Finite Group Databasebeta

← #215566

Subgroups — #215566

Order 512.

924 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35281C211yesmaximal, characteristic
2#21922C211yesmaximal, characteristic
3#20812C211yesmaximal, characteristic
4#9702C211yesmaximal, characteristic, complemented
5#9701C211yesmaximal, characteristic, complemented
6#9697C212yesmaximal, complemented
7#9693C211yesmaximal, characteristic, complemented
8#9691C212yesmaximal, complemented
9#9691C211yesmaximal, characteristic, complemented
10#8343C212yesmaximal
11#8342C212yesmaximal
12C2×#533C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C2×#433—21no—
15#2868C2211yescharacteristic
16#2868C2211yescharacteristic
17#2862C2211yescharacteristic
18#2862C2211yescharacteristic
19#2660—21no—
20#2660—21no—
21#2567—22no—
22#2546C414yes—
23#2502—22no—
24#2434C414yes—
25#2317—21no—
26#2317—21no—
27C4×#117C2211yescharacteristic
28#1923—22no—
29#1916—22no—
30#1842C2212yes—
31#1842C2211yescharacteristic
32#1832—21no—
33#1831C2211yescharacteristic
34#1798—21no—
35#1797C2212yes—
36#1797C2212yes—
37#1768—21no—
38#1767C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41C2×#333C2212yes—
42C2×#333C2212yes—
43C2×#333C2212yes—
44C2×#333C2212yes—
45C2×#333C2212yes—
46C2×#333C2212yes—
47C2×#332C2212yes—
48C2×#332C2212yes—
49C23×Q8C2×C412yes—
50#533—42no—
51#533—42no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8C2×C412yes—
57C2×C4⋊Q8C2×C414yes—
58C2×C4⋊Q8C2×C414yes—
59C2×C4⋊Q8C2311yescharacteristic
60C2×C4⋊Q8C2311yescharacteristic
61C2×C4⋊Q8C2311yescharacteristic
62C2×C4⋊Q8C2311yescharacteristic
63C2×#125—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8C2×C412yes—
66C2×C4×Q8—21no—
67C2×C4×Q8C2×C412yes—
68C2×#117—21no—
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C42C2311yescharacteristic
74#433—44no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×#97—21no—
85C2×C4×C8—21no—
86C2×C4×C8—22no—
87#398—24no—
88#398—24no—
89#395—24no—
90#395—24no—
91#391—22no—
92#391—22no—
93#391C2×C414yes—
94#391C2×C414yes—
95#389—24no—
96#389—24no—
97#385—24no—
98#385—24no—
99#384—22no—
100#384—22no—
101#384—21no—
102#384—21no—
103#384—21no—
104#384—21no—
105#382—21no—
106#382—21no—
107#382—21no—
108#382—21no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4D412yes—
111C4×C22⋊C4C2×C412yes—
112C4×C22⋊C4D412yes—
113#376D411yescharacteristic
114#376D411yescharacteristic
115C43C2311yescharacteristic
116C43C2311yescharacteristic
117#336—22no—
118#336—22no—
119#336—21no—
120#336—21no—
121#336—22no—
122#336—22no—
123#333—42no—
124#333—42no—
125#333—42no—
126#333—42no—
127#333—42no—
128#333—42no—
129#332—42no—
130#332—42no—
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—44no—
134C22×Q8—84no—
135C22×Q8C22×C412yes—
136C22×Q8C22×C412yes—
137C22×Q8C22×C412yes—
138C23×C4—22no—
139C23×C4C22×C411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—44no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—44no—
158C4⋊Q8—84no—
159C4⋊Q8—84no—
160#125—44no—
161C22⋊Q8—44no—
162C22⋊Q8—42no—
163C22⋊Q8—42no—
164C4×Q8—44no—
165C4×Q8—42no—
166C4×Q8—42no—
167#117—24no—
168#117—44no—
169#117SD1614yes—
170#117Q1614yes—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4C22⋊C412yes—
194C2×C22⋊C4C22⋊C412yes—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C412yes—
200C2×C42C22×C412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2411yescharacteristic
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—44no—
219C4⋊C8—44no—
220C4⋊C8—82no—
221C22⋊C8—44no—
222C22⋊C8—82no—
223#97—42no—
224C4×C8—42no—
225C4×C8—82no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95C22⋊C412yes—
231#95C22⋊C412yes—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234C24C2×C22⋊C411yescharacteristic
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—44no—
239C2×Q8—44no—
240C2×Q8—44no—
241C2×Q8—44no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—84no—
247C2×Q8—816no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—84no—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×SD1612yes—
270C22×C4C2×Q1612yes—
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—84no—
303C4⋊C4—84no—
304C4⋊C4—84no—
305C4⋊C4—84no—
306C4⋊C4#10318yes—
307C22⋊C4—24no—
308C22⋊C4—44no—
309C22⋊C4#10318yes—
310C42—22no—
311C42—21no—
312C42—21no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—42no—
320C42—42no—
321C42—44no—
322C42C2×SD1612yes—
323C42C2×SD1612yes—
324C42C2×Q1612yes—
325C42C2×Q1612yes—
326C42#13714yes—
327C42H(5,𝔽2)12yes—
328C42H(5,𝔽2)12yes—
329C23—21no—
330C23—22no—
331C23—21no—
332C23—22no—
333C23C2×#10312yes—
334C23C22×C22⋊C411yescharacteristic
335Q8—84no—
336Q8—84no—
337Q8—84no—
338Q8—88no—
339Q8—81no—
340Q8—81no—
341Q8—82no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—44no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—88no—
373C2×C4C2×#10314yes—
374C2×C4C2×#10314yes—
375C2×C4C2×#10312yes—
376C2×C4#41914yes—
377C2×C4#51812yes—
378C2×C4#53512yes—
379C2×C4#53511yescharacteristic
380C2×C4#53511yescharacteristic
381C2×C4C22×SD1611yescharacteristic
382C2×C4C22×Q1611yescharacteristic
383C2×C4C2×#13712yes—
384C2×C4#57712yes—
385C2×C4#57812yes—
386C8—82no—
387C8—82no—
388C8—82no—
389C8—82no—
390C22—22no—
391C22—24no—
392C22—41no—
393C22—41no—
394C22—42no—
395C22#230911yescharacteristic
396C22#288812yes—
397C22C22×#10311yescharacteristic
398C22C2×#41911yescharacteristic
399C22#289812yes—
400C4—22no—
401C4—22no—
402C4—24no—
403C4—24no—
404C4—24no—
405C4—41no—
406C4—41no—
407C4—42no—
408C4—82no—
409C4—82no—
410C4—82no—
411C4—82no—
412C4#320612yes—
413C4#320912yes—
414C4#322111yescharacteristic
415C4#322311yescharacteristic
416C4#322512yes—
417C2—42no—
418C2#1772212yes—
419C2#1776012yes—
420C2#2180111yescharacteristic
421C2#2180411yescharacteristic
422C2#2181411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9702C2semidirect
5. #9701C2semidirect
6. #9697C2semidirect
7. #9693C2semidirect
8. #9691C2semidirect
9. #9691C2semidirect

Maximal subgroups

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

subgroupindex
#352812
#219222
#208122
#97022
#97012
#96972
#96932
#96912
#83432
#83422