Finite Group Databasebeta

← #191976

Subgroups — #191976

Order 512.

1206 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2935C211yesmaximal, characteristic
2#21386C211yesmaximal, characteristic
3#21146C211yesmaximal, characteristic, complemented
4C2×#1571C211yesmaximal, characteristic, complemented
5C2×#1492C211yesmaximal, characteristic
6#8847C214yesmaximal, complemented
7#8843C214yesmaximal, complemented
8C2×#1279C211yesmaximal, characteristic, complemented
9#7848C211yesmaximal, characteristic, complemented
10C22×#131—21no—
11C2×#532C2211yescharacteristic
12#2935—24no—
13#2935—24no—
14C2×#436C2212yes—
15C2×#435—22no—
16C2×C8×D4C2212yes—
17C2×#431C2211yescharacteristic
18C2×#423—21no—
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C2×#408—21no—
22C2×C4×M16C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1988C411yescharacteristic
26#1988C411yescharacteristic
27#1982C411yescharacteristic
28#1982C411yescharacteristic
29#1940—22no—
30#1939C2214yes—
31#1874C414yes—
32C4×#102C2214yes—
33#1728—21no—
34#1571—24no—
35C2×#335C2212yes—
36C2×#335C2212yes—
37#1492—24no—
38C2×D4⋊C8C2214yes—
39#1279C2212yes—
40#1279C2212yes—
41#1273C2214yes—
42C22×#41—21no—
43C2×#131—28no—
44C2×#131—44no—
45C2×#131—44no—
46C22×M16—21no—
47C22×M16—22no—
48C22×M16D411yescharacteristic
49C22×M16D411yescharacteristic
50C23×C8—21no—
51C23×C8—22no—
52C23×C8—21no—
53#532—22no—
54#532—22no—
55#532D414yes—
56#532Q814yes—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#127C2×C411yescharacteristic
59C2×#124—21no—
60C2×#121—21no—
61C2×#121C2×C412yes—
62C2×C4×D4—21no—
63C2×C4×D4C2312yes—
64C22×C4⋊C4C2×C412yes—
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67#436—24no—
68#436—24no—
69#435—216no—
70C8×D4—24no—
71C8×D4—24no—
72#431—22no—
73#431—22no—
74#423—22no—
75#423—22no—
76#423—44no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2×C411yescharacteristic
82C2×C4⋊C8C2×C411yescharacteristic
83C2×C4⋊C8D414yes—
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×#102C2×C414yes—
88C2×#102C2×C414yes—
89#408—28no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8C2×C412yes—
93C4×M16—22no—
94C4×M16—22no—
95C2×#97D412yes—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8—21no—
99C2×C4×C8C2312yes—
100C2×C4×C8C2312yes—
101C2×C4×C8C2312yes—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4C2×C411yescharacteristic
105C4×C4⋊C4C2×C411yescharacteristic
106C4×C4⋊C4C2312yes—
107C2×#95—21no—
108#340C2×C414yes—
109#340C2×C414yes—
110#336—24no—
111#335—24no—
112#335—24no—
113D4⋊C8—28no—
114C2×#41—24no—
115C2×#41—24no—
116C2×#41—42no—
117C2×#41—42no—
118C2×#41—42no—
119C2×#41—42no—
120C22×Q8—21no—
121C22×D4—22no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4C22×C412yes—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4C22×C411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4#4111yescharacteristic
130#131—432no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—28no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—44no—
138C2×M16—42no—
139C2×M16—42no—
140C2×M16—42no—
141C2×M16—44no—
142C2×M16D814yes—
143C22×C8—28no—
144C22×C8—24no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—44no—
150C22×C8—44no—
151C22×C8—42no—
152C22×C8—42no—
153C22×C8—42no—
154C22×C8—44no—
155C22×C8C22×C412yes—
156C22×C8C22×C412yes—
157C22×C8C2×D412yes—
158C22×C8C2×D412yes—
159C4⋊Q8—22no—
160C4⋊Q8C4⋊C414yes—
161#127—22no—
162#127C4⋊C414yes—
163#124—24no—
164#124—24no—
165#121—24no—
166#121—24no—
167#121—24no—
168#121C4⋊C418yes—
169C4×D4—24no—
170C4×D4—24no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4C2×D414yes—
174C4×D4C2×Q814yes—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C412yes—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42C22×C411yescharacteristic
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×Q812yes—
194C2×C42#4112yes—
195C2×C42#4111yescharacteristic
196C2×C42C2411yescharacteristic
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—24no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—48no—
205C4⋊C8—42no—
