Finite Group Databasebeta

← #161193

Subgroups — #161193

Order 512.

1210 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2293C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic, complemented
3#20778C211yesmaximal, characteristic, complemented
4#7642C214yesmaximal, complemented
5#7627C212yesmaximal, complemented
6#7626C212yesmaximal
7#7594C214yesmaximal, complemented
8C22×C4×D4C411yescharacteristic
9C22×#117—21no—
10#2293—21no—
11#2293—21no—
12#2293—21no—
13#2293—21no—
14#2293C412yes—
15#2293C412yes—
16#2293C2212yes—
17#2293C2212yes—
18C2×#390C411yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#389C411yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#388—21no—
23C2×#388—21no—
24C2×C4×C4⋊C4C411yescharacteristic
25C2×C4×C22⋊C4—21no—
26C2×C43C2211yescharacteristic
27#1839C2212yescomplemented
28#1839C2212yescomplemented
29#1831C2212yescomplemented
30#1831C2212yescomplemented
31#1817—21no—
32#1817—21no—
33#1797—21no—
34#1797—21no—
35#1767—21no—
36#1767—21no—
37#1749C2212yes—
38#1749C2212yes—
39#1745—21no—
40#1745—21no—
41#1271C2218yescomplemented
42#1270C2218yes—
43C23×D4C2×C411yescharacteristic
44C24×C4—21no—
45C2×C4×D4—28no—
46C2×C4×D4—21no—
47C2×C4×D4—21no—
48C2×C4×D4C2×C412yes—
49C2×C4×D4C2×C412yes—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—48no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C42—21no—
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42C2311yescharacteristic
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×#97—22no—
78C2×#97—22no—
79#390—21no—
80#390—21no—
81#390—21no—
82#390—21no—
83#390C2×C412yes—
84#390C2×C412yes—
85#390C2×C412yes—
86#390C2312yes—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2312yes—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4C2×C412yes—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C43C2×C411yescharacteristic
115C43C2×C411yescharacteristic
116C43D411yescharacteristic
117C43D411yescharacteristic
118C43D411yescharacteristic
119C43D411yescharacteristic
120C43C2311yescharacteristic
121C43C2311yescharacteristic
122#336—22no—
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336C2314yes—
129#336C2314yes—
130C25—21no—
131C22×D4—24no—
132C22×D4—21no—
133C22×D4—42no—
134C22×D4—42no—
135C22×D4C22×C412yes—
136C23×C4—28no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—44no—
143C23×C4—44no—
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C22×C8—24no—
152C22×C8—24no—
153C4×D4—28no—
154C4×D4—28no—
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—24no—
158C4×D4M1618yes—
159#117—48no—
160#117—48no—
161#117—48no—
162#117—48no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—44no—
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22⋊C411yescharacteristic
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22⋊C411yescharacteristic
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22⋊C411yescharacteristic
209C2×C42C22⋊C411yescharacteristic
210C2×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2411yescharacteristic
231C4⋊C8—44no—
232C4⋊C8—44no—
233C22⋊C8—44no—
234C22⋊C8—44no—
235#97—44no—
236#97—44no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243C24—24no—
244C24—22no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24—42no—
249C24—42no—
250C24—42no—
251C24C2×C22⋊C411yescharacteristic
252C2×D4—24no—
253C2×D4—22no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—416no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4C2×M1614yes—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—28no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—28no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—44no—
288C22×C4—432no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C23×C411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C2×C8—44no—
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—44no—
318C2×C8—44no—
319C2×C8—44no—
320C4⋊C4—24no—
321C4⋊C4—22no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C4⋊C4—44no—
330C4⋊C4C2×M1614yes—
331C22⋊C4—28no—
332C22⋊C4—28no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—44no—
368C42—44no—
369C42C2×M1614yes—
370C42AGL(1,ℤ/8)14yes—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—24no—
376C23—48no—
377C23—48no—
378C23—48no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—84no—
386C23—84no—
387C23—84no—
388C23—84no—
389C23#41312yes—
390C23#42112yes—
391C23#42112yes—
392C23C22×C22⋊C411yescharacteristic
393D4—44no—
394D4—44no—
395D4—48no—
396C2×C4—28no—
397C2×C4—28no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—24no—
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—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—416no—
438C2×C4—416no—
439C2×C4—48no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4#40712yes—
452C2×C4#40712yes—
453C2×C4#41012yes—
454C2×C4#41812yes—
455C2×C4#41812yes—
456C2×C4C22×M1612yes—
457C2×C4C2×AGL(1,ℤ/8)12yes—
458C2×C4#57512yes—
459C8—84no—
460C8—84no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—44no—
468C22—44no—
469C22—44no—
470C22—816no—
471C22—82no—
472C22—82no—
473C22—82no—
474C22—82no—
475C22#153514yes—
476C22C2×#40711yescharacteristic
477C22C2×#41011yescharacteristic
478C22C2×#41311yescharacteristic
479C22C2×#41811yescharacteristic
480C22#289611yescharacteristic
481C22C2×#42111yescharacteristic
482C22C2×#42111yescharacteristic
483C4—22no—
484C4—22no—
485C4—22no—
486C4—22no—
487C4—22no—
488C4—22no—
489C4—22no—
490C4—22no—
491C4—44no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—41no—
496C4—41no—
497C4—41no—
498C4—82no—
499C4—82no—
500C2—24no—
501C2—84no—
502C2#1735711yescharacteristic
503C2#1744811yescharacteristic
504C2#1745211yescharacteristic
505C2#1757911yescharacteristic
506C2#1758311yescharacteristic
507C2C2×#153511yescharacteristic
508C2#1773111yescharacteristic
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. #20778C2semidirect
3. #20778C2semidirect
4. #7642C2semidirect
5. #7627C2semidirect
7. #7594C2semidirect
27. #1839C22semidirect
28. #1839C22semidirect
29. #1831C22semidirect
30. #1831C22semidirect
41. #1271C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22932
#207782
#76422
#76272
#76262
#75942