Finite Group Databasebeta

← #201751

Subgroups — #201751

Order 512.

1207 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60282C211yesmaximal, characteristic
2#34952C211yesmaximal, characteristic
3#20727C212yesmaximal, complemented
4#17366C211yesmaximal, characteristic
5#9509C214yesmaximal, complemented
6#9505C214yesmaximal, complemented
7#8109C212yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2990—24no—
11#2983—21no—
12#2973C2214yes—
13#2972—21no—
14#2970—24no—
15#2920C2212yes—
16#2920C2212yes—
17C2×#432—22no—
18#2582—21no—
19#2581—21no—
20#2580—21no—
21#2579C414yes—
22#2538—21no—
23#2487C2212yes—
24#2409C2212yes—
25#2396—22no—
26C4×C22⋊Q8—22no—
27#2297C412yes—
28#2296C412yes—
29#2285C2211yescharacteristic
30#1821C2214yes—
31#1801—24no—
32#1784C2214yes—
33C4×#101—24no—
34#1465C2214yes—
35#1464—22no—
36#1461—21no—
37#1460—21no—
38#1305C2218yes—
39C22×#41C2×C411yescharacteristic
40C23×Q8C2311yescharacteristic
41C2×C22⋊Q8—22no—
42C4×#41—22no—
43C4×#41—22no—
44C4×#41—22no—
45C4×#41—22no—
46C2×C4×Q8—22no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—22no—
51C2×C4×Q8—21no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×D4C2×C412yes—
57C2×C4×D4D411yescharacteristic
58C2×C4×D4D411yescharacteristic
59C2×C4×D4C2312yes—
60C2×#117C2×C412yes—
61C2×#117C2×C411yescharacteristic
62C2×#117D412yes—
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C42C2×C412yes—
66C22×C42D412yes—
67C22×C42C2311yescharacteristic
68#446—28no—
69#435—216no—
70C8×D4—28no—
71#432—24no—
72#432—24no—
73#432—22no—
74#432—24no—
75#432—22no—
76#432—22no—
77#432—22no—
78#431—22no—
79#431—22no—
80#431—22no—
81#431—22no—
82#423—22no—
83#423D414yes—
84C2×C4⋊C8—24no—
85C2×#101—28no—
86#407C2314yes—
87#407C2314yes—
88C2×C22⋊C8—24no—
89C4×M16—22no—
90C4×M16D414yes—
91C2×#97—24no—
92#400—22no—
93#399—22no—
94#398—24no—
95#396—24no—
96#395—22no—
97#393—22no—
98#391—22no—
99#391—22no—
100#390—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#387—24no—
105#386C2×C414yes—
106#386C2×C414yes—
107#385—24no—
108#385C2×C414yes—
109#385C2×C414yes—
110#384—21no—
111#382—21no—
112#380—22no—
113#380C2×C412yes—
114#380C2312yes—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C22⋊C4—24no—
119#343—28no—
120#324—24no—
121#323—24no—
122C2×#41—24no—
123C2×#41—22no—
124C2×#41—22no—
125C2×#41—42no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—44no—
131C22×Q8C22⋊C411yescharacteristic
132C22×D4C22⋊C411yescharacteristic
133C22×D4C22×C412yes—
134C23×C4C22⋊C412yes—
135C23×C4C22⋊C412yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C412yes—
138C23×C4C22×C412yes—
139C23×C4C22×C411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2411yescharacteristic
142C2×M16—24no—
143C2×M16C2×D418yes—
144C22×C8—28no—
145C22⋊Q8—48no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—42no—
151C4×Q8—41no—
152C4×Q8—41no—
153C4×Q8—42no—
154C4×Q8—42no—
155C4×Q8—42no—
156C4×Q8—42no—
157C4×Q8—41no—
158C4×Q8—41no—
159C4×Q8—42no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168#117—22no—
169#117—22no—
170#117—24no—
171#117—24no—
172#117—44no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4C22⋊C412yes—
193C2×C22⋊C4C22×C412yes—
194C2×C22⋊C4C22×C414yes—
195C2×C22⋊C4C2×D412yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42C22⋊C412yes—
206C2×C42C22⋊C412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42#4112yes—
