Finite Group Databasebeta

← #195420

Subgroups — #195420

Order 512.

1037 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60943C211yesmaximal, characteristic
2#60192C211yesmaximal, characteristic
3#21302C211yesmaximal, characteristic
4C2×#1542C211yesmaximal, characteristic
5#17486C211yesmaximal, characteristic
6#9048C218yesmaximal, complemented
7C2×#1301C211yesmaximal, characteristic
8#7918C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#533C2211yescharacteristic
11#3249—22no—
12#3230—21no—
13#3230C412yes—
14#3209—21no—
15#3209C412yes—
16#3189C414yes—
17#3170—22no—
18#3104—22no—
19#3104C2214yes—
20Q8×M16—22no—
21Q8×M16—22no—
22#2962—22no—
23#2962—22no—
24#2961C2214yes—
25#2960—22no—
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28#2905—21no—
29C2×#423C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31#2873—21no—
32C2×C4×M16C2211yescharacteristic
33#1985C2214yes—
34#1984—22no—
35#1729—21no—
36#1729—21no—
37#1542—22no—
38#1542C2214yes—
39C2×#333—21no—
40C2×#332—21no—
41#1493—22no—
42#1487C2214yes—
43#1481—22no—
44#1301C2214yes—
45#1281C2214yes—
46C22×M16—21no—
47C22×M16D411yescharacteristic
48C22×M16D411yescharacteristic
49C23×C8—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60#533—21no—
61#533—21no—
62#533C2×C412yes—
63#533D412yes—
64#533D412yes—
65#533C2312yes—
66C2×C4⋊Q8C2×C411yescharacteristic
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×C4⋊Q8D411yescharacteristic
69C2×C4⋊Q8D411yescharacteristic
70C2×#125—21no—
71C2×#125—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8C2×C412yes—
75C2×C22⋊Q8C2×C412yes—
76C2×C4×Q8—21no—
77C2×C4×Q8—21no—
78C2×C4×Q8D412yes—
79C2×C4×Q8C2312yes—
80C2×#117—21no—
81C2×#117—21no—
82C22×C42C2311yescharacteristic
83#500—22no—
84#494—22no—
85#487—22no—
86#487—22no—
87#484—24no—
88#484C2×C418yes—
89#477—24no—
90#477—24no—
91#467—28no—
92#462—24no—
93#462—24no—
94#451—24no—
95#451C2×C418yes—
96#446—24no—
97#446—24no—
98#446—28no—
99#446—28no—
100C8×Q8—24no—
101C8×Q8—24no—
102C4×Q16—28no—
103#431—21no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431C2×C412yes—
109#431C2312yes—
110#431C2312yes—
111#424—24no—
112#424—24no—
113#423—24no—
114#423—21no—
115#423—21no—
116#423D412yes—
117#423C2312yes—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8D412yes—
127C2×C4⋊C8C2312yes—
128C2×C22⋊C8—21no—
129C2×C22⋊C8C2×C412yes—
130#405—24no—
131#405—24no—
132C4×M16—24no—
133C4×M16—21no—
134C4×M16—21no—
135C4×M16—21no—
136C4×M16—21no—
137C2×#97C2312yes—
138C2×C4×C8—21no—
139C2×C4×C8—21no—
140C2×C4×C8D412yes—
141#336—24no—
142#336—24no—
143#333—24no—
144#332—24no—
145Q8⋊C8—28no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8C22⋊C412yes—
149C22×Q8C22×C412yes—
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4#4111yescharacteristic
153C2×Q16—28no—
154C2×M16—21no—
155C2×M16—21no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—42no—
164C2×M16—42no—
165C2×M16Q1614yescomplemented
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—41no—
170C22×C8—41no—
171C22×C8—41no—
172C22×C8—42no—
173C22×C8—42no—
174C22×C8C2×D412yes—
175C22×C8C2×D412yes—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8C22⋊C412yes—
189C4⋊Q8C22⋊C412yes—
190C4⋊Q8C2×D412yes—
191C4⋊Q8C2×D412yes—
192#125—22no—
193#125—22no—
194#125—22no—
195#125—22no—
196#125—41no—
197#125—41no—
198#125—41no—
199#125—41no—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204C22⋊Q8—22no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—41no—
211C22⋊Q8—41no—
212C22⋊Q8—41no—
213C22⋊Q8—41no—
214C22⋊Q8C22⋊C414yes—
215C22⋊Q8C22×C414yes—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—41no—
227C4×Q8—41no—
228C4×Q8—41no—
229C4×Q8—41no—
230C4×Q8C22⋊C414yes—
231C4×Q8C2×D414yes—
232#117—22no—
233#117—22no—
234#117—22no—
235#117—22no—
236#117—41no—
237#117—41no—
238#117—41no—
239#117—41no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4C22⋊C412yes—
251C2×C4⋊C4C22×C412yes—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42C22×C411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42#4111yescharacteristic
265C2×C42#4111yescharacteristic
266C2×C42#4111yescharacteristic
267C2×C42C2411yescharacteristic
268#107—28no—
269C4⋊C8—28no—
270C4⋊C8—28no—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—22no—
279C4⋊C8—41no—
280C4⋊C8—41no—
