Finite Group Databasebeta

← #189840

Subgroups — #189840

Order 512.

2762 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#380C211yesmaximal, characteristic
2C2×#1779C211yesmaximal, characteristic
3#14680C211yesmaximal, characteristic
4#8712C218yesmaximal, complemented
5#8023C214yesmaximal, complemented
6C25×C4C2211yescharacteristic
7C23×C4⋊C4C412yescomplemented
8C23×C22⋊C4—21no—
9C22×C22⋊C8—22no—
10C22×C22⋊C8C2212yes—
11C2×#380—22no—
12C2×#380—21no—
13C2×#380—22no—
14C2×#380—21no—
15C2×#380—22no—
16C2×#380C414yescomplemented
17C2×#380C412yescomplemented
18C2×#380C2212yes—
19C2×#380C2214yes—
20C2×#380C2212yes—
21C2×#379—21no—
22C22×#95—21no—
23#1779—21no—
24#1779—21no—
25#1779C2214yes—
26C22⋊#95—21no—
27C22⋊#95C2214yes—
28#1294C22116yescomplemented
29C26D411yescharacteristic
30C24×C4—28no—
31C24×C4—22no—
32C24×C4—48no—
33C24×C4C2×C411yescharacteristic
34C24×C4C2×C411yescharacteristic
35C24×C4C2×C412yes—
36C24×C4D412yes—
37C24×C4D412yes—
38C24×C4D411yescharacteristic
39C24×C4D411yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4C2312yes—
42C24×C4C2311yescharacteristic
43C23×C8—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—48no—
49C22×C4⋊C4C2×C414yes—
50C22×C4⋊C4C2×C414yes—
51C22×C4⋊C4C2×C414yes—
52C22×C22⋊C4—24no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—24no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—24no—
58C22×C22⋊C4—44no—
59C22×C22⋊C4—44no—
60C2×C22⋊C8—28no—
61C2×C22⋊C8—24no—
62C2×C22⋊C8—24no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—48no—
65C2×C22⋊C8C2314yes—
66#380—24no—
67#380—24no—
68#380—24no—
69#380—24no—
70#380—22no—
71#380—24no—
72#380—22no—
73#380—28no—
74#380—48no—
75#380C2×C418yes—
76#380C2×C414yes—
77#380C2314yes—
78#379—44no—
79C2×#95—24no—
80C2×#95—22no—
81C2×#95—24no—
82C2×#95—22no—
83C2×#95—24no—
84C2×#95C2314yes—
85C25—24no—
86C25—24no—
87C25—44no—
88C25—44no—
89C25C22⋊C411yescharacteristic
90C25C22⋊C412yes—
91C25C22⋊C411yescharacteristic
92C25D814yes—
93C25SD1614yes—
94C25C2×D412yes—
95C25C2×D411yescharacteristic
96C23×C4—28no—
97C23×C4—24no—
98C23×C4—24no—
99C23×C4—28no—
100C23×C4—28no—
101C23×C4—28no—
102C23×C4—24no—
103C23×C4—24no—
104C23×C4—28no—
105C23×C4—28no—
106C23×C4—24no—
107C23×C4—28no—
108C23×C4—24no—
109C23×C4—24no—
110C23×C4—24no—
111C23×C4—24no—
112C23×C4—44no—
113C23×C4—44no—
114C23×C4—44no—
115C23×C4—44no—
116C23×C4—44no—
117C23×C4—44no—
118C23×C4—44no—
119C23×C4—44no—
120C23×C4—44no—
121C23×C4—44no—
122C23×C4—44no—
123C23×C4—464no—
124C23×C4—48no—
125C23×C4—48no—
126C23×C4—48no—
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C412yes—
130C23×C4C22⋊C412yes—
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22×C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C412yes—
137C23×C4C22×C411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4#4118yes—
145C23×C4C2411yescharacteristic
146C22×C8—24no—
147C22×C8—48no—
148C22×C8—44no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—48no—
153C2×C4⋊C4—48no—
154C2×C4⋊C4—48no—
155C2×C4⋊C4—48no—
156C2×C4⋊C4—48no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4—48no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4C22×C418yes—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—416no—
165C2×C22⋊C4—48no—
166C2×C22⋊C4—48no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4—48no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—48no—
171C2×C22⋊C4—48no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—48no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C22⋊C8—28no—
177C22⋊C8—48no—
178C22⋊C8—48no—
179C22⋊C8—44no—
180#95—28no—
181#95—24no—
182#95—24no—
183#95—28no—
184#95—48no—
185C24—22no—
186C24—24no—
187C24—22no—
188C24—24no—
189C24—24no—
190C24—24no—
191C24—24no—
192C24—22no—
193C24—24no—
194C24—24no—
195C24—24no—
196C24—22no—
197C24—24no—
198C24—24no—
199C24—216no—
200C24—216no—
201C24—42no—
202C24—42no—
203C24—42no—
204C24—42no—
205C24—42no—
206C24—42no—
207C24—416no—
208C24—416no—
209C24—42no—
210C24—42no—
211C24—416no—
212C24—416no—
213C24—432no—
214C24#10214yes—
