Finite Group Databasebeta

← #185773

Subgroups — #185773

Order 512.

2494 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic, complemented
3#17742C212yesmaximal, complemented
4#17448C212yesmaximal
5#17365C211yesmaximal, characteristic, complemented
6#8398C218yesmaximal, complemented
7C2×#554—24no—
8C2×#554—24no—
9C2×#554—22no—
10C2×#554—22no—
11C2×#554—24no—
12C2×#554—24no—
13C2×#554C414yes—
14C2×#554C2214yes—
15C22×C4⋊Q8C2211yescharacteristic
16C22×#125C412yes—
17C22×C22⋊Q8—21no—
18C22×C22⋊Q8—22no—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C412yes—
21C22×C4×D4C2211yescharacteristic
22C23×C4⋊C4—21no—
23#2920C2212yes—
24#2920C2212yes—
25#1821—21no—
26#1801C2214yescomplemented
27#1784—21no—
28C4×#101—22no—
29#1556—24no—
30#1549C2218yescomplemented
31#1546—24no—
32#1490—24no—
33#1486C2218yes—
34#1478—24no—
35#1467—22no—
36#1463—22no—
37#1461C2214yescomplemented
38C23×Q8—21no—
39C23×Q8D411yescharacteristic
40C23×D4C2×C411yescharacteristic
41C24×C4—21no—
42C24×C4—22no—
43#554—216no—
44#554—216no—
45#554—216no—
46#554—216no—
47#554—216no—
48#554—216no—
49#554—216no—
50#554—216no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8C2×C412yes—
56C2×C4⋊Q8D414yes—
57C2×C4⋊Q8D414yes—
58C2×C4⋊Q8C2312yes—
59C2×#125—24no—
60C2×#125—24no—
61C2×#125—24no—
62C2×#125—24no—
63C2×#125—24no—
64C2×#125C2×C418yes—
65C2×C22⋊Q8—28no—
66C2×C22⋊Q8—28no—
67C2×C22⋊Q8—216no—
68C2×C22⋊Q8—216no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—24no—
71C2×C22⋊Q8—28no—
72C2×C22⋊Q8—28no—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8C2×C412yes—
78C2×C4×Q8D412yes—
79C2×C4×Q8D412yes—
80C2×C4×Q8C2312yes—
81C2×C4×D4—24no—
82C2×C4×D4—24no—
83C2×C4×D4—24no—
84C2×C4×D4—24no—
85C2×C4×D4—24no—
86C2×C4×D4—22no—
87C2×C4×D4—22no—
88C2×C4×D4—21no—
89C2×C4×D4—21no—
90C2×C4×D4C2×C418yes—
91C2×C4×D4C2×C412yes—
92C2×C4×D4D414yes—
93C2×C4×D4D414yes—
94C2×C4×D4C2312yes—
95C22×C4⋊C4—28no—
96C22×C4⋊C4—28no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4C2×C412yes—
103C22×C4⋊C4C2×C411yescharacteristic
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4D412yes—
106C22×C22⋊C4—22no—
107C22×C22⋊C4—21no—
108C22×C42C2×C412yes—
109C22×C42C2311yescharacteristic
110#432—22no—
111#432—22no—
112#432—22no—
113#432—22no—
114#431—22no—
115#431—22no—
116#431—22no—
117#431C2314yes—
118#423—22no—
119#423C2314yes—
120C2×#101—24no—
121#407—22no—
122#407—22no—
123C4×M16—22no—
124C4×M16—22no—
125#343—24no—
126#332—216no—
127D4⋊C8—216no—
128#324—28no—
129C25—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—24no—
134C22×Q8—24no—
135C22×Q8—22no—
136C22×Q8—44no—
137C22×Q8—84no—
138C22×Q8C22⋊C412yes—
139C22×Q8C2×D412yes—
140C22×D4—28no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4C22⋊C412yes—
145C22×D4C22⋊C412yes—
146C22×D4C22×C412yes—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—28no—
150C23×C4—28no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4—28no—
156C23×C4—48no—
157C23×C4—88no—
158C23×C4C22⋊C412yes—
159C23×C4C22⋊C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C2×M16—24no—
164C2×M16—42no—
165C4⋊Q8—24no—
166C4⋊Q8—24no—
167C4⋊Q8—24no—
168C4⋊Q8—24no—
169C4⋊Q8—24no—
170C4⋊Q8—24no—
171C4⋊Q8C22⋊C418yes—
172C4⋊Q8C2×D418yes—
173#125—216no—
174#125—216no—
175#125—216no—
176#125—216no—
177C22⋊Q8—232no—
178C22⋊Q8—232no—
179C22⋊Q8—264no—
180C22⋊Q8—264no—
181C4×Q8—24no—
182C4×Q8—24no—
183C4×Q8—24no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×D4—216no—
190C4×D4—216no—
191C4×D4—216no—
192C4×D4—216no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—24no—
197C4×D4—24no—
198C4×D4—24no—
199C4×D4C22⋊C418yes—
200C4×D4C2×D418yes—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—232no—
204C2×C4⋊C4—232no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4C22⋊C412yes—
224C2×C4⋊C4C22⋊C414yes—
225C2×C4⋊C4C22⋊C414yes—
226C2×C4⋊C4C22⋊C414yes—
