Finite Group Databasebeta

← #185758

Subgroups — #185758

Order 512.

3330 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Q8C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic, complemented
3#17742C212yesmaximal, complemented
4#17448C212yesmaximal
5#17365C211yesmaximal, characteristic, complemented
6#8388C218yesmaximal, complemented
7C24×Q8—21no—
8C2×D4×Q8—24no—
9C2×D4×Q8—24no—
10C2×D4×Q8—22no—
11C2×D4×Q8—22no—
12C2×D4×Q8—24no—
13C2×D4×Q8—24no—
14C2×D4×Q8C414yes—
15C2×D4×Q8C2214yes—
16C22×C4⋊Q8C412yes—
17C22×C4⋊Q8C2211yescharacteristic
18C22×C22⋊Q8—21no—
19C22×C22⋊Q8—22no—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C412yes—
22C22×C4×D4C2211yescharacteristic
23#2920C2212yes—
24#2920C2212yes—
25#1821—21no—
26#1801C2214yescomplemented
27#1784—21no—
28C4×#101—22no—
29#1556—24no—
30#1549—24no—
31#1546C2218yescomplemented
32#1490—24no—
33#1486C2218yes—
34#1478—24no—
35#1467—22no—
36#1463—22no—
37#1461C2214yescomplemented
38C23×Q8—28no—
39C23×Q8—28no—
40C23×Q8—24no—
41C23×Q8—24no—
42C23×Q8—22no—
43C23×Q8—21no—
44C23×Q8—416no—
45C23×Q8D411yescharacteristic
46C23×D4C2×C411yescharacteristic
47C24×C4—22no—
48C24×C4—21no—
49D4×Q8—216no—
50D4×Q8—216no—
51D4×Q8—216no—
52D4×Q8—216no—
53D4×Q8—216no—
54D4×Q8—216no—
55D4×Q8—216no—
56D4×Q8—216no—
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8—24no—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8—24no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8C2×C418yes—
67C2×C4⋊Q8C2×C412yes—
68C2×C4⋊Q8D414yes—
69C2×C4⋊Q8D414yes—
70C2×C4⋊Q8C2312yes—
71C2×C22⋊Q8—28no—
72C2×C22⋊Q8—28no—
73C2×C22⋊Q8—216no—
74C2×C22⋊Q8—216no—
75C2×C22⋊Q8—24no—
76C2×C22⋊Q8—28no—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8—28no—
79C2×C4×Q8—22no—
80C2×C4×Q8—22no—
81C2×C4×Q8—22no—
82C2×C4×Q8—22no—
83C2×C4×Q8C2×C412yes—
84C2×C4×Q8D412yes—
85C2×C4×Q8D412yes—
86C2×C4×Q8C2312yes—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4—22no—
93C2×C4×D4—22no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4C2×C418yes—
97C2×C4×D4C2×C412yes—
98C2×C4×D4D414yes—
99C2×C4×D4D414yes—
100C2×C4×D4C2312yes—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2×C412yes—
104C22×C4⋊C4C2×C411yescharacteristic
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4D412yes—
107C22×C22⋊C4—22no—
108C22×C22⋊C4—21no—
109C22×C42C2×C412yes—
110C22×C42C2311yescharacteristic
111#432—22no—
112#432—22no—
113#432—22no—
114#432—22no—
115#431—22no—
116#431—22no—
117#431—22no—
118#431C2314yes—
119#423—22no—
120#423C2314yes—
121C2×#101—24no—
122#407—22no—
123#407—22no—
124C4×M16—22no—
125C4×M16—22no—
126#343—24no—
127#332—216no—
128D4⋊C8—216no—
129#324—28no—
130C25—21no—
131C22×Q8—28no—
132C22×Q8—28no—
133C22×Q8—28no—
134C22×Q8—232no—
135C22×Q8—232no—
136C22×Q8—22no—
137C22×Q8—28no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—24no—
141C22×Q8—24no—
142C22×Q8—22no—
143C22×Q8—48no—
144C22×Q8—416no—
145C22×Q8—464no—
146C22×Q8—464no—
147C22×Q8—416no—
148C22×Q8—432no—
149C22×Q8—416no—
150C22×Q8C22⋊C412yes—
151C22×Q8C2×D412yes—
152C22×D4—28no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4C22⋊C412yes—
157C22×D4C22⋊C412yes—
158C22×D4C22×C412yes—
159C23×C4—28no—
160C23×C4—28no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—28no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—48no—
169C23×C4—416no—
170C23×C4C22⋊C412yes—
171C23×C4C22⋊C412yes—
172C23×C4C22×C411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C2×M16—24no—
176C2×M16—42no—
177C4⋊Q8—216no—
178C4⋊Q8—216no—
179C4⋊Q8—216no—
180C4⋊Q8—216no—
181C4⋊Q8—24no—
182C4⋊Q8—24no—
183C4⋊Q8—24no—
184C4⋊Q8—24no—
185C4⋊Q8—24no—
186C4⋊Q8—24no—
187C4⋊Q8C22⋊C418yes—
188C4⋊Q8C2×D418yes—
189C22⋊Q8—232no—
190C22⋊Q8—232no—
191C22⋊Q8—264no—
192C22⋊Q8—264no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×Q8—24no—
201C4×D4—216no—
202C4×D4—216no—
203C4×D4—216no—
204C4×D4—216no—
205C4×D4—24no—
206C4×D4—24no—
207C4×D4—24no—
208C4×D4—24no—
209C4×D4—24no—
210C4×D4—24no—
211C4×D4C22⋊C418yes—
212C4×D4C2×D418yes—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4C22⋊C412yes—
231C2×C4⋊C4C22⋊C414yes—
232C2×C4⋊C4C22⋊C414yes—
233C2×C4⋊C4C22⋊C414yes—
234C2×C4⋊C4C22⋊C412yes—
235C2×C4⋊C4C22⋊C412yes—
236C2×C4⋊C4C22×C414yes—
