Finite Group Databasebeta

← #214792

Subgroups — #214792

Order 512.

1042 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35702C211yesmaximal, characteristic
2#20638C211yesmaximal, characteristic
3#17350C211yesmaximal, characteristic
4#9483C212yesmaximal, complemented
5#9391C212yesmaximal, complemented
6#9191C212yesmaximal, complemented
7#9033C212yesmaximal, complemented
8#8765C212yesmaximal, complemented
9#8391C212yesmaximal, complemented
10C2×D4×Q8C2211yescharacteristic
11C2×#518C2211yescharacteristic
12#2972C2212yes—
13#2905C2212yes—
14C22×#100—21no—
15#2628C2211yescharacteristic
16#2626C412yes—
17#2580C2211yescharacteristic
18#2564C412yes—
19#2563C2211yescharacteristic
20#2452—21no—
21#2450—22no—
22#2439—21no—
23#2437—22no—
24#2409C412yes—
25#2396C2211yescharacteristic
26C2×#393C412yes—
27#2349—21no—
28#2349C2211yescharacteristic
29D4×C22⋊C4—21no—
30#1800—21no—
31#1800—21no—
32#1800C2212yes—
33#1800C2212yes—
34#1742—21no—
35#1548C2214yescomplemented
36#1487C2214yescomplemented
37#1460C2212yes—
38#1459—22no—
39#1458C2212yes—
40C2×#323—22no—
41#1303C2214yescomplemented
42#1286C2214yescomplemented
43C23×Q8C2×C412yes—
44C23×D4D411yescharacteristic
45C22×M16—22no—
46D4×Q8—44no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8C2311yescharacteristic
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8C2×C412yes—
52#518—44no—
53C2×C4×Q8C2311yescharacteristic
54C2×C4×D4—21no—
55C2×C4×D4—21no—
56C2×C4×D4C2×C412yes—
57C2×C4×D4D412yes—
58C2×C4×D4D411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4C2311yescharacteristic
61C2×C4×D4C2311yescharacteristic
62C2×#117D411yescharacteristic
63C2×#117C2311yescharacteristic
64C22×C22⋊C4—22no—
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4D412yes—
67C8×D4—44no—
68#431C2314yes—
69#424—44no—
70#423—22no—
71#423C2314yes—
72C2×#100—22no—
73C2×#100—22no—
74C2×#100—44no—
75C2×C22⋊C8—24no—
76#400—21no—
77#400C2×C411yescharacteristic
78#399—21no—
79#399C2×C412yes—
80#399C2×C411yescharacteristic
81#398—21no—
82#396—22no—
83#396—22no—
84#396—21no—
85#395C2×C412yes—
86#395C2×C411yescharacteristic
87#393—22no—
88#393—22no—
89#393C2×C412yes—
90#393C2×C412yes—
91#393C2×C412yes—
92#393C2×C412yes—
93#393C2×C411yescharacteristic
94#391—22no—
95#390—21no—
96#390—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#387—22no—
101#386—22no—
102#386—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#382—21no—
109#382C2311yescharacteristic
110#380—22no—
111#380—21no—
112#380—22no—
113#380C2×C412yes—
114#379—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2311yescharacteristic
117C4×C22⋊C4—21no—
118C2×#95—22no—
119C2×#95C2×C412yes—
120#343—22no—
121#343—22no—
122#333—44no—
123Q8⋊C8—44no—
124#324—22no—
125#323—22no—
126#323—24no—
127C25C22⋊C412yes—
128C22×Q8—22no—
129C22×Q8—44no—
130C22×Q8—84no—
131C22×Q8C22×C411yescharacteristic
132C22×Q8C22×C412yes—
133C22×D4—42no—
134C22×D4—42no—
135C22×D4C22⋊C412yes—
136C22×D4SD1614yes—
137C22×D4Q1614yes—
138C22×D4C2×D411yescharacteristic
139C22×D4C2×D411yescharacteristic
140C23×C4—22no—
141C23×C4—22no—
142C23×C4C22⋊C412yes—
143C23×C4C22⋊C412yes—
144C23×C4C22×C412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C2×M16—24no—
148C2×M16—42no—
149C2×M16—44no—
150C2×M16—44no—
151C22×C8—42no—
152C22×C8—44no—
153C4⋊Q8—24no—
154C4⋊Q8—44no—
155C22⋊Q8—48no—
156C22⋊Q8—44no—
157C22⋊Q8—84no—
158C4×Q8—24no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—42no—
162C4×D4—42no—
163C4×D4—44no—
164C4×D4—44no—
165C4×D4—44no—
166C4×D4—44no—
167#117—24no—
168#117—42no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C22⋊C4—21no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—84no—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C22×C411yescharacteristic
