Finite Group Databasebeta

← #387407

Subgroups — #387407

Order 512.

1476 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic
2#13870C212yesmaximal
3#11381C212yesmaximal
4#11197C214yesmaximal
5#10801C212yesmaximal
6#10736C214yesmaximal
7C22×C22⋊Q8C2211yescharacteristic
8C22×#117C2212yes—
9C23×C4⋊C4C2211yescharacteristic
10#2503—24no—
11#2503—24no—
12C2×#400C414yescomplemented
13C2×#391C412yescomplemented
14C2×#382C412yescomplemented
15C2×#381C2214yes—
16C2×#381C2212yes—
17C2×#381C2214yes—
18C2×C4×C22⋊C4C2211yescharacteristic
19C2×#342C2214yes—
20C2×#342C2212yes—
21C2×#342C2212yes—
22C2×#342C2212yes—
23C2×#342C2214yes—
24C2×#342C2212yes—
25#1441—24no—
26#1436—24no—
27C22⋊#95C2214yes—
28C23×Q8C2×C411yescharacteristic
29C24×C4C2312yes—
30C24×C4C2311yescharacteristic
31C2×C22⋊Q8—24no—
32C2×C22⋊Q8—42no—
33C2×C22⋊Q8—41no—
34C2×C22⋊Q8—41no—
35C2×#117—28no—
36C2×#117—44no—
37C2×#117—42no—
38C2×#117—42no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—41no—
41C22×C4⋊C4—42no—
42C22×C4⋊C4—41no—
43C22×C4⋊C4C2×C412yes—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C412yes—
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4D414yes—
48C22×C4⋊C4D414yes—
49C22×C22⋊C4C2314yes—
50C22×C22⋊C4C2312yes—
51C22×C22⋊C4C2312yes—
52C22×C22⋊C4C2314yes—
53C22×C42—22no—
54C22×C42C2×C412yes—
55#400—24no—
56#400—28no—
57#400—24no—
58#391—22no—
59#391—22no—
60#391—24no—
61#382—24no—
62#382—22no—
63#382—22no—
64#381—24no—
65#381—28no—
66#381—24no—
67#381—28no—
68#381—44no—
69#381—42no—
70#381—42no—
71C4×C22⋊C4—24no—
72C2×#95—28no—
73C2×#95—22no—
74C2×#95—28no—
75C2×#95—22no—
76C2×#95C2×C412yes—
77C2×#95C2×C418yes—
78C2×#95C2×C418yes—
79#342—28no—
80#342—42no—
81#342—42no—
82#342—44no—
83#342—42no—
84#342—42no—
85C25C2411yescharacteristic
86C22×Q8—22no—
87C22×Q8—41no—
88C22×Q8—41no—
89C22×Q8—44no—
90C23×C4—24no—
91C23×C4—24no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—24no—
100C23×C4—24no—
101C23×C4—24no—
102C23×C4—24no—
103C23×C4—44no—
104C23×C4C22×C411yescharacteristic
105C23×C4C22×C414yes—
106C23×C4C22×C412yes—
107C23×C4C22×C412yes—
108C23×C4C22×C414yes—
109C23×C4C22×C411yescharacteristic
110C23×C4C2×D412yes—
111C23×C4C2×D414yes—
112C23×C4C2×D412yes—
113C23×C4C2×D414yes—
114C22⋊Q8—48no—
115C22⋊Q8—48no—
116#117—48no—
117#117—416no—
118#117—48no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—48no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4C22⋊C412yes—
136C2×C4⋊C4C22⋊C414yes—
137C2×C4⋊C4C22⋊C412yes—
138C2×C4⋊C4C22⋊C412yes—
139C2×C4⋊C4C22⋊C414yes—
140C2×C4⋊C4C22⋊C412yes—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4#4118yes—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—48no—
164C2×C42—44no—
165C2×C42—44no—
166#95—24no—
167#95—28no—
168#95—28no—
169#95—28no—
170#95—28no—
171#95—416no—
172#95—416no—
173#95—44no—
174#95—44no—
175C24—42no—
176C24—42no—
177C24C23×C411yescharacteristic
178C24C22×D412yes—
179C24C2×#4114yes—
180C24H(5,𝔽2)12yes—
181C24H(5,𝔽2)11yescharacteristic
182C24H(5,𝔽2)11yescharacteristic
183C24H(5,𝔽2)12yes—
184C242−1+412yes—
185C2×Q8—44no—
186C2×Q8—44no—
187C2×Q8—44no—
188C2×Q8—84no—
189C2×Q8—84no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—24no—
193C22×C4—28no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—28no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—28no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—28no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—28no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—28no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—48no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—48no—
240C22×C4—48no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—48no—
248C22×C4—48no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4C2×C22⋊C414yes—
253C22×C4C2×C22⋊C414yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—44no—
261C4⋊C4—42no—
262C4⋊C4—44no—
263C4⋊C4—48no—
264C4⋊C4—416no—
265C22⋊C4—28no—
266C22⋊C4—28no—
267C22⋊C4—44no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C42—44no—
278C42—416no—
279C42—44no—
280C23—22no—
281C23—24no—
282C23—22no—
283C23—24no—
284C23—42no—
285C23—41no—
286C23—41no—
287C23—48no—
288C23—44no—
289C23—42no—
290C23—41no—
291C23—41no—
292C23—44no—
293C23C22×C22⋊C411yescharacteristic
294C23C4×#4112yes—
295C23#51811yescharacteristic
296C23#51811yescharacteristic
297C23#52012yes—
298C23#53412yes—
299C23#53412yes—
300C23#53814yes—
301C23#54214yes—
302C23#54614yes—
303C23#54714yes—
304C23#55112yes—
305C23#55512yes—
306C23H(3,𝔽4)11yescharacteristic
307C23#56212yes—
308C23#56311yescharacteristic
309Q8—88no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—28no—
318C2×C4—28no—
319C2×C4—24no—
320C2×C4—48no—
321C2×C4—48no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—48no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—48no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—416no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—48no—
345C2×C4—88no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—88no—
355C2×C4—88no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4#41014yes—
360C22—28no—
361C22—24no—
362C22—24no—
363C22—28no—
364C22—21no—
365C22—21no—
366C22—22no—
367C22—22no—
368C22—21no—
369C22—21no—
370C22—22no—
371C22—22no—
372C22—44no—
373C22—44no—
374C22—42no—
375C22—42no—
376C22—41no—
377C22—42no—
378C22—41no—
379C22#230811yescharacteristic
380C22#233712yes—
381C22#233912yes—
382C22#237211yescharacteristic
383C22#237911yescharacteristic
384C22#268412yes—
385C22#268614yes—
386C22#271712yes—
387C22C2×#41012yes—
388C22#288412yes—
389C4—24no—
390C4—44no—
391C4—44no—
392C4—42no—
393C4—88no—
394C4—84no—
395C4—82no—
396C4—82no—
397C4—164no—
398C2—22no—
399C2—24no—
400C2—22no—
401C2—21no—
402C2—21no—
403C2—22no—
404C2#1447311yescharacteristic
405C2#2174711yescharacteristic
406C2#2175111yescharacteristic
407C2#2247814yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. C2×#400C4semidirect
13. C2×#391C4semidirect
14. C2×#382C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
#138702
#113812
#111972
#108012
#107362