Finite Group Databasebeta

← #200421

Subgroups — #200421

Order 512.

1166 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
1#60277C211yesmaximal, characteristic
2#35334C211yesmaximal, characteristic
3#20632C211yesmaximal, characteristic
4#17575C212yesmaximal, complemented
5#9385C218yesmaximal, complemented
6#8026C212yesmaximal, complemented
7C2×#533C2211yescharacteristic
8C2×C4×#41C2211yescharacteristic
9#2989—24no—
10#2983—21no—
11#2975—21no—
12#2973C2214yes—
13#2965—24no—
14#2920—22no—
15C2×#431C2212yes—
16C2×#410—21no—
17#2851—22no—
18#2628C412yescomplemented
19#2628C412yescomplemented
20#2581—21no—
21#2581C412yescomplemented
22#2580—21no—
23#2580C412yescomplemented
24#2564—22no—
25#2564C2214yes—
26#2316—21no—
27#2311—21no—
28#2285C2212yes—
29C2×#382C2211yescharacteristic
30#1799—22no—
31#1799C2214yes—
32#1740—21no—
33#1520C2218yes—
34#1518—24no—
35#1503—24no—
36#1294C2218yes—
37C22×#41D411yescharacteristic
38#533—22no—
39#533—22no—
40#533—22no—
41#533—22no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8C2312yes—
47C4×#41—22no—
48C4×#41—22no—
49C4×#41—22no—
50C4×#41—22no—
51C2×C4×Q8—21no—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×D4C2×C411yescharacteristic
56C2×C4×D4D412yes—
57C2×C4×D4D411yescharacteristic
58C2×C4×D4C2312yes—
59C2×#117—21no—
60C2×#117—22no—
61C2×#117C2×C412yes—
62C2×#117D411yescharacteristic
63C22×C4⋊C4—21no—
64C22×C42D412yes—
65C22×C42C2311yescharacteristic
66#446—28no—
67#436—28no—
68#435—216no—
69#432—24no—
70#432—24no—
71#431—24no—
72#431—22no—
73#431—22no—
74#431D414yes—
75#431D414yes—
76#423—24no—
77C2×C4⋊C8—22no—
78#410—44no—
79C2×#99—24no—
80#407—24no—
81C2×C22⋊C8C2314yes—
82C4×M16—24no—
83C2×C4×C8—22no—
84#400—22no—
85#400C2×C414yes—
86#399—22no—
87#399C2×C414yes—
88#395—22no—
89#395C2×C414yes—
90#393—22no—
91#393C2×C414yes—
92#391—22no—
93#390—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#387—24no—
99#387—24no—
100#386—24no—
101#385—24no—
102#385—24no—
103#384—22no—
104#382—21no—
105#382—21no—
106#382—22no—
107#382C2×C412yes—
108#382C2×C412yes—
109#380—22no—
110#380—22no—
111#380—24no—
112#380—22no—
113#380C2314yes—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C2×#95—21no—
117C2×#95C2312yes—
118#342—24no—
119#328—216no—
120C2×#41—22no—
121C2×#41—22no—
122C2×#41—22no—
123C2×#41—22no—
124C2×#41—24no—
125C22×Q8—21no—
126C22×Q8C22⋊C411yescharacteristic
127C22×Q8C22×C412yes—
128C22×D4C22⋊C411yescharacteristic
129C22×D4C2×D412yes—
130C23×C4—22no—
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C2×M16—28no—
138C22×C8—24no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143#125—41no—
144#125—41no—
145#125—41no—
146#125—41no—
147C22⋊Q8—24no—
148C22⋊Q8—24no—
149C22⋊Q8—42no—
150C22⋊Q8—42no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167#117—22no—
168#117—22no—
169#117—24no—
170#117—24no—
171#117—42no—
172#117—42no—
173#117—44no—
174#117—44no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22×C412yes—
200C2×C22⋊C4C2×D412yes—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—42no—
207C2×C42C22⋊C412yes—
208C2×C42C22⋊C412yes—
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42#4112yes—
217C2×C42#4112yes—
218C4⋊C8—28no—
219C4⋊C8—24no—
