Finite Group Databasebeta

← #214804

Subgroups — #214804

Order 512.

1089 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36020C211yesmaximal, characteristic
2#20640C211yesmaximal, characteristic, complemented
3#17353C211yesmaximal, characteristic, complemented
4#9486C212yesmaximal, complemented
5#9394C212yesmaximal, complemented
6#9195C212yesmaximal, complemented
7#9035C212yesmaximal, complemented
8#8764C212yesmaximal, complemented
9#8387C212yesmaximal, complemented
10C2×#547—21no—
11C2×#547C2211yescharacteristic
12C2×#520C2211yescharacteristic
13#2982C2212yes—
14#2905C2212yes—
15C2×#413—21no—
16#2621C2211yescharacteristic
17#2618C412yes—
18#2546C2211yescharacteristic
19#2543C412yes—
20#2536C2211yescharacteristic
21#2486C412yes—
22#2429—21no—
23#2425—22no—
24#2417—21no—
25#2413—22no—
26#2392C2211yescharacteristic
27C2×#398C412yes—
28#2349C2211yescharacteristic
29#2339—21no—
30#1801—21no—
31#1801C2212yes—
32#1800—21no—
33#1800C2212yes—
34C2×#343—21no—
35#1546C2214yes—
36#1486C2214yes—
37#1460—22no—
38#1460C2212yes—
39#1458—22no—
40#1458C2212yes—
41#1303C2214yes—
42#1286C2214yes—
43C22×#41—21no—
44C22×#41D411yescharacteristic
45C22×M16—22no—
46#547—44no—
47#547—44no—
48#547—44no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8C2311yescharacteristic
51C2×#123—22no—
52C2×#123—22no—
53C2×#123—22no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×#121—22no—
58C2×#121C2×C412yes—
59#520—44no—
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4C2×C412yes—
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D412yes—
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117D412yes—
70C2×#117C2311yescharacteristic
71C22×C4⋊C4—22no—
72C22×C4⋊C4C2×C412yes—
73C22×C4⋊C4C2×C412yes—
74C22×C4⋊C4D411yescharacteristic
75C8×Q8—44no—
76#431—24no—
77#431C2314yes—
78#424—44no—
79#423—22no—
80#423C2314yes—
81#413—44no—
82C2×#101—22no—
83C2×#100—22no—
84#398—22no—
85#398—22no—
86#398—21no—
87#398C2×C412yes—
88#398C2×C412yes—
89#398C2×C411yescharacteristic
90#398C2×C412yes—
91#398C2×C412yes—
92#397—21no—
93#397C2×C412yes—
94#397C2×C411yescharacteristic
95#396—22no—
96#396—22no—
97#396—21no—
98#394—21no—
99#394C2×C411yescharacteristic
100#394C2×C412yes—
101#392C2×C411yescharacteristic
102#391—21no—
103#391—21no—
104#389—21no—
105#389—22no—
106#389—22no—
107#388—22no—
108#388—22no—
109#386—22no—
110#386—22no—
111#386—22no—
112#385—22no—
113#382C2311yescharacteristic
114#381—22no—
115#380—22no—
116#380—21no—
117#380C2×C412yes—
118C4×C4⋊C4C2311yescharacteristic
119C4×C22⋊C4—21no—
120C2×#95—22no—
121C2×#95C2×C412yes—
122#343—22no—
123#343—22no—
124#332—44no—
125D4⋊C8—44no—
126#324—22no—
127#323—24no—
128#323—22no—
129C2×#41—42no—
130C2×#41—42no—
131C2×#41—44no—
132C2×#41—44no—
133C2×#41—42no—
134C2×#41—42no—
135C2×#41D814yes—
136C2×#41SD1614yes—
137C22×Q8—21no—
138C22×Q8C2×D411yescharacteristic
139C22×D4—22no—
140C22×D4—21no—
141C22×D4C22⋊C412yes—
142C22×D4C22×C411yescharacteristic
143C22×D4C2×D411yescharacteristic
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4C22⋊C412yes—
148C23×C4C22⋊C412yes—
149C23×C4C22⋊C412yes—
150C23×C4C22×C412yes—
151C23×C4C2×D411yescharacteristic
152C2×M16—24no—
153C2×M16—42no—
154C2×M16—44no—
155C2×M16—44no—
156C22×C8—42no—
157C4⋊Q8—24no—
158C4⋊Q8—42no—
159C4⋊Q8—42no—
160#123—48no—
161#123—48no—
162#123—48no—
163C22⋊Q8—44no—
164C22⋊Q8—42no—
165C22⋊Q8—42no—
166C22⋊Q8—42no—
167C22⋊Q8—42no—
168#121—44no—
169#121—48no—
170C4×Q8—24no—
171C4×Q8—42no—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—44no—
175C4×D4—42no—
176C4×D4—44no—
177C4×D4—42no—
178C4×D4—42no—
179C4×D4—42no—
180C4×D4—42no—
181#117—24no—
