Finite Group Databasebeta

← #195574

Subgroups — #195574

Order 512.

1238 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#34145C211yesmaximal, characteristic
3#21313C211yesmaximal, characteristic
4C2×#1987C211yesmaximal, characteristic
5C2×#1550C211yesmaximal, characteristic
6#9066C214yesmaximal, complemented
7#9065C214yesmaximal, complemented
8#7901C211yesmaximal, characteristic
9#7891C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3209—24no—
13#3209C418yescomplemented
14C2×#494C2212yes—
15C2×#487C2212yes—
16C2×#462—22no—
17C2×#451—22no—
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2370—22no—
22#2369C2212yes—
23#2368—22no—
24#2365C2212yes—
25#2360—22no—
26#2296—21no—
27#2295—21no—
28#2295—22no—
29C42×Q8—21no—
30C4×#117—21no—
31C4×#117C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1988—22no—
34#1987—21no—
35#1987—21no—
36#1987C2214yes—
37#1987C2214yes—
38#1736—21no—
39#1736—21no—
40C2×#340—22no—
41#1550—24no—
42C2×#333C2212yes—
43C2×#332C2212yes—
44#1279C2214yes—
45#1279C2214yes—
46C23×Q8—21no—
47C22×Q16—24no—
48C22×M16—21no—
49C23×C8—21no—
50#533—22no—
51#533—22no—
52#533C2×C414yes—
53#533D414yes—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8C2×C414yes—
61C2×C4⋊Q8C2312yes—
62C2×C4⋊Q8C2311yescharacteristic
63C2×C4⋊Q8C2311yescharacteristic
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8D412yes—
67C2×C4×Q8—22no—
68C2×C4×Q8—21no—
69C2×C4×Q8D412yes—
70C2×#117—21no—
71C2×#117D411yescharacteristic
72C2×#117D411yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D412yes—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78#494—24no—
79#494C2×C418yescomplemented
80#487—24no—
81#487C2×C418yescomplemented
82#462—216no—
83#451—216no—
84#431—22no—
85#431C2×C414yes—
86#423—22no—
87#423—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2312yes—
94C2×C4⋊C8C2312yes—
95C2×#103—24no—
96C2×C22⋊C8—21no—
97C2×C4×C8C2312yes—
98#391—24no—
99#391—22no—
100#389—24no—
101#389—22no—
102#389—22no—
103#389—22no—
104#387—24no—
105#387—24no—
106#387—24no—
107#385—24no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4D412yes—
117C4×C4⋊C4D411yescharacteristic
118C4×C4⋊C4D411yescharacteristic
119C4×C4⋊C4C2312yes—
120C4×C4⋊C4C2312yes—
121C4×C22⋊C4—24no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124#376—22no—
125#376—21no—
126C2×#95—21no—
127C43—22no—
128C43—21no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—24no—
133#333—24no—
134#332—24no—
135C22×Q8—21no—
136C22×Q8—24no—
137C22×Q8—22no—
138C22×Q8—41no—
139C22×Q8—42no—
140C22×Q8—84no—
141C22×Q8C2×D412yes—
142C23×C4—22no—
143C23×C4—41no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×Q16—216no—
148C2×Q16—88no—
149C2×M16—42no—
150C2×M16—42no—
151C2×M16—42no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—44no—
156C22×C8—42no—
157C4⋊Q8—24no—
158C4⋊Q8—24no—
159C4⋊Q8—24no—
160C4⋊Q8—22no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169C4⋊Q8—42no—
170C4⋊Q8C22×C414yes—
171C4⋊Q8C22×C412yes—
172C4⋊Q8C22×C412yes—
173C4⋊Q8C2×D412yes—
174C4⋊Q8C2×D412yes—
175#125—24no—
176#125—24no—
177C22⋊Q8—24no—
178C22⋊Q8—24no—
179C22⋊Q8—24no—
180C22⋊Q8C22⋊C418yes—
181C4×Q8—24no—
182C4×Q8—24no—
183C4×Q8—24no—
184C4×Q8—44no—
185C4×Q8—44no—
186C4×Q8C22⋊C418yes—
187#117—24no—
188#117—24no—
189#117—22no—
190#117—22no—
191#117—22no—
192#117Q1614yescomplemented
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—24no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42C22×C412yes—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D412yes—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42#4112yes—
237C2×C42#4112yes—
238C2×C42C2411yescharacteristic
