Finite Group Databasebeta

← #201119

Subgroups — #201119

Order 512.

1212 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60124C211yesmaximal, characteristic
2#35311C211yesmaximal, characteristic
3C2×#1820C211yesmaximal, characteristic
4C2×#1511C212yesmaximal
5#9444C218yesmaximal, complemented
6#7943C212yesmaximal
7C22×C22⋊Q8C2211yescharacteristic
8C22×#117C2211yescharacteristic
9#2924—24no—
10#2924—24no—
11#2921C2214yes—
12#2920—22no—
13#2905—21no—
14C2×#407C2212yes—
15#2873—21no—
16#2855—22no—
17#2625—21no—
18#2625—21no—
19#2503C412yescomplemented
20#2502C412yescomplemented
21#2499C2214yes—
22#2484—21no—
23#2484—21no—
24#2483—22no—
25#2317C412yescomplemented
26#2310C412yescomplemented
27#2280C2211yescharacteristic
28C2×#381C2212yes—
29#1820—21no—
30#1820—21no—
31#1820C2214yes—
32C2×#342—22no—
33#1511—24no—
34#1511C2218yes—
35C2×#328—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×#117C2×C412yes—
53C2×#117C2×C412yes—
54C2×#117D411yescharacteristic
55C2×#117D411yescharacteristic
56C2×#117D411yescharacteristic
57C2×#117D411yescharacteristic
58C2×#117C2312yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C22⋊C4—21no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4D412yes—
66C22×C42D412yes—
67#433—216no—
68#432—24no—
69#432—24no—
70#431—24no—
71#431—24no—
72#424—28no—
73#423—24no—
74#407—22no—
75#407—22no—
76#407D414yes—
77#407C2314yes—
78C2×C22⋊C8—22no—
79C2×C22⋊C8D414yes—
80#405—28no—
81C4×M16—24no—
82#400—22no—
83#400—22no—
84#398—22no—
85#398—22no—
86#397—24no—
87#397—24no—
88#391—22no—
89#391—22no—
90#391C2×C414yescomplemented
91#391C2×C414yescomplemented
92#389—22no—
93#389—22no—
94#387—24no—
95#387—24no—
96#385—24no—
97#385—22no—
98#385—22no—
99#384C2×C414yescomplemented
100#382C2×C414yescomplemented
101#381—24no—
102#381—21no—
103#381—22no—
104#381—22no—
105#381—22no—
106#381—22no—
107#381—22no—
108#381C2314yes—
109#380—21no—
110#380—21no—
111#379—21no—
112C4×C22⋊C4C2×C412yes—
113C4×C22⋊C4C2312yes—
114C2×#95—22no—
115C2×#95—22no—
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×C4—42no—
129C23×C4C22⋊C412yes—
130C23×C4C22⋊C412yes—
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4#4112yes—
139C23×C4#4112yes—
140C23×C4C2411yescharacteristic
141C2×M16—24no—
142C2×M16—24no—
143C2×M16—48no—
144C22×C8—24no—
145C22⋊Q8—24no—
146C22⋊Q8—24no—
147C22⋊Q8—24no—
148C22⋊Q8—24no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C22⋊Q8—42no—
152C22⋊Q8—42no—
153C22⋊Q8—41no—
154C22⋊Q8—41no—
155#117—24no—
156#117—24no—
157#117—22no—
158#117—22no—
159#117—22no—
160#117—22no—
161#117—22no—
162#117—22no—
163#117—22no—
164#117—22no—
165#117—24no—
166#117—24no—
167#117—44no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—48no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4C22⋊C412yes—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C2×D412yes—
204C2×C22⋊C4C2×D412yes—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42C22⋊C412yes—
210C2×C42C22⋊C412yes—
211C2×C42C22×C414yes—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C4⋊C8—28no—
215C4⋊C8—28no—
216C22⋊C8—24no—
217C22⋊C8—24no—
218C22⋊C8—24no—
219C22⋊C8C2×D418yes—
220#97—28no—
221C4×C8—28no—
222#95—22no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—22no—
230#95—24no—
231#95—24no—
232#95—42no—
233#95—42no—
234C24—24no—
235C24—42no—
236C24—81no—
237C24C2×C22⋊C411yescharacteristic
238C24C2×C22⋊C411yescharacteristic
239C24#12112yes—
240C24#12112yes—
241C24C22×D411yescharacteristic
242C2×Q8—24no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—84no—
248C2×Q8—84no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—84no—
274C22×C4—84no—
275C22×C4—84no—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C4×D412yes—
283C22×C4C4×D412yes—
284C22×C4C4×D412yes—
285C22×C4C4×D412yes—
286C22×C4#12012yes—
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4#12112yes—
290C22×C4#12112yes—
291C22×C4#12312yes—
292C22×C4#12311yescharacteristic
293C22×C4#12311yescharacteristic
294C22×C4C23×C411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D412yes—
297C22×C4C2×#4112yes—
298M16—48no—
299M16—48no—
300C2×C8—28no—
301C2×C8—48no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—24no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—44no—
315C4⋊C4—84no—
316C4⋊C4—84no—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—44no—
326C22⋊C4—48no—
327C22⋊C4—82no—
328C22⋊C4—84no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—44no—
338C23—22no—
339C23—24no—
340C23—22no—
341C23—24no—
342C23—22no—
343C23—24no—
344C23—41no—
345C23—41no—
346C23—42no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—41no—
351C23—42no—
352C23—88no—
353C23#38611yescharacteristic
354C23#38612yes—
355C23#38611yescharacteristic
356C23#41014yes—
357C23C22×C22⋊C411yescharacteristic
358C23C2×#12112yes—
359Q8—84no—
360Q8—82no—
361Q8—82no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4—168no—
406C2×C4#38611yescharacteristic
407C2×C4#38611yescharacteristic
408C2×C4#38612yes—
409C2×C4C2×C4×D412yes—
410C2×C4C2×#12011yescharacteristic
411C2×C4C2×#12311yescharacteristic
412C8—48no—
413C22—21no—
414C22—22no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—21no—
419C22—22no—
420C22—21no—
421C22—21no—
422C22—22no—
423C22—22no—
424C22—21no—
425C22—44no—
426C22—48no—
427C22—44no—
428C22—82no—
429C22—84no—
430C22#189914yes—
431C22#190912yes—
432C22C2×#38611yescharacteristic
433C22C2×#41012yes—
434C4—21no—
435C4—22no—
436C4—21no—
437C4—22no—
438C4—21no—
439C4—21no—
440C4—42no—
441C4—42no—
442C4—42no—
443C4—42no—
444C4—84no—
445C4—82no—
446C4—82no—
447C4—164no—
448C2—22no—
449C2—22no—
450C2—24no—
451C2—42no—
452C2—41no—
453C2#2094711yescharacteristic
454C2C2×#189911yescharacteristic
455C2C2×#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
90. #391C2×C4semidirect
91. #391C2×C4semidirect
99. #384C2×C4semidirect
100. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#601242
#353112
C2×#18202
C2×#15112
#94442
#79432