Finite Group Databasebeta

← #195678

Subgroups — #195678

Order 512.

1254 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic
3#21839C211yesmaximal, characteristic
4#21392C211yesmaximal, characteristic
5C2×#1556C211yesmaximal, characteristic
6#9065C214yesmaximal, complemented
7#9058C214yesmaximal, complemented
8#7902C211yesmaximal, characteristic
9#7900C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#519—21no—
13#3230—24no—
14#3230C418yescomplemented
15C2×#500C2212yes—
16C2×#487C2212yes—
17C2×#484—22no—
18C2×#477—22no—
19C2×#432C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2365—22no—
23#2365C2214yes—
24#2344—22no—
25#2344—22no—
26#2327—22no—
27#2310—21no—
28C4×C4⋊Q8—22no—
29#2285C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1992—22no—
32#1988C2212yes—
33#1987C2212yes—
34#1987C2212yes—
35#1981C2212yes—
36#1873—21no—
37#1873—21no—
38#1825—22no—
39#1763—22no—
40#1556—24no—
41C2×#332C2214yes—
42#1279C2214yes—
43#1279C2214yes—
44C23×Q8—21no—
45#533—22no—
46#533—22no—
47#533—22no—
48#533C2×C414yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C414yes—
54C2×C4⋊Q8D414yes—
55C2×C4⋊Q8D412yes—
56C2×C4⋊Q8C2311yescharacteristic
57C2×C4⋊Q8C2311yescharacteristic
58C2×C4⋊Q8C2312yes—
59C2×#125—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62#519—48no—
63C2×C4×Q8—24no—
64C2×C4×Q8—24no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×#117—24no—
68C2×#117—21no—
69C2×#117—21no—
70C2×#117D411yescharacteristic
71C2×#117D411yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4D412yes—
74C22×C42C2311yescharacteristic
75#500—24no—
76#500C2×C418yes—
77#487—24no—
78#487C2×C418yes—
79#484—216no—
80#477—216no—
81#432—22no—
82#432—22no—
83#432—22no—
84#432—22no—
85#431—22no—
86#431C2×C414yes—
87C2×#107—24no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×#103—24no—
92C2×#103—22no—
93C2×#103—22no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×#97—21no—
98C2×#97—21no—
99C2×C4×C8C2312yes—
100#391—24no—
101#391—24no—
102#389—24no—
103#389—24no—
104#387—28no—
105#387—28no—
106#384D412yes—
107#382—21no—
108#381—24no—
109#380—21no—
110#380D412yes—
111C4×C4⋊C4—24no—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4D412yes—
115C4×C4⋊C4C2312yes—
116C4×C4⋊C4C2312yes—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119#376—21no—
120C43—21no—
121#340—24no—
122#340—24no—
123#340—22no—
124#340—22no—
125#332—28no—
126C22×Q8—24no—
127C22×Q8—22no—
128C22×Q8—24no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—44no—
133C22×Q8—44no—
134C23×C4—22no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C4⋊Q8—24no—
142C4⋊Q8—24no—
143C4⋊Q8—24no—
144C4⋊Q8—24no—
145C4⋊Q8—24no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8C22⋊C418yes—
155C4⋊Q8C22×C412yes—
156C4⋊Q8C22×C412yes—
157C4⋊Q8C22×C414yes—
158C4⋊Q8C2×D414yes—
159#125—24no—
160#125—42no—
161C22⋊Q8—24no—
162C22⋊Q8—24no—
163C22⋊Q8—42no—
164C22⋊Q8—42no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—416no—
168C4×Q8—48no—
169C4×Q8—48no—
170C4×Q8—42no—
171C4×Q8—42no—
172#117—24no—
173#117—24no—
174#117—416no—
175#117—42no—
176#117—44no—
177#117—42no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4C22⋊C418yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D414yes—
196C2×C4⋊C4C2×D414yes—
197C2×C4⋊C4C2×D412yes—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—41no—
202C2×C42—21no—
203C2×C42—28no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—41no—
209C2×C42C22×C412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42#4112yes—
215C2×C42#4112yes—
216C2×C42C2411yescharacteristic
217#107—216no—
218C4⋊C8—24no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223#103—216no—
224#103—44no—
225#103—44no—
226C22⋊C8—24no—
227C22⋊C8—42no—
228C22⋊C8—42no—
229#97—42no—
230#97—42no—
231C4×C8—22no—
232C4×C8C22×C414yes—
233#95—28no—
234#95—28no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239C24#12011yescharacteristic
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—28no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—416no—
248C2×Q8—416no—
249C2×Q8—44no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—44no—
253C2×Q8—44no—
254C2×Q8—42no—
255C2×Q8—42no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—28no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4C2×C22⋊C412yes—
270C22×C4C4×D412yes—
271C22×C4C4×D412yes—
272C22×C4#12011yescharacteristic
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12112yes—
276C22×C4#12112yes—
277C22×C4#12312yes—
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C2×C8—24no—
282C2×C8—44no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—44no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—22no—
292C4⋊C4—28no—
293C4⋊C4—28no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—28no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—416no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4C2×C22⋊C418yes—
311C4⋊C4AGL(1,ℤ/8)14yes—
312C22⋊C4—24no—
313C22⋊C4—42no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C42—24no—
317C42—44no—
318C42—42no—
319C42—416no—
320C42—44no—
321C42—44no—
322C42—44no—
323C42C2×C22⋊C412yes—
324C42C4×D412yes—
325C42C4×D412yes—
326C42#12112yes—
327C42#12112yes—
328C42#12312yes—
329C42C23×C411yescharacteristic
330C42C22×D411yescharacteristic
331C42C2×#4111yescharacteristic
332C42C2×#4111yescharacteristic
333C23—21no—
334C23—21no—
335C23—22no—
336C23—22no—
337C23#38612yes—
338C23C2×#12011yescharacteristic
339Q8—416no—
340Q8—44no—
341Q8—48no—
342Q8—48no—
343Q8—44no—
344Q8—82no—
345Q8—82no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—28no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—416no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4#38612yes—
376C2×C4#38612yes—
377C2×C4#38612yes—
378C2×C4C22×C22⋊C411yescharacteristic
379C2×C4C2×C4×D411yescharacteristic
380C2×C4C2×C4×D411yescharacteristic
381C2×C4C2×#12111yescharacteristic
382C2×C4C2×#12111yescharacteristic
383C2×C4C2×#12311yescharacteristic
384C2×C4C2×AGL(1,ℤ/8)12yes—
385C2×C4#57512yes—
386C2×C4#57814yes—
387C8—44no—
388C8—82no—
389C8—82no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—42no—
394C22—42no—
395C22C2×#38611yescharacteristic
396C22#300311yescharacteristic
397C22#300511yescharacteristic
398C22#301112yes—
399C22#302012yes—
400C4—22no—
401C4—22no—
402C4—21no—
403C4—21no—
404C4—22no—
405C4—44no—
406C4—48no—
407C4—84no—
408C4—82no—
409C4—82no—
410C4—82no—
411C4C4×AGL(1,ℤ/8)12yes—
412C4#295012yes—
413C4#303612yes—
414C4#304412yes—
415C2—42no—
416C2#1882712yes—
417C2#1883112yes—
418C2#2092511yescharacteristic
419C2#2093511yescharacteristic
420C2#2095411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
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
6. #9065C2semidirect
7. #9058C2semidirect
14. #3230C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
#342092
#218392
#213922
C2×#15562
#90652
#90582
#79022
#79002