Finite Group Databasebeta

← #6477752

Subgroups — #6477752

Order 512.

1762 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3189C211yesmaximal, characteristic
2C2×#3095C212yesmaximal
3C2×#2924C211yesmaximal, characteristic
4#33124C214yesmaximal, complemented
5#29789C214yesmaximal, complemented
6#24622C218yesmaximal, complemented
7#24522C218yesmaximal, complemented
8#21948C211yesmaximal, characteristic
9#21753C212yesmaximal
10C22×C22⋊Q8C2212yes—
11C22×#117C2211yescharacteristic
12#3189—21no—
13#3189—21no—
14#3189—22no—
15#3189—24no—
16#3095—24no—
17#3095—22no—
18#3095—22no—
19#3095C2218yes—
20#3095C2214yes—
21#3095C2214yes—
22C2×#484—22no—
23C2×#484C2214yes—
24C2×#485—24no—
25C2×#483—22no—
26C2×#467C2214yes—
27C2×#451C2214yes—
28#2924—24no—
29#2924C2218yes—
30C2×#433—22no—
31C2×#432—21no—
32C2×#431—21no—
33C2×#428C2214yes—
34C2×#424C2212yes—
35C2×#419C2214yes—
36C2×#407C2212yes—
37C2×#407C2211yescharacteristic
38C22×C22⋊C8C2211yescharacteristic
39C2×#405C2212yes—
40#2089C2218yescomplemented
41#2064C22116yescomplemented
42#2047C2218yescomplemented
43#1979C2218yescomplemented
44#1952C2218yescomplemented
45#1930C22116yescomplemented
46#1901C2218yes—
47#1901C2218yes—
48#1855C2218yes—
49#1821C2214yes—
50#1810C2214yes—
51#1784C2214yes—
52C23×Q8C2312yes—
53C24×C4C2311yescharacteristic
54C22×Q16C2314yes—
55C22×M16C2312yes—
56C22×M16C2314yes—
57C23×C8C2312yes—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8C2314yes—
69C2×C22⋊Q8C2314yes—
70C2×#117—21no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117—21no—
74C2×#117D414yes—
75C2×#117D411yescharacteristic
76C2×#117D411yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2314yes—
80C22×C4⋊C4—22no—
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—22no—
86C22×C42—21no—
87#485—28no—
88#485—28no—
89#485—28no—
90#485—28no—
91#484—28no—
92#484—24no—
93#484—24no—
94#484—24no—
95#484—24no—
96#484—28no—
97#483—28no—
98#483—24no—
99#483—24no—
100#467—24no—
101#467—24no—
102#467—24no—
103#467—24no—
104#451—24no—
105#451—24no—
106#451—28no—
107#433—216no—
108#432—28no—
109#431—28no—
110C2×#108C2318yes—
111#428—24no—
112#428—24no—
113#428C2318yes—
114#428C2318yes—
115C2×#107—24no—
116C2×#107C2314yes—
117C2×#106—24no—
118#424—24no—
119#424C2318yes—
120C2×C4⋊C8—22no—
121C2×C4⋊C8—22no—
122#419—24no—
123#419—24no—
124#419C2318yes—
125#419C2318yes—
126C2×#103—24no—
127C2×#103—24no—
128C2×#103C2318yes—
129C2×#101C23116yes—
130#407—22no—
131#407—22no—
132#407—22no—
133#407C2314yes—
134#407C2314yes—
135#407C2314yes—
136C2×C22⋊C8—22no—
137C2×C22⋊C8—22no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8D411yescharacteristic
141C2×C22⋊C8D411yescharacteristic
142C2×C22⋊C8D412yes—
143C2×C22⋊C8D414yes—
144C2×C22⋊C8C2314yes—
145#405—24no—
146#405C2318yes—
147C2×#97—22no—
148C2×C4×C8—22no—
149#343C23116yes—
150#341C2318yes—
151C25#4111yescharacteristic
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—48no—
158C22×Q8C2414yes—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—44no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4#4112yes—
171C23×C4#4111yescharacteristic
172C23×C4C2412yes—
173C23×C4C2411yescharacteristic
174C2×Q16—24no—
175C2×Q16—24no—
176C2×Q16—48no—
177C2×M16—22no—
178C2×M16—24no—
179C2×M16—44no—
180C2×M16—48no—
181C2×M16C2418yes—
182C2×M16C2414yes—
183C22×C8—22no—
184C22×C8—24no—
185C22×C8—44no—
186C22×C8C2×D412yes—
187C22×C8C2×D412yes—
188C22×C8C2×D414yes—
189C22×C8C2414yes—
190C22⋊Q8—24no—
191C22⋊Q8—24no—
192C22⋊Q8—24no—
193C22⋊Q8—24no—
194C22⋊Q8—24no—
195C22⋊Q8—24no—
196C22⋊Q8—24no—
197C22⋊Q8—24no—
198C22⋊Q8—42no—
199C22⋊Q8—42no—
200C22⋊Q8—42no—
201C22⋊Q8—42no—
202C22⋊Q8—42no—
203C22⋊Q8—42no—
204C22⋊Q8—42no—
205C22⋊Q8—42no—
206#117—24no—
207#117—22no—
208#117—22no—
209#117—24no—
210#117—21no—
211#117—21no—
212#117—24no—
213#117—21no—
214#117—21no—
215#117—24no—
216#117C2×D414yes—
217#117C2×D414yes—
218#117C2×D412yes—
219#117C2×D412yes—
220#117C2412yes—
