Finite Group Databasebeta

← #192359

Subgroups — #192359

Order 512.

1372 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34206C211yesmaximal, characteristic
2#21395C211yesmaximal, characteristic
3#21365C211yesmaximal, characteristic, complemented
4C2×#1505C211yesmaximal, characteristic
5#13856C211yesmaximal, characteristic, complemented
6#8873C214yesmaximal, complemented
7#8866C214yesmaximal, complemented
8#8022C211yesmaximal, characteristic, complemented
9#7912C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C2×#520—21no—
12C2×#432C2211yescharacteristic
13C2×#432C2211yescharacteristic
14#2367—22no—
15#2366—21no—
16#2365—21no—
17#2354—21no—
18#2352—22no—
19D4×C4⋊C4—21no—
20#2325C2214yes—
21#2311—21no—
22C4×#121—22no—
23#2285C2211yescharacteristic
24C2×C4×C4⋊C4—22no—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×#376C2211yescharacteristic
28#1987—21no—
29#1982—21no—
30#1948—21no—
31#1947—21no—
32#1874C2212yes—
33#1874C2212yes—
34#1853C2212yes—
35#1852C2212yes—
36#1776—22no—
37#1762—22no—
38#1505C418yescomplemented
39C2×#327C2214yes—
40#1444—21no—
41#1444—21no—
42#1444C2212yes—
43#1444C2212yes—
44#1294C2214yes—
45#1280C2214yes—
46C22×#41—21no—
47#532—22no—
48#532—22no—
49#532C2×C414yes—
50#532Q814yes—
51C2×C4⋊Q8D411yescharacteristic
52C2×#127D411yescharacteristic
53C2×#124—21no—
54C2×#121—21no—
55C2×#121C2312yes—
56#520—48no—
57C2×C4×Q8—22no—
58C2×C4×D4—22no—
59C2×C4×D4—24no—
60C2×C4×D4—22no—
61C2×C4×D4—21no—
62C2×C4×D4D412yes—
63C2×#117—22no—
64C2×#117—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C42—21no—
73C22×C42—21no—
74C22×C42D412yes—
75C22×C42C2311yescharacteristic
76#432—22no—
77#432—22no—
78#432—22no—
79#432C2×C414yescomplemented
80C2×#107—22no—
81C2×#106—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×#102—22no—
85C2×#102—22no—
86C2×C22⋊C8C2312yes—
87C2×C22⋊C8C2312yes—
88C2×#97—21no—
89C2×#97—21no—
90#391—22no—
91#390—22no—
92#390—22no—
93#389—22no—
94#388—24no—
95#388—24no—
96#387—24no—
97#386—24no—
98#386—24no—
99#385—24no—
100#384D411yescharacteristic
101#384D411yescharacteristic
102#382—21no—
103#380—22no—
104#380—22no—
105#380—21no—
106#380C2312yes—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—48no—
118C4×C4⋊C4D412yes—
119C4×C22⋊C4—24no—
120#376—22no—
121#376C2×C414yescomplemented
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95C2312yes—
127C2×#95C2312yes—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#327C2×C4116yescomplemented
133C2×#41—24no—
134C2×#41—24no—
135C2×#41—44no—
136C2×#41—44no—
137C22×Q8—21no—
138C22×D4—22no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4C2×D412yes—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—24no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2411yescharacteristic
152C22×C8—22no—
153C22×C8—22no—
154C4⋊Q8—22no—
155C4⋊Q8C4⋊C414yescomplemented
156#127—22no—
157#127C4⋊C414yescomplemented
158#124—24no—
159#124—24no—
160#121—24no—
161#121—24no—
162#121—22no—
163#121—22no—
164#121C22×C414yes—
165#121C2×Q814yes—
166C4×Q8—48no—
167C4×D4—24no—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—48no—
171C4×D4—48no—
172C4×D4—48no—
173C4×D4—84no—
174C4×D4C4⋊C418yescomplemented
175#117—48no—
176#117—44no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—48no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—44no—
219C2×C42—44no—
220C2×C42—41no—
221C2×C42—41no—
222C2×C42C22×C412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×Q812yes—
228C2×C42#4112yes—
229C2×C42#4112yes—
230#107—44no—
231#106—44no—
232C4⋊C8—24no—
233C4⋊C8—42no—
234#102—44no—
235#102—44no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8C22×C414yes—
240#97—24no—
