Finite Group Databasebeta

← #195659

Subgroups — #195659

Order 512.

1268 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3141C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#21893C211yesmaximal, characteristic
4#21390C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic, complemented
6#9065C214yesmaximal, complemented
7#9062C214yesmaximal, complemented
8#7903C211yesmaximal, characteristic, complemented
9#7896C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3141—24no—
13#3141C418yescomplemented
14C2×#489—22no—
15C2×#487C2212yes—
16C2×#486C2212yes—
17C2×#431C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C2×#417—22no—
20C2×C4×M16C2211yescharacteristic
21#2369—22no—
22#2365C2214yes—
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#1987C2212yes—
32#1987C2212yes—
33#1982C2212yes—
34#1982C2212yes—
35#1895—22no—
36#1875—21no—
37#1875—21no—
38#1802—22no—
39C4×#102—22no—
40#1546—24no—
41C2×#332C2214yes—
42#1279C2214yes—
43#1279C2214yes—
44C22×#41—21no—
45C23×Q8—21no—
46C22×M16—21no—
47C22×M16—21no—
48#532—22no—
49#532—22no—
50#532—22no—
51#532C2×C414yes—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2×C414yes—
57C2×C4⋊Q8D414yes—
58C2×C4⋊Q8D412yes—
59C2×C4⋊Q8C2311yescharacteristic
60C2×C4⋊Q8C2312yes—
61C2×#127C2311yescharacteristic
62C2×#124—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×C4×Q8—24no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×#117—21no—
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4D412yes—
73C22×C42C2311yescharacteristic
74#489—216no—
75#487—24no—
76#487C2×C418yes—
77#486—24no—
78#486C2×C418yes—
79#431—22no—
80#431—22no—
81#423—22no—
82#423—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2312yes—
86#417—216no—
87C2×#103—24no—
88C2×#102—24no—
89C2×#102—22no—
90C2×#102—22no—
91C2×#102—24no—
92C2×C22⋊C8—21no—
93C4×M16—22no—
94C4×M16C2×C414yes—
95C2×#97—21no—
96C2×C4×C8—21no—
97C2×C4×C8C2312yes—
98#391—28no—
99#391—24no—
100#389—24no—
101#389—24no—
102#389—24no—
103#387—28no—
104#384D412yes—
105#382—21no—
106#380—24no—
107#380—21no—
108#380D412yes—
109C4×C4⋊C4—24no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—24no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4C2312yes—
115C4×C4⋊C4C2312yes—
116C4×C22⋊C4—24no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119#376—21no—
120C43—21no—
121#340—24no—
122#340—22no—
123#340—22no—
124#332—28no—
125C2×#41—24no—
126C2×#41—48no—
127C2×#41—42no—
128C22×Q8—24no—
129C22×Q8—22no—
130C22×Q8—24no—
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—82no—
134C22×Q8—82no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—41no—
139C23×C4—22no—
140C23×C4—41no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×M16—24no—
145C2×M16—44no—
146C2×M16—42no—
147C2×M16—82no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C4⋊Q8—24no—
152C4⋊Q8—24no—
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8—24no—
156C4⋊Q8—21no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8C22⋊C418yes—
162C4⋊Q8C22×C412yes—
163C4⋊Q8C22×C414yes—
164C4⋊Q8C2×D414yes—
165#127—21no—
166#127—21no—
167#127—21no—
168#127C22×C412yes—
169#124—24no—
170#124—42no—
171#121—24no—
172#121—24no—
173#121—42no—
174#121—42no—
175C4×Q8—84no—
176C4×Q8—84no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—42no—
180C4×D4—42no—
181#117—24no—
182#117—42no—
183#117—44no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4C22⋊C418yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D414yes—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2×D412yes—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—28no—
211C2×C42C22×C412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42#4112yes—
217C2×C42#4112yes—
