Finite Group Databasebeta

← #368979

Subgroups — #368979

Order 512.

1236 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60192C211yesmaximal, characteristic
2#35281C211yesmaximal, characteristic
3#21337C211yesmaximal, characteristic
4C2×#1997C211yesmaximal, characteristic
5C2×#1997C211yesmaximal, characteristic
6#11206C214yesmaximal
7#10916C214yesmaximal
8#10856C211yesmaximal, characteristic
9C4×#344C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12Q8×M16—22no—
13Q8×M16—22no—
14#2962—22no—
15#2962—22no—
16#2961—22no—
17#2960—22no—
18#2905C2212yes—
19C2×#423C2211yescharacteristic
20C2×#423C2211yescharacteristic
21#2873C2212yes—
22C2×C4×M16C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24#2868—21no—
25#2862—21no—
26#2660—22no—
27#2567—22no—
28#2546—22no—
29#2502C414yes—
30#2434C414yes—
31#2317C2214yes—
32C4×#117C2211yescharacteristic
33#1998—22no—
34#1997—22no—
35#1997—22no—
36#1997—22no—
37#1997—22no—
38#1997—21no—
39#1997—21no—
40C2×#344C2214yes—
41C2×#344C2212yes—
42C2×#344C2212yes—
43#1742C2214yes—
44C2×#343C2212yes—
45C2×#343C2214yes—
46C2×#343C2212yes—
47C23×Q8C2×C412yes—
48C22×M16C2312yes—
49C22×M16C2311yescharacteristic
50C22×M16C2312yes—
51C22×M16C2312yes—
52C22×M16C2311yescharacteristic
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60#533—21no—
61C2×C4⋊Q8—24no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8D411yescharacteristic
75C2×C4⋊Q8D411yescharacteristic
76C2×#125D412yes—
77C2×C22⋊Q8C2×C412yes—
78C2×C22⋊Q8C2×C412yes—
79C2×C4×Q8D412yes—
80C2×C4×Q8D412yes—
81C2×#117C2312yes—
82C2×#117C2312yes—
83C22×C4⋊C4C2×C412yes—
84C22×C4⋊C4C2312yes—
85C22×C42C2311yescharacteristic
86#446—24no—
87#446—24no—
88#446—28no—
89#446—28no—
90C8×Q8—24no—
91C8×Q8—24no—
92#424—24no—
93#424—24no—
94#423—21no—
95#423—21no—
96#423—24no—
97#423—21no—
98#423—21no—
99#423—21no—
100#423—21no—
101#423D412yes—
102#423D412yes—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107#405—24no—
108#405—24no—
109C4×M16—21no—
110C4×M16—21no—
111C4×M16—24no—
112C4×M16—21no—
113C4×M16—21no—
114C4×M16—21no—
115C4×M16—21no—
116C4×M16D412yes—
117C4×M16D412yes—
118C2×#97—21no—
119C2×#97—21no—
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122#398—28no—
123#395—28no—
124#391—24no—
125#391—24no—
126#389—28no—
127#385C2×C4116yes—
128#384—24no—
129#384—24no—
130#382—24no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C22⋊C4C2×C412yes—
134C4×C22⋊C4C2×C412yes—
135#376—21no—
136C43—21no—
137#344—24no—
138#344—28no—
139#344Q818yescomplemented
140#343—28no—
141#343—24no—
142#343—24no—
143#343—24no—
144#343—24no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—84no—
152C22×Q8C22⋊C412yes—
153C22×Q8C22⋊C412yes—
154C23×C4C22×C414yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C2411yescharacteristic
158C2×M16—21no—
159C2×M16—22no—
160C2×M16—21no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—24no—
166C2×M16—22no—
167C2×M16—24no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—24no—
174C2×M16—24no—
175C2×M16C2×D412yes—
176C2×M16C2×D414yes—
177C2×M16C2×D412yes—
178C2×M16C2×Q814yes—
179C2×M16C2×Q814yes—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—41no—
194C4⋊Q8—41no—
195C4⋊Q8—41no—
196C4⋊Q8—41no—
197C4⋊Q8—41no—
198C4⋊Q8—41no—
199C4⋊Q8—41no—
200C4⋊Q8—41no—
201C4⋊Q8—41no—
202C4⋊Q8—41no—
203C4⋊Q8—41no—
204C4⋊Q8—41no—
205C4⋊Q8—41no—
206C4⋊Q8—41no—
207C4⋊Q8—41no—
208C4⋊Q8—41no—
209C4⋊Q8—88no—
210#125—22no—
211#125—22no—
212#125—22no—
213#125—22no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C22⋊Q8—22no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—22no—
230#117—22no—
231#117—22no—
232#117—22no—
233#117—22no—
234#117—22no—
235#117—22no—
236#117C2×Q814yes—
237#117#4114yes—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—216no—
252C2×C4⋊C4C22⋊C414yes—
253C2×C4⋊C4C22⋊C414yes—
254C2×C4⋊C4C22⋊C412yes—
255C2×C4⋊C4C22⋊C412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C22⋊C4C22×C414yes—
259C2×C22⋊C4C22×C414yes—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42C2×D414yes—
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42#4111yescharacteristic
273C2×C42#4111yescharacteristic
274C2×C42#4111yescharacteristic
275C2×C42#4111yescharacteristic
276C4⋊C8—28no—
277C4⋊C8—28no—
