Finite Group Databasebeta

← #236102

Subgroups — #236102

Order 512.

1390 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2C22×#477C211yesmaximal, characteristic
3#34953C211yesmaximal, characteristic
4#21313C211yesmaximal, characteristic
5C2×#1550C211yesmaximal, characteristic
6C2×#1487C211yesmaximal, characteristic
7#10059C214yesmaximal, complemented
8#10057C214yesmaximal, complemented
9#7891C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13#3209—24no—
14#3209—24no—
15C2×#494C2212yes—
16C2×#487C2212yes—
17C2×#477—24no—
18C2×#477—24no—
19C2×#477—22no—
20C2×#477C2214yes—
21C2×#462—22no—
22C2×#451—22no—
23C2×#431C2211yescharacteristic
24C22×#107—21no—
25C2×#423C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×#103—21no—
28#2584—22no—
29#2583—22no—
30#2567C2212yes—
31#2546C2212yes—
32#2502—21no—
33#2434—21no—
34#2399—22no—
35C4×C4⋊Q8—22no—
36#2296—22no—
37C4×#117C2211yescharacteristic
38#1988—22no—
39#1987C2214yes—
40#1736—22no—
41#1550—24no—
42C2×#333C2212yes—
43C2×#332C2212yes—
44#1487—24no—
45C2×Q8⋊C8C2214yes—
46#1279C2214yes—
47C23×Q8—21no—
48C23×Q8D411yescharacteristic
49C22×Q16—24no—
50C22×M16—21no—
51C23×C8—21no—
52#533—22no—
53#533—22no—
54#533—22no—
55#533—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8D414yes—
64C2×C4⋊Q8C2311yescharacteristic
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2311yescharacteristic
67C2×#125—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8D412yes—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8D412yes—
78C2×C4×Q8D412yes—
79C2×C4×Q8C2312yes—
80C2×#117—21no—
81C2×#117D412yes—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4D411yescharacteristic
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88#494—24no—
89#494—24no—
90#487—24no—
91#487—24no—
92#477—216no—
93#477—28no—
94#477—28no—
95#462—216no—
96#451—216no—
97#431—22no—
98#431D414yes—
99C2×#107—28no—
100C2×#107—24no—
101#423—22no—
102#423—22no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8D414yes—
108C2×C4⋊C8C2312yes—
109C2×C4⋊C8C2312yes—
110C2×#103—28no—
111C2×#103—24no—
112C2×#103—24no—
113C2×C22⋊C8—21no—
114C2×C4×C8C2312yes—
115#398—24no—
116#398—24no—
117#395—24no—
118#395—24no—
119#391—22no—
120#391—22no—
121#389—28no—
122#389—24no—
123#389—24no—
124#385—24no—
125#385—24no—
126#384—22no—
127C4×C4⋊C4—28no—
128C4×C4⋊C4D412yes—
129C4×C4⋊C4C2312yes—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132#376—21no—
133C43—21no—
134#340—28no—
135#333—24no—
136#332—24no—
137Q8⋊C8—28no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—24no—
141C22×Q8—22no—
142C22×Q8—24no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—42no—
146C22×Q8—41no—
147C22×Q8—42no—
148C22×Q8—84no—
149C22×Q8C2×D412yes—
150C22×Q8C2×D412yes—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—41no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C2×Q16—216no—
158C2×Q16—88no—
159C2×M16—42no—
160C2×M16—44no—
161C22×C8—24no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—44no—
166C4⋊Q8—24no—
167C4⋊Q8—24no—
168C4⋊Q8—24no—
169C4⋊Q8—22no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—22no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—22no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—44no—
181C4⋊Q8—44no—
182C4⋊Q8—42no—
183C4⋊Q8—42no—
184C4⋊Q8—42no—
185C4⋊Q8C2×D414yes—
186#125—42no—
187#125—42no—
188C22⋊Q8—24no—
189C22⋊Q8—24no—
190C22⋊Q8—42no—
191C22⋊Q8—42no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—24no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×Q8—42no—
206#117—24no—
207#117—42no—
208#117—42no—
209#117Q1618yescomplemented
