Finite Group Databasebeta

← #6481946

Subgroups — #6481946

Order 512.

1568 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2C2×#2356C211yesmaximal, characteristic
3#33148C212yesmaximal
4#33146C212yesmaximal
5#29953C212yesmaximal
6#29951C212yesmaximal
7#26104C218yesmaximal, complemented
8#26038C218yesmaximal, complemented
9#21986C212yesmaximal
10#21955C211yesmaximal, characteristic
11#20740C212yesmaximal
12C2×#544—21no—
13C22×C4×Q8C2211yescharacteristic
14C2×#432C2212yes—
15#2857C2212yes—
16#2857C2212yes—
17#2659—21no—
18#2659—21no—
19#2552C2214yes—
20#2551C2214yes—
21#2550—24no—
22#2356—24no—
23#2356C2218yes—
24#2316—21no—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#387—21no—
28C2×#387—21no—
29#2285C2212yes—
30C2×#383—21no—
31C2×#382C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×#376—21no—
36#2096C2214yes—
37#2095C2214yes—
38#2068C2218yes—
39#2066C2218yes—
40#2024C2214yes—
41#2024C2214yes—
42#1988C2214yes—
43#1987C2214yes—
44#1949C2218yes—
45#1948C2214yes—
46#1947C2214yes—
47#1940C2218yes—
48#1920C2218yes—
49#1918C2218yes—
50#1853C2214yes—
51#1852C2214yes—
52#1847C2214yes—
53#1836C2214yes—
54#1823C2214yes—
55#1803C2214yes—
56#1803C2214yes—
57#1797C2212yes—
58#1797C2212yes—
59#1787C2214yes—
60#1763C2218yes—
61C23×Q8C2311yescharacteristic
62#544—48no—
63C2×C4⋊Q8—24no—
64C2×C4⋊Q8C2312yes—
65C2×C4⋊Q8C2312yes—
66C2×#125—24no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—24no—
71C2×C4×Q8Q818yescomplemented
72C2×C4×Q8C2314yes—
73C2×#117—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C42—21no—
81C22×C42—21no—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#432—28no—
87C2×#107C2318yes—
88C2×#106C2318yes—
89C2×C4⋊C8C2314yes—
90C2×C4⋊C8C2314yes—
91C2×C4⋊C8C2314yes—
92C2×#103C2318yes—
93C2×#103C2318yes—
94C2×C22⋊C8C2314yes—
95C2×#97C2314yes—
96#398—24no—
97#398—24no—
98#395—24no—
99#395—24no—
100#391—22no—
101#391—22no—
102#391—22no—
103#391—22no—
104#391C2314yes—
105#391C2314yes—
106#389—24no—
107#389—24no—
108#389—22no—
109#389—22no—
110#389C2314yes—
111#389C2314yes—
112#387—28no—
113#387—28no—
114#384C2312yes—
115#384C2312yes—
116#384C2312yes—
117#384C2312yes—
118#383—28no—
119#382—22no—
120#382—22no—
121#382C2314yes—
122#380—22no—
123#380—22no—
124#380—22no—
125#380C2314yes—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4D414yes—
130C4×C4⋊C4C2314yes—
131C4×C4⋊C4C2314yes—
132C4×C4⋊C4C2314yes—
133C4×C4⋊C4C2314yes—
134#376—28no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95D411yescharacteristic
144C2×#95D411yescharacteristic
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#340C2318yes—
149#336C2314yes—
150#336C2314yes—
151C22×Q8—21no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×Q8C2×Q814yes—
158C22×Q8C2412yes—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C22×C8C2418yes—
171C4⋊Q8—44no—
172C4⋊Q8—44no—
173C4⋊Q8—416no—
174#125—416no—
175C22⋊Q8—48no—
176C22⋊Q8—48no—
177C4×Q8—216no—
178C4×Q8—28no—
179C4×Q8C2×Q8116yes—
180#117—48no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
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—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2×D414yes—
205C2×C4⋊C4C2×Q814yes—
206C2×C4⋊C4C2412yes—
207C2×C4⋊C4C2412yes—
208C2×C4⋊C4C2414yes—
209C2×C4⋊C4C2414yes—