237C2×C4⋊C4C22×C412yes—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×D412yes—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42C22⋊C414yes—
252C2×C42C22×C414yes—
253C2×C42C22×C411yescharacteristic
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2411yescharacteristic
261C4⋊C8—24no—
262C4⋊C8—24no—
263C4⋊C8—24no—
264C4⋊C8—42no—
265#101—48no—
266C22⋊C8—24no—
267C22⋊C8—42no—
268C22⋊C8—42no—
269C22⋊C8—42no—
270#97—42no—
271#97—42no—
272C4×C8—24no—
273C4×C8—42no—
274C24—28no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—44no—
279C24—44no—
280C24C2×C22⋊C411yescharacteristic
281C2×Q8—216no—
282C2×Q8—216no—
283C2×Q8—232no—
284C2×Q8—24no—
285C2×Q8—28no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—28no—
289C2×Q8—28no—
290C2×Q8—432no—
291C2×Q8—432no—
292C2×Q8—48no—
293C2×Q8—416no—
294C2×Q8—416no—
295C2×Q8—416no—
296C2×Q8—4128no—
297C2×Q8—4128no—
298C2×Q8—48no—
299C2×Q8—416no—
300C2×Q8AGL(1,ℤ/8)18yes—
301C2×D4—28no—
302C2×D4—28no—
303C2×D4—28no—
304C2×D4—28no—
305C2×D4—28no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—24no—
310C2×D4#10118yescomplemented
311C2×D4C2×C22⋊C414yes—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—28no—
317C22×C4—232no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—42no—
329C22×C4—432no—
330C22×C4—432no—
331C22×C4—48no—
332C22×C4—416no—
333C22×C4—416no—
334C22×C4—416no—
335C22×C4—44no—
336C22×C4—48no—
337C22×C4—48no—
338C22×C4—416no—
339C22×C4—44no—
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C414yes—
346C22×C4C2×C22⋊C414yes—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4#12012yes—
350C22×C4#12012yes—
351C22×C4#12012yes—
352C22×C4#12012yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356M16—48no—
357M16—84no—
358C2×C8—44no—
359C2×C8—44no—
360C4⋊C4—232no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—28no—
364C4⋊C4—28no—
365C4⋊C4—24no—
366C4⋊C4—28no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—28no—
372C4⋊C4—24no—
373C4⋊C4C2×C22⋊C418yes—
374C4⋊C4C2×C22⋊C414yes—
375C22⋊C4—232no—
376C22⋊C4—28no—
377C22⋊C4—28no—
378C42—24no—
379C42—24no—
380C42—24no—
381C42—24no—
382C42C2×C22⋊C412yes—
383C42C2×C22⋊C412yes—
384C42#12012yes—
385C42#12012yes—
386C42#12012yes—
387C42#12012yes—
388C42C22×D412yes—
389C42C22×D412yes—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—28no—
394C23—28no—
395C23—24no—
396C23—28no—
397C23—44no—
398C23—44no—
399C23—432no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23#37911yescharacteristic
405C23#37911yescharacteristic
406C23C22×C22⋊C411yescharacteristic
407Q8—216no—
408Q8—216no—
409Q8—416no—
410Q8—464no—
411Q8—416no—
412D4—28no—
413D4—28no—
414D4—24no—
415D4—28no—
416D4C2×#10118yes—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—28no—
432C2×C4—24no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—416no—
437C2×C4—464no—
438C2×C4—44no—
439C2×C4—48no—
440C2×C4—48no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—48no—
446C2×C4—416no—
447C2×C4—44no—
448C2×C4#37911yescharacteristic
449C2×C4#37912yes—
450C2×C4#37912yes—
451C2×C4#37911yescharacteristic
452C2×C4C2×#10114yes—
453C2×C4#41818yes—
454C2×C4C22×C22⋊C411yescharacteristic
455C2×C4C22×C22⋊C411yescharacteristic
456C2×C4C2×#12012yes—
457C2×C4C2×#12012yes—
458C2×C4C2×AGL(1,ℤ/8)14yes—
459C8—84no—
460C8—84no—
461C22—28no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—28no—
469C22—416no—
470C22—416no—
471C22—416no—
472C22—44no—
473C22—44no—
474C22#178412yes—
475C22#179612yes—
476C22C2×#37911yescharacteristic
477C22C2×#41812yes—
478C4—21no—
479C4—21no—
480C4—22no—
481C4—28no—
482C4—22no—
483C4—22no—
484C4—48no—
485C4—44no—
486C4—44no—
487C4#178412yes—
488C4C22×#10112yes—
489C4#300314yes—
490C2—22no—
491C2—22no—
492C2—22no—
493C2—48no—
494C2C2×#178411yescharacteristic
495C2#2059811yescharacteristic
496C2C2×#179611yescharacteristic
subgroupG/H
center, socleC22C2×#379
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20727C2semidirect
3. #17742C2semidirect
5. #17365C2semidirect
6. #8388C2semidirect
26. #1801C22semidirect
31. #1546C22semidirect
37. #1461C22semidirect
310. C2×D4#101semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Q82
#207272
#177422
#174482
#173652
#83882