Finite Group Databasebeta

← #193241

Subgroups — #193241

Order 512.

1088 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34236C211yesmaximal, characteristic
2#21985C211yesmaximal, characteristic, complemented
3#21805C211yesmaximal, characteristic
4#21358C211yesmaximal, characteristic, complemented
5C2×#1543C211yesmaximal, characteristic
6#8929C212yesmaximal, complemented
7#8927C212yesmaximal, complemented
8#8921C212yesmaximal, complemented
9#8920C212yesmaximal, complemented
10#7894C211yesmaximal, characteristic, complemented
11#7892C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2369C2212yes—
17#2366C2212yes—
18#2361—22no—
19#2357—21no—
20#2349—21no—
21#2339—22no—
22#2337—22no—
23#2319—21no—
24C4×#121—22no—
25#2285C2211yescharacteristic
26#2283C2211yescharacteristic
27#2275—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1983—22no—
30#1957C2212yes—
31#1956C2212yes—
32#1948C2212yes—
33#1947C2212yes—
34#1947C2212yes—
35#1947C2212yes—
36#1854—22no—
37#1852—22no—
38#1846—21no—
39#1835—21no—
40#1788—22no—
41C4×#107—21no—
42C4×#107—21no—
43#1543C418yescomplemented
44C2×#333C2211yescharacteristic
45C2×#333C2211yescharacteristic
46C2×#331C2211yescharacteristic
47C2×#331C2211yescharacteristic
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52C22×#41D412yes—
53#532—22no—
54#532—22no—
55#532C2×C414yes—
56#532Q814yes—
57C2×C4⋊Q8C2311yescharacteristic
58C2×#127C2311yescharacteristic
59C2×#124D412yes—
60C2×#121—21no—
61C2×#121—21no—
62C2×C4×D4—24no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×#117—21no—
68C22×C4⋊C4D412yes—
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71#431—22no—
72#431—22no—
73#431—22no—
74#431—22no—
75#431C2×C414yescomplemented
76#431C2×C414yescomplemented
77C2×#107—22no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#106—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×#103—24no—
87C2×#102—24no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93C2×C4×C8C2312yes—
94#391—24no—
95#390—22no—
96#390—22no—
97#389—22no—
98#389—22no—
99#388—24no—
100#388—24no—
101#386—24no—
102#386—24no—
103#385—24no—
104#384C2312yes—
105#383—21no—
106#382—21no—
107#382D411yescharacteristic
108#382D411yescharacteristic
109#381—22no—
110#381—21no—
111#380—22no—
112#380—21no—
113#380—21no—
114#380D412yes—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4C2312yes—
124C4×C4⋊C4C2312yes—
125C4×C22⋊C4—24no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—21no—
128#376—21no—
129C2×#95—22no—
130C2×#95—21no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#333C2×C414yescomplemented
136#333C2×C414yescomplemented
137#331C2×C414yescomplemented
138#331C2×C414yescomplemented
139C2×#41—24no—
140C2×#41—84no—
141C2×#41C4⋊C418yes—
142C22×Q8C2×D412yes—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4C2×D412yes—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8C22×C412yes—
161C4⋊Q8C2×Q812yes—
162#127—21no—
163#127—21no—
164#127C22×C412yes—
165#127C2×Q812yes—
166#124—24no—
167#124C4⋊C418yes—
168#121—24no—
169#121—24no—
170#121—24no—
171#121—24no—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—44no—
177C4×D4—44no—
178C4×D4—82no—
179C4×D4—82no—
180C4×D4—88no—
181#117—81no—
182#117—81no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—41no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—24no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—41no—
209C2×C42—81no—
210C2×C42—81no—
211C2×C42C22×C412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×Q812yes—
216C2×C42#4112yes—
217C2×C42#4112yes—
218C2×C42C2411yescharacteristic
219#107—44no—
220#107—44no—
221#107—82no—
222#106—82no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—42no—
228C4⋊C8C22×C412yes—
229C4⋊C8C22×C412yes—
230C4⋊C8C22×C412yes—
231C4⋊C8C22×C412yes—
