Finite Group Databasebeta

← #368959

Subgroups — #368959

Order 512.

1374 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic
2C2×#2330C211yesmaximal, characteristic
3C2×#1991C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic
5#14282C211yesmaximal, characteristic
6#11220C212yesmaximal
7#11219C212yesmaximal
8#10908C212yesmaximal
9#10907C212yesmaximal
10#10843C211yesmaximal, characteristic
11#10694C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C2×#501—24no—
15C2×#501—24no—
16C2×#501C418yescomplemented
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×#97C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2330—24no—
22#2330—24no—
23C2×#389—21no—
24C2×#389—21no—
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#387—21no—
28C2×#387—21no—
29C2×#384C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×#376C2211yescharacteristic
34#1991—24no—
35#1990—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1446—21no—
45#1446—21no—
46#1445—21no—
47#1445—21no—
48C4⋊#95C2212yes—
49C4⋊#95C2212yes—
50C23×Q8—21no—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yes—
59C2×#125—22no—
60C2×#125—22no—
61C2×#125—22no—
62C2×#125C2×C414yes—
63C2×#125D414yes—
64C2×#125D414yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C42—21no—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#501—232no—
84C2×#107—24no—
85C2×#107—24no—
86C2×#107C2×C418yes—
87C2×#106—24no—
88C2×#106—24no—
89C2×#106C2×C418yes—
90C2×#97—22no—
91C2×#97—22no—
92C2×#97C2×C414yes—
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95C2×C4×C8—22no—
96#389—28no—
97#389—28no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#387—28no—
103#387—28no—
104#384—22no—
105#384D414yes—
106#382—22no—
107#382—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112#376—22no—
113#376—22no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95D411yescharacteristic
125C2×#95D411yescharacteristic
126C2×#95D411yescharacteristic
127C2×#95D411yescharacteristic
128#340—24no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—24no—
134#336—24no—
135#336Q818yescomplemented
136C22×Q8—24no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—82no—
140C22×Q8—82no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8C22×C414yes—
165C22×C8C2×Q814yes—
166C22×C8C2×Q814yes—
167C22×C8#4114yes—
168C4⋊Q8—28no—
169C4⋊Q8—28no—
170C4⋊Q8—44no—
171C4⋊Q8—44no—
172#125—28no—
173#125—28no—
174#125—28no—
175#125C22⋊C4116yes—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
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—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4C22×C414yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—82no—
235C2×C42—82no—
236C2×C42C22×C412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×Q814yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243#107—216no—
244#106—216no—
245#97—28no—
246C4×C8—44no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—42no—
256#95—42no—
257#95—82no—
258#95—82no—
259#95—82no—
260#95—82no—
261C24C22×D411yescharacteristic
262C2×Q8—48no—
263C2×Q8—88no—
264C2×Q8—84no—
265C2×Q8—84no—
266C2×Q8—88no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4C4×D412yes—
324C22×C4C4×D412yes—
325C22×C4#12012yes—
326C22×C4#12012yes—
327C22×C4#12112yes—
328C22×C4#12112yes—
329C22×C4C4⋊Q812yes—
330C22×C4C4⋊Q812yes—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×Q812yes—
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4112yes—
338C22×C4C2×#4111yescharacteristic
339C22×C42−1+412yes—
340C22×C42−1+412yes—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8C4×Q818yes—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—216no—
362C4⋊C4—216no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—48no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C4⋊C4—84no—
380C4⋊C4C2×C22⋊C414yes—
381C4⋊C4C2×C22⋊C414yes—
382C42—24no—
383C42—24no—
384C42—42no—
385C42—42no—
386C42—84no—
387C42—84no—
388C42C2×C22⋊C414yes—
389C42C4×D414yes—
390C42#12114yes—
391C42#12314yes—
392C23C2×C4×D411yescharacteristic
393C23C2×#12011yescharacteristic
394C23C2×#12111yescharacteristic
395C23C2×C4⋊Q811yescharacteristic
396C23#53211yescharacteristic
397C23#53611yescharacteristic
398C23#53611yescharacteristic
399C23#57612yes—
400C23#57712yes—
401C23#57712yes—
402C23#57812yes—
403Q8—816no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—44no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4#38614yes—
467C2×C4#38614yes—
468C2×C4C22×C22⋊C411yescharacteristic
469C2×C4C2×C4×D411yescharacteristic
470C2×C4C2×C4×Q812yes—
471C2×C4#51912yes—
472C2×C4C2×#12111yescharacteristic
473C2×C4C2×#12311yescharacteristic
474C2×C4#53312yes—
475C2×C4#54112yes—
476C2×C4#54712yes—
477C2×C4#54812yes—
478C2×C4#54812yes—
479C2×C4D4×Q812yes—
480C2×C4D4×Q812yes—
481C2×C4D4×Q812yes—
482C2×C4#55612yes—
483C2×C4#55712yes—
484C8—28no—
485C8—44no—
486C22C2×#38611yescharacteristic
487C22C4×C4⋊Q811yescharacteristic
488C22#232711yescharacteristic
489C22#243211yescharacteristic
490C22#243411yescharacteristic
491C22#254511yescharacteristic
492C22#254611yescharacteristic
493C22#294612yes—
494C22#294712yes—
495C22#300811yescharacteristic
496C22#300911yescharacteristic
497C22#301011yescharacteristic
498C22#301111yescharacteristic
499C22#304512yes—
500C22#304612yes—
501C22#316312yes—
502C22#316412yes—
503C22#317812yes—
504C22#317912yes—
505C22#324112yes—
506C22#324212yes—
507C22#324312yes—
508C22#324412yes—
509C4—24no—
510C4—24no—
511C4—44no—
512C4—44no—
513C4—42no—
514C4—42no—
515C4—81no—
516C4—82no—
517C4—82no—
518C4—81no—
519C4—84no—
520C4—162no—
521C4Q8×C22⋊C412yes—
522C4#236512yes—
523C4#242912yes—
524C4#244612yes—
525C2#1402411yescharacteristic
526C2#2093111yescharacteristic
527C2#2093211yescharacteristic
528C2#2173012yes—
529C2#2225512yes—
530C2#2568511yescharacteristic
531C2#2568611yescharacteristic
532C2#2582011yescharacteristic
533C2#2582111yescharacteristic
534C2#2866812yes—
535C2#2869812yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. C2×#501C4semidirect
135. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#23302
C2×#19912
C2×#19902
#142822
#112202
#112192
#109082
#109072
#108432
#106942