Finite Group Databasebeta

← #6651608

Subgroups — #6651608

Order 512.

1798 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35178C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic, complemented
3#33095C212yesmaximal, complemented
4#32693C214yesmaximal, complemented
5#32646C214yesmaximal, complemented
6#32589C218yesmaximal, complemented
7#29638C212yesmaximal, complemented
8#27613C212yesmaximal, complemented
9#24608C214yesmaximal, complemented
10#24060C212yesmaximal
11#21822C211yesmaximal, characteristic, complemented
12C2×#536—21no—
13C2×#532C2211yescharacteristic
14C22×C4⋊Q8—21no—
15C22×C4⋊Q8C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17C2×#480C2214yes—
18C2×#432C2212yes—
19#2854C2212yes—
20#2854C2212yes—
21#2618—21no—
22#2618—21no—
23#2548C2212yes—
24#2546—22no—
25#2546—22no—
26#2546C2214yes—
27#2546C2212yes—
28#2546C2214yes—
29#2545—22no—
30#2538C2214yes—
31#2536—22no—
32C2×#398—21no—
33C2×#398—21no—
34#2311—21no—
35C2×#391—21no—
36C2×#389C2212yes—
37C2×#389C2212yes—
38#2285C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40#2100C2214yes—
41#2087C2218yes—
42#2084C2214yes—
43#2065C2218yes—
44#2057C2218yes—
45#2057C2214yes—
46#2041C2214yes—
47#2023C2218yes—
48#1985C2214yes—
49#1985C2214yes—
50#1944C2218yes—
51#1936C2218yes—
52#1926C2218yes—
53#1919C2218yes—
54#1919C2218yes—
55#1874C2214yes—
56#1862C2218yes—
57#1853C2214yes—
58#1823C2214yes—
59#1817C2212yes—
60#1817C2212yes—
61C22×#41—21no—
62C23×Q8—21no—
63C23×Q8D411yescharacteristic
64#536—48no—
65#532—44no—
66C2×C4⋊Q8—24no—
67C2×C4⋊Q8—24no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—44no—
71C2×C4⋊Q8—48no—
72C2×C4⋊Q8C2311yescharacteristic
73C2×C4⋊Q8C2314yes—
74C2×#127C2311yescharacteristic
75C2×#124—22no—
76C2×#124—21no—
77C2×#123—22no—
78C2×#123—22no—
79C2×C22⋊Q8—24no—
80C2×#121—21no—
81C2×#121C2312yes—
82C2×C4×Q8—44no—
83C2×C4×Q8D412yes—
84C2×C4×Q8D412yes—
85C2×C4×Q8C2312yes—
86C2×C4×Q8C2312yes—
87C2×C4×D4—21no—
88C2×C4×D4C2312yes—
89C2×#117—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4D411yescharacteristic
94C22×C4⋊C4D411yescharacteristic
95C22×C4⋊C4D411yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C42—21no—
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102#480—48no—
103#432—44no—
104C2×#107C2318yes—
105C2×#106C2318yes—
106C2×C4⋊C8C2314yes—
107C2×#102C2318yes—
108C2×#102C2318yes—
109C2×C22⋊C8C2314yes—
110C2×#97C2314yes—
111#398—22no—
112#398—22no—
113#398—24no—
114#398—24no—
115#398—24no—
116#398—24no—
117#396—24no—
118#396—24no—
119#394—22no—
120#394—22no—
121#391—22no—
122#391—24no—
123#391—24no—
124#390—22no—
125#390C2314yes—
126#390C2314yes—
127#389—22no—
128#389—22no—
129#389—24no—
130#389C2314yes—
131#389C2314yes—
132#389C2318yes—
133#389C2314yes—
134#389C2314yes—
135#388—24no—
136#388—24no—
137#386—22no—
138#386—22no—
139#384C2312yes—
140#382—21no—
141#380—21no—
142#380—24no—
143#380C2312yes—
144C4×C4⋊C4—22no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4C2312yes—
147C4×C4⋊C4C2314yes—
148C2×#95—21no—
149C2×#95—22no—
150C2×#95—22no—
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155#336C2314yes—
156#336C2318yes—
157#336C2314yes—
158C2×#41—44no—
159C2×#41—44no—
160C2×#41—44no—
161C22×Q8—24no—
162C22×Q8—24no—
163C22×Q8—21no—
164C22×Q8—24no—
165C22×Q8—42no—
166C22×Q8—44no—
167C22×Q8—44no—
168C22×Q8—44no—
169C22×Q8C2×D412yes—
170C22×Q8C2×D412yes—
171C22×D4—22no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4C2412yes—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2411yescharacteristic
185C22×C8C2418yes—
186C4⋊Q8—44no—
187C4⋊Q8—44no—
188C4⋊Q8—42no—
189C4⋊Q8—48no—
190C4⋊Q8—416no—
191C4⋊Q8—416no—
192C4⋊Q8—48no—
193#127—42no—
194#124—44no—
195#124—48no—
196#123—48no—
197#123—48no—
198C22⋊Q8—416no—
199#121—28no—
200#121—44no—
201C4×Q8—44no—
202C4×Q8—44no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×Q8—42no—
206C4×Q8—42no—
207C4×D4—44no—
208C4×D4—44no—