232#103—84no—
233#102—84no—
234C22⋊C8—24no—
235C22⋊C8—24no—
236C22⋊C8—42no—
237C4×C8—24no—
238C4×C8—24no—
239C4×C8—22no—
240C4×C8—22no—
241#95—24no—
242#95—24no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—22no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—22no—
251#95—22no—
252#95—41no—
253#95—41no—
254C24—22no—
255C24#12011yescharacteristic
256#41—88no—
257#41—88no—
258C2×Q8—22no—
259C2×Q8—44no—
260C2×Q8C2×C4⋊C414yes—
261C2×D4—28no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—44no—
268C2×D4—48no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4C2×C4⋊C414yes—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—28no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—48no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—84no—
294C22×C4—82no—
295C22×C4—84no—
296C22×C4—84no—
297C22×C4C2×C4⋊C412yes—
298C22×C4C4×D412yes—
299C22×C4C4×D412yes—
300C22×C4#12011yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4C22⋊Q812yes—
304C22×C4C22⋊Q812yes—
305C22×C4#12312yes—
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—84no—
340C4⋊C4#13514yes—
341C22⋊C4—28no—
342C22⋊C4—28no—
343C22⋊C4—48no—
344C22⋊C4—44no—
345C22⋊C4—84no—
346C22⋊C4—84no—
347C22⋊C4—82no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42—42no—
352C42—81no—
353C42—81no—
354C42—82no—
355C42—84no—
356C42C2×C4⋊C412yes—
357C42C4×D412yes—
358C42C4×D412yes—
359C42C22⋊Q812yes—
360C42C22⋊Q812yes—
361C42#12312yes—
362C42C23×C411yescharacteristic
363C42C22×Q811yescharacteristic
364C42C2×#4111yescharacteristic
365C42C2×#4111yescharacteristic
366C23—22no—
367C23—21no—
368C23—21no—
369C23—22no—
370C23—24no—
371C23—44no—
372C23—42no—
373C23—42no—
374C23—44no—
375C23#38512yes—
376C23C2×#12011yescharacteristic
377Q8—44no—
378Q8—82no—
379D4—44no—
380D4—416no—
381D4—44no—
382D4—42no—
383D4—42no—
384D4—42no—
385D4—42no—
386D4—42no—
387D4—42no—
388D4—82no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—88no—
418C2×C4—88no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—168no—
423C2×C4#38512yes—
424C2×C4#38512yes—
425C2×C4#38512yes—
426C2×C4C22×C4⋊C411yescharacteristic
427C2×C4C2×C4×D411yescharacteristic
428C2×C4C2×C4×D411yescharacteristic
429C2×C4C2×C22⋊Q811yescharacteristic
430C2×C4C2×C22⋊Q811yescharacteristic
431C2×C4C2×#12311yescharacteristic
432C2×C4C2×#13512yes—
433C2×C4#57512yes—
434C2×C4#57611yescharacteristic
435C2×C4#57611yescharacteristic
436C2×C4#57811yescharacteristic
437C2×C4#57811yescharacteristic
438C8—42no—
439C8—42no—
440C8—42no—
441C8—42no—
442C8—82no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—42no—
447C22—42no—
448C22—44no—
449C22—44no—
450C22—44no—
451C22—84no—
452C22—84no—
453C22C2×#38511yescharacteristic
454C22#300212yes—
455C22#300811yescharacteristic
456C22#301111yescharacteristic
457C22#301912yes—
458C4—21no—
459C4—22no—
460C4—21no—
461C4—22no—
462C4—22no—
463C4—44no—
464C4—44no—
465C4—82no—
466C4—82no—
467C4—82no—
468C4—162no—
469C4C4×#13511yescharacteristic
470C4#294311yescharacteristic
471C4#295111yescharacteristic
472C4#295111yescharacteristic
473C4#307612yes—
474C4#308311yescharacteristic
475C4#308311yescharacteristic
476C2—42no—
477C2—84no—
478C2#1886612yes—
479C2#1886912yes—
480C2#2085611yescharacteristic
481C2#2087211yescharacteristic
482C2#2088111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21985C2semidirect
4. #21358C2semidirect
6. #8929C2semidirect
7. #8927C2semidirect
8. #8921C2semidirect
9. #8920C2semidirect
10. #7894C2semidirect
11. #7892C2semidirect
43. #1543C4semidirect
75. #431C2×C4semidirect
76. #431C2×C4semidirect
135. #333C2×C4semidirect
136. #333C2×C4semidirect
137. #331C2×C4semidirect
138. #331C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342362
#219852
#218052
#213582
C2×#15432
#89292
#89272
#89212
#89202
#78942
#78922