Finite Group Databasebeta

← #192752

Subgroups — #192752

Order 512.

1230 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic
3#21170C211yesmaximal, characteristic
4#20734C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic
6#8892C214yesmaximal, complemented
7#8888C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C22×C4×Q8—21no—
11C22×C4×Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#428—21no—
15C2×#423C2211yescharacteristic
16Q8×C4⋊C4—22no—
17Q8×C4⋊C4—22no—
18Q8×C4⋊C4—22no—
19Q8×C4⋊C4C2214yes—
20C2×#389—22no—
21C2×#389—21no—
22#2285C2211yescharacteristic
23C2×#384—21no—
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27#1949—22no—
28#1948—22no—
29#1948C2212yes—
30#1947C2212yes—
31#1853—22no—
32#1844C2212yes—
33#1834C2212yes—
34#1824—21no—
35#1803C2212yes—
36#1803C2212yes—
37#1787—21no—
38C4×#103—22no—
39#1736—21no—
40#1487C418yescomplemented
41C2×Q8⋊C8C2214yes—
42#1296C2214yes—
43#1279C2214yes—
44C23×Q8D411yescharacteristic
45C22×M16—21no—
46C2×C4×Q8—24no—
47C2×C4×Q8—24no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—48no—
53C2×C4×Q8C2×C414yes—
54C2×C4×Q8D412yes—
55C2×C4×Q8Q814yes—
56C2×C4×Q8C2312yes—
57C2×#117—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D412yes—
67C22×C42—21no—
68C22×C42—22no—
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71#431—22no—
72#431—22no—
73#431—22no—
74#431C2×C414yes—
75#428—44no—
76C2×#107—22no—
77C2×#106—22no—
78C2×#106—24no—
79#423—22no—
80#423C2×C414yes—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2312yes—
85C2×#103—22no—
86C2×#103—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8C2312yes—
91#389—24no—
92#389—28no—
93#389—24no—
94#389—28no—
95#384—24no—
96#384—24no—
97#384—21no—
98#384—21no—
99#384D412yes—
100#384C2311yescharacteristic
101#384C2311yescharacteristic
102#384C2312yes—
103#382—21no—
104#380—21no—
105#380—21no—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4C2312yes—
114C2×#95—22no—
115C2×#95—21no—
116#340—22no—
117#340—22no—
118#340—24no—
119#340—22no—
120#340—22no—
121Q8⋊C8C2×C4116yescomplemented
122C22×Q8—24no—
123C22×Q8—24no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8C4⋊C414yes—
127C22×Q8C2×D412yes—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C2×M16—24no—
137C2×M16—42no—
138C2×M16—42no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C4×Q8—24no—
143C4×Q8—24no—
144C4×Q8—24no—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—416no—
150C4×Q8—48no—
151C4×Q8—48no—
152C4×Q8C4⋊C418yes—
153C4×Q8C22×C414yes—
154C4×Q8C2×Q814yes—
155#117—42no—
156#117—42no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—84no—
187C2×C4⋊C4C4⋊C414yes—
188C2×C4⋊C4SD1614yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D414yes—
193C2×C22⋊C4—41no—
194C2×C42—24no—
195C2×C42—22no—
196C2×C42—24no—
197C2×C42—22no—
198C2×C42—28no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—48no—
202C2×C42—44no—
203C2×C42—41no—
204C2×C42C22×C412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×Q812yes—
209C2×C42#4112yes—
210C2×C42#4112yes—
211C2×C42C2411yescharacteristic
212#107—44no—
213#106—44no—
214#106—84no—
215C4⋊C8—24no—
216C4⋊C8—24no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8C22×C414yes—
221#103—44no—
222#103—44no—
223C22⋊C8—24no—
224C22⋊C8—42no—
225C4×C8—22no—
226C4×C8—42no—
227C4×C8C22×C414yes—
228#95—28no—
229#95—22no—
230#95—22no—
231#95—41no—
232#95—41no—
