Finite Group Databasebeta

← #206892

Subgroups — #206892

Order 512.

922 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2863C211yesmaximal, characteristic
2#20810C211yesmaximal, characteristic
3C4×#431C211yesmaximal, characteristic
4#9880C214yesmaximal
5#9879C212yesmaximal
6#9878C212yesmaximal
7#7916C214yesmaximal
8C2×#432C2212yes—
9C2×#431C2212yes—
10#2863—21no—
11#2863—21no—
12#2863—21no—
13#2863—21no—
14#2863—21no—
15#2863—21no—
16#2863—21no—
17#2863—21no—
18C2×#391C411yescharacteristic
19C2×#391C411yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#389C411yescharacteristic
22C2×#389C411yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#387C412yes—
25C2×#387C412yes—
26C2×#387C2212yes—
27C2×#383C2212yes—
28C2×C43C2211yescharacteristic
29#1842—21no—
30#1831—21no—
31#1797C2212yes—
32#1797C2212yes—
33#1767—21no—
34#1767—21no—
35C4×C4⋊C8—21no—
36C4×C4⋊C8—21no—
37C4×C22⋊C8C2212yes—
38C4×C22⋊C8C2212yes—
39#1744—21no—
40C42×C8—21no—
41#1280C2218yes—
42#1280C2218yes—
43C23×Q8C2×C411yescharacteristic
44C22×C4⋊C4C2×C412yes—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C42C2×C411yescharacteristic
51C22×C42C2×C411yescharacteristic
52C22×C42C2×C411yescharacteristic
53C22×C42C2×C411yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57#432—28no—
58#431—24no—
59#431Q818yescomplemented
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C22⋊C8C2314yes—
65C2×C22⋊C8C2314yes—
66C2×#97—22no—
67C2×C4×C8—22no—
68C2×C4×C8—22no—
69C2×C4×C8—22no—
70#391—21no—
71#391—21no—
72#391—21no—
73#391—21no—
74#391—21no—
75#391—21no—
76#391—21no—
77#391—21no—
78#391—21no—
79#391—21no—
80#391—21no—
81#391—21no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#383—22no—
107#383—22no—
108#383—22no—
109#383—22no—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113C2×#95C2×C412yes—
114C2×#95C2×C412yes—
115C2×#95C2×C412yes—
116C2×#95C2×C412yes—
117C2×#95C2312yes—
118C2×#95C2312yes—
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125C43D411yescharacteristic
126C43D411yescharacteristic
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—24no—
134C22×Q8—42no—
135C22×Q8—42no—
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C412yes—
141C23×C4C22×C412yes—
142C23×C4C22×C412yes—
143C23×C4C22×C412yes—
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C2411yescharacteristic
147C22×C8—24no—
148C22×C8—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—44no—
167C2×C42—24no—
168C2×C42—24no—
169C2×C42—42no—
170C2×C42—42no—
171C2×C42—42no—
172C2×C42—42no—
173C2×C42C22⋊C411yescharacteristic
174C2×C42C22⋊C411yescharacteristic
175C2×C42C22⋊C411yescharacteristic
176C2×C42C22⋊C411yescharacteristic
177C2×C42C22⋊C411yescharacteristic
178C2×C42C22⋊C411yescharacteristic
179C2×C42C22⋊C411yescharacteristic
180C2×C42C22⋊C411yescharacteristic
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×Q814yes—
202C2×C42#4114yes—
203C4⋊C8—28no—
204C4⋊C8—44no—
205C4⋊C8—44no—
206C4⋊C8—44no—
207C22⋊C8—24no—
208C22⋊C8—28no—
209C22⋊C8C2×Q818yes—
210#97—44no—
211C4×C8—28no—
212C4×C8—44no—
213C4×C8—44no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232C24C23×C411yescharacteristic
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—88no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
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—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C4×Q814yes—
285C22×C4C4×Q814yes—
286C22×C4#13114yes—
287C22×C4#13114yes—
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×Q812yes—
291C22×C4C2×#4112yes—
292C2×C8—28no—
293C2×C8—44no—
294C2×C8—44no—
295C2×C8—44no—
296C2×C8—44no—
297C2×C8—44no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—42no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42—41no—
333C42—41no—
334C42C22⋊Q814yes—
335C42C22⋊Q814yes—
336C42#12412yes—
337C42#12412yes—
338C42C4⋊Q812yes—
339C42C4⋊Q812yes—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23C22×C22⋊C411yescharacteristic
345C23C2×C4×Q812yes—
346C23C2×#13112yes—
347C23C2×#13112yes—
348Q8—84no—
349Q8—84no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—28no—
373C2×C4—28no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—84no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4#39112yes—
403C2×C4#39112yes—
404C2×C4#39112yes—
405C2×C4#39112yes—
406C2×C4#40812yes—
407C2×C4#40812yes—
408C2×C4#40812yes—
409C2×C4#40812yes—
410C2×C4#41012yes—
411C2×C4#41012yes—
412C2×C4#41312yes—
413C2×C4#41312yes—
414C2×C4C2×C22⋊Q812yes—
415C2×C4C2×#12411yescharacteristic
416C2×C4C2×C4⋊Q811yescharacteristic
417C8—48no—
418C8—84no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22C2×#39111yescharacteristic
426C22C2×#40811yescharacteristic
427C22C2×#40811yescharacteristic
428C22C2×#41011yescharacteristic
429C22C2×#41011yescharacteristic
430C22C2×#41311yescharacteristic
431C22C2×#41311yescharacteristic
432C22#296014yes—
433C4—22no—
434C4—22no—
435C4—22no—
436C4—21no—
437C4—21no—
438C4—22no—
439C4—22no—
440C4—22no—
441C4—22no—
442C4—22no—
443C4—22no—
444C4—22no—
445C4—41no—
446C4—41no—
447C4—82no—
448C4—82no—
449C4—82no—
450C4—82no—
451C4#147314yes—
452C2—24no—
453C2C2×#147311yescharacteristic
454C2#1738011yescharacteristic
455C2#1738211yescharacteristic
456C2#1738311yescharacteristic
457C2#2130011yescharacteristic
458C2#2133211yescharacteristic
459C2#2133711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
59. #431Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28632
#208102
C4×#4312
#98802
#98792
#98782
#79162