Finite Group Databasebeta

← #193244

Subgroups — #193244

Order 512.

944 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34235C211yesmaximal, characteristic
2#21985C211yesmaximal, characteristic
3#21835C211yesmaximal, characteristic
4#21359C211yesmaximal, characteristic
5C2×#1544C211yesmaximal, characteristic
6#8926C212yesmaximal, complemented
7#8925C212yesmaximal, complemented
8#8925C212yesmaximal, complemented
9#8923C212yesmaximal, complemented
10#7894C211yesmaximal, characteristic
11#7892C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2365C2212yes—
17#2365C2212yes—
18#2360—22no—
19#2356—21no—
20#2351—21no—
21#2343—22no—
22#2336—22no—
23#2320—21no—
24C4×C22⋊Q8—22no—
25#2285C2211yescharacteristic
26#2283C2211yescharacteristic
27#2276—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1990C2212yes—
30#1990C2212yes—
31#1989—22no—
32#1948C2212yes—
33#1948C2212yes—
34#1948C2212yes—
35#1947C2212yes—
36#1854—22no—
37#1853—22no—
38#1846—21no—
39#1835—21no—
40#1825—22no—
41C4×#106—21no—
42C4×#106—21no—
43#1544C418yescomplemented
44C2×#332C2211yescharacteristic
45C2×#332C2211yescharacteristic
46C2×#332C2211yescharacteristic
47C2×#332C2211yescharacteristic
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#533—22no—
53#533—22no—
54#533C2×C414yes—
55#533Q814yes—
56C2×C4⋊Q8C2311yescharacteristic
57C2×C4⋊Q8C2311yescharacteristic
58C2×#125D412yes—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×#117—21no—
66C2×#117D412yes—
67C22×C4⋊C4D412yes—
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70#431—22no—
71#431—22no—
72#431—22no—
73#431—22no—
74#431C2×C414yes—
75#431C2×C414yes—
76C2×#107—22no—
77C2×#106—22no—
78C2×#106—22no—
79C2×#106—22no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8C2312yes—
91#391—22no—
92#391—22no—
93#389—24no—
94#389—22no—
95#389—22no—
96#387—24no—
97#387—24no—
98#387—24no—
99#387—24no—
100#385—24no—
101#384C2312yes—
102#383—22no—
103#383—21no—
104#382—22no—
105#382—21no—
106#382D411yescharacteristic
107#382D411yescharacteristic
108#381—21no—
109#380—21no—
110#380—21no—
111#380D412yes—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4C2312yes—
123C4×C4⋊C4C2312yes—
124C4×C22⋊C4—24no—
125C4×C22⋊C4—21no—
126#376—22no—
127#376—21no—
128C2×#95—21no—
129#340—24no—
130#340—24no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#332C2×C414yescomplemented
136#332C2×C414yescomplemented
137#332C2×C414yescomplemented
138#332C2×C414yescomplemented
139C22×Q8—21no—
140C22×Q8—21no—
141C23×C4—22no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8C22×C412yes—
155C4⋊Q8C22×C412yes—
156C4⋊Q8C2×Q812yes—
157C4⋊Q8C2×Q812yes—
158#125—24no—
159#125C4⋊C418yes—
160C22⋊Q8—24no—
161C22⋊Q8—24no—
162C22⋊Q8—24no—
163C22⋊Q8—24no—
164C4×Q8—24no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—44no—
169C4×Q8—44no—
170C4×Q8—82no—
171C4×Q8—82no—
172#117—24no—
173#117—81no—
174#117—81no—
175#117C4⋊C418yes—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—41no—
197C2×C42—22no—
198C2×C42—24no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—24no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—41no—
205C2×C42—81no—
206C2×C42—81no—
207C2×C42C22×C412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×Q812yes—
212C2×C42#4112yes—
213C2×C42#4112yes—
214C2×C42C2411yescharacteristic
215#107—82no—
216#106—44no—
217#106—44no—
218#106—82no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—42no—