233C24#12011yescharacteristic
234C2×Q8—24no—
235C2×Q8—216no—
236C2×Q8—24no—
237C2×Q8—216no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—24no—
242C2×Q8C2×C4⋊C414yes—
243C2×Q8AGL(1,ℤ/8)14yes—
244C2×Q8#13714yes—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4C2×C4⋊C412yes—
272C22×C4C4×D412yes—
273C22×C4C4×D412yes—
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4#12011yescharacteristic
277C22×C4C22⋊Q812yes—
278C22×C4C22⋊Q812yes—
279C22×C4#12312yes—
280C22×C4C2×SD1612yes—
281C22×C4#13512yes—
282C22×C4#13712yes—
283C22×C4#13712yes—
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287M16—48no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—28no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—44no—
307C4⋊C4—416no—
308C4⋊C4—44no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—82no—
323C4⋊C4—88no—
324C4⋊C4C2×C4⋊C414yes—
325C4⋊C4C2×SD1614yes—
326C22⋊C4—82no—
327C42—24no—
328C42—24no—
329C42—28no—
330C42—416no—
331C42—44no—
332C42—44no—
333C42—82no—
334C42C2×C4⋊C412yes—
335C42C4×D412yes—
336C42C4×D412yes—
337C42C22⋊Q812yes—
338C42C22⋊Q812yes—
339C42#12312yes—
340C42C23×C411yescharacteristic
341C42C22×Q811yescharacteristic
342C42C2×#4111yescharacteristic
343C42C2×#4111yescharacteristic
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23#38512yes—
349C23#44812yes—
350C23#45212yes—
351C23C2×#12011yescharacteristic
352Q8—216no—
353Q8—24no—
354Q8—24no—
355Q8—24no—
356Q8#42818yescomplemented
357C2×C4—216no—
358C2×C4—24no—
359C2×C4—21no—
360C2×C4—24no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—84no—
386C2×C4—88no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4#38512yes—
394C2×C4#38512yes—
395C2×C4#38512yes—
396C2×C4C4×SD1614yes—
397C2×C4#44114yes—
398C2×C4#44812yes—
399C2×C4#45212yes—
400C2×C4#47514yes—
401C2×C4#47914yes—
402C2×C4C22×C4⋊C411yescharacteristic
403C2×C4C2×C4×D411yescharacteristic
404C2×C4C2×C4×D411yescharacteristic
405C2×C4C2×C22⋊Q811yescharacteristic
406C2×C4C2×C22⋊Q811yescharacteristic
407C2×C4C2×#12311yescharacteristic
408C2×C4C22×SD1611yescharacteristic
409C2×C4C2×#13511yescharacteristic
410C2×C4C2×AGL(1,ℤ/8)12yes—
411C2×C4C2×#13711yescharacteristic
412C2×C4C2×#13711yescharacteristic
413C2×C4C2×#13712yes—
414C8—44no—
415C8—44no—
416C8—82no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22#186514yes—
424C22C2×#38511yescharacteristic
425C22C2×#44811yescharacteristic
426C22C2×#45211yescharacteristic
427C22#300311yescharacteristic
428C22#300411yescharacteristic
429C22#301911yescharacteristic
430C22#302011yescharacteristic
431C4—28no—
432C4—22no—
433C4—21no—
434C4—21no—
435C4—22no—
436C4—44no—
437C4—44no—
438C4—82no—
439C4—84no—
440C4—161no—
441C4—161no—
442C4#186514yes—
443C4C2×#42812yes—
444C4C2×C4×SD1611yescharacteristic
445C4C2×#44111yescharacteristic
446C4C2×#47511yescharacteristic
447C4C2×#47911yescharacteristic
448C4#309212yes—
449C2—24no—
450C2#1875811yescharacteristic
451C2#1876111yescharacteristic
452C2#1876811yescharacteristic
453C2#1877111yescharacteristic
454C2#2084411yescharacteristic
455C2#2086411yescharacteristic
456C2C2×#186511yescharacteristic
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. #8892C2semidirect
7. #8888C2semidirect
40. #1487C4semidirect
121. Q8⋊C8C2×C4semidirect
356. Q8#428semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C4⋊C42
#219812
#211702
#207342
C2×#14872
#88922
#88882
#80432
#78922