Finite Group Databasebeta

← #369241

Subgroups — #369241

Order 512.

1370 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35341C211yesmaximal, characteristic
2#34217C211yesmaximal, characteristic, complemented
3C2×#1998C211yesmaximal, characteristic, complemented
4#21252C211yesmaximal, characteristic
5#21209C211yesmaximal, characteristic
6#11072C214yesmaximal, complemented
7#10949C214yesmaximal, complemented
8#10836C211yesmaximal, characteristic, complemented
9C4×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×#125C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C2×#421C2212yes—
14C2×#421C2214yes—
15C2×#421C2212yes—
16C2×C4×M16C2211yescharacteristic
17#2861C414yescomplemented
18#2642C412yescomplemented
19#2641C412yescomplemented
20#2591—24no—
21#2562—24no—
22#2370—24no—
23#2345—24no—
24#2330—22no—
25#2316C2212yes—
26#2311C2214yes—
27C4×#125—22no—
28#2285C2211yescharacteristic
29#2274C2212yes—
30C4×#117C2211yescharacteristic
31C4×#117C2211yescharacteristic
32#1998—24no—
33#1980—24no—
34#1963—22no—
35#1962—22no—
36C2×#343C2214yes—
37C2×H(3,ℤ/4)C2212yes—
38C2×H(3,ℤ/4)C2212yes—
39C2×H(3,ℤ/4)C2214yes—
40C22×#41C2312yes—
41C22×M16C2312yes—
42C22×M16C2312yes—
43#532—21no—
44#532—21no—
45#532—22no—
46#532—21no—
47#532—21no—
48#532—22no—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#127C2×C411yescharacteristic
51C2×#125—21no—
52C2×#125—21no—
53C2×#125—22no—
54C2×#125—21no—
55C2×#125—21no—
56C2×#125—22no—
57C2×#125C2×C412yes—
58C2×#125C2×C412yes—
59C2×#125D414yes—
60C2×#124C2×C412yes—
61C2×#121—22no—
62C2×C4×D4—22no—
63C2×#117C2312yes—
64C2×#117C2311yescharacteristic
65C2×#117C2312yes—
66C2×#117C2311yescharacteristic
67C22×C4⋊C4D414yes—
68C22×C4⋊C4C2312yes—
69C22×C42C2311yescharacteristic
70#423—21no—
71#423—21no—
72#423—22no—
73C2×C4⋊C8—22no—
74#421—28no—
75#421—28no—
76#421—44no—
77#421—44no—
78C2×#104—24no—
79C2×#104—24no—
80C2×#104—28no—
81C4×M16—22no—
82C4×M16—22no—
83C2×#97—21no—
84C2×C4×C8—21no—
85#400—28no—
86#394—28no—
87#391C2×C418yes—
88#390C2×C418yes—
89#389—28no—
90#389—24no—
91#389—24no—
92#388—216no—
93#387—28no—
94#387—28no—
95#386—28no—
96#384—21no—
97#382—21no—
98#380—24no—
99#380—22no—
100#380—28no—
101C4×C4⋊C4—28no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4D414yes—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—24no—
110#376C2×C412yes—
111#376D412yes—
112C43C2×C412yes—
113C43D412yes—
114#343—28no—
115H(3,ℤ/4)—24no—
116H(3,ℤ/4)—24no—
117H(3,ℤ/4)—44no—
118C2×#41—22no—
119C2×#41—22no—
120C2×#41—22no—
121C2×#41—22no—
122C2×#41—24no—
123C2×#41—24no—
124C2×#41—24no—
125C2×#41—24no—
126C22×Q8C22×C412yes—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4C22×C412yes—
130C23×C4—22no—
131C23×C4C2×D414yes—
132C23×C4C2×D412yes—
133C23×C4C2411yescharacteristic
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—24no—
142C2×M16—24no—
143C2×M16#4114yes—
144C22×C8—22no—
145C22×C8—22no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—22no—
149#127—21no—
150#127—21no—
151#127—22no—
152#125—22no—
153#125—22no—
154#125—24no—
155#125—22no—
156#125—22no—
157#125—24no—
158#125—24no—
159#125—24no—
160#125—24no—
161#125—24no—
162#124—22no—
163#124—22no—
164#124—24no—
165#121—42no—
166#121—42no—
167#121—42no—
168#121—42no—
169C4×D4—42no—
170C4×D4—42no—
171C4×D4—42no—
172C4×D4—42no—
173#117—22no—
174#117—22no—
175#117—22no—
176#117—22no—
177#117—24no—
178#117—22no—
179#117—24no—
180#117—22no—
181#117—22no—
182#117#4114yes—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4C22⋊C418yes—
199C2×C4⋊C4C22⋊C418yes—
200C2×C4⋊C4C22×C414yes—
201C2×C4⋊C4C2×D414yes—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—216no—
208C2×C42—28no—
209C2×C42—24no—
210C2×C42—24no—
211C2×C42C22×C414yes—
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42C2×D414yes—
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42#4111yescharacteristic
