Finite Group Databasebeta

← #401659

Subgroups — #401659

Order 512.

1442 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
1C22×#424C211yesmaximal, characteristic
2C2×#2088C212yesmaximal, complemented
3C2×#1809C211yesmaximal, characteristic
4#14264C212yesmaximal
5#11232C212yesmaximal, complemented
6#11228C212yesmaximal, complemented
7#11227C212yesmaximal, complemented
8#10722C211yesmaximal, characteristic
9#10721C212yesmaximal
10C23×M16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C2×#424—28no—
16C2×#424—24no—
17C22×C4⋊C8C2212yes—
18C22×C4⋊C8C2212yes—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#380C2212yes—
22C2×#380C2212yes—
23#2088—28no—
24#1809—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33#1444—22no—
34#1442—22no—
35#1439—22no—
36#1439—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C24×C4C2311yescharacteristic
40C22×M16—24no—
41C22×M16—24no—
42C22×M16—41no—
43C22×M16—41no—
44C22×M16D414yes—
45C23×C8—22no—
46C23×C8—24no—
47C23×C8—24no—
48C23×C8—41no—
49C23×C8—41no—
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—24no—
56C2×#117—24no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4D412yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4D411yescharacteristic
67C22×C22⋊C4D411yescharacteristic
68C22×C42C2312yes—
69C2×#107—22no—
70C2×#107—41no—
71C2×#107—41no—
72C2×#106—22no—
73C2×#106—24no—
74#424—232no—
75C2×C4⋊C8—24no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8D418yes—
81#389—24no—
82#389—24no—
83#389—24no—
84#389—24no—
85#380—24no—
86#380—24no—
87#380—24no—
88#380Q818yes—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—41no—
102#340—41no—
103#340—42no—
104#340—42no—
105C25C2×D411yescharacteristic
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D412yes—
120C23×C4C2×Q814yes—
121C23×C4#4112yes—
122C23×C4#4114yes—
123C23×C4C2411yescharacteristic
124C2×M16—28no—
125C2×M16—28no—
126C2×M16—44no—
127C2×M16—48no—
128C2×M16—48no—
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—28no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—48no—
144C22×C8—48no—
145C22×C8—44no—
146C22×C8—44no—
147C22×C8C2×D414yes—
148C22×C8C2×D412yes—
149C22×C8C2×D414yes—
150C22×C8C2×D412yes—
151#117—216no—
152#117—28no—
153#117—28no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4C2×Q814yes—
186C2×C4⋊C4C2×Q814yes—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4—82no—
195C2×C22⋊C4—82no—
196C2×C22⋊C4—82no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42C2×D414yes—
202C2×C42#4114yes—
203#107—44no—
204#106—44no—
205C4⋊C8—216no—
206C4⋊C8—216no—
207#95—28no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—82no—
212#95—82no—
213#95—82no—
214#95—82no—
215C24—22no—
216C24—21no—
217C24—21no—
218C24—22no—
219C24—22no—
220C24—22no—
221C24—22no—
222C24#12112yes—
223C24C22⋊Q814yes—
224C24C22×D411yescharacteristic
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4#12114yes—
284C22×C4#12114yes—
285C22×C4#12112yes—
286C22×C4C22⋊Q814yes—
287C22×C4#12314yes—
288C22×C4#12518yes—
289C22×C4C4⋊Q814yes—
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D412yes—
292C22×C4C22×Q812yes—
293C22×C4C2×#4112yes—
294C22×C4C2×#4111yescharacteristic
295C22×C4C2×#4112yes—
296C22×C4H(5,𝔽2)11yescharacteristic
297C22×C4H(5,𝔽2)11yescharacteristic
298C22×C42−1+411yescharacteristic
299C22×C42−1+411yescharacteristic
300M16—416no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—416no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8#12014yes—
319C4⋊C4—216no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—88no—
337C4⋊C4—88no—
338C22⋊C4—28no—
339C22⋊C4—28no—
340C22⋊C4—88no—
341C22⋊C4—88no—
342C42—24no—
343C42—24no—
344C42—24no—
345C42—24no—
346C23—24no—
347C23—24no—
348C23—24no—
349C23—24no—
350C23—22no—
351C23—22no—
352C23—24no—
353C23—24no—
354C23—24no—
355C23—24no—
356C23—42no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23#39814yes—
367C23C2×#12111yescharacteristic
368C23C2×#12112yes—
369C23C2×C22⋊Q812yes—
370C23#53711yescharacteristic
371C23#53711yescharacteristic
372C23#57612yes—
373C23#57712yes—
374C23#57712yes—
375C23#57812yes—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—28no—
384C2×C4—28no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4#39814yes—
429C2×C4C2×#12011yescharacteristic
430C2×C4C2×#12311yescharacteristic
431C2×C4#52511yescharacteristic
432C2×C4#52512yes—
433C2×C4C2×#12512yes—
434C2×C4C2×C4⋊Q811yescharacteristic
435C2×C4#53411yescharacteristic
436C2×C4#53611yescharacteristic
437C2×C4#53911yescharacteristic
438C2×C4#53911yescharacteristic
439C2×C4#54612yes—
440C2×C4#54612yes—
441C2×C4D4×Q812yes—
442C2×C4D4×Q812yes—
443C2×C4#55212yes—
444C2×C4#55212yes—
445C2×C4#55412yes—
446C2×C4#55412yes—
447C8—24no—
448C8—44no—
449C8—42no—
450C22—28no—
451C22—24no—
452C22—44no—
453C22—44no—
454C22—44no—
455C22—44no—
456C22—42no—
457C22—42no—
458C22—44no—
459C22C2×#39811yescharacteristic
460C22#241712yes—
461C22#241712yes—
462C22#261811yescharacteristic
463C22#284711yescharacteristic
464C22#305712yes—
465C22#306112yes—
466C22#306212yes—
467C22#306212yes—
468C22#306312yes—
469C22#306312yes—
470C22#308112yes—
471C22#308212yes—
472C22#308312yes—
473C22#308412yes—
474C22#325712yes—
475C22#325812yes—
476C22#325911yescharacteristic
477C22#326111yescharacteristic
478C22#326311yescharacteristic
479C22#326311yescharacteristic
480C4—24no—
481C4—44no—
482C4—44no—
483C4—84no—
484C4—84no—
485C4—162no—
486C4—162no—
487C4#240511yescharacteristic
488C4#243411yescharacteristic
489C4#247312yes—
490C4#247312yes—
491C4#248411yescharacteristic
492C4#258311yescharacteristic
493C2—44no—
494C2#1482111yescharacteristic
495C2#2367112yes—
496C2#2484812yes—
497C2#2485012yes—
498C2#2485112yes—
499C2#2485312yes—
500C2#2978511yescharacteristic
501C2#2978611yescharacteristic
502C2#3293812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2088C2semidirect
5. #11232C2semidirect
6. #11228C2semidirect
7. #11227C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4242
C2×#20882
C2×#18092
#142642
#112322
#112282
#112272
#107222
#107212