Finite Group Databasebeta

← #365488

Subgroups — #365488

Order 512.

1166 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34215C211yesmaximal, characteristic
2C2×#1995C211yesmaximal, characteristic
3#21322C211yesmaximal, characteristic
4#21171C211yesmaximal, characteristic
5#14354C211yesmaximal, characteristic
6#11340C212yesmaximal
7#11339C212yesmaximal
8#10892C212yesmaximal
9#10891C212yesmaximal
10#10858C211yesmaximal, characteristic
11#10701C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#428C2212yes—
14C2×#423C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2368—24no—
17#2355—22no—
18#2355—22no—
19#2331—22no—
20#2317C2211yescharacteristic
21#2317C2211yescharacteristic
22C4×#125—22no—
23C2×#384C2211yescharacteristic
24C2×#384C2212yes—
25C2×#382C2212yes—
26C2×#382C2211yescharacteristic
27#2274C2211yescharacteristic
28#2274C2211yescharacteristic
29C4×#117C2211yescharacteristic
30C2×#376C2211yescharacteristic
31#1995C418yescomplemented
32#1992—22no—
33#1990—22no—
34#1948—22no—
35#1947—22no—
36C2×#343C2212yes—
37C2×#343C2212yes—
38#1736C2212yes—
39#1736C2212yes—
40#1736C2212yes—
41C2×#339C2212yes—
42C2×#339C2212yes—
43C2×#339C2212yes—
44C2×#339C2212yes—
45#1444—24no—
46C22×M16C2312yes—
47C22×M16C2312yes—
48#533—22no—
49#533—22no—
50#533C2×C414yes—
51#533Q814yes—
52C2×C4⋊Q8D412yes—
53C2×#125D412yes—
54C2×C22⋊Q8—22no—
55C2×C4×Q8—22no—
56C2×#117C2311yescharacteristic
57C2×#117C2311yescharacteristic
58C2×#117C2311yescharacteristic
59C2×#117C2311yescharacteristic
60C22×C4⋊C4—24no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67#428—42no—
68#428—42no—
69C2×#107—22no—
70C2×#106—22no—
71#423—22no—
72#423C2×C414yes—
73C2×C4⋊C8—22no—
74C4×M16—22no—
75C4×M16—22no—
76C2×#97—21no—
77C2×C4×C8—21no—
78#391—24no—
79#391—24no—
80#389—22no—
81#389—22no—
82#387—24no—
83#387—24no—
84#387—24no—
85#387—24no—
86#385—22no—
87#385—22no—
88#384—21no—
89#384—21no—
90#384—22no—
91#384—21no—
92#384—21no—
93#384—24no—
94#384—42no—
95#382—21no—
96#382—21no—
97#382—21no—
98#382—21no—
99#382—22no—
100#382—24no—
101#382—42no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—24no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4D412yes—
109C4×C4⋊C4D412yes—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113#376—21no—
114#376—22no—
115#376C2×C414yes—
116C2×#95—22no—
117C2×#95—24no—
118C2×#95D414yes—
119C43—21no—
120#343—42no—
121#343—42no—
122#340—24no—
123#340—24no—
124#340—24no—
125#339—24no—
126#339C2×C418yes—
127#339C2×C418yes—
128#339Q818yescomplemented
129C22×Q8—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4C2411yescharacteristic
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16C22×C414yes—
147C2×M16C2×Q814yes—
148C22×C8—22no—
149C22×C8—22no—
150C4⋊Q8—24no—
151C4⋊Q8C4⋊C418yescomplemented
152#125—24no—
153#125C4⋊C418yescomplemented
154C22⋊Q8—28no—
155C22⋊Q8—28no—
156C4×Q8—28no—
157C4×Q8—28no—
158#117—21no—
159#117—21no—
160#117—21no—
161#117—21no—
162#117—22no—
163#117—22no—
164#117—22no—
165#117C22×C412yes—
166#117C22×C412yes—
167#117C2×Q812yes—
168#117C2×Q812yes—
169#117C2×Q814yes—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—48no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—84no—
193C2×C4⋊C4—88no—
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42C22×C414yes—
215C2×C42C22×C412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×Q814yes—
220C2×C42C2×Q812yes—
