Finite Group Databasebeta

← #380014

Subgroups — #380014

Order 512.

1934 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2626C211yesmaximal, characteristic
2C2×#2312C211yesmaximal, characteristic
3C2×#1996C212yesmaximal, complemented
4#14338C211yesmaximal, characteristic
5#11203C214yesmaximal, complemented
6#11196C214yesmaximal
7#10854C212yesmaximal, complemented
8C22×C22⋊Q8C2211yescharacteristic
9C23×C22⋊C4C2211yescharacteristic
10#2626—24no—
11#2626—24no—
12C2×#399C412yescomplemented
13C2×#396C414yescomplemented
14C2×#393C412yescomplemented
15#2312—28no—
16C2×#388—24no—
17C2×#385C2212yes—
18C2×#380C2212yes—
19C2×#380C2212yes—
20C2×#380C2211yescharacteristic
21C2×#379C2212yes—
22C2×#379C2212yes—
23C2×#379C2212yes—
24C2×C4×C22⋊C4C2212yes—
25#1996—28no—
26C2×#342C2218yes—
27C2×#342C2218yes—
28#1439—24no—
29C22⋊#95C2212yes—
30C26C2311yescharacteristic
31C23×Q8C2×C411yescharacteristic
32C24×C4C2311yescharacteristic
33C24×C4C2311yescharacteristic
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—22no—
36C2×C22⋊Q8—24no—
37C2×C22⋊Q8—24no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4C2×C412yes—
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C412yes—
43C22×C4⋊C4C2×C414yes—
44C22×C22⋊C4—24no—
45C22×C22⋊C4—24no—
46C22×C22⋊C4—22no—
47C22×C22⋊C4—22no—
48C22×C22⋊C4—24no—
49C22×C22⋊C4C2×C412yes—
50C22×C22⋊C4C2×C412yes—
51C22×C22⋊C4C2×C414yes—
52C22×C22⋊C4C2×C414yes—
53C22×C22⋊C4D414yes—
54C22×C22⋊C4D414yes—
55C22×C22⋊C4C2314yes—
56C22×C22⋊C4C2312yes—
57C22×C22⋊C4C2312yes—
58C22×C22⋊C4C2314yes—
59C22×C42—22no—
60#399—28no—
61#396—216no—
62#393—28no—
63#388—232no—
64#385—24no—
65#385—24no—
66#380—22no—
67#380—22no—
68#380—28no—
69#380—24no—
70#380—24no—
71#379—24no—
72#379—24no—
73#379—28no—
74#379—44no—
75C4×C22⋊C4—24no—
76C4×C22⋊C4D418yes—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—24no—
81C2×#95C2×C414yes—
82C2×#95C2×C412yes—
83#342—216no—
84#342—48no—
85C25—22no—
86C25—22no—
87C25—22no—
88C25—24no—
89C25—22no—
90C25—82no—
91C25—82no—
92C25C22×C412yes—
93C25C2×D412yes—
94C25C2×D412yes—
95C25C2411yescharacteristic
96C22×Q8—22no—
97C22×Q8—24no—
98C22×Q8—82no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—24no—
102C23×C4—24no—
103C23×C4—82no—
104C23×C4—82no—
105C23×C4C22×C412yes—
106C23×C4C22×C414yes—
107C23×C4C22×C412yes—
108C23×C4C22×C412yes—
109C23×C4C22×C411yescharacteristic
110C23×C4C22×C411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D414yes—
116C23×C4#4114yes—
117C22⋊Q8—44no—
118C22⋊Q8—44no—
119C22⋊Q8—44no—
120C22⋊Q8—44no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—84no—
130C2×C4⋊C4—84no—
131C2×C4⋊C4—82no—
132C2×C4⋊C4—84no—
133C2×C22⋊C4—28no—
134C2×C22⋊C4—28no—
135C2×C22⋊C4—28no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—28no—
138C2×C22⋊C4—28no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
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—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—28no—
155C2×C22⋊C4—28no—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—28no—
158C2×C22⋊C4—28no—
159C2×C22⋊C4—28no—
160C2×C22⋊C4—48no—
161C2×C22⋊C4—84no—
162C2×C22⋊C4—84no—
163C2×C22⋊C4—84no—
164C2×C22⋊C4—84no—
165C2×C22⋊C4—84no—
166C2×C22⋊C4—84no—
167C2×C22⋊C4C22⋊C418yes—
168C2×C22⋊C4C22⋊C418yes—
169C2×C22⋊C4C2×D418yes—
170C2×C42—28no—
171C2×C42—84no—
172#95—28no—
173#95—24no—
174#95—28no—
175#95—28no—
176#95—44no—
177#95—84no—
178C24—24no—
179C24—22no—
180C24—22no—
181C24—22no—
182C24—22no—
183C24—24no—
184C24—24no—
185C24—24no—
186C24—24no—
187C24—24no—
188C24—24no—
189C24—42no—
190C24—42no—
191C24—44no—
192C24—44no—
193C24—44no—
194C24—44no—
195C24—44no—
196C24—44no—
197C24—44no—
198C24—44no—
199C24—44no—
200C24—44no—
201C24—84no—
202C24—84no—
203C24—84no—
204C24—84no—
205C24—88no—
206C24—816no—
207C24—88no—
208C24C2×C22⋊C414yes—
209C24C23×C411yescharacteristic
