Finite Group Databasebeta

← #416524

Subgroups — #416524

Order 512.

1354 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#486C211yesmaximal, characteristic
2C2×#1997C211yesmaximal, characteristic, complemented
3#21324C211yesmaximal, characteristic, complemented
4C2×#1965C211yesmaximal, characteristic
5#21203C211yesmaximal, characteristic
6#21143C211yesmaximal, characteristic, complemented
7#11163C214yesmaximal, complemented
8#11061C214yesmaximal, complemented
9C4×#341C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×#127C2211yescharacteristic
12C2×#486—22no—
13C2×#486—22no—
14C2×#486—24no—
15C2×#486—24no—
16C2×#424C2212yes—
17C2×#423C2211yescharacteristic
18C2×#418C2214yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×#100C2214yes—
22C2×C4×M16C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2285C2211yescharacteristic
26#1997—24no—
27#1993—22no—
28#1990—22no—
29#1965—24no—
30#1961—22no—
31#1957—22no—
32#1939—22no—
33#1939—22no—
34C2×#343C2214yes—
35C2×#341C2212yes—
36C2×#341C2212yes—
37#1736C2214yes—
38#1729C2212yes—
39C23×Q8—21no—
40C23×D4—21no—
41C23×D4—21no—
42C23×D4C2312yes—
43C22×M16C2312yes—
44C22×M16C2312yes—
45C22×M16C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8D414yes—
56C2×#127—22no—
57C2×#127—21no—
58C2×#127—21no—
59C2×#127—21no—
60C2×#127—21no—
61C2×#127—22no—
62C2×#127—22no—
63C2×#127D414yes—
64C2×#117C2312yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2312yes—
67C22×C42C2311yescharacteristic
68#486—216no—
69#486—216no—
70#424—44no—
71#423—22no—
72#423—22no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76#418—28no—
77#418—44no—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—28no—
85C2×#100—28no—
86C2×#100—28no—
87C2×#100—44no—
88C4×M16—22no—
89C4×M16—22no—
90C4×M16—22no—
91C4×M16—22no—
92C2×#97—21no—
93C2×#97—21no—
94C2×C4×C8—22no—
95C2×C4×C8—22no—
96C2×C4×C8—21no—
97C2×C4×C8—21no—
98C2×C4×C8D412yes—
99C2×C4×C8D412yes—
100#384D412yes—
101#382D412yes—
102#380—21no—
103#380—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106#343—28no—
107#341—24no—
108#341—24no—
109#340—28no—
110#336—24no—
111C25—22no—
112C22×Q8—24no—
113C22×Q8—41no—
114C22×Q8—41no—
115C22×Q8—42no—
116C22×Q8—84no—
117C22×D4—24no—
118C22×D4—24no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—22no—
126C22×D4—24no—
127C22×D4—24no—
128C22×D4—41no—
129C22×D4—41no—
130C22×D4—41no—
131C22×D4—41no—
132C22×D4—42no—
133C22×D4—42no—
134C22×D4—84no—
135C22×D4—84no—
136C22×D4C2×D414yes—
137C23×C4—22no—
138C23×C4#4111yescharacteristic
139C23×C4#4111yescharacteristic
140C23×C4C2411yescharacteristic
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—24no—
154C2×M16—24no—
155C2×M16#4114yes—
156C2×M16#4114yes—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8C2×D412yes—
169C22×C8C2×D412yes—
170C4⋊Q8—24no—
171C4⋊Q8—24no—
172C4⋊Q8—24no—
173C4⋊Q8—24no—
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176C4⋊Q8—42no—
177C4⋊Q8—42no—
178#127—24no—
179#127—24no—
180#127—24no—
181#127—24no—
182#127—42no—
183#127—42no—
184#127—42no—
185#127—42no—
186#117—24no—
187#117—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4C2×D414yes—
200C2×C4⋊C4C2×D414yes—
201C2×C22⋊C4—22no—
202C2×C42—22no—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42#4112yes—
210C2×C42#4112yes—
211C4⋊C8—48no—
212C4⋊C8—42no—
213C4⋊C8—42no—
214#102—216no—
215#102—216no—
216#102—416no—
217#100—416no—
218#97—42no—
219#97—42no—
220C4×C8—24no—
221C4×C8—24no—
222C4×C8—42no—
223C4×C8—42no—
224#95—22no—
225#95—22no—
226C24—24no—
227C24—42no—