222C2×C42#4111yescharacteristic
223C2×C42#4112yes—
224C4⋊C8—42no—
225C4⋊C8—42no—
226#104—416no—
227#104—48no—
228#104—48no—
229#97—42no—
230C4×C8—42no—
231#95—28no—
232#95—28no—
233#95—24no—
234#95—216no—
235#95—24no—
236#95—24no—
237C24—22no—
238C24C22×D411yescharacteristic
239#41—42no—
240#41—44no—
241#41—42no—
242#41—42no—
243#41—42no—
244#41—44no—
245#41—44no—
246#41—44no—
247#41—44no—
248#41—44no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—84no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—82no—
276C22×C4—84no—
277C22×C4C2×C22⋊C418yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4#12012yes—
281C22×C4#12012yes—
282C22×C4#12312yes—
283C22×C4#12312yes—
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C2×#4111yescharacteristic
289C22×C4C2×#4111yescharacteristic
290C22×C4C2×#4112yes—
291C22×C4C2×#4112yes—
292C22×C4H(5,𝔽2)12yes—
293C22×C42−1+412yes—
294M16—24no—
295M16—24no—
296M16—42no—
297M16—42no—
298M16—42no—
299M16—42no—
300M16—44no—
301C2×C8—24no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—44no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—28no—
312C4⋊C4—28no—
313C4⋊C4—28no—
314C4⋊C4—28no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—84no—
323C4⋊C4—88no—
324C22⋊C4—24no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—816no—
332C42—22no—
333C42—24no—
334C42—24no—
335C42—22no—
336C42—22no—
337C42—28no—
338C42—84no—
339C42—84no—
340C42—88no—
341C42C4×D411yescharacteristic
342C42C4×D411yescharacteristic
343C42C4×D412yes—
344C42C4×D411yescharacteristic
345C42C4×D411yescharacteristic
346C42C4×D412yes—
347C42#12112yes—
348C42#12112yes—
349C42#12112yes—
350C42#12112yes—
351C42#12414yes—
352C23—42no—
353C23—48no—
354C23—44no—
355C23—42no—
356C23C22×C22⋊C411yescharacteristic
357C23C2×#12011yescharacteristic
358C23C2×#12311yescharacteristic
359C23#54812yes—
360C23#55012yes—
361Q8—42no—
362Q8—44no—
363Q8—42no—
364D4—42no—
365D4—44no—
366D4—42no—
367D4—84no—
368D4—84no—
369D4—82no—
370D4—82no—
371D4—84no—
372C2×C4—24no—
373C2×C4—28no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—48no—
382C2×C4—48no—
383C2×C4—44no—
384C2×C4—48no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—84no—
393C2×C4—84no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4#38612yes—
397C2×C4#38612yes—
398C2×C4#38612yes—
399C2×C4#38612yes—
400C2×C4C2×C4×D411yescharacteristic
401C2×C4C2×C4×D411yescharacteristic
402C2×C4C4×#4112yes—
403C2×C4#52012yes—
404C2×C4C2×#12111yescharacteristic
405C2×C4C2×#12111yescharacteristic
406C2×C4C2×#12411yescharacteristic
407C2×C4#52912yes—
408C2×C4#53211yescharacteristic
409C2×C4#53511yescharacteristic
410C2×C4#53611yescharacteristic
411C2×C4#54212yes—
412C2×C4#54612yes—
413C2×C4#54812yes—
414C2×C4#54812yes—
415C2×C4#55012yes—
416C2×C4#55612yes—
417C2×C4#55912yes—
418C8—44no—
419C8—82no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—42no—
424C22—42no—
425C22—88no—
426C22—84no—
427C22—82no—
428C22C2×#38611yescharacteristic
429C22#234112yes—
430C22#243111yescharacteristic
431C22#243611yescharacteristic
432C22#244812yes—
433C4—22no—
434C4—22no—
435C4—22no—
436C4—22no—
437C4—44no—
438C4—48no—
439C4—84no—
440C4—82no—
441C4—84no—
442C4—82no—
443C4C4×#12411yescharacteristic
444C4#232611yescharacteristic
445C4#236712yes—
446C4#242612yes—
447C4#254511yescharacteristic
448C4#254811yescharacteristic
449C2—21no—
450C2—21no—
451C2—22no—
452C2—22no—
453C2—162no—
454C2#1402611yescharacteristic
455C2#2101212yes—
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34217C2semidirect
3. C2×#1998C2semidirect
6. #11072C2semidirect
7. #10949C2semidirect
8. #10836C2semidirect
9. C4×H(3,ℤ/4)C2semidirect
17. #2861C4semidirect
18. #2642C4semidirect
19. #2641C4semidirect

Maximal subgroups

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

subgroupindex
#353412
#342172
C2×#19982
#212522
#212092
#110722
#109492
#108362
C4×H(3,ℤ/4)2