Finite Group Databasebeta

← #106955

Subgroups — #106955

Order 512.

900 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34187C211yesmaximal, characteristic
2C2×#1805C211yesmaximal, characteristic
3C2×#1479C211yesmaximal, characteristic
4#8039C211yesmaximal, characteristic
5#8026C211yesmaximal, characteristic
6#8026C211yesmaximal, characteristic
7#7893C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C22×#103—21no—
13#2356—22no—
14#2356—22no—
15#2355—22no—
16#2355C414yescomplemented
17#2313—22no—
18#2310—21no—
19#2285C2211yescharacteristic
20#2285C2211yescharacteristic
21#2283—22no—
22C2×#382C2211yescharacteristic
23#2274—21no—
24#1805—22no—
25#1805C414yescomplemented
26C2×#340—21no—
27#1479—22no—
28#1479C414yescomplemented
29C2×Q8⋊C8—22no—
30#1296—22no—
31#1294—22no—
32#1294—22no—
33#1279—22no—
34C23×Q8D411yescharacteristic
35C23×C8—21no—
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8C2×C412yes—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8C2×C412yes—
45C2×C4×Q8D412yes—
46C2×C4×Q8D412yes—
47C2×C4×Q8Q812yes—
48C2×#117—21no—
49C2×#117—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4D411yescharacteristic
53C22×C42—21no—
54C22×C42C2311yescharacteristic
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431C2×C412yes—
62#431C2×C412yes—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2×C412yes—
71C2×#103—24no—
72C2×#103—42no—
73C2×#103—42no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78#391—24no—
79#391—24no—
80#387—28no—
81#387—28no—
82#384C2×C412yes—
83#384D412yes—
84#383—24no—
85#382—24no—
86#382—21no—
87#382—21no—
88#382—21no—
89#382—21no—
90#382—21no—
91#382C2×C412yes—
92#381—24no—
93#380—21no—
94#380—21no—
95#380D412yes—
96#380D412yes—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4C2×C412yes—
103C4×C4⋊C4D412yes—
104C4×C22⋊C4—24no—
105#376—24no—
106C2×#95—21no—
107C2×#95—21no—
108#340—24no—
109Q8⋊C8—28no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8C22⋊C412yes—
122C22×Q8C22⋊C412yes—
123C22×Q8C4⋊C412yes—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—41no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C22×C8—22no—
131C22×C8—41no—
132C22×C8—41no—
133C22×C8—41no—
134C22×C8—41no—
135C22×C8—41no—
136C22×C8—41no—
137C22×C8—41no—
138C22×C8—41no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8C4214yes—
152C4×Q8C22⋊C414yes—
153C4×Q8C4⋊C414yes—
154C4×Q8C4⋊C414yes—
155#117—41no—
156#117—41no—
157#117—41no—
158#117—41no—
159#117—41no—
160#117—41no—
161#117—41no—
162#117—41no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C414yes—
185C2×C4⋊C4C22⋊C414yes—
186C2×C4⋊C4C4⋊C412yes—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—28no—
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×Q811yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42#4111yescharacteristic
208C2×C42#4111yescharacteristic
209C2×C42#4111yescharacteristic
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—41no—
215C4⋊C8—41no—
216C4⋊C8—41no—
217C4⋊C8—41no—
218#103—48no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—41no—
222C22⋊C8—41no—
223C4×C8—22no—
224C4×C8—22no—
225C4×C8—41no—
226C4×C8—41no—
227#95—28no—
228#95—28no—
229#95—28no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—41no—
237#95—41no—
238C24#12011yescharacteristic
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8#9514yes—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C4⋊C411yescharacteristic
292C22×C4C4×D411yescharacteristic
293C22×C4C4×D411yescharacteristic
294C22×C4C4×D411yescharacteristic
295C22×C4C4×D411yescharacteristic
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4#12111yescharacteristic
301C22×C4C22⋊Q811yescharacteristic
302C22×C4C22⋊Q811yescharacteristic
303C22×C4#12311yescharacteristic
304C22×C4#12311yescharacteristic
305C22×C4#13511yescharacteristic
306C22×C4#13511yescharacteristic
