Finite Group Databasebeta

← #193234

Subgroups — #193234

Order 512.

1032 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34216C211yesmaximal, characteristic
2#21959C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic
4#21169C211yesmaximal, characteristic
5C22×#332C211yesmaximal, characteristic
6#8925C214yesmaximal, complemented
7#8923C214yesmaximal, complemented
8#7893C211yesmaximal, characteristic
9#7891C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#427—21no—
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15#2369—22no—
16#2365C2214yes—
17#2345—22no—
18Q8×C22⋊C4—22no—
19#2331—22no—
20#2316—21no—
21C4×C4⋊Q8—22no—
22#2285C2211yescharacteristic
23#2285C2211yescharacteristic
24#2274—21no—
25C4×#117C2211yescharacteristic
26#1990C2214yes—
27#1987—22no—
28#1949—22no—
29#1948—22no—
30#1948C2212yes—
31#1948C2214yes—
32#1947C2212yes—
33#1824—22no—
34#1807—22no—
35#1735—21no—
36C2×#332C418yescomplemented
37C2×#332C2212yes—
38C2×#332C2212yes—
39#1279C2214yes—
40#1279C2212yes—
41#1279C2212yes—
42C23×Q8—21no—
43C23×C8—21no—
44C23×C8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8C2×C414yes—
50C2×C4⋊Q8D412yes—
51C2×C4⋊Q8Q814yes—
52C2×C4⋊Q8C2312yes—
53C2×C4×Q8—24no—
54C2×#117—21no—
55C2×#117—21no—
56C2×#117D411yescharacteristic
57C2×#117D411yescharacteristic
58C22×C4⋊C4—21no—
59C22×C4⋊C4D412yes—
60C22×C42C2311yescharacteristic
61#431—22no—
62#431—22no—
63#427—44no—
64C2×#107—22no—
65C2×#106—22no—
66C2×#106—24no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8C2×C414yes—
73C2×C4⋊C8C2×C414yes—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2311yescharacteristic
77C2×C22⋊C8—21no—
78C2×C4×C8C2312yes—
79#391—28no—
80#391—24no—
81#389—24no—
82#389—24no—
83#389—24no—
84#387—28no—
85#384D412yes—
86#384C2312yes—
87#382—21no—
88#382—21no—
89#380—24no—
90#380—21no—
91#380—21no—
92#380—21no—
93#380D412yes—
94C4×C4⋊C4—24no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4D412yes—
101C4×C4⋊C4C2312yes—
102C4×C4⋊C4C2312yes—
103C4×C22⋊C4—24no—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—21no—
106#376—21no—
107C43—21no—
108#340—24no—
109#340—28no—
110#340—22no—
111#340—22no—
112#332C2×C418yes—
113#332C2×C418yes—
114C22×Q8—24no—
115C22×Q8—22no—
116C22×Q8—24no—
117C22×Q8—22no—
118C22×Q8—82no—
119C22×Q8—82no—
120C23×C4—22no—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C22×C8—24no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—24no—
129C22×C8—22no—
130C22×C8—42no—
131C22×C8—42no—
132C22×C8—82no—
133C4⋊Q8—24no—
134C4⋊Q8—24no—
135C4⋊Q8—24no—
136C4⋊Q8—24no—
137C4⋊Q8—24no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8C4⋊C418yes—
141C4⋊Q8C22×C414yes—
142C4⋊Q8C2×Q814yes—
143C4×Q8—84no—
144C4×Q8—84no—
145#117—22no—
146#117—22no—
147#117—22no—
148#117—42no—
149#117—42no—
150#117—42no—
151#117—42no—
152#117SD1614yes—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4C4⋊C418yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D414yes—
168C2×C4⋊C4C2×D414yes—
169C2×C4⋊C4C2×D412yes—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—41no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—28no—
180C2×C42—41no—
181C2×C42C22×C412yes—
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×Q812yes—
186C2×C42#4112yes—
187C2×C42#4112yes—
188C2×C42C2411yescharacteristic
189#107—44no—
190#106—44no—
191#106—84no—
192C4⋊C8—24no—
193C4⋊C8—24no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—22no—
197C4⋊C8—42no—
198C4⋊C8C22×C412yes—
199C4⋊C8C22×C412yes—
200C4⋊C8C22×C414yes—
201C22⋊C8—42no—
202C4×C8—22no—
203C4×C8—22no—
204#95—28no—
