Finite Group Databasebeta

← #160979

Subgroups — #160979

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2293C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic, complemented
3#20752C211yesmaximal, characteristic, complemented
4#7642C214yesmaximal, complemented
5#7610C212yesmaximal, complemented
6#7606C212yesmaximal
7#7574C214yesmaximal, complemented
8C22×C4×D4C411yescharacteristic
9C22×#117—21no—
10C2×#423—21no—
11#2293—21no—
12#2293—21no—
13#2293—21no—
14#2293—21no—
15#2293C412yes—
16#2293C412yes—
17#2293C2212yes—
18#2293C2212yes—
19C2×#390C411yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#389C411yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#388—21no—
24C2×#388—21no—
25C2×C4×C4⋊C4C411yescharacteristic
26C2×C4×C22⋊C4—21no—
27C2×C43C2211yescharacteristic
28#1839C2212yescomplemented
29#1831—21no—
30#1831—21no—
31#1831—21no—
32#1831C2212yescomplemented
33#1831C2212yescomplemented
34#1817C2212yes—
35#1797—21no—
36C4×C4⋊C8C2212yescomplemented
37C4×C22⋊C8—21no—
38#1744—21no—
39#1728—21no—
40#1728C2212yes—
41#1271C2218yescomplemented
42#1270C2218yes—
43C23×D4C2×C411yescharacteristic
44C24×C4—21no—
45C22×M16—22no—
46C2×C4×D4—28no—
47C2×C4×D4—21no—
48C2×C4×D4—21no—
49C2×C4×D4C2×C412yes—
50C2×C4×D4C2×C412yes—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—44no—
56C2×#117—44no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C42—21no—
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42C2311yescharacteristic
75#423—44no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C4×C8—22no—
81#390—21no—
82#390—21no—
83#390—21no—
84#390—21no—
85#390C2×C412yes—
86#390C2×C412yes—
87#390C2×C412yes—
88#390C2312yes—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2312yes—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4C2×C412yes—
108C4×C4⋊C4C2×C412yes—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C43C2×C411yescharacteristic
117C43C2×C411yescharacteristic
118C43D411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43C2311yescharacteristic
123C43C2311yescharacteristic
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336C2314yes—
131#336C2314yes—
132C25—21no—
133C22×D4—24no—
134C22×D4—21no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4C22×C412yes—
138C23×C4—28no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C2×M16—44no—
154C2×M16—44no—
155C2×M16—44no—
156C22×C8—24no—
157C22×C8—24no—
158C4×D4—28no—
159C4×D4—28no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4M1618yescomplemented
164#117—48no—
165#117—48no—
166#117—44no—
167#117—44no—
168#117—44no—
169#117—44no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C42—24no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42—44no—
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42C22⋊C411yescharacteristic
216C2×C42C22⋊C411yescharacteristic
217C2×C42C22⋊C411yescharacteristic
218C2×C42C22⋊C411yescharacteristic
219C2×C42C22⋊C411yescharacteristic
220C2×C42C22⋊C411yescharacteristic
221C2×C42C22⋊C411yescharacteristic
222C2×C42C22⋊C411yescharacteristic
223C2×C42D814yes—
224C2×C42SD1614yes—
225C2×C42C22×C411yescharacteristic
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2411yescharacteristic
241C4⋊C8—44no—
242C4⋊C8—44no—
243C4⋊C8—44no—
244C22⋊C8—44no—
245C4×C8—44no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252C24—24no—
253C24—22no—
254C24—42no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24—42no—
260C24C2×C22⋊C411yescharacteristic
261C2×D4—24no—
262C2×D4—22no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—416no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4C2×M1614yes—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—28no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—28no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—44no—
294C22×C4—432no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4#10214yes—
307C22×C4#10214yes—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×D812yes—
320C22×C4C2×SD1612yes—
321C22×C4C23×C411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324M16—88no—
325C2×C8—44no—
326C2×C8—44no—
327C2×C8—44no—
328C2×C8—44no—
329C2×C8—44no—
330C2×C8—44no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4C2×M1614yes—
343C22⋊C4—28no—
344C22⋊C4—28no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—44no—
378C42—44no—
379C42#10214yes—
380C42#10214yes—
381C42C2×M1614yes—
382C42C2×D812yes—
383C42C2×D812yes—
384C42C2×SD1612yes—
385C42C2×SD1612yes—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—24no—
391C23—48no—
392C23—48no—
393C23—48no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—84no—
401C23—84no—
402C23—84no—
403C23—84no—
404C23#41012yes—
405C23C2×#10212yes—
406C23#42112yes—
407C23C22×C22⋊C411yescharacteristic
408D4—44no—
409D4—44no—
410D4—48no—
411C2×C4—28no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—28no—
433C2×C4—21no—
434C2×C4—24no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—416no—
450C2×C4—44no—
451C2×C4—48no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—416no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—84no—
462C2×C4—84no—
463C2×C4#40712yes—
464C2×C4#40712yes—
465C2×C4#41312yes—
466C2×C4C2×#10212yes—
467C2×C4C2×#10212yes—
468C2×C4C2×#10212yes—
469C2×C4C2×#10212yes—
470C2×C4C2×#10212yes—
471C2×C4#42112yes—
472C2×C4C22×M1612yes—
473C2×C4C22×D811yescharacteristic
474C2×C4C22×SD1611yescharacteristic
475C2×C4#57512yes—
476C8—84no—
477C8—84no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—44no—
485C22—44no—
486C22—44no—
487C22—816no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22#151714yes—
493C22C2×#40711yescharacteristic
494C22C2×#41011yescharacteristic
495C22C2×#41311yescharacteristic
496C22C22×#10211yescharacteristic
497C22#289611yescharacteristic
498C22C2×#42111yescharacteristic
499C22C2×#42111yescharacteristic
500C4—22no—
501C4—22no—
502C4—22no—
503C4—22no—
504C4—21no—
505C4—21no—
506C4—22no—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—41no—
511C4—41no—
512C4—44no—
513C4—41no—
514C4—41no—
515C4—82no—
516C4—82no—
517C4#148614yes—
518C2—24no—
519C2—84no—
520C2#1735711yescharacteristic
521C2C2×#148611yescharacteristic
522C2#1745211yescharacteristic
523C2C2×#151711yescharacteristic
524C2#1758011yescharacteristic
525C2#1767011yescharacteristic
526C2#1770411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20777C2semidirect
3. #20752C2semidirect
4. #7642C2semidirect
5. #7610C2semidirect
7. #7574C2semidirect
28. #1839C22semidirect
32. #1831C22semidirect
33. #1831C22semidirect
36. C4×C4⋊C8C22semidirect
41. #1271C22semidirect
163. C4×D4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#22932
#207772
#207522
#76422
#76102
#76062
#75742