Finite Group Databasebeta

← #6482294

Subgroups — #6482294

Order 512.

1542 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2C2×#2370C211yesmaximal, characteristic
3#33148C212yesmaximal
4#33147C212yesmaximal
5#29969C212yesmaximal
6#29966C212yesmaximal
7#26522C218yesmaximal, complemented
8#26456C218yesmaximal, complemented
9#21955C211yesmaximal, characteristic
10#21838C212yesmaximal
11#21365C212yesmaximal
12C2×#544—21no—
13C22×#125C2211yescharacteristic
14C2×#432C2212yes—
15#2857C2212yes—
16#2857C2212yes—
17#2659—21no—
18#2659—21no—
19#2552C2214yes—
20#2551C2214yes—
21#2550—24no—
22#2370—24no—
23#2370C2218yes—
24#2316—21no—
25C2×#389—21no—
26C2×#389C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#387—21no—
29C2×#387—21no—
30#2285C2212yes—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#2100C2214yes—
37#2099C2214yes—
38#2087C2218yes—
39#2086C2218yes—
40#2057C2214yes—
41#2057C2214yes—
42#1992C2214yes—
43#1992C2214yes—
44#1989C2218yes—
45#1988C2214yes—
46#1987C2214yes—
47#1948C2214yes—
48#1948C2214yes—
49#1948C2214yes—
50#1947C2214yes—
51#1940C2218yes—
52#1920C2218yes—
53#1919C2218yes—
54#1853C2214yes—
55#1852C2214yes—
56#1824C2214yes—
57#1823C2214yes—
58#1797C2212yes—
59#1797C2212yes—
60#1776C2218yes—
61#544—48no—
62C2×C4⋊Q8—24no—
63C2×C4⋊Q8C2312yes—
64C2×C4⋊Q8C2312yes—
65C2×#125—24no—
66C2×#125—22no—
67C2×#125—24no—
68C2×#125Q818yescomplemented
69C2×#125C2314yes—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×#117—22no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C42—21no—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#432—28no—
90C2×#107C2318yes—
91C2×#106C2318yes—
92C2×C4⋊C8C2314yes—
93C2×C4⋊C8C2314yes—
94C2×C4⋊C8C2314yes—
95C2×C22⋊C8C2314yes—
96C2×#97C2314yes—
97#398—24no—
98#398—24no—
99#395—24no—
100#395—24no—
101#391—22no—
102#391—22no—
103#389—24no—
104#389—24no—
105#389—28no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389C2314yes—
111#389C2314yes—
112#389C2314yes—
113#389C2314yes—
114#387—28no—
115#387—28no—
116#384C2312yes—
117#384C2312yes—
118#384C2312yes—
119#384C2312yes—
120#382—22no—
121#380—22no—
122#380—22no—
123#380—22no—
124#380C2314yes—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—28no—
130C4×C4⋊C4D414yes—
131C4×C4⋊C4C2314yes—
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—
141C2×#95D411yescharacteristic
142C2×#95D411yescharacteristic
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#340C2318yes—
149#336C2314yes—
150#336C2314yes—
151C22×Q8—22no—
152C22×Q8—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2411yescharacteristic
164C22×C8C2418yes—
165C4⋊Q8—44no—
166C4⋊Q8—44no—
167C4⋊Q8—416no—
168#125—216no—
169#125—28no—
170#125—416no—
171#125C2×Q8116yes—
172C22⋊Q8—48no—
173C22⋊Q8—48no—
174#117—48no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4C2×D414yes—
207C2×C4⋊C4C2×Q814yes—
208C2×C4⋊C4C2×Q814yes—
209C2×C4⋊C4C2412yes—
210C2×C4⋊C4C2414yes—
211C2×C4⋊C4C2414yes—
212C2×C4⋊C4C2412yes—
213C2×C4⋊C4C2412yes—
214C2×C4⋊C4C2412yes—
215C2×C22⋊C4—24no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—24no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42—44no—
