Finite Group Databasebeta

← #195599

Subgroups — #195599

Order 512.

1188 conjugacy classes of subgroups.

The subgroup lattice has 485 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×#1988C211yesmaximal, characteristic
5C2×#1542C211yesmaximal, characteristic
6#9060C214yesmaximal, complemented
7#9058C214yesmaximal, complemented
8#7893C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3230—24no—
13#3230C418yescomplemented
14C2×#500C2212yes—
15C2×#487C2212yes—
16C2×#484—22no—
17C2×#477—22no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2370—22no—
22#2369C2212yes—
23#2365C2212yes—
24#2355—21no—
25#2355—22no—
26Q8×C4⋊C4—21no—
27#2329—22no—
28#2317—21no—
29C4×C22⋊Q8—22no—
30#2285C2211yescharacteristic
31#2274—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1988—22no—
34#1988C2214yes—
35#1981C2214yes—
36#1956—21no—
37#1956—21no—
38#1787—22no—
39C2×#340—21no—
40C2×#340—21no—
41#1542—24no—
42C2×#333C2212yes—
43C2×#332C2212yes—
44#1279C2214yes—
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⋊Q8C2312yes—
60C2×C4⋊Q8C2311yescharacteristic
61C2×C4⋊Q8C2311yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8D412yes—
65C2×C4×Q8—22no—
66C2×C4×Q8—21no—
67C2×C4×Q8D412yes—
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—24no—
77#500C2×C418yescomplemented
78#487—24no—
79#487C2×C418yescomplemented
80#484—216no—
81#477—216no—
82#431—22no—
83#431—22no—
84#431—22no—
85#431C2×C414yes—
86C2×#107—24no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2312yes—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8C2312yes—
100#391—24no—
101#391—22no—
102#389—24no—
103#389—22no—
104#389—22no—
105#389—22no—
106#387—24no—
107#387—24no—
108#387—24no—
109#385—24no—
110#384—22no—
111#384D411yescharacteristic
112#384D411yescharacteristic
113#382—22no—
114#382—21no—
115#380—21no—
116#380D412yes—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—24no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4D412yes—
127C4×C4⋊C4C2312yes—
128C4×C4⋊C4C2312yes—
129C4×C22⋊C4—24no—
130C2×#95—21no—
131#340—24no—
132#340—24no—
133#333—24no—
134#332—24no—
135C22×Q8—21no—
136C22×Q8—24no—
137C22×Q8—22no—
138C22×Q8—41no—
139C22×Q8—42no—
140C22×Q8—82no—
141C22×Q8—82no—
142C22×Q8C2×D412yes—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—41no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—42no—
153C22×C8—42no—
154C22×C8—42no—
155C4⋊Q8—24no—
156C4⋊Q8—24no—
157C4⋊Q8—24no—
158C4⋊Q8—22no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4⋊Q8—42no—
168C4⋊Q8C22×C414yes—
169C4⋊Q8C22×C412yes—
170C4⋊Q8C22×C412yes—
171C4⋊Q8C2×D412yes—
172C4⋊Q8C2×D412yes—
173#125—24no—
174#125—24no—
175C22⋊Q8—24no—
176C22⋊Q8—24no—
177C22⋊Q8—24no—
178C22⋊Q8C22⋊C418yes—
179C4×Q8—24no—
180C4×Q8—24no—
181C4×Q8—24no—
182C4×Q8—84no—
183C4×Q8C22⋊C418yes—
184#117—24no—
185#117—24no—
186#117—42no—
187#117—42no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4Q1614yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—82no—
227C2×C42C22×C412yes—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4112yes—
233C2×C42#4112yes—
234C2×C42C2411yescharacteristic
235#107—216no—
236C4⋊C8—24no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—42no—
242#103—216no—
243#103—84no—
244#103—84no—
245C22⋊C8—24no—
246C22⋊C8—42no—
247C4×C8—22no—
248C4×C8—42no—
249C4×C8C22×C414yes—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258C24#12011yescharacteristic
