Finite Group Databasebeta

← #204562

Subgroups — #204562

Order 512.

1156 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34955C211yesmaximal, characteristic
2#21350C211yesmaximal, characteristic, complemented
3#20470C211yesmaximal, characteristic, complemented
4C2×#1496C211yesmaximal, characteristic
5C2×#1484C211yesmaximal, characteristic
6C2×#1491C211yesmaximal, characteristic, complemented
7C2×#1479C211yesmaximal, characteristic, complemented
8#9676C214yesmaximal, complemented
9#9675C214yesmaximal, complemented
10C2×#532C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2592—22no—
17#2591—22no—
18#2568C412yes—
19#2548C412yes—
20#2504C412yes—
21#2436C412yes—
22#2398—21no—
23#2398—21no—
24C4×#124C2214yes—
25#2297—22no—
26C4×#117C2211yescharacteristic
27#1940—22no—
28#1851—22no—
29#1767C2212yes—
30C4×C4⋊C8C2212yes—
31C4×C4⋊C8C2214yes—
32#1750—22no—
33#1746—21no—
34#1726—21no—
35#1496—24no—
36#1491—24no—
37#1484—24no—
38#1479—24no—
39C2×Q8⋊C8C2212yes—
40C2×Q8⋊C8C2212yes—
41C2×Q8⋊C8C2214yes—
42C2×D4⋊C8C2212yes—
43C2×D4⋊C8C2212yes—
44C2×D4⋊C8C2214yes—
45C22×#41—21no—
46C23×Q8C2×C411yescharacteristic
47C22×M16—21no—
48#532—22no—
49#532—22no—
50#532—22no—
51#532—22no—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#127C2×C411yescharacteristic
54C2×#124—21no—
55C2×#124C2×C414yes—
56C2×#121—21no—
57C2×#121C2×C412yes—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8C2×C412yes—
65C2×C4×Q8C2312yes—
66C2×C4×D4—21no—
67C2×C4×D4C2312yes—
68C2×#117D412yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73#432—22no—
74#432D414yes—
75#431—22no—
76#431—22no—
77#431—22no—
78#431D414yes—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2312yes—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C4×M16—22no—
88C4×M16—22no—
89C2×#97—21no—
90C2×#97—21no—
91C2×#97—22no—
92C2×C4×C8—21no—
93C2×C4×C8—22no—
94C2×C4×C8C2312yes—
95C2×C4×C8C2312yes—
96#400—22no—
97#400—22no—
98#397—22no—
99#397—22no—
100#394—22no—
101#394—22no—
102#393—22no—
103#393—22no—
104#391—24no—
105#391C2×C414yes—
106#390—24no—
107#390C2×C414yes—
108#388—24no—
109#388—24no—
110#386—24no—
111#386—24no—
112#382C2×C414yes—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4D412yes—
115C4×C22⋊C4—24no—
116C4×C22⋊C4—24no—
117C4×C22⋊C4C2×C412yes—
118C4×C22⋊C4D412yes—
119#376D411yescharacteristic
120#376D411yescharacteristic
121C43C2312yes—
122#336—24no—
123#336—22no—
124#336—22no—
125Q8⋊C8—24no—
126Q8⋊C8—24no—
127Q8⋊C8—28no—
128D4⋊C8—24no—
129D4⋊C8—24no—
130D4⋊C8—28no—
131C2×#41—44no—
132C2×#41—44no—
133C2×#41—42no—
134C2×#41—42no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—24no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—42no—
141C22×Q8C22×C412yes—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4C22×C412yes—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4C22×C411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C2×M16—44no—
153C2×M16—42no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161#127—22no—
162#127—22no—
163#124—28no—
164#124—44no—
165#124—42no—
166#124—42no—
167#121—24no—
168#121—24no—
169#121—42no—
170#121—42no—
171C4×Q8—24no—
172C4×Q8—24no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—42no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×Q8—42no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—42no—
186C4×D4—42no—
187#117—42no—
188#117—42no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4C22⋊C414yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4C22⋊C414yes—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42C22×C414yes—
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D414yes—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42C2411yescharacteristic
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—22no—
226C4⋊C8—48no—
