Finite Group Databasebeta

← #215581

Subgroups — #215581

Order 512.

837 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35322C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic
3#20812C211yesmaximal, characteristic
4#9708C212yesmaximal, complemented
5#9705C211yesmaximal, characteristic, complemented
6#9701C212yesmaximal, complemented
7#9701C211yesmaximal, characteristic, complemented
8#9694C211yesmaximal, characteristic, complemented
9#9693C211yesmaximal, characteristic, complemented
10#8345C212yesmaximal
11#8343C212yesmaximal
12C2×#533C2211yescharacteristic
13C2×#533C2211yescharacteristic
14C2×#433—21no—
15#2868C2211yescharacteristic
16#2862C2211yescharacteristic
17#2862C2211yescharacteristic
18#2862C2211yescharacteristic
19#2621—21no—
20#2621—21no—
21#2565—22no—
22#2565C414yes—
23#2499—22no—
24#2499C414yes—
25#2327—21no—
26#2327—21no—
27C4×#117C2211yescharacteristic
28#1924—22no—
29#1917—22no—
30#1842C2212yes—
31#1842C2211yescharacteristic
32#1832—21no—
33#1831C2211yescharacteristic
34#1798—21no—
35#1797C2212yes—
36#1797C2212yes—
37#1768—21no—
38#1767C2211yescharacteristic
39#1767C2212yes—
40C4×C4⋊C8C2211yescharacteristic
41C2×#333C2212yes—
42C2×#333C2212yes—
43C2×#332C2212yes—
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47C2×#332C2212yes—
48C2×#332C2212yes—
49#533—42no—
50#533—42no—
51#533—42no—
52#533—42no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8C2311yescharacteristic
56C2×C4⋊Q8C2311yescharacteristic
57C2×C4⋊Q8C2311yescharacteristic
58C2×C4⋊Q8C2311yescharacteristic
59C2×#125—21no—
60C2×#125C2×C412yes—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8C2×C412yes—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8C2×C412yes—
69C2×#117—21no—
70C2×#117C2×C412yes—
71C2×#117D411yescharacteristic
72C2×#117D411yescharacteristic
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×#97—22no—
86C2×C4×C8—21no—
87#398—24no—
88#398—24no—
89#397—24no—
90#397—24no—
91#391—22no—
92#391—22no—
93#391—21no—
94#391—21no—
95#391—21no—
96#391—21no—
97#391C2×C414yes—
98#391C2×C414yes—
99#387—24no—
100#387—24no—
101#387—24no—
102#387—24no—
103#384—22no—
104#384C2×C414yes—
105#382—22no—
106#382—21no—
107#382—21no—
108#382C2×C414yes—
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#332—42no—
126#332—42no—
127#332—42no—
128#332—42no—
129#332—42no—
130#332—42no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8C22×C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—44no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153#125—44no—
154#125—42no—
155#125—42no—
156C22⋊Q8—44no—
157C22⋊Q8—42no—
158C22⋊Q8—42no—
159C22⋊Q8—44no—
160C22⋊Q8—44no—
161C4×Q8—44no—
162C4×Q8—44no—
163C4×Q8—44no—
164C4×Q8—42no—
165C4×Q8—42no—
166#117—22no—
167#117—22no—
168#117—42no—
169#117—44no—
170#117—42no—
171#117—42no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4C22⋊C412yes—
191C2×C22⋊C4C22⋊C412yes—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C412yes—
198C2×C42C22×C412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2411yescharacteristic
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—44no—
217C4⋊C8—44no—
218C4⋊C8—82no—
219C22⋊C8—44no—
220C22⋊C8—82no—
221#97—42no—
222#97—82no—
223C4×C8—42no—
224#95—24no—
225#95—24no—
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—44no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—42no—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C412yes—
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C2×C8—42no—
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—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—48no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—48no—
304C22⋊C4—24no—
305C22⋊C4—42no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C42—22no—
309C42—22no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—42no—
321C42—42no—
322C42—44no—
323C42—44no—
324C42AGL(1,ℤ/8)12yes—
325C42#13712yes—
326C42H(5,𝔽2)12yes—
327C42H(5,𝔽2)12yes—
328C23—21no—
329C23—22no—
330C23—21no—
331C23—21no—
332C23—22no—
333C23C22×C22⋊C411yescharacteristic
334Q8—84no—
335Q8—82no—
336Q8—82no—
337Q8—82no—
338Q8—82no—
339Q8—81no—
340Q8—81no—
341Q8—82no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4#41714yes—
378C2×C4#51812yes—
379C2×C4#53512yes—
380C2×C4#53511yescharacteristic
381C2×C4#53511yescharacteristic
382C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
383C2×C4C2×#13711yescharacteristic
384C2×C4#57512yes—
385C2×C4#57712yes—
386C2×C4#57812yes—
387C8—82no—
388C8—82no—
389C8—82no—
390C8—82no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—42no—
395C22—42no—
396C22#230911yescharacteristic
397C22#288912yes—
398C22C2×#41711yescharacteristic
399C22#289611yescharacteristic
400C22#289812yes—
401C4—22no—
402C4—24no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—42no—
407C4—42no—
408C4—42no—
409C4—82no—
410C4—82no—
411C4—82no—
412C4—82no—
413C4#321412yes—
414C4#321612yes—
415C4#322111yescharacteristic
416C4#322311yescharacteristic
417C4#322512yes—
418C2—81no—
419C2#1776312yes—
420C2#1776712yes—
421C2#2180211yescharacteristic
422C2#2180911yescharacteristic
423C2#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. #9708C2semidirect
5. #9705C2semidirect
6. #9701C2semidirect
7. #9701C2semidirect
8. #9694C2semidirect
9. #9693C2semidirect

Maximal subgroups

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

subgroupindex
#353222
#219262
#208122
#97082
#97052
#97012
#96942
#96932
#83452
#83432