209#117—44no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—48no—
228C2×C4⋊C4—48no—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2414yes—
236C2×C4⋊C4C2412yes—
237C2×C4⋊C4C2414yes—
238C2×C4⋊C4C2414yes—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—44no—
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2411yescharacteristic
257C2×C42C2411yescharacteristic
258C2×C42C2412yes—
259C2×C42C2412yes—
260#107—48no—
261#106—216no—
262C4⋊C8—44no—
263#102—216no—
264#102—48no—
265C22⋊C8—28no—
266#97—44no—
267#95—24no—
268#95—28no—
269#95—24no—
270#95—24no—
271#95—22no—
272#95—22no—
273#95—24no—
274#95—24no—
275C24—22no—
276C24C22×D411yescharacteristic
277#41—816no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—416no—
285C2×Q8—42no—
286C2×Q8—48no—
287C2×Q8—48no—
288C2×Q8—42no—
289C2×Q8—48no—
290C2×Q8—42no—
291C2×Q8—816no—
292C2×D4—24no—
293C2×D4—28no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—42no—
301C2×D4—82no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—28no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4AGL(1,ℤ/8)14yes—
330C22×C4AGL(1,ℤ/8)14yes—
331C22×C4C22×D412yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D412yes—
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D412yes—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D412yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4H(5,𝔽2)14yes—
341C22×C4C2511yescharacteristic
342C2×C8—28no—
343C2×C8—28no—
344C2×C8—28no—
345C4⋊C4—28no—
346C4⋊C4—24no—
347C4⋊C4—28no—
348C4⋊C4—24no—
349C4⋊C4—28no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—48no—
357C4⋊C4—416no—
358C4⋊C4—44no—
359C4⋊C4—42no—
360C4⋊C4—48no—
361C4⋊C4—44no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—82no—
366C4⋊C42−1+418yes—
367C22⋊C4—44no—
368C22⋊C4—48no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—48no—
372C22⋊C4—84no—
373C22⋊C4—84no—
374C42—22no—
375C42—24no—
376C42—22no—
377C42—24no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—48no—
385C42—44no—
386C23—22no—
387C23—22no—
388C23—44no—
389C23—42no—
390C23—44no—
391C23—42no—
392C23—42no—
393C23#53412yes—
394C23C2×AGL(1,ℤ/8)12yes—
395C23C2×AGL(1,ℤ/8)12yes—
396C23C23×D411yescharacteristic
397Q8—48no—
398Q8—48no—
399Q8—816no—
400Q8—84no—
401D4—48no—
402D4—84no—
403D4—84no—
404D4—88no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—28no—
409C2×C4—28no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—28no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—28no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—44no—
428C2×C4—416no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—88no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—88no—
445C2×C4#53412yes—
446C2×C4C2×AGL(1,ℤ/8)14yes—
447C2×C4C2×AGL(1,ℤ/8)12yes—
448C2×C4C2×AGL(1,ℤ/8)12yes—
449C2×C4C2×H(5,𝔽2)12yes—
450C2×C4C2×2−1+414yes—
451C8—48no—
452C22—22no—
453C22—24no—
454C22—24no—
455C22—22no—
456C22—82no—
457C22—82no—
458C22—84no—
459C22—84no—
460C22—82no—
461C22#319214yes—
462C22C2×#53411yescharacteristic
463C22#348612yes—
464C22C22×AGL(1,ℤ/8)11yescharacteristic
465C22C22×AGL(1,ℤ/8)11yescharacteristic
466C22#358612yes—
467C4—22no—
468C4—22no—
469C4—24no—
470C4—22no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—41no—
475C4—41no—
476C4—48no—
477C4—82no—
478C4—84no—
479C4—84no—
480C4#354414yes—
481C2—24no—
482C2—162no—
483C2#3725811yescharacteristic
484C2C2×#319211yescharacteristic
485C2#6075411yescharacteristic
486C2#6182712yes—
487C2#6182812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2546C2semidirect
3. #33095C2semidirect
4. #32693C2semidirect
5. #32646C2semidirect
6. #32589C2semidirect
7. #29638C2semidirect
8. #27613C2semidirect
9. #24608C2semidirect
11. #21822C2semidirect

Maximal subgroups

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

subgroupindex
#351782
C2×#25462
#330952
#326932
#326462
#325892
#296382
#276132
#246082
#240602
#218222