Finite Group Databasebeta

← #99857

Subgroups — #99857

Order 512.

668 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#459C211yesmaximal, characteristic
2C4×C8×D4C211yesmaximal, characteristic, complemented
3#9638C211yesmaximal, characteristic, complemented
4#8563C211yesmaximal, characteristic, complemented
5#7569C211yesmaximal, characteristic, complemented
6#7031C211yesmaximal, characteristic, complemented
7#7015C211yesmaximal, characteristic, complemented
8C2×#459C411yescharacteristic
9C2×C4×D8—21no—
10C2×C8×D4—21no—
11C22×C4×C8—21no—
12C4×#127C2211yescharacteristic
13C42×D4C2211yescharacteristic
14#1982C411yescharacteristic
15#1947C411yescharacteristic
16#1925—21no—
17#1879—21no—
18#1842C2211yescharacteristic
19#1802C411yescharacteristic
20C8×C4⋊C4—21no—
21C4×C4⋊C8C2211yescharacteristic
22C4×C4⋊C8C2211yescharacteristic
23C4×C4⋊C8C2211yescharacteristic
24C4×#102—21no—
25C8×C22⋊C4—22no—
26C4×C22⋊C8—21no—
27C42×C8C2211yescharacteristic
28C2×D4⋊C8—21no—
29C2×D4⋊C8—21no—
30C2×D4⋊C8—21no—
31#1279—21no—
32#1270—22no—
33C22×D8—21no—
34C23×C8—22no—
35C2×#127C2×C411yescharacteristic
36C2×C4×D4—21no—
37C2×C4×D4—21no—
38C2×C4×D4—21no—
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C22×C42—21no—
43#459—22no—
44#459C814yescomplemented
45C4×D8—42no—
46C4×D8—42no—
47C8×D4—24no—
48C8×D4—24no—
49C2×#107—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8C2×C411yescharacteristic
56C2×C4⋊C8C2×C411yescharacteristic
57C2×#102—21no—
58C2×#102—21no—
59C2×#102—21no—
60C2×#102—21no—
61C2×C22⋊C8—22no—
62C2×C4×C8—21no—
63C2×C4×C8—22no—
64C2×C4×C8—21no—
65C2×C4×C8—21no—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
68C2×C4×C8—42no—
69C2×C4×C8—42no—
70C2×C4×C8D411yescharacteristic
71C2×C4×C8D411yescharacteristic
72#390—21no—
73#390—21no—
74#390—21no—
75#384C2×C411yescharacteristic
76C4×C4⋊C4C2×C411yescharacteristic
77C4×C4⋊C4D411yescharacteristic
78C4×C4⋊C4D411yescharacteristic
79C4×C22⋊C4—21no—
80C4×C22⋊C4—21no—
81C4×C22⋊C4—21no—
82C43C2311yescharacteristic
83#340—21no—
84#340—21no—
85#336—22no—
86#336—21no—
87#336—21no—
88#336D412yes—
89D4⋊C8—42no—
90D4⋊C8—42no—
91D4⋊C8—42no—
92C22×D4—21no—
93C22×D4—21no—
94C22×D4—21no—
95C22×D4C22⋊C411yescharacteristic
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—41no—
100C2×D8—24no—
101C2×D8—42no—
102C2×D8—42no—
103C22×C8—21no—
104C22×C8—28no—
105C22×C8—21no—
106C22×C8—22no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—44no—
110C22×C8—44no—
111C22×C8—41no—
112C22×C8—41no—
113#127—21no—
114#127—21no—
115#127—21no—
116#127C2×C812yescomplemented
117C4×D4—22no—
118C4×D4—22no—
119C4×D4—21no—
120C4×D4—21no—
121C4×D4—21no—
122C4×D4—21no—
123C4×D4—41no—
124C4×D4—41no—
125C4×D4—41no—
126C4×D4—41no—
127C4×D4—41no—
128C4×D4—41no—
129C4×D4—41no—
130C4×D4—41no—
131C4×D4—81no—
132C4×D4—81no—
133C4×D4C2×C812yes—
134C4×D4M1612yes—
135C4×D4D812yes—
136C4×D4SD1612yes—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4C22⋊C411yescharacteristic
141C2×C4⋊C4C22⋊C411yescharacteristic
142C2×C4⋊C4C22⋊C411yescharacteristic
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—41no—
153C2×C42—41no—
154C2×C42—41no—
155C2×C42—41no—
156C2×C42C22×C411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163#107—42no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—41no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175C4⋊C8—81no—
176C4⋊C8C2×C812yescomplemented
177#102—24no—
178#102—42no—
179#102—42no—
180#102—42no—
181C22⋊C8—28no—
182C4×C8—22no—
183C4×C8—22no—
184C4×C8—21no—
185C4×C8—21no—
186C4×C8—21no—
187C4×C8—42no—
188C4×C8—44no—
189C4×C8—41no—
190C4×C8—41no—
191C4×C8—41no—
192C4×C8—41no—
193C4×C8—41no—
194C4×C8—41no—
195C4×C8D812yescomplemented
196#95—21no—
197#95—21no—
198#95—21no—
199C24—21no—
200C24—41no—
201C2×D4—22no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—81no—
219C2×D4—81no—
220C2×D4C22⋊C812yes—
221C2×D4#10212yes—
222C2×D4#10412yes—
223C22×C4—24no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4#12011yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12311yescharacteristic
