Finite Group Databasebeta

← #199155

Subgroups — #199155

Order 512.

1081 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2#34365C211yesmaximal, characteristic
3#21845C211yesmaximal, characteristic
4#17742C211yesmaximal, characteristic
5#17590C211yesmaximal, characteristic
6#9298C218yesmaximal, complemented
7#7956C211yesmaximal, characteristic
8#7858C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#3230—22no—
12#3230—22no—
13#3230C414yescomplemented
14#3230C2214yes—
15C2×#500C412yescomplemented
16C2×#487C412yescomplemented
17C2×#484—22no—
18C2×#477—22no—
19#2920C2211yescharacteristic
20#2920C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2381—22no—
23#2369—22no—
24#2344—22no—
25#2344—22no—
26#2331—22no—
27#2331C2214yes—
28#2310—22no—
29#2283C2211yescharacteristic
30#2283C2211yescharacteristic
31#1995C2214yes—
32#1986—22no—
33#1826—22no—
34#1556—22no—
35#1549C2214yes—
36#1546—22no—
37#1518C2214yes—
38#1511—22no—
39#1506—22no—
40#1286C2214yes—
41#1274C2214yes—
42C23×Q8—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533C2×C412yes—
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⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×C4⋊Q8C2×C412yes—
60C2×C4⋊Q8C2×C412yes—
61C2×C4⋊Q8D412yes—
62C2×C4⋊Q8D412yes—
63C2×C4⋊Q8D412yes—
64C2×C4⋊Q8C2312yes—
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×#117—24no—
71C2×#117—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D412yes—
74C22×C42C2311yescharacteristic
75#500—24no—
76#500C2×C418yes—
77#487—24no—
78#487C2×C418yes—
79#484—28no—
80#484—28no—
81#477—28no—
82#477—28no—
83#432—21no—
84#432—21no—
85#432—21no—
86#432—21no—
87#431—21no—
88#431—21no—
89#431—21no—
90#431—21no—
91#431—21no—
92#431C2×C412yes—
93#431C2312yes—
94#431C2312yes—
95C2×#107—24no—
96#423—21no—
97#423C2312yes—
98C2×C4⋊C8—21no—
99C2×#103—24no—
100#407—21no—
101#407—21no—
102C2×C22⋊C8—21no—
103C4×M16—21no—
104C4×M16—21no—
105C2×C4×C8C2×C412yes—
106#391—28no—
107#391—28no—
108#389—24no—
109#389—24no—
110#387—28no—
111#387—28no—
112#383—21no—
113#383—21no—
114#382—21no—
115#382C2312yes—
116#381—24no—
117#381—21no—
118#381D412yes—
119#380—21no—
120#380—21no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4D412yes—
125C4×C22⋊C4—21no—
126C4×C22⋊C4D412yes—
127#376D412yes—
128#376C2312yes—
129#339—28no—
130#332—28no—
131#328—28no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—84no—
141C23×C4—22no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C2×M16—22no—
146C2×M16—41no—
147C22×C8—22no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8C22⋊C414yes—
167C4⋊Q8C22⋊C414yes—
168C4⋊Q8C22×C412yes—
169C4⋊Q8C22×C412yes—
170C4⋊Q8C22×C414yes—
171C4⋊Q8C2×D414yes—
172#125—22no—
173#125—22no—
174#125—41no—
175#125—41no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—41no—
181C22⋊Q8—41no—
182C22⋊Q8—41no—
183C22⋊Q8—41no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—41no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×Q8—41no—
192#117—22no—
193#117—22no—
194#117—41no—
195#117—41no—
196#117—88no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
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—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4C22⋊C414yes—
216C2×C4⋊C4C22⋊C414yes—
217C2×C4⋊C4C2×D414yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—41no—
221C2×C42—22no—
222C2×C42—28no—
223C2×C42—41no—
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226C2×C42C2×D414yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42#4111yescharacteristic
235C2×C42#4111yescharacteristic
236C2×C42C2411yescharacteristic
237#107—216no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—41no—
244#103—216no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—41no—
249C22⋊C8—41no—
250C22⋊C8—41no—
251#97—41no—
252#97—41no—
253C4×C8—22no—
254C4×C8—22no—
255C4×C8—41no—
256C4×C8C22×C414yes—
257#95—28no—
258#95—28no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95C2×D414yes—
267C24#12011yescharacteristic
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—44no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—84no—
278C2×Q8—84no—
279C2×Q8—82no—
280C2×Q8—84no—
281C2×Q8—82no—
282C2×Q8—82no—
283C2×Q8—84no—
284C2×Q8—84no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—48no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4C4×D411yescharacteristic
301C22×C4C4×D411yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12311yescharacteristic
310C22×C4#12311yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314M16—44no—
315M16—82no—
316C2×C8—24no—
317C2×C8—44no—
318C2×C8—42no—
319C2×C8—42no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—44no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—44no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—88no—
342C4⋊C4C2×C22⋊C418yes—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C42—28no—
348C42—44no—
349C42—42no—
350C42—42no—
351C42—88no—
352C42C2×C22⋊C411yescharacteristic
353C42C2×C22⋊C411yescharacteristic
354C42C4×D411yescharacteristic
355C42C4×D411yescharacteristic
356C42C4×D411yescharacteristic
357C42C4×D411yescharacteristic
358C42#12111yescharacteristic
359C42#12111yescharacteristic
360C42#12111yescharacteristic
361C42#12111yescharacteristic
362C42#12311yescharacteristic
363C42#12311yescharacteristic
364C42C23×C411yescharacteristic
365C42C22×D411yescharacteristic
366C42C2×#4111yescharacteristic
367C42C2×#4111yescharacteristic
368C23—22no—
369C23—22no—
370C23—41no—
371C23#38611yescharacteristic
372C23#38611yescharacteristic
373C23C2×#12011yescharacteristic
374Q8—44no—
375Q8—84no—
376Q8—84no—
377Q8—84no—
378Q8—84no—
379Q8—82no—
380Q8—82no—
381Q8—82no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—28no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—48no—
403C2×C4—48no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4#38611yescharacteristic
415C2×C4#38611yescharacteristic
416C2×C4#38611yescharacteristic
417C2×C4#38611yescharacteristic
418C2×C4#38611yescharacteristic
419C2×C4#38611yescharacteristic
420C2×C4C22×C22⋊C411yescharacteristic
421C2×C4C2×C4×D411yescharacteristic
422C2×C4C2×C4×D411yescharacteristic
423C2×C4C2×#12111yescharacteristic
424C2×C4C2×#12111yescharacteristic
425C2×C4C2×#12311yescharacteristic
426C2×C4#57814yes—
427C8—44no—
428C8—82no—
429C8—82no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—44no—
438C22#190412yes—
439C22C2×#38611yescharacteristic
440C22#301112yes—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—48no—
451C4—84no—
452C4—84no—
453C4—82no—
454C4—82no—
455C4#191212yes—
456C4#295012yes—
457C4#304412yes—
458C2—22no—
459C2—22no—
460C2—41no—
461C2#2093511yescharacteristic
462C2C2×#190411yescharacteristic
463C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
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. #9298C2semidirect
13. #3230C4semidirect
15. C2×#500C4semidirect
16. C2×#487C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
#343652
#218452
#177422
#175902
#92982
#79562
#78582