Finite Group Databasebeta

← #194048

Subgroups — #194048

Order 512.

1270 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic
2#34196C211yesmaximal, characteristic
3#21840C211yesmaximal, characteristic
4#21202C211yesmaximal, characteristic
5C2×#1491C211yesmaximal, characteristic
6#8975C214yesmaximal, complemented
7#8974C214yesmaximal, complemented
8#8044C211yesmaximal, characteristic
9#7895C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#477—24no—
13C2×#477—21no—
14C2×#477—21no—
15C2×#477C418yescomplemented
16C2×#477C2212yes—
17C2×#477C2212yes—
18C2×#432C2211yescharacteristic
19C22×#107—21no—
20C22×C4⋊C8C2211yescharacteristic
21C2×#419—22no—
22C22×#103—21no—
23C2×C4×M16C2211yescharacteristic
24#2355—22no—
25#2355C2212yes—
26Q8×C4⋊C4C2212yes—
27#2344—22no—
28#2344—21no—
29Q8×C22⋊C4—21no—
30#2320—22no—
31#2320—21no—
32#2285—21no—
33#2285C2211yescharacteristic
34#2283—21no—
35#2283C2211yescharacteristic
36#2276—21no—
37#1993—22no—
38#1987C2214yes—
39#1960—22no—
40#1956C2214yes—
41#1823—21no—
42#1823—21no—
43#1491—24no—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46#1296C2214yes—
47#1279C2214yes—
48C23×Q8—21no—
49C23×Q8D411yescharacteristic
50C22×M16—21no—
51C23×C8—21no—
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×Q8—22no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8C2×C414yes—
66C2×C4×Q8D414yes—
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×#117—22no—
72C2×#117D412yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D411yescharacteristic
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79#477—28no—
80#477—28no—
81#477—24no—
82#477—24no—
83#477C2×C418yes—
84#477C2×C418yes—
85#432—22no—
86#432—22no—
87C2×#107—28no—
88C2×#107—22no—
89C2×#107—22no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8C2×C414yes—
93C2×C4⋊C8C2312yes—
94C2×C4⋊C8C2312yes—
95#419—48no—
96C2×#103—28no—
97C2×#103—22no—
98C2×#103—22no—
99C2×#103—28no—
100C2×C22⋊C8—21no—
101C4×M16—22no—
102C4×M16—22no—
103C2×#97—21no—
104C2×#97—21no—
105C2×C4×C8C2312yes—
106#391—24no—
107#391—22no—
108#389—22no—
109#389—24no—
110#387—24no—
111#387—24no—
112#387—24no—
113#384—22no—
114#384C2312yes—
115#383—22no—
116#383—21no—
117#382—22no—
118#382—22no—
119#382D412yes—
120#382D412yes—
121#381—22no—
122#381—21no—
123#380—22no—
124#380—21no—
125#380—21no—
126#380D411yescharacteristic
127#380D411yescharacteristic
128C4×C4⋊C4—22no—
129C4×C4⋊C4—24no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4C2312yes—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—21no—
138#376—22no—
139#376—21no—
140#340—24no—
141#340—24no—
142Q8⋊C8—24no—
143Q8⋊C8—24no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—24no—
147C22×Q8—24no—
148C22×Q8—22no—
149C22×Q8—42no—
150C22×Q8—44no—
151C22×Q8—44no—
152C22×Q8—41no—
153C22×Q8—42no—
154C22×Q8C22⋊C414yes—
155C22×Q8C2×D411yescharacteristic
156C22×Q8C2×D411yescharacteristic
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—41no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C2×M16—24no—
164C2×M16—42no—
165C2×M16—42no—
166C22×C8—24no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—44no—
171C4⋊Q8—24no—
172C4⋊Q8—24no—
173C4⋊Q8—24no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—42no—
178C4⋊Q8—42no—
179C4⋊Q8—42no—
180C4⋊Q8C22×C414yes—
181C4×Q8—24no—
182C4×Q8—24no—
183C4×Q8—24no—
184C4×Q8—24no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—44no—
192C4×Q8—44no—
193C4×Q8C22⋊C414yes—
194C4×Q8C22⋊C414yes—
195C4×Q8C22×C412yes—
196C4×Q8C22×C412yes—
197C4×Q8C2×D412yes—
198C4×Q8C2×D412yes—
199#117—44no—
200#117—84no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4C22⋊C414yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D414yes—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C42—24no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—24no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—24no—
