Finite Group Databasebeta

← #201105

Subgroups — #201105

Order 512.

1198 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2921C211yesmaximal, characteristic
2#35311C211yesmaximal, characteristic
3#20688C211yesmaximal, characteristic
4#17578C212yesmaximal
5#9444C218yesmaximal, complemented
6#7946C212yesmaximal
7C22×C22⋊Q8C2211yescharacteristic
8C22×#117C2211yescharacteristic
9#2921—24no—
10#2921—22no—
11#2921C2214yes—
12C2×#433—24no—
13C2×#423—21no—
14C2×#407C2212yes—
15C2×C4×M16—21no—
16#2855—21no—
17#2855—21no—
18#2625—22no—
19#2503C412yescomplemented
20#2502C412yescomplemented
21#2499C2214yes—
22#2484—22no—
23#2483—21no—
24#2483—21no—
25#2317C412yescomplemented
26#2310C412yescomplemented
27#2280C2212yes—
28#2280C2211yescharacteristic
29#1820C2214yes—
30#1819—21no—
31#1819—21no—
32#1740—22no—
33#1520—24no—
34#1511C2218yes—
35#1508—24no—
36#1284C2218yes—
37C23×Q8C2×C411yescharacteristic
38C24×C4C2311yescharacteristic
39C22×M16—22no—
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8C2×C412yes—
47C2×C22⋊Q8C2312yes—
48C2×#117—22no—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117C2×C412yes—
54C2×#117C2×C412yes—
55C2×#117D412yes—
56C2×#117D412yes—
57C2×#117C2312yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C22⋊C4—21no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67#433—216no—
68#433—216no—
69#423—24no—
70#423—24no—
71C2×C4⋊C8—24no—
72#407—22no—
73#407—22no—
74#407D414yes—
75#407C2314yes—
76C2×C22⋊C8—22no—
77C2×C22⋊C8D414yes—
78C4×M16—24no—
79C4×M16—24no—
80C2×#97—22no—
81C2×C4×C8—22no—
82#400—24no—
83#398—24no—
84#397—24no—
85#397—24no—
86#391—22no—
87#391—22no—
88#391C2×C414yes—
89#391C2×C414yes—
90#389—22no—
91#389—22no—
92#387—24no—
93#387—24no—
94#385—22no—
95#385—22no—
96#385—22no—
97#385—22no—
98#384C2×C414yes—
99#382C2×C414yes—
100#381—22no—
101#381—22no—
102#381—21no—
103#381—22no—
104#381—22no—
105#381C2314yes—
106#380—21no—
107#380—21no—
108#380—22no—
109#380—22no—
110#379—21no—
111#379—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4C2×C412yes—
115C4×C22⋊C4C2312yes—
116#342—24no—
117#342—24no—
118#328—216no—
119C25C2×D411yescharacteristic
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—44no—
124C22×Q8C22×C412yes—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—44no—
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4#4112yes—
140C23×C4#4112yes—
141C23×C4C2411yescharacteristic
142C2×M16—24no—
143C2×M16—24no—
144C2×M16—48no—
145C22×C8—24no—
146C22⋊Q8—24no—
147C22⋊Q8—24no—
148C22⋊Q8—24no—
149C22⋊Q8—24no—
150C22⋊Q8—41no—
151C22⋊Q8—41no—
152C22⋊Q8—42no—
153C22⋊Q8—42no—
154C22⋊Q8—41no—
155C22⋊Q8—41no—
156#117—24no—
157#117—24no—
158#117—24no—
159#117—24no—
160#117—24no—
161#117—24no—
162#117—24no—
163#117—24no—
164#117—44no—
165#117—48no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C22⋊C412yes—
202C2×C22⋊C4C2×D412yes—
203C2×C22⋊C4C2×D412yes—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—42no—
213C2×C42C22⋊C412yes—
214C2×C42C22⋊C412yes—
215C2×C42C22×C414yes—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C4⋊C8—216no—
219C22⋊C8—24no—
220C22⋊C8—24no—
221C22⋊C8—24no—
222C22⋊C8C2×D418yes—
223#97—28no—
224C4×C8—28no—
225#95—22no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—22no—
233#95—24no—
234#95—24no—
235C24—42no—
236C24—42no—
237C24—81no—
238C24C2×C22⋊C411yescharacteristic
239C24C2×C22⋊C411yescharacteristic
240C24#12112yes—
241C24#12112yes—
242C24C22×D411yescharacteristic
243C2×Q8—24no—
244C2×Q8—44no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—44no—
248C2×Q8—84no—
249C2×Q8—84no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4—82no—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C4×D412yes—
287C22×C4C4×D412yes—
288C22×C4C4×D412yes—
289C22×C4C4×D412yes—
290C22×C4#12011yescharacteristic
291C22×C4#12011yescharacteristic
292C22×C4#12012yes—
293C22×C4#12112yes—
294C22×C4#12112yes—
295C22×C4#12311yescharacteristic
296C22×C4#12311yescharacteristic
297C22×C4#12312yes—
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D412yes—
301C22×C4C2×#4112yes—
302M16—48no—
303M16—48no—
304C2×C8—28no—
305C2×C8—48no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—84no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—48no—
330C22⋊C4—82no—
331C22⋊C4—84no—
332C22⋊C4—84no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—44no—
342C42—84no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—24no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—42no—
352C23—41no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—41no—
357C23—42no—
358C23—88no—
359C23#38612yes—
360C23#38611yescharacteristic
361C23#38611yescharacteristic
362C23C22×C22⋊C411yescharacteristic
363C23C2×#12112yes—
364Q8—84no—
365Q8—82no—
366Q8—82no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
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—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4#38612yes—
413C2×C4#38611yescharacteristic
414C2×C4#38611yescharacteristic
415C2×C4#41014yes—
416C2×C4C2×C4×D412yes—
417C2×C4C2×#12011yescharacteristic
418C2×C4C2×#12311yescharacteristic
419C8—48no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—22no—
425C22—21no—
426C22—22no—
427C22—22no—
428C22—21no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—21no—
433C22—44no—
434C22—44no—
435C22—82no—
436C22—84no—
437C22—84no—
438C22#190912yes—
439C22C2×#38611yescharacteristic
440C22C2×#41012yes—
441C4—21no—
442C4—22no—
443C4—21no—
444C4—22no—
445C4—42no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—84no—
450C4—82no—
451C4—82no—
452C4—164no—
453C4#189912yes—
454C4#189912yes—
455C2—22no—
456C2—22no—
457C2—42no—
458C2—41no—
459C2—42no—
460C2#2094711yescharacteristic
461C2C2×#189911yescharacteristic
462C2C2×#190911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9444C2semidirect
19. #2503C4semidirect
20. #2502C4semidirect
25. #2317C4semidirect
26. #2310C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29212
#353112
#206882
#175782
#94442
#79462