Finite Group Databasebeta

← #199038

Subgroups — #199038

Order 512.

1031 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C211yesmaximal, characteristic
2#60251C211yesmaximal, characteristic
3#21811C211yesmaximal, characteristic
4#17742C211yesmaximal, characteristic
5#17584C211yesmaximal, characteristic
6#9292C218yesmaximal, complemented
7#7956C211yesmaximal, characteristic
8#7864C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3229—22no—
12#3229—22no—
13#3229C414yescomplemented
14#3229C2214yes—
15C2×#498C412yescomplemented
16C2×#487C412yescomplemented
17C2×#483—22no—
18C2×#475—22no—
19#2987—22no—
20#2985—22no—
21#2984—22no—
22#2983—22no—
23#2962—22no—
24#2961C2214yes—
25#2924—22no—
26#2920C2211yescharacteristic
27#2920C2211yescharacteristic
28C2×#432C2211yescharacteristic
29C2×#431C2211yescharacteristic
30#2283C2211yescharacteristic
31#1981—22no—
32#1960C2214yes—
33#1787—22no—
34#1556—22no—
35#1549—22no—
36#1546C2214yes—
37#1518C2214yes—
38#1515—22no—
39#1508—22no—
40#1286C2214yes—
41#1275C2214yes—
42C23×Q8—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533C2×C412yes—
48#533D412yes—
49#533D412yes—
50#533C2312yes—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8C2×C412yes—
61C2×C4⋊Q8C2×C412yes—
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2312yes—
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8D412yes—
68C2×C4×Q8—21no—
69C2×C4×Q8D412yes—
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C42C2311yescharacteristic
74#498—24no—
75#498C2×C418yes—
76#487—24no—
77#487C2×C418yes—
78#483—28no—
79#483—28no—
80#475—28no—
81#475—28no—
82#446—28no—
83#446—24no—
84#446—28no—
85#446—28no—
86C8×Q8—24no—
87#433—28no—
88#432—24no—
89#432—21no—
90#432—21no—
91#432—21no—
92#432—21no—
93#432—21no—
94#432—21no—
95#432—21no—
96#432D412yes—
97#431—24no—
98#431—21no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431C2×C412yes—
103#431D412yes—
104#431C2312yes—
105#431C2312yes—
106C2×#106—24no—
107#424—24no—
108#423—21no—
109#423C2312yes—
110C2×C4⋊C8—21no—
111C2×C4⋊C8D412yes—
112C2×#103—24no—
113C2×#103—28no—
114#407—21no—
115#407—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8D412yes—
118#405—24no—
119C4×M16—21no—
120C4×M16—21no—
121C2×#97C2312yes—
122C2×C4×C8C2×C412yes—
123#383—21no—
124#382C2312yes—
125#381—21no—
126#380—21no—
127C4×C4⋊C4—21no—
128C4×C22⋊C4—21no—
129#376—21no—
130#332—28no—
131#328—28no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—41no—
137C22×Q8—41no—
138C22×Q8—41no—
139C22×Q8—84no—
140C22×Q8C22⋊C412yes—
141C23×C4—41no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4#4111yescharacteristic
145C2×M16—22no—
146C2×M16—41no—
147C22×C8—22no—
148C22×C8C2×D414yes—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—41no—
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8C22⋊C412yes—
169C4⋊Q8C22⋊C412yes—
170C4⋊Q8C22×C414yes—
171C4⋊Q8C22×C412yes—
172C4⋊Q8C22×C412yes—
173#125—22no—
174#125—22no—
175#125—22no—
176#125—22no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8C22⋊C414yes—
184C22⋊Q8C2×D414yes—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8C22⋊C414yes—
192C4×Q8C2×D414yes—
193#117—22no—
194#117—22no—
195#117—22no—
196#117—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4C22⋊C412yes—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—41no—
218C2×C42—22no—
219C2×C42—41no—
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42#4111yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42C2411yescharacteristic
232#106—216no—
233C4⋊C8—28no—
234C4⋊C8—28no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—41no—
242C4⋊C8C2×D414yes—
243#103—216no—
244#103—88no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—41no—
253#97—28no—
254#97—22no—
255#97—22no—
256#97—22no—
257#97—41no—
258C4×C8—28no—
259C4×C8—22no—
260C4×C8—22no—
261C4×C8—22no—
262C4×C8C22×C414yes—
263#95—22no—
264#95—22no—
265#95—41no—
266#95—41no—
267C24#12111yescharacteristic
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—24no—
272C2×Q8—44no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—48no—
277C2×Q8—82no—
278C2×Q8—84no—
279C2×Q8—84no—
280C2×Q8—84no—
281C2×Q8—84no—
282C2×Q8—84no—
283C2×Q8C2×C22⋊C414yes—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—84no—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C4×D411yescharacteristic
297C22×C4C4×D411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4#12111yescharacteristic
303C22×C4#12111yescharacteristic
304C22×C4#12111yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12311yescharacteristic
308C22×C4#12311yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C2×#4111yescharacteristic
312M16—44no—
313M16—82no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—28no—
318C2×C8—44no—
319C2×C8—42no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—44no—
340C4⋊C4C2×C22⋊C414yes—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—82no—
344C42—24no—
345C42—24no—
346C42—82no—
347C42C2×C22⋊C411yescharacteristic
348C42C2×C22⋊C411yescharacteristic
349C42C4×D411yescharacteristic
350C42C4×D411yescharacteristic
351C42C4×D411yescharacteristic
352C42C4×D411yescharacteristic
353C42#12011yescharacteristic
354C42#12011yescharacteristic
355C42#12111yescharacteristic
356C42#12111yescharacteristic
357C42#12311yescharacteristic
358C42#12311yescharacteristic
359C42C23×C411yescharacteristic
360C42C22×D411yescharacteristic
361C42C22×D411yescharacteristic
362C42C2×#4111yescharacteristic
363C23—22no—
364C23—22no—
365C23—41no—
366C23#38611yescharacteristic
367C23#38611yescharacteristic
368C23C2×#12111yescharacteristic
369Q8—44no—
370Q8—44no—
371Q8—48no—
372Q8—82no—
373Q8—82no—
374Q8—84no—
375Q8—84no—
376Q8—84no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—82no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4#38611yescharacteristic
406C2×C4#38611yescharacteristic
407C2×C4#38611yescharacteristic
408C2×C4#38611yescharacteristic
409C2×C4#38611yescharacteristic
410C2×C4#38611yescharacteristic
411C2×C4C22×C22⋊C411yescharacteristic
412C2×C4C2×C4×D411yescharacteristic
413C2×C4C2×C4×D411yescharacteristic
414C2×C4C2×#12011yescharacteristic
415C2×C4C2×#12111yescharacteristic
416C2×C4C2×#12311yescharacteristic
417C2×C4#57714yes—
418C8—44no—
419C8—44no—
420C8—44no—
421C8—82no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—44no—
430C22#190212yes—
431C22C2×#38611yescharacteristic
432C22#304312yes—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—44no—
443C4—44no—
444C4—84no—
445C4—162no—
446C4#190912yes—
447C4#294912yes—
448C4#301012yes—
449C2—22no—
450C2—22no—
451C2—41no—
452C2#2093611yescharacteristic
453C2C2×#190211yescharacteristic
454C2C2×#190911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ 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
6. #9292C2semidirect
13. #3229C4semidirect
15. C2×#498C4semidirect
16. C2×#487C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
#602512
#218112
#177422
#175842
#92922
#79562
#78642