237C2×C42—22no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42C22×C412yes—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D412yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42#4112yes—
246C2×C42#4112yes—
247C2×C42C2411yescharacteristic
248#107—28no—
249#107—28no—
250C4⋊C8—24no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8C22×C414yes—
255#103—28no—
256#103—28no—
257#103—416no—
258C22⋊C8—42no—
259#97—42no—
260#97—42no—
261C4×C8—22no—
262C4×C8—22no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—21no—
268#95—22no—
269#95—21no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274C24#12011yescharacteristic
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—22no—
278C2×Q8—24no—
279C2×Q8—28no—
280C2×Q8—24no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—44no—
284C2×Q8—44no—
285C2×Q8—48no—
286C2×Q8—44no—
287C2×Q8—416no—
288C2×Q8—44no—
289C2×Q8—42no—
290C2×Q8—44no—
291C2×Q8—88no—
292C2×Q8—82no—
293C2×Q8—82no—
294C2×Q8C2×C22⋊C412yes—
295C2×Q8C2×C22⋊C412yes—
296C2×Q8#13714yes—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4C2×C22⋊C412yes—
319C22×C4C4×D412yes—
320C22×C4C4×D412yes—
321C22×C4#12011yescharacteristic
322C22×C4#12011yescharacteristic
323C22×C4#12011yescharacteristic
324C22×C4#12112yes—
325C22×C4#12112yes—
326C22×C4#12312yes—
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330M16—48no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—48no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—44no—
337C2×C8—42no—
338C2×C8—44no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—44no—
353C4⋊C4—48no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—84no—
362C4⋊C4C2×C22⋊C412yes—
363C4⋊C4C2×C22⋊C412yes—
364C22⋊C4—44no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—24no—
372C42—24no—
373C42—48no—
374C42—42no—
375C42—42no—
376C42—82no—
377C42—84no—
378C42C2×C22⋊C412yes—
379C42C4×D412yes—
380C42C4×D412yes—
381C42#12112yes—
382C42#12112yes—
383C42#12312yes—
384C42C23×C411yescharacteristic
385C42C22×D411yescharacteristic
386C42C2×#4111yescharacteristic
387C42C2×#4111yescharacteristic
388C23—22no—
389C23—22no—
390C23#38612yes—
391C23#45312yes—
392C23#45412yes—
393C23C2×#12011yescharacteristic
394Q8—24no—
395Q8—22no—
396Q8—22no—
397Q8—22no—
398Q8—48no—
399Q8—48no—
400Q8—42no—
401Q8—84no—
402Q8—88no—
403Q8#41914yescomplemented
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—48no—
423C2×C4—48no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—88no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4#38612yes—
443C2×C4#38612yes—
444C2×C4#38612yes—
445C2×C4C22×C22⋊C411yescharacteristic
446C2×C4C2×C4×D411yescharacteristic
447C2×C4C2×C4×D411yescharacteristic
448C2×C4C2×#12111yescharacteristic
449C2×C4C2×#12111yescharacteristic
450C2×C4C2×#12311yescharacteristic
451C2×C4C2×#13712yes—
452C2×C4#57512yes—
453C2×C4#57814yes—
454C8—44no—
455C8—44no—
456C8—82no—
457C22—24no—
458C22—24no—
459C22—22no—
460C22—22no—
461C22#188314yes—
462C22C2×#38611yescharacteristic
463C22#300712yes—
464C22#301112yes—
465C22C2×#45311yescharacteristic
466C22C2×#45411yescharacteristic
467C4—22no—
468C4—24no—
469C4—22no—
470C4—21no—
471C4—21no—
472C4—22no—
473C4—42no—
474C4—42no—
475C4—42no—
476C4—82no—
477C4—84no—
478C4—82no—
479C4—84no—
480C4C2×#41911yescharacteristic
481C4#289611yescharacteristic
482C4#295012yes—
483C4#304412yes—
484C4#309512yes—
485C2—24no—
486C2#1856512yes—
487C2#1858112yes—
488C2#2091111yescharacteristic
489C2#2093511yescharacteristic
490C2C2×#188311yescharacteristic
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
15. C2×#477C4semidirect
403. Q8#419semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
#341962
#218402
#212022
C2×#14912
#89752
#89742
#80442
#78952