Finite Group Databasebeta

← #380536

Subgroups — #380536

Order 512.

1828 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×#2618C211yesmaximal, characteristic
2C2×#1899C214yesmaximal, complemented
3#14339C211yesmaximal, characteristic, complemented
4#13795C211yesmaximal, characteristic, complemented
5#11353C212yesmaximal, complemented
6#11043C214yesmaximal, complemented
7#10980C212yesmaximal
8C22×#123C2212yes—
9C22×#121C2211yescharacteristic
10C22×#117C2212yes—
11C23×C4⋊C4C2211yescharacteristic
12C22×#99C2218yes—
13#2618—24no—
14#2618—24no—
15C2×#398C412yescomplemented
16C2×#396C414yescomplemented
17C2×#394C412yescomplemented
18C2×#386C2212yes—
19C2×#381C2212yes—
20C2×#380C2212yes—
21C2×#380C2211yescharacteristic
22#1899—216no—
23C2×#342C2218yes—
24#1441—24no—
25C4⋊#95—24no—
26C22⋊#95C2212yes—
27C22⋊#95C2212yes—
28C22⋊#95C2212yes—
29C23×D4C2×C411yescharacteristic
30C23×D4C2312yes—
31C24×C4C2311yescharacteristic
32C24×C4C2311yescharacteristic
33C24×C4C2311yescharacteristic
34C2×#123—24no—
35C2×#123—24no—
36C2×#123—28no—
37C2×#123—28no—
38C2×#121—22no—
39C2×#121—22no—
40C2×#121—44no—
41C2×#117—24no—
42C2×#117—44no—
43C2×#117—44no—
44C2×#117D418yescomplemented
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—44no—
48C22×C4⋊C4C2×C414yes—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4D414yes—
54C22×C4⋊C4D414yes—
55C22×C22⋊C4C2×C414yes—
56C22×C22⋊C4C2×C412yes—
57C22×C22⋊C4C2×C414yes—
58C22×C22⋊C4C2314yes—
59C22×C22⋊C4C2312yes—
60C22×C22⋊C4C2314yes—
61C22×C42—22no—
62C2×#99—216no—
63C2×#99—48no—
64C2×#99—48no—
65#398—28no—
66#396—216no—
67#394—28no—
68#386—24no—
69#386—24no—
70#381—28no—
71#380—22no—
72#380—22no—
73#380—44no—
74C2×#95—22no—
75C2×#95—24no—
76C2×#95—24no—
77C2×#95—24no—
78C2×#95—24no—
79C2×#95C2×C412yes—
80C2×#95C2×C414yes—
81#342—216no—
82C25C22×C412yes—
83C25C2411yescharacteristic
84C22×D4—22no—
85C22×D4—22no—
86C22×D4—24no—
87C22×D4—24no—
88C22×D4—28no—
89C22×D4—24no—
90C22×D4—24no—
91C22×D4—22no—
92C22×D4—42no—
93C22×D4—82no—
94C23×C4—24no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—24no—
98C23×C4—24no—
99C23×C4—24no—
100C23×C4—24no—
101C23×C4—82no—
102C23×C4—82no—
103C23×C4C22×C412yes—
104C23×C4C22×C412yes—
105C23×C4C22×C414yes—
106C23×C4C22×C411yescharacteristic
107C23×C4C22×C412yes—
108C23×C4C22×C411yescharacteristic
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2×D414yes—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4#4114yes—
117#123—48no—
118#123—48no—
119#123—48no—
120#123—48no—
121#121—48no—
122#121—48no—
123#117—416no—
124#117—416no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—216no—
128C2×C4⋊C4—28no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—44no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—84no—
141C2×C4⋊C4—82no—
142C2×C4⋊C4C22⋊C418yes—
143C2×C4⋊C4C22⋊C418yes—
144C2×C22⋊C4—28no—
145C2×C22⋊C4—28no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—28no—
153C2×C22⋊C4—216no—
154C2×C22⋊C4—28no—
155C2×C22⋊C4—28no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—28no—
159C2×C22⋊C4—44no—
160C2×C22⋊C4—84no—
161C2×C22⋊C4C2×D418yes—
162C2×C42—28no—
163C2×C42—44no—
164C2×C42—44no—
165#99—432no—
166#95—24no—
167#95—28no—
168#95—44no—
169#95—48no—
170#95—84no—
171#95—84no—
172#95—84no—
173C24—22no—
174C24—24no—
175C24—24no—
176C24—24no—
177C24—42no—
178C24—42no—
179C24—44no—
180C24—44no—
181C24C23×C411yescharacteristic
182C24C22×D411yescharacteristic
183C24C22×D411yescharacteristic
184C24C22×D412yes—
185C24C2×#4112yes—
186C24H(5,𝔽2)14yes—
187C24H(5,𝔽2)12yes—
