Finite Group Databasebeta

← #382782

Subgroups — #382782

Order 512.

1532 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2322C211yesmaximal, characteristic
2#14342C211yesmaximal, characteristic, complemented
3#14339C211yesmaximal, characteristic, complemented
4#11203C212yesmaximal, complemented
5#11197C212yesmaximal, complemented
6#11043C212yesmaximal
7#10966C212yesmaximal
8#10801C212yesmaximal, complemented
9#10761C212yesmaximal
10C22×#123C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×#99C2214yes—
14#2322—24no—
15#2322—24no—
16C2×#389C412yescomplemented
17C2×#388C412yescomplemented
18C2×#387C412yescomplemented
19C2×#386C2211yescharacteristic
20C2×#385C412yescomplemented
21C2×#385C2211yescharacteristic
22C2×#381C2212yes—
23C2×#381C2212yes—
24C2×#381C2211yescharacteristic
25C2×#380C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27C2×#342C2214yes—
28C2×#342C2214yes—
29C2×#342C2214yes—
30#1445—24no—
31#1441—24no—
32C22⋊#95C2212yes—
33C22⋊#95C2212yes—
34C22⋊#95C2212yes—
35C23×D4C2311yescharacteristic
36C24×C4C2×C411yescharacteristic
37C24×C4C2311yescharacteristic
38C24×C4C2311yescharacteristic
39C24×C4C2311yescharacteristic
40C2×#123—24no—
41C2×#123—22no—
42C2×#123—22no—
43C2×#123—24no—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C2×C4×D4—44no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—44no—
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4D414yes—
55C22×C4⋊C4D414yes—
56C22×C22⋊C4C2×C412yes—
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C412yes—
59C22×C22⋊C4C2314yes—
60C22×C22⋊C4C2314yes—
61C22×C22⋊C4C2311yescharacteristic
62C22×C22⋊C4C2311yescharacteristic
63C22×C22⋊C4C2311yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C412yes—
66C2×#99—28no—
67C2×#99—44no—
68C2×#99—44no—
69#389—28no—
70#388—28no—
71#387—28no—
72#386—22no—
73#386—22no—
74#385—22no—
75#385—22no—
76#385—28no—
77#381—22no—
78#381—22no—
79#381—44no—
80#381—44no—
81#380—44no—
82C4×C22⋊C4—22no—
83C4×C22⋊C4—22no—
84C2×#95—22no—
85C2×#95—24no—
86C2×#95—24no—
87C2×#95—24no—
88C2×#95—24no—
89C2×#95—24no—
90C2×#95C2×C412yes—
91C2×#95C2×C412yes—
92C2×#95C2×C412yes—
93C2×#95C2×C412yes—
94C2×#95C2×C412yes—
95#342—28no—
96#342—44no—
97#342—44no—
98C25C22×C411yescharacteristic
99C25C2411yescharacteristic
100C22×D4—21no—
101C22×D4—21no—
102C22×D4—22no—
103C22×D4—24no—
104C22×D4—22no—
105C22×D4—22no—
106C22×D4#4114yes—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—24no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—24no—
113C23×C4—24no—
114C23×C4—22no—
115C23×C4—24no—
116C23×C4—24no—
117C23×C4—42no—
118C23×C4—82no—
119C23×C4—82no—
120C23×C4—82no—
121C23×C4C22×C412yes—
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C412yes—
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C412yes—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136#123—216no—
137#123—216no—
138C4×D4—48no—
139C4×D4—48no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—48no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—84no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4C22⋊C418yes—
155C2×C4⋊C4C22⋊C418yes—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4#4114yes—
186C2×C42—22no—
187C2×C42—24no—
188C2×C42—42no—
189C2×C42—82no—
190#99—416no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95—44no—
197#95—48no—
198#95—48no—
199#95—84no—
200#95—84no—
201#95—84no—
202C24—21no—
203C24—21no—
204C24—22no—
205C24—22no—
206C24—42no—
207C24—42no—
208C24—42no—
209C24—42no—
210C24#11712yes—
211C24C23×C411yescharacteristic
212C24C22×D411yescharacteristic
213C24C22×D411yescharacteristic
214C24C2×#4112yes—
215C24C2×#4112yes—
216C24H(5,𝔽2)14yes—
217C24H(5,𝔽2)12yes—
218C242−1+412yes—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—28no—
222C2×D4—28no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—42no—
228C2×D4—44no—
229C2×D4—44no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—48no—
281C22×C4—48no—
282C22×C4—48no—
283C22×C4—48no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—84no—
291C22×C4—88no—
292C22×C4—88no—
293C22×C4—88no—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C414yes—
296C22×C4C2×C22⋊C414yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4#11712yes—
299C22×C4#11714yes—
300C4⋊C4—216no—
301C4⋊C4—48no—
302C4⋊C4—48no—
303C4⋊C4—44no—
304C4⋊C4—84no—
305C4⋊C4—88no—
306C22⋊C4—216no—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—44no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—42no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—84no—
324C42—44no—
325C42—84no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—24no—
334C23—44no—
335C23—44no—
336C23—44no—
337C23—44no—
338C23—44no—
339C23—44no—
340C23—44no—
341C23—44no—
342C23—41no—
343C23—41no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—82no—
348C23—84no—
349C23—82no—
350C23C22×C22⋊C411yescharacteristic
351C23C2×#11711yescharacteristic
352C23C2×#11711yescharacteristic
353C23#51812yes—
354C23#52012yes—
355C23#52512yes—
356C23#52512yes—
357C23#53412yes—
358C23#53412yes—
359C23#53511yescharacteristic
360C23#53611yescharacteristic
361C23#53712yes—
362C23#53814yes—
363C23#53914yes—
364C23#54014yes—
365C23#54214yes—
366D4—48no—
367D4—44no—
368D4—44no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—48no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4—88no—
410C2×C4—84no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—84no—
414C2×C4—88no—
415C2×C4—88no—
416C2×C4—88no—
417C2×C4—88no—
418C2×C4—84no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4—84no—
422C2×C4—84no—
423C22—22no—
424C22—24no—
425C22—24no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—24no—
430C22—24no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—44no—
438C22—44no—
439C22—44no—
440C22—44no—
441C22—42no—
442C22—44no—
443C22—84no—
444C22—84no—
445C22—82no—
446C22—81no—
447C22—81no—
448C22#228011yescharacteristic
449C22#230811yescharacteristic
450C22#231111yescharacteristic
451C22#231812yes—
452C22#231912yes—
453C22#263014yes—
454C22#263512yes—
455C22#263812yes—
456C22#288312yes—
457C22#288412yes—
458C4—44no—
459C4—42no—
460C4—44no—
461C4—42no—
462C4—44no—
463C4—42no—
464C4—82no—
465C4—81no—
466C4—81no—
467C4—164no—
468C4—164no—
469C2—22no—
470C2—24no—
471C2—22no—
472C2—22no—
473C2—22no—
474C2—82no—
475C2#1434411yescharacteristic
476C2#2072512yes—
477C2#2174911yescharacteristic
478C2#2175111yescharacteristic
479C2#2187712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
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. #14342C2semidirect
3. #14339C2semidirect
4. #11203C2semidirect
5. #11197C2semidirect
8. #10801C2semidirect
16. C2×#389C4semidirect
17. C2×#388C4semidirect
18. C2×#387C4semidirect
20. C2×#385C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23222
#143422
#143392
#112032
#111972
#110432
#109662
#108012
#107612