182#117—44no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—48no—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4C22⋊C412yes—
217C2×C22⋊C4C22×C411yescharacteristic
218C2×C22⋊C4C2×D412yes—
219C2×C22⋊C4C2×D411yescharacteristic
220C2×C42—21no—
221C2×C42C22⋊C412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2411yescharacteristic
225C4⋊C8—24no—
226C4⋊C8—24no—
227C4⋊C8—24no—
228C4⋊C8—44no—
229C4⋊C8—44no—
230#101—44no—
231#100—44no—
232C22⋊C8—24no—
233C4×C8—24no—
234C4×C8—44no—
235#95—21no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—21no—
241#95—22no—
242#95—22no—
243#95—21no—
244#95C22×C411yescharacteristic
245#95C22×C412yes—
246#95C22×C411yescharacteristic
247C24—21no—
248C24C2×C22⋊C411yescharacteristic
249C24C2×C22⋊C412yes—
250#41—48no—
251#41—48no—
252#41—88no—
253#41—88no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—42no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—42no—
260C2×Q8C2×D812yes—
261C2×Q8C2×SD1612yes—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—42no—
271C2×D4—44no—
272C2×D4—81no—
273C2×D4—81no—
274C2×D4#10218yes—
275C2×D4C2×D812yes—
276C2×D4C2×SD1612yes—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4#10218yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×D812yes—
310C22×C4C2×SD1612yes—
311C22×C4C23×C411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314M16—44no—
315M16—44no—
316M16—84no—
317C2×C8—44no—
318C2×C8—44no—
319C2×C8—84no—
320C4⋊C4—22no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—44no—
331C4⋊C4—42no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—82no—
340C22⋊C4—22no—
341C22⋊C4—22no—
342C22⋊C4—24no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—42no—
348C22⋊C4—44no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—44no—
352C22⋊C4—81no—
353C22⋊C4—82no—
354C42—24no—
355C42—22no—
356C42—42no—
357C42—41no—
358C42—41no—
359C42H(5,𝔽2)14yes—
360C23—21no—
361C23—22no—
362C23—22no—
363C23—21no—
364C23—42no—
365C23—42no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23C2×#10212yes—
376C23C2×#10214yes—
377C23C22×C22⋊C411yescharacteristic
378Q8—24no—
379Q8—24no—
380Q8—82no—
381Q8—82no—
382D4—24no—
383D4—24no—
384D4—48no—
385D4—44no—
386D4—84no—
387D4—88no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—48no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—84no—
421C2×C4—88no—
422C2×C4—88no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—162no—
426C2×C4—162no—
427C2×C4C2×#10212yes—
428C2×C4C2×#10214yes—
429C2×C4#51812yes—
430C2×C4#53412yes—
431C2×C4#53412yes—
432C2×C4C22×D811yescharacteristic
433C2×C4C22×SD1611yescharacteristic
434C2×C4#57512yes—
435C8—84no—
436C8—84no—
437C22—21no—
438C22—21no—
439C22—22no—
440C22—24no—
441C22—44no—
442C22—42no—
443C22—44no—
444C22—42no—
445C22—41no—
446C22—42no—
447C22—41no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22#230811yescharacteristic
457C22C22×#10211yescharacteristic
458C22#289611yescharacteristic
459C4—22no—
460C4—22no—
461C4—22no—
462C4—24no—
463C4—42no—
464C4—42no—
465C4—41no—
466C4—42no—
467C4—41no—
468C4—81no—
469C4—82no—
470C4—81no—
471C4—82no—
472C4#318612yes—
473C4#318712yes—
474C2—21no—
475C2—21no—
476C2—44no—
477C2—42no—
478C2—161no—
479C2#2175811yescharacteristic
480C2#2177911yescharacteristic
481C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36020C2
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
2. #20640C2semidirect
3. #17353C2semidirect
4. #9486C2semidirect
5. #9394C2semidirect
6. #9195C2semidirect
7. #9035C2semidirect
8. #8764C2semidirect
9. #8387C2semidirect

Maximal subgroups

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

subgroupindex
#360202
#206402
#173532
#94862
#93942
#91952
#90352
#87642
#83872