Finite Group Databasebeta

← #192624

Subgroups — #192624

Order 512.

1254 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic
2#21365C211yesmaximal, characteristic
3#20800C211yesmaximal, characteristic
4#20740C211yesmaximal, characteristic
5C2×#1491C211yesmaximal, characteristic
6#8888C214yesmaximal, complemented
7#8884C214yesmaximal, complemented
8#8040C211yesmaximal, characteristic
9#7904C211yesmaximal, characteristic
10C22×C4×Q8—21no—
11C22×C4×Q8C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#428—22no—
15C2×C4×M16C2211yescharacteristic
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#1948C2212yes—
28#1947C2212yes—
29#1853—21no—
30#1852—21no—
31#1847—22no—
32#1845C2212yes—
33#1844C2212yes—
34#1823—21no—
35#1803C2212yes—
36#1803C2212yes—
37#1787—21no—
38#1776—22no—
39#1763—22no—
40#1491C418yescomplemented
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#432—22no—
72#432—22no—
73#432—22no—
74#432C2×C414yes—
75#428—48no—
76C2×#107—24no—
77C2×#107—22no—
78C2×#106—24no—
79C2×#106—22no—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×#103—22no—
83C2×#103—22no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C4×M16—22no—
87C4×M16C2×C414yes—
88C2×#97—21no—
89C2×#97—21no—
90C2×#97—21no—
91C2×C4×C8C2312yes—
92#389—24no—
93#389—28no—
94#389—24no—
95#389—28no—
96#384—24no—
97#384—24no—
98#384—21no—
99#384—21no—
100#384D412yes—
101#384C2311yescharacteristic
102#384C2311yescharacteristic
103#384C2312yes—
104#382—21no—
105#380—21no—
106#380—21no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4C2312yes—
115C2×#95—22no—
116C2×#95—21no—
117#340—22no—
118#340—22no—
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—22no—
132C23×C4—21no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C2×M16—24no—
137C2×M16—24no—
138C2×M16—24no—
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—44no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—84no—
184C2×C4⋊C4C4⋊C414yes—
185C2×C4⋊C4C2×D412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D414yes—
189C2×C22⋊C4—41no—
190C2×C42—24no—
191C2×C42—22no—
192C2×C42—24no—
193C2×C42—22no—
194C2×C42—28no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—48no—
198C2×C42—44no—
199C2×C42—41no—
200C2×C42C22×C412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×Q812yes—
205C2×C42#4112yes—
206C2×C42#4112yes—
207C2×C42C2411yescharacteristic
208#107—44no—
209#107—48no—
210#106—48no—
211#106—44no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8C22×C414yes—
216#103—44no—
217#103—44no—
218C22⋊C8—24no—
219C22⋊C8—42no—
220#97—24no—
221#97—24no—
222#97—42no—
223C4×C8—22no—
224C4×C8C22×C414yes—
225#95—28no—
226#95—22no—
227#95—22no—
228#95—41no—
229#95—41no—
230C24#12011yescharacteristic
231C2×Q8—24no—
232C2×Q8—216no—
233C2×Q8—24no—
234C2×Q8—216no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—24no—
239C2×Q8C2×C4⋊C414yes—
240C2×Q8AGL(1,ℤ/8)14yes—
241C2×Q8#13714yes—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4C2×C4⋊C412yes—
271C22×C4C4×D412yes—
272C22×C4C4×D412yes—
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4C22⋊Q812yes—
277C22×C4C22⋊Q812yes—
278C22×C4#12312yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282M16—216no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—28no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—416no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—82no—
315C4⋊C4—88no—
316C4⋊C4C2×C4⋊C414yes—
317C4⋊C4#13714yes—
318C22⋊C4—82no—
319C42—24no—
320C42—24no—
321C42—28no—
322C42—416no—
323C42—44no—
324C42—44no—
325C42—82no—
326C42C2×C4⋊C412yes—
327C42C4×D412yes—
328C42C4×D412yes—
329C42C22⋊Q812yes—
330C42C22⋊Q812yes—
331C42#12312yes—
332C42C23×C411yescharacteristic
333C42C22×Q811yescharacteristic
334C42C2×#4111yescharacteristic
335C42C2×#4111yescharacteristic
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23#38512yes—
341C23#45312yes—
342C23#45612yes—
343C23C2×#12011yescharacteristic
344Q8—216no—
345Q8—24no—
346Q8—24no—
347Q8—24no—
348Q8#42818yescomplemented
349C2×C4—216no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—24no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—48no—
370C2×C4—48no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—88no—
382C2×C4—84no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4#38512yes—
390C2×C4#38512yes—
391C2×C4#38512yes—
392C2×C4C22×C4⋊C411yescharacteristic
393C2×C4C2×C4×D411yescharacteristic
394C2×C4C2×C4×D411yescharacteristic
395C2×C4C2×C22⋊Q811yescharacteristic
396C2×C4C2×C22⋊Q811yescharacteristic
397C2×C4C2×#12311yescharacteristic
398C2×C4C2×AGL(1,ℤ/8)12yes—
399C2×C4C2×#13712yes—
400C2×C4C2×#13712yes—
401C2×C4#57512yes—
402C8—28no—
403C8—44no—
404C8—82no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22#185914yes—
412C22C2×#38511yescharacteristic
413C22#300311yescharacteristic
414C22#300411yescharacteristic
415C22#300411yescharacteristic
416C22#300511yescharacteristic
417C22C2×#45311yescharacteristic
418C22C2×#45611yescharacteristic
419C4—28no—
420C4—22no—
421C4—21no—
422C4—21no—
423C4—22no—
424C4—22no—
425C4—44no—
426C4—44no—
427C4—82no—
428C4—84no—
429C4—161no—
430C4—161no—
431C4C2×#42812yes—
432C4C4×#13712yes—
433C4#307812yes—
434C4#309212yes—
435C2—24no—
436C2#1854011yescharacteristic
437C2#1854111yescharacteristic
438C2#1854511yescharacteristic
439C2#1854611yescharacteristic
440C2#2084411yescharacteristic
441C2C2×#185911yescharacteristic
442C2#2087411yescharacteristic
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. #8888C2semidirect
7. #8884C2semidirect
40. #1491C4semidirect
121. Q8⋊C8C2×C4semidirect
348. 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
#213652
#208002
#207402
C2×#14912
#88882
#88842
#80402
#79042