Finite Group Databasebeta

← #361871

Subgroups — #361871

Order 512.

1342 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic
2#20701C211yesmaximal, characteristic
3#20700C211yesmaximal, characteristic
4C2×#1822C211yesmaximal, characteristic
5C2×#1821C211yesmaximal, characteristic
6C2×#1811C211yesmaximal, characteristic
7#20386C211yesmaximal, characteristic
8#11366C212yesmaximal, complemented
9#11365C212yesmaximal, complemented
10#10870C212yesmaximal, complemented
11#10869C212yesmaximal, complemented
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#108C2212yes—
16C2×#428—24no—
17C2×#428—22no—
18C2×#428—22no—
19C2×#428C418yescomplemented
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C2×#407C2211yescharacteristic
25C2×#407C2211yescharacteristic
26C2×#407C2211yescharacteristic
27C2×#405C2212yes—
28#2280C2211yescharacteristic
29#1825—22no—
30#1825—22no—
31#1824—22no—
32#1823—22no—
33#1822—24no—
34#1821—24no—
35#1811—24no—
36#1753—22no—
37#1753—22no—
38C2×#343C2212yes—
39C2×#343C2212yes—
40C2×#343C2212yes—
41C2×#343C2212yes—
42C2×#341C2212yes—
43#1736C2212yes—
44#1736C2212yes—
45#1736C2212yes—
46#1736C2212yes—
47#1729C2212yes—
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C22×M16—28no—
51C22×M16C2×C414yes—
52C22×M16D414yes—
53C22×M16D412yes—
54C22×M16D412yes—
55C22×M16C2312yes—
56C22×M16C2312yes—
57C22×M16C2312yes—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C2×#117—22no—
61C2×#117—21no—
62C2×#117—21no—
63C2×#117—21no—
64C2×#117—21no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117C2×C414yes—
68C2×#117C2312yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4C2×C414yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C22⋊C4—21no—
83C22×C42—21no—
84C2×#108—24no—
85C2×#108—28no—
86#428—216no—
87#428—216no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107—22no—
92C2×#107C2×C414yes—
93C2×#107C2×C414yes—
94C2×#106—22no—
95C2×#106—22no—
96C2×#106—22no—
97C2×#106—22no—
98C2×#106C2×C414yes—
99C2×#106C2×C414yes—
100#407—22no—
101#407—22no—
102#407—22no—
103#407—22no—
104#407—22no—
105#407D414yes—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112#405—28no—
113C2×#97—22no—
114C2×C4×C8—22no—
115#381—21no—
116#380D412yes—
117#380D412yes—
118#379—21no—
119C4×C22⋊C4—21no—
120C4×C22⋊C4—21no—
121#343—24no—
122#343—24no—
123#343—24no—
124#343—24no—
125#341—24no—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—24no—
130#336—24no—
131C25#4111yescharacteristic
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—41no—
140C23×C4—41no—
141C23×C4—42no—
142C23×C4—84no—
143C23×C4C22×C412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4#4111yescharacteristic
151C23×C4#4112yes—
152C23×C4C2411yescharacteristic
153C2×M16—22no—
154C2×M16—216no—
155C2×M16—24no—
156C2×M16—24no—
157C2×M16—24no—
158C2×M16—24no—
159C2×M16—216no—
160C2×M16—24no—
161C2×M16—24no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—24no—
167C2×M16—24no—
168C2×M16—24no—
169C2×M16—24no—
170C2×M16C22⋊C418yes—
171C2×M16C22⋊C418yes—
172C2×M16C2×D414yes—
173C22×C8—24no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8C22×C412yes—
178C22×C8C22×C412yes—
179C22×C8C2×D412yes—
180C22×C8C2×D412yes—
181C22×C8C2×D412yes—
182C22×C8C2×D412yes—
183C22×C8#4118yes—
184#117—24no—
185#117—24no—
186#117—24no—
187#117—24no—
188#117—24no—
189#117—24no—
190#117—42no—
191#117—42no—
192#117—42no—
193#117—42no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
