Finite Group Databasebeta

← #112076

Subgroups — #112076

Order 512.

952 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2C2×#1995C211yesmaximal, characteristic
3C2×#1548C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic
5#14282C211yesmaximal, characteristic
6C2×#1286C211yesmaximal, characteristic
7C2×#1274C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10#3209—22no—
11#3209—22no—
12#3209—22no—
13#3209—22no—
14C2×#494C412yescomplemented
15C2×#487C412yescomplemented
16C2×#462—22no—
17C2×#451—22no—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#376C2211yescharacteristic
23#1995—22no—
24#1995—22no—
25C2×#339—22no—
26#1548—22no—
27#1548—22no—
28C2×#333—22no—
29#1518—22no—
30#1518—22no—
31C2×#328—22no—
32#1446—22no—
33#1445—22no—
34C4⋊#95—22no—
35#1286—22no—
36#1274—22no—
37C23×Q8D412yes—
38C22×Q16—24no—
39C22×M16—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533D412yes—
47#533D412yes—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
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⋊Q8—21no—
58C2×C4⋊Q8C2×C412yes—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×C4⋊Q8D412yes—
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×#117—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C42D412yes—
72C22×C42C2311yescharacteristic
73#494—24no—
74#494—24no—
75#487—24no—
76#487—24no—
77#462—28no—
78#462—28no—
79#451—28no—
80#451—28no—
81#431—21no—
82#431—21no—
83#431—21no—
84#431—21no—
85#431—21no—
86#431D412yes—
87#431D412yes—
88#431D412yes—
89#423—21no—
90#423—21no—
91#423—21no—
92#423D412yes—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×#103—24no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8C2×C412yes—
102#382—21no—
103#382—21no—
104#382—21no—
105#382—21no—
106#376—21no—
107#376—21no—
108#376—21no—
109#376—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—28no—
114C2×#95D412yes—
115#339—28no—
116#333—28no—
117#328—28no—
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—84no—
126C22×Q8C22⋊C414yes—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C2×Q16—216no—
134C2×Q16—88no—
135C2×M16—22no—
136C2×M16—41no—
137C2×M16—42no—
138C2×M16—42no—
139C22×C8—22no—
140C22×C8—41no—
141C22×C8—41no—
142C4⋊Q8—21no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—21no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—41no—
157C4⋊Q8—41no—
158C4⋊Q8—41no—
159C4⋊Q8—41no—
160C4⋊Q8—41no—
161C4⋊Q8—41no—
162C4⋊Q8—41no—
163C4⋊Q8—41no—
164C4⋊Q8C22⋊C412yes—
165C4⋊Q8C22⋊C412yes—
166#125—22no—
167#125—22no—
168#125—41no—
169#125—41no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173C22⋊Q8—22no—
174C22⋊Q8—41no—
175C22⋊Q8—41no—
176C22⋊Q8—41no—
177C22⋊Q8—41no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×Q8—41no—
185C4×Q8—41no—
186#117—22no—
187#117—22no—
188#117—41no—
189#117—41no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C22⋊C4—41no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—41no—
219C2×C42Q1614yescomplemented
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42#4111yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42#4111yescharacteristic
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—41no—
238C4⋊C8—41no—
239C4⋊C8—41no—
240#103—216no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—22no—
244C22⋊C8—41no—
245C4×C8—22no—
246C4×C8—22no—
247C4×C8—41no—
248C4×C8C22⋊C414yes—
249#95—22no—
250#95—22no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—88no—
258C24#12011yescharacteristic
259C2×Q8—24no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—24no—
263C2×Q8—48no—
264C2×Q8—44no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—84no—
270C2×Q8—82no—
271C2×Q8—84no—
