Finite Group Databasebeta

← #412909

Subgroups — #412909

Order 512.

858 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
1C2×#2089C212yesmaximal
2#21170C211yesmaximal, characteristic
3C2×#1947C211yesmaximal, characteristic
4#21145C211yesmaximal, characteristic
5#20800C211yesmaximal, characteristic
6#11331C212yesmaximal
7#11290C212yesmaximal
8#11216C212yesmaximal
9#10857C211yesmaximal, characteristic
10#10853C211yesmaximal, characteristic
11C4×#343C211yesmaximal, characteristic
12C2×#428C2212yes—
13C2×#428C2212yes—
14C2×#428C2212yes—
15C22×#107C2212yes—
16C2×#424C2212yes—
17C2×#423C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21#2285C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#2089—28no—
25#1949—22no—
26#1948—21no—
27#1947—22no—
28#1947—22no—
29#1947—21no—
30#1940—22no—
31#1939—22no—
32#1847—22no—
33#1845—21no—
34#1844—21no—
35C2×#343C2211yescharacteristic
36C2×#343C2212yes—
37C2×#343C2211yescharacteristic
38C2×#343C2212yes—
39C2×#343C2211yescharacteristic
40C2×#343C2211yescharacteristic
41C2×#341C2212yes—
42C2×#341C2212yes—
43#1736C2212yes—
44C2×#340C2212yes—
45#1729C2212yes—
46C22×M16C2311yescharacteristic
47C22×M16C2311yescharacteristic
48C22×M16C2312yes—
49C22×M16C2311yescharacteristic
50C22×M16C2311yescharacteristic
51C23×C8C2312yes—
52C2×#117C2312yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60#428—24no—
61#428—24no—
62#428—24no—
63#428—42no—
64#428—42no—
65#428—42no—
66C2×#107—24no—
67C2×#107—24no—
68C2×#107—22no—
69C2×#107—22no—
70C2×#107—22no—
71C2×#107—42no—
72C2×#106—22no—
73C2×#106—22no—
74C2×#106—22no—
75#424—24no—
76#424—42no—
77#423—22no—
78#423Q814yes—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8D412yes—
85C2×C4⋊C8D412yes—
86C4×M16—22no—
87C4×M16—22no—
88C4×M16—22no—
89C4×M16Q814yes—
90C2×#97—21no—
91C2×#97—21no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384D411yescharacteristic
99#384D411yescharacteristic
100#382D412yes—
101#380—21no—
102#380—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C2×#95—21no—
110#343—24no—
111#343—22no—
112#343—22no—
113#343—42no—
114#343D414yes—
115#343Q814yes—
116#341—24no—
117#341—24no—
118#340—24no—
119#340—22no—
120#340—22no—
121#336—24no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4#4111yescharacteristic
126C23×C4#4111yescharacteristic
127C23×C4C2411yescharacteristic
128C2×M16—21no—
129C2×M16—21no—
130C2×M16—21no—
131C2×M16—21no—
132C2×M16—21no—
133C2×M16—21no—
134C2×M16—22no—
135C2×M16—21no—
136C2×M16—21no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—21no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—21no—
145C2×M16—21no—
146C2×M16—22no—
147C2×M16—21no—
148C2×M16—22no—
149C2×M16C2×D412yes—
150C2×M16C2×D412yes—
151C2×M16C2×Q812yes—
152C2×M16C2×Q814yes—
153C2×M16C2×Q812yes—
154C2×M16C2×Q812yes—
155C2×M16C2×Q812yes—
156C2×M16#4112yes—
157C2×M16#4112yes—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8C2×D414yes—
170#117—22no—
171#117—22no—
172#117—22no—
173#117—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×Q812yes—
204C2×C22⋊C4—22no—
205C2×C42—22no—
206C2×C42—41no—
207C2×C42—41no—
208C2×C42—82no—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×Q812yes—
214C2×C42C2×Q812yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217#107—44no—
218#107—48no—
219#107—44no—
220#107—44no—
221#106—44no—
222#106—44no—
223#106—44no—
224C4⋊C8—24no—
225C4⋊C8—24no—
226C4⋊C8—24no—
227C4⋊C8—24no—
228C4⋊C8—48no—
229#97—24no—
230#97—42no—
231C4×C8—24no—
232C4×C8—42no—
233#95—22no—
234#95—22no—
235#95—41no—
236#95—41no—
237C24C2×#4111yescharacteristic
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4#12111yescharacteristic
260C22×C4#12111yescharacteristic
261C22×C4#12111yescharacteristic
262C22×C4#12111yescharacteristic
263C22×C4#12512yes—
264C22×C4#12512yes—
265C22×C4#12512yes—
266C22×C4#12512yes—
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×Q811yescharacteristic
271C22×C4C22×Q811yescharacteristic
