no—
162C22×C8—41no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C4⋊Q8—41no—
182C4⋊Q8—41no—
183C4⋊Q8—41no—
184C4⋊Q8—41no—
185C4⋊Q8—41no—
186C4⋊Q8—41no—
187C4⋊Q8—84no—
188C4⋊Q8—84no—
189#125—22no—
190#125—22no—
191#125—22no—
192#125—22no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—22no—
209#117—28no—
210#117—22no—
211#117—22no—
212#117—22no—
213#117—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—216no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22⋊C412yes—
230C2×C4⋊C4C22⋊C414yes—
231C2×C4⋊C4C22×C414yes—
232C2×C4⋊C4C22×C414yes—
233C2×C4⋊C4C2×D412yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4C22⋊C414yes—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D414yes—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42#4111yescharacteristic
248C2×C42#4111yescharacteristic
249C2×C42#4111yescharacteristic
250C2×C42#4111yescharacteristic
251C2×C42C2411yescharacteristic
252C4⋊C8—28no—
253C4⋊C8—28no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—41no—
261C4⋊C8—41no—
262C4⋊C8C2×D412yes—
263C4⋊C8C2×D412yes—
264C4⋊C8C2×D414yes—
265#101—84no—
266C22⋊C8—22no—
267C22⋊C8—22no—
268C22⋊C8—21no—
269C22⋊C8—21no—
270C22⋊C8—22no—
271C22⋊C8—41no—
272C22⋊C8—41no—
273#97—28no—
274#97—22no—
275#97—41no—
276C4×C8—28no—
277C4×C8—22no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—22no—
281C4×C8—41no—
282#95—22no—
283#95—216no—
284#95C22×C414yes—
285C24C2×C22⋊C411yescharacteristic
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—44no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—44no—
296C2×Q8—44no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—84no—
300C2×Q8—82no—
301C2×Q8—82no—
302C2×Q8—82no—
303C2×Q8—82no—
304C2×Q8—84no—
305C2×Q8—84no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—84no—
315C22×C4C2×C22⋊C414yes—
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C414yes—
320C22×C4C4×D411yescharacteristic
321C22×C4C4×D411yescharacteristic
322C22×C4C4×D411yescharacteristic
323C22×C4C4×D411yescharacteristic
324C22×C4#12011yescharacteristic
325C22×C4#12011yescharacteristic
326C22×C4#12111yescharacteristic
327C22×C4#12111yescharacteristic
328C22×C4#12111yescharacteristic
329C22×C4#12111yescharacteristic
330C22×C4#12311yescharacteristic
331C22×C4#12311yescharacteristic
332C22×C4C23×C411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335M16—44no—
336M16—42no—
337M16—44no—
338M16—42no—
339M16—82no—
340C2×C8—22no—
341C2×C8—24no—
342C2×C8—22no—
343C2×C8—44no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—82no—
347C4⋊C4—28no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—816no—
369C22⋊C4—28no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C42—28no—
373C42—24no—
374C42—24no—
375C42—44no—
376C42—42no—
377C42—42no—
378C42C4×D411yescharacteristic
379C42C4×D411yescharacteristic
380C42C4×D411yescharacteristic
381C42C4×D411yescharacteristic
382C42#12011yescharacteristic
383C42#12011yescharacteristic
384C42#12111yescharacteristic
385C42#12111yescharacteristic
386C42#12111yescharacteristic
387C42#12111yescharacteristic
388C42#12311yescharacteristic
389C42#12311yescharacteristic
390C42C22×D411yescharacteristic
391C42C22×D411yescharacteristic
392C42C2×#4111yescharacteristic
393C42C2×#4111yescharacteristic
394C23—22no—
395C23—22no—
396C23—41no—
397C23#38611yescharacteristic
398C23#38611yescharacteristic
399C23C22×C22⋊C411yescharacteristic
400Q8—44no—
401Q8—42no—
402Q8—42no—
403Q8—44no—
404Q8—82no—
405Q8—84no—
406Q8—84no—
407Q8—82no—
408Q8—82no—
409Q8—84no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—28no—
426C2×C4—28no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—48no—
436C2×C4—44no—
437C2×C4—84no—
438C2×C4—84no—
439C2×C4#38611yescharacteristic
440C2×C4#38611yescharacteristic
441C2×C4#38611yescharacteristic
442C2×C4#38611yescharacteristic
443C2×C4#38611yescharacteristic
444C2×C4#38611yescharacteristic
445C2×C4#45614yes—
446C2×C4C2×C4×D411yescharacteristic
447C2×C4C2×C4×D411yescharacteristic
448C2×C4C2×#12011yescharacteristic
449C2×C4C2×#12111yescharacteristic
450C2×C4C2×#12111yescharacteristic
451C2×C4C2×#12311yescharacteristic
452C2×C4#57612yes—
453C2×C4#57712yes—
454C8—42no—
455C8—42no—
456C8—44no—
457C8—82no—
458C8—82no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—44no—
467C22#188612yes—
468C22C2×#38611yescharacteristic
469C22#289712yes—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—48no—
478C4—44no—
479C4—44no—
480C4—84no—
481C4#188612yes—
482C4#191512yes—
483C4C2×#45612yes—
484C4#305711yescharacteristic
485C4#305811yescharacteristic
486C4#306212yes—
487C2—22no—
488C2—22no—
489C2—41no—
490C2C2×#188611yescharacteristic
491C2#2097511yescharacteristic
492C2C2×#191511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9402C2semidirect
7. #9394C2semidirect

Maximal subgroups

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

subgroupindex
#602502
#352812
#218792
#177422
C2×#15462
#94022
#93942
#80972
#80922