Finite Group Databasebeta

← #201873

Subgroups — #201873

Order 512.

1181 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60284C211yesmaximal, characteristic
2#34952C211yesmaximal, characteristic
3#17366C211yesmaximal, characteristic
4#17365C212yesmaximal, complemented
5#9518C218yesmaximal, complemented
6#8109C212yesmaximal, complemented
7C2×C4×#41C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9#2991—24no—
10#2982—21no—
11#2975—21no—
12#2975C2214yes—
13#2971—24no—
14#2920—22no—
15#2920C2212yes—
16#2920C2212yes—
17#2582—21no—
18#2581—21no—
19#2580—21no—
20#2579C2214yes—
21#2538—21no—
22#2487C412yes—
23#2409C412yes—
24#2396—22no—
25C4×C22⋊Q8—22no—
26#2297C412yes—
27#2296C412yes—
28#2285C2211yescharacteristic
29#1467—24no—
30#1465C2214yes—
31#1464—22no—
32#1463C2218yes—
33#1461—24no—
34#1461—21no—
35#1460—21no—
36#1305C2218yes—
37C22×#41C2×C411yescharacteristic
38C23×Q8C2×C411yescharacteristic
39C2×C22⋊Q8—22no—
40C4×#41—22no—
41C4×#41—22no—
42C4×#41—22no—
43C4×#41—22no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—21no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8D411yescharacteristic
54C2×C4×D4C2×C412yes—
55C2×C4×D4D411yescharacteristic
56C2×C4×D4D411yescharacteristic
57C2×C4×D4C2312yes—
58C2×#117C2×C412yes—
59C2×#117C2×C411yescharacteristic
60C2×#117D412yes—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C42C2×C412yes—
64C22×C42D412yes—
65C22×C42C2311yescharacteristic
66C8×Q8—28no—
67#436—28no—
68#436—216no—
69#432—24no—
70#432—22no—
71#432—22no—
72#432—22no—
73#432D414yes—
74#431—24no—
75#431—24no—
76#431—22no—
77#431—22no—
78#431—22no—
79#431D414yes—
80#423—24no—
81#423—22no—
82#423—22no—
83#407—24no—
84#407C2314yes—
85#407C2314yes—
86C4×M16—24no—
87C4×M16—22no—
88C4×M16—22no—
89#400—22no—
90#399—22no—
91#398—24no—
92#396—24no—
93#395—22no—
94#393—22no—
95#391—22no—
96#391—22no—
97#390—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#387—24no—
102#386C2×C414yes—
103#386C2×C414yes—
104#385—24no—
105#385C2×C414yes—
106#385C2×C414yes—
107#384—21no—
108#382—21no—
109#380—22no—
110#380C2×C412yes—
111#380C2312yes—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C22⋊C4—24no—
116#324—24no—
117#324—216no—
118#323—24no—
119C2×#41—24no—
120C2×#41—22no—
121C2×#41—22no—
122C2×#41—42no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—44no—
128C22×Q8C22⋊C411yescharacteristic
129C22×D4C22⋊C411yescharacteristic
130C22×D4C22×C412yes—
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C412yes—
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C412yes—
135C23×C4C22×C412yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2411yescharacteristic
139C2×M16—28no—
140C2×M16—24no—
141C2×M16—24no—
142C22⋊Q8—48no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—42no—
148C4×Q8—41no—
149C4×Q8—41no—
150C4×Q8—42no—
151C4×Q8—42no—
152C4×Q8—42no—
153C4×Q8—42no—
154C4×Q8—41no—
155C4×Q8—41no—
156C4×Q8—42no—
157C4×D4—24no—
158C4×D4—24no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—22no—
165#117—22no—
166#117—22no—
167#117—24no—
168#117—24no—
169#117—44no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C2×D412yes—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C22×C412yes—
191C2×C22⋊C4C22×C414yes—
192C2×C22⋊C4C2×D412yes—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42C22⋊C412yes—
203C2×C42C22⋊C412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42#4112yes—
212C2×C42#4112yes—
213C4⋊C8—28no—
214C4⋊C8—24no—
215C4⋊C8—24no—
216C4⋊C8—42no—
217C4⋊C8—42no—
218C22⋊C8—28no—
219C22⋊C8—24no—
220C22⋊C8—24no—
221C22⋊C8—24no—
222C22⋊C8C2×D418yes—
223#97—24no—
224#97—42no—
225C4×C8—28no—
226C4×C8—24no—
227C4×C8—42no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—22no—
232#95—24no—
233#95—22no—
234C24C2×C22⋊C412yes—
235C24C23×C411yescharacteristic
236#41—48no—
237#41—48no—
238C2×Q8—24no—
239C2×Q8—28no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—42no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8—48no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—41no—
249C2×Q8#10114yes—
250C2×D4—24no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—24no—
254C2×D4—22no—
255C2×D4—42no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—81no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C4×D412yes—
285C22×C4C4×D412yes—
286C22×C4C4×D412yes—
287C22×C4C4×D412yes—
288C22×C4C22×D412yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C2×#4112yes—
292M16—84no—
293M16—88no—
294M16—84no—
295C2×C8—28no—
296C2×C8—44no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—44no—
302C4⋊C4—48no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—44no—
310C4⋊C4—82no—
311C4⋊C4—84no—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C22⋊C4—48no—
316C22⋊C4—44no—
317C22⋊C4—84no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—44no—
325C42—42no—
326C42—42no—
327C42—44no—
328C42#12012yes—
329C42#12011yescharacteristic
330C42#12011yescharacteristic
331C42#12112yes—
332C42#12112yes—
333C42#12112yes—
334C42#12112yes—
335C42#12311yescharacteristic
336C42#12311yescharacteristic
337C42#12312yes—
338C23—21no—
339C23—21no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—41no—
344C23—41no—
345C23—42no—
346C23—42no—
347C23C22×C22⋊C411yescharacteristic
348C23C2×C4×D412yes—
349Q8—28no—
350Q8—44no—
351Q8—48no—
352Q8—44no—
353D4—48no—
354D4—44no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—28no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—84no—
391C2×C4#38611yescharacteristic
392C2×C4#38611yescharacteristic
393C2×C4#38611yescharacteristic
394C2×C4#38612yes—
395C2×C4#38611yescharacteristic
396C2×C4#38612yes—
397C2×C4C2×#10114yes—
398C2×C4C2×#10112yes—
399C2×C4#41312yes—
400C2×C4C2×#12011yescharacteristic
401C2×C4C2×#12112yes—
402C2×C4C2×#12311yescharacteristic
403C8—48no—
404C8—84no—
405C22—21no—
406C22—21no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—42no—
411C22—42no—
412C22—44no—
413C22—42no—
414C22—42no—
415C22—42no—
416C22—82no—
417C22C2×#38611yescharacteristic
418C22C22×#10111yescharacteristic
419C22C2×#41311yescharacteristic
420C4—21no—
421C4—22no—
422C4—21no—
423C4—21no—
424C4—22no—
425C4—21no—
426C4—21no—
427C4—24no—
428C4—42no—
429C4—44no—
430C4—42no—
431C4—84no—
432C4—82no—
433C4#189112yes—
434C4#190114yes—
435C2—21no—
436C2—21no—
437C2—42no—
438C2—82no—
439C2C2×#189111yescharacteristic
440C2C2×#190111yescharacteristic
441C2#2101411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ 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. #17365C2semidirect
5. #9518C2semidirect
6. #8109C2semidirect

Maximal subgroups

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

subgroupindex
#602842
#349522
#173662
#173652
#95182
#81092