272C2×Q8—84no—
273C2×Q8—84no—
274C2×Q8—84no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—48no—
281C22×C4—44no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C4×D411yescharacteristic
294C22×C4C4×D411yescharacteristic
295C22×C4#12011yescharacteristic
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4#12111yescharacteristic
301C22×C4#12111yescharacteristic
302C22×C4#12111yescharacteristic
303C22×C4#12111yescharacteristic
304C22×C4#12111yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12311yescharacteristic
308C22×C4#12711yescharacteristic
309C22×C4#12711yescharacteristic
310C22×C4C2×Q1612yes—
311C22×C4#13512yes—
312C22×C4#13712yes—
313C22×C4#13712yes—
314Q16—816no—
315M16—44no—
316M16—81no—
317M16—81no—
318C2×C8—24no—
319C2×C8—44no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—82no—
323C2×C8—82no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—82no—
345C4⋊C4—81no—
346C4⋊C4—81no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C42—24no—
350C42—24no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—42no—
355C42C2×C22⋊C411yescharacteristic
356C42C2×C22⋊C411yescharacteristic
357C42C4×D411yescharacteristic
358C42C4×D411yescharacteristic
359C42#12011yescharacteristic
360C42#12011yescharacteristic
361C42#12011yescharacteristic
362C42#12111yescharacteristic
363C42#12111yescharacteristic
364C42#12111yescharacteristic
365C42#12111yescharacteristic
366C42#12311yescharacteristic
367C42#12311yescharacteristic
368C42#12311yescharacteristic
369C42#12411yescharacteristic
370C42#12411yescharacteristic
371C23—22no—
372C23—22no—
373C23#38611yescharacteristic
374C23#39211yescharacteristic
375C23#39211yescharacteristic
376C23#45112yes—
377C23#45212yes—
378Q8—44no—
379Q8—48no—
380Q8—84no—
381Q8—82no—
382Q8—82no—
383Q8—82no—
384Q8—84no—
385Q8—84no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—48no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—82no—
405C2×C4—84no—
406C2×C4—82no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—88no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4#37911yescharacteristic
416C2×C4#38611yescharacteristic
417C2×C4#38611yescharacteristic
418C2×C4#38611yescharacteristic
419C2×C4#39011yescharacteristic
420C2×C4#39011yescharacteristic
421C2×C4#39211yescharacteristic
422C2×C4#39211yescharacteristic
423C2×C4#39411yescharacteristic
424C2×C4#39411yescharacteristic
425C2×C4#39411yescharacteristic
426C2×C4#39411yescharacteristic
427C2×C4C4×Q1612yes—
428C2×C4#44112yes—
429C2×C4#45112yes—
430C2×C4#45212yes—
431C2×C4#46212yes—
432C2×C4#46212yes—
433C2×C4#46412yes—
434C2×C4#46412yes—
435C2×C4#46712yes—
436C2×C4#47012yes—
437C2×C4#49411yescharacteristic
438C2×C4#49511yescharacteristic
439C2×C4#49711yescharacteristic
440C2×C4#49711yescharacteristic
441C8—44no—
442C8—82no—
443C8—82no—
444C22—21no—
445C22—21no—
446C22—24no—
447C22—24no—
448C22—21no—
449C22—21no—
450C22#143811yescharacteristic
451C22#187812yes—
452C22#190512yes—
453C22#191112yes—
454C22#200112yes—
455C22#201511yescharacteristic
456C22#201611yescharacteristic
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—84no—
466C4—82no—
467C4—84no—
468C4—82no—
469C4—162no—
470C4#187812yes—
471C4#196911yescharacteristic
472C4#197111yescharacteristic
473C4#201511yescharacteristic
474C4#201611yescharacteristic
475C4#205612yes—
476C2—22no—
477C2—21no—
478C2—21no—
479C2#929714yes—
480C2#1098911yescharacteristic
481C2#1101311yescharacteristic
482C2#1102411yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. C2×#494C4semidirect
15. C2×#487C4semidirect
219. C2×C42Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
C2×#19952
C2×#15482
C2×#15182
#142822
C2×#12862
C2×#12742