Finite Group Databasebeta

← #240470

Subgroups — #240470

Order 512.

978 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#21885C211yesmaximal, characteristic
3#21324C211yesmaximal, characteristic
4#21302C211yesmaximal, characteristic
5C2×#1565C211yesmaximal, characteristic
6C22×#332C211yesmaximal, characteristic
7#10479C214yesmaximal, complemented
8#10478C214yesmaximal, complemented
9#8039C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3209—24no—
13#3209—24no—
14C2×#494—21no—
15C2×#487—21no—
16C2×#462C2212yes—
17C2×#462C2212yes—
18C2×#451—22no—
19C2×#431C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23#2285C2211yescharacteristic
24#1993—22no—
25#1990C2214yes—
26#1987—21no—
27#1985—22no—
28#1984C2214yes—
29#1981—21no—
30#1892—22no—
31#1803C2214yes—
32#1736—21no—
33#1736—21no—
34#1729—21no—
35#1729—21no—
36#1565—24no—
37C2×#335C2212yes—
38C2×#334C2212yes—
39C2×#332—24no—
40C2×#332C2214yes—
41#1296C2212yes—
42#1296C2212yes—
43C23×Q8—21no—
44C22×Q16—22no—
45C22×Q16—22no—
46C22×M16D411yescharacteristic
47C22×M16D411yescharacteristic
48C23×C8—21no—
49C23×C8—21no—
50#533—22no—
51#533—22no—
52#533—22no—
53#533—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D414yes—
62C2×C4⋊Q8D411yescharacteristic
63C2×C4⋊Q8C2312yes—
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8D412yes—
67C2×C4×Q8—21no—
68C2×C4×Q8C2312yes—
69C2×#117—21no—
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C42C2311yescharacteristic
74#494—28no—
75#487—28no—
76#462—24no—
77#462—24no—
78#462—24no—
79#462—24no—
80#451—216no—
81#431—22no—
82#431D414yes—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—24no—
96C2×C22⋊C8—21no—
97C4×M16—22no—
98C4×M16—22no—
99C2×#97D412yes—
100C2×C4×C8—21no—
101C2×C4×C8C2312yes—
102#384C2312yes—
103#382—21no—
104#380—21no—
105#380—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108#340—24no—
109#340—24no—
110#340—24no—
111#336—24no—
112#336—24no—
113#335—24no—
114#334—24no—
115#332—28no—
116C22×Q8—21no—
117C22×Q8—24no—
118C22×Q8—22no—
119C22×Q8—41no—
120C22×Q8—42no—
121C22×Q8—82no—
122C22×Q8—82no—
123C22×Q8C2×D412yes—
124C23×C4—41no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4#4111yescharacteristic
128C2×Q16—28no—
129C2×Q16—28no—
130C2×Q16—84no—
131C2×Q16—84no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16Q1614yescomplemented
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—42no—
143C22×C8—42no—
144C22×C8—42no—
145C22×C8—42no—
146C22×C8—82no—
147C22×C8C2×D412yes—
148C22×C8C2×D412yes—
149C4⋊Q8—22no—
150C4⋊Q8—24no—
151C4⋊Q8—22no—
152C4⋊Q8—24no—
153C4⋊Q8—24no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8C2×D414yes—
163#125—42no—
164#125—42no—
165C22⋊Q8—24no—
166C22⋊Q8—24no—
167C22⋊Q8—42no—
168C22⋊Q8—42no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—42no—
174C4×Q8—42no—
175#117—42no—
176#117—42no—
177#117—42no—
178#117—42no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4C2×D412yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—41no—
198C2×C42—22no—
199C2×C42—41no—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42#4112yes—
205C2×C42#4112yes—
206C2×C42#4111yescharacteristic
207C2×C42C2411yescharacteristic
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—42no—
