Finite Group Databasebeta

← #216199

Subgroups — #216199

Order 512.

1000 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2334C211yesmaximal, characteristic
2#21951C211yesmaximal, characteristic, complemented
3#21912C211yesmaximal, characteristic, complemented
4#9879C212yesmaximal, complemented
5#9651C212yesmaximal
6#9082C212yesmaximal, complemented
7#9080C212yesmaximal
8#8559C212yesmaximal
9#8558C212yesmaximal, complemented
10C22×#126C411yescharacteristic
11C2×#431C2212yes—
12#2334—24no—
13#2334—24no—
14C2×#389C411yescharacteristic
15C2×#389C2211yescharacteristic
16C2×#388C411yescharacteristic
17C2×#388C2211yescharacteristic
18C2×#388C2211yescharacteristic
19C2×#387C411yescharacteristic
20C2×#387C411yescharacteristic
21C2×#387C2211yescharacteristic
22C2×#385C411yescharacteristic
23C2×#385C411yescharacteristic
24C2×#382C411yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28#1939—24no—
29#1916—24no—
30#1817C2212yes—
31#1797C2212yes—
32#1797C2212yes—
33#1797C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#324C2214yes—
36C2×#323C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C24×C4C2×C411yescharacteristic
40C2×#126—24no—
41C2×#126—24no—
42C2×#126C818yescomplemented
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2311yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65#431—44no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8C2314yes—
71C2×C22⋊C8C2314yes—
72C2×C4×C8—22no—
73C2×C4×C8—22no—
74#389—24no—
75#389—22no—
76#389—22no—
77#388—22no—
78#388—22no—
79#388—22no—
80#388—22no—
81#388—24no—
82#387—24no—
83#387—22no—
84#387—22no—
85#387—24no—
86#385—24no—
87#385—24no—
88#382—24no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4D414yes—
92C4×C4⋊C4D414yes—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C2×#95C2×C411yescharacteristic
96C2×#95C2×C411yescharacteristic
97C2×#95C2×C411yescharacteristic
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2311yescharacteristic
106C2×#95C2311yescharacteristic
107#336—22no—
108#336—22no—
109#336—22no—
110#336—24no—
111#336—22no—
112#324—44no—
113#323—44no—
114C25C22×C411yescharacteristic
115C23×C4—22no—
116C23×C4—42no—
117C23×C4—81no—
118C23×C4—81no—
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C2411yescharacteristic
133C22×C8—24no—
134C22×C8—24no—
135#126—232no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—81no—
152C2×C4⋊C4—81no—
153C2×C4⋊C4—81no—
154C2×C4⋊C4—81no—
155C2×C4⋊C4C22⋊C414yes—
156C2×C4⋊C4C22⋊C414yes—
157C2×C4⋊C4C2×C814yes—
158C2×C4⋊C4C2×C814yes—
159C2×C4⋊C4C2×C814yes—
160C2×C4⋊C4M1614yes—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—81no—
174C2×C22⋊C4—81no—
175C2×C22⋊C4C2×C814yes—
176C2×C22⋊C4C2×C814yes—
177C2×C22⋊C4C2×C814yes—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—81no—
185C2×C42—81no—
186C2×C42—81no—
187C2×C42—81no—
188C2×C42C2×C814yes—
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C4⋊C8—44no—
196C4⋊C8—82no—
197C4⋊C8—82no—
198C4⋊C8—82no—
199C22⋊C8—24no—
200C22⋊C8—24no—
201C22⋊C8—24no—
202C22⋊C8—24no—
203C4×C8—44no—
204C4×C8—82no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95C22⋊C414yes—
218#95C22⋊C414yes—
219C24—21no—
220C24—21no—
221C24—22no—
222C24—22no—
223C24—41no—
224C24—41no—
225C24—41no—
226C24—41no—
227C24C22×C812yes—
228C24C23×C411yescharacteristic
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—44no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—88no—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C22×C812yes—
285C22×C4C22×C812yes—
286C22×C4C22×C812yes—
287C22×C4C22×C812yes—
288C22×C4C22×C812yes—
289C22×C4C22×C812yes—
290C22×C4C2×M1612yes—
291C22×C4C2×M1612yes—
292C22×C4C2×M1612yes—
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4H(5,𝔽2)12yes—
296C22×C42−1+412yes—
297C2×C8—44no—
298C2×C8—44no—
299C2×C8—44no—
300C2×C8—44no—
301C2×C8—44no—
302C2×C8—44no—
303C4⋊C4—28no—
304C4⋊C4—28no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C4⋊C4C22⋊C8116yes—
315C22⋊C4—28no—
316C22⋊C4—28no—
317C22⋊C4—28no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—42no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C42—22no—
324C42—22no—
325C42—24no—
326C42—28no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—42no—
331C42—82no—
332C42—82no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—22no—
338C23—24no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—82no—
351C23—82no—
352C23—82no—
353C23—82no—
354C23C22×C22⋊C411yescharacteristic
355C23#52012yes—
356C23C23×C811yescharacteristic
357C23C22×M1611yescharacteristic
358C23#56812yes—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—88no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4C2×C22⋊C814yes—
422C2×C4C2×C22⋊C814yes—
423C2×C4C2×C22⋊C814yes—
424C2×C4#53511yescharacteristic
425C2×C4#53611yescharacteristic
426C2×C4#53712yes—
427C8—84no—
428C8—84no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—22no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—84no—
441C22—84no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22#231111yescharacteristic
447C22C22×C22⋊C811yescharacteristic
448C22#287911yescharacteristic
449C22#288311yescharacteristic
450C22#288411yescharacteristic
451C22#288811yescharacteristic
452C22#288911yescharacteristic
453C22#298812yes—
454C22#298912yes—
455C4—24no—
456C4—21no—
457C4—22no—
458C4—21no—
459C4—24no—
460C4—24no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—81no—
468C4—81no—
469C4—81no—
470C4—81no—
471C2—22no—
472C2—22no—
473C2—82no—
474C2#1735111yescharacteristic
475C2#1735311yescharacteristic
476C2#1740511yescharacteristic
477C2#1741011yescharacteristic
478C2#2184711yescharacteristic
479C2#2187711yescharacteristic
480C2#2188111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21951C2semidirect
3. #21912C2semidirect
4. #9879C2semidirect
6. #9082C2semidirect
9. #8558C2semidirect
42. C2×#126C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23342
#219512
#219122
#98792
#96512
#90822
#90802
#85592
#85582