Finite Group Databasebeta

← #127127

Subgroups — #127127

Order 512.

749 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21786C211yesmaximal, characteristic
2#20722C211yesmaximal, characteristic
3#17362C211yesmaximal, characteristic
4#7205C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C2×#431C411yescharacteristic, complemented
7C2×#431C411yescharacteristic, complemented
8C2×#407—21no—
9#2280—21no—
10#1954—21no—
11#1954—21no—
12#1954C412yescomplemented
13#1954C2212yes—
14#1819—21no—
15#1799—21no—
16#1799C412yes—
17#1781C412yes—
18#1779—21no—
19#1779—21no—
20C4×#99—21no—
21C4×#99C2212yes—
22#1465—21no—
23#1465C412yescomplemented
24#1463C412yescomplemented
25#1462—21no—
26#1459—21no—
27#1459C2212yes—
28C23×D4C2×C411yescharacteristic
29C24×C4D411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—41no—
32C2×C4×D4—41no—
33C2×C4×D4—41no—
34C2×C4×D4—41no—
35C2×C4×D4—41no—
36C2×C4×D4C2×C411yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4D411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4Q811yescharacteristic
43C2×C4×D4C2311yescharacteristic
44C2×#117—41no—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C22⋊C4D411yescharacteristic
47C22×C22⋊C4Q811yescharacteristic
48C22×C42C2×C411yescharacteristic
49#431—41no—
50#431—41no—
51#431—41no—
52#431—41no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8C2×C412yes—
55C2×#99—22no—
56C2×#99C2×C414yes—
57#407—22no—
58#407—22no—
59#407—41no—
60#407—41no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—41no—
69C2×C4×C8—21no—
70C2×C4×C8C2×C412yes—
71#381—41no—
72#380—22no—
73#380—41no—
74#379—22no—
75C4×C22⋊C4—22no—
76C4×C22⋊C4—41no—
77#342—22no—
78#342—22no—
79#324—22no—
80#324C2×C414yes—
81#323—22no—
82#323—22no—
83C25C22⋊C411yescharacteristic
84C25C4⋊C411yescharacteristic
85C22×D4—24no—
86C22×D4—42no—
87C22×D4—41no—
88C22×D4—41no—
89C22×D4C4211yescharacteristic
90C22×D4C22⋊C411yescharacteristic
91C22×D4C4⋊C411yescharacteristic
92C22×D4C22×C411yescharacteristic
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×C4C4⋊C411yescharacteristic
111C23×C4C4⋊C411yescharacteristic
112C23×C4C4⋊C411yescharacteristic
113C23×C4C4⋊C411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C2×M16—42no—
117C2×M16—42no—
118C2×M16—84no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—42no—
122C4×D4—41no—
123C4×D4—41no—
124C4×D4—41no—
125C4×D4—41no—
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—
138#117—84no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4C4211yescharacteristic
143C2×C4⋊C4C22⋊C411yescharacteristic
144C2×C4⋊C4C4⋊C411yescharacteristic
145C2×C4⋊C4C22×C411yescharacteristic
146C2×C22⋊C4—24no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—42no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4C22⋊C411yescharacteristic
154C2×C22⋊C4C22⋊C411yescharacteristic
155C2×C22⋊C4C4⋊C411yescharacteristic
156C2×C22⋊C4C4⋊C411yescharacteristic
157C2×C22⋊C4C4⋊C411yescharacteristic
158C2×C22⋊C4C4⋊C411yescharacteristic
159C2×C22⋊C4C2×D411yescharacteristic
160C2×C22⋊C4C2×Q811yescharacteristic
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—42no—
166C2×C42C4211yescharacteristic
167C2×C42C22⋊C411yescharacteristic
168C2×C42C4⋊C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170C4⋊C8—42no—
171C4⋊C8—42no—
172#99—28no—
173#99—44no—
174C22⋊C8—24no—
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—
186C4×C8—42no—
187C4×C8—42no—
188#95—42no—
189#95—42no—
190C24—21no—
191C24—21no—
192C24—21no—
193C24—21no—
194C24—22no—
195C24—22no—
196C24—41no—
197C24—41no—
198C24—81no—
199C24—81no—
200C24—81no—
201C24—81no—
202C24#9511yescharacteristic
203C24#9511yescharacteristic
204C24#9511yescharacteristic
205C24C2×C22⋊C411yescharacteristic
206C24C2×C4⋊C411yescharacteristic
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—22no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—48no—
219C2×D4—48no—
220C2×D4#13114yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
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—21no—
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—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—81no—
268C22×C4#9511yescharacteristic
269C22×C4#9511yescharacteristic
270C22×C4#9511yescharacteristic
271C22×C4#9511yescharacteristic
272C22×C4#9511yescharacteristic
273C22×C4#9911yescharacteristic
274C22×C4#9911yescharacteristic
275C22×C4#9911yescharacteristic
276C22×C4#9911yescharacteristic
277C22×C4C2×C4211yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C4⋊C411yescharacteristic
281C22×C4C2×C4⋊C411yescharacteristic
282M16—84no—
283M16—84no—
284C2×C8—44no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—84no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—84no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—44no—
306C22⋊C4—44no—
307C22⋊C4—84no—
308C22⋊C4—84no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—84no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—41no—
331C23—41no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—84no—
353C23—84no—
354C23—84no—
355C23—84no—
356C23#34211yescharacteristic
357C23#34211yescharacteristic
358C23C2×#9511yescharacteristic
359C23#40812yes—
360C23#42412yes—
361D4—44no—
362D4—44no—
363D4—44no—
364D4—44no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
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—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
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—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—84no—
412C2×C4—84no—
413C2×C4—84no—
414C2×C4#34211yescharacteristic
415C2×C4#34211yescharacteristic
416C2×C4#34412yes—
417C2×C4#40512yes—
418C2×C4C2×#9911yescharacteristic
419C2×C4C2×#9911yescharacteristic
420C8—82no—
421C8—84no—
422C8—82no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—22no—
432C22—22no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—84no—
454C22—84no—
455C22—84no—
456C22—84no—
457C22#172911yescharacteristic
458C22C2×#34211yescharacteristic
459C22C2×#34411yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—22no—
464C4—22no—
465C4—41no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—164no—
472C4#212712yes—
473C2—21no—
474C2—21no—
475C2—42no—
476C2—41no—
477C2—41no—
478C2—42no—
479C2—81no—
480C2—81no—
481C2#792711yescharacteristic
482C2#1258211yescharacteristic
483C2#1258911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. #7205C2semidirect
6. C2×#431C4semidirect
7. C2×#431C4semidirect
12. #1954C4semidirect
23. #1465C4semidirect
24. #1463C4semidirect

Maximal subgroups

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

subgroupindex
#217862
#207222
#173622
#72052