Finite Group Databasebeta

← #190212

Subgroups — #190212

Order 512.

1317 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34181C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic
4#8761C218yesmaximal, complemented
5#8023C212yesmaximal, complemented
6C2×#1289C212yesmaximal, complemented
7C2×#518—21no—
8C22×C4×D4C2211yescharacteristic
9C22×#99—21no—
10C22×C22⋊C8C2212yes—
11#2349C414yescomplemented
12#2349C2214yes—
13D4×C4⋊C4—22no—
14D4×C4⋊C4—22no—
15#2316—21no—
16#2315—21no—
17#2315—21no—
18#2285C412yescomplemented
19C2×#380C412yescomplemented
20C2×#380C2212yes—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23#1799—22no—
24#1799C2214yes—
25C2×#342—21no—
26#1459—22no—
27#1459C2214yes—
28C2×#324—21no—
29C2×#323—21no—
30#1294C2218yescomplemented
31#1289C2218yescomplemented
32C23×D4D411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C24×C4C2311yescharacteristic
35C23×C8—22no—
36#518—48no—
37C2×C4×D4—22no—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4—22no—
41C2×C4×D4—24no—
42C2×C4×D4—24no—
43C2×C4×D4C2×C412yes—
44C2×C4×D4D412yes—
45C2×C4×D4D412yes—
46C2×C4×D4C2312yes—
47C2×#117—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4D411yescharacteristic
53C22×C22⋊C4—22no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4D411yescharacteristic
58C22×C22⋊C4D411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C2×#99—24no—
61C2×#99—44no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8C2314yes—
68#389—24no—
69#389—24no—
70#385—24no—
71#385—24no—
72#385—24no—
73#385—24no—
74#384—22no—
75#382C2×C414yescomplemented
76#380—22no—
77#380—22no—
78#380—22no—
79#380—21no—
80#380—21no—
81#380—21no—
82#380—21no—
83#380—22no—
84#380—22no—
85#380—22no—
86#380C2×C412yes—
87#380C2×C412yes—
88#380C2×C412yes—
89#380C2×C414yescomplemented
90#380C2314yes—
91#380C2312yes—
92C4×C4⋊C4—22no—
93C4×C4⋊C4C2×C414yescomplemented
94C2×#95—22no—
95C2×#95C2×C412yes—
96C2×#95C2×C414yescomplemented
97C2×#95C2312yes—
98#342—24no—
99#324—24no—
100#323—24no—
101C25C22⋊C411yescharacteristic
102C25C2×D411yescharacteristic
103C22×D4—22no—
104C22×D4—22no—
105C22×D4—24no—
106C22×D4C22⋊C412yes—
107C22×D4SD1614yes—
108C22×D4Q1614yes—
109C22×D4C2×D412yes—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—24no—
116C23×C4—22no—
117C23×C4—42no—
118C23×C4—82no—
119C23×C4—82no—
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22⋊C412yes—
122C23×C4C22⋊C411yescharacteristic
123C23×C4C22⋊C411yescharacteristic
124C23×C4C22×C412yes—
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4#4112yes—
131C23×C4#4112yes—
132C23×C4#4112yes—
133C23×C4#4112yes—
134C23×C4C2411yescharacteristic
135C22×C8—24no—
136C22×C8—42no—
137C22×C8—42no—
138C22×C8—84no—
139C4×D4—24no—
140C4×D4—24no—
141C4×D4—24no—
142C4×D4—24no—
143C4×D4—24no—
144C4×D4—24no—
145C4×D4—24no—
146C4×D4—24no—
147C4×D4—416no—
148C4×D4—416no—
149#117—48no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—84no—
164C2×C4⋊C4—84no—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C2×D412yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—48no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4C22⋊C412yes—
184C2×C22⋊C4C22⋊C412yes—
185C2×C22⋊C4C2×D412yes—
186C2×C22⋊C4C2×D412yes—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—24no—
192C2×C42C22⋊C412yes—
193C2×C42C22×C412yes—
194#99—48no—
195C22⋊C8—24no—
196C22⋊C8—24no—
197C22⋊C8—42no—
198C22⋊C8—42no—
199#95—22no—
200#95—24no—
201#95—24no—
202#95—22no—
203#95—24no—
204#95—42no—
205#95C22×C414yes—
206#95C22×C414yes—
207C24—21no—
208C24—21no—
209C24—22no—
210C24—81no—
211C24—81no—
212C24—81no—
213C24—81no—
214C24#10314yes—
215C24C2×C22⋊C412yes—
216C24C2×C22⋊C411yescharacteristic
217C24C2×C22⋊C411yescharacteristic
218C24#12312yes—
219C24#12312yes—
220C24C2×SD1612yes—
221C24C2×Q1612yes—
222C24AGL(1,ℤ/8)14yes—
223C24C22×D411yescharacteristic
