Finite Group Databasebeta

← #386721

Subgroups — #386721

Order 512.

1158 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1974C211yesmaximal, characteristic
2#20937C212yesmaximal
3#20847C212yesmaximal, complemented
4#11249C212yesmaximal, complemented
5#11113C212yesmaximal, complemented
6#11107C212yesmaximal, complemented
7#10862C211yesmaximal, characteristic, complemented
8#10818C211yesmaximal, characteristic, complemented
9#10781C212yesmaximal, complemented
10C22×#135C2211yescharacteristic
11C2×#520C2212yes—
12C22×#108C2211yescharacteristic
13C2×#428C2212yes—
14C22×#107C2212yes—
15C22×#104C2212yes—
16C2×#417C2212yes—
17C2×#417C2212yes—
18C2×#408C2212yes—
19C22×C4×C8C2211yescharacteristic
20#2311C2212yes—
21#1974C418yescomplemented
22#1880—22no—
23#1879—22no—
24#1878—22no—
25#1877—22no—
26#1854—22no—
27#1854—22no—
28#1852—22no—
29#1852—22no—
30C2×#343C2212yes—
31C2×#343C2212yes—
32C2×#341C2211yescharacteristic
33#1736C2212yes—
34#1735C2211yescharacteristic
35C2×#340C2212yes—
36C2×#339C2212yes—
37#1729C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C22×#41C2312yes—
41C2×#135—42no—
42C2×#135—42no—
43C2×#135C2×C414yes—
44C2×#135Q814yes—
45C22×Q16D411yescharacteristic
46C22×SD16D412yes—
47C22×D8D411yescharacteristic
48C22×M16C2312yes—
49C22×M16C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52#520—24no—
53#520—42no—
54#520—42no—
55#520—42no—
56C2×C4×Q8—22no—
57C2×C4×D4—22no—
58C2×C4×D4—42no—
59C2×#117—22no—
60C2×#117C2312yes—
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C42C2311yescharacteristic
65C2×#108—42no—
66C2×#108C2×C414yescomplemented
67#428—42no—
68#428—42no—
69C2×#107—22no—
70C2×#107—22no—
71C2×#107—22no—
72C2×#107—24no—
73C2×#107—42no—
74C2×#106—22no—
75C2×#104—42no—
76C2×#104—42no—
77C2×#104C2×C418yescomplemented
78#417—24no—
79#417—42no—
80#417—42no—
81#417—42no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#102—22no—
85C2×#102—22no—
86#408—24no—
87#408C2×C418yescomplemented
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C4×C8—42no—
91C2×C4×C8C2×C414yescomplemented
92#391—22no—
93#390—22no—
94#386—42no—
95#380—22no—
96#380—22no—
97C2×#95—41no—
98C2×#95—41no—
99#343—42no—
100#343—42no—
101#341—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340D411yescharacteristic
107#340D412yes—
108#340D411yescharacteristic
109#339—42no—
110#336—22no—
111#336—22no—
112C2×#41—22no—
113C2×#41—22no—
114C2×#41—22no—
115C2×#41—22no—
116C2×#41—22no—
117C2×#41—22no—
118C2×#41—22no—
119C2×#41—22no—
120C2×#41—22no—
121C2×#41—22no—
122C2×#41C22×C414yes—
123C2×#41C2×Q814yes—
124C22×Q8C2×D412yes—
125C22×D4—22no—
126C22×D4C2×D412yes—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—42no—
132C23×C4C2411yescharacteristic
133#135—44no—
134#135—44no—
135#135—48no—
136C2×Q16—42no—
137C2×Q16C4⋊C414yes—
138C2×SD16—42no—
139C2×SD16—42no—
140C2×SD16C4⋊C418yes—
141C2×D8—42no—
142C2×D8C4⋊C414yes—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16C22×C414yes—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8C22×C412yes—
162C22×C8C22×C412yes—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×Q812yes—
168C22×C8#4112yes—
169C4×Q8—44no—
170C4×Q8—44no—
171C4×D4—44no—
172C4×D4—48no—
173C4×D4—44no—
174C4×D4—48no—
175#117—22no—
176#117—22no—
177#117—22no—
178#117—22no—
179#117—44no—
180#117—44no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
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—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4#4114yes—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—42no—
207C2×C42C22×C412yes—
208#108—44no—
209#107—44no—
210#107—44no—
211#107—44no—
212#106—44no—
213#104—48no—
214#103—44no—
215#103—44no—
216#102—44no—
217#102—44no—
218C22⋊C8—28no—
219C22⋊C8—28no—
220C4×C8—44no—
221#95—42no—
222#95—42no—
