Finite Group Databasebeta

← #194282

Subgroups — #194282

Order 512.

1190 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic
4C2×#1987C211yesmaximal, characteristic
5C2×#1481C211yesmaximal, characteristic
6#8975C214yesmaximal, complemented
7#8974C214yesmaximal, complemented
8#8039C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3230—24no—
13#3230C418yescomplemented
14C2×#500—21no—
15C2×#487—21no—
16C2×#484—22no—
17C2×#477C2214yes—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2370—22no—
22#2369—21no—
23#2365—21no—
24#2355—22no—
25#2355C2212yes—
26Q8×C4⋊C4C2212yes—
27#2329—22no—
28#2317—21no—
29C4×C22⋊Q8—22no—
30#2285C2211yescharacteristic
31#2274—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1987—22no—
34#1987C2214yes—
35#1981—22no—
36#1956C2212yes—
37#1956C2212yes—
38#1787—22no—
39C2×#340—21no—
40C2×#340—21no—
41#1481—24no—
42C2×Q8⋊C8C2214yes—
43#1296C2212yes—
44#1296C2212yes—
45#1279C2214yes—
46C23×Q8—21no—
47C23×C8—21no—
48#533—22no—
49#533—22no—
50#533C2×C414yes—
51#533D414yes—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yes—
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8C2312yes—
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8D412yes—
65C2×C4×Q8—22no—
66C2×C4×Q8—21no—
67C2×C4×Q8C2312yes—
68C2×#117—21no—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#500—28no—
77#487—28no—
78#484—216no—
79#477—28no—
80#477C2×C4116yescomplemented
81#431—22no—
82#431—22no—
83#431—22no—
84#431C2×C414yes—
85C2×#107—24no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C4×C8—21no—
98C2×C4×C8C2312yes—
99#391—24no—
100#391—22no—
101#389—22no—
102#389—22no—
103#389—24no—
104#389—22no—
105#387—24no—
106#387—24no—
107#387—24no—
108#385—24no—
109#384—22no—
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#382—22no—
113#382—21no—
114#380—21no—
115#380D412yes—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D412yes—
127C4×C4⋊C4C2312yes—
128C4×C22⋊C4—24no—
129C2×#95—21no—
130#340—24no—
131#340—24no—
132Q8⋊C8—28no—
133C22×Q8—21no—
134C22×Q8—24no—
135C22×Q8—22no—
136C22×Q8—41no—
137C22×Q8—42no—
138C22×Q8—82no—
139C22×Q8—82no—
140C22×Q8C2×D412yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—41no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—42no—
151C22×C8—42no—
152C22×C8—42no—
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8—24no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—42no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8C22⋊C414yes—
165C4⋊Q8C22⋊C414yes—
166C4⋊Q8C22×C414yes—
167#125—24no—
168#125—24no—
169C22⋊Q8—24no—
170C22⋊Q8—24no—
171C22⋊Q8—24no—
172C22⋊Q8C22⋊C418yes—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—84no—
178C4×Q8C22×C414yes—
179C4×Q8C2×D414yes—
180#117—24no—
181#117—24no—
182#117—42no—
183#117—42no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4Q1614yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—22no—
222C2×C42—82no—
223C2×C42C22×C412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D412yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42#4112yes—
229C2×C42#4112yes—
230C2×C42C2411yescharacteristic
231#107—216no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—42no—
236C4⋊C8—42no—
237C4⋊C8C22×C414yes—
238#103—216no—
239#103—84no—
240#103—84no—
241C22⋊C8—24no—
242C22⋊C8—42no—
243C4×C8—24no—
244C4×C8—22no—
245C4×C8—22no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254C24#12011yescharacteristic
255C2×Q8—22no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8—42no—
263C2×Q8—44no—
264C2×Q8—81no—
265C2×Q8—81no—
266C2×Q8—82no—
267C2×Q8—82no—
268C2×Q8—88no—
269C2×Q8—88no—
270C2×Q8C2×C22⋊C414yes—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4C2×C22⋊C412yes—
292C22×C4C4×D412yes—
293C22×C4C4×D412yes—
294C22×C4#12011yescharacteristic
295C22×C4#12011yescharacteristic
296C22×C4#12011yescharacteristic
297C22×C4#12112yes—
298C22×C4#12112yes—
299C22×C4#12312yes—
300C22×C4C2×Q1612yes—
301C22×C4#13512yes—
302C22×C4#13712yes—
303C22×C4#13712yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C2×C8—24no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—84no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—28no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—42no—
343C4⋊C4—84no—
344C4⋊C4—84no—
345C4⋊C4C2×C22⋊C414yes—
346C4⋊C4C2×Q1614yes—
347C22⋊C4—28no—
348C22⋊C4—44no—
349C42—24no—
350C42—24no—
351C42—44no—
352C42—44no—
353C42—82no—
354C42—84no—
355C42—84no—
356C42C2×C22⋊C412yes—
357C42C4×D412yes—
358C42C4×D412yes—
359C42#12112yes—
360C42#12112yes—
361C42#12312yes—
362C42C23×C411yescharacteristic
363C42C22×D411yescharacteristic
364C42C2×#4111yescharacteristic
365C42C2×#4111yescharacteristic
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23#38612yes—
371C23#45112yes—
372C23#45212yes—
373C23C2×#12011yescharacteristic
374Q8—28no—
375Q8—48no—
376Q8—44no—
377Q8—84no—
378Q8—84no—
379Q8—88no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4—88no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4#38612yes—
415C2×C4#38612yes—
416C2×C4#38612yes—
417C2×C4C4×Q1614yes—
418C2×C4#44114yes—
419C2×C4#45112yes—
420C2×C4#45212yes—
421C2×C4#46214yes—
422C2×C4#46414yes—
423C2×C4C22×C22⋊C411yescharacteristic
424C2×C4C2×C4×D411yescharacteristic
425C2×C4C2×C4×D411yescharacteristic
426C2×C4C2×#12111yescharacteristic
427C2×C4C2×#12111yescharacteristic
428C2×C4C2×#12311yescharacteristic
429C2×C4C22×Q1611yescharacteristic
430C2×C4C2×#13511yescharacteristic
431C2×C4C2×#13711yescharacteristic
432C2×C4C2×#13711yescharacteristic
433C2×C4#57712yes—
434C2×C4#57812yes—
435C8—44no—
436C8—82no—
437C8—82no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22#187814yes—
445C22C2×#38611yescharacteristic
446C22C2×#45111yescharacteristic
447C22C2×#45211yescharacteristic
448C22#300911yescharacteristic
449C22#301111yescharacteristic
450C22#302111yescharacteristic
451C22#302111yescharacteristic
452C4—22no—
453C4—21no—
454C4—21no—
455C4—22no—
456C4—44no—
457C4—44no—
458C4—44no—
459C4—82no—
460C4—84no—
461C4—82no—
462C4—82no—
463C4#187814yes—
464C4#289812yes—
465C4C2×C4×Q1611yescharacteristic
466C4C2×#44111yescharacteristic
467C4C2×#46211yescharacteristic
468C4C2×#46411yescharacteristic
469C4#309712yes—
470C2—24no—
471C2#1889511yescharacteristic
472C2#1889711yescharacteristic
473C2#1891511yescharacteristic
474C2#1891711yescharacteristic
475C2#2091311yescharacteristic
476C2C2×#187811yescharacteristic
477C2#2094511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8975C2semidirect
7. #8974C2semidirect
13. #3230C4semidirect
80. #477C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
#342132
#218042
C2×#19872
C2×#14812
#89752
#89742
#80392
#78922