221#117C2412yes—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2414yes—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—24no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—24no—
262#108—48no—
263#107—24no—
264#107—24no—
265#107—28no—
266#107—28no—
267#106—28no—
268#106—28no—
269C4⋊C8—28no—
270C4⋊C8—28no—
271#103—28no—
272#103—28no—
273#103—28no—
274#103—28no—
275#103—28no—
276#103—28no—
277#101—416no—
278C22⋊C8—24no—
279C22⋊C8—24no—
280C22⋊C8—24no—
281C22⋊C8—24no—
282C22⋊C8—24no—
283C22⋊C8—24no—
284C22⋊C8C2×D414yes—
285C22⋊C8C2×D414yes—
286C22⋊C8C2×D418yes—
287#97—28no—
288C4×C8—28no—
289C24—24no—
290C24—42no—
291C24—81no—
292C24#12112yes—
293C24#12111yescharacteristic
294C24#12111yescharacteristic
295C24C2×#4111yescharacteristic
296C24C2×#4112yes—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—216no—
300C2×Q8—48no—
301C2×Q8—48no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—84no—
305C2×Q8—84no—
306C2×Q8—84no—
307C22×C4—24no—
308C22×C4—28no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—84no—
331C22×C4—82no—
332C22×C4—84no—
333C22×C4#12112yes—
334C22×C4#12111yescharacteristic
335C22×C4#12112yes—
336C22×C4#12111yescharacteristic
337C22×C4#12112yes—
338C22×C4#12112yes—
339C22×C4#12112yes—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D412yes—
343C22×C4C22×D412yes—
344C22×C4C22×D412yes—
345C22×C4C22×D412yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4112yes—
350C22×C4C2511yescharacteristic
351Q16—84no—
352Q16—84no—
353M16—28no—
354M16—216no—
355M16—48no—
356M16—44no—
357C2×C8—28no—
358C2×C8—24no—
359C2×C8—28no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—44no—
363C2×C8—48no—
364C2×C8—44no—
365C2×C8C22×D414yes—
366C2×C8C22×D414yes—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4C22×D412yes—
382C4⋊C4C22×D412yes—
383C4⋊C4C22×D412yes—
384C4⋊C4C22×D412yes—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C42—22no—
396C42—22no—
397C42—24no—
398C42—24no—
399C42—22no—
400C42—22no—
401C23—24no—
402C23—22no—
403C23—22no—
404C23—24no—
405C23—24no—
406C23—22no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—41no—
411C23—42no—
412C23—41no—
413C23—41no—
414C23—42no—
415C23—88no—
416C23#47314yes—
417C23C2×#12111yescharacteristic
418C23C2×#12112yes—
419C23C2×#12111yescharacteristic
420C23C2×#12112yes—
421C23C22×#4111yescharacteristic
422Q8—48no—
423Q8—48no—
424Q8—84no—
425Q8—84no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—48no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—48no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4—84no—
458C2×C4—84no—
459C2×C4—84no—
460C2×C4—84no—
461C2×C4C2×#12111yescharacteristic
462C2×C4C2×#12112yes—
463C2×C4C2×#12111yescharacteristic
464C2×C4C2×#12112yes—
465C2×C4C23×D411yescharacteristic
466C2×C4C23×D411yescharacteristic
467C2×C4#58514yes—
468C8—24no—
469C8—24no—
470C8—48no—
471C8—44no—
472C22—22no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—22no—
477C22—21no—
478C22—22no—
479C22—21no—
480C22—21no—
481C22—22no—
482C22—22no—
483C22—21no—
484C22—44no—
485C22—48no—
486C22—44no—
487C22—84no—
488C22—82no—
489C22C2×#47312yes—
490C22C2×#47314yes—
491C22C22×#12111yescharacteristic
492C22#349112yes—
493C4—21no—
494C4—21no—
495C4—22no—
496C4—22no—
497C4—42no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—84no—
502C4—84no—
503C4#344912yes—
504C4#348512yes—
505C2—22no—
506C2—24no—
507C2—22no—
508C2—42no—
509C2—41no—
510C2#3508911yescharacteristic
511C2C22×#47311yescharacteristic
512C2#6046911yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC22×C4C25
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #33124C2semidirect
5. #29789C2semidirect
6. #24622C2semidirect
7. #24522C2semidirect
40. #2089C22semidirect
41. #2064C22semidirect
42. #2047C22semidirect
43. #1979C22semidirect
44. #1952C22semidirect
45. #1930C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
C2×#30952
C2×#29242
#331242
#297892
#246222
#245222
#219482
#217532