Finite Group Databasebeta

← #189883

Subgroups — #189883

Order 512.

1262 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34186C211yesmaximal, characteristic
2#21742C211yesmaximal, characteristic
3#13960C211yesmaximal, characteristic
4#8717C212yesmaximal, complemented
5#8716C212yesmaximal, complemented
6#8715C214yesmaximal, complemented
7#8020C212yesmaximal, complemented
8#8016C212yesmaximal, complemented
9C22×C4×D4C2211yescharacteristic
10C2×#407C2212yes—
11#2357—22no—
12#2357—22no—
13#2354C414yescomplemented
14#2354C414yescomplemented
15#2312—21no—
16#2312—21no—
17#2309—21no—
18C2×#386—21no—
19#2285C2211yescharacteristic
20#2285C2211yescharacteristic
21#2280C2212yes—
22#2280C2212yes—
23C2×#380C2211yescharacteristic
24#2274—21no—
25C22×#95C2211yescharacteristic
26#1870—22no—
27#1870—22no—
28#1817C2212yes—
29#1779C2212yes—
30#1441—21no—
31#1439—21no—
32#1434C2212yes—
33C22⋊#95—21no—
34C22⋊#95C2212yes—
35#1294C2214yes—
36#1294C2214yes—
37#1294C2214yes—
38#1294C2214yes—
39C23×D4D411yescharacteristic
40C24×C4—21no—
41C24×C4C2311yescharacteristic
42C24×C4C2311yescharacteristic
43C22×M16—22no—
44C2×C4×D4—24no—
45C2×C4×D4—24no—
46C2×C4×D4C2×C412yes—
47C2×C4×D4C2×C412yes—
48C2×C4×D4D412yes—
49C2×C4×D4D412yes—
50C2×#117—21no—
51C2×#117—21no—
52C2×#117C2×C414yes—
53C2×#117C2×C414yescomplemented
54C22×C4⋊C4—21no—
55C22×C4⋊C4D411yescharacteristic
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4D411yescharacteristic
59C22×C22⋊C4D411yescharacteristic
60C22×C42C2311yescharacteristic
61#407—24no—
62#407—24no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—24no—
65C2×C22⋊C8C2314yes—
66#390—24no—
67#390—24no—
68#388—24no—
69#388—24no—
70#388—24no—
71#388—24no—
72#386—44no—
73#384C2×C412yescomplemented
74#384C2×C412yescomplemented
75#382—21no—
76#382—21no—
77#381—22no—
78#381—22no—
79#380—21no—
80#380—21no—
81#380—24no—
82#380C2×C414yes—
83#380C2×C414yescomplemented
84#380C2312yes—
85#380C2312yes—
86#379—22no—
87#379—22no—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—24no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4C2×C412yescomplemented
93C4×C4⋊C4C2×C412yescomplemented
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
98C2×#95—24no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95C2312yes—
104C2×#95C2312yes—
105#336—22no—
106#336—22no—
107C25C2×D411yescharacteristic
108C25C2×D411yescharacteristic
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—41no—
114C22×D4—41no—
115C22×D4C22⋊C412yes—
116C22×D4C22⋊C412yes—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—44no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—82no—
126C23×C4—82no—
127C23×C4—82no—
128C23×C4C22×C412yes—
129C23×C4C22×C412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4#4114yes—
137C23×C4#4114yes—
138C23×C4C2411yescharacteristic
139C2×M16—44no—
140C2×M16—84no—
141C22×C8—24no—
142C4×D4—28no—
143C4×D4—28no—
144C4×D4—24no—
145C4×D4—24no—
146C4×D4—24no—
147C4×D4—24no—
148#117—42no—
149#117—42no—
150#117—42no—
151#117—42no—
152#117—84no—
153#117—84no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22×C414yes—
165C2×C4⋊C4C22×C414yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—82no—
182C2×C22⋊C4C22⋊C412yes—
183C2×C22⋊C4C22⋊C412yes—
184C2×C22⋊C4C22⋊C412yes—
185C2×C22⋊C4C22⋊C412yes—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C22⋊C8—24no—
197C22⋊C8—24no—
198C22⋊C8—44no—
199C22⋊C8—84no—
200#95—24no—
201#95—24no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—24no—
207#95—24no—
208#95—44no—
209#95—42no—
210#95—44no—
211#95—82no—
212C24—21no—
213C24—21no—
214C24—21no—
215C24—21no—
216C24—21no—
217C24—21no—
218C24—21no—