272C22×C4C22×Q811yescharacteristic
273C22×C4C2×#4111yescharacteristic
274C22×C4C2×#4111yescharacteristic
275C22×C4C2×#4111yescharacteristic
276C22×C4C2×#4111yescharacteristic
277C22×C4H(5,𝔽2)11yescharacteristic
278C22×C4H(5,𝔽2)11yescharacteristic
279C22×C42−1+411yescharacteristic
280C22×C42−1+411yescharacteristic
281M16—22no—
282M16—22no—
283M16—22no—
284M16—22no—
285M16—22no—
286M16—22no—
287M16—22no—
288M16—22no—
289M16—22no—
290M16—24no—
291M16—22no—
292M16—24no—
293M16—42no—
294M16—41no—
295M16—41no—
296M16—42no—
297M16—41no—
298M16—41no—
299M16#12514yes—
300M16C4⋊Q814yes—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—22no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—22no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—41no—
320C2×C8—42no—
321C2×C8—41no—
322C2×C8—42no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—81no—
342C4⋊C4—81no—
343C4⋊C4—82no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C42—42no—
350C42—42no—
351C42—82no—
352C42—82no—
353C42C22⋊Q812yes—
354C42C22⋊Q812yes—
355C42C22⋊Q812yes—
356C42C22⋊Q812yes—
357C42#12312yes—
358C42#12312yes—
359C42C4⋊Q812yes—
360C42C4⋊Q812yes—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23C2×#12111yescharacteristic
366C23C2×#12511yescharacteristic
367C23C2×#12511yescharacteristic
368C23#53211yescharacteristic
369C23#53311yescharacteristic
370C23#54011yescharacteristic
371C23#54111yescharacteristic
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—162no—
402C2×C4—162no—
403C2×C4#39812yes—
404C2×C4#39812yes—
405C2×C4#39812yes—
406C2×C4#39812yes—
407C2×C4#47212yes—
408C2×C4#47312yes—
409C2×C4C2×C22⋊Q811yescharacteristic
410C2×C4C2×C22⋊Q811yescharacteristic
411C2×C4C2×#12311yescharacteristic
412C2×C4C2×#12511yescharacteristic
413C2×C4C2×C4⋊Q811yescharacteristic
414C2×C4C2×C4⋊Q811yescharacteristic
415C2×C4#53211yescharacteristic
416C2×C4#53311yescharacteristic
417C2×C4#54211yescharacteristic
418C2×C4#54211yescharacteristic
419C2×C4#54411yescharacteristic
420C2×C4#54411yescharacteristic
421C2×C4D4211yescharacteristic
422C2×C4D4211yescharacteristic
423C2×C4#54611yescharacteristic
424C2×C4#54711yescharacteristic
425C2×C4#54711yescharacteristic
426C2×C4#54811yescharacteristic
427C2×C4D4×Q811yescharacteristic
428C2×C4D4×Q811yescharacteristic
429C2×C4D4×Q811yescharacteristic
430C2×C4#55011yescharacteristic
431C2×C4#55211yescharacteristic
432C2×C4#55311yescharacteristic
433C2×C4#55411yescharacteristic
434C2×C4#55411yescharacteristic
435C2×C4#55711yescharacteristic
436C2×C4Q8211yescharacteristic
437C8—24no—
438C8—24no—
439C8—44no—
440C8—42no—
441C8—42no—
442C8—44no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22C2×#39811yescharacteristic
456C22#242811yescharacteristic
457C22#242911yescharacteristic
458C22#250611yescharacteristic
459C22#250711yescharacteristic
460C22#264711yescharacteristic
461C22#285911yescharacteristic
462C22C2×#47211yescharacteristic
463C22C2×#47311yescharacteristic
464C22#306811yescharacteristic
465C22#306911yescharacteristic
466C4—22no—
467C4—22no—
468C4—22no—
469C4—82no—
470C4—82no—
471C4—82no—
472C4—82no—
473C4—162no—
474C4#208914yescomplemented
475C4#242211yescharacteristic
476C4#242311yescharacteristic
477C4#244011yescharacteristic
478C4#244211yescharacteristic
479C4#245111yescharacteristic
480C4#245211yescharacteristic
481C4#255111yescharacteristic
482C4#255211yescharacteristic
483C2—21no—
484C2—21no—
485C2—21no—
486C2—21no—
487C2#1513711yescharacteristic
488C2C2×#208911yescharacteristic
489C2#2367511yescharacteristic
490C2#2540411yescharacteristic
491C2#2540611yescharacteristic
492C2#2548911yescharacteristic
493C2#2549111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×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
474. C4#2089semidirect

Maximal subgroups

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

subgroupindex
C2×#20892
#211702
C2×#19472
#211452
#208002
#113312
#112902
#112162
#108572
#108532
C4×#3432