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—22no—
282C4⋊C8—41no—
283C4⋊C8—41no—
284C4⋊C8—41no—
285C4⋊C8—41no—
286#97—28no—
287#97—22no—
288#97—22no—
289#97—41no—
290#97—41no—
291C4×C8—28no—
292C4×C8—22no—
293C4×C8—22no—
294C4×C8—41no—
295C4×C8—41no—
296#95—22no—
297#95—22no—
298#95—216no—
299C24C23×C411yescharacteristic
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—44no—
305C2×Q8—44no—
306C2×Q8—48no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—84no—
312C2×Q8—84no—
313C2×Q8—84no—
314C2×Q8—84no—
315C2×Q8—84no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—84no—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C414yes—
326C22×C4C2×C22⋊C414yes—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C4×D411yescharacteristic
330C22×C4C4×D411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4C4×D411yescharacteristic
333C22×C4C4×D411yescharacteristic
334C22×C4C4×D411yescharacteristic
335C22×C4C4×D411yescharacteristic
336C22×C4C4×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×Q812yes—
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4112yes—
345C22×C42−1+412yes—
346C22×C42−1+412yes—
347M16—22no—
348M16—24no—
349M16—22no—
350M16—22no—
351M16—28no—
352M16—24no—
353M16—24no—
354M16—42no—
355M16—42no—
356M16—44no—
357M16—42no—
358M16—42no—
359M16—42no—
360M16—44no—
361M16C4⋊Q814yescomplemented
362C2×C8—22no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—22no—
366C2×C8—24no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—44no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—816no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4C4×Q818yes—
401C42—24no—
402C42—24no—
403C42—24no—
404C42—24no—
405C42—44no—
406C42—44no—
407C42#12011yescharacteristic
408C42#12011yescharacteristic
409C42#12011yescharacteristic
410C42#12011yescharacteristic
411C42#12111yescharacteristic
412C42#12111yescharacteristic
413C42#12111yescharacteristic
414C42#12111yescharacteristic
415C42#12111yescharacteristic
416C42#12111yescharacteristic
417C42#12111yescharacteristic
418C42#12111yescharacteristic
419C42#12311yescharacteristic
420C42#12311yescharacteristic
421C42#12311yescharacteristic
422C42#12311yescharacteristic
423C23—41no—
424C23—41no—
425C23C22×C22⋊C411yescharacteristic
426C23C2×C4×D411yescharacteristic
427C23C2×C4×D411yescharacteristic
428C23C2×C4×Q812yes—
429C23#51912yes—
430Q8—44no—
431Q8—44no—
432Q8—82no—
433Q8—82no—
434Q8—84no—
435Q8—84no—
436Q8—84no—
437Q8—84no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—28no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—48no—
458C2×C4—44no—
459C2×C4—84no—
460C2×C4—84no—
461C2×C4#38611yescharacteristic
462C2×C4#38611yescharacteristic
463C2×C4#38611yescharacteristic
464C2×C4#38611yescharacteristic
465C2×C4#38611yescharacteristic
466C2×C4#38611yescharacteristic
467C2×C4#38611yescharacteristic
468C2×C4#38611yescharacteristic
469C2×C4C2×#12011yescharacteristic
470C2×C4C2×#12111yescharacteristic
471C2×C4C2×#12111yescharacteristic
472C2×C4C2×#12311yescharacteristic
473C2×C4C2×C4⋊Q811yescharacteristic
474C2×C4#53211yescharacteristic
475C2×C4#53312yes—
476C2×C4#53611yescharacteristic
477C2×C4#53611yescharacteristic
478C2×C4#54112yes—
479C2×C4#54712yes—
480C2×C4#54812yes—
481C2×C4#54812yes—
482C2×C4D4×Q812yes—
483C2×C4D4×Q812yes—
484C2×C4D4×Q812yes—
485C2×C4#55612yes—
486C2×C4#55712yes—
487C8—24no—
488C8—44no—
489C8—44no—
490C8—42no—
491C8—82no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—42no—
499C22—42no—
500C22C2×#38611yescharacteristic
501C22C4×C4⋊Q811yescharacteristic
502C22#232711yescharacteristic
503C22Q8×C22⋊C412yes—
504C22#236512yes—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—48no—
512C4—44no—
513C4—44no—
514C4—84no—
515C4#191412yes—
516C4#191512yes—
517C4#242912yes—
518C4#243211yescharacteristic
519C4#243411yescharacteristic
520C4#244612yes—
521C4#254511yescharacteristic
522C4#254611yescharacteristic
523C2—22no—
524C2—21no—
525C2—21no—
526C2—22no—
527C2#1402411yescharacteristic
528C2C2×#191411yescharacteristic
529C2C2×#191511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
139. #344Q8semidirect
361. M16C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#601922
#352812
#213372
C2×#19972
#112062
#109162
#108562
C4×#3442