218C2×C42C2411yescharacteristic
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—42no—
224#103—216no—
225#102—216no—
226#102—44no—
227#102—44no—
228#102—84no—
229C22⋊C8—42no—
230#97—24no—
231C4×C8—22no—
232C4×C8—42no—
233C4×C8C22×C414yes—
234#95—28no—
235#95—28no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240C24—41no—
241C24#12011yescharacteristic
242#41—416no—
243#41—88no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—28no—
247C2×Q8—22no—
248C2×Q8—44no—
249C2×Q8—41no—
250C2×Q8—44no—
251C2×Q8—44no—
252C2×Q8—84no—
253C2×Q8—88no—
254C2×Q8—84no—
255C2×Q8—84no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—84no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—28no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—41no—
280C22×C4—44no—
281C22×C4—84no—
282C22×C4—84no—
283C22×C4C2×C22⋊C412yes—
284C22×C4C4×D412yes—
285C22×C4C4×D412yes—
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4#12112yes—
290C22×C4#12112yes—
291C22×C4#12312yes—
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295M16—48no—
296M16—84no—
297C2×C8—24no—
298C2×C8—44no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—44no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—28no—
309C4⋊C4—28no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—44no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—44no—
320C4⋊C4—88no—
321C4⋊C4C2×C22⋊C418yes—
322C4⋊C4#13714yes—
323C22⋊C4—24no—
324C22⋊C4—42no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C42—24no—
329C42—42no—
330C42—44no—
331C42—42no—
332C42—42no—
333C42—44no—
334C42—88no—
335C42C2×C22⋊C412yes—
336C42C4×D412yes—
337C42C4×D412yes—
338C42#12112yes—
339C42#12112yes—
340C42#12312yes—
341C42C23×C411yescharacteristic
342C42C22×D411yescharacteristic
343C42C2×#4111yescharacteristic
344C42C2×#4111yescharacteristic
345C23—21no—
346C23—21no—
347C23—22no—
348C23—22no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23—82no—
353C23—82no—
354C23#38612yes—
355C23C2×#12011yescharacteristic
356Q8—44no—
357Q8—48no—
358Q8—44no—
359Q8—44no—
360Q8—88no—
361Q8—82no—
362D4—44no—
363D4—44no—
364D4—44no—
365D4—82no—
366D4—88no—
367D4—82no—
368D4—81no—
369D4—81no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—28no—
383C2×C4—42no—
384C2×C4—48no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—88no—
394C2×C4—82no—
395C2×C4—88no—
396C2×C4#38612yes—
397C2×C4#38612yes—
398C2×C4#38612yes—
399C2×C4#45414yes—
400C2×C4C22×C22⋊C411yescharacteristic
401C2×C4C2×C4×D411yescharacteristic
402C2×C4C2×C4×D411yescharacteristic
403C2×C4C2×#12111yescharacteristic
404C2×C4C2×#12111yescharacteristic
405C2×C4C2×#12311yescharacteristic
406C2×C4C2×#13712yes—
407C2×C4#57512yes—
408C2×C4#57612yes—
409C2×C4#57712yes—
410C8—44no—
411C8—82no—
412C8—82no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—42no—
417C22—42no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22—162no—
422C22—162no—
423C22C2×#38611yescharacteristic
424C22#300411yescharacteristic
425C22#300511yescharacteristic
426C22#301212yes—
427C22C2×#45412yes—
428C4—22no—
429C4—22no—
430C4—21no—
431C4—21no—
432C4—44no—
433C4—48no—
434C4—82no—
435C4—84no—
436C4—82no—
437C4#188814yes—
438C4C4×#13712yes—
439C4#294612yes—
440C4#303712yes—
441C4#304111yescharacteristic
442C4#304211yescharacteristic
443C2—42no—
444C2—162no—
445C2#1858312yes—
446C2#1858512yes—
447C2#2092611yescharacteristic
448C2#2092911yescharacteristic
449C2C2×#188811yescharacteristic
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×#1546C2semidirect
6. #9065C2semidirect
7. #9062C2semidirect
8. #7903C2semidirect
9. #7896C2semidirect
13. #3141C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31412
#342162
#218932
#213902
C2×#15462
#90652
#90622
#79032
#78962