221C2×C42#4114yes—
222C2×C42#4112yes—
223C2×C42#4112yes—
224#107—44no—
225#106—44no—
226C4⋊C8—28no—
227#97—42no—
228C4×C8—42no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—28no—
234#95—24no—
235#95—24no—
236#95—22no—
237#95—22no—
238#95—48no—
239C24C22×D411yescharacteristic
240C2×Q8—28no—
241C2×Q8—82no—
242C2×Q8—82no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—48no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4C4×D414yes—
266C22×C4#12012yes—
267C22×C4#12012yes—
268C22×C4C22⋊Q814yes—
269C22×C4C23×C411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×Q811yescharacteristic
273C22×C4C22×Q812yes—
274C22×C4C2×#4111yescharacteristic
275C22×C4C2×#4112yes—
276C22×C4C2×#4111yescharacteristic
277C22×C4H(5,𝔽2)12yes—
278C22×C42−1+412yes—
279M16—24no—
280M16—24no—
281M16—24no—
282M16—24no—
283M16—42no—
284M16—42no—
285M16—42no—
286M16—42no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—28no—
297C4⋊C4—28no—
298C4⋊C4—28no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C4⋊C4—84no—
316C4⋊C4—84no—
317C4⋊C4—816no—
318C4⋊C4C2×C4⋊C414yes—
319C4⋊C4C2×C4⋊C414yes—
320C4⋊C4C4⋊Q814yes—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—24no—
326C22⋊C4—42no—
327C42—22no—
328C42—22no—
329C42—24no—
330C42—22no—
331C42—22no—
332C42—24no—
333C42—24no—
334C42—24no—
335C42—42no—
336C42—82no—
337C42—82no—
338C42—84no—
339C42—84no—
340C42C2×C4⋊C412yes—
341C42C2×C4⋊C412yes—
342C42C4×D414yes—
343C42C4×Q818yes—
344C42C22⋊Q814yes—
345C42#12312yes—
346C42#12312yes—
347C23—22no—
348C23—22no—
349C23C2×C4×D411yescharacteristic
350C23C2×#12011yescharacteristic
351C23C2×C22⋊Q811yescharacteristic
352C23#54612yes—
353C23D4×Q812yes—
354Q8—88no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—28no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—48no—
367C2×C4—48no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—82no—
388C2×C4—84no—
389C2×C4#38514yes—
390C2×C4#38514yes—
391C2×C4C22×C4⋊C411yescharacteristic
392C2×C4C2×C4×D411yescharacteristic
393C2×C4C2×C4×Q812yes—
394C2×C4#52012yes—
395C2×C4C2×C22⋊Q811yescharacteristic
396C2×C4C2×#12311yescharacteristic
397C2×C4C2×C4⋊Q811yescharacteristic
398C2×C4#53211yescharacteristic
399C2×C4#53312yes—
400C2×C4#53511yescharacteristic
401C2×C4#53611yescharacteristic
402C2×C4#54212yes—
403C2×C4D4×Q812yes—
404C2×C4#55012yes—
405C2×C4#55212yes—
406C2×C4#55412yes—
407C2×C4#55712yes—
408C2×C4Q8212yes—
409C8—44no—
410C8—44no—
411C22—22no—
412C22—22no—
413C22—24no—
414C22—22no—
415C22—22no—
416C22C2×#38511yescharacteristic
417C22#236012yes—
418C22#243111yescharacteristic
419C22#243411yescharacteristic
420C22#245112yes—
421C22#302112yes—
422C22#302212yes—
423C4—24no—
424C4—24no—
425C4—48no—
426C4—44no—
427C4—44no—
428C4—44no—
429C4—81no—
430C4—82no—
431C4—81no—
432C4—82no—
433C4—162no—
434C4C4×C4⋊Q811yescharacteristic
435C4#232611yescharacteristic
436C4Q8×C4⋊C412yes—
437C4#242312yes—
438C4#256511yescharacteristic
439C4#256711yescharacteristic
440C2—22no—
441C2—22no—
442C2#1397611yescharacteristic
443C2#2086512yes—
444C2#2559511yescharacteristic
445C2#2559611yescharacteristic
446C2#2583312yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
31. #1995C4semidirect
128. #339Q8semidirect
151. C4⋊Q8C4⋊C4semidirect
153. #125C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342152
C2×#19952
#213222
#211712
#143542
#113402
#113392
#108922
#108912
#108582
#107012