223#95—82no—
224#95—82no—
225C24—22no—
226C24H(5,𝔽2)11yescharacteristic
227#41—24no—
228#41—24no—
229#41—24no—
230#41—24no—
231#41—24no—
232#41—24no—
233#41—24no—
234#41—24no—
235#41—42no—
236#41—42no—
237#41—42no—
238#41—42no—
239#41—42no—
240#41—42no—
241#41—42no—
242#41—42no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8C2×C4⋊C414yes—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4C2×C4⋊C414yes—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—48no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—162no—
287C22×C4—162no—
288C22×C4C23×C411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×Q811yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C22×C4C2×#4112yes—
294C22×C4H(5,𝔽2)12yes—
295C22×C4H(5,𝔽2)12yes—
296C22×C4H(5,𝔽2)12yes—
297C22×C42−1+411yescharacteristic
298Q16—44no—
299SD16—48no—
300D8—44no—
301M16—24no—
302M16—24no—
303M16—42no—
304M16—42no—
305M16—42no—
306M16—42no—
307M16—42no—
308M16—42no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—24no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8C2×C4⋊C412yes—
326C2×C8C2×C4⋊C412yes—
327C2×C8C4×D412yes—
328C2×C8C4×D412yes—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—48no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C22⋊C4—48no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—162no—
346C22⋊C4—162no—
347C42—22no—
348C42—22no—
349C42—48no—
350C42—42no—
351C42—42no—
352C42—42no—
353C23—22no—
354C23—21no—
355C23—21no—
356C23—24no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23#52011yescharacteristic
364C23#53812yes—
365C23D4211yescharacteristic
366C23#55411yescharacteristic
367C23#56012yes—
368Q8—24no—
369Q8—24no—
370Q8—42no—
371Q8—42no—
372D4—24no—
373D4—24no—
374D4—44no—
375D4—44no—
376D4—44no—
377D4—44no—
378D4—42no—
379D4—42no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—84no—
402C2×C4—84no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—162no—
407C2×C4—168no—
408C2×C4—162no—
409C2×C4C22×C4⋊C411yescharacteristic
410C2×C4C2×C4×D411yescharacteristic
411C2×C4C4×#4111yescharacteristic
412C2×C4#51811yescharacteristic
413C2×C4#51811yescharacteristic
414C2×C4#51811yescharacteristic
415C2×C4#53412yes—
416C2×C4#53512yes—
417C2×C4#54012yes—
418C2×C4#54212yes—
419C2×C4#54312yes—
420C2×C4#54612yes—
421C2×C4#54612yes—
422C2×C4#54711yescharacteristic
423C2×C4D4×Q811yescharacteristic
424C2×C4#55312yes—
425C2×C4#55412yes—
426C2×C4#56212yes—
427C8—22no—
428C8—22no—
429C8—22no—
430C8—22no—
431C8—44no—
432C8—82no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—44no—
442C22—44no—
443C22—44no—
444C22—82no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22#233711yescharacteristic
449C22D4×C4⋊C411yescharacteristic
450C22#237711yescharacteristic
451C22#267812yes—
452C22#278012yes—
453C22#329012yes—
454C22#329211yescharacteristic
455C22#329311yescharacteristic
456C4—44no—
457C4—44no—
458C4—82no—
459C4—84no—
460C4—82no—
461C4—162no—
462C4#231511yescharacteristic
463C4#232811yescharacteristic
464C4#234911yescharacteristic
465C4#236411yescharacteristic
466C4#269212yes—
467C4#270112yes—
468C2—21no—
469C2—21no—
470C2—22no—
471C2—84no—
472C2#1443811yescharacteristic
473C2#2269111yescharacteristic
474C2#2269211yescharacteristic
475C2#3104812yes—
476C2#3104912yes—
subgroupG/H
center, socleC23D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#135C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20847C2semidirect
4. #11249C2semidirect
5. #11113C2semidirect
6. #11107C2semidirect
7. #10862C2semidirect
8. #10818C2semidirect
9. #10781C2semidirect
21. #1974C4semidirect
66. C2×#108C2×C4semidirect
77. C2×#104C2×C4semidirect
87. #408C2×C4semidirect
91. C2×C4×C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19742
#209372
#208472
#112492
#111132
#111072
#108622
#108182
#107812