Finite Group Databasebeta

← #274237

Subgroups — #274237

Order 512.

787 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33146C211yesmaximal, characteristic
2#25722C211yesmaximal, characteristic
3#22603C211yesmaximal, characteristic
4C2×#1988C211yesmaximal, characteristic
5#10588C212yesmaximal, complemented
6#10538C212yesmaximal, complemented
7#9742C212yesmaximal, complemented
8#9569C212yesmaximal, complemented
9#9129C211yesmaximal, characteristic
10#8830C211yesmaximal, characteristic
11#8340C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#494C2212yes—
14C2×#451—21no—
15C2×#451—21no—
16C2×#432—21no—
17C22×C4⋊C8C2211yescharacteristic
18#2857C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#2016—22no—
21#1988—21no—
22#1988—21no—
23#1988C2211yescharacteristic
24#1988C2212yes—
25#1988C2211yescharacteristic
26#1988C2212yes—
27#1956C2212yes—
28#1947C2211yescharacteristic
29#1947C2211yescharacteristic
30#1939C2212yes—
31#1918—21no—
32#1918—21no—
33#1917—22no—
34#1852—21no—
35#1823—21no—
36#1817—21no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1770C2212yes—
40C4×C22⋊C8C2211yescharacteristic
41C2×#340—21no—
42C2×#340—21no—
43C2×#334C2212yes—
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×Q8—21no—
50C22×Q16—22no—
51C22×Q16—22no—
52C23×C8—21no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8C2311yescharacteristic
59C2×C4⋊Q8C2312yes—
60C2×C22⋊Q8—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67#494—24no—
68#494—42no—
69#451—44no—
70#451—44no—
71#432—44no—
72C2×#107—21no—
73C2×#107—21no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—42no—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2312yes—
81C2×#103—22no—
82C2×#103—22no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8D411yescharacteristic
86C2×C22⋊C8D411yescharacteristic
87C2×#97—22no—
88C2×C4×C8C2312yes—
89#391—21no—
90#389—21no—
91#389—21no—
92#384C2311yescharacteristic
93#380—21no—
94#380—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4D411yescharacteristic
98C4×C4⋊C4D411yescharacteristic
99C4×C4⋊C4C2311yescharacteristic
100C4×C4⋊C4C2311yescharacteristic
101C2×#95—21no—
102#340—21no—
103#340—21no—
104#340—21no—
105#340—21no—
106#340—21no—
107#340—21no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#336—22no—
113#336—22no—
114#336—21no—
115#336—21no—
116#336D412yes—
117#334—42no—
118#332—24no—
119#332—42no—
120#332—42no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—44no—
124C22×Q8—41no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—82no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—41no—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4#4111yescharacteristic
134C2×Q16—44no—
135C2×Q16—44no—
136C2×Q16—44no—
137C2×Q16—44no—
138C2×Q16—82no—
139C2×Q16—82no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—42no—
145C22×C8—42no—
146C22×C8—42no—
147C22×C8C2×D412yes—
148C22×C8C2×D412yes—
149C4⋊Q8—24no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—44no—
156C4⋊Q8—44no—
157C4⋊Q8—42no—
158C4⋊Q8#4114yes—
159C22⋊Q8—42no—
160C22⋊Q8—42no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4C2×D411yescharacteristic
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C22⋊C4—41no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—42no—
192C2×C42—42no—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42C2411yescharacteristic
197#107—42no—
198#107—42no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—42no—
206C4⋊C8—44no—
207#103—44no—
208#103—44no—
209C22⋊C8—22no—
210C22⋊C8—42no—
211C22⋊C8—42no—
212C22⋊C8D814yes—
213#97—44no—
214C4×C8—22no—
215C4×C8—22no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—41no—
221#95—41no—
222C24#12111yescharacteristic
223C2×Q8—24no—
224C2×Q8—44no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—41no—
229C2×Q8—42no—
230C2×Q8—41no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—88no—
235C2×Q8—88no—
236C2×Q8—82no—
237C2×Q8—82no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—81no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4#12111yescharacteristic
264C22×C4#12111yescharacteristic
265C22×C4#12111yescharacteristic
266C22×C4C2×D812yes—
267C22×C4AGL(1,ℤ/8)12yes—
268C22×C4#13712yes—
269C22×C4#13712yes—
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C2×#4111yescharacteristic
273Q16—88no—
274Q16—88no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—84no—
293C2×C8C2×D814yes—
294C4⋊C4—21no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—81no—
317C4⋊C4—82no—
318C4⋊C4#13712yes—
319C22⋊C4—82no—
320C42—21no—
321C42—21no—
322C42—22no—
323C42—22no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42—42no—
329C42—42no—
330C42C2×#4112yes—
331C42H(5,𝔽2)12yes—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23#45912yes—
337C23#47012yes—
338C23#47312yes—
339C23C2×#12111yescharacteristic
340Q8—44no—
341Q8—44no—
342Q8—88no—
343Q8—82no—
344Q8—82no—
345Q8—82no—
346Q8—82no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—44no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—44no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—84no—
377C2×C4—88no—
378C2×C4—84no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4#45912yes—
388C2×C4#47012yes—
389C2×C4#52511yescharacteristic
390C2×C4#53511yescharacteristic
391C2×C4#54612yes—
392C2×C4#55112yes—
393C2×C4C22×D811yescharacteristic
394C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
395C2×C4C2×#13711yescharacteristic
396C2×C4C2×#13711yescharacteristic
397C2×C4C2×#13711yescharacteristic
398C2×C4#57511yescharacteristic
399C2×C4#57712yes—
400C8—24no—
401C8—42no—
402C8—81no—
403C8—82no—
404C8—81no—
405C8—82no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22#168914yes—
413C22#241111yescharacteristic
414C22C2×#45911yescharacteristic
415C22#304311yescharacteristic
416C22C2×#47011yescharacteristic
417C22#305611yescharacteristic
418C22C2×#47311yescharacteristic
419C22#306911yescharacteristic
420C4—22no—
421C4—22no—
422C4—22no—
423C4—41no—
424C4—42no—
425C4—84no—
426C4—81no—
427C4—81no—
428C4—81no—
429C4—161no—
430C4—161no—
431C4#311311yescharacteristic
432C4#311711yescharacteristic
433C4#321511yescharacteristic
434C4#321811yescharacteristic
435C4#329412yes—
436C4#331012yes—
437C2—24no—
438C2C2×#168911yescharacteristic
439C2#1916311yescharacteristic
440C2#1943511yescharacteristic
441C2#1947111yescharacteristic
442C2#2447611yescharacteristic
443C2#2450811yescharacteristic
444C2#2452211yescharacteristic
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. #10588C2semidirect
6. #10538C2semidirect
7. #9742C2semidirect
8. #9569C2semidirect

Maximal subgroups

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

subgroupindex
#331462
#257222
#226032
C2×#19882
#105882
#105382
#97422
#95692
#91292
#88302
#83402