241#97—42no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—44no—
251#95—44no—
252#95—41no—
253#95—41no—
254#95C22×C414yes—
255C24—22no—
256C24C22×D411yescharacteristic
257#41—416no—
258#41—416no—
259C2×Q8—24no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—22no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—44no—
272C2×D4—44no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4C2×C4⋊C414yes—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—28no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—48no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—84no—
316C22×C4—84no—
317C22×C4C4×D412yes—
318C22×C4C4×D412yes—
319C22×C4#12011yescharacteristic
320C22×C4#12011yescharacteristic
321C22×C4#12011yescharacteristic
322C22×C4#12011yescharacteristic
323C22×C4C22⋊Q812yes—
324C22×C4C22⋊Q812yes—
325C22×C4AGL(1,ℤ/8)14yes—
326C22×C4C23×C411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×Q811yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C2×C8—24no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—48no—
356C4⋊C4—42no—
357C4⋊C4—416no—
358C4⋊C4—416no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4C2×C4⋊C414yes—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—28no—
365C22⋊C4—48no—
366C22⋊C4—44no—
367C22⋊C4—84no—
368C22⋊C4—84no—
369C42—216no—
370C42—44no—
371C42—48no—
372C42—44no—
373C42—82no—
374C42—82no—
375C42—82no—
376C42C2×C4⋊C412yes—
377C42C2×C4⋊C412yes—
378C42C4×D412yes—
379C42C4×D412yes—
380C42C22⋊Q812yes—
381C42C22⋊Q812yes—
382C42#12312yes—
383C42#12312yes—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—24no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23C2×C4×D411yescharacteristic
396C23C2×#12011yescharacteristic
397C23C2×C22⋊Q811yescharacteristic
398C23C2×AGL(1,ℤ/8)12yes—
399C23#57512yes—
400Q8—48no—
401D4—44no—
402D4—48no—
403D4—48no—
404D4—48no—
405D4—48no—
406D4—82no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—28no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—416no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—168no—
463C2×C4#38512yes—
464C2×C4#38512yes—
465C2×C4#38512yes—
466C2×C4#38512yes—
467C2×C4#45312yes—
468C2×C4#45612yes—
469C2×C4C22×C4⋊C411yescharacteristic
470C2×C4C2×C4×D411yescharacteristic
471C2×C4C2×C22⋊Q811yescharacteristic
472C2×C4C2×#12311yescharacteristic
473C8—44no—
474C8—82no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—44no—
484C22—44no—
485C22—44no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22C2×#38511yescharacteristic
491C22C4×AGL(1,ℤ/8)12yes—
492C22#300311yescharacteristic
493C22#300511yescharacteristic
494C22C2×#45311yescharacteristic
495C22C2×#45611yescharacteristic
496C22#302511yescharacteristic
497C22#302511yescharacteristic
498C22#307712yes—
499C4—22no—
500C4—22no—
501C4—22no—
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—48no—
507C4—44no—
508C4—44no—
509C4—82no—
510C4—82no—
511C4—82no—
512C4—82no—
513C4—82no—
514C4#185914yes—
515C2—22no—
516C2—22no—
517C2—84no—
518C2#1817811yescharacteristic
519C2#1818311yescharacteristic
520C2#1818711yescharacteristic
521C2#1819011yescharacteristic
522C2C2×#185911yescharacteristic
523C2#2087311yescharacteristic
524C2#2088811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21365C2semidirect
5. #13856C2semidirect
6. #8873C2semidirect
7. #8866C2semidirect
8. #8022C2semidirect
9. #7912C2semidirect
38. #1505C4semidirect
79. #432C2×C4semidirect
121. #376C2×C4semidirect
132. #327C2×C4semidirect
155. C4⋊Q8C4⋊C4semidirect
157. #127C4⋊C4semidirect
174. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342062
#213952
#213652
C2×#15052
#138562
#88732
#88662
#80222
#79122