205#95—28no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—41no—
211#95—41no—
212C24#12011yescharacteristic
213C2×Q8—24no—
214C2×Q8—24no—
215C2×Q8—28no—
216C2×Q8—44no—
217C2×Q8—44no—
218C2×Q8—84no—
219C2×Q8—88no—
220C2×Q8—84no—
221C2×Q8—84no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—28no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—84no—
232C22×C4C2×C4⋊C412yes—
233C22×C4C4×D412yes—
234C22×C4C4×D412yes—
235C22×C4#12011yescharacteristic
236C22×C4#12011yescharacteristic
237C22×C4#12011yescharacteristic
238C22×C4C22⋊Q812yes—
239C22×C4C22⋊Q812yes—
240C22×C4#12312yes—
241C22×C4C2×SD1612yes—
242C22×C4#13512yes—
243C22×C4#13712yes—
244C22×C4#13712yes—
245C22×C4C22×D411yescharacteristic
246C22×C4C22×D411yescharacteristic
247C22×C4C22×D411yescharacteristic
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—24no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—44no—
256C2×C8—48no—
257C2×C8—44no—
258C2×C8—42no—
259C2×C8—84no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—22no—
264C4⋊C4—28no—
265C4⋊C4—28no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—88no—
278C4⋊C4C2×C4⋊C418yes—
279C4⋊C4C2×SD1614yes—
280C22⋊C4—24no—
281C22⋊C4—42no—
282C22⋊C4—44no—
283C22⋊C4—82no—
284C42—24no—
285C42—44no—
286C42—42no—
287C42—42no—
288C42—42no—
289C42—42no—
290C42—42no—
291C42—88no—
292C42—82no—
293C42C2×C4⋊C412yes—
294C42C4×D412yes—
295C42C4×D412yes—
296C42C22⋊Q812yes—
297C42C22⋊Q812yes—
298C42#12312yes—
299C42C23×C411yescharacteristic
300C42C22×Q811yescharacteristic
301C42C2×#4111yescharacteristic
302C42C2×#4111yescharacteristic
303C23—22no—
304C23—22no—
305C23#38512yes—
306C23#44812yes—
307C23#45212yes—
308C23C2×#12011yescharacteristic
309Q8—48no—
310Q8—44no—
311Q8—44no—
312Q8—88no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—28no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—48no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—44no—
332C2×C4—88no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4#38512yes—
337C2×C4#38512yes—
338C2×C4#38512yes—
339C2×C4C4×SD1614yes—
340C2×C4#44114yes—
341C2×C4#44812yes—
342C2×C4#45212yes—
343C2×C4#47514yes—
344C2×C4#47914yes—
345C2×C4C22×C4⋊C411yescharacteristic
346C2×C4C2×C4×D411yescharacteristic
347C2×C4C2×C4×D411yescharacteristic
348C2×C4C2×C22⋊Q811yescharacteristic
349C2×C4C2×C22⋊Q811yescharacteristic
350C2×C4C2×#12311yescharacteristic
351C2×C4C22×SD1611yescharacteristic
352C2×C4C2×#13511yescharacteristic
353C2×C4C2×#13711yescharacteristic
354C2×C4C2×#13711yescharacteristic
355C2×C4#57612yes—
356C2×C4#57712yes—
357C8—42no—
358C8—42no—
359C8—44no—
360C8—82no—
361C22—22no—
362C22—22no—
363C22—24no—
364C22—24no—
365C22#186514yes—
366C22C2×#38511yescharacteristic
367C22C2×#44811yescharacteristic
368C22C2×#45211yescharacteristic
369C22#301212yes—
370C22#302012yes—
371C4—21no—
372C4—21no—
373C4—22no—
374C4—22no—
375C4—48no—
376C4—44no—
377C4—82no—
378C4—84no—
379C4—162no—
380C4#186514yes—
381C4C2×C4×SD1611yescharacteristic
382C4C2×#44111yescharacteristic
383C4#294811yescharacteristic
384C4#294911yescharacteristic
385C4C2×#47511yescharacteristic
386C4C2×#47911yescharacteristic
387C4#308112yes—
388C2—24no—
389C2#1888412yes—
390C2#1888912yes—
391C2#2085711yescharacteristic
392C2C2×#186511yescharacteristic
393C2#2087911yescharacteristic
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. #8925C2semidirect
7. #8923C2semidirect
36. C2×#332C4semidirect

Maximal subgroups

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

subgroupindex
#342162
#219592
#218342
#211692
C22×#3322
#89252
#89232
#78932
#78912