256D8—48no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—24no—
262C2×C8—22no—
263C2×C8—416no—
264C2×C8—41no—
265C2×C8—42no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—42no—
269C2×C8—41no—
270C2×C8—42no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—81no—
275C2×C8—81no—
276C2×C8—81no—
277C2×C8—81no—
278C2×C8—81no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—81no—
287C4⋊C4C22⋊C812yes—
288C4⋊C4#10212yes—
289C4⋊C4#10412yes—
290C22⋊C4—24no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—82no—
296C22⋊C4—81no—
297C22⋊C4—81no—
298C22⋊C4—81no—
299C22⋊C4—81no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—41no—
307C42—41no—
308C42—41no—
309C42—41no—
310C42—41no—
311C42—41no—
312C42—41no—
313C42C22×C811yescharacteristic
314C42C2×M1611yescharacteristic
315C42#13111yescharacteristic
316C42#13111yescharacteristic
317C42C2×D811yescharacteristic
318C42C2×D811yescharacteristic
319C42C2×SD1611yescharacteristic
320C42#13511yescharacteristic
321C42#13511yescharacteristic
322C42#13511yescharacteristic
323C42AGL(1,ℤ/8)11yescharacteristic
324C42AGL(1,ℤ/8)11yescharacteristic
325C23—22no—
326C23—22no—
327C23—22no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—42no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23#38611yescharacteristic
340D4—22no—
341D4—22no—
342D4—22no—
343D4—42no—
344D4—42no—
345D4—42no—
346D4—42no—
347D4—82no—
348D4—82no—
349D4—81no—
350D4—81no—
351D4—81no—
352D4—81no—
353D4D4⋊C814yescomplemented
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—161no—
386C2×C4—161no—
387C2×C4—161no—
388C2×C4—161no—
389C2×C4C2×C22⋊C811yescharacteristic
390C2×C4#40811yescharacteristic
391C2×C4C2×#10211yescharacteristic
392C2×C4#41611yescharacteristic
393C2×C4C2×#10411yescharacteristic
394C2×C4C2×#10411yescharacteristic
395C2×C4C8×D411yescharacteristic
396C2×C4C8×D411yescharacteristic
397C2×C4#43511yescharacteristic
398C2×C4#43511yescharacteristic
399C2×C4C4×D811yescharacteristic
400C2×C4#44211yescharacteristic
401C2×C4#44311yescharacteristic
402C2×C4#44411yescharacteristic
403C2×C4#44711yescharacteristic
404C2×C4#44911yescharacteristic
405C2×C4#45911yescharacteristic
406C2×C4#46311yescharacteristic
407C2×C4#46511yescharacteristic
408C2×C4#46511yescharacteristic
409C2×C4#46611yescharacteristic
410C2×C4#46611yescharacteristic
411C2×C4#47111yescharacteristic
412C2×C4#47211yescharacteristic
413C8—24no—
414C8—42no—
415C8—44no—
416C8—42no—
417C8—82no—
418C8—81no—
419C8—161no—
420C22—24no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—161no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22#186911yescharacteristic
435C22#187611yescharacteristic
436C22#187911yescharacteristic
437C22#189311yescharacteristic
438C22#189611yescharacteristic
439C22#190611yescharacteristic
440C22#190711yescharacteristic
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—41no—
451C4—161no—
452C4C2×D4⋊C811yescharacteristic
453C4#147611yescharacteristic
454C4#149211yescharacteristic
455C4#149411yescharacteristic
456C4C8×D811yescharacteristic
457C4#158011yescharacteristic
458C4#158111yescharacteristic
459C4#158611yescharacteristic
460C4#166511yescharacteristic
461C4#166711yescharacteristic
462C4#167711yescharacteristic
463C4#168111yescharacteristic
464C2—44no—
465C2—162no—
466C2#894711yescharacteristic
467C2#907211yescharacteristic
468C2#909511yescharacteristic
469C2#920811yescharacteristic
470C2#921111yescharacteristic
471C2#922111yescharacteristic
472C2#923311yescharacteristic
subgroupG/H
centerC22×C4#121
commutatorC42C22×C8
Frattini, cosocleC43C23
socleC23#386
prime radicalC2×#459C4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×C8×D4C2semidirect
3. #9638C2semidirect
4. #8563C2semidirect
5. #7569C2semidirect
6. #7031C2semidirect
7. #7015C2semidirect
44. #459C8semidirect
116. #127C2×C8semidirect
176. C4⋊C8C2×C8semidirect
195. C4×C8D8semidirect
353. D4D4⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×#4592
C4×C8×D42
#96382
#85632
#75692
#70312
#70152