Finite Group Databasebeta

← #386242

Subgroups — #386242

Order 512.

914 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21053C212yesmaximal
2#20660C211yesmaximal, characteristic
3#11291C212yesmaximal, complemented
4#11221C212yesmaximal, complemented
5#11201C212yesmaximal
6#10909C212yesmaximal, complemented
7#10809C212yesmaximal, complemented
8#10724C211yesmaximal, characteristic
9#10715C211yesmaximal, characteristic
10C23×M16C2211yescharacteristic
11C2×#427C2211yescharacteristic
12C2×#427C2211yescharacteristic
13C2×#424C2212yes—
14C2×#407C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16C2×#405C2212yes—
17#2280C2212yes—
18C2×C4×C22⋊C4C2211yescharacteristic
19#1924—22no—
20#1924—22no—
21#1923—22no—
22#1923—22no—
23#1809C411yescharacteristic, complemented
24#1809C411yescharacteristic, complemented
25#1809C412yescomplemented
26#1809C412yescomplemented
27#1809C411yescharacteristic, complemented
28#1809C411yescharacteristic, complemented
29C2×#341C2212yes—
30C2×#341C2211yescharacteristic
31C2×#341C2211yescharacteristic
32#1735C2212yes—
33C2×#339C2212yes—
34C2×#339C2212yes—
35C2×#339C2212yes—
36C2×#339C2212yes—
37#1729C2212yes—
38#1729C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41C24×C4C2311yescharacteristic
42C22×M16—24no—
43C22×M16—42no—
44C22×M16—41no—
45C22×M16—41no—
46C22×M16C2×C411yescharacteristic
47C22×M16C2×C411yescharacteristic
48C22×M16C2×C411yescharacteristic
49C22×M16C2×C411yescharacteristic
50C22×M16C2312yes—
51C22×M16C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C2×#117C2312yes—
56C2×#117C2312yes—
57C22×C22⋊C4—21no—
58C22×C42C2312yes—
59#427—41no—
60#427—42no—
61#427—41no—
62C2×#107C2×C412yes—
63C2×#107C2×C411yescharacteristic
64C2×#107C2×C411yescharacteristic
65C2×#106C2×C411yescharacteristic
66C2×#106C2×C411yescharacteristic
67C2×#106C2×C412yes—
68#424—42no—
69#424—42no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72#407—24no—
73#407—42no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—41no—
77C2×C22⋊C8—41no—
78C2×C22⋊C8D411yescharacteristic
79C2×C22⋊C8D412yes—
80C2×C22⋊C8D411yescharacteristic
81#405—24no—
82#405—42no—
83C2×#97—22no—
84C2×C4×C8—22no—
85#381—22no—
86#380C2×C412yes—
87#380C2×C412yes—
88#380C2×C412yes—
89#380C2×C412yes—
90#379—22no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—41no—
94C4×C22⋊C4—41no—
95C2×#95—21no—
96#341—24no—
97#341—41no—
98#341—41no—
99#340C2×C412yes—
100#340C2×C412yes—
101#340C2×C412yes—
102#340C2×C412yes—
103#339—24no—
104#339—42no—
105#339—42no—
106#339—42no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111#336—24no—
112#336—42no—
113C25#4111yescharacteristic
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4#4111yescharacteristic
128C23×C4C2411yescharacteristic
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—41no—
145C2×M16—41no—
146C2×M16—41no—
147C2×M16—41no—
148C2×M16—41no—
149C2×M16—41no—
150C2×M16—41no—
151C2×M16—41no—
152C2×M16—48no—
153C2×M16—48no—
154C2×M16#4114yes—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8C22×C411yescharacteristic
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C2×D412yes—
175C22×C8C2×D412yes—
176C22×C8#4114yes—
177#117—22no—
178#117—22no—
179#117—22no—
180#117—22no—
181#117—22no—
182#117—22no—
183#117—22no—
184#117—22no—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—42no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205#107—41no—
