Finite Group Databasebeta

← #190214

Subgroups — #190214

Order 512.

1151 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34185C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic
4#8763C218yesmaximal, complemented
5#8023C212yesmaximal, complemented
6C2×#1289C212yesmaximal, complemented
7C22×C4×D4C2211yescharacteristic
8C22×#99—21no—
9C22×C22⋊C8C2212yes—
10#2353C414yescomplemented
11#2353C2214yes—
12#2352—22no—
13#2352—22no—
14#2313—21no—
15#2312—21no—
16#2308—21no—
17#2283C412yescomplemented
18#2280C412yescomplemented
19C2×#380C2212yes—
20C2×#380C2211yescharacteristic
21C2×#380C2211yescharacteristic
22#2273—21no—
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—
36C2×C4×D4—22no—
37C2×C4×D4—22no—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4C2×C412yes—
41C2×C4×D4D412yes—
42C2×C4×D4D412yes—
43C2×C4×D4C2312yes—
44C2×#117—22no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4D411yescharacteristic
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4D411yescharacteristic
53C22×C22⋊C4D411yescharacteristic
54C22×C42C2×C411yescharacteristic
55C2×#99—24no—
56C2×#99—44no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8C2314yes—
63#388—24no—
64#388—24no—
65#387—24no—
66#387—24no—
67#386—24no—
68#386—24no—
69#383C2×C414yescomplemented
70#382—22no—
71#381—22no—
72#381C2×C414yescomplemented
73#380—22no—
74#380—21no—
75#380—21no—
76#380—21no—
77#380—21no—
78#380—22no—
79#380—22no—
80#380—22no—
81#380C2×C412yes—
82#380C2×C412yes—
83#380C2×C412yes—
84#380C2314yes—
85#380C2312yes—
86#379—22no—
87C4×C4⋊C4—22no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—24no—
91C4×C22⋊C4—24no—
92C4×C22⋊C4C2×C414yescomplemented
93#376C2×C414yescomplemented
94C2×#95—22no—
95C2×#95C2×C412yes—
96C2×#95C2312yes—
97#342—24no—
98#324—24no—
99#323—24no—
100C25C22⋊C411yescharacteristic
101C25C2×D411yescharacteristic
102C22×D4—22no—
103C22×D4—22no—
104C22×D4—24no—
105C22×D4—22no—
106C22×D4—22no—
107C22×D4C22⋊C412yes—
108C22×D4C2×D412yes—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—42no—
116C23×C4—82no—
117C23×C4—82no—
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C412yes—
120C23×C4C22⋊C411yescharacteristic
121C23×C4C22⋊C411yescharacteristic
122C23×C4C22×C412yes—
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4#4112yes—
129C23×C4#4112yes—
130C23×C4#4112yes—
131C23×C4#4112yes—
132C23×C4C2411yescharacteristic
133C22×C8—24no—
134C22×C8—42no—
135C22×C8—42no—
136C22×C8—84no—
137C4×D4—24no—
138C4×D4—24no—
139C4×D4—24no—
140C4×D4—24no—
141C4×D4—24no—
142C4×D4—24no—
143C4×D4—24no—
144C4×D4—24no—
145#117—84no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—84no—
157C2×C4⋊C4C22⋊C412yes—
158C2×C4⋊C4C2×D412yes—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22⋊C412yes—
178C2×C22⋊C4C2×D412yes—
179C2×C22⋊C4C2×D412yes—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42C22⋊C412yes—
187C2×C42C22×C412yes—
188#99—48no—
189C22⋊C8—24no—
190C22⋊C8—24no—
191C22⋊C8—42no—
192C22⋊C8—42no—
193#95—22no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—22no—
199#95—24no—
200#95—42no—
201#95C22×C414yes—
202#95C22×C414yes—
203C24—21no—
204C24—21no—
205C24—22no—
206C24—22no—
207C24—81no—
208C24—81no—
209C24—81no—
210C24—81no—
211C24C2×C22⋊C412yes—
212C24C2×C22⋊C411yescharacteristic
213C24C2×C22⋊C411yescharacteristic
214C24#12312yes—
215C24#12312yes—
216C24#13514yes—
217C24AGL(1,ℤ/8)12yes—
218C24#13712yes—
219C24C22×D411yescharacteristic