210C2×C4⋊C4C2412yes—
211C2×C22⋊C4—24no—
212C2×C42—28no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—44no—
220C2×C42—44no—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×Q814yes—
225C2×C42#4114yes—
226C2×C42C2412yes—
227C2×C42C2412yes—
228C2×C42C2412yes—
229#107—48no—
230#106—48no—
231C4⋊C8—28no—
232C4⋊C8—44no—
233C4⋊C8—44no—
234#103—48no—
235#103—48no—
236C22⋊C8—28no—
237#97—28no—
238#95—24no—
239#95—24no—
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#95—24no—
250#95C2×D414yes—
251#95C2×D414yes—
252C24C22×D411yescharacteristic
253C2×Q8—24no—
254C2×Q8—22no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—48no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—48no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—82no—
264C2×Q8—82no—
265C2×Q8—82no—
266C2×Q8—82no—
267C2×Q8C22×Q814yes—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4C22⋊Q814yes—
303C22×C4C22⋊Q814yes—
304C22×C4AGL(1,ℤ/8)14yes—
305C22×C4#13714yes—
306C22×C4C22×D412yes—
307C22×C4C22×D412yes—
308C22×C4C22×D412yes—
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D412yes—
311C22×C4C22×D412yes—
312C22×C4C22×D412yes—
313C22×C4C22×Q812yes—
314C22×C4C2×#4112yes—
315C22×C4C2511yescharacteristic
316C2×C8—28no—
317C2×C8—28no—
318C2×C8—28no—
319C4⋊C4—216no—
320C4⋊C4—24no—
321C4⋊C4—28no—
322C4⋊C4—28no—
323C4⋊C4—22no—
324C4⋊C4—28no—
325C4⋊C4—28no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—48no—
334C4⋊C4—44no—
335C4⋊C4—48no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4C22×Q814yes—
343C22⋊C4—48no—
344C42—216no—
345C42—24no—
346C42—24no—
347C42—42no—
348C42—42no—
349C42—48no—
350C42—48no—
351C42—48no—
352C42C22⋊Q818yes—
353C42C22×Q814yes—
354C42C2×#4114yes—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23C2×C22⋊Q812yes—
360C23C2×AGL(1,ℤ/8)12yes—
361C23C2×#13712yes—
362C23C23×D411yescharacteristic
363Q8—28no—
364Q8—48no—
365Q8—44no—
366Q8—164no—
367Q8—164no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—28no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—84no—
413C2×C4—84no—
414C2×C4C2×C22⋊Q814yes—
415C2×C4C2×C22⋊Q812yes—
416C2×C4C2×C22⋊Q814yes—
417C2×C4C2×AGL(1,ℤ/8)14yes—
418C2×C4C2×#13714yes—
419C2×C4C23×Q811yescharacteristic
420C2×C4C22×#4111yescharacteristic
421C2×C4#58512yes—
422C2×C4#58512yes—
423C8—48no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22#307914yes—
431C22C22×C22⋊Q811yescharacteristic
432C22#348511yescharacteristic
433C22#348711yescharacteristic
434C22C22×AGL(1,ℤ/8)11yescharacteristic
435C22C22×#13711yescharacteristic
436C22#358611yescharacteristic
437C22#358611yescharacteristic
438C4—24no—
439C4—24no—
440C4—24no—
441C4—24no—
442C4—22no—
443C4—42no—
444C4—42no—
445C4—44no—
446C4—44no—
447C4—84no—
448C4—84no—
449C4—82no—
450C4—82no—
451C4#345812yes—
452C4#349312yes—
453C2—24no—
454C2#3515111yescharacteristic
455C2C2×#307911yescharacteristic
456C2#6049111yescharacteristic
457C2#6170411yescharacteristic
458C2#6170511yescharacteristic
459C2#6188811yescharacteristic
460C2#6188911yescharacteristic
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. #26104C2semidirect
8. #26038C2semidirect
71. C2×C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#23562
#331482
#331462
#299532
#299512
#261042
#260382
#219862
#219552
#207402