Finite Group Databasebeta

← #380534

Subgroups — #380534

Order 512.

1932 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2C2×#1899C214yesmaximal, complemented
3#14685C211yesmaximal, characteristic, complemented
4#13756C211yesmaximal, characteristic, complemented
5#11353C212yesmaximal, complemented
6#11043C214yesmaximal, complemented
7#10980C212yesmaximal
8C22×#123C2212yes—
9C22×#121C2211yescharacteristic
10C22×#117C2212yes—
11C23×C42C2211yescharacteristic
12C22×#99C2218yes—
13#2854—22no—
14#2854—22no—
15#2854—22no—
16#2854—22no—
17C2×#390C412yescomplemented
18C2×#389C412yescomplemented
19C2×#388C414yescomplemented
20C2×#386C2212yes—
21C2×#381C2212yes—
22C2×#380C2212yes—
23C2×#380C2211yescharacteristic
24#1899—216no—
25C2×#342C2218yes—
26C4⋊#95—22no—
27C4⋊#95—22no—
28#1434—22no—
29C22⋊#95C2212yes—
30C22⋊#95C2212yes—
31C22⋊#95C2212yes—
32C4×#95—22no—
33C23×D4C2×C411yescharacteristic
34C23×D4C2312yes—
35C24×C4C2311yescharacteristic
36C24×C4C2311yescharacteristic
37C24×C4C2311yescharacteristic
38C2×#123—22no—
39C2×#123—22no—
40C2×#123—24no—
41C2×#123—28no—
42C2×#123—28no—
43C2×#121—22no—
44C2×#121—22no—
45C2×#121—44no—
46C2×#117—24no—
47C2×#117—44no—
48C2×#117—44no—
49C2×#117D418yescomplemented
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2×C414yes—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C412yes—
55C22×C22⋊C4C2×C414yes—
56C22×C22⋊C4C2×C412yes—
57C22×C22⋊C4C2×C414yes—
58C22×C22⋊C4C2314yes—
59C22×C22⋊C4C2312yes—
60C22×C22⋊C4C2314yes—
61C22×C42—22no—
62C22×C42—44no—
63C22×C42C2×C412yes—
64C22×C42C2×C412yes—
65C22×C42D412yes—
66C22×C42D412yes—
67C22×C42D412yes—
68C22×C42D412yes—
69C2×#99—216no—
70C2×#99—48no—
71C2×#99—48no—
72#390—22no—
73#390—22no—
74#390—22no—
75#390—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#388—24no—
81#388—24no—
82#388—24no—
83#388—24no—
84#386—22no—
85#386—24no—
86#386—22no—
87#381—28no—
88#380—22no—
89#380—22no—
90#380—44no—
91C2×#95—24no—
92C2×#95—24no—
93C2×#95—24no—
94C2×#95—22no—
95C2×#95—24no—
96C2×#95C2×C412yes—
97C2×#95C2×C414yes—
98#342—216no—
99C25C22×C412yes—
100C25C2411yescharacteristic
101C22×D4—22no—
102C22×D4—22no—
103C22×D4—24no—
104C22×D4—24no—
105C22×D4—28no—
106C22×D4—24no—
107C22×D4—24no—
108C22×D4—22no—
109C22×D4—42no—
110C22×D4—44no—
111C23×C4—22no—
112C23×C4—24no—
113C23×C4—22no—
114C23×C4—24no—
115C23×C4—24no—
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—44no—
119C23×C4—44no—
120C23×C4C22×C412yes—
121C23×C4C22×C411yescharacteristic
122C23×C4C22×C412yes—
123C23×C4C22×C412yes—
124C23×C4C22×C414yes—
125C23×C4C22×C411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D414yes—
134C23×C4C2×D411yescharacteristic
135C23×C4#4114yes—
136#123—48no—
137#123—48no—
138#123—48no—
139#123—48no—
140#121—48no—
141#121—48no—
142#117—416no—
143#117—416no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—216no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—48no—
156C2×C4⋊C4—44no—
157C2×C22⋊C4—28no—
158C2×C22⋊C4—28no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—216no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—48no—
176C2×C22⋊C4C2×D418yes—
177C2×C42—28no—
178C2×C42—44no—
179C2×C42—44no—
180C2×C42—44no—
181C2×C42—44no—
182C2×C42—44no—
183C2×C42—44no—
184C2×C42—44no—
185C2×C42—44no—
186C2×C42C22⋊C412yes—
187C2×C42C22⋊C412yes—
188C2×C42C22⋊C412yes—
189C2×C42C22⋊C412yes—
190C2×C42C22⋊C412yes—
191C2×C42C22⋊C412yes—
192C2×C42C22⋊C412yes—
193C2×C42C22⋊C412yes—
194#99—432no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—48no—
199#95—48no—
200#95—48no—
201#95—44no—
202#95—48no—
203C24—22no—
204C24—24no—
205C24—24no—
206C24—24no—
207C24—42no—
208C24—42no—
209C24—44no—
210C24—44no—
211C24C23×C411yescharacteristic
212C24C22×D411yescharacteristic
213C24C22×D411yescharacteristic
