Finite Group Databasebeta

← #6466244

Subgroups — #6466244

Order 512.

1826 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic
2C2×#2961C211yesmaximal, characteristic
3#35335C211yesmaximal, characteristic
4#35281C211yesmaximal, characteristic
5#28686C218yesmaximal, complemented
6#28685C218yesmaximal, complemented
7#23668C212yesmaximal
8#23667C212yesmaximal
9#21314C211yesmaximal, characteristic
10#21313C211yesmaximal, characteristic
11#21156C212yesmaximal
12#21155C212yesmaximal
13#20451C211yesmaximal, characteristic
14C2×#533C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×#125C2211yescharacteristic
17C2×#501—28no—
18C2×#501C2214yes—
19C2×#501C2214yes—
20#2961—28no—
21C2×#446C2214yes—
22C2×#446C2214yes—
23C2×#427C2212yes—
24C22×#107C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C2×#424C2212yes—
29C2×#423C2211yescharacteristic
30C2×#405C2212yes—
31C22×#97C2211yescharacteristic
32#2868—21no—
33#2862—21no—
34#2859—22no—
35#2660—21no—
36#2660—21no—
37#2660—22no—
38#2584C2214yes—
39#2584C2214yes—
40#2583—22no—
41#2583—22no—
42#2567C2214yes—
43#2546C2214yes—
44#2502—22no—
45#2434—22no—
46#2317C2212yes—
47#2317C2212yes—
48#2317C2212yes—
49#2317C2212yes—
50C4×#117C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2088C2218yes—
53#2088C2218yes—
54#2087C2218yes—
55#2086C2218yes—
56#1989C2214yes—
57#1989C2214yes—
58#1988C2214yes—
59#1987C2214yes—
60#1944C2218yes—
61#1943C2218yes—
62#1942C2218yes—
63#1941C2218yes—
64#1776C2214yes—
65#1776C2214yes—
66#1736C2212yes—
67#1736C2212yes—
68#1736C2212yes—
69#1736C2212yes—
70#1735C2212yes—
71C23×Q8—21no—
72C22×M16C2312yes—
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75#533—24no—
76#533—24no—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8D411yescharacteristic
83C2×C4⋊Q8D411yescharacteristic
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2312yes—
86C2×#125—21no—
87C2×#125—21no—
88C2×#125—24no—
89C2×#125—24no—
90C2×#125D412yes—
91C2×#125C2312yes—
92C2×#125C2312yes—
93C2×C22⋊Q8D412yes—
94C2×C22⋊Q8D412yes—
95C2×C4×Q8C2312yes—
96C2×C4×Q8C2312yes—
97C2×#117C2312yes—
98C2×#117C2312yes—
99C2×#117C2312yes—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C42C2311yescharacteristic
108#501—216no—
109#501—216no—
110#446—216no—
111#446—216no—
112#427Q8116yescomplemented
113C2×#107—24no—
114C2×#107—24no—
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#106—24no—
119C2×#106—24no—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×#106C2314yes—
123#424—28no—
124#423—24no—
125C2×C4⋊C8C2314yes—
126C2×C4⋊C8C2314yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2312yes—
129#405—28no—
130C2×#97D418yes—
131C2×#97C2314yes—
132C2×#97C2312yes—
133C2×#97C2312yes—
134C2×C4×C8C2314yes—
135#398—24no—
136#398—24no—
137#398—28no—
138#395—24no—
139#395—24no—
140#395—28no—
141#391—24no—
142#391—24no—
143#389—28no—
144#389—28no—
145#389C2318yes—
146#389C2318yes—
147#389C2318yes—
148#389C2318yes—
149#385—24no—
150#385—24no—
151#385—24no—
152#385—24no—
153#384—24no—
154#384—22no—
155#384—22no—
156#384—22no—
157#384—22no—
158#382—22no—
159#382—22no—
160#382—22no—
161#382—22no—
162C4×C4⋊C4C2312yes—
163C4×C4⋊C4C2312yes—
164C4×C4⋊C4C2312yes—
165C4×C4⋊C4C2312yes—
166C4×C22⋊C4—21no—
167C4×C22⋊C4—21no—
168C4×C22⋊C4—21no—
169C4×C22⋊C4—21no—
170#376—21no—
171#376—21no—
172C43—21no—
173C43—21no—
174#340C2314yes—
175#340C2318yes—
176#340C2318yes—
177#340C2314yes—
178#340C2318yes—
179#340C2318yes—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—44no—
183C22×Q8—84no—
184C22×Q8C2×D412yes—
185C22×Q8C2×D412yes—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2411yescharacteristic
193C2×M16—24no—
194C2×M16—28no—
195C22×C8—24no—
196C22×C8C2×D414yes—
197C22×C8C2×D414yes—
198C22×C8C2×Q818yes—
