Finite Group Databasebeta

← #386112

Subgroups — #386112

Order 512.

1066 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2C2×#1998C212yesmaximal
3#21023C212yesmaximal
4#11383C212yesmaximal
5#11354C212yesmaximal
6#11204C212yesmaximal
7#10913C212yesmaximal
8#10802C211yesmaximal, characteristic
9#10802C211yesmaximal, characteristic
10C22×#125C2212yes—
11C2×#423C2212yes—
12C2×C4×M16C2212yes—
13#2552—22no—
14#2552—22no—
15#2552—22no—
16#2552—22no—
17C2×#398C414yes—
18C2×#389C412yes—
19C2×#389C2212yes—
20C2×#384C412yes—
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1998—28no—
25#1917—22no—
26#1917—22no—
27#1916—22no—
28#1916—22no—
29C2×#343C2212yes—
30C2×#343C2211yescharacteristic
31C2×#343C2212yes—
32C2×#343C2211yescharacteristic
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2211yescharacteristic
38C2×#343C2211yescharacteristic
39#1728C2212yes—
40#1728C2212yes—
41#1728C2212yes—
42#1728C2212yes—
43C22×M16C2312yes—
44C22×M16C2312yes—
45C22×M16C2312yes—
46C22×M16C2312yes—
47C2×#125—24no—
48C2×#125—24no—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125—42no—
54C2×#125—42no—
55C22×C4⋊C4C2×C414yes—
56C22×C4⋊C4C2×C414yes—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2×C414yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C42C2×C412yes—
66C22×C42C2312yes—
67C22×C42C2311yescharacteristic
68#423—24no—
69#423—42no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C4×M16—24no—
73C4×M16—42no—
74C2×#97—22no—
75C2×C4×C8—22no—
76#398—24no—
77#398—24no—
78#398—24no—
79#398—24no—
80#389—24no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#384—24no—
88#384—22no—
89#384—22no—
90#384—22no—
91#384—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4D412yes—
95C4×C4⋊C4D412yes—
96C4×C4⋊C4D412yes—
97C4×C4⋊C4D412yes—
98C2×#95C2×C412yes—
99C2×#95C2×C414yes—
100C2×#95C2×C412yes—
101#343—24no—
102#343—24no—
103#343—24no—
104#343—22no—
105#343—24no—
106#343—22no—
107#343—42no—
108#343—42no—
109#343D414yes—
110#343Q814yes—
111#336—22no—
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C414yes—
121C23×C4C22×C412yes—
122C23×C4C22×C412yes—
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C412yes—
125C23×C4C22×C412yes—
126C23×C4C2411yescharacteristic
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16C2×D414yes—
150C2×M16C2×Q814yes—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155#125—44no—
156#125—44no—
157#125—44no—
158#125—44no—
159#125—44no—
160#125—44no—
161#125—44no—
162#125—44no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—84no—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×Q812yes—
199C2×C4⋊C4#4112yes—
200C2×C4⋊C4#4112yes—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—44no—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C4⋊C8—44no—
218C4⋊C8—44no—
219#97—44no—
220C4×C8—44no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95C22⋊C412yes—
225#95C22⋊C412yes—
226#95C22⋊C412yes—
227#95C22⋊C412yes—
228C24C23×C411yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C4×D414yes—
266C22×C4C4×Q814yes—
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D412yes—
269C22×C4C22×Q811yescharacteristic
270C22×C4C2×#4111yescharacteristic
271C22×C4C2×#4111yescharacteristic
272C22×C4H(5,𝔽2)12yes—
273C22×C42−1+412yes—
274C22×C42−1+411yescharacteristic
275C22×C42−1+411yescharacteristic
276C22×C42−1+412yes—
277M16—24no—
278M16—24no—
279M16—24no—
280M16—24no—
281M16—42no—
282M16—42no—
283M16—42no—
284M16—42no—
285M16—42no—
286M16—42no—
287M16—42no—
288M16—42no—
289M16—42no—
290M16—42no—
291M16—42no—
292M16—42no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—48no—
316C4⋊C4—48no—
317C4⋊C4—48no—
318C4⋊C4—48no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—44no—
325C4⋊C4—84no—
326C4⋊C4—84no—
327C4⋊C4—84no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—44no—
341C42—44no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23C22×C22⋊C411yescharacteristic
347C23C2×C4×D411yescharacteristic
348C23C2×C4×Q811yescharacteristic
349C23#51811yescharacteristic
350C23#51911yescharacteristic
351C23#51911yescharacteristic
352C23#51911yescharacteristic
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—84no—
394C2×C4—84no—
395C2×C4#41312yes—
396C2×C4#41312yes—
397C2×C4#53612yes—
398C2×C4#53612yes—
399C2×C4#53612yes—
400C2×C4#54112yes—
401C2×C4#54212yes—
402C2×C4#54412yes—
403C2×C4D4212yes—
404C2×C4#54612yes—
405C2×C4#54711yescharacteristic
406C2×C4#54812yes—
407C2×C4D4×Q811yescharacteristic
408C2×C4D4×Q812yes—
409C2×C4#55611yescharacteristic
410C2×C4#55711yescharacteristic
411C2×C4#56312yes—
412C2×C4#56312yes—
413C8—44no—
414C8—44no—
415C8—82no—
416C8—82no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22#231011yescharacteristic
430C22#232711yescharacteristic
431C22#233111yescharacteristic
432C22D4×C22⋊C411yescharacteristic
433C22Q8×C22⋊C411yescharacteristic
434C22#236511yescharacteristic
435C22#238111yescharacteristic
436C22C2×#41311yescharacteristic
437C22C2×#41311yescharacteristic
438C22#288811yescharacteristic
439C22#288911yescharacteristic
440C4—22no—
441C4—22no—
442C4—42no—
443C4—42no—
444C4—42no—
445C4—44no—
446C4—42no—
447C4—42no—
448C4—44no—
449C4—84no—
450C4#267412yes—
451C4#267712yes—
452C4#271312yes—
453C4#271612yes—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—21no—
458C2#1442011yescharacteristic
459C2#2183011yescharacteristic
460C2#2183211yescharacteristic
461C2#2241111yescharacteristic
462C2#2241211yescharacteristic
463C2#2251511yescharacteristic
464C2#2251711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#19982
#210232
#113832
#113542
#112042
#109132
#108022