Finite Group Databasebeta

← #195685

Subgroups — #195685

Order 512.

1264 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3206C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#21809C211yesmaximal, characteristic
4#21393C211yesmaximal, characteristic
5C2×#1558C211yesmaximal, characteristic, complemented
6#9063C214yesmaximal, complemented
7#9060C214yesmaximal, complemented
8#7902C211yesmaximal, characteristic, complemented
9#7900C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3206—24no—
13#3206C418yescomplemented
14C2×#492C2212yes—
15C2×#487C2212yes—
16C2×#461—22no—
17C2×#448—22no—
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#431C2211yescharacteristic
21#2369C2214yes—
22#2365—22no—
23#2345—22no—
24Q8×C22⋊C4—22no—
25#2331—22no—
26#2316—21no—
27C4×C4⋊Q8—22no—
28#2285C2211yescharacteristic
29#2274—21no—
30C4×#117C2211yescharacteristic
31#1988C2212yes—
32#1982C2212yes—
33#1982C2212yes—
34#1981C2212yes—
35#1959—22no—
36#1875—21no—
37#1875—21no—
38#1788—22no—
39#1762—22no—
40#1558—24no—
41C2×#333C2214yes—
42#1279C2214yes—
43#1279C2214yes—
44C22×#41—21no—
45C23×Q8D412yes—
46C22×SD16—24no—
47#532—22no—
48#532—22no—
49#532—22no—
50#532C2×C414yes—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8C2×C414yes—
56C2×C4⋊Q8D412yes—
57C2×C4⋊Q8D414yes—
58C2×C4⋊Q8C2311yescharacteristic
59C2×C4⋊Q8C2312yes—
60C2×#127C2311yescharacteristic
61C2×#124—21no—
62C2×#121—21no—
63C2×#121—21no—
64C2×C4×Q8—24no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×#117—21no—
68C2×#117D411yescharacteristic
69C2×#117D411yescharacteristic
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C42C2311yescharacteristic
73#492—24no—
74#492C2×C418yes—
75#487—24no—
76#487C2×C418yes—
77#461—216no—
78#448—216no—
79#432—22no—
80#432—22no—
81#432—22no—
82#432—22no—
83#431—22no—
84#431C2×C414yes—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2312yes—
88C2×#103—24no—
89C2×#103—24no—
90C2×#102—22no—
91C2×#102—22no—
92C2×#102—24no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×#97—21no—
97C2×#97—21no—
98C2×C4×C8C2312yes—
99#391—24no—
100#391—28no—
101#389—24no—
102#389—24no—
103#389—24no—
104#387—28no—
105#384D412yes—
106#382—21no—
107#380—24no—
108#380—21no—
109#380D412yes—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—24no—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4D412yes—
115C4×C4⋊C4C2312yes—
116C4×C4⋊C4C2312yes—
117C4×C22⋊C4—24no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120#376—21no—
121C43—21no—
122#340—22no—
123#340—22no—
124#333—28no—
125C2×#41—24no—
126C2×#41—42no—
127C2×#41—84no—
128C22×Q8—22no—
129C22×Q8—24no—
130C22×Q8—21no—
131C22×Q8—84no—
132C22×Q8C22⋊C418yes—
133C22×Q8C2×D414yes—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—41no—
138C23×C4—22no—
139C23×C4—41no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C2×SD16—216no—
144C2×SD16—88no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C4⋊Q8—24no—
149C4⋊Q8—24no—
150C4⋊Q8—24no—
151C4⋊Q8—24no—
152C4⋊Q8—24no—
153C4⋊Q8—21no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8C22⋊C418yes—
159C4⋊Q8C22×C412yes—
160C4⋊Q8C22×C414yes—
161C4⋊Q8C2×D414yes—
162#127—21no—
163#127—21no—
164#127—21no—
165#127C22×C412yes—
166#124—24no—
167#124—42no—
168#121—24no—
169#121—24no—
170#121—42no—
171#121—42no—
172C4×Q8—88no—
173C4×D4—24no—
174C4×D4—24no—
175C4×D4—42no—
176C4×D4—42no—
177#117—24no—
178#117—42no—
179#117—44no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D414yes—
195C2×C4⋊C4C2×D412yes—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—28no—
207C2×C42C22×C412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42#4112yes—