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—28no—
229C2×D4—28no—
230C2×D4—28no—
231C2×D4—28no—
232C2×D4#10318yes—
233C2×D4C2×SD1614yes—
234C2×D4C2×Q1614yes—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—48no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—81no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—81no—
268C22×C4—82no—
269C22×C4—81no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—84no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—84no—
276C22×C4—84no—
277C22×C4—84no—
278C22×C4—84no—
279C22×C4#10314yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4#11711yescharacteristic
286C22×C4#11711yescharacteristic
287C22×C4#11711yescharacteristic
288C22×C4#11711yescharacteristic
289C22×C4#11712yes—
290C22×C4#11712yes—
291C22×C4#12312yes—
292C22×C4#12312yes—
293C22×C4#12312yes—
294C22×C4#12312yes—
295C22×C4#12312yes—
296C22×C4#12312yes—
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C2×#4112yes—
300C22×C4C2×#4112yes—
301C2×C8—44no—
302C2×C8—84no—
303C2×C8—84no—
304C2×C8—84no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—48no—
310C4⋊C4—84no—
311C4⋊C4—84no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4—84no—
315C4⋊C4—84no—
316C22⋊C4—24no—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C22⋊C4—24no—
320C22⋊C4—48no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—84no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—48no—
333C23—22no—
334C23—22no—
335C23—24no—
336C23—24no—
337C23—41no—
338C23—42no—
339C23—41no—
340C23—41no—
341C23—42no—
342C23—41no—
343C23—42no—
344C23—41no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—82no—
354C23—82no—
355C23—84no—
356C23—84no—
357C23—84no—
358C23#38112yes—
359C23#38111yescharacteristic
360C23#38111yescharacteristic
361C23C2×#10314yes—
362C23C2×#10312yes—
363C23C2×#10312yes—
364C23#41812yes—
365C23#41812yes—
366C23#48114yes—
367C23#48114yes—
368C23#48414yes—
369C23#48414yes—
370C23C22×C22⋊C411yescharacteristic
371C23C2×#12312yes—
372C23C22×SD1611yescharacteristic
373C23C22×Q1611yescharacteristic
374C23C2×AGL(1,ℤ/8)12yes—
375D4—28no—
376D4—28no—
377D4—216no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—84no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4—164no—
422C2×C4—164no—
423C2×C4—168no—
424C2×C4—168no—
425C2×C4#38111yescharacteristic
426C2×C4#38112yes—
427C2×C4#38111yescharacteristic
428C2×C4C2×#10314yes—
429C2×C4C2×#11711yescharacteristic
430C2×C4C2×#11711yescharacteristic
431C2×C4C2×#12312yes—
432C8—84no—
433C22—21no—
434C22—22no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—21no—
439C22—22no—
440C22—21no—
441C22—21no—
442C22—22no—
443C22—24no—
444C22—42no—
445C22—42no—
446C22—44no—
447C22—42no—
448C22—42no—
449C22—41no—
450C22—41no—
451C22—42no—
452C22—44no—
453C22—44no—
454C22—44no—
455C22—44no—
456C22—44no—
457C22—42no—
458C22—42no—
459C22—84no—
460C22—84no—
461C22#182412yes—
462C22#182412yes—
463C22#182414yes—
464C22C2×#38111yescharacteristic
465C22C22×#10311yescharacteristic
466C22C2×#41811yescharacteristic
467C22C2×#48112yes—
468C22C2×#48412yes—
469C4—24no—
470C4—22no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—22no—
475C4—42no—
476C4—42no—
477C4—84no—
478C4—84no—
479C4—164no—
480C4#182812yes—
481C2—21no—
482C2—22no—
483C2—21no—
484C2—22no—
485C2—22no—
486C2—22no—
487C2—24no—
488C2—44no—
489C2C2×#182411yescharacteristic
490C2#2070811yescharacteristic
491C2C2×#182811yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8761C2semidirect
5. #8023C2semidirect
6. C2×#1289C2semidirect
11. #2349C4semidirect
18. #2285C4semidirect
19. C2×#380C4semidirect
30. #1294C22semidirect
31. #1289C22semidirect
75. #382C2×C4semidirect
89. #380C2×C4semidirect
93. C4×C4⋊C4C2×C4semidirect
96. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341812
C2×#17992
C2×#14592
#87612
#80232
C2×#12892