Finite Group Databasebeta

← #468161

Subgroups — #468161

Order 512.

1096 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35173C211yesmaximal, characteristic
2#23554C212yesmaximal
3#14049C211yesmaximal, characteristic
4#11386C212yesmaximal
5#11342C212yesmaximal
6#11342C212yesmaximal
7#11296C212yesmaximal
8#11210C212yesmaximal
9#10802C211yesmaximal, characteristic
10C2×#533C2212yes—
11C2×#428C2214yes—
12#2857—22no—
13#2659—22no—
14#2584—24no—
15#2567—22no—
16#2546—22no—
17#2316C2212yes—
18#2316C2212yes—
19C2×#389C2211yescharacteristic
20C2×#387—22no—
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#2059—24no—
24#2058—22no—
25#2057—22no—
26C2×#343C2212yes—
27C2×#343C2212yes—
28#1736C2214yes—
29C2×#339C2214yes—
30C2×#339C2214yes—
31#1728C2212yes—
32#1445—22no—
33C4⋊#95C2211yescharacteristic
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2211yescharacteristic
36C4×#95C2211yescharacteristic
37C22×M16C2314yes—
38#533—44no—
39#533—44no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8D412yes—
42C2×C4⋊Q8D412yes—
43C2×#125D414yes—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46C2×C4×Q8—22no—
47C2×C4×Q8—22no—
48C2×#117C2314yes—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C42D412yes—
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59#428—44no—
60#428—44no—
61C2×#107—24no—
62C2×#106—24no—
63#398—28no—
64#395—28no—
65#391—24no—
66#389—24no—
67#389—24no—
68#389—24no—
69#389—24no—
70#389—24no—
71#389—24no—
72#389—22no—
73#389—22no—
74#387—48no—
75#384—22no—
76#384—22no—
77#384Q814yescomplemented
78#380—24no—
79#380—24no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—21no—
88C2×#95D412yes—
89#343—24no—
90#343Q818yescomplemented
91#340—24no—
92#340—24no—
93#339—44no—
94#339—44no—
95#336—22no—
96#336—22no—
97C22×Q8—22no—
98C22×Q8—22no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D412yes—
106C23×C4C2×D412yes—
107C23×C4C2411yescharacteristic
108C2×M16—24no—
109C2×M16—24no—
110C2×M16—24no—
111C2×M16—24no—
112C2×M16—24no—
113C2×M16C2×Q818yes—
114C22×C8—24no—
115C4⋊Q8—44no—
116C4⋊Q8—44no—
117C4⋊Q8—84no—
118#125—44no—
119#125—44no—
120C22⋊Q8—44no—
121C22⋊Q8—44no—
122C22⋊Q8—44no—
123C22⋊Q8—44no—
124C4×Q8—48no—
125C4×Q8—48no—
126#117—24no—
127#117—24no—
128#117—24no—
129#117—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4C2×D414yes—
155C2×C4⋊C4C2×D414yes—
156C2×C4⋊C4C2×Q812yes—
157C2×C4⋊C4C2×Q812yes—
158C2×C4⋊C4C2×Q814yes—
159C2×C4⋊C4#4112yes—
160C2×C22⋊C4—24no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—24no—
167C2×C42—21no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—44no—
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2×Q812yes—
176#107—48no—
177#106—48no—
178#95—24no—
179#95—28no—
180#95—28no—
181#95—24no—
182#95—42no—
183#95—44no—
184#95—44no—
185#95—44no—
186#95—42no—
187C24C22×D411yescharacteristic
188C2×Q8—44no—
189C2×Q8—44no—
190C2×Q8—44no—
191C2×Q8—44no—
192C2×Q8—82no—
193C2×Q8—82no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4#12012yes—
237C22×C4#12011yescharacteristic
238C22×C4#12011yescharacteristic
239C22×C4C22⋊Q814yes—
240C22×C4C4⋊Q814yes—
241C22×C4C22×D412yes—
242C22×C4C22×Q812yes—
243C22×C4C22×Q811yescharacteristic
244C22×C4C2×#4111yescharacteristic
245C22×C4H(5,𝔽2)12yes—
246C22×C4H(5,𝔽2)12yes—
247C22×C42−1+412yes—
248C22×C42−1+412yes—
249M16—28no—
250M16—28no—
251M16—44no—
252M16—44no—
253C2×C8—28no—
254C2×C8—44no—
255C2×C8—44no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—28no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—416no—
265C4⋊C4—48no—
266C4⋊C4—48no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—82no—
272C4⋊C4—84no—
273C4⋊C4—84no—
274C4⋊C4—84no—
275C4⋊C4—84no—
276C4⋊C4—82no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C42—24no—
280C42—24no—
281C42—24no—
282C42—44no—
283C42—44no—
284C42—42no—
285C42—44no—
286C42—44no—
287C42—82no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23C2×#12011yescharacteristic
293C23C2×C22⋊Q811yescharacteristic
294C23C2×C4⋊Q811yescharacteristic
295C23#53511yescharacteristic
296C23#53511yescharacteristic
297C23#53611yescharacteristic
298C23#53611yescharacteristic
299Q8—88no—
300Q8—88no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—24no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—82no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4#45312yes—
354C2×C4#53414yes—
355C2×C4#54212yes—
356C2×C4#54414yes—
357C2×C4#54412yes—
358C2×C4#54812yes—
359C2×C4D4×Q814yes—
360C2×C4D4×Q812yes—
361C2×C4#55412yes—
362C2×C4Q8212yes—
363C2×C4#56314yes—
364C8—48no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22#240911yescharacteristic
378C22#243411yescharacteristic
379C22#256711yescharacteristic
380C22#261311yescharacteristic
381C22#262811yescharacteristic
382C22#266011yescharacteristic
383C22#282111yescharacteristic
384C22C2×#45311yescharacteristic
385C22#301811yescharacteristic
386C22#302011yescharacteristic
387C22#302211yescharacteristic
388C4—22no—
389C4—44no—
390C4—41no—
391C4—42no—
392C4—44no—
393C4—44no—
394C4—82no—
395C4—84no—
396C4—84no—
397C4—82no—
398C4—161no—
399C4#272914yes—
400C4#273414yes—
401C2—21no—
402C2—21no—
403C2—21no—
404C2—21no—
405C2#1660311yescharacteristic
406C2#2431211yescharacteristic
407C2#2431911yescharacteristic
408C2#2582411yescharacteristic
409C2#2582711yescharacteristic
410C2#2961611yescharacteristic
411C2#2962111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×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
77. #384Q8semidirect
90. #343Q8semidirect

Maximal subgroups

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

subgroupindex
#351732
#235542
#140492
#113862
#113422
#112962
#112102
#108022