206#107—41no—
207#107—42no—
208#106—42no—
209#106—41no—
210#106—41no—
211C4⋊C8—44no—
212C4⋊C8—44no—
213C22⋊C8—41no—
214C22⋊C8—42no—
215C22⋊C8—41no—
216C22⋊C8—44no—
217C22⋊C8—44no—
218#97—44no—
219C4×C8—44no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—42no—
225C24—21no—
226C24—21no—
227C24—22no—
228C24—22no—
229C24—42no—
230C24—42no—
231C24C4×D411yescharacteristic
232C24C4×D411yescharacteristic
233C24C2×#4111yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
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×C4C4×D411yescharacteristic
278C22×C4C4×D411yescharacteristic
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C2×#4112yes—
282C22×C4C2×#4112yes—
283C22×C4H(5,𝔽2)11yescharacteristic
284C22×C4H(5,𝔽2)11yescharacteristic
285C22×C4H(5,𝔽2)12yes—
286C22×C4H(5,𝔽2)11yescharacteristic
287C22×C42−1+412yes—
288C22×C42−1+411yescharacteristic
289M16—24no—
290M16—24no—
291M16—44no—
292M16—44no—
293M16—44no—
294M16—44no—
295M16—42no—
296M16—42no—
297M16—42no—
298M16—42no—
299M16—42no—
300M16—42no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—24no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8#11712yes—
332C2×C8#11712yes—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—48no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—42no—
358C23—22no—
359C23—22no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—82no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23C2×C4×D411yescharacteristic
381C23#53811yescharacteristic
382C23#53911yescharacteristic
383C23#55212yes—
384C23#55312yes—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4C2×#11711yescharacteristic
425C2×C4C4×#4111yescharacteristic
426C2×C4#51811yescharacteristic
427C2×C4#51811yescharacteristic
428C2×C4#51911yescharacteristic
429C2×C4#52011yescharacteristic
430C2×C4#53411yescharacteristic
431C2×C4#53711yescharacteristic
432C2×C4#53812yes—
433C2×C4#53812yes—
434C2×C4#53912yes—
435C2×C4#54212yes—
436C2×C4#54612yes—
437C2×C4#54712yes—
438C2×C4#55212yes—
439C2×C4#55512yes—
440C2×C4#56212yes—
441C2×C4#56312yes—
442C8—22no—
443C8—22no—
444C8—41no—
445C8—41no—
446C8—42no—
447C8—42no—
448C8—44no—
449C8—82no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—42no—
459C22—42no—
460C22—41no—
461C22—41no—
462C22—42no—
463C22—42no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—84no—
469C22—84no—
470C22#232511yescharacteristic
471C22#234411yescharacteristic
472C22#236411yescharacteristic
473C22#277012yes—
474C22#278112yes—
475C22#295512yes—
476C22#295612yes—
477C4—41no—
478C4—41no—
479C4—84no—
480C4—82no—
481C4—82no—
482C4—82no—
483C4—82no—
484C4#231211yescharacteristic
485C4#233211yescharacteristic
486C4#234611yescharacteristic
487C4#237811yescharacteristic
488C4#268612yes—
489C4#269512yes—
490C2—22no—
491C2—22no—
492C2—81no—
493C2—81no—
494C2#1442211yescharacteristic
495C2#2220511yescharacteristic
496C2#2220711yescharacteristic
497C2#2260812yes—
498C2#2314812yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11291C2semidirect
4. #11221C2semidirect
6. #10909C2semidirect
7. #10809C2semidirect
23. #1809C4semidirect
24. #1809C4semidirect
25. #1809C4semidirect
26. #1809C4semidirect
27. #1809C4semidirect
28. #1809C4semidirect

Maximal subgroups

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

subgroupindex
#210532
#206602
#112912
#112212
#112012
#109092
#108092
#107242
#107152