214C24C22×D412yes—
215C24C2×#4112yes—
216C24H(5,𝔽2)14yes—
217C24H(5,𝔽2)12yes—
218C242−1+412yes—
219C2×D4—44no—
220C2×D4—44no—
221C2×D4—44no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—48no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—44no—
230C2×D4—48no—
231C2×D4—48no—
232C2×D4—48no—
233C2×D4—44no—
234C2×D4—88no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—28no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—24no—
248C22×C4—28no—
249C22×C4—28no—
250C22×C4—28no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—28no—
269C22×C4—28no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—416no—
277C22×C4—416no—
278C22×C4—48no—
279C22×C4—48no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C4×D418yes—
296C22×C4#12114yes—
297C22×C4#12114yes—
298C22×C4#12414yes—
299C22×C4#12714yes—
300C4⋊C4—44no—
301C4⋊C4—416no—
302C4⋊C4—44no—
303C4⋊C4—48no—
304C4⋊C4—48no—
305C4⋊C4—44no—
306C4⋊C4—88no—
307C22⋊C4—216no—
308C22⋊C4—48no—
309C22⋊C4—48no—
310C22⋊C4—44no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—48no—
314C22⋊C4—48no—
315C22⋊C4—48no—
316C22⋊C4—48no—
317C22⋊C4—48no—
318C22⋊C4—48no—
319C42—42no—
320C42—42no—
321C42—42no—
322C42—42no—
323C42—42no—
324C42—42no—
325C42—42no—
326C42—42no—
327C42—416no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—44no—
335C23—44no—
336C23—44no—
337C23—44no—
338C23—44no—
339C23—44no—
340C23—42no—
341C23—42no—
342C23—44no—
343C23—44no—
344C23—42no—
345C23—48no—
346C23—84no—
347C23—88no—
348C23—88no—
349C23—84no—
350C23C22×C22⋊C411yescharacteristic
351C23C2×C4×D412yes—
352C23#51812yes—
353C23#52012yes—
354C23C2×#12112yes—
355C23C2×#12411yescharacteristic
356C23C2×#12711yescharacteristic
357C23#53412yes—
358C23#53412yes—
359C23#53414yes—
360C23#53511yescharacteristic
361C23#53611yescharacteristic
362C23#53712yes—
363C23#53714yes—
364C23#53814yes—
365C23#53914yes—
366D4—88no—
367D4—84no—
368D4—84no—
369D4—84no—
370D4—84no—
371C2×C4—28no—
372C2×C4—28no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—216no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—28no—
387C2×C4—28no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—48no—
395C2×C4—48no—
396C2×C4—48no—
397C2×C4—48no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—416no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—48no—
404C2×C4—48no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—88no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4—88no—
413C2×C4—84no—
414C2×C4—88no—
415C2×C4—88no—
416C2×C4—84no—
417C2×C4—84no—
418C2×C4#39014yes—
419C2×C4#39014yes—
420C2×C4#41014yes—
421C2×C4#41014yes—
422C22—22no—
423C22—24no—
424C22—24no—
425C22—22no—
426C22—24no—
427C22—22no—
428C22—24no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—44no—
437C22—44no—
438C22—44no—
439C22—44no—
440C22—84no—
441C22—82no—
442C22—84no—
443C22—82no—
444C22—82no—
445C22—168no—
446C22C2×#39011yescharacteristic
447C22#230811yescharacteristic
448C22#231111yescharacteristic
449C22#232514yes—
450C22#261712yes—
451C22#261812yes—
452C22#263912yes—
453C22#265612yes—
454C22C2×#41012yes—
455C22C2×#41012yes—
456C4—24no—
457C4—24no—
458C4—24no—
459C4—24no—
460C4—42no—
461C4—42no—
462C4—88no—
463C4—82no—
464C4—84no—
465C4—164no—
466C2—22no—
467C2—22no—
468C2—24no—
469C2—22no—
470C2—22no—
471C2—162no—
472C2#1427111yescharacteristic
473C2#2121212yes—
474C2#2174711yescharacteristic
475C2#2174811yescharacteristic
476C2#2187512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1899C2semidirect
3. #14685C2semidirect
4. #13756C2semidirect
5. #11353C2semidirect
6. #11043C2semidirect
17. C2×#390C4semidirect
18. C2×#389C4semidirect
19. C2×#388C4semidirect
49. C2×#117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C2×#18992
#146852
#137562
#113532
#110432
#109802