Finite Group Databasebeta

← #193247

Subgroups — #193247

Order 512.

930 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34231C211yesmaximal, characteristic
2#21962C211yesmaximal, characteristic
3#21864C211yesmaximal, characteristic
4#21362C211yesmaximal, characteristic
5C2×#1542C211yesmaximal, characteristic
6#8927C212yesmaximal, complemented
7#8927C212yesmaximal, complemented
8#8925C212yesmaximal, complemented
9#8922C212yesmaximal, complemented
10#7894C211yesmaximal, characteristic
11#7891C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16#2369C2212yes—
17#2365C2212yes—
18#2362—22no—
19#2356—22no—
20#2347—21no—
21#2346—21no—
22#2332—22no—
23#2320—21no—
24C4×#125—22no—
25#2285C2211yescharacteristic
26#2283C2211yescharacteristic
27#2276—21no—
28C4×#117C2211yescharacteristic
29#1990C2212yes—
30#1956C2212yes—
31#1949—22no—
32#1948C2212yes—
33#1948C2212yes—
34#1947C2212yes—
35#1947C2212yes—
36#1873—22no—
37#1854—22no—
38#1808—21no—
39#1807—21no—
40#1805—22no—
41C4×#107—21no—
42C4×#106—21no—
43#1542C418yescomplemented
44C2×#333C2211yescharacteristic
45C2×#333C2211yescharacteristic
46C2×#332C2211yescharacteristic
47C2×#332C2211yescharacteristic
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52C23×C8—21no—
53#533—22no—
54#533—22no—
55#533C2×C414yes—
56#533Q814yes—
57C2×C4⋊Q8C2311yescharacteristic
58C2×C4⋊Q8C2311yescharacteristic
59C2×#125—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8D412yes—
62C2×C4×Q8—22no—
63C2×C4×Q8—21no—
64C2×C4×Q8D412yes—
65C2×#117—21no—
66C2×#117—21no—
67C2×#117D412yes—
68C22×C42C2311yescharacteristic
69#431—22no—
70#431—22no—
71#431—22no—
72#431C2×C414yes—
73C2×#107—22no—
74C2×#107—22no—
75C2×#106—22no—
76C2×#106—22no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8C2×C414yes—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×#103—24no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8C2312yes—
90#391—24no—
91#391—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#388—24no—
96#388—24no—
97#387—24no—
98#387—24no—
99#387—24no—
100#384C2312yes—
101#383—22no—
102#383—21no—
103#382—22no—
104#382—21no—
105#382D412yes—
106#381—21no—
107#380—21no—
108#380—21no—
109#380D411yescharacteristic
110#380D411yescharacteristic
111C4×C4⋊C4—24no—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4D411yescharacteristic
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4C2312yes—
122C4×C4⋊C4C2312yes—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126#376—22no—
127#376—21no—
128#376—21no—
129C43—21no—
130#340—24no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#333C2×C414yes—
136#333C2×C414yes—
137#332C2×C414yes—
138#332C2×C414yes—
139C22×Q8—21no—
140C22×Q8C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—24no—
149C22×C8—22no—
150C22×C8—82no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8C22×C412yes—
156C4⋊Q8C22×C412yes—
157C4⋊Q8C2×Q812yes—
158C4⋊Q8C2×Q812yes—
159#125—24no—
160#125—24no—
161C22⋊Q8—24no—
162C22⋊Q8—24no—
163C22⋊Q8—24no—
164C22⋊Q8C4⋊C418yes—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—44no—
169C4×Q8—44no—
170C4×Q8C4⋊C418yes—
171#117—24no—
172#117—24no—
173#117—24no—
174#117—42no—
175#117—82no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—41no—
196C2×C42—22no—
197C2×C42—24no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—24no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—41no—
205C2×C42C22×C412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×Q812yes—
210C2×C42#4112yes—
211C2×C42#4112yes—
212C2×C42C2411yescharacteristic
213#107—44no—
214#107—82no—
215#106—44no—
216#106—82no—
217C4⋊C8—24no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—42no—
223C4⋊C8C22×C412yes—
224C4⋊C8C22×C412yes—
225C4⋊C8C22×C412yes—
226C4⋊C8C22×C412yes—
227#103—84no—
228C22⋊C8—24no—
229C22⋊C8—42no—
230C4×C8—24no—
231C4×C8—22no—
232C4×C8—22no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—41no—
246#95—41no—
247C24#12011yescharacteristic
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8C2×C4⋊C414yes—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4C2×C4⋊C412yes—
270C22×C4C4×D412yes—
271C22×C4C4×D412yes—
272C22×C4#12011yescharacteristic
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4C22⋊Q812yes—
276C22×C4C22⋊Q812yes—
277C22×C4#12312yes—
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—84no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—28no—
302C4⋊C4—28no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—48no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—84no—
316C4⋊C4C2×C4⋊C414yes—
317C4⋊C4#13514yes—
318C22⋊C4—28no—
319C22⋊C4—44no—
320C22⋊C4—81no—
321C22⋊C4—81no—
322C22⋊C4—82no—
323C42—28no—
324C42—44no—
325C42—48no—
326C42—42no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—81no—
331C42—81no—
332C42—82no—
333C42—84no—
334C42C2×C4⋊C412yes—
335C42C4×D412yes—
336C42C4×D412yes—
337C42C22⋊Q812yes—
338C42C22⋊Q812yes—
339C42#12312yes—
340C42C23×C411yescharacteristic
341C42C22×Q811yescharacteristic
342C42C2×#4111yescharacteristic
343C42C2×#4111yescharacteristic
344C23—22no—
345C23—22no—
346C23—21no—
347C23—21no—
348C23#38512yes—
349C23C2×#12011yescharacteristic
350Q8—44no—
351Q8—44no—
352Q8—42no—
353Q8—42no—
354Q8—82no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4#38512yes—
390C2×C4#38512yes—
391C2×C4#38512yes—
392C2×C4C22×C4⋊C411yescharacteristic
393C2×C4C2×C4×D411yescharacteristic
394C2×C4C2×C4×D411yescharacteristic
395C2×C4C2×C22⋊Q811yescharacteristic
396C2×C4C2×C22⋊Q811yescharacteristic
397C2×C4C2×#12311yescharacteristic
398C2×C4C2×#13512yes—
399C2×C4#57512yes—
400C2×C4#57712yes—
401C2×C4#57812yes—
402C8—42no—
403C8—42no—
404C8—42no—
405C8—42no—
406C8—82no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—42no—
411C22—42no—
412C22C2×#38511yescharacteristic
413C22#300212yes—
414C22#301312yes—
415C22#302111yescharacteristic
416C22#302211yescharacteristic
417C4—21no—
418C4—21no—
419C4—22no—
420C4—22no—
421C4—22no—
422C4—44no—
423C4—44no—
424C4—44no—
425C4—82no—
426C4—82no—
427C4—82no—
428C4—162no—
429C4C4×#13511yescharacteristic
430C4#294311yescharacteristic
431C4#294911yescharacteristic
432C4#295011yescharacteristic
433C4#307612yes—
434C4#308212yes—
435C2—42no—
436C2#1889112yes—
437C2#1892112yes—
438C2#2085811yescharacteristic
439C2#2087211yescharacteristic
440C2#2088011yescharacteristic
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. #8927C2semidirect
7. #8927C2semidirect
8. #8925C2semidirect
9. #8922C2semidirect
43. #1542C4semidirect

Maximal subgroups

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

subgroupindex
#342312
#219622
#218642
#213622
C2×#15422
#89272
#89252
#89222
#78942
#78912