281C4⋊C8—41no—
282C4⋊C8C2×D414yes—
283#103—28no—
284#103—28no—
285C22⋊C8—22no—
286C22⋊C8—22no—
287C22⋊C8—41no—
288C22⋊C8C22×C414yes—
289#97—28no—
290#97—22no—
291#97—22no—
292C4×C8—28no—
293C4×C8—22no—
294C4×C8—22no—
295C4×C8—22no—
296C4×C8—22no—
297C4×C8—22no—
298C4×C8—41no—
299C24C4×D411yescharacteristic
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—44no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—82no—
309C2×Q8—82no—
310C2×Q8—82no—
311C2×Q8C2×C22⋊C414yes—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—42no—
317C22×C4—81no—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C4×D411yescharacteristic
321C22×C4C4×D411yescharacteristic
322C22×C4C4×D411yescharacteristic
323C22×C4C4×D411yescharacteristic
324C22×C4C4×D411yescharacteristic
325C22×C4#12011yescharacteristic
326C22×C4#12011yescharacteristic
327C22×C4#12111yescharacteristic
328C22×C4#12111yescharacteristic
329C22×C4#12111yescharacteristic
330C22×C4#12111yescharacteristic
331C22×C4#12311yescharacteristic
332C22×C4#12311yescharacteristic
333C22×C4C2×Q1612yes—
334C22×C4#13512yes—
335C22×C4#13712yes—
336C22×C4#13712yes—
337C22×C4C23×C411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C2×#4111yescharacteristic
340Q16—88no—
341M16—24no—
342M16—24no—
343M16—42no—
344M16—42no—
345M16—42no—
346M16—42no—
347M16—44no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—24no—
351C2×C8—22no—
352C2×C8—24no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—44no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—82no—
361C2×C8—82no—
362C2×C8—82no—
363C2×C8—82no—
364C2×C8C2×Q1614yes—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4C2×C22⋊C414yes—
389C22⋊C4—24no—
390C22⋊C4—24no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C42—24no—
394C42—24no—
395C42—42no—
396C42—42no—
397C42C2×C22⋊C411yescharacteristic
398C42C2×C22⋊C411yescharacteristic
399C42C4×D411yescharacteristic
400C42C4×D411yescharacteristic
401C42#12011yescharacteristic
402C42#12011yescharacteristic
403C42#12111yescharacteristic
404C42#12111yescharacteristic
405C42#12111yescharacteristic
406C42#12111yescharacteristic
407C42#12311yescharacteristic
408C42#12311yescharacteristic
409C42C22×D411yescharacteristic
410C42C22×D411yescharacteristic
411C42C22×D411yescharacteristic
412C42C2×#4111yescharacteristic
413C23—41no—
414C23—41no—
415C23#38611yescharacteristic
416C23#38611yescharacteristic
417C23C4×Q1612yes—
418C23#44112yes—
419C23C2×C4×D411yescharacteristic
420Q8—44no—
421Q8—44no—
422Q8—44no—
423Q8—44no—
424Q8—44no—
425Q8—82no—
426Q8—82no—
427Q8—82no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4#38611yescharacteristic
449C2×C4#38611yescharacteristic
450C2×C4#38611yescharacteristic
451C2×C4#38611yescharacteristic
452C2×C4#38611yescharacteristic
453C2×C4#38611yescharacteristic
454C2×C4C4×Q1612yes—
455C2×C4#44112yes—
456C2×C4#45112yes—
457C2×C4#45112yes—
458C2×C4#45212yes—
459C2×C4#45212yes—
460C2×C4#46212yes—
461C2×C4#46212yes—
462C2×C4#46412yes—
463C2×C4#46412yes—
464C2×C4C22×C22⋊C411yescharacteristic
465C2×C4C2×C4×D411yescharacteristic
466C2×C4C2×#12011yescharacteristic
467C2×C4C2×#12111yescharacteristic
468C2×C4C2×#12111yescharacteristic
469C2×C4C2×#12311yescharacteristic
470C2×C4C22×Q1611yescharacteristic
471C2×C4C2×#13511yescharacteristic
472C2×C4C2×#13711yescharacteristic
473C2×C4C2×#13711yescharacteristic
474C8—24no—
475C8—44no—
476C8—42no—
477C8—44no—
478C8—82no—
479C8—82no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—42no—
487C22—42no—
488C22#187812yes—
489C22#187812yes—
490C22C2×#38611yescharacteristic
491C22C2×C4×Q1611yescharacteristic
492C22C2×#44111yescharacteristic
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—44no—
501C4—44no—
502C4—82no—
503C4—82no—
504C4#187812yes—
505C4#187812yes—
506C4#188112yes—
507C4C2×#45111yescharacteristic
508C4C2×#45211yescharacteristic
509C4C2×#46211yescharacteristic
510C4C2×#46411yescharacteristic
511C2—22no—
512C2—22no—
513C2—21no—
514C2—21no—
515C2C2×#187811yescharacteristic
516C2C2×#188111yescharacteristic
517C2#2096411yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9048C2semidirect
165. C2×M16Q16semidirect

Maximal subgroups

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

subgroupindex
#609432
#601922
#213022
C2×#15422
#174862
#90482
C2×#13012
#79182