206C4⋊C8C2×D412yes—
207C4⋊C8C2×D412yes—
208#102—28no—
209#102—44no—
210C22⋊C8—28no—
211C22⋊C8—28no—
212C22⋊C8—24no—
213#97—24no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—22no—
218C4×C8—22no—
219C4×C8—22no—
220C4×C8—42no—
221#95—21no—
222#95—21no—
223#95—22no—
224C24—22no—
225C24C4×D411yescharacteristic
226#41—48no—
227#41—48no—
228#41—48no—
229#41—48no—
230C2×Q8—24no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×D4—28no—
234C2×D4—22no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—24no—
239C2×D4—44no—
240C2×D4—44no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—82no—
244C2×D4C2×C4⋊C414yes—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—82no—
260C22×C4C2×C4⋊C412yes—
261C22×C4C4×D412yes—
262C22×C4C4×D411yescharacteristic
263C22×C4C4×D411yescharacteristic
264C22×C4C4×D411yescharacteristic
265C22×C4#12012yes—
266C22×C4C22⋊Q812yes—
267C22×C4C22⋊Q812yes—
268C22×C4#12312yes—
269C22×C4C2×D812yes—
270C22×C4#13512yes—
271C22×C4AGL(1,ℤ/8)12yes—
272C22×C4#13712yes—
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C2×#4111yescharacteristic
276M16—28no—
277M16—416no—
278M16—44no—
279M16—84no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—24no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—416no—
293C2×C8—42no—
294C2×C8—48no—
295C2×C8—42no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—48no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8C2×D814yes—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—24no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—44no—
311C4⋊C4C2×C4⋊C414yes—
312C4⋊C4#13114yescomplemented
313C22⋊C4—24no—
314C22⋊C4—24no—
315C22⋊C4—28no—
316C42—42no—
317C42—42no—
318C42—42no—
319C42—42no—
320C42C2×C4⋊C412yes—
321C42C4×D412yes—
322C42#12012yes—
323C42C22⋊Q812yes—
324C42C22⋊Q812yes—
325C42#12312yes—
326C42C22×D411yescharacteristic
327C42C22×D411yescharacteristic
328C42C22×Q811yescharacteristic
329C42C2×#4111yescharacteristic
330C23—22no—
331C23—22no—
332C23—24no—
333C23—44no—
334C23—42no—
335C23—42no—
336C23—44no—
337C23#38512yes—
338C23C4×D812yes—
339C23#44012yes—
340C23C2×C4×D411yescharacteristic
341Q8—48no—
342D4—28no—
343D4—416no—
344D4—48no—
345D4—42no—
346D4—42no—
347D4—84no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—88no—
370C2×C4#38512yes—
371C2×C4#38512yes—
372C2×C4#38512yes—
373C2×C4C4×D812yes—
374C2×C4#44012yes—
375C2×C4#44714yes—
376C2×C4#45214yes—
377C2×C4#47414yes—
378C2×C4#47814yes—
379C2×C4C22×C4⋊C411yescharacteristic
380C2×C4C2×C4×D411yescharacteristic
381C2×C4C2×#12011yescharacteristic
382C2×C4C2×C22⋊Q811yescharacteristic
383C2×C4C2×C22⋊Q811yescharacteristic
384C2×C4C2×#12311yescharacteristic
385C2×C4C2×#13112yes—
386C2×C4#56812yes—
387C2×C4C22×D811yescharacteristic
388C2×C4C2×#13511yescharacteristic
389C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
390C2×C4C2×#13711yescharacteristic
391C8—24no—
392C8—44no—
393C8—42no—
394C8—42no—
395C8—42no—
396C8—42no—
397C8—42no—
398C8—82no—
399C22—24no—
400C22—24no—
401C22—22no—
402C22—22no—
403C22—44no—
404C22—44no—
405C22—44no—
406C22—84no—
407C22—84no—
408C22#186214yes—
409C22C2×#38511yescharacteristic
410C22#293312yes—
411C22C2×C4×D811yescharacteristic
412C22C2×#44011yescharacteristic
413C22#295612yes—
414C4—22no—
415C4—21no—
416C4—21no—
417C4—22no—
418C4—44no—
419C4—44no—
420C4—82no—
421C4#186214yes—
422C4#290612yes—
423C4#293111yescharacteristic
424C4#293211yescharacteristic
425C4C2×#44711yescharacteristic
426C4C2×#45211yescharacteristic
427C4C2×#47411yescharacteristic
428C4C2×#47811yescharacteristic
429C2—24no—
430C2—84no—
431C2#1799412yes—
432C2#1800412yes—
433C2#2083911yescharacteristic
434C2C2×#186211yescharacteristic
435C2#2088511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21146C2semidirect
4. C2×#1571C2semidirect
6. #8847C2semidirect
7. #8843C2semidirect
8. C2×#1279C2semidirect
9. #7848C2semidirect
312. C4⋊C4#131semidirect

Maximal subgroups

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

subgroupindex
C2×#29352
#213862
#211462
C2×#15712
C2×#14922
#88472
#88432
C2×#12792
#78482