223C2×C42—82no—
224C2×C42—82no—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×Q814yes—
229C2×C42#4114yes—
230C2×C42C2412yes—
231C2×C42C2412yes—
232C2×C42C2412yes—
233#107—48no—
234#106—216no—
235C4⋊C8—28no—
236C4⋊C8—44no—
237C4⋊C8—44no—
238C22⋊C8—28no—
239#97—28no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95C2×D414yes—
250#95C2×D414yes—
251C24C22×D411yescharacteristic
252C2×Q8—44no—
253C2×Q8—44no—
254C2×Q8—82no—
255C2×Q8—82no—
256C2×Q8—82no—
257C2×Q8—82no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4C22⋊Q814yes—
293C22×C4C22⋊Q814yes—
294C22×C4AGL(1,ℤ/8)14yes—
295C22×C4#13714yes—
296C22×C4C22×D412yes—
297C22×C4C22×D412yes—
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D412yes—
300C22×C4C22×D412yes—
301C22×C4C22×D412yes—
302C22×C4C22×D412yes—
303C22×C4C22×Q812yes—
304C22×C4C2×#4112yes—
305C22×C4C2511yescharacteristic
306C2×C8—28no—
307C2×C8—28no—
308C2×C8—28no—
309C4⋊C4—24no—
310C4⋊C4—216no—
311C4⋊C4—216no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—28no—
315C4⋊C4—28no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—28no—
320C4⋊C4—24no—
321C4⋊C4—28no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—48no—
327C4⋊C4—48no—
328C4⋊C4—44no—
329C4⋊C4—48no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—84no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4C22×Q814yes—
338C4⋊C4C22×Q814yes—
339C4⋊C42−1+414yes—
340C22⋊C4—48no—
341C42—24no—
342C42—24no—
343C42—42no—
344C42—42no—
345C42—48no—
346C42—48no—
347C42—84no—
348C42—84no—
349C42C22⋊Q818yes—
350C42C22×Q814yes—
351C42C2×#4114yes—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23C2×C22⋊Q812yes—
357C23C2×AGL(1,ℤ/8)12yes—
358C23C2×#13712yes—
359C23C23×D411yescharacteristic
360Q8—164no—
361Q8—164no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—28no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—48no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—84no—
408C2×C4C2×C22⋊Q812yes—
409C2×C4C2×C22⋊Q814yes—
410C2×C4C2×C22⋊Q814yes—
411C2×C4C2×AGL(1,ℤ/8)14yes—
412C2×C4C2×#13714yes—
413C2×C4C23×Q811yescharacteristic
414C2×C4C22×#4111yescharacteristic
415C2×C4C2×2−1+412yes—
416C2×C4#58512yes—
417C8—48no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22#307914yes—
425C22C22×C22⋊Q811yescharacteristic
426C22#348511yescharacteristic
427C22#348611yescharacteristic
428C22C22×AGL(1,ℤ/8)11yescharacteristic
429C22C22×#13711yescharacteristic
430C22#358511yescharacteristic
431C22#358611yescharacteristic
432C4—24no—
433C4—24no—
434C4—24no—
435C4—24no—
436C4—22no—
437C4—42no—
438C4—42no—
439C4—44no—
440C4—44no—
441C4—44no—
442C4—82no—
443C4—84no—
444C4—84no—
445C4#346112yes—
446C4#350212yes—
447C2—24no—
448C2#3515411yescharacteristic
449C2C2×#307911yescharacteristic
450C2#6049211yescharacteristic
451C2#6170511yescharacteristic
452C2#6170711yescharacteristic
453C2#6182611yescharacteristic
454C2#6182811yescharacteristic
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
7. #26522C2semidirect
8. #26456C2semidirect
68. C2×#125Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#23702
#331482
#331472
#299692
#299662
#265222
#264562
#219552
#218382
#213652