Finite Group Databasebeta

← #404747

Subgroups — #404747

Order 512.

859 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2058C211yesmaximal, characteristic
2#21313C211yesmaximal, characteristic
3#21034C211yesmaximal, characteristic
4#20368C211yesmaximal, characteristic
5#11382C211yesmaximal, characteristic
6#11326C211yesmaximal, characteristic
7#11271C212yesmaximal
8#11249C212yesmaximal
9#11204C211yesmaximal, characteristic
10#10869C212yesmaximal
11#10827C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#108C2212yes—
14C2×#428C2211yescharacteristic
15C22×#106C2212yes—
16C2×#423C2211yescharacteristic
17C2×#405C2212yes—
18C2×#389C2211yescharacteristic
19C4×#117C2211yescharacteristic
20#2058—21no—
21#2058—21no—
22#2058—22no—
23#1988—21no—
24#1988—21no—
25#1987—21no—
26#1987—21no—
27#1921—21no—
28#1920—21no—
29#1919—21no—
30#1918—21no—
31#1746—22no—
32#1745—21no—
33#1744—21no—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#341C2212yes—
39#1736C2211yescharacteristic
40#1736C2212yes—
41#1736C2211yescharacteristic
42#1736C2211yescharacteristic
43#1736C2212yes—
44C2×#340C2212yes—
45#1729C2212yes—
46#1728C2211yescharacteristic
47#1728C2211yescharacteristic
48C2×#336C2211yescharacteristic
49C23×Q8—21no—
50C22×M16C2312yes—
51C22×M16C2311yescharacteristic
52C22×M16C2312yes—
53C22×M16C2311yescharacteristic
54C23×C8C2312yes—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—42no—
61C2×#117C2311yescharacteristic
62C2×#117C2311yescharacteristic
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×#108—24no—
70C2×#108—24no—
71C2×#108—42no—
72#428—22no—
73#428—22no—
74C2×#107—21no—
75C2×#106—21no—
76C2×#106—22no—
77C2×#106—22no—
78C2×#106—24no—
79C2×#106—42no—
80#423—42no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83#405—24no—
84#405Q818yescomplemented
85C2×#97—22no—
86C2×C4×C8—22no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91C4×C4⋊C4D411yescharacteristic
92C4×C4⋊C4D411yescharacteristic
93C4×C4⋊C4D411yescharacteristic
94C4×C4⋊C4D411yescharacteristic
95C4×C22⋊C4—21no—
96C4×C22⋊C4—21no—
97#376—21no—
98C2×#95—21no—
99C43—21no—
100#343—24no—
101#343—24no—
102#343—42no—
103#343—42no—
104#341—24no—
105#340—22no—
106#340—22no—
107#340—21no—
108#340—21no—
109#340—21no—
110#340—21no—
111#340—21no—
112#340—21no—
113#340—22no—
114#340—22no—
115#340—22no—
116#340—22no—
117#336—22no—
118#336—22no—
119#336—21no—
120#336—21no—
121#336—21no—
122#336—21no—
123#336D411yescharacteristic
124#336D411yescharacteristic
125#336D412yes—
126C22×Q8—44no—
127C22×Q8—41no—
128C22×Q8—41no—
129C22×Q8—82no—
130C22×Q8—82no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4#4111yescharacteristic
135C23×C4#4111yescharacteristic
136C23×C4C2411yescharacteristic
137C2×M16—22no—
138C2×M16—21no—
139C2×M16—22no—
140C2×M16—22no—
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—21no—
154C2×M16—21no—
155C2×M16C2×Q814yes—
156C2×M16#4112yes—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8C2×D412yes—
166C22×C8C2×D412yes—
167C22×C8C2×Q814yes—
168C22×C8#4114yes—
169C4⋊Q8—24no—
170C4⋊Q8—24no—
171C4⋊Q8—44no—
172C4⋊Q8—44no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175#117—22no—
176#117—21no—
177#117—21no—
178#117—21no—
179#117—21no—
180#117C2×Q814yes—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4#4114yes—
206C2×C4⋊C4#4112yes—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C42—21no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221#108—48no—
222#107—24no—
223#106—24no—
224#106—44no—