228C24—42no—
229C24—44no—
230C24—44no—
231C24—84no—
232C24—84no—
233C24C2×#4111yescharacteristic
234C2×Q8—44no—
235C2×Q8—44no—
236C2×Q8—88no—
237C2×Q8—84no—
238C2×Q8—88no—
239C2×Q8—84no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—28no—
243C2×D4—28no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—44no—
262C2×D4—84no—
263C2×D4—84no—
264C2×D4—88no—
265C2×D4—88no—
266C2×D4—88no—
267C2×D4—88no—
268C2×D4—84no—
269C2×D4—84no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—84no—
277C22×C4#12111yescharacteristic
278C22×C4#12111yescharacteristic
279C22×C4#12111yescharacteristic
280C22×C4#12111yescharacteristic
281C22×C4#12312yes—
282C22×C4#12312yes—
283C22×C4#12312yes—
284C22×C4#12312yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C2×#4111yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C22×C4C2×#4112yes—
292C22×C4C2×#4112yes—
293C22×C4H(5,𝔽2)12yes—
294C22×C42−1+412yes—
295M16—24no—
296M16—24no—
297M16—24no—
298M16—24no—
299M16—42no—
300M16—42no—
301M16—42no—
302M16—42no—
303M16—44no—
304M16—44no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—28no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C22⋊C4—44no—
334C42—42no—
335C42—42no—
336C42#12012yes—
337C42#12012yes—
338C42#12112yes—
339C42#12112yes—
340C42#12112yes—
341C42#12112yes—
342C42#12412yes—
343C42#12412yes—
344C23—22no—
345C23—22no—
346C23—44no—
347C23—48no—
348C23—84no—
349C23—84no—
350C23—84no—
351C23—84no—
352C23—84no—
353C23—82no—
354C23—82no—
355C23—84no—
356C23—84no—
357C23—168no—
358C23—168no—
359C23C2×#12111yescharacteristic
360C23C2×#12311yescharacteristic
361C23C2×#12311yescharacteristic
362C23#55612yes—
363C23#55912yes—
364Q8—88no—
365Q8—88no—
366D4—48no—
367D4—44no—
368D4—44no—
369D4—88no—
370D4—88no—
371D4—82no—
372D4—82no—
373D4—84no—
374D4—88no—
375D4—88no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4#39412yes—
390C2×C4#39412yes—
391C2×C4#39412yes—
392C2×C4#39412yes—
393C2×C4#47114yes—
394C2×C4#47214yes—
395C2×C4C2×#12011yescharacteristic
396C2×C4C2×#12111yescharacteristic
397C2×C4C2×#12111yescharacteristic
398C2×C4C2×#12411yescharacteristic
399C2×C4#52912yes—
400C2×C4#52912yes—
401C2×C4#53211yescharacteristic
402C2×C4#53211yescharacteristic
403C2×C4#53511yescharacteristic
404C2×C4#53611yescharacteristic
405C2×C4#54212yes—
406C2×C4#54212yes—
407C2×C4#54612yes—
408C2×C4#54612yes—
409C2×C4#54712yes—
410C2×C4#54812yes—
411C2×C4#54812yes—
412C2×C4#55012yes—
413C8—44no—
414C8—44no—
415C8—42no—
416C8—42no—
417C8—82no—
418C22—24no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—24no—
424C22—84no—
425C22—84no—
426C22—84no—
427C22—168no—
428C22—168no—
429C22—164no—
430C22—164no—
431C22C2×#39411yescharacteristic
432C22#244712yes—
433C22#244812yes—
434C22#254411yescharacteristic
435C22#254511yescharacteristic
436C22C2×#47112yes—
437C22C2×#47212yes—
438C4—22no—
439C4—22no—
440C4—84no—
441C4—84no—
442C4#204514yes—
443C4#204614yes—
444C4#242612yes—
445C4#242712yes—
446C4#243111yescharacteristic
447C4#243211yescharacteristic
448C4#264011yescharacteristic
449C4#286111yescharacteristic
450C2—22no—
451C2—22no—
452C2—164no—
453C2#1522611yescharacteristic
454C2C2×#204511yescharacteristic
455C2C2×#204611yescharacteristic
456C2#2863212yes—
457C2#2867012yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1997C2semidirect
3. #21324C2semidirect
6. #21143C2semidirect
7. #11163C2semidirect
8. #11061C2semidirect
9. C4×#341C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4862
C2×#19972
#213242
C2×#19652
#212032
#211432
#111632
#110612
C4×#3412