215C2×C42#4112yes—
216C4⋊C8—28no—
217C4⋊C8—24no—
218C4⋊C8—24no—
219C4⋊C8—42no—
220C4⋊C8—42no—
221#101—416no—
222C22⋊C8—28no—
223C22⋊C8—24no—
224C22⋊C8—24no—
225C22⋊C8—24no—
226C22⋊C8—24no—
227#97—28no—
228#97—24no—
229#97—42no—
230C4×C8—24no—
231C4×C8—42no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—22no—
236#95—24no—
237#95—22no—
238C24C2×C22⋊C412yes—
239C24C23×C411yescharacteristic
240#41—48no—
241#41—48no—
242C2×Q8—28no—
243C2×Q8—24no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—42no—
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—48no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—41no—
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—42no—
259C2×D4#10114yescomplemented
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—81no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C4×D412yes—
289C22×C4C4×D412yes—
290C22×C4C4×D412yes—
291C22×C4C4×D412yes—
292C22×C4C22×D412yes—
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C2×#4112yes—
296M16—48no—
297M16—48no—
298C2×C8—28no—
299C2×C8—44no—
300C2×C8—88no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—44no—
306C4⋊C4—48no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—82no—
315C4⋊C4—84no—
316C22⋊C4—24no—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C22⋊C4—48no—
320C22⋊C4—44no—
321C22⋊C4—84no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—44no—
329C42—42no—
330C42—42no—
331C42—44no—
332C42#12012yes—
333C42#12011yescharacteristic
334C42#12011yescharacteristic
335C42#12112yes—
336C42#12112yes—
337C42#12112yes—
338C42#12112yes—
339C42#12311yescharacteristic
340C42#12311yescharacteristic
341C42#12312yes—
342C23—21no—
343C23—21no—
344C23—22no—
345C23—22no—
346C23—41no—
347C23—41no—
348C23—42no—
349C23—42no—
350C23C2×#10114yes—
351C23C22×C22⋊C411yescharacteristic
352C23C2×C4×D412yes—
353Q8—48no—
354Q8—44no—
355Q8—44no—
356Q8—44no—
357D4—28no—
358D4—48no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—28no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—44no—
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—82no—
394C2×C4—82no—
395C2×C4—84no—
396C2×C4#38611yescharacteristic
397C2×C4#38611yescharacteristic
398C2×C4#38611yescharacteristic
399C2×C4#38611yescharacteristic
400C2×C4#38612yes—
401C2×C4#38612yes—
402C2×C4C2×#10112yes—
403C2×C4#41312yes—
404C2×C4C2×#12011yescharacteristic
405C2×C4C2×#12112yes—
406C2×C4C2×#12311yescharacteristic
407C8—48no—
408C8—84no—
409C22—21no—
410C22—21no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—24no—
415C22—44no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—82no—
421C22C2×#38611yescharacteristic
422C22C22×#10111yescharacteristic
423C22C2×#41311yescharacteristic
424C4—21no—
425C4—21no—
426C4—22no—
427C4—22no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—42no—
432C4—44no—
433C4—42no—
434C4—42no—
435C4—84no—
436C4—82no—
437C4#190114yes—
438C4#302512yes—
439C2—21no—
440C2—21no—
441C2—42no—
442C2—44no—
443C2#2095911yescharacteristic
444C2C2×#190111yescharacteristic
445C2#2101711yescharacteristic
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
3. #20727C2semidirect
5. #9509C2semidirect
6. #9505C2semidirect
7. #8109C2semidirect
259. C2×D4#101semidirect

Maximal subgroups

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

subgroupindex
#602822
#349522
#207272
#173662
#95092
#95052
#81092