Finite Group Databasebeta

← #368968

Subgroups — #368968

Order 512.

1336 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2660C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3C2×#1997C211yesmaximal, characteristic
4#21325C211yesmaximal, characteristic
5#21314C211yesmaximal, characteristic
6#11211C212yesmaximal
7#11210C212yesmaximal
8#10892C212yesmaximal
9#10891C212yesmaximal
10#10840C211yesmaximal, characteristic
11#10701C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2660—24no—
17#2660C418yescomplemented
18C2×#398—22no—
19C2×#395—22no—
20#2369—22no—
21#2365—22no—
22#2345—22no—
23Q8×C22⋊C4—22no—
24#2331—22no—
25#2316C2212yes—
26C4×C4⋊Q8—22no—
27#2285C2211yescharacteristic
28C2×#384C2212yes—
29C2×#382C2212yes—
30#2274C2212yes—
31C4×#117C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1997—24no—
34#1992—22no—
35#1991—22no—
36#1989—22no—
37#1989—22no—
38C2×#343C2212yes—
39C2×#343C2212yes—
40#1736C2211yescharacteristic
41#1736C2211yescharacteristic
42#1736C2211yescharacteristic
43#1736C2211yescharacteristic
44#1736C2212yes—
45#1736C2212yes—
46C2×#339C2212yes—
47C2×#339C2212yes—
48C2×#339C2212yes—
49C2×#339C2212yes—
50C23×Q8D412yes—
51C22×M16C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C22×M16C2312yes—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8C2×C414yes—
61C2×C4⋊Q8D414yes—
62C2×#125—22no—
63C2×#125—21no—
64C2×#125—21no—
65C2×#125—21no—
66C2×#125—21no—
67C2×#125—22no—
68C2×#125—22no—
69C2×#125C2×C414yes—
70C2×C4×Q8—24no—
71C2×#117C2311yescharacteristic
72C2×#117C2312yes—
73C2×#117C2311yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—24no—
77C22×C4⋊C4D412yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83#423—22no—
84#423—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C4×M16—22no—
88C4×M16—22no—
89C2×#97—21no—
90C2×C4×C8—21no—
91#398—216no—
92#395—216no—
93#391—24no—
94#391—28no—
95#389—24no—
96#389—24no—
97#389—24no—
98#387—28no—
99#384—24no—
100#384C2×C418yes—
101#384D412yes—
102#382—24no—
103#382C2×C418yes—
104#382D412yes—
105#380—24no—
106#380—21no—
107#380—21no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—24no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4D412yes—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—24no—
117C4×C22⋊C4—21no—
118#376—21no—
119#376—22no—
120#376C2×C414yes—
121C2×#95—21no—
122C2×#95—24no—
123C2×#95—21no—
124C43—21no—
125#343—24no—
126#343—24no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—24no—
132#340—24no—
133#339—24no—
134#339—24no—
135#339—24no—
136#339Q818yescomplemented
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—24no—
140C22×Q8—84no—
141C22×Q8C22⋊C418yes—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4C2×D414yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
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×M16C2×Q814yes—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—21no—
166C4⋊Q8—24no—
167C4⋊Q8—24no—
168C4⋊Q8—24no—
169C4⋊Q8—24no—
170C4⋊Q8—24no—
171C4⋊Q8—24no—
172C4⋊Q8—24no—
173C4⋊Q8—24no—
174#125—24no—
175#125—24no—
176#125—24no—
177#125—24no—
178#125—42no—
179#125—42no—
180#125—42no—
181#125—42no—
182C4×Q8—88no—
183#117—22no—
184#117—22no—
185#117—22no—
186#117—22no—
187#117—22no—
188#117—22no—
189#117—22no—
190#117C2×Q814yes—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—216no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—88no—
221C2×C4⋊C4C22⋊C418yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C414yes—
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4C2×D412yes—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—28no—
241C2×C42C22×C414yes—
