Finite Group Databasebeta

← #274102

Subgroups — #274102

Order 512.

761 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33108C211yesmaximal, characteristic
2#25811C211yesmaximal, characteristic
3#22599C211yesmaximal, characteristic
4C2×#1987C211yesmaximal, characteristic
5#10597C212yesmaximal, complemented
6#10544C212yesmaximal, complemented
7#9729C212yesmaximal, complemented
8#9567C212yesmaximal, complemented
9#9131C211yesmaximal, characteristic
10#8830C211yesmaximal, characteristic
11#8342C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#498C2212yes—
14C2×#483—21no—
15C2×#431—21no—
16C22×C4⋊C8C2211yescharacteristic
17#2857C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#2058—22no—
20#1990C2212yes—
21#1987—21no—
22#1987—21no—
23#1987C2212yes—
24#1987C2212yes—
25#1981C2211yescharacteristic
26#1981C2211yescharacteristic
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1939C2212yes—
30#1927—21no—
31#1927—21no—
32#1916—22no—
33#1872—21no—
34#1853—21no—
35#1853—21no—
36#1817—21no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C8×C4⋊C4C2212yes—
40C4×C22⋊C8C2211yescharacteristic
41C2×#340—21no—
42C2×#340—21no—
43C2×#334C2212yes—
44C2×#333C2212yes—
45C2×#333C2212yes—
46C2×#332C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×Q8—21no—
50C23×C8—21no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8C2311yescharacteristic
57C2×C4⋊Q8C2312yes—
58C2×C22⋊Q8—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C42—21no—
64C22×C42C2311yescharacteristic
65#498—24no—
66#498—42no—
67#483—44no—
68#431—44no—
69C2×#106—22no—
70C2×#106—22no—
71C2×#106—21no—
72C2×#106—21no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—42no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2311yescharacteristic
80C2×#103—21no—
81C2×#103—21no—
82C2×#103—21no—
83C2×#103—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8D411yescharacteristic
87C2×C22⋊C8D411yescharacteristic
88C2×C4×C8—22no—
89C2×C4×C8C2312yes—
90#391—21no—
91#389—21no—
92#389—21no—
93#384C2311yescharacteristic
94#380—21no—
95#380—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4D411yescharacteristic
99C4×C4⋊C4D411yescharacteristic
100C4×C4⋊C4C2311yescharacteristic
101C4×C4⋊C4C2311yescharacteristic
102C2×#95—21no—
103#340—22no—
104#340—22no—
105#340—21no—
106#340—21no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#336—22no—
112#336—22no—
113#336—21no—
114#336—21no—
115#336D412yes—
116#334—42no—
117#333—24no—
118#333—42no—
119#332—42no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—44no—
123C22×Q8—41no—
124C22×Q8—82no—
125C22×Q8—82no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—41no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4#4111yescharacteristic
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—42no—
137C22×C8—42no—
138C22×C8—42no—
139C22×C8C2×D412yes—
140C22×C8C2×D412yes—
141C4⋊Q8—24no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—44no—
148C4⋊Q8—44no—
149C4⋊Q8—42no—
150C4⋊Q8#4114yes—
151C22⋊Q8—42no—
152C22⋊Q8—42no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4SD1614yes—
172C2×C4⋊C4C2×D411yescharacteristic
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C4⋊C4C2×D411yescharacteristic
175C2×C4⋊C4C2×D411yescharacteristic
176C2×C22⋊C4—41no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42C2411yescharacteristic
189#106—44no—
190#106—44no—
191#106—42no—
192#106—42no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—42no—
200C4⋊C8—44no—
201#103—42no—
202#103—42no—
203#103—42no—
204#103—42no—
205C22⋊C8—22no—
206C22⋊C8—42no—
207C22⋊C8—42no—
208C22⋊C8SD1614yes—
209C4×C8—22no—
210C4×C8—22no—
211C4×C8—44no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—41no—
217#95—41no—
218C24#12111yescharacteristic
219C2×Q8—24no—
220C2×Q8—44no—
221C2×Q8—42no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—82no—
229C2×Q8—82no—
230C2×Q8—82no—
231C2×Q8—82no—
232C2×Q8—816no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—81no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4#12111yescharacteristic
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4C2×SD1612yes—
262C22×C4C2×SD1612yes—
263C22×C4#13512yes—
264C22×C4#13712yes—
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C2×#4111yescharacteristic
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—41no—
274C2×C8—42no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—42no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—84no—
286C2×C8C2×SD1614yes—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—81no—
306C4⋊C4—82no—
307C4⋊C4C2×SD1612yes—
308C4⋊C4C2×SD1612yes—
309C4⋊C4#13712yes—
310C22⋊C4—82no—
311C42—21no—
312C42—21no—
313C42—22no—
314C42—22no—
315C42—41no—
316C42—41no—
317C42—41no—
318C42—41no—
319C42—42no—
320C42—42no—
321C42C2×#4112yes—
322C42H(5,𝔽2)12yes—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23#46112yes—
328C23#46512yes—
329C23#47112yes—
330C23C2×#12111yescharacteristic
331Q8—84no—
332Q8—84no—
333Q8—88no—
334Q8—81no—
335Q8—81no—
336Q8—81no—
337Q8—81no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—44no—
357C2×C4—48no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—44no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—84no—
369C2×C4—88no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4#46112yes—
379C2×C4#46512yes—
380C2×C4#52511yescharacteristic
381C2×C4#53511yescharacteristic
382C2×C4#54612yes—
383C2×C4#55112yes—
384C2×C4C22×SD1611yescharacteristic
385C2×C4C22×SD1611yescharacteristic
386C2×C4C2×#13511yescharacteristic
387C2×C4C2×#13711yescharacteristic
388C2×C4C2×#13711yescharacteristic
389C2×C4#57511yescharacteristic
390C2×C4#57812yes—
391C8—24no—
392C8—42no—
393C8—81no—
394C8—81no—
395C8—82no—
396C8—82no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22#167114yes—
404C22#241111yescharacteristic
405C22C2×#46111yescharacteristic
406C22#304411yescharacteristic
407C22C2×#46511yescharacteristic
408C22#305611yescharacteristic
409C22C2×#47111yescharacteristic
410C22#306811yescharacteristic
411C4—22no—
412C4—22no—
413C4—22no—
414C4—42no—
415C4—42no—
416C4—41no—
417C4—42no—
418C4—84no—
419C4—81no—
420C4—161no—
421C4—161no—
422C4#310311yescharacteristic
423C4#310811yescharacteristic
424C4#320611yescharacteristic
425C4#321111yescharacteristic
426C4#329812yes—
427C4#331212yes—
428C2—24no—
429C2C2×#167111yescharacteristic
430C2#1916511yescharacteristic
431C2#1940311yescharacteristic
432C2#1947511yescharacteristic
433C2#2448011yescharacteristic
434C2#2450311yescharacteristic
435C2#2452111yescharacteristic
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. #10597C2semidirect
6. #10544C2semidirect
7. #9729C2semidirect
8. #9567C2semidirect

Maximal subgroups

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

subgroupindex
#331082
#258112
#225992
C2×#19872
#105972
#105442
#97292
#95672
#91312
#88302
#83422