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—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C2×D414yes—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—42no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42—42no—
243#108—48no—
244#107—28no—
245#107—28no—
246#106—28no—
247#106—28no—
248C22⋊C8—24no—
249C22⋊C8—24no—
250C22⋊C8—42no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253C22⋊C8—42no—
254#97—44no—
255C4×C8—44no—
256#95—22no—
257#95—22no—
258C24—24no—
259C24—42no—
260C24—42no—
261C24C4×D412yes—
262C24#12111yescharacteristic
263C24#12111yescharacteristic
264C24#12412yes—
265C24C2×#4111yescharacteristic
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—28no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—28no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—84no—
295C22×C4—84no—
296C22×C4—88no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—88no—
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4C4×D412yes—
305C22×C4C4×D412yes—
306C22×C4C4×D412yes—
307C22×C4#12112yes—
308C22×C4#12112yes—
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12412yes—
312C22×C4#12712yes—
313C22×C4#12712yes—
314C22×C4C23×C411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C2×#4114yes—
320C22×C4C2×#4111yescharacteristic
321C22×C42−1+414yes—
322M16—216no—
323M16—216no—
324M16—28no—
325M16—44no—
326M16—42no—
327M16—42no—
328M16—42no—
329M16—42no—
330M16—44no—
331M16—44no—
332C2×C8—24no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—44no—
342C2×C8—44no—
343C2×C8C2×C22⋊C414yes—
344C2×C8C2×C22⋊C414yes—
345C2×C8#11718yes—
346C2×C8#12314yes—
347C2×C8#12314yes—
348C2×C8#12318yes—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—28no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—82no—
372C22⋊C4—82no—
373C42—42no—
374C42—42no—
375C42—44no—
376C42—44no—
377C23—22no—
378C23—22no—
379C23—24no—
380C23—22no—
381C23—22no—
382C23—24no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—416no—
392C23#39012yes—
393C23#39012yes—
394C23C2×C4×D411yescharacteristic
395C23C2×#12111yescharacteristic
396C23C2×#12411yescharacteristic
397C23#55214yes—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—88no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—88no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4#39012yes—
434C2×C4#39012yes—
435C2×C4C22×C22⋊C411yescharacteristic
436C2×C4C2×#11712yes—
437C2×C4C2×C4×D411yescharacteristic
438C2×C4#51912yes—
439C2×C4C2×#12111yescharacteristic
440C2×C4C2×#12312yes—
441C2×C4C2×#12312yes—
442C2×C4C2×#12711yescharacteristic
443C2×C4#53612yes—
444C2×C4#53612yes—
445C2×C4#54714yes—
446C2×C4#54714yes—
447C2×C4#55214yes—
448C8—24no—
449C8—44no—
450C8—82no—
451C8—82no—
452C8#38118yes—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—28no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—48no—
465C22—48no—
466C22—44no—
467C22—44no—
468C22C2×#39011yescharacteristic
469C22#234412yes—
470C22#248612yes—
471C22#250312yes—
472C22#306812yes—
473C22#306912yes—
474C4—22no—
475C4—22no—
476C4—82no—
477C4—82no—
478C4—82no—
479C4—82no—
480C4—84no—
481C4C2×#38111yescharacteristic
482C4#231011yescharacteristic
483C4#234412yes—
484C4#248612yes—
485C4#250112yes—
486C2—22no—
487C2—22no—
488C2—42no—
489C2—42no—
490C2#1390411yescharacteristic
491C2#2124911yescharacteristic
492C2#2125011yescharacteristic
493C2#2767614yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #11366C2semidirect
9. #11365C2semidirect
10. #10870C2semidirect
11. #10869C2semidirect
19. C2×#428C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
#207012
#207002
C2×#18222
C2×#18212
C2×#18112
#203862
#113662
#113652
#108702
#108692