199C22×C8C2414yes—
200C22×C8C2412yes—
201C22×C8C2412yes—
202C4⋊Q8—24no—
203C4⋊Q8—24no—
204C4⋊Q8—24no—
205C4⋊Q8—24no—
206C4⋊Q8—24no—
207C4⋊Q8—24no—
208C4⋊Q8—44no—
209C4⋊Q8—44no—
210C4⋊Q8—88no—
211#125—28no—
212#125—24no—
213#125—24no—
214#125—24no—
215#125—24no—
216#125—44no—
217#125—44no—
218C22⋊Q8—28no—
219C22⋊Q8—28no—
220C4×Q8—24no—
221C4×Q8—24no—
222C4×Q8—24no—
223C4×Q8—24no—
224#117—24no—
225#117—24no—
226#117—24no—
227#117—24no—
228#117C2×Q818yes—
229#117C2×Q818yes—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2414yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4C2×D414yes—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D414yes—
279C2×C42#4114yes—
280C2×C42C2414yes—
281C2×C42C2411yescharacteristic
282C2×C42C2411yescharacteristic
283#107—28no—
284#107—28no—
285#107C2×Q8116yescomplemented
286#106—28no—
287#106—28no—
288#106C2×Q8116yescomplemented
289C4⋊C8—28no—
290C4⋊C8—28no—
291C4⋊C8—24no—
292C4⋊C8—24no—
293#97—28no—
294#97C2×D418yes—
295#97C2×D418yes—
296C4×C8—28no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305C24C22×D411yescharacteristic
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—48no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8—84no—
314C2×Q8—816no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4#12114yes—
332C22×C4#12114yes—
333C22×C4C22×D414yes—
334C22×C4C22×D414yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D412yes—
341C22×C4C22×Q814yes—
342C22×C4C2×#4112yes—
343C22×C42−1+414yes—
344C22×C4C2511yescharacteristic
345M16—48no—
346C2×C8—28no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—28no—
350C2×C8C4⋊Q8116yescomplemented
351C2×C8C22×D414yes—
352C2×C8C22×D414yes—
353C2×C8C22×Q814yes—
354C2×C8C22×Q814yes—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—28no—
358C4⋊C4—28no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—88no—
382C4⋊C4—88no—
383C4⋊C4—88no—
384C4⋊C4—88no—
385C4⋊C4C22×Q818yes—
386C4⋊C4C22×Q818yes—
387C22⋊C4—28no—
388C22⋊C4—28no—
389C22⋊C4—28no—
390C42—28no—
391C42—28no—
392C42—24no—
393C42—24no—
394C42—44no—
395C42—44no—
396C42—82no—
397C42—82no—
398C42#12114yes—
399C42#12114yes—
400C42C22×D412yes—
401C42C22×D412yes—
402C42C2×#4112yes—
403C42C2×#4112yes—
404C23—22no—
405C23—22no—
406C23C2×#12112yes—
407C23D4×Q814yes—
408C23C23×D411yescharacteristic
409Q8—84no—
410Q8—84no—
411Q8—88no—
412Q8—88no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—28no—
422C2×C4—28no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—84no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4—84no—
441C2×C4C2×#12112yes—
442C2×C4C2×#12112yes—
443C2×C4C2×#12112yes—
444C2×C4C2×#12112yes—
445C2×C4C2×#12112yes—
446C2×C4C2×C4⋊Q814yes—
447C2×C4#53614yes—
448C2×C4D4×Q814yes—
449C2×C4#55718yes—
450C2×C4C23×D411yescharacteristic
451C2×C4C23×Q812yes—
452C2×C4C22×#4111yescharacteristic
453C2×C4C2×2−1+412yes—
454C8—24no—
455C8—44no—
456C8C2×C4⋊Q818yes—
457C22—22no—
458C22—22no—
459C22—28no—
460C22#254614yes—
461C22C22×#12111yescharacteristic
462C22C2×D4×Q812yes—
463C22#358512yes—
464C22#358612yes—
465C4—22no—
466C4—22no—
467C4—22no—
468C4—22no—
469C4—44no—
470C4—44no—
471C4—48no—
472C4—48no—
473C4—82no—
474C4—82no—
475C4—82no—
476C4—82no—
477C4#254614yes—
478C4C22×C4⋊Q811yescharacteristic
479C4C2×#53611yescharacteristic
480C4C2×#55712yes—
481C2—24no—
482C2C2×#254611yescharacteristic
483C2#6045411yescharacteristic
484C2#6045511yescharacteristic
485C2#6138014yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #28686C2semidirect
6. #28685C2semidirect
112. #427Q8semidirect
285. #107C2×Q8semidirect
288. #106C2×Q8semidirect
350. C2×C8C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#29612
#353352
#352812
#286862
#286852
#236682
#236672
#213142
#213132
#211562
#211552
#204512