Finite Group Databasebeta

← #197422

Subgroups — #197422

Order 512.

1146 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2921C211yesmaximal, characteristic
2#34634C211yesmaximal, characteristic
3#17708C212yesmaximal, complemented
4C2×#1458C211yesmaximal, characteristic
5#9194C214yesmaximal, complemented
6#9192C214yesmaximal, complemented
7#7957C212yesmaximal, complemented
8C2×#533C2211yescharacteristic
9C22×#117C2211yescharacteristic
10#2921—24no—
11#2921—22no—
12#2921C2214yes—
13C2×#433—24no—
14C2×#431C2212yes—
15C2×#423C2212yes—
16C2×#407—21no—
17C2×C4×M16—21no—
18#2629—22no—
19#2628C2212yes—
20#2628C2212yes—
21#2627—21no—
22#2627C412yes—
23#2626—21no—
24#2626C412yes—
25#2396—22no—
26#2396C414yes—
27#2296—21no—
28#2285C2211yescharacteristic
29#2280—21no—
30#1550—24no—
31#1549C2214yescomplemented
32#1548C2214yescomplemented
33#1542—24no—
34#1458—22no—
35#1458C2214yes—
36C2×#324—21no—
37C2×#323—21no—
38#1286C2218yescomplemented
39C24×C4D411yescharacteristic
40C22×M16—22no—
41#533—22no—
42#533—22no—
43#533—22no—
44#533—22no—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8C2×C412yes—
50C2×C4×Q8—21no—
51C2×C4×Q8C2×C412yes—
52C2×#117—21no—
53C2×#117—21no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117C2×C412yes—
59C2×#117D412yes—
60C2×#117D412yes—
61C2×#117C2312yes—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68#433—216no—
69#433—216no—
70#431—22no—
71#431—22no—
72#431—22no—
73#431C2314yes—
74#423—22no—
75#423—22no—
76#423D414yes—
77#423C2314yes—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8D414yes—
81#407—24no—
82#407—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—22no—
85C4×M16—24no—
86C4×M16—24no—
87C2×#97—22no—
88C2×C4×C8—22no—
89C2×C4×C8—22no—
90#400—22no—
91#400C2×C414yes—
92#399—22no—
93#399C2×C414yes—
94#398—24no—
95#398—24no—
96#396—24no—
97#396—24no—
98#395—22no—
99#395C2×C414yes—
100#393—22no—
101#393C2×C414yes—
102#389—22no—
103#389—22no—
104#385—22no—
105#385—22no—
106#384—21no—
107#382C2312yes—
108#381—22no—
109#380—21no—
110#380—22no—
111#380—22no—
112#380C2×C412yes—
113#379—22no—
114C4×C4⋊C4—24no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2×C412yes—
117C4×C22⋊C4—24no—
118#333—28no—
119#332—28no—
120#324—24no—
121#323—24no—
122C25C22⋊C411yescharacteristic
123C22×Q8—21no—
124C22×Q8C22×C412yes—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C2×M16—24no—
140C2×M16—24no—
141C2×M16—44no—
142C2×M16—44no—
143C22×C8—24no—
144C22×C8—42no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149#125—42no—
150#125—42no—
151C22⋊Q8—24no—
152C22⋊Q8—24no—
153C22⋊Q8—42no—
154C22⋊Q8—42no—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—42no—
158C4×Q8—42no—
159#117—24no—
160#117—24no—
161#117—24no—
162#117—24no—
163#117—24no—
164#117—24no—
165#117—24no—
166#117—24no—
167#117—41no—
168#117—41no—
169#117—41no—
170#117—41no—
171#117—44no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C22×C412yes—
202C2×C22⋊C4C2×D412yes—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42C22⋊C412yes—
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42#4112yes—
217C2×C42#4112yes—
218C2×C42C2411yescharacteristic
219C4⋊C8—24no—
220C4⋊C8—24no—
221C4⋊C8—24no—
222C4⋊C8—24no—
223C4⋊C8—42no—
224C4⋊C8C2×D418yes—
225C22⋊C8—216no—
226C22⋊C8—24no—
227C22⋊C8—42no—
228#97—28no—
229C4×C8—28no—
230C4×C8—24no—
231C4×C8—42no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—22no—
237#95C22×C414yes—
238C24—21no—
239C24—21no—
240C24—22no—
241C24—22no—
242C24—81no—
243C24—81no—
244C24C2×C22⋊C412yes—
245C24C2×C22⋊C411yescharacteristic
246C2×Q8—24no—
247C2×Q8—42no—
248C2×Q8—82no—
249C2×Q8—82no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—81no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4—81no—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C4×D412yes—
284C22×C4C4×D412yes—
285C22×C4#12011yescharacteristic
286C22×C4#12011yescharacteristic
287C22×C4#12112yes—
288C22×C4#12112yes—
289C22×C4#12311yescharacteristic
290C22×C4#12311yescharacteristic
291C22×C4#13514yes—
292C22×C4#13714yes—
293C22×C4C23×C411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296M16—44no—
297M16—44no—
298M16—48no—
299C2×C8—28no—
300C2×C8—48no—
301C2×C8—44no—
302C2×C8—84no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—44no—
327C22⋊C4—84no—
328C22⋊C4—84no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—42no—
334C42—42no—
335C42—44no—
336C42C4×D412yes—
337C42C4×D412yes—
338C42#12012yes—
339C42#12112yes—
340C42#12112yes—
341C42#12312yes—
342C42C22×D412yes—
343C42C2×#4112yes—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—81no—
356C23—81no—
357C23—84no—
358C23—84no—
359C23#38611yescharacteristic
360C23#38611yescharacteristic
361C23#41612yes—
362C23#41912yes—
363C23C22×C22⋊C411yescharacteristic
364Q8—84no—
365Q8—84no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4#38612yes—
399C2×C4#38612yes—
400C2×C4#38611yescharacteristic
401C2×C4#38611yescharacteristic
402C2×C4#41612yes—
403C2×C4#41912yes—
404C2×C4#45214yes—
405C2×C4#45214yes—
406C2×C4#48514yes—
407C2×C4#48514yes—
408C2×C4C2×C4×D412yes—
409C2×C4C2×#12011yescharacteristic
410C2×C4C2×#12112yes—
411C2×C4C2×#12311yescharacteristic
412C2×C4C2×#13512yes—
413C2×C4C2×#13712yes—
414C8—48no—
415C8—84no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—42no—
423C22—42no—
424C22—44no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—82no—
429C22—84no—
430C22—84no—
431C22#187312yes—
432C22#187312yes—
433C22C2×#38611yescharacteristic
434C22C2×#41611yescharacteristic
435C22C2×#41911yescharacteristic
436C4—22no—
437C4—22no—
438C4—22no—
439C4—22no—
440C4—42no—
441C4—42no—
442C4—42no—
443C4—84no—
444C4—82no—
445C4—82no—
446C4#187312yes—
447C4#187312yes—
448C4C2×#45212yes—
449C4C2×#48512yes—
450C2—21no—
451C2—21no—
452C2—22no—
453C2—22no—
454C2—82no—
455C2C2×#187311yescharacteristic
456C2#2100911yescharacteristic
457C2#2101711yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17708C2semidirect
5. #9194C2semidirect
6. #9192C2semidirect
7. #7957C2semidirect
31. #1549C22semidirect
32. #1548C22semidirect
38. #1286C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29212
#346342
#177082
C2×#14582
#91942
#91922
#79572