307C22×C4#13511yescharacteristic
308C22×C4#13511yescharacteristic
309C22×C4AGL(1,ℤ/8)11yescharacteristic
310C22×C4AGL(1,ℤ/8)11yescharacteristic
311C22×C4#13711yescharacteristic
312C22×C4#13711yescharacteristic
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—82no—
319C2×C8—82no—
320C2×C8—81no—
321C2×C8—81no—
322C2×C8—81no—
323C2×C8—81no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—82no—
341C4⋊C4—81no—
342C4⋊C4—81no—
343C4⋊C4#9514yes—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C42—24no—
349C42—24no—
350C42—24no—
351C42—24no—
352C42—42no—
353C42—42no—
354C42—42no—
355C42—42no—
356C42—88no—
357C42C2×C4211yescharacteristic
358C42C2×C22⋊C411yescharacteristic
359C42C2×C4⋊C411yescharacteristic
360C42C2×C4⋊C411yescharacteristic
361C42#11711yescharacteristic
362C42#11711yescharacteristic
363C42C4×D411yescharacteristic
364C42C4×D411yescharacteristic
365C42C4×D411yescharacteristic
366C42C4×D411yescharacteristic
367C42#12111yescharacteristic
368C42#12111yescharacteristic
369C42C22⋊Q811yescharacteristic
370C42C22⋊Q811yescharacteristic
371C42#12311yescharacteristic
372C42#12311yescharacteristic
373C23—22no—
374C23—22no—
375C23#37911yescharacteristic
376C23#38511yescharacteristic
377C23#38611yescharacteristic
378C23#44911yescharacteristic
379C23#44911yescharacteristic
380C23#45211yescharacteristic
381C23#45211yescharacteristic
382Q8—24no—
383Q8—24no—
384Q8—42no—
385Q8—42no—
386Q8—42no—
387Q8—42no—
388Q8—42no—
389Q8—42no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—162no—
423C2×C4—162no—
424C2×C4C2×#9511yescharacteristic
425C2×C4C4×C22⋊C411yescharacteristic
426C2×C4C4×C22⋊C411yescharacteristic
427C2×C4#38011yescharacteristic
428C2×C4#38011yescharacteristic
429C2×C4#38111yescharacteristic
430C2×C4#38511yescharacteristic
431C2×C4#38511yescharacteristic
432C2×C4#38511yescharacteristic
433C2×C4#38611yescharacteristic
434C2×C4#38611yescharacteristic
435C2×C4#38611yescharacteristic
436C2×C4#41611yescharacteristic
437C2×C4#41611yescharacteristic
438C2×C4#41611yescharacteristic
439C2×C4#41611yescharacteristic
440C2×C4#41711yescharacteristic
441C2×C4#41711yescharacteristic
442C2×C4#41811yescharacteristic
443C2×C4#41911yescharacteristic
444C2×C4#42711yescharacteristic
445C2×C4#42711yescharacteristic
446C2×C4#42811yescharacteristic
447C2×C4#42811yescharacteristic
448C2×C4#44911yescharacteristic
449C2×C4#44911yescharacteristic
450C2×C4#45211yescharacteristic
451C2×C4#45211yescharacteristic
452C2×C4#48211yescharacteristic
453C2×C4#48211yescharacteristic
454C2×C4#48211yescharacteristic
455C2×C4#48211yescharacteristic
456C2×C4#48511yescharacteristic
457C2×C4#48511yescharacteristic
458C2×C4#48511yescharacteristic
459C2×C4#48511yescharacteristic
460C8—82no—
461C8—82no—
462C8—82no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22C22⋊#9511yescharacteristic
472C22#178811yescharacteristic
473C22#178811yescharacteristic
474C22#179412yes—
475C22#179612yes—
476C22#185411yescharacteristic
477C22#185411yescharacteristic
478C22#187311yescharacteristic
479C22#187511yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—44no—
489C4—44no—
490C4—84no—
491C4—84no—
492C4—162no—
493C4#173511yescharacteristic
494C4#173611yescharacteristic
495C4#182511yescharacteristic
496C4#182511yescharacteristic
497C4#185411yescharacteristic
498C4#185411yescharacteristic
499C4#187311yescharacteristic
500C4#187511yescharacteristic
501C2—21no—
502C2—21no—
503C2—21no—
504C2—21no—
505C2#843112yes—
506C2#843512yes—
507C2#1073111yescharacteristic
508C2#1077711yescharacteristic
509C2#1077911yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #2355C4semidirect
25. #1805C4semidirect
28. #1479C4semidirect

Maximal subgroups

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

subgroupindex
#341872
C2×#18052
C2×#14792
#80392
#80262
#78932