Finite Group Databasebeta

← #368932

Subgroups — #368932

Order 512.

1166 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3166C211yesmaximal, characteristic
2C2×#2962C211yesmaximal, characteristic
3C2×#1998C211yesmaximal, characteristic
4#21312C211yesmaximal, characteristic
5C2×#1989C211yesmaximal, characteristic
6#11216C212yesmaximal
7#11216C212yesmaximal
8#10896C212yesmaximal
9#10896C212yesmaximal
10#10853C211yesmaximal, characteristic
11C4×#343C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×#125C2211yescharacteristic
14#3166—24no—
15#3166C418yescomplemented
16C2×#501—21no—
17C2×#501—21no—
18C2×#499—22no—
19#2962—24no—
20#2962—24no—
21C2×#446—21no—
22C2×#446—22no—
23C2×C8×Q8—21no—
24C2×#428C2212yes—
25C2×#428C2212yes—
26C2×#424C2212yes—
27C2×#423C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C2×#405C2212yes—
30C2×C4×M16C2211yescharacteristic
31C2×C4×M16C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1998—24no—
34#1989—21no—
35#1989—21no—
36#1989—22no—
37#1989—22no—
38#1988—21no—
39#1987—21no—
40C2×#343C2211yescharacteristic
41C2×#343C2211yescharacteristic
42C2×#343C2211yescharacteristic
43C2×#343C2211yescharacteristic
44C2×#343C2212yes—
45C2×#343C2212yes—
46C2×#341C2212yes—
47C2×#341C2212yes—
48#1736C2212yes—
49#1736C2212yes—
50C2×#340C2212yes—
51C2×#340C2212yes—
52C22×M16C2311yescharacteristic
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C22×M16C2312yes—
56C22×M16C2311yescharacteristic
57C23×C8C2312yes—
58#533—22no—
59#533—22no—
60#533C2×C414yes—
61#533D414yes—
62C2×C4⋊Q8D411yescharacteristic
63C2×C4⋊Q8D411yescharacteristic
64C2×#125—22no—
65C2×#125—21no—
66C2×#125—21no—
67C2×#125—22no—
68C2×#125—21no—
69C2×#125—21no—
70C2×#125—22no—
71C2×#125C2×C414yes—
72C2×#125D412yes—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×#117C2312yes—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#501—28no—
86#501—28no—
87#499—216no—
88#446—216no—
89#446—28no—
90C8×Q8—28no—
91#428—24no—
92#428—24no—
93#428C2×C418yes—
94#428C2×C418yes—
95C2×#107—22no—
96C2×#107—22no—
97C2×#106—22no—
98C2×#106—22no—
99#424—24no—
100#424—24no—
101#423—22no—
102#423—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8—22no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8D411yescharacteristic
110C2×C4⋊C8D411yescharacteristic
111C2×C4⋊C8D412yes—
112#405—24no—
113#405—24no—
114C4×M16—22no—
115C4×M16—22no—
116C4×M16C2×C414yes—
117C4×M16D414yes—
118C2×#97—22no—
119C2×#97—21no—
120C2×#97—21no—
121C2×C4×C8—22no—
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C2×#95—21no—
129#343—22no—
130#343—22no—
131#343—22no—
132#343—22no—
133#343—24no—
134#343Q818yescomplemented
135#341—24no—
136#341—24no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145C22×Q8—21no—
146C22×Q8—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4#4111yescharacteristic
151C23×C4#4111yescharacteristic
152C23×C4C2411yescharacteristic
153C2×M16—21no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—21no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—21no—
162C2×M16—21no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16C22×C414yes—
170C2×M16C2×D412yes—
171C2×M16C2×D412yes—
172C2×M16C2×Q814yes—
173C2×M16C2×Q814yes—
174C2×M16#4114yes—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8C2×D412yes—
187C22×C8C2×D412yes—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8C22⋊C414yes—
191C4⋊Q8C22⋊C414yes—
192#125—24no—
193#125—24no—
194#125—24no—
195#125—24no—
196#125—24no—
197#125—42no—
198#125—42no—
199#125—42no—
200#125—42no—
201#125C22⋊C418yes—
202C22⋊Q8—24no—
203C22⋊Q8—24no—
204C22⋊Q8—24no—
205C22⋊Q8—24no—
206C4×Q8—24no—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—24no—
210#117—22no—
211#117—22no—