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22×C414yes—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D414yes—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42C22⋊C414yes—
245C2×C42C22×C414yes—
246C2×C42C22×C411yescharacteristic
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2411yescharacteristic
254C4⋊C8—24no—
255C4⋊C8—24no—
256C4⋊C8—24no—
257C4⋊C8—42no—
258#101—48no—
259C22⋊C8—24no—
260C22⋊C8—42no—
261C22⋊C8—42no—
262C22⋊C8—42no—
263#97—42no—
264#97—42no—
265C4×C8—24no—
266C4×C8—42no—
267C24—28no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—44no—
272C24—44no—
273C24C2×C22⋊C411yescharacteristic
274C2×Q8—24no—
275C2×Q8—28no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—28no—
280C2×Q8—44no—
281C2×Q8—48no—
282C2×Q8—48no—
283C2×Q8—88no—
284C2×Q8—84no—
285C2×Q8—88no—
286C2×Q8—84no—
287C2×D4—28no—
288C2×D4—28no—
289C2×D4—28no—
290C2×D4—28no—
291C2×D4—28no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—24no—
296C2×D4#10118yescomplemented
297C2×D4C2×C22⋊C414yes—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—232no—
311C22×C4—28no—
312C22×C4—28no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—48no—
316C22×C4—416no—
317C22×C4—416no—
318C22×C4—44no—
319C22×C4—48no—
320C22×C4—48no—
321C22×C4—44no—
322C22×C4—816no—
323C22×C4—816no—
324C22×C4—816no—
325C22×C4C2×C22⋊C414yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C414yes—
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4#12012yes—
335C22×C4#12012yes—
336C22×C4#12012yes—
337C22×C4#12012yes—
338C22×C4C23×C411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341M16—48no—
342M16—84no—
343C2×C8—44no—
344C2×C8—44no—
345C4⋊C4—232no—
346C4⋊C4—232no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—28no—
351C4⋊C4—28no—
352C4⋊C4—24no—
353C4⋊C4—28no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—28no—
359C4⋊C4C2×C22⋊C418yes—
360C4⋊C4C2×C22⋊C414yes—
361C22⋊C4—232no—
362C22⋊C4—28no—
363C22⋊C4—28no—
364C42—24no—
365C42—24no—
366C42—24no—
367C42—24no—
368C42C2×C22⋊C412yes—
369C42C2×C22⋊C412yes—
370C42#12012yes—
371C42#12012yes—
372C42#12012yes—
373C42#12012yes—
374C42C22×D412yes—
375C42C22×D412yes—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—28no—
380C23—28no—
381C23—24no—
382C23—28no—
383C23—44no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—432no—
388C23—44no—
389C23—44no—
390C23#37911yescharacteristic
391C23#37911yescharacteristic
392C23C22×C22⋊C411yescharacteristic
393Q8—48no—
394Q8—48no—
395Q8—88no—
396Q8—88no—
397D4—28no—
398D4—28no—
399D4—24no—
400D4—28no—
401D4C2×#10118yes—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—28no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—416no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—48no—
425C2×C4—48no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—48no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—416no—
432C2×C4—44no—
433C2×C4—832no—
434C2×C4#37911yescharacteristic
435C2×C4#37912yes—
436C2×C4#37912yes—
437C2×C4#37911yescharacteristic
438C2×C4C2×#10114yes—
439C2×C4C22×C22⋊C411yescharacteristic
440C2×C4C22×C22⋊C411yescharacteristic
441C2×C4C2×#12012yes—
442C2×C4C2×#12012yes—
443C2×C4#57514yes—
444C8—84no—
445C8—84no—
446C22—28no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—28no—
454C22—416no—
455C22—416no—
456C22—416no—
457C22—44no—
458C22—44no—
459C22#178412yes—
460C22#179512yes—
461C22C2×#37911yescharacteristic
462C22#289612yes—
463C4—21no—
464C4—21no—
465C4—22no—
466C4—22no—
467C4—22no—
468C4—44no—
469C4—44no—
470C4—48no—
471C4—44no—
472C4#178412yes—
473C4C22×#10112yes—
474C4#300514yes—
475C2—22no—
476C2—22no—
477C2—22no—
478C2—48no—
479C2C2×#178411yescharacteristic
480C2#2060011yescharacteristic
481C2C2×#179511yescharacteristic
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. #8398C2semidirect
26. #1801C22semidirect
30. #1549C22semidirect
37. #1461C22semidirect
296. C2×D4#101semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
#207272
#177422
#174482
#173652
#83982