220C4⋊C8—24no—
221#99—48no—
222C22⋊C8—28no—
223C22⋊C8—24no—
224C22⋊C8C2×D418yes—
225#97—28no—
226C4×C8—24no—
227C4×C8—24no—
228#95—22no—
229#95—22no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—22no—
235#95—24no—
236#95—42no—
237#95C22×C414yes—
238C24C2×C22⋊C412yes—
239C24C22×D411yescharacteristic
240#41—44no—
241#41—44no—
242#41—44no—
243#41—44no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—42no—
249C2×Q8—82no—
250C2×Q8—82no—
251C2×D4—24no—
252C2×D4—22no—
253C2×D4—24no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—81no—
273C22×C4—81no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4#12112yes—
286C22×C4#12112yes—
287C22×C4#12112yes—
288C22×C4#12112yes—
289C22×C4AGL(1,ℤ/8)12yes—
290C22×C4#13712yes—
291C22×C4#13714yes—
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D412yes—
294C22×C4C22×D411yescharacteristic
295C22×C4C2×#4112yes—
296M16—88no—
297C2×C8—28no—
298C2×C8—84no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—84no—
316C4⋊C4—84no—
317C4⋊C4—84no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—84no—
327C22⋊C4—84no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—44no—
335C42—84no—
336C42C4×D412yes—
337C42C4×D412yes—
338C42C4×D412yes—
339C42C4×D412yes—
340C42#12011yescharacteristic
341C42#12012yes—
342C42#12011yescharacteristic
343C42#12311yescharacteristic
344C42#12311yescharacteristic
345C42#12312yes—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—41no—
350C23—41no—
351C23—42no—
352C23—42no—
353C23C22×C22⋊C411yescharacteristic
354C23C2×#12112yes—
355C23C2×AGL(1,ℤ/8)11yescharacteristic
356C23C2×#13712yes—
357C23C2×#13711yescharacteristic
358Q8—44no—
359Q8—84no—
360Q8—84no—
361D4—44no—
362D4—44no—
363D4—44no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4—162no—
398C2×C4—162no—
399C2×C4#38611yescharacteristic
400C2×C4#38611yescharacteristic
401C2×C4#38612yes—
402C2×C4#38611yescharacteristic
403C2×C4#38611yescharacteristic
404C2×C4#38612yes—
405C2×C4#41712yes—
406C2×C4#41712yes—
407C2×C4#41912yes—
408C2×C4#41912yes—
409C2×C4#46814yes—
410C2×C4#46814yes—
411C2×C4#47014yes—
412C2×C4#47014yes—
413C2×C4C2×C4×D412yes—
414C2×C4C2×#12011yescharacteristic
415C2×C4C2×#12311yescharacteristic
416C8—48no—
417C22—21no—
418C22—22no—
419C22—21no—
420C22—22no—
421C22—41no—
422C22—41no—
423C22—42no—
424C22—44no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—82no—
429C22C2×#38611yescharacteristic
430C22C2×#41911yescharacteristic
431C22C2×#41711yescharacteristic
432C22C2×#46812yes—
433C22C2×#47012yes—
434C4—21no—
435C4—21no—
436C4—22no—
437C4—21no—
438C4—21no—
439C4—22no—
440C4—42no—
441C4—42no—
442C4—42no—
443C4—84no—
444C4—82no—
445C4—82no—
446C4—164no—
447C4#189412yes—
448C4#189412yes—
449C4#189414yes—
450C2—21no—
451C2—22no—
452C2—21no—
453C2—22no—
454C2—82no—
455C2#2094711yescharacteristic
456C2C2×#189411yescharacteristic
457C2#2101411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17575C2semidirect
5. #9385C2semidirect
6. #8026C2semidirect
18. #2628C4semidirect
19. #2628C4semidirect
21. #2581C4semidirect
23. #2580C4semidirect

Maximal subgroups

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

subgroupindex
#602772
#353342
#206322
#175752
#93852
#80262