188C242−1+412yes—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—48no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—44no—
199C2×D4—48no—
200C2×D4—48no—
201C2×D4—48no—
202C2×D4—84no—
203C2×D4—88no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—28no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—28no—
214C22×C4—28no—
215C22×C4—28no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—28no—
230C22×C4—28no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—48no—
239C22×C4—48no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—48no—
245C22×C4—48no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—88no—
250C22×C4—88no—
251C22×C4C2×C22⋊C414yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C414yes—
255C22×C4C4×D418yes—
256C22×C4#12114yes—
257C22×C4#12114yes—
258C22×C4#12414yes—
259C22×C4#12714yes—
260C4⋊C4—48no—
261C4⋊C4—48no—
262C4⋊C4—416no—
263C4⋊C4—44no—
264C4⋊C4—416no—
265C4⋊C4—84no—
266C4⋊C4—88no—
267C22⋊C4—216no—
268C22⋊C4—48no—
269C22⋊C4—44no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—48no—
273C22⋊C4—416no—
274C22⋊C4—48no—
275C22⋊C4—48no—
276C22⋊C4—88no—
277C42—416no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—44no—
285C23—44no—
286C23—44no—
287C23—44no—
288C23—44no—
289C23—44no—
290C23—42no—
291C23—42no—
292C23—44no—
293C23—44no—
294C23—42no—
295C23—48no—
296C23—84no—
297C23—88no—
298C23—88no—
299C23—84no—
300C23C22×C22⋊C411yescharacteristic
301C23C2×C4×D412yes—
302C23#51812yes—
303C23#52012yes—
304C23C2×#12112yes—
305C23C2×#12411yescharacteristic
306C23C2×#12711yescharacteristic
307C23#53412yes—
308C23#53412yes—
309C23#53414yes—
310C23#53511yescharacteristic
311C23#53611yescharacteristic
312C23#53712yes—
313C23#53714yes—
314C23#53814yes—
315C23#53914yes—
316D4—88no—
317D4—84no—
318D4—84no—
319D4—84no—
320D4—84no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—216no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—28no—
329C2×C4—28no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—28no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—416no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—48no—
344C2×C4—48no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—84no—
350C2×C4—84no—
351C2×C4—84no—
352C2×C4—84no—
353C2×C4—88no—
354C2×C4—84no—
355C2×C4—84no—
356C2×C4—88no—
357C2×C4—84no—
358C2×C4—88no—
359C2×C4—88no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4#39018yes—
363C22—22no—
364C22—24no—
365C22—24no—
366C22—22no—
367C22—24no—
368C22—22no—
369C22—24no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—44no—
378C22—44no—
379C22—44no—
380C22—44no—
381C22—84no—
382C22—82no—
383C22—84no—
384C22—82no—
385C22—82no—
386C22—168no—
387C22C2×#39011yescharacteristic
388C22#230811yescharacteristic
389C22#231111yescharacteristic
390C22#232514yes—
391C22#261712yes—
392C22#261812yes—
393C22#263912yes—
394C22#265612yes—
395C22#288312yes—
396C22#288412yes—
397C4—28no—
398C4—42no—
399C4—44no—
400C4—42no—
401C4—88no—
402C4—82no—
403C4—84no—
404C4—164no—
405C2—22no—
406C2—22no—
407C2—24no—
408C2—22no—
409C2—22no—
410C2—162no—
411C2#1427111yescharacteristic
412C2#2121412yes—
413C2#2174911yescharacteristic
414C2#2175111yescharacteristic
415C2#2187712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC22×#121C22
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
2. C2×#1899C2semidirect
3. #14339C2semidirect
4. #13795C2semidirect
5. #11353C2semidirect
6. #11043C2semidirect
15. C2×#398C4semidirect
16. C2×#396C4semidirect
17. C2×#394C4semidirect
44. C2×#117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26182
C2×#18992
#143392
#137952
#113532
#110432
#109802