227C4⋊C8—42no—
228C4⋊C8C2×D412yes—
229C4⋊C8C2×D412yes—
230C4⋊C8C2×D414yes—
231C22⋊C8—24no—
232C22⋊C8—24no—
233C22⋊C8—42no—
234#97—24no—
235#97—44no—
236#97—42no—
237C4×C8—24no—
238C4×C8—22no—
239C4×C8—22no—
240C4×C8—22no—
241C4×C8—22no—
242C4×C8—44no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95C22⋊C412yes—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251C24—22no—
252C24C2×C22⋊C411yescharacteristic
253#41—84no—
254#41—84no—
255#41—88no—
256C2×Q8—22no—
257C2×Q8—28no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—44no—
263C2×Q8—48no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8#13114yes—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—44no—
274C2×D4—48no—
275C2×D4—44no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—82no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—82no—
295C22×C4C2×C22⋊C414yes—
296C22×C4C2×C22⋊C414yes—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C4×D412yes—
301C22×C4C4×D412yes—
302C22×C4#12112yes—
303C22×C4#12112yes—
304C22×C4#12412yes—
305C22×C4#12712yes—
306C22×C4C23×C411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309M16—48no—
310C2×C8—24no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—24no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—44no—
321C2×C8—42no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—44no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—44no—
331C4⋊C4—42no—
332C22⋊C4—44no—
333C22⋊C4—48no—
334C22⋊C4—48no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—44no—
338C22⋊C4—84no—
339C22⋊C4—84no—
340C42—24no—
341C42—24no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—44no—
347C42—42no—
348C42—42no—
349C42—44no—
350C42C4×D412yes—
351C42C4×D412yes—
352C42#12112yes—
353C42#12112yes—
354C42#12412yes—
355C42#12712yes—
356C42C22×D411yescharacteristic
357C42C22×D411yescharacteristic
358C42C2×#4111yescharacteristic
359C42C2×#4111yescharacteristic
360C23—22no—
361C23—22no—
362C23—22no—
363C23—44no—
364C23—44no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23#39012yes—
369C23C22×C22⋊C411yescharacteristic
370Q8—24no—
371Q8—24no—
372Q8—48no—
373Q8—42no—
374Q8—42no—
375Q8—84no—
376D4—42no—
377D4—42no—
378D4—44no—
379D4—84no—
380D4—84no—
381D4—88no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—48no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—84no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—84no—
412C2×C4#39012yes—
413C2×C4#39012yes—
414C2×C4#39012yes—
415C2×C4C2×C4×D411yescharacteristic
416C2×C4C2×C4×D411yescharacteristic
417C2×C4C2×#12111yescharacteristic
418C2×C4C2×#12111yescharacteristic
419C2×C4C2×#12411yescharacteristic
420C2×C4C2×#12711yescharacteristic
421C2×C4C2×#13112yes—
422C2×C4#56812yes—
423C2×C4#57512yes—
424C2×C4#57512yes—
425C8—44no—
426C8—42no—
427C8—42no—
428C8—44no—
429C8—82no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—44no—
434C22—82no—
435C22—82no—
436C22—84no—
437C22—84no—
438C22—82no—
439C22C2×#39011yescharacteristic
440C22#288012yes—
441C22#289612yes—
442C22#290112yes—
443C4—21no—
444C4—21no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—44no—
449C4—44no—
450C4—44no—
451C4—82no—
452C4—82no—
453C4#292911yescharacteristic
454C4#293011yescharacteristic
455C4#293312yes—
456C4#304011yescharacteristic
457C4#304011yescharacteristic
458C4#304012yes—
459C2—42no—
460C2—162no—
461C2#1751614yes—
462C2#2119111yescharacteristic
463C2#2125311yescharacteristic
464C2#2126111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21350C2semidirect
3. #20470C2semidirect
6. C2×#1491C2semidirect
7. C2×#1479C2semidirect
8. #9676C2semidirect
9. #9675C2semidirect

Maximal subgroups

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

subgroupindex
#349552
#213502
#204702
C2×#14962
C2×#14842
C2×#14912
C2×#14792
#96762
#96752