215C4⋊C8—42no—
216C4⋊C8C2×D412yes—
217C4⋊C8C2×D412yes—
218#103—28no—
219#103—28no—
220#103—84no—
221C22⋊C8—24no—
222#97—24no—
223C4×C8—24no—
224C4×C8—22no—
225C4×C8—22no—
226#95—41no—
227#95—41no—
228C24#12111yescharacteristic
229C2×Q8—24no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—82no—
240C2×Q8—84no—
241C2×Q8—84no—
242C2×Q8—84no—
243C2×Q8—88no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—84no—
253C22×C4#12012yes—
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12112yes—
257C22×C4#12112yes—
258C22×C4#12111yescharacteristic
259C22×C4#12312yes—
260C22×C4#12312yes—
261C22×C4#12412yes—
262C22×C4C2×Q1612yes—
263C22×C4#13512yes—
264C22×C4#13712yes—
265C22×C4#13712yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C2×#4111yescharacteristic
269Q16—88no—
270Q16—88no—
271M16—28no—
272M16—44no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—44no—
287C2×C8—84no—
288C2×C8—84no—
289C2×C8C2×Q1614yes—
290C4⋊C4—22no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—82no—
309C22⋊C4—44no—
310C22⋊C4—82no—
311C42—42no—
312C42—42no—
313C42—82no—
314C42#12012yes—
315C42#12112yes—
316C42#12112yes—
317C42#12312yes—
318C42#12312yes—
319C42#12412yes—
320C42C22×D411yescharacteristic
321C42C22×D411yescharacteristic
322C42C2×#4111yescharacteristic
323C42C2×#4111yescharacteristic
324C23—22no—
325C23—22no—
326C23#39412yes—
327C23#46212yes—
328C23#46412yes—
329C23C2×#12111yescharacteristic
330Q8—44no—
331Q8—42no—
332Q8—44no—
333Q8—42no—
334Q8—44no—
335Q8—84no—
336Q8—88no—
337Q8—84no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—88no—
352C2×C4—84no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4#39412yes—
357C2×C4#39412yes—
358C2×C4#39412yes—
359C2×C4#46212yes—
360C2×C4#46214yes—
361C2×C4#46412yes—
362C2×C4#46414yes—
363C2×C4#46714yes—
364C2×C4#47014yes—
365C2×C4C2×#12011yescharacteristic
366C2×C4C2×#12111yescharacteristic
367C2×C4C2×#12111yescharacteristic
368C2×C4C2×#12311yescharacteristic
369C2×C4C2×#12311yescharacteristic
370C2×C4C2×#12411yescharacteristic
371C2×C4C22×Q1611yescharacteristic
372C2×C4C2×#13511yescharacteristic
373C2×C4C2×#13711yescharacteristic
374C2×C4C2×#13711yescharacteristic
375C2×C4#57612yes—
376C2×C4#57712yes—
377C8—24no—
378C8—42no—
379C8—42no—
380C8—42no—
381C8—82no—
382C8—82no—
383C22—24no—
384C22—24no—
385C22—22no—
386C22—22no—
387C22#205614yes—
388C22C2×#39411yescharacteristic
389C22C2×#46211yescharacteristic
390C22C2×#46411yescharacteristic
391C22#306212yes—
392C22#306912yes—
393C4—22no—
394C4—21no—
395C4—21no—
396C4—22no—
397C4—82no—
398C4—84no—
399C4—82no—
400C4—162no—
401C4#205614yes—
402C4#300811yescharacteristic
403C4#300911yescharacteristic
404C4C2×#46211yescharacteristic
405C4C2×#46411yescharacteristic
406C4C2×#46711yescharacteristic
407C4C2×#47011yescharacteristic
408C4#309612yes—
409C2—24no—
410C2#1924012yes—
411C2#1924512yes—
412C2#2348311yescharacteristic
413C2#2353211yescharacteristic
414C2C2×#205611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×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
7. #10479C2semidirect
8. #10478C2semidirect
137. C2×M16Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#218852
#213242
#213022
C2×#15652
C22×#3322
#104792
#104782
#80392