219C24—21no—
220C24—21no—
221C24—21no—
222C24—42no—
223C24—42no—
224C24—42no—
225C24—42no—
226C24C2×C22⋊C412yes—
227C24C2×C22⋊C412yes—
228C24#12312yes—
229C24#12312yes—
230C24C22×D411yescharacteristic
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4—44no—
240C2×D4—44no—
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—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—48no—
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—84no—
278C22×C4—84no—
279C22×C4—84no—
280C22×C4—84no—
281C22×C4—88no—
282C22×C4—88no—
283C22×C4—84no—
284C22×C4—84no—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4#11718yes—
292C22×C4#12314yes—
293C22×C4#12314yes—
294C22×C4#12312yes—
295C22×C4#12312yes—
296C22×C4AGL(1,ℤ/8)12yes—
297C22×C4#13712yes—
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C2×#4112yes—
301C22×C4C2×#4112yes—
302M16—88no—
303C2×C8—48no—
304C2×C8—44no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—84no—
314C22⋊C4—24no—
315C22⋊C4—24no—
316C22⋊C4—24no—
317C22⋊C4—24no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—84no—
323C22⋊C4—84no—
324C22⋊C4—84no—
325C22⋊C4—84no—
326C22⋊C4—88no—
327C22⋊C4—84no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—44no—
333C42—44no—
334C42—84no—
335C42—84no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—24no—
343C23—22no—
344C23—22no—
345C23—24no—
346C23—24no—
347C23—22no—
348C23—22no—
349C23—24no—
350C23—22no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—44no—
356C23—44no—
357C23—44no—
358C23—44no—
359C23—44no—
360C23—44no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—81no—
366C23—81no—
367C23—82no—
368C23—82no—
369C23—81no—
370C23—81no—
371C23#38114yes—
372C23C22×C22⋊C411yescharacteristic
373C23C2×#12311yescharacteristic
374C23C2×#12311yescharacteristic
375C23C2×AGL(1,ℤ/8)11yescharacteristic
376C23C2×#13711yescharacteristic
377C23#57511yescharacteristic
378C23#57511yescharacteristic
379D4—48no—
380D4—48no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—48no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—48no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—816no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4#38114yes—
434C2×C4#41712yes—
435C2×C4#41712yes—
436C2×C4C2×#11712yes—
437C2×C4C2×#12312yes—
438C8—84no—
439C22—21no—
440C22—21no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—21no—
446C22—21no—
447C22—24no—
448C22—24no—
449C22—24no—
450C22—44no—
451C22—44no—
452C22—44no—
453C22—44no—
454C22—44no—
455C22—44no—
456C22—42no—
457C22—44no—
458C22—44no—
459C22—42no—
460C22—42no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22C2×#38111yescharacteristic
466C22#288312yes—
467C22C2×#41711yescharacteristic
468C22#289611yescharacteristic
469C22#290112yes—
470C22#309411yescharacteristic
471C22#309411yescharacteristic
472C22#309511yescharacteristic
473C22#309511yescharacteristic
474C4—22no—
475C4—22no—
476C4—24no—
477C4—22no—
478C4—22no—
479C4—42no—
480C4—41no—
481C4—41no—
482C4—84no—
483C4—84no—
484C4—84no—
485C4—162no—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—21no—
490C2—44no—
491C2—82no—
492C2#1762912yes—
493C2#1763012yes—
494C2#2068911yescharacteristic
495C2#2070211yescharacteristic
496C2#2070611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8717C2semidirect
5. #8716C2semidirect
6. #8715C2semidirect
7. #8020C2semidirect
8. #8016C2semidirect
13. #2354C4semidirect
14. #2354C4semidirect
53. C2×#117C2×C4semidirect
73. #384C2×C4semidirect
74. #384C2×C4semidirect
83. #380C2×C4semidirect
92. C4×C4⋊C4C2×C4semidirect
93. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341862
#217422
#139602
#87172
#87162
#87152
#80202
#80162