Finite Group Databasebeta

← #192178

Subgroups — #192178

Order 512.

1286 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34379C211yesmaximal, characteristic
2#21944C211yesmaximal, characteristic, complemented
3#21940C211yesmaximal, characteristic, complemented
4C2×#1471C211yesmaximal, characteristic
5#14659C211yesmaximal, characteristic
6#8858C214yesmaximal, complemented
7#8853C214yesmaximal, complemented
8#8024C211yesmaximal, characteristic, complemented
9#8015C211yesmaximal, characteristic, complemented
10C22×#121C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2377—22no—
14#2361—22no—
15#2345—21no—
16D4×C22⋊C4—21no—
17D4×C22⋊C4—22no—
18#2325—22no—
19#2324—22no—
20#2315C2212yes—
21#2312C2212yes—
22#2280C2211yescharacteristic
23C2×#381C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26#1854—22no—
27#1853C2212yes—
28#1852—22no—
29#1852C2212yes—
30#1852C2214yes—
31#1809—22no—
32#1808—22no—
33#1471C418yescomplemented
34C2×#323C2214yes—
35#1445—21no—
36#1439C2214yes—
37#1438—22no—
38#1438—21no—
39#1294C2214yes—
40#1294C2212yes—
41#1294C2212yes—
42C23×D4D412yes—
43C23×D4D411yescharacteristic
44C24×C4C2311yescharacteristic
45C22×M16—21no—
46C23×C8—21no—
47C2×#121—21no—
48C2×#121—21no—
49C2×#121—21no—
50C2×#121—21no—
51C2×#121—22no—
52C2×#121—22no—
53C2×#121C2×C414yes—
54C2×#121Q814yes—
55C2×C4×D4—22no—
56C2×C4×D4—24no—
57C2×#117D412yes—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D412yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—24no—
66C22×C22⋊C4C2312yes—
67C2×#107—24no—
68C2×#107—22no—
69C2×#106—22no—
70#407—22no—
71#407C2×C414yescomplemented
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8C2×C414yescomplemented
76C2×C22⋊C8C2312yes—
77C2×C22⋊C8C2312yes—
78#390—22no—
79#390—24no—
80#389—22no—
81#388—24no—
82#388—24no—
83#387—24no—
84#386—24no—
85#386—24no—
86#386—24no—
87#385—24no—
88#385—24no—
89#381—21no—
90#381—22no—
91#381—22no—
92#380—22no—
93#380—21no—
94#380—21no—
95#380—21no—
96#380—21no—
97#380—21no—
98#380—21no—
99#380D412yes—
100#380D412yes—
101#380C2311yescharacteristic
102#380C2312yes—
103#380C2311yescharacteristic
104#379—22no—
105#379—21no—
106C4×C4⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—24no—
114C2×#95C2312yes—
115#340—24no—
116#340—22no—
117#340—22no—
118#323C2×C4116yescomplemented
119C25C2×D412yes—
120C25C2×D411yescharacteristic
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—24no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—82no—
128C22×D4—84no—
129C22×D4C4⋊C418yes—
130C22×D4C4⋊C414yes—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—82no—
136C23×C4C22×C412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×Q812yes—
144C23×C4#4112yes—
145C23×C4#4112yes—
146C23×C4C2411yescharacteristic
147C2×M16—24no—
148C2×M16—82no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—24no—
152C22×C8—82no—
153#121—24no—
154#121—24no—
155#121—24no—
156#121—24no—
157#121—24no—
158#121—24no—
159#121—24no—
160#121—24no—
161C4×D4—84no—
162C4×D4—88no—
163#117—82no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4C4⋊C414yes—
174C2×C4⋊C4C2×D414yes—
175C2×C4⋊C4C2×D412yes—
176C2×C4⋊C4C2×D412yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4—84no—
195C2×C22⋊C4—82no—
196C2×C22⋊C4—84no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4—84no—
199C2×C22⋊C4—84no—
200C2×C22⋊C4C22×C414yes—
201C2×C22⋊C4C2×Q814yes—
202C2×C42—22no—
203C2×C42—24no—
204#107—84no—
205#107—82no—
206#106—82no—
207C22⋊C8—24no—
208C22⋊C8—24no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8C22×C414yes—
212C22⋊C8C22×C414yes—
213#95—24no—
