Finite Group Databasebeta

← #192836

Subgroups — #192836

Order 512.

960 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34214C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3#21870C211yesmaximal, characteristic
4#21364C211yesmaximal, characteristic
5C2×#1497C211yesmaximal, characteristic
6#8892C212yesmaximal, complemented
7#8891C212yesmaximal, complemented
8#8884C212yesmaximal, complemented
9#8883C212yesmaximal, complemented
10#8040C211yesmaximal, characteristic
11#7902C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2360—22no—
17#2360—22no—
18#2355—21no—
19#2355C2212yes—
20Q8×C4⋊C4—21no—
21Q8×C4⋊C4C2212yes—
22#2330—22no—
23#2317—21no—
24C4×C4⋊Q8—21no—
25C4×C4⋊Q8—21no—
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28#2274—21no—
29C4×#117C2211yescharacteristic
30#1993—21no—
31#1960—21no—
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947—22no—
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1873—22no—
38#1854—21no—
39#1854—21no—
40#1852—22no—
41#1845C2212yes—
42#1834C2212yes—
43#1803C2212yes—
44#1803C2212yes—
45#1776—21no—
46#1776—21no—
47#1497C418yescomplemented
48C2×Q8⋊C8C2212yes—
49C2×Q8⋊C8C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54#533—22no—
55#533—22no—
56#533C2×C414yes—
57#533Q814yes—
58C2×C4⋊Q8D411yescharacteristic
59C2×C4⋊Q8D411yescharacteristic
60C2×#125—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8D412yes—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—21no—
66C2×C4×Q8C2312yes—
67C2×#117—21no—
68C2×#117—21no—
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D412yes—
74C22×C42C2311yescharacteristic
75#432—22no—
76#432—22no—
77#432—22no—
78#432C2×C414yes—
79#431—22no—
80#431C2×C414yes—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—24no—
84C2×#106—21no—
85C2×#106—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#103—22no—
91C2×#103—22no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×#97—21no—
96C2×#97—21no—
97C2×C4×C8C2312yes—
98#391—22no—
99#391—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—24no—
104#389—22no—
105#389—24no—
106#387—24no—
107#387—24no—
108#385—24no—
109#384—22no—
110#384—21no—
111#384—21no—
112#384D412yes—
113#384C2311yescharacteristic
114#384C2311yescharacteristic
115#384C2312yes—
116#382—22no—
117#382—21no—
118#380—21no—
119#380—21no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C22⋊C4—24no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135#376—21no—
136C43—21no—
137#340—22no—
138#340—22no—
139#340—24no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144Q8⋊C8C2×C418yescomplemented
145Q8⋊C8C2×C418yescomplemented
146C22×Q8—21no—
147C22×Q8C2×D412yes—
148C23×C4—22no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8C4⋊C414yes—
159C4⋊Q8C4⋊C414yes—
160#125—24no—
161#125—24no—
162C22⋊Q8—24no—
163C22⋊Q8—24no—
164C22⋊Q8—24no—
165C22⋊Q8C4⋊C418yes—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—82no—
171C4×Q8—82no—
172C4×Q8—84no—
173C4×Q8C22×C414yes—
174C4×Q8C2×Q814yes—
175#117—24no—
176#117—24no—
177#117—22no—
178#117—22no—
179#117—41no—
180#117—41no—
181#117—82no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D414yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—41no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—41no—
224C2×C42C22×C412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×Q812yes—
229C2×C42#4112yes—
230C2×C42#4112yes—
231C2×C42C2411yescharacteristic
232#107—42no—
233#107—42no—
234#107—84no—
235#106—42no—
236#106—42no—
237C4⋊C8—24no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8C22×C412yes—
243C4⋊C8C22×C412yes—
244#103—82no—
245#103—82no—
246C22⋊C8—24no—
247C22⋊C8—24no—
248C22⋊C8—42no—
249#97—24no—
250#97—42no—
251C4×C8—22no—
252C4×C8C22×C414yes—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—41no—
262#95—41no—
263C24#12011yescharacteristic
264C2×Q8—22no—
265C2×Q8—24no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8C2×C4⋊C414yes—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—24no—
285C22×C4—21no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—44no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4C2×C4⋊C412yes—
294C22×C4C4×D412yes—
295C22×C4C4×D412yes—
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4C22⋊Q812yes—
300C22×C4C22⋊Q812yes—
301C22×C4#12312yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C2×C8—24no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—28no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—82no—
346C4⋊C4—84no—
347C4⋊C4—84no—
348C4⋊C4C2×C4⋊C414yes—
349C4⋊C4#13712yes—
350C4⋊C4#13712yes—
351C22⋊C4—24no—
352C22⋊C4—28no—
353C22⋊C4—42no—
354C22⋊C4—82no—
355C42—24no—
356C42—24no—
357C42—24no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—82no—
364C42—84no—
365C42—84no—
366C42C2×C4⋊C412yes—
367C42C4×D412yes—
368C42C4×D412yes—
369C42C22⋊Q812yes—
370C42C22⋊Q812yes—
371C42#12312yes—
372C42C23×C411yescharacteristic
373C42C22×Q811yescharacteristic
374C42C2×#4111yescharacteristic
375C42C2×#4111yescharacteristic
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23#38512yes—
383C23C2×#12011yescharacteristic
384Q8—24no—
385Q8—24no—
386Q8—48no—
387Q8—42no—
388Q8—42no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—44no—
420C2×C4—41no—
421C2×C4—44no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—44no—
425C2×C4—41no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—164no—
433C2×C4#38512yes—
434C2×C4#38512yes—
435C2×C4#38512yes—
436C2×C4C22×C4⋊C411yescharacteristic
437C2×C4C2×C4×D411yescharacteristic
438C2×C4C2×C4×D411yescharacteristic
439C2×C4C2×C22⋊Q811yescharacteristic
440C2×C4C2×C22⋊Q811yescharacteristic
441C2×C4C2×#12311yescharacteristic
442C2×C4C2×#13711yescharacteristic
443C2×C4C2×#13711yescharacteristic
444C2×C4#57511yescharacteristic
445C2×C4#57511yescharacteristic
446C2×C4#57612yes—
447C2×C4#57812yes—
448C8—44no—
449C8—42no—
450C8—42no—
451C8—82no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22C2×#38511yescharacteristic
460C22#300411yescharacteristic
461C22#300511yescharacteristic
462C22#300811yescharacteristic
463C22#301111yescharacteristic
464C22#301911yescharacteristic
465C22#302011yescharacteristic
466C4—22no—
467C4—21no—
468C4—21no—
469C4—22no—
470C4—22no—
471C4—44no—
472C4—44no—
473C4—42no—
474C4—42no—
475C4—81no—
476C4—84no—
477C4—81no—
478C4—161no—
479C4—161no—
480C4#291312yes—
481C4C4×#13711yescharacteristic
482C4C4×#13711yescharacteristic
483C4#307811yescharacteristic
484C4#307811yescharacteristic
485C4#309812yes—
486C2—42no—
487C2#1881711yescharacteristic
488C2#1881811yescharacteristic
489C2#1882811yescharacteristic
490C2#1883111yescharacteristic
491C2#2084711yescharacteristic
492C2#2086411yescharacteristic
493C2#2087411yescharacteristic
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. #8892C2semidirect
7. #8891C2semidirect
8. #8884C2semidirect
9. #8883C2semidirect
47. #1497C4semidirect
144. Q8⋊C8C2×C4semidirect
145. Q8⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342142
#219822
#218702
#213642
C2×#14972
#88922
#88912
#88842
#88832
#80402
#79022