Finite Group Databasebeta

← #390996

Subgroups — #390996

Order 512.

974 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2030C211yesmaximal, characteristic
2#11394C212yesmaximal
3#11318C212yesmaximal
4#11070C211yesmaximal, characteristic
5#10936C212yesmaximal
6#10915C212yesmaximal
7#10829C212yesmaximal
8#10736C212yesmaximal
9#10726C211yesmaximal, characteristic
10C2×#543C2211yescharacteristic
11C22×#117C2211yescharacteristic
12#2315C2211yescharacteristic
13C2×#381C2212yes—
14C2×#381C2211yescharacteristic
15C2×#381C2212yes—
16C2×#381C2211yescharacteristic
17#2273C2212yes—
18#2030C418yescomplemented
19C2×#342C2212yes—
20C2×#342C2212yes—
21C2×#342C2211yescharacteristic
22C2×#342C2212yes—
23C2×#342C2212yes—
24C2×#342C2212yes—
25C2×#342C2212yes—
26C2×#342C2212yes—
27C2×#342C2212yes—
28C2×#342C2212yes—
29C2×#342C2212yes—
30C2×#342C2212yes—
31C2×#342C2211yescharacteristic
32C24×C4C2311yescharacteristic
33#543—41no—
34#543—42no—
35#543C2×C414yes—
36C2×C22⋊Q8—42no—
37C2×C22⋊Q8—41no—
38C2×#117—24no—
39C2×#117—42no—
40C2×#117—41no—
41C2×#117—41no—
42C22×C4⋊C4—21no—
43C22×C22⋊C4C2312yes—
44C22×C22⋊C4C2312yes—
45C22×C22⋊C4C2311yescharacteristic
46C22×C22⋊C4C2312yes—
47C22×C22⋊C4C2311yescharacteristic
48C22×C22⋊C4C2312yes—
49C22×C22⋊C4C2312yes—
50C22×C22⋊C4C2312yes—
51C22×C42—21no—
52#385—24no—
53#381—22no—
54#381—22no—
55#381—42no—
56#381—41no—
57#381—41no—
58#381—42no—
59#381C2×C414yes—
60#381C2×C418yes—
61#381Q814yescomplemented
62#380—41no—
63C4×C22⋊C4—24no—
64C4×C22⋊C4—42no—
65C4×C22⋊C4—42no—
66C2×#95—24no—
67C2×#95—41no—
68C2×#95—42no—
69C2×#95—41no—
70#342—24no—
71#342—24no—
72#342—42no—
73#342—42no—
74#342—42no—
75#342—42no—
76#342—42no—
77#342—42no—
78#342—42no—
79#342—42no—
80#342—41no—
81#342C2×C418yes—
82#342C2×C414yes—
83C25C2411yescharacteristic
84C22×Q8—41no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—21no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4C22×C412yes—
101C23×C4C2×Q812yes—
102C23×C4#4111yescharacteristic
103C23×C4#4111yescharacteristic
104C23×C4#4114yes—
105C22⋊Q8—44no—
106C22⋊Q8—48no—
107C22⋊Q8—48no—
108C22⋊Q8—44no—
109#117—44no—
110#117—48no—
111#117—44no—
112C2×C4⋊C4—44no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—43no—
115C2×C4⋊C4—42no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4C22×C414yes—
155C2×C22⋊C4C22×C412yes—
156C2×C22⋊C4C22×C412yes—
157C2×C22⋊C4C22×C414yes—
158C2×C22⋊C4C2×Q814yes—
159C2×C22⋊C4#4114yes—
160C2×C42—44no—
161C2×C42—42no—
162C2×C42—44no—
163#95—44no—
164#95—48no—
165#95—44no—
166#95—44no—
167#95—44no—
168#95—42no—
169#95—44no—
170C24—82no—
171C24C23×C411yescharacteristic
172C24C22×Q811yescharacteristic
173C24C2×#4111yescharacteristic
174C24C2×#4112yes—
175C24C2×#4112yes—
176C24H(5,𝔽2)12yes—
177C24H(5,𝔽2)11yescharacteristic
178C24H(5,𝔽2)11yescharacteristic
179C24H(5,𝔽2)12yes—
180C24H(5,𝔽2)11yescharacteristic
181C242−1+411yescharacteristic
182C2×Q8—44no—
183C2×Q8—162no—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—84no—
235C22×C4—84no—
236C22×C4—84no—
237C22×C4—82no—
238C22×C4—84no—
239C22×C4—82no—
240C22×C4—84no—
241C22×C4—84no—
242C22×C4C4×Q812yes—
243C22×C4C4×Q812yes—
244C4⋊C4—48no—
245C4⋊C4—412no—
246C4⋊C4—162no—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C22⋊C4—24no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—24no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C42—48no—
276C42—164no—
277C23—44no—
278C23—42no—
279C23—42no—
280C23—44no—
281C23—42no—
282C23—81no—
283C23—81no—
284C23—82no—
285C23—81no—
286C23—82no—
287C23—81no—
288C23C2×C4×Q811yescharacteristic
289C23C4×#4112yes—
290C23#51811yescharacteristic
291C23#51812yes—
292C23#52011yescharacteristic
293C23#53811yescharacteristic
294C23#53812yes—
295C23#53812yes—
296C23#53911yescharacteristic
297C23#54212yes—
298C23#54212yes—
299C23#54311yescharacteristic
300C23#54411yescharacteristic
301C23#55112yes—
302C23#55111yescharacteristic
303C23#55112yes—
304C23#55111yescharacteristic
305C23#55412yes—
306C23#55412yes—
307C23#55512yes—
308C23#56011yescharacteristic
309C23#56012yes—
310C23H(3,𝔽4)11yescharacteristic
311Q8—164no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—84no—
336C2×C4—84no—
337C2×C4—82no—
338C2×C4—84no—
339C2×C4—82no—
340C2×C4—84no—
341C2×C4—84no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—84no—
347C2×C4—82no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4—82no—
351C2×C4—84no—
352C2×C4—82no—
353C2×C4—162no—
354C2×C4—162no—
355C2×C4—161no—
356C2×C4—162no—
357C2×C4—161no—
358C2×C4—162no—
359C2×C4—162no—
360C22—24no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—24no—
365C22—42no—
366C22—41no—
367C22—41no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—41no—
373C22—42no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—42no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—42no—
382C22—42no—
383C22#232911yescharacteristic
384C22#233712yes—
385C22#236812yes—
386C22#237211yescharacteristic
387C22#237511yescharacteristic
388C22#273611yescharacteristic
389C22#273712yes—
390C22#274211yescharacteristic
391C22#276011yescharacteristic
392C22#276311yescharacteristic
393C22#278412yes—
394C4—42no—
395C4—42no—
396C4—84no—
397C4—84no—
398C4—82no—
399C4—84no—
400C4—82no—
401C4—84no—
402C4—84no—
403C4—162no—
404C2—21no—
405C2—21no—
406C2—22no—
407C2—21no—
408C2—22no—
409C2—22no—
410C2—21no—
411C2—22no—
412C2—21no—
413C2—21no—
414C2#1459611yescharacteristic
415C2#3270112yes—
subgroupG/H
center, socleC22#2760
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #2030C4semidirect
61. #381Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20302
#113942
#113182
#110702
#109362
#109152
#108292
#107362
#107262