242C2×C42C22×C412yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D412yes—
248C2×C42C2×Q814yes—
249C2×C42#4114yes—
250C2×C42#4112yes—
251C2×C42#4112yes—
252C4⋊C8—42no—
253C4⋊C8—42no—
254#97—42no—
255C4×C8—42no—
256#95—28no—
257#95—28no—
258#95—24no—
259#95—24no—
260#95—216no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265C24C22×D411yescharacteristic
266C2×Q8—28no—
267C2×Q8—28no—
268C2×Q8—48no—
269C2×Q8—84no—
270C2×Q8—88no—
271C2×Q8—88no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—48no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4C2×C22⋊C418yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12312yes—
301C22×C4#12312yes—
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×Q812yes—
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4112yes—
310C22×C42−1+412yes—
311C22×C42−1+412yes—
312M16—24no—
313M16—24no—
314M16—42no—
315M16—42no—
316M16—42no—
317M16—42no—
318M16—42no—
319M16—42no—
320C2×C8—24no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—28no—
335C4⋊C4—28no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—816no—
355C4⋊C4—88no—
356C4⋊C4C4⋊Q814yes—
357C22⋊C4—24no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C42—24no—
362C42—24no—
363C42—24no—
364C42—24no—
365C42—44no—
366C42—44no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—88no—
371C42C4×D412yes—
372C42C4×D412yes—
373C42C4×D412yes—
374C42C4×D412yes—
375C42C4×Q818yescomplemented
376C42#12112yes—
377C42#12112yes—
378C42#12112yes—
379C42#12112yes—
380C23—22no—
381C23—22no—
382C23C22×C22⋊C411yescharacteristic
383C23C2×#12011yescharacteristic
384C23C2×#12311yescharacteristic
385C23#54812yes—
386C23D4×Q812yes—
387Q8—416no—
388Q8—88no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—28no—
394C2×C4—24no—
395C2×C4—28no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—48no—
402C2×C4—44no—
403C2×C4—48no—
404C2×C4—42no—
405C2×C4—48no—
406C2×C4—42no—
407C2×C4—48no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—84no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4#38612yes—
424C2×C4#38612yes—
425C2×C4#38612yes—
426C2×C4#38612yes—
427C2×C4C2×C4×D411yescharacteristic
428C2×C4C2×C4×D411yescharacteristic
429C2×C4C2×C4×Q812yes—
430C2×C4#51912yes—
431C2×C4C2×#12111yescharacteristic
432C2×C4C2×#12111yescharacteristic
433C2×C4C2×C4⋊Q811yescharacteristic
434C2×C4#53211yescharacteristic
435C2×C4#53312yes—
436C2×C4#53611yescharacteristic
437C2×C4#53611yescharacteristic
438C2×C4#54112yes—
439C2×C4#54712yes—
440C2×C4#54812yes—
441C2×C4D4×Q812yes—
442C2×C4D4×Q812yes—
443C2×C4#55612yes—
444C2×C4#55712yes—
445C8—44no—
446C8—81no—
447C8—81no—
448C22—24no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—24no—
454C22C2×#38611yescharacteristic
455C22Q8×C22⋊C412yes—
456C22#243211yescharacteristic
457C22#243411yescharacteristic
458C22#244612yes—
459C22#301912yes—
460C22#302012yes—
461C4—22no—
462C4—22no—
463C4—22no—
464C4—22no—
465C4—48no—
466C4—48no—
467C4—44no—
468C4—82no—
469C4—84no—
470C4—82no—
471C4—82no—
472C4C4×C4⋊Q811yescharacteristic
473C4#232711yescharacteristic
474C4#236512yes—
475C4#242912yes—
476C4#254511yescharacteristic
477C4#254611yescharacteristic
478C2—22no—
479C2—22no—
480C2#1402411yescharacteristic
481C2#2095311yescharacteristic
482C2#2095411yescharacteristic
483C2#2566912yes—
484C2#2583212yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
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
17. #2660C4semidirect
136. #339Q8semidirect
375. C42C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26602
#342162
C2×#19972
#213252
#213142
#112112
#112102
#108922
#108912
#108402
#107012