210C24C22×D411yescharacteristic
211C24C22×D411yescharacteristic
212C24C22×D412yes—
213C24C2×#4112yes—
214C24H(5,𝔽2)12yes—
215C242−1+414yes—
216C242−1+412yes—
217C2×Q8—44no—
218C2×Q8—84no—
219C2×Q8—88no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—28no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—28no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—42no—
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—44no—
245C22×C4—44no—
246C22×C4—48no—
247C22×C4—48no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—84no—
252C22×C4—84no—
253C22×C4—84no—
254C22×C4—84no—
255C22×C4C2×C22⋊C414yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C4×D414yes—
259C22×C4C4×D414yes—
260C22×C4#12014yes—
261C22×C4#12114yes—
262C22×C4#12114yes—
263C22×C4#12314yes—
264C4⋊C4—44no—
265C4⋊C4—48no—
266C4⋊C4—84no—
267C4⋊C4—88no—
268C4⋊C4—88no—
269C4⋊C4—88no—
270C22⋊C4—216no—
271C22⋊C4—216no—
272C22⋊C4—28no—
273C22⋊C4—28no—
274C22⋊C4—42no—
275C22⋊C4—44no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—44no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C22⋊C4—48no—
283C22⋊C4—48no—
284C22⋊C4—88no—
285C22⋊C4—88no—
286C22⋊C4—88no—
287C22⋊C4—88no—
288C22⋊C4—816no—
289C42—88no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—24no—
297C23—24no—
298C23—28no—
299C23—28no—
300C23—24no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—44no—
306C23—48no—
307C23—44no—
308C23—48no—
309C23—44no—
310C23—44no—
311C23—416no—
312C23—48no—
313C23—44no—
314C23—44no—
315C23—48no—
316C23—44no—
317C23—44no—
318C23—44no—
319C23—44no—
320C23—88no—
321C23—88no—
322C23—88no—
323C23—88no—
324C23—84no—
325C23—84no—
326C23—84no—
327C23—84no—
328C23—816no—
329C23—816no—
330C23—816no—
331C23—816no—
332C23C22×C22⋊C411yescharacteristic
333C23C2×C4×D412yes—
334C23#51912yes—
335C23#52012yes—
336C23C2×#12011yescharacteristic
337C23C2×#12112yes—
338C23C2×#12311yescharacteristic
339C23#53511yescharacteristic
340C23#53612yes—
341C23#53612yes—
342C23#53611yescharacteristic
343C23#53614yes—
344C23#53712yes—
345C23#53714yes—
346C23#54114yes—
347C23#54214yes—
348Q8—88no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—28no—
352C2×C4—24no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—48no—
362C2×C4—48no—
363C2×C4—44no—
364C2×C4—48no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—88no—
374C2×C4—88no—
375C2×C4—88no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—88no—
379C2×C4—84no—
380C2×C4—88no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4#38614yes—
384C2×C4#38614yes—
385C22—24no—
386C22—24no—
387C22—24no—
388C22—22no—
389C22—24no—
390C22—24no—
391C22—22no—
392C22—21no—
393C22—22no—
394C22—22no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—22no—
399C22—22no—
400C22—44no—
401C22—44no—
402C22—44no—
403C22—44no—
404C22—48no—
405C22—48no—
406C22—44no—
407C22—44no—
408C22—44no—
409C22—88no—
410C22—88no—
411C22—816no—
412C22—88no—
413C22—88no—
414C22C2×#38611yescharacteristic
415C22#231011yescharacteristic
416C22#231111yescharacteristic
417C22#232314yes—
418C22#261612yes—
419C22#262012yes—
420C22#262112yes—
421C22#262512yes—
422C22#288312yes—
423C22#288312yes—
424C4—24no—
425C4—24no—
426C4—44no—
427C4—88no—
428C4—84no—
429C4—82no—
430C4—82no—
431C4—164no—
432C2—22no—
433C2—22no—
434C2—24no—
435C2—22no—
436C2—22no—
437C2—84no—
438C2#1425811yescharacteristic
439C2#2098411yescharacteristic
440C2#2098511yescharacteristic
441C2#2182712yes—
442C2#2187612yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC26C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C22⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1996C2semidirect
5. #11203C2semidirect
7. #10854C2semidirect
12. C2×#399C4semidirect
13. C2×#396C4semidirect
14. C2×#393C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26262
C2×#23122
C2×#19962
#143382
#112032
#111962
#108542