214#95—24no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—21no—
219#95—21no—
220#95—22no—
221#95—22no—
222#95—41no—
223#95—41no—
224#95—42no—
225#95—84no—
226C24—22no—
227C24—22no—
228C24—22no—
229C24—22no—
230C24—24no—
231C24—24no—
232C24—42no—
233C24—42no—
234C24—44no—
235C24—44no—
236C24C2×C4⋊C414yes—
237C24C2×C4⋊C412yes—
238C24#12011yescharacteristic
239C24#12011yescharacteristic
240C24#12312yes—
241C24C22×D411yescharacteristic
242C2×D4—28no—
243C2×D4—28no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—44no—
247C2×D4—48no—
248C2×D4—82no—
249C2×D4—84no—
250C2×D4—84no—
251C2×D4—88no—
252C2×D4—84no—
253C2×D4—88no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—84no—
284C22×C4—84no—
285C22×C4—84no—
286C22×C4C2×C4⋊C414yes—
287C22×C4C2×C4⋊C412yes—
288C22×C4C4×D412yes—
289C22×C4C4×D412yes—
290C22×C4C4×D412yes—
291C22×C4C4×D412yes—
292C22×C4#12011yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4C22⋊Q812yes—
295C22×C4C22⋊Q812yes—
296C22×C4C22⋊Q812yes—
297C22×C4C22⋊Q812yes—
298C22×C4#12312yes—
299C22×C4C23×C411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×Q811yescharacteristic
303C22×C4C2×#4111yescharacteristic
304C22×C4C2×#4111yescharacteristic
305M16—84no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—42no—
309C2×C8—44no—
310C2×C8—42no—
311C2×C8—84no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—84no—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—84no—
333C22⋊C4—84no—
334C22⋊C4—84no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—84no—
338C22⋊C4—816no—
339C42—44no—
340C42—84no—
341C23—22no—
342C23—22no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—24no—
352C23—24no—
353C23—24no—
354C23—28no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—44no—
360C23—44no—
361C23—42no—
362C23—42no—
363C23—44no—
364C23—44no—
365C23—44no—
366C23—84no—
367C23—84no—
368C23—82no—
369C23—82no—
370C23—84no—
371C23—84no—
372C23—88no—
373C23#38512yes—
374C23#38512yes—
375C23C22×C4⋊C411yescharacteristic
376C23C2×#12011yescharacteristic
377C23C2×#12311yescharacteristic
378C23#57612yes—
379C23#57812yes—
380D4—48no—
381D4—416no—
382D4—84no—
383D4—88no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—82no—
415C2×C4—84no—
416C2×C4—82no—
417C2×C4—84no—
418C2×C4—84no—
419C2×C4—82no—
420C2×C4—84no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—164no—
426C2×C4—168no—
427C2×C4#38512yes—
428C2×C4#38512yes—
429C2×C4C2×C4×D411yescharacteristic
430C2×C4C2×C4×D411yescharacteristic
431C2×C4C2×C22⋊Q811yescharacteristic
432C2×C4C2×C22⋊Q811yescharacteristic
433C8—44no—
434C8—44no—
435C22—24no—
436C22—24no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—44no—
452C22—44no—
453C22—44no—
454C22—84no—
455C22—84no—
456C22—88no—
457C22—84no—
458C22—84no—
459C22—164no—
460C22C2×#38511yescharacteristic
461C22#291312yes—
462C22#300811yescharacteristic
463C22#301111yescharacteristic
464C22#301912yes—
465C22#302612yes—
466C22#309812yes—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—22no—
472C4—22no—
473C4—44no—
474C4—84no—
475C4—82no—
476C4—82no—
477C4—162no—
478C4—162no—
479C2—22no—
480C2—22no—
481C2—42no—
482C2—42no—
483C2—84no—
484C2#1845012yes—
485C2#1845312yes—
486C2#2084711yescharacteristic
487C2#2085611yescharacteristic
488C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21944C2semidirect
3. #21940C2semidirect
6. #8858C2semidirect
7. #8853C2semidirect
8. #8024C2semidirect
9. #8015C2semidirect
33. #1471C4semidirect
71. #407C2×C4semidirect
75. C2×C22⋊C8C2×C4semidirect
118. #323C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343792
#219442
#219402
C2×#14712
#146592
#88582
#88532
#80242
#80152