Finite Group Databasebeta

← #329450

Subgroups — #329450

Order 512.

768 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33043C211yesmaximal, characteristic
2#32954C211yesmaximal, characteristic
3#32906C211yesmaximal, characteristic
4C2×#1984C211yesmaximal, characteristic
5#10626C212yesmaximal, complemented
6#10487C212yesmaximal, complemented
7#10478C212yesmaximal, complemented
8#10272C212yesmaximal, complemented
9#9131C211yesmaximal, characteristic
10#9129C211yesmaximal, characteristic
11#8596C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#487C2211yescharacteristic
14C2×#475C2212yes—
15C2×#462C2212yes—
16C2×#451—21no—
17C22×C4⋊C8C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2857C2211yescharacteristic
20#2857C2211yescharacteristic
21#1990C2211yescharacteristic
22#1984—21no—
23#1984—21no—
24#1984C2212yes—
25#1984C2212yes—
26#1956C2211yescharacteristic
27#1933—21no—
28#1933—21no—
29#1892—21no—
30#1892—21no—
31#1878—22no—
32#1872—21no—
33#1865—22no—
34#1853—21no—
35#1834C2211yescharacteristic
36#1803C2212yes—
37#1803C2212yes—
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40#1787—21no—
41C2×#336—21no—
42C2×#336—21no—
43C2×#335C2212yes—
44C2×#332C2212yes—
45C2×#332C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×Q8D411yescharacteristic
50C22×Q16—22no—
51C23×C8—21no—
52C23×C8D411yescharacteristic
53C23×C8D411yescharacteristic
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4⋊Q8C2311yescharacteristic
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—42no—
61C2×C4×Q8—42no—
62C2×C4×Q8D412yes—
63C2×C4×Q8C2312yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C42—21no—
67C22×C42C2311yescharacteristic
68#487—42no—
69#475—42no—
70#475—42no—
71#462—42no—
72#462—42no—
73#451—44no—
74C2×#106—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—42no—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—21no—
84C2×#103—21no—
85C2×#103—21no—
86C2×#103—21no—
87C2×#103—22no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C4×C8—42no—
91C2×C4×C8D411yescharacteristic
92C2×C4×C8D411yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#391—21no—
96#391—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#384C2311yescharacteristic
102#384C2311yescharacteristic
103#380—21no—
104#380—21no—
105#380—21no—
106#380—21no—
107#340—21no—
108#340—21no—
109#340—21no—
110#340—21no—
111#340—22no—
112#340—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—21no—
118#336—21no—
119#335—42no—
120#332—42no—
121#332—42no—
122C22×Q8—24no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—42no—
129C22×Q8C2×D412yes—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4#4111yescharacteristic
135C2×Q16—44no—
136C2×Q16—44no—
137C2×Q16—44no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—41no—
145C22×C8—41no—
146C22×C8—41no—
147C22×C8—42no—
148C22×C8—42no—
149C22×C8—42no—
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C22×C8C2×D411yescharacteristic
153C22×C8C2×D411yescharacteristic
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C22⋊Q8—42no—
159C22⋊Q8—42no—
160C22⋊Q8—42no—
161C22⋊Q8—42no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—42no—
167C4×Q8—42no—
168C4×Q8—42no—
169C4×Q8—42no—
170C4×Q8—44no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4C2×D412yes—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42C2411yescharacteristic
201#106—44no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—42no—
209#103—44no—
210#103—42no—
211#103—42no—
212#103—42no—
213#103—44no—
214#103—44no—
215#103—44no—
216#103—42no—
217C22⋊C8—42no—
218C22⋊C8—42no—
219C4×C8—22no—
220C4×C8—22no—
221C4×C8—41no—
222C4×C8—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227C24#12111yescharacteristic
228C2×Q8—24no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—48no—
235C2×Q8—48no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—41no—
239C2×Q8—42no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—42no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8#13514yes—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4#12111yescharacteristic
273C22×C4#13512yes—
274C22×C4AGL(1,ℤ/8)12yes—
275C22×C4#13712yes—
276C22×C4#13712yes—
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C2×#4111yescharacteristic
280Q16—88no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—44no—
293C2×C8—44no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—84no—
304C2×C8#13712yes—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—81no—
324C4⋊C4—81no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C42—21no—
330C42—22no—
331C42—21no—
332C42—22no—
333C42—44no—
334C42—42no—
335C42—42no—
336C42H(5,𝔽2)12yes—
337C42H(5,𝔽2)12yes—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23#46412yes—
343C23#46812yes—
344C23#47312yes—
345C23C2×#12111yescharacteristic
346Q8—24no—
347Q8—24no—
348Q8—48no—
349Q8—42no—
350Q8—42no—
351Q8—82no—
352Q8—81no—
353Q8—82no—
354Q8—81no—
355Q8—81no—
356Q8—81no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—24no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—44no—
374C2×C4—48no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—84no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4#46412yes—
394C2×C4#46812yes—
395C2×C4#53412yes—
396C2×C4#53411yescharacteristic
397C2×C4#53511yescharacteristic
398C2×C4#53812yes—
399C2×C4C2×#13512yes—
400C2×C4C2×#13511yescharacteristic
401C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
402C2×C4C2×#13711yescharacteristic
403C2×C4C2×#13711yescharacteristic
404C2×C4C2×#13711yescharacteristic
405C2×C4#57511yescharacteristic
406C8—22no—
407C8—41no—
408C8—41no—
409C8—41no—
410C8—82no—
411C8—82no—
412C8—82no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22#170014yes—
420C22#261711yescharacteristic
421C22C2×#46411yescharacteristic
422C22#303711yescharacteristic
423C22#305011yescharacteristic
424C22C2×#46811yescharacteristic
425C22C2×#47311yescharacteristic
426C22#306711yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—44no—
431C4—41no—
432C4—82no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4—161no—
437C4#318812yes—
438C4#319011yescharacteristic
439C4#319012yes—
440C4#319611yescharacteristic
441C4#321111yescharacteristic
442C4#321811yescharacteristic
443C2—24no—
444C2#1912311yescharacteristic
445C2C2×#170011yescharacteristic
446C2#1926511yescharacteristic
447C2#1928511yescharacteristic
448C2#2971911yescharacteristic
449C2#2973511yescharacteristic
450C2#2974911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×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
5. #10626C2semidirect
6. #10487C2semidirect
7. #10478C2semidirect
8. #10272C2semidirect

Maximal subgroups

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

subgroupindex
#330432
#329542
#329062
C2×#19842
#106262
#104872
#104782
#102722
#91312
#91292
#85962