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—44no—
228C2×D4—44no—
229C2×D4—44no—
230C2×D4—44no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—81no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—81no—
264C22×C4—82no—
265C22×C4—81no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—84no—
271C22×C4—84no—
272C22×C4—84no—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4#11711yescharacteristic
279C22×C4#11711yescharacteristic
280C22×C4#11711yescharacteristic
281C22×C4#11711yescharacteristic
282C22×C4#11712yes—
283C22×C4#11712yes—
284C22×C4#12312yes—
285C22×C4#12312yes—
286C22×C4#12312yes—
287C22×C4#12312yes—
288C22×C4#12312yes—
289C22×C4#12312yes—
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C2×#4112yes—
293C22×C4C2×#4112yes—
294C2×C8—44no—
295C2×C8—84no—
296C2×C8—84no—
297C2×C8—84no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—84no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—84no—
306C22⋊C4—24no—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—84no—
318C22⋊C4—84no—
319C22⋊C4—84no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—84no—
325C42—84no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—41no—
330C23—42no—
331C23—41no—
332C23—41no—
333C23—42no—
334C23—41no—
335C23—42no—
336C23—41no—
337C23—41no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—84no—
348C23—82no—
349C23—82no—
350C23—84no—
351C23—84no—
352C23#38112yes—
353C23#38111yescharacteristic
354C23#38111yescharacteristic
355C23#41612yes—
356C23#41612yes—
357C23#41712yes—
358C23#41712yes—
359C23#48214yes—
360C23#48214yes—
361C23#48514yes—
362C23#48514yes—
363C23C22×C22⋊C411yescharacteristic
364C23C2×#12312yes—
365C23C2×#13512yes—
366C23C2×AGL(1,ℤ/8)11yescharacteristic
367C23C2×#13711yescharacteristic
368D4—44no—
369D4—44no—
370D4—48no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—84no—
413C2×C4—84no—
414C2×C4—84no—
415C2×C4—164no—
416C2×C4—164no—
417C2×C4—168no—
418C2×C4#38111yescharacteristic
419C2×C4#38112yes—
420C2×C4#38111yescharacteristic
421C2×C4C2×#11711yescharacteristic
422C2×C4C2×#11711yescharacteristic
423C2×C4C2×#12312yes—
424C8—84no—
425C22—21no—
426C22—22no—
427C22—21no—
428C22—21no—
429C22—22no—
430C22—21no—
431C22—22no—
432C22—21no—
433C22—21no—
434C22—22no—
435C22—42no—
436C22—42no—
437C22—44no—
438C22—42no—
439C22—42no—
440C22—41no—
441C22—41no—
442C22—42no—
443C22—44no—
444C22—44no—
445C22—44no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—82no—
450C22—82no—
451C22—84no—
452C22—84no—
453C22#182512yes—
454C22#182512yes—
455C22#182514yes—
456C22C2×#38111yescharacteristic
457C22C2×#41611yescharacteristic
458C22C2×#41711yescharacteristic
459C22C2×#48212yes—
460C22C2×#48512yes—
461C4—22no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—22no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—84no—
470C4—84no—
471C4—164no—
472C4#182812yes—
473C2—21no—
474C2—21no—
475C2—22no—
476C2—22no—
477C2—22no—
478C2—22no—
479C2—24no—
480C2—82no—
481C2C2×#182511yescharacteristic
482C2#2070811yescharacteristic
483C2C2×#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. #8763C2semidirect
5. #8023C2semidirect
6. C2×#1289C2semidirect
10. #2353C4semidirect
17. #2283C4semidirect
18. #2280C4semidirect
30. #1294C22semidirect
31. #1289C22semidirect
69. #383C2×C4semidirect
72. #381C2×C4semidirect
92. C4×C22⋊C4C2×C4semidirect
93. #376C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341852
C2×#17992
C2×#14592
#87632
#80232
C2×#12892