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—24no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—44no—
269C2×Q8—44no—
270C2×Q8—44no—
271C2×Q8—88no—
272C2×Q8—82no—
273C2×Q8—82no—
274C2×Q8—88no—
275C2×Q8C2×C22⋊C414yes—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—81no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4C2×C22⋊C412yes—
297C22×C4C4×D412yes—
298C22×C4C4×D412yes—
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4#12112yes—
303C22×C4#12112yes—
304C22×C4#12312yes—
305C22×C4C2×Q1612yes—
306C22×C4#13512yes—
307C22×C4#13712yes—
308C22×C4#13712yes—
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C2×C8—24no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—84no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—28no—
331C4⋊C4—28no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—81no—
347C4⋊C4—81no—
348C4⋊C4—84no—
349C4⋊C4—84no—
350C4⋊C4C2×C22⋊C414yes—
351C4⋊C4C2×Q1614yes—
352C22⋊C4—28no—
353C22⋊C4—44no—
354C42—28no—
355C42—42no—
356C42—42no—
357C42—44no—
358C42—82no—
359C42—84no—
360C42—84no—
361C42C2×C22⋊C412yes—
362C42C4×D412yes—
363C42C4×D412yes—
364C42#12112yes—
365C42#12112yes—
366C42#12312yes—
367C42C23×C411yescharacteristic
368C42C22×D411yescharacteristic
369C42C2×#4111yescharacteristic
370C42C2×#4111yescharacteristic
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23#38612yes—
376C23#45112yes—
377C23#45212yes—
378C23C2×#12011yescharacteristic
379Q8—44no—
380Q8—44no—
381Q8—44no—
382Q8—48no—
383Q8—82no—
384Q8—88no—
385Q8—84no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
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—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—84no—
416C2×C4—84no—
417C2×C4—88no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4#38612yes—
421C2×C4#38612yes—
422C2×C4#38612yes—
423C2×C4C4×Q1614yes—
424C2×C4#44114yes—
425C2×C4#45112yes—
426C2×C4#45212yes—
427C2×C4#46214yes—
428C2×C4#46414yes—
429C2×C4C22×C22⋊C411yescharacteristic
430C2×C4C2×C4×D411yescharacteristic
431C2×C4C2×C4×D411yescharacteristic
432C2×C4C2×#12111yescharacteristic
433C2×C4C2×#12111yescharacteristic
434C2×C4C2×#12311yescharacteristic
435C2×C4C22×Q1611yescharacteristic
436C2×C4C2×#13511yescharacteristic
437C2×C4C2×#13711yescharacteristic
438C2×C4C2×#13711yescharacteristic
439C2×C4#57612yes—
440C2×C4#57712yes—
441C8—44no—
442C8—82no—
443C8—82no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22#187814yes—
451C22C2×#38611yescharacteristic
452C22C2×#45111yescharacteristic
453C22C2×#45211yescharacteristic
454C22#300811yescharacteristic
455C22#301011yescharacteristic
456C22#302211yescharacteristic
457C22#302211yescharacteristic
458C4—22no—
459C4—21no—
460C4—21no—
461C4—22no—
462C4—44no—
463C4—44no—
464C4—44no—
465C4—84no—
466C4—82no—
467C4—82no—
468C4—82no—
469C4#187814yes—
470C4C2×C4×Q1611yescharacteristic
471C4C2×#44111yescharacteristic
472C4#294612yes—
473C4C2×#46211yescharacteristic
474C4C2×#46411yescharacteristic
475C4#304512yes—
476C2—24no—
477C2#1890511yescharacteristic
478C2#1890911yescharacteristic
479C2#1892511yescharacteristic
480C2#1892911yescharacteristic
481C2C2×#187811yescharacteristic
482C2#2093211yescharacteristic
483C2#2095511yescharacteristic
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. #9060C2semidirect
7. #9058C2semidirect
13. #3230C4semidirect
77. #500C2×C4semidirect
79. #487C2×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×#19882
C2×#15422
#90602
#90582
#78932
#78922