Finite Group Databasebeta

← #6532183

Subgroups — #6532183

Order 512.

1630 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2584C211yesmaximal, characteristic
2#34182C211yesmaximal, characteristic
3#27614C212yesmaximal
4#27613C212yesmaximal
5#27611C212yesmaximal
6#27610C212yesmaximal
7#26290C218yesmaximal, complemented
8#26288C218yesmaximal, complemented
9#21838C212yesmaximal
10#21822C211yesmaximal, characteristic
11#21365C212yesmaximal
12C22×#125C2211yescharacteristic
13C2×#519—21no—
14C22×C4×Q8C2211yescharacteristic
15C2×#432C2212yes—
16#2584—24no—
17#2584C2218yes—
18C2×#398—21no—
19C2×#395—21no—
20#2351—24no—
21Q8×C4⋊C4C2214yes—
22Q8×C4⋊C4C2214yes—
23#2316—21no—
24#2316—21no—
25#2316—21no—
26C2×#389—21no—
27C2×#389—21no—
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32#2285C2212yes—
33#2285C2212yes—
34#2285C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36#2100C2214yes—
37#2100C2214yes—
38#2099C2214yes—
39#2099C2214yes—
40#2084C2214yes—
41#2084C2214yes—
42#2083C2214yes—
43#2083C2214yes—
44#1992C2214yes—
45#1992C2214yes—
46#1989C2218yes—
47#1985C2218yes—
48#1948C2214yes—
49#1947C2214yes—
50#1921C2218yes—
51#1920C2218yes—
52#1919C2218yes—
53#1918C2218yes—
54#1853C2214yes—
55#1852C2214yes—
56#1824C2214yes—
57#1823C2214yes—
58#1817C2212yes—
59#1817C2212yes—
60#1776C2218yes—
61C23×Q8D411yescharacteristic
62C2×#125—22no—
63C2×#125—28no—
64C2×#125C2314yes—
65#519—48no—
66C2×C4×Q8—24no—
67C2×C4×Q8—24no—
68C2×C4×Q8—28no—
69C2×C4×Q8D414yes—
70C2×C4×Q8C2314yes—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#432—28no—
92C2×#107C2318yes—
93C2×#106C2318yes—
94C2×C4⋊C8C2314yes—
95C2×C22⋊C8C2314yes—
96C2×#97C2314yes—
97#398—28no—
98#395—28no—
99#389—24no—
100#389—24no—
101#389—24no—
102#389—24no—
103#389—24no—
104#389—24no—
105#389—28no—
106#389—28no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389—22no—
111#389C2314yes—
112#389C2314yes—
113#389C2314yes—
114#389C2314yes—
115#384C2312yes—
116#384C2312yes—
117#384C2312yes—
118#384C2312yes—
119#384C2312yes—
120#384C2312yes—
121#382—22no—
122#382—22no—
123#382—22no—
124#380—22no—
125#380—22no—
126#380—22no—
127#380C2314yes—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4C2314yes—
133C4×C4⋊C4C2314yes—
134C4×C4⋊C4C2314yes—
135C4×C4⋊C4C2314yes—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141#340C2318yes—
142#340C2318yes—
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#336C2314yes—
148#336C2314yes—
149#336C2314yes—
150#336C2314yes—
151C22×Q8—24no—
152C22×Q8—24no—
153C22×Q8—24no—
154C22×Q8C2×D414yes—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2411yescharacteristic
166C22×C8C2418yes—
167#125—28no—
168#125—28no—
169#125—48no—
170C4×Q8—216no—
171C4×Q8—28no—
172C4×Q8—28no—
173C4×Q8—416no—
174C4×Q8—416no—
175#117—48no—
176#117—48no—
177#117—48no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4C2412yes—
217C2×C4⋊C4C2412yes—
218C2×C4⋊C4C2414yes—
219C2×C4⋊C4C2414yes—
220C2×C4⋊C4C2412yes—
221C2×C4⋊C4C2412yes—
222C2×C22⋊C4—24no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—82no—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42#4118yes—
234C2×C42C2412yes—
235C2×C42C2412yes—
236C2×C42C2412yes—
237#107—216no—
238#106—216no—
239C4⋊C8—28no—
240C22⋊C8—28no—
241#97—28no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251C24C22×D411yescharacteristic
252C2×Q8—232no—
253C2×Q8—28no—
254C2×Q8—28no—
255C2×Q8—28no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4#12314yes—
291C22×C4#12314yes—
292C22×C4AGL(1,ℤ/8)14yes—
293C22×C4#13714yes—
294C22×C4C22×D412yes—
295C22×C4C22×D412yes—
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D412yes—
298C22×C4C22×D412yes—
299C22×C4C22×D412yes—
300C22×C4C22×D412yes—
301C22×C4C2×#4114yes—
302C22×C4C2511yescharacteristic
303C2×C8—28no—
304C2×C8—28no—
305C2×C8—28no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—28no—
309C4⋊C4—28no—
310C4⋊C4—28no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—28no—
316C4⋊C4—28no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—48no—
324C4⋊C4—48no—
325C4⋊C4—48no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—48no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—84no—
336C4⋊C4—84no—
337C4⋊C42−1+414yes—
338C4⋊C42−1+414yes—
339C22⋊C4—48no—
340C42—24no—
341C42—24no—
342C42—24no—
343C42—24no—
344C42—48no—
345C42—48no—
346C42—48no—
347C42—84no—
348C42#12318yes—
349C42C2×#4114yes—
350C42C2×#4114yes—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23C2×#12312yes—
356C23C2×AGL(1,ℤ/8)12yes—
357C23C2×#13712yes—
358C23C23×D411yescharacteristic
359Q8—232no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—28no—
363C2×C4—28no—
364C2×C4—28no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—28no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—44no—
383C2×C4—48no—
384C2×C4—48no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4C2×#12312yes—
406C2×C4C2×#12314yes—
407C2×C4C2×#12314yes—
408C2×C4C2×AGL(1,ℤ/8)14yes—
409C2×C4C2×#13714yes—
410C2×C4C22×#4112yes—
411C2×C4C2×2−1+412yes—
412C2×C4C2×2−1+412yes—
413C8—48no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22#309214yes—
421C22C22×#12311yescharacteristic
422C22#348611yescharacteristic
423C22#348611yescharacteristic
424C22C22×AGL(1,ℤ/8)11yescharacteristic
425C22C22×#13711yescharacteristic
426C22#358511yescharacteristic
427C22#358611yescharacteristic
428C4—22no—
429C4—24no—
430C4—24no—
431C4—24no—
432C4—24no—
433C4—42no—
434C4—42no—
435C4—44no—
436C4—44no—
437C4—82no—
438C4—82no—
439C4—84no—
440C4—84no—
441C4#350914yes—
442C2—24no—
443C2#3567411yescharacteristic
444C2C2×#309211yescharacteristic
445C2#6052011yescharacteristic
446C2#6182511yescharacteristic
447C2#6182611yescharacteristic
448C2#6182811yescharacteristic
449C2#6182911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #26290C2semidirect
8. #26288C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25842
#341822
#276142
#276132
#276112
#276102
#262902
#262882
#218382
#218222
#213652