225#106—44no—
226C4⋊C8—42no—
227C4⋊C8—42no—
228#97—28no—
229C4×C8—28no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—41no—
235#95—41no—
236C24C2×#4111yescharacteristic
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—82no—
240C2×Q8—82no—
241C2×Q8—82no—
242C2×Q8—816no—
243C2×Q8—82no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4#12111yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×Q812yes—
275C22×C4C2×#4112yes—
276C22×C4C2×#4112yes—
277C22×C4C2×#4111yescharacteristic
278C22×C4C2×#4111yescharacteristic
279C22×C4H(5,𝔽2)12yes—
280C22×C42−1+411yescharacteristic
281C22×C42−1+411yescharacteristic
282M16—24no—
283M16—24no—
284M16—24no—
285M16—42no—
286M16—42no—
287M16—42no—
288M16—42no—
289M16—42no—
290M16—42no—
291M16—42no—
292M16—42no—
293M16—42no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—22no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—41no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—41no—
314C2×C8C22⋊Q814yes—
315C2×C8C22⋊Q814yes—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—24no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4C4⋊Q812yes—
338C4⋊C4C4⋊Q812yes—
339C22⋊C4—24no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C42—24no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—22no—
348C42—22no—
349C42—41no—
350C42—41no—
351C42—42no—
352C42—42no—
353C42—81no—
354C42—81no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23C2×#12111yescharacteristic
360C23#53312yes—
361C23#55112yes—
362C23#55211yescharacteristic
363C23#55611yescharacteristic
364Q8—88no—
365Q8—88no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—44no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—82no—
393C2×C4—84no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—84no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—162no—
404C2×C4#47112yes—
405C2×C4#47112yes—
406C2×C4C2×C22⋊Q812yes—
407C2×C4#52511yescharacteristic
408C2×C4#52511yescharacteristic
409C2×C4C2×C4⋊Q811yescharacteristic
410C2×C4#53211yescharacteristic
411C2×C4#53312yes—
412C2×C4#53312yes—
413C2×C4#53511yescharacteristic
414C2×C4#53611yescharacteristic
415C2×C4#54412yes—
416C2×C4#54412yes—
417C2×C4#54612yes—
418C2×C4#54711yescharacteristic
419C2×C4#54811yescharacteristic
420C2×C4#55112yes—
421C2×C4#55212yes—
422C2×C4#55512yes—
423C2×C4#55612yes—
424C8—24no—
425C8—24no—
426C8—42no—
427C8—44no—
428C8—81no—
429C8—82no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22#241111yescharacteristic
439C22#241511yescharacteristic
440C22#260512yes—
441C22#261212yes—
442C22#285711yescharacteristic
443C22C2×#47111yescharacteristic
444C22C2×#47111yescharacteristic
445C22#306811yescharacteristic
446C22#306911yescharacteristic
447C4—22no—
448C4—22no—
449C4—42no—
450C4—42no—
451C4—41no—
452C4—41no—
453C4—81no—
454C4—82no—
455C4—81no—
456C4—84no—
457C4—161no—
458C4#239911yescharacteristic
459C4#243812yes—
460C4#243912yes—
461C4#260411yescharacteristic
462C4#261011yescharacteristic
463C4#266011yescharacteristic
464C2—22no—
465C2—22no—
466C2#1490111yescharacteristic
467C2#2452111yescharacteristic
468C2#2452311yescharacteristic
469C2#2472111yescharacteristic
470C2#2472311yescharacteristic
471C2#3322011yescharacteristic
472C2#3322311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
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
84. #405Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20582
#213132
#210342
#203682
#113822
#113262
#112712
#112492
#112042
#108692
#108272