201C2×C22⋊C4C22×C412yes—
202C2×C22⋊C4C2×D411yescharacteristic
203C2×C22⋊C4C2×D412yes—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42C22⋊C412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2411yescharacteristic
209C4⋊C8—24no—
210C4⋊C8—24no—
211C4⋊C8—24no—
212C4⋊C8—44no—
213#100—44no—
214#100—44no—
215C22⋊C8—24no—
216C22⋊C8—44no—
217C4×C8—24no—
218#95—21no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—22no—
227#95—22no—
228#95—21no—
229#95C22×C411yescharacteristic
230#95C22×C412yes—
231#95C22×C411yescharacteristic
232C24—42no—
233C24—42no—
234C24—82no—
235C24—82no—
236C24#10318yes—
237C24C2×C22⋊C411yescharacteristic
238C24C2×C22⋊C412yes—
239C24C2×C22⋊C412yes—
240C2×Q8—22no—
241C2×Q8—24no—
242C2×Q8—42no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8—82no—
246C2×Q8—82no—
247C2×Q8—88no—
248C2×Q8—88no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—48no—
254C2×D4—48no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4#10318yes—
258C2×D4C2×SD1612yes—
259C2×D4C2×SD1612yes—
260C2×D4C2×Q1612yes—
261C2×D4C2×Q1612yes—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×SD1612yes—
290C22×C4C2×Q1612yes—
291C22×C4C23×C411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294M16—44no—
295M16—44no—
296M16—84no—
297C2×C8—44no—
298C2×C8—44no—
299C2×C8—84no—
300C2×C8—164no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—81no—
313C4⋊C4—81no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C22⋊C4—22no—
318C22⋊C4—24no—
319C22⋊C4—22no—
320C22⋊C4—24no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—82no—
329C22⋊C4—81no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C42—22no—
334C42—42no—
335C42—42no—
336C42—42no—
337C42H(5,𝔽2)14yes—
338C23—21no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—21no—
343C23—22no—
344C23—48no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—41no—
349C23—42no—
350C23—41no—
351C23—88no—
352C23—162no—
353C23—162no—
354C23—164no—
355C23C2×#10312yes—
356C23C2×#10314yes—
357C23C2×#10314yes—
358C23C22×C22⋊C411yescharacteristic
359Q8—24no—
360Q8—42no—
361Q8—84no—
362Q8—88no—
363D4—24no—
364D4—24no—
365D4—24no—
366D4—24no—
367D4—48no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4—88no—
403C2×C4—162no—
404C2×C4—162no—
405C2×C4—164no—
406C2×C4C2×#10312yes—
407C2×C4#51812yes—
408C2×C4#53412yes—
409C2×C4#53412yes—
410C2×C4C22×SD1611yescharacteristic
411C2×C4C22×Q1611yescharacteristic
412C2×C4#57512yes—
413C8—84no—
414C8—84no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—24no—
419C22—24no—
420C22—42no—
421C22—44no—
422C22—44no—
423C22—44no—
424C22—42no—
425C22—42no—
426C22—41no—
427C22—42no—
428C22—42no—
429C22—41no—
430C22—82no—
431C22—81no—
432C22—168no—
433C22#230811yescharacteristic
434C22C22×#10311yescharacteristic
435C22#289611yescharacteristic
436C4—22no—
437C4—22no—
438C4—22no—
439C4—42no—
440C4—42no—
441C4—41no—
442C4—42no—
443C4—41no—
444C4—81no—
445C4—82no—
446C4—81no—
447C4—82no—
448C4—81no—
449C4#318712yes—
450C4#318912yes—
451C2—21no—
452C2—21no—
453C2—44no—
454C2—44no—
455C2—42no—
456C2#2175911yescharacteristic
457C2#2177811yescharacteristic
458C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9483C2semidirect
5. #9391C2semidirect
6. #9191C2semidirect
7. #9033C2semidirect
8. #8765C2semidirect
9. #8391C2semidirect
35. #1548C22semidirect
36. #1487C22semidirect
41. #1303C22semidirect
42. #1286C22semidirect

Maximal subgroups

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

subgroupindex
#357022
#206382
#173502
#94832
#93912
#91912
#90332
#87652
#83912