Finite Group Databasebeta

← #390609

Subgroups — #390609

Order 512.

894 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23638C211yesmaximal, characteristic
2#21050C212yesmaximal
3#11291C212yesmaximal, complemented
4#11258C212yesmaximal, complemented
5#11097C211yesmaximal, characteristic
6#10909C212yesmaximal, complemented
7#10887C212yesmaximal, complemented
8#10720C212yesmaximal
9#10713C211yesmaximal, characteristic
10C22×#117C2211yescharacteristic
11C22×#108C2211yescharacteristic
12C2×#427C2211yescharacteristic
13C2×#424C2212yes—
14C2×#407C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C2×#405C2212yes—
18#2280C2212yes—
19#2077C412yescomplemented
20#2077C412yescomplemented
21#2076C411yescharacteristic, complemented
22#2076C411yescharacteristic, complemented
23#2075C411yescharacteristic, complemented
24#2075C411yescharacteristic, complemented
25#1924—22no—
26#1924—22no—
27#1923—22no—
28#1923—22no—
29C2×#341C2212yes—
30C2×#341C2211yescharacteristic
31#1735C2211yescharacteristic
32#1735C2212yes—
33C2×#339C2212yes—
34C2×#339C2212yes—
35C2×#339C2212yes—
36C2×#339C2212yes—
37#1729C2212yes—
38#1729C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41C24×C4C2311yescharacteristic
42C22×M16C2312yes—
43C22×M16C2312yes—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C2×#117—24no—
48C2×#117—41no—
49C2×#117—41no—
50C2×#117—42no—
51C2×#117C2312yes—
52C2×#117C2312yes—
53C22×C4⋊C4—21no—
54C22×C22⋊C4—21no—
55C22×C42C2312yes—
56C2×#108—42no—
57C2×#108—41no—
58#427—41no—
59#427—41no—
60C2×#107C2×C411yescharacteristic
61C2×#107C2×C411yescharacteristic
62C2×#106C2×C412yes—
63#424—24no—
64#424—42no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67#407—24no—
68#407—42no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—41no—
72C2×C22⋊C8—41no—
73C2×C22⋊C8—41no—
74C2×C22⋊C8—41no—
75C2×C22⋊C8C2×C411yescharacteristic
76C2×C22⋊C8C2×C411yescharacteristic
77C2×C22⋊C8C2×C411yescharacteristic
78C2×C22⋊C8C2×C411yescharacteristic
79C2×C22⋊C8D411yescharacteristic
80C2×C22⋊C8D412yes—
81C2×C22⋊C8D411yescharacteristic
82#405—42no—
83#405—42no—
84C2×#97—22no—
85C2×C4×C8—22no—
86#381—22no—
87#380C2×C412yes—
88#380C2×C412yes—
89#380C2×C412yes—
90#380C2×C412yes—
91#379—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94#341—24no—
95#341—41no—
96#340C2×C411yescharacteristic
97#340C2×C411yescharacteristic
98#340C2×C412yes—
99#340C2×C412yes—
100#340C2×C412yes—
101#340C2×C412yes—
102#340C2×C412yes—
103#339—24no—
104#339—42no—
105#339—42no—
106#339—42no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111#336—24no—
112#336—42no—
113C25#4111yescharacteristic
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4#4111yescharacteristic
128C23×C4C2411yescharacteristic
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16#4114yes—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8C22×C411yescharacteristic
158C22×C8C22×C411yescharacteristic
159C22×C8C22×C411yescharacteristic
160C22×C8C22×C411yescharacteristic
161C22×C8C2×D412yes—
162C22×C8C2×D412yes—
163#117—22no—
164#117—22no—
165#117—22no—
166#117—22no—
167#117—22no—
168#117—22no—
169#117—22no—
170#117—22no—
171#117—44no—
172#117—48no—
173#117—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—42no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42#4114yes—
196#108—44no—
197#107—41no—
198#107—41no—
199#106—42no—
200C4⋊C8—44no—
