Finite Group Databasebeta

← #127487

Subgroups — #127487

Order 512.

704 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21788C211yesmaximal, characteristic
2#17362C211yesmaximal, characteristic
3C2×#1465C211yesmaximal, characteristic
4#7249C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C2×#432C411yescharacteristic
7C2×#431C411yescharacteristic
8C2×#407—21no—
9C2×#407—21no—
10#1955—21no—
11#1955—21no—
12#1954C412yes—
13#1954C2212yes—
14#1779—21no—
15#1779—21no—
16#1465—21no—
17#1465—21no—
18#1465—21no—
19#1465C412yes—
20#1465C2212yes—
21#1465C2212yes—
22#1463—21no—
23#1462C412yes—
24#1459—21no—
25#1459C412yes—
26C2×#324—21no—
27C2×#323C412yes—
28C23×D4Q811yescharacteristic
29C24×C4D411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—41no—
32C22×M16—41no—
33C2×C4×D4—41no—
34C2×C4×D4—41no—
35C2×C4×D4—41no—
36C2×C4×D4—41no—
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4D411yescharacteristic
43C2×C4×D4Q811yescharacteristic
44C2×C4×D4C2311yescharacteristic
45C22×C4⋊C4D411yescharacteristic
46C22×C22⋊C4C2×C411yescharacteristic
47C22×C22⋊C4C2×C411yescharacteristic
48C22×C42C2×C411yescharacteristic
49#432—41no—
50#432—41no—
51#431—41no—
52#431—41no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8C2×C412yes—
55#407—22no—
56#407—22no—
57#407—22no—
58#407—22no—
59#407—41no—
60#407—41no—
61#407—41no—
62#407—41no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—41no—
72C2×C22⋊C8—41no—
73C2×#97—21no—
74C2×C4×C8C2×C412yes—
75#324—22no—
76#324—22no—
77#324—22no—
78#324—22no—
79#323—22no—
80#323—22no—
81#323C2×C414yes—
82#323C2×C414yes—
83C25C4⋊C411yescharacteristic
84C25C4⋊C411yescharacteristic
85C22×D4—42no—
86C22×D4—41no—
87C22×D4—41no—
88C22×D4—42no—
89C22×D4C4⋊C411yescharacteristic
90C22×D4C4⋊C411yescharacteristic
91C22×D4C4⋊C411yescharacteristic
92C22×D4C2×Q811yescharacteristic
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—82no—
102C23×C4—82no—
103C23×C4C4211yescharacteristic
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22⋊C411yescharacteristic
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C4⋊C411yescharacteristic
112C23×C4C4⋊C411yescharacteristic
113C23×C4C4⋊C411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C2×M16—42no—
117C2×M16—42no—
118C2×M16—42no—
119C2×M16—42no—
120C2×M16—84no—
121C2×M16—84no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—42no—
125C22×C8—42no—
126C4×D4—41no—
127C4×D4—41no—
128C4×D4—41no—
129C4×D4—41no—
130C4×D4—41no—
131C4×D4—41no—
132C4×D4—41no—
133C4×D4—41no—
134C4×D4—41no—
135C4×D4—41no—
136C4×D4—41no—
137C4×D4—41no—
138C4×D4—41no—
139C4×D4—41no—
140C4×D4—41no—
141C4×D4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4C22⋊C411yescharacteristic
145C2×C4⋊C4C22⋊C411yescharacteristic
146C2×C4⋊C4C4⋊C411yescharacteristic
147C2×C4⋊C4C2×D411yescharacteristic
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4C4211yescharacteristic
153C2×C22⋊C4C4211yescharacteristic
154C2×C22⋊C4C22⋊C411yescharacteristic
155C2×C22⋊C4C22⋊C411yescharacteristic
156C2×C22⋊C4C4⋊C411yescharacteristic
157C2×C22⋊C4C4⋊C411yescharacteristic
158C2×C22⋊C4C22×C411yescharacteristic
159C2×C22⋊C4C22×C411yescharacteristic
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42C4211yescharacteristic
165C2×C42C22⋊C411yescharacteristic
166C2×C42C4⋊C411yescharacteristic
167C2×C42C22×C411yescharacteristic
168C4⋊C8—42no—
169C4⋊C8—42no—
170C22⋊C8—24no—
171C22⋊C8—24no—
172C22⋊C8—42no—
173C22⋊C8—42no—
174C22⋊C8—42no—
175C22⋊C8—42no—
176C22⋊C8—42no—
177C22⋊C8—42no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182C22⋊C8—82no—
183C22⋊C8—82no—
184C22⋊C8—82no—
185C22⋊C8—82no—
186#97—42no—
187C4×C8—42no—
188C24—21no—
189C24—21no—
190C24—21no—
191C24—22no—
192C24—21no—
193C24—41no—
194C24—41no—
195C24—41no—
196C24—42no—
197C24—42no—
198C24—81no—
199C24—81no—
200C24#9511yescharacteristic
201C24#9511yescharacteristic
202C24#9511yescharacteristic
203C24C2×C4⋊C411yescharacteristic
204C24C2×C4⋊C411yescharacteristic
205C2×D4—22no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—48no—
218C2×D4—48no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4#9511yescharacteristic
266C22×C4#9511yescharacteristic
267C22×C4#9511yescharacteristic
268C22×C4#9511yescharacteristic
269C22×C4#9511yescharacteristic
270C22×C4#9911yescharacteristic
271C22×C4#9911yescharacteristic
272C22×C4#9911yescharacteristic
273C22×C4#9911yescharacteristic
274C22×C4C2×C4211yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C4⋊C411yescharacteristic
279M16—84no—
280M16—84no—
281M16—84no—
282M16—84no—
283C2×C8—44no—
284C2×C8—44no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—84no—
292C2×C8—84no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C23—22no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—41no—
328C23—42no—
329C23—42no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—84no—
349C23—84no—
350C23—84no—
351C23—84no—
352C23#33812yes—
353C23#34211yescharacteristic
354C23#34211yescharacteristic
355C23C2×#9511yescharacteristic
356D4—44no—
357D4—44no—
358D4—44no—
359D4—44no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4#34211yescharacteristic
408C2×C4#34211yescharacteristic
409C2×C4C2×#9911yescharacteristic
410C2×C4C2×#9911yescharacteristic
411C2×C4#43012yes—
412C8—82no—
413C8—84no—
414C8—84no—
415C8—82no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—22no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—81no—
442C22—84no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—84no—
447C22—84no—
448C22—84no—
449C22#139212yes—
450C22C2×#33811yescharacteristic
451C22#173811yescharacteristic
452C22C2×#34211yescharacteristic
453C4—21no—
454C4—21no—
455C4—21no—
456C4—22no—
457C4—41no—
458C4—41no—
459C4—41no—
460C4—41no—
461C4—41no—
462C4—41no—
463C4—41no—
464C4#212712yes—
465C2—21no—
466C2—21no—
467C2—22no—
468C2—41no—
469C2—42no—
470C2—81no—
471C2—81no—
472C2—81no—
473C2#800911yescharacteristic
474C2C2×#139211yescharacteristic
475C2#1258211yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
prime radicalC23×D4Q8
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7249C2semidirect

Maximal subgroups

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

subgroupindex
#217882
#173622
C2×#14652
#72492