212#117C22×C414yes—
213#117C2×D414yes—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4C22×C412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×Q814yes—
246C2×C22⋊C4—22no—
247C2×C42—22no—
248C2×C42—41no—
249C2×C42—41no—
250C2×C42—82no—
251C2×C42C22×C412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D412yes—
258C2×C42#4112yes—
259#107—28no—
260#107—28no—
261#106—28no—
262#106—28no—
263C4⋊C8—28no—
264C4⋊C8—28no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—24no—
268C4⋊C8—42no—
269C4⋊C8—42no—
270#97—28no—
271#97—24no—
272#97—24no—
273C4×C8—28no—
274C4×C8—24no—
275C4×C8—24no—
276#95—41no—
277#95—41no—
278C24C2×#4111yescharacteristic
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—44no—
282C2×Q8—81no—
283C2×Q8—81no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—81no—
302C22×C4—81no—
303C22×C4C4×D412yes—
304C22×C4C4×D412yes—
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12312yes—
310C22×C4#12312yes—
311C22×C4C23×C411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×Q812yes—
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4112yes—
319C22×C42−1+412yes—
320C22×C42−1+412yes—
321M16—24no—
322M16—22no—
323M16—24no—
324M16—24no—
325M16—24no—
326M16—22no—
327M16—22no—
328M16—42no—
329M16—41no—
330M16—41no—
331M16—41no—
332M16—41no—
333M16—42no—
334M16—42no—
335M16C4×Q818yes—
336M16C4⋊Q814yes—
337C2×C8—24no—
338C2×C8—22no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—42no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—28no—
361C4⋊C4—28no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4C2×C22⋊C414yes—
381C4⋊C4C2×C22⋊C414yes—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C42—24no—
385C42—24no—
386C42—82no—
387C42—82no—
388C42C2×C22⋊C412yes—
389C42C2×C22⋊C412yes—
390C42C4×D412yes—
391C42C4×D412yes—
392C42#12012yes—
393C42#12012yes—
394C42#12112yes—
395C42#12112yes—
396C23—22no—
397C23—21no—
398C23—21no—
399C23C2×C4×D411yescharacteristic
400C23C2×#12111yescharacteristic
401C23C2×#12311yescharacteristic
402C23#53312yes—
403C23#54112yes—
404Q8—48no—
405Q8—84no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—84no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—84no—
435C2×C4#38612yes—
436C2×C4#38612yes—
437C2×C4#38612yes—
438C2×C4#38612yes—
439C2×C4C22×C22⋊C411yescharacteristic
440C2×C4C2×C4×D411yescharacteristic
441C2×C4C2×C4×Q812yes—
442C2×C4#51912yes—
443C2×C4C2×#12011yescharacteristic
444C2×C4C2×#12111yescharacteristic
445C2×C4C2×C4⋊Q811yescharacteristic
446C2×C4#53211yescharacteristic
447C2×C4#53611yescharacteristic
448C2×C4#53611yescharacteristic
449C2×C4#54712yes—
450C2×C4#54812yes—
451C2×C4#54812yes—
452C2×C4D4×Q812yes—
453C2×C4D4×Q812yes—
454C2×C4D4×Q812yes—
455C2×C4#55612yes—
456C2×C4#55712yes—
457C8—28no—
458C8—24no—
459C8—42no—
460C8—42no—
461C8—42no—
462C8—81no—
463C8—81no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22C2×#38611yescharacteristic
473C22C4×C4⋊Q811yescharacteristic
474C22#232711yescharacteristic
475C22#242912yes—
476C22#244612yes—
477C22#306712yes—
478C22#306811yescharacteristic
479C22#306911yescharacteristic
480C4—22no—
481C4—22no—
482C4—22no—
483C4—22no—
484C4—44no—
485C4—44no—
486C4—82no—
487C4—82no—
488C4—84no—
489C4Q8×C22⋊C412yes—
490C4#236512yes—
491C4#243211yescharacteristic
492C4#243411yescharacteristic
493C4#254511yescharacteristic
494C4#254611yescharacteristic
495C2—21no—
496C2—21no—
497C2—22no—
498C2#1402411yescharacteristic
499C2#2100311yescharacteristic
500C2#2100411yescharacteristic
501C2#2549312yes—
502C2#2549412yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #3166C4semidirect
134. #343Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31662
C2×#29622
C2×#19982
#213122
C2×#19892
#112162
#108962
#108532
C4×#3432