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D414yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D411yescharacteristic
251C2×C42#4112yes—
252C2×C42C2411yescharacteristic
253#107—216no—
254C4⋊C8—24no—
255C4⋊C8—24no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—42no—
260C4⋊C8C2×D414yes—
261#103—216no—
262#103—216no—
263C22⋊C8—24no—
264C4×C8—22no—
265C4×C8C2×D414yes—
266#95—28no—
267#95—28no—
268#95—22no—
269#95—22no—
270C24#12011yescharacteristic
271C2×Q8—22no—
272C2×Q8—28no—
273C2×Q8—22no—
274C2×Q8—24no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—44no—
281C2×Q8—48no—
282C2×Q8—44no—
283C2×Q8—44no—
284C2×Q8—48no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—44no—
292C2×Q8—84no—
293C2×Q8—88no—
294C2×Q8—88no—
295C2×Q8#13714yes—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—82no—
310C22×C4—84no—
311C22×C4#12012yes—
312C22×C4#12011yescharacteristic
313C22×C4#12011yescharacteristic
314C22×C4#12012yes—
315C22×C4#12011yescharacteristic
316C22×C4#12112yes—
317C22×C4#12112yes—
318C22×C4#12112yes—
319C22×C4#12712yes—
320C22×C4C2×Q1614yes—
321C22×C4#13714yes—
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325Q16—816no—
326M16—84no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—48no—
330C2×C8—44no—
331C2×C8—44no—
332C2×C8—42no—
333C2×C8—44no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—416no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—88no—
359C4⋊C4C2×Q1618yes—
360C22⋊C4—28no—
361C22⋊C4—44no—
362C42—28no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42—42no—
367C42—44no—
368C42—44no—
369C42—82no—
370C42#12012yes—
371C42#12012yes—
372C42#12112yes—
373C42#12112yes—
374C42#12112yes—
375C42#12712yes—
376C42C22×D411yescharacteristic
377C42C22×D411yescharacteristic
378C42C22×D411yescharacteristic
379C42C2×#4111yescharacteristic
380C23—22no—
381C23—22no—
382C23#39212yes—
383C23#45114yes—
384C23C2×#12011yescharacteristic
385Q8—24no—
386Q8—24no—
387Q8—48no—
388Q8—44no—
389Q8—48no—
390Q8—42no—
391Q8—42no—
392Q8—82no—
393Q8—84no—
394Q8—88no—
395Q8—81no—
396Q8—81no—
397Q8—82no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—28no—
405C2×C4—24no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—48no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—84no—
421C2×C4—88no—
422C2×C4—84no—
423C2×C4#39212yes—
424C2×C4#39212yes—
425C2×C4#39212yes—
426C2×C4#45114yes—
427C2×C4#46218yes—
428C2×C4#49414yes—
429C2×C4#49714yes—
430C2×C4C2×#12011yescharacteristic
431C2×C4C2×#12011yescharacteristic
432C2×C4C2×#12111yescharacteristic
433C2×C4C2×#12111yescharacteristic
434C2×C4C2×#12111yescharacteristic
435C2×C4C2×#12711yescharacteristic
436C2×C4C22×Q1612yes—
437C2×C4C2×#13712yes—
438C2×C4C2×#13712yes—
439C2×C4#57512yes—
440C8—44no—
441C8—44no—
442C8—82no—
443C22—22no—
444C22—28no—
445C22—22no—
446C22#201614yes—
447C22C2×#39211yescharacteristic
448C22C2×#45112yes—
449C22#300712yes—
450C22#302112yes—
451C4—22no—
452C4—21no—
453C4—21no—
454C4—22no—
455C4—44no—
456C4—48no—
457C4—82no—
458C4—82no—
459C4—82no—
460C4—84no—
461C4#201614yes—
462C4#300411yescharacteristic
463C4#300511yescharacteristic
464C4C2×#46212yes—
465C4#305612yes—
466C4C2×#49411yescharacteristic
467C4C2×#49711yescharacteristic
468C2—24no—
469C2#1878914yes—
470C2#2339211yescharacteristic
471C2#2340011yescharacteristic
472C2C2×#201611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×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
7. #10059C2semidirect
8. #10057C2semidirect
209. #117Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
C22×#4772
#349532
#213132
C2×#15502
C2×#14872
#100592
#100572
#78912