Finite Group Databasebeta

← #6528429

Subgroups — #6528429

Order 512.

1446 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2C2×#2570C211yesmaximal, characteristic
3#29302C212yesmaximal
4#29296C212yesmaximal
5#28837C218yesmaximal, complemented
6#28801C218yesmaximal, complemented
7#27956C212yesmaximal
8#27948C212yesmaximal
9#21986C212yesmaximal
10#21365C212yesmaximal
11#20534C211yesmaximal, characteristic
12C2×#544—21no—
13C2×#432C2212yes—
14#2857C2212yes—
15#2857C2212yes—
16#2659—21no—
17#2659—21no—
18#2570—24no—
19#2570C2218yes—
20#2552C2214yes—
21#2551C2214yes—
22#2550—24no—
23C2×#400—21no—
24C2×#398—21no—
25#2316—21no—
26C2×#389C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#387—21no—
29C2×#387—21no—
30#2285C2212yes—
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#2085C2218yes—
37#2084C2218yes—
38#2084C2214yes—
39#2083C2214yes—
40#2057C2214yes—
41#2057C2214yes—
42#1993C2214yes—
43#1990C2214yes—
44#1949C2218yes—
45#1948C2214yes—
46#1947C2214yes—
47#1944C2218yes—
48#1942C2218yes—
49#1853C2214yes—
50#1853C2214yes—
51#1852C2214yes—
52#1852C2214yes—
53#1847C2214yes—
54#1847C2214yes—
55#1844C2214yes—
56#1836C2214yes—
57#1776C2218yes—
58#1771C2218yes—
59#1749C2212yes—
60C4×C22⋊C8C2212yes—
61#544—48no—
62C2×C4⋊Q8—24no—
63C2×C4⋊Q8C2312yes—
64C2×C4⋊Q8C2312yes—
65C2×#125—24no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×#117—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#432Q8116yescomplemented
84C2×#107C2318yes—
85C2×#107C2318yes—
86C2×#106C2318yes—
87C2×#106C2318yes—
88C2×C4⋊C8C2314yes—
89C2×C22⋊C8C2314yes—
90C2×#97C2314yes—
91C2×#97C2314yes—
92C2×C4×C8C2314yes—
93#400—28no—
94#398—28no—
95#398—24no—
96#398—24no—
97#395—24no—
98#395—24no—
99#391—22no—
100#391—22no—
101#389—24no—
102#389—24no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389C2314yes—
108#389C2314yes—
109#389C2314yes—
110#389C2314yes—
111#387—28no—
112#387—28no—
113#384—22no—
114#384C2314yes—
115#384C2312yes—
116#384C2312yes—
117#384C2312yes—
118#384C2312yes—
119#382—22no—
120#382—22no—
121#382—22no—
122#382C2314yes—
123#382C2314yes—
124#380—22no—
125#380—22no—
126#380—22no—
127#380C2314yes—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4D414yes—
131C4×C4⋊C4C2314yes—
132C4×C4⋊C4C2314yes—
133C4×C4⋊C4C2314yes—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95D411yescharacteristic
142C2×#95D411yescharacteristic
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#336C2314yes—
148#336C2314yes—
149C22×Q8—22no—
150C22×Q8—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8C2418yes—
163C4⋊Q8—44no—
164C4⋊Q8—44no—
165C4⋊Q8—416no—
166#125—416no—
167C22⋊Q8—48no—
168C22⋊Q8—48no—
169#117—48no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4C2×D414yes—
202C2×C4⋊C4C2412yes—
203C2×C4⋊C4C2412yes—
204C2×C4⋊C4C2412yes—
205C2×C4⋊C4C2412yes—
206C2×C4⋊C4C2414yes—
207C2×C4⋊C4C2414yes—
208C2×C22⋊C4—24no—
209C2×C42—24no—
210C2×C42—24no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42—82no—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×Q818yes—
218C2×C42C2412yes—
219C2×C42C2412yes—
220C2×C42C2412yes—
221#107—48no—
222#107—48no—
223#106—48no—
224#106—48no—
225C4⋊C8C2×Q8116yes—
226C22⋊C8C2×Q8116yes—
227#97—44no—
228#97C2×Q8116yes—
229C4×C8—44no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95C2×D414yes—
242#95C2×D414yes—
243C24C22×D411yescharacteristic
244C2×Q8—44no—
245C2×Q8—44no—
246C2×Q8—82no—
247C2×Q8—82no—
248C2×Q8—82no—
249C2×Q8—82no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4C4⋊Q814yes—
285C22×C4C4⋊Q814yes—
286C22×C4AGL(1,ℤ/8)14yes—
287C22×C4#13714yes—
288C22×C4C22×D412yes—
289C22×C4C22×D412yes—
290C22×C4C22×D412yes—
291C22×C4C22×D412yes—
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D412yes—
294C22×C4C22×D412yes—
295C22×C4C22×Q814yes—
296C22×C4C2511yescharacteristic
297C2×C8—28no—
298C2×C8—28no—
299C2×C8C22×Q8116yes—
300C4⋊C4—28no—
301C4⋊C4—28no—
302C4⋊C4—28no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—28no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—48no—
316C4⋊C4—48no—
317C4⋊C4—44no—
318C4⋊C4—48no—
319C4⋊C4—48no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—84no—
325C4⋊C4—84no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C22⋊C4—48no—
329C42—24no—
330C42—24no—
331C42—42no—
332C42—42no—
333C42—48no—
334C42—84no—
335C42C4⋊Q818yes—
336C42C22×Q814yes—
337C42C22×Q814yes—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23C2×C4⋊Q812yes—
343C23C2×AGL(1,ℤ/8)12yes—
344C23C2×#13712yes—
345C23C23×D411yescharacteristic
346Q8—164no—
347Q8—164no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—28no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—28no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4C2×C4⋊Q812yes—
394C2×C4C2×C4⋊Q814yes—
395C2×C4C2×C4⋊Q814yes—
396C2×C4C2×AGL(1,ℤ/8)14yes—
397C2×C4C2×#13714yes—
398C2×C4C23×Q812yes—
399C2×C4#58512yes—
400C2×C4#58512yes—
401C8—216no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22#316514yes—
409C22C22×C4⋊Q811yescharacteristic
410C22#348511yescharacteristic
411C22#348711yescharacteristic
412C22C22×AGL(1,ℤ/8)11yescharacteristic
413C22C22×#13711yescharacteristic
414C22#358511yescharacteristic
415C22#358611yescharacteristic
416C4—24no—
417C4—24no—
418C4—22no—
419C4—24no—
420C4—24no—
421C4—42no—
422C4—42no—
423C4—82no—
424C4—84no—
425C4—84no—
426C4—82no—
427C4—82no—
428C4—82no—
429C4#350414yes—
430C2—24no—
431C2#3564011yescharacteristic
432C2C2×#316511yescharacteristic
433C2#6066411yescharacteristic
434C2#6170511yescharacteristic
435C2#6170711yescharacteristic
436C2#6188811yescharacteristic
437C2#6189211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #28837C2semidirect
6. #28801C2semidirect
83. #432Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#25702
#293022
#292962
#288372
#288012
#279562
#279482
#219862
#213652
#205342