213C2×C42#4112yes—
214C2×C42C2411yescharacteristic
215C4⋊C8—24no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220#103—216no—
221#103—84no—
222#102—44no—
223#102—44no—
224#102—84no—
225C22⋊C8—24no—
226C22⋊C8—42no—
227C22⋊C8—42no—
228#97—42no—
229#97—42no—
230C4×C8—22no—
231C4×C8C22×C414yes—
232#95—28no—
233#95—28no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238C24—41no—
239C24#12011yescharacteristic
240#41—88no—
241#41—88no—
242C2×Q8—24no—
243C2×Q8—28no—
244C2×Q8—22no—
245C2×Q8—48no—
246C2×Q8—48no—
247C2×Q8—41no—
248C2×Q8—44no—
249C2×Q8—88no—
250C2×Q8—88no—
251C2×Q8C2×C22⋊C418yes—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—44no—
263C2×D4—84no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—28no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—44no—
275C22×C4—84no—
276C22×C4—84no—
277C22×C4C2×C22⋊C412yes—
278C22×C4C4×D412yes—
279C22×C4C4×D412yes—
280C22×C4#12011yescharacteristic
281C22×C4#12011yescharacteristic
282C22×C4#12011yescharacteristic
283C22×C4#12112yes—
284C22×C4#12112yes—
285C22×C4#12312yes—
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289SD16—816no—
290C2×C8—24no—
291C2×C8—44no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—44no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—28no—
303C4⋊C4—28no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—88no—
315C4⋊C4#13714yes—
316C22⋊C4—24no—
317C22⋊C4—42no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C42—24no—
322C42—42no—
323C42—44no—
324C42—44no—
325C42—44no—
326C42—88no—
327C42C2×C22⋊C412yes—
328C42C4×D412yes—
329C42C4×D412yes—
330C42#12112yes—
331C42#12112yes—
332C42#12312yes—
333C42C23×C411yescharacteristic
334C42C22×D411yescharacteristic
335C42C2×#4111yescharacteristic
336C42C2×#4111yescharacteristic
337C23—21no—
338C23—21no—
339C23—22no—
340C23—22no—
341C23—42no—
342C23—81no—
343C23—81no—
344C23—82no—
345C23—82no—
346C23#38612yes—
347C23C2×#12011yescharacteristic
348Q8—48no—
349Q8—416no—
350Q8—44no—
351Q8—84no—
352Q8—82no—
353D4—44no—
354D4—44no—
355D4—44no—
356D4—82no—
357D4—88no—
358D4—82no—
359D4—82no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—28no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—48no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—88no—
384C2×C4—88no—
385C2×C4—82no—
386C2×C4#38612yes—
387C2×C4#38612yes—
388C2×C4#38612yes—
389C2×C4C22×C22⋊C411yescharacteristic
390C2×C4C2×C4×D411yescharacteristic
391C2×C4C2×C4×D411yescharacteristic
392C2×C4C2×#12111yescharacteristic
393C2×C4C2×#12111yescharacteristic
394C2×C4C2×#12311yescharacteristic
395C2×C4C2×#13712yes—
396C2×C4#57512yes—
397C2×C4#57714yes—
398C8—44no—
399C8—82no—
400C8—82no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—42no—
405C22—42no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—162no—
410C22—162no—
411C22C2×#38611yescharacteristic
412C22#300411yescharacteristic
413C22#300511yescharacteristic
414C22#301012yes—
415C22#301912yes—
416C4—22no—
417C4—22no—
418C4—21no—
419C4—21no—
420C4—22no—
421C4—44no—
422C4—48no—
423C4—82no—
424C4—84no—
425C4—82no—
426C4C4×#13712yes—
427C4#294912yes—
428C4#303712yes—
429C4#304312yes—
430C2—42no—
431C2—162no—
432C2#1881612yes—
433C2#1882012yes—
434C2#2092611yescharacteristic
435C2#2093611yescharacteristic
436C2#2095311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
2. #34216C2semidirect
5. C2×#1558C2semidirect
6. #9063C2semidirect
7. #9060C2semidirect
8. #7902C2semidirect
9. #7900C2semidirect
13. #3206C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32062
#342162
#218092
#213932
C2×#15582
#90632
#90602
#79022
#79002