239C4⋊C8—24no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246#103—216no—
247C22⋊C8—24no—
248C4×C8—22no—
249C4×C8C22×C414yes—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—21no—
255#95—21no—
256#95—22no—
257#95—22no—
258#95—22no—
259C24#12011yescharacteristic
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—24no—
264C2×Q8—44no—
265C2×Q8—48no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—44no—
270C2×Q8—84no—
271C2×Q8—88no—
272C2×Q8—88no—
273C2×Q8C2×C22⋊C414yes—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—82no—
288C22×C4—84no—
289C22×C4C2×C22⋊C412yes—
290C22×C4C4×D412yes—
291C22×C4C4×D412yes—
292C22×C4#12011yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4#12011yescharacteristic
295C22×C4#12112yes—
296C22×C4#12112yes—
297C22×C4#12312yes—
298C22×C4C2×Q1612yes—
299C22×C4#13512yes—
300C22×C4#13712yes—
301C22×C4#13712yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305Q16—816no—
306M16—84no—
307C2×C8—24no—
308C2×C8—44no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—44no—
314C2×C8—84no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—28no—
325C4⋊C4—28no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—44no—
333C4⋊C4—48no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—84no—
339C4⋊C4C2×C22⋊C414yes—
340C4⋊C4C2×Q1614yes—
341C4⋊C4#13714yes—
342C22⋊C4—28no—
343C22⋊C4—24no—
344C22⋊C4—42no—
345C42—28no—
346C42—24no—
347C42—48no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—42no—
355C42—82no—
356C42—84no—
357C42C2×C22⋊C412yes—
358C42C4×D412yes—
359C42C4×D412yes—
360C42#12112yes—
361C42#12112yes—
362C42#12312yes—
363C42C23×C411yescharacteristic
364C42C22×D411yescharacteristic
365C42C2×#4111yescharacteristic
366C42C2×#4111yescharacteristic
367C23—22no—
368C23—22no—
369C23#38612yes—
370C23#45112yes—
371C23#45212yes—
372C23C2×#12011yescharacteristic
373Q8—44no—
374Q8—48no—
375Q8—44no—
376Q8—42no—
377Q8—42no—
378Q8—82no—
379Q8—84no—
380Q8—88no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—88no—
407C2×C4—88no—
408C2×C4—84no—
409C2×C4#38612yes—
410C2×C4#38612yes—
411C2×C4#38612yes—
412C2×C4C4×Q1614yes—
413C2×C4#44114yes—
414C2×C4#45112yes—
415C2×C4#45212yes—
416C2×C4#46214yes—
417C2×C4#46414yes—
418C2×C4C22×C22⋊C411yescharacteristic
419C2×C4C2×C4×D411yescharacteristic
420C2×C4C2×C4×D411yescharacteristic
421C2×C4C2×#12111yescharacteristic
422C2×C4C2×#12111yescharacteristic
423C2×C4C2×#12311yescharacteristic
424C2×C4C22×Q1611yescharacteristic
425C2×C4C2×#13511yescharacteristic
426C2×C4C2×#13711yescharacteristic
427C2×C4C2×#13711yescharacteristic
428C2×C4C2×#13712yes—
429C2×C4#57512yes—
430C8—44no—
431C8—82no—
432C8—82no—
433C22—24no—
434C22—24no—
435C22—22no—
436C22—22no—
437C22#187814yes—
438C22C2×#38611yescharacteristic
439C22C2×#45111yescharacteristic
440C22C2×#45211yescharacteristic
441C22#300712yes—
442C22#302112yes—
443C4—22no—
444C4—21no—
445C4—21no—
446C4—22no—
447C4—44no—
448C4—44no—
449C4—44no—
450C4—44no—
451C4—82no—
452C4—82no—
453C4—84no—
454C4#187814yes—
455C4C2×C4×Q1611yescharacteristic
456C4C2×#44111yescharacteristic
457C4C4×#13712yes—
458C4C2×#46211yescharacteristic
459C4C2×#46411yescharacteristic
460C4#303911yescharacteristic
461C4#304011yescharacteristic
462C2—24no—
463C2#1878912yes—
464C2#1879312yes—
465C2C2×#187811yescharacteristic
466C2#2092711yescharacteristic
467C2#2095611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9066C2semidirect
7. #9065C2semidirect
13. #3209C4semidirect
79. #494C2×C4semidirect
81. #487C2×C4semidirect
192. #117Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#341452
#213132
C2×#19872
C2×#15502
#90662
#90652
#79012
#78912