224C4⋊C8C22×C412yes—
225C4⋊C8C22×C412yes—
226C4⋊C8C22×C412yes—
227C4⋊C8C22×C412yes—
228C22⋊C8—24no—
229C22⋊C8—24no—
230C22⋊C8—42no—
231C4×C8—24no—
232C4×C8—24no—
233C4×C8—22no—
234C4×C8—22no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—22no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—22no—
247#95—22no—
248#95—41no—
249#95—41no—
250C24#12011yescharacteristic
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—82no—
278C22×C4C2×C4⋊C412yes—
279C22×C4C4×D412yes—
280C22×C4C4×D412yes—
281C22×C4#12011yescharacteristic
282C22×C4#12011yescharacteristic
283C22×C4#12011yescharacteristic
284C22×C4C22⋊Q812yes—
285C22×C4C22⋊Q812yes—
286C22×C4#12312yes—
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—44no—
301C2×C8—42no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—28no—
310C4⋊C4—28no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—44no—
318C4⋊C4—48no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—84no—
325C4⋊C4C2×C4⋊C414yes—
326C4⋊C4C2×C4⋊C414yes—
327C4⋊C4#13514yes—
328C22⋊C4—28no—
329C22⋊C4—44no—
330C22⋊C4—82no—
331C42—28no—
332C42—48no—
333C42—42no—
334C42—42no—
335C42—42no—
336C42—42no—
337C42—81no—
338C42—81no—
339C42—82no—
340C42—84no—
341C42C2×C4⋊C412yes—
342C42C4×D412yes—
343C42C4×D412yes—
344C42C22⋊Q812yes—
345C42C22⋊Q812yes—
346C42#12312yes—
347C42C23×C411yescharacteristic
348C42C22×Q811yescharacteristic
349C42C2×#4111yescharacteristic
350C42C2×#4111yescharacteristic
351C23—22no—
352C23—21no—
353C23—21no—
354C23—22no—
355C23#38512yes—
356C23C2×#12011yescharacteristic
357Q8—44no—
358Q8—42no—
359Q8—42no—
360Q8—42no—
361Q8—42no—
362Q8—42no—
363Q8—42no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
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—88no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4#38512yes—
400C2×C4#38512yes—
401C2×C4#38512yes—
402C2×C4C22×C4⋊C411yescharacteristic
403C2×C4C2×C4×D411yescharacteristic
404C2×C4C2×C4×D411yescharacteristic
405C2×C4C2×C22⋊Q811yescharacteristic
406C2×C4C2×C22⋊Q811yescharacteristic
407C2×C4C2×#12311yescharacteristic
408C2×C4C2×#13512yes—
409C2×C4#57512yes—
410C2×C4#57711yescharacteristic
411C2×C4#57711yescharacteristic
412C2×C4#57711yescharacteristic
413C2×C4#57711yescharacteristic
414C8—42no—
415C8—42no—
416C8—42no—
417C8—42no—
418C8—82no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—42no—
423C22—42no—
424C22C2×#38511yescharacteristic
425C22#300212yes—
426C22#300911yescharacteristic
427C22#301011yescharacteristic
428C22#302012yes—
429C4—21no—
430C4—21no—
431C4—22no—
432C4—22no—
433C4—22no—
434C4—44no—
435C4—44no—
436C4—44no—
437C4—82no—
438C4—82no—
439C4—82no—
440C4—162no—
441C4C4×#13511yescharacteristic
442C4#294311yescharacteristic
443C4#295211yescharacteristic
444C4#295211yescharacteristic
445C4#307612yes—
446C4#308411yescharacteristic
447C4#308411yescharacteristic
448C2—42no—
449C2#1887912yes—
450C2#1888012yes—
451C2#2085711yescharacteristic
452C2#2087211yescharacteristic
453C2#2088211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
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
6. #8926C2semidirect
7. #8925C2semidirect
8. #8925C2semidirect
9. #8923C2semidirect
43. #1544C4semidirect
135. #332C2×C4semidirect
136. #332C2×C4semidirect
137. #332C2×C4semidirect
138. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342352
#219852
#218352
#213592
C2×#15442
#89262
#89252
#89232
#78942
#78922