201C4⋊C8—44no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204C22⋊C8—41no—
205C22⋊C8—41no—
206C22⋊C8—41no—
207C22⋊C8—42no—
208C22⋊C8—41no—
209C22⋊C8—44no—
210C22⋊C8—44no—
211#97—44no—
212C4×C8—44no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217C24—21no—
218C24—21no—
219C24—22no—
220C24—22no—
221C24—42no—
222C24—41no—
223C24—41no—
224C24C4×D411yescharacteristic
225C24C4×D411yescharacteristic
226C24C2×#4111yescharacteristic
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4C4×D411yescharacteristic
269C22×C4C4×D411yescharacteristic
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C2×#4112yes—
273C22×C4C2×#4112yes—
274C22×C4H(5,𝔽2)11yescharacteristic
275C22×C4H(5,𝔽2)11yescharacteristic
276C22×C4H(5,𝔽2)11yescharacteristic
277C22×C4H(5,𝔽2)12yes—
278C22×C42−1+411yescharacteristic
279C22×C42−1+412yes—
280M16—24no—
281M16—24no—
282M16—42no—
283M16—42no—
284M16—42no—
285M16—42no—
286M16—42no—
287M16—42no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—24no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—41no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—48no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—48no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C42—24no—
337C42—24no—
338C42—42no—
339C42—42no—
340C42—42no—
341C42—42no—
342C42—42no—
343C42—42no—
344C23—22no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—41no—
362C23—42no—
363C23—41no—
364C23—82no—
365C23—82no—
366C23—82no—
367C23—82no—
368C23C2×C4×D411yescharacteristic
369C23#53811yescharacteristic
370C23#53911yescharacteristic
371C23#55112yes—
372C23#55212yes—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4C4×#4112yes—
412C2×C4#51811yescharacteristic
413C2×C4#52011yescharacteristic
414C2×C4#52012yes—
415C2×C4#53411yescharacteristic
416C2×C4#53711yescharacteristic
417C2×C4#53812yes—
418C2×C4#53912yes—
419C2×C4#54112yes—
420C2×C4#54212yes—
421C2×C4#54612yes—
422C2×C4#54712yes—
423C2×C4#55111yescharacteristic
424C2×C4#55312yes—
425C2×C4#55911yescharacteristic
426C2×C4H(3,𝔽4)11yescharacteristic
427C2×C4#56212yes—
428C2×C4#56311yescharacteristic
429C8—41no—
430C8—41no—
431C8—41no—
432C8—41no—
433C8—42no—
434C8—42no—
435C8—44no—
436C8—82no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—82no—
456C22—84no—
457C22—82no—
458C22#232511yescharacteristic
459C22#235912yes—
460C22#273611yescharacteristic
461C22#273912yes—
462C22#277111yescharacteristic
463C22#295512yes—
464C22#295711yescharacteristic
465C22#295711yescharacteristic
466C4—41no—
467C4—41no—
468C4—84no—
469C4—82no—
470C4—82no—
471C4—82no—
472C4—82no—
473C4#233412yes—
474C4#237411yescharacteristic
475C4#237911yescharacteristic
476C4#270812yes—
477C4#271211yescharacteristic
478C4#272211yescharacteristic
479C2—21no—
480C2—22no—
481C2—21no—
482C2—81no—
483C2—81no—
484C2#1457711yescharacteristic
485C2#2220511yescharacteristic
486C2#2220611yescharacteristic
487C2#2294912yes—
488C2#2295112yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #11291C2semidirect
4. #11258C2semidirect
6. #10909C2semidirect
7. #10887C2semidirect
19. #2077C4semidirect
20. #2077C4semidirect
21. #2076C4semidirect
22. #2076C4semidirect
23. #2075C4semidirect
24. #2075C4semidirect

Maximal subgroups

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

subgroupindex
#236382
#210502
#112912
#112582
#110972
#109092
#108872
#107202
#107132