215C24#10218yes—
216C24#10214yes—
217C24C2×C22⋊C412yes—
218C24C2×C22⋊C411yescharacteristic
219C24C2×C22⋊C411yescharacteristic
220C24C2×C22⋊C412yes—
221C24#12314yes—
222C24C2×D814yes—
223C24C2×D812yes—
224C24C2×SD1614yes—
225C24C2×SD1612yes—
226C24#13714yes—
227C24C22×D411yescharacteristic
228C22×C4—232no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—28no—
257C22×C4—416no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—432no—
271C22×C4—432no—
272C22×C4—432no—
273C22×C4—416no—
274C22×C4—432no—
275C22×C4—416no—
276C22×C4—416no—
277C22×C4—416no—
278C22×C4—48no—
279C22×C4—48no—
280C22×C4—48no—
281C22×C4—48no—
282C22×C4—48no—
283C22×C4—48no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—48no—
287C22×C4—48no—
288C22×C4—48no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—832no—
293C22×C4#9918yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4#11714yes—
299C22×C4#11714yes—
300C22×C4#12314yes—
301C22×C4#12314yes—
302C22×C4#12314yes—
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C2×#4114yes—
306C2×C8—48no—
307C2×C8—44no—
308C2×C8—88no—
309C4⋊C4—48no—
310C4⋊C4—48no—
311C4⋊C4—48no—
312C4⋊C4—48no—
313C4⋊C4—88no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—416no—
317C22⋊C4—48no—
318C22⋊C4—816no—
319C22⋊C4—88no—
320C23—28no—
321C23—216no—
322C23—28no—
323C23—28no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—216no—
335C23—28no—
336C23—28no—
337C23—216no—
338C23—28no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—28no—
346C23—216no—
347C23—28no—
348C23—44no—
349C23—48no—
350C23—48no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—44no—
355C23—44no—
356C23—48no—
357C23—48no—
358C23—48no—
359C23—48no—
360C23—48no—
361C23—48no—
362C23—48no—
363C23—48no—
364C23—48no—
365C23—44no—
366C23—464no—
367C23—464no—
368C23—48no—
369C23—44no—
370C23#38112yes—
371C23#38112yes—
372C23C2×#9914yes—
373C23C2×#10214yes—
374C23C2×#10212yes—
375C23C2×#10212yes—
376C23C2×#10214yes—
377C23#41912yes—
378C23#41912yes—
379C23#42114yes—
380C23#48018yes—
381C23#48318yes—
382C23C22×C22⋊C411yescharacteristic
383C23C2×#12312yes—
384C23C22×D811yescharacteristic
385C23C22×SD1611yescharacteristic
386C23C2×#13712yes—
387C2×C4—216no—
388C2×C4—28no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—28no—
395C2×C4—28no—
396C2×C4—216no—
397C2×C4—28no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—48no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—48no—
408C2×C4—48no—
409C2×C4—48no—
410C2×C4—48no—
411C2×C4—44no—
412C2×C4—48no—
413C2×C4—48no—
414C2×C4—48no—
415C2×C4—48no—
416C2×C4—816no—
417C2×C4—816no—
418C2×C4—816no—
419C2×C4—88no—
420C2×C4—84no—
421C2×C4—88no—
422C2×C4—84no—
423C2×C4—84no—
424C2×C4#38112yes—
425C2×C4#38112yes—
426C2×C4C2×#9914yes—
427C2×C4C2×#9914yes—
428C2×C4C2×#11712yes—
429C2×C4C2×#12312yes—
430C8—84no—
431C22—232no—
432C22—22no—
433C22—24no—
434C22—24no—
435C22—22no—
436C22—24no—
437C22—24no—
438C22—24no—
439C22—24no—
440C22—24no—
441C22—24no—
442C22—22no—
443C22—24no—
444C22—22no—
445C22—22no—
446C22—24no—
447C22—22no—
448C22—22no—
449C22—24no—
450C22—22no—
451C22—24no—
452C22—24no—
453C22—24no—
454C22—48no—
455C22—48no—
456C22—416no—
457C22—416no—
458C22—432no—
459C22—416no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22#151918yes—
465C22#182314yes—
466C22#182314yes—
467C22C2×#38111yescharacteristic
468C22C22×#9912yes—
469C22C22×#10211yescharacteristic
470C22C2×#41911yescharacteristic
471C22C2×#42112yes—
472C22C2×#48012yes—
473C22C2×#48312yes—
474C4—22no—
475C4—22no—
476C4—24no—
477C4—22no—
478C4—42no—
479C4—88no—
480C4—164no—
481C4#181914yes—
482C2—22no—
483C2—24no—
484C2—28no—
485C2—22no—
486C2—24no—
487C2—44no—
488C2C2×#151912yes—
489C2#1757812yes—
490C2C2×#181911yescharacteristic
491C2C2×#182311yescharacteristic
492C2#2070511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8712C2semidirect
5. #8023C2semidirect
7. C23×C4⋊C4C4semidirect
16. C2×#380C4semidirect
17. C2×#380C4semidirect
28. #1294C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3802
C2×#17792
#146802
#87122
#80232