Finite Group Databasebeta

← #214801

Subgroups — #214801

Order 512.

1059 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35700C211yesmaximal, characteristic
2#20639C211yesmaximal, characteristic, complemented
3#17352C211yesmaximal, characteristic, complemented
4#9486C212yesmaximal, complemented
5#9394C212yesmaximal, complemented
6#9195C212yesmaximal, complemented
7#9035C212yesmaximal
8#8766C212yesmaximal, complemented
9#8388C212yesmaximal, complemented
10C2×D4×Q8—21no—
11C2×D4×Q8C2211yescharacteristic
12C2×#519C2211yescharacteristic
13#2982C2212yes—
14#2905C2212yes—
15C22×#101—21no—
16#2621C412yes—
17#2621C2211yescharacteristic
18#2546C412yes—
19#2546C2211yescharacteristic
20#2536C412yes—
21#2536C2211yescharacteristic
22#2429—21no—
23#2429—22no—
24#2415—21no—
25#2415—22no—
26#2392C412yes—
27#2392C2211yescharacteristic
28#2351C2211yescharacteristic
29Q8×C22⋊C4—21no—
30#1801—21no—
31#1801—21no—
32#1801C2212yes—
33#1801C2212yes—
34#1742—21no—
35#1546C2214yes—
36#1486C2214yes—
37#1461—22no—
38#1461C2212yes—
39#1458—22no—
40#1458C2212yes—
41#1303C2214yes—
42#1286C2214yes—
43C23×Q8—21no—
44C23×Q8D411yescharacteristic
45C22×M16—22no—
46D4×Q8—44no—
47D4×Q8—44no—
48D4×Q8—44no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8C2×C412yes—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—22no—
59#519—44no—
60C2×C4×Q8—21no—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D412yes—
65C2×C4×Q8C2311yescharacteristic
66C2×C4×Q8C2311yescharacteristic
67C2×C4×D4—22no—
68C2×C4×D4—21no—
69C2×C4×D4C2×C412yes—
70C2×C4×D4C2311yescharacteristic
71C2×#117C2×C412yes—
72C2×#117D411yescharacteristic
73C2×#117D412yes—
74C2×#117C2311yescharacteristic
75C8×Q8—44no—
76#431—24no—
77#431C2314yes—
78#424—44no—
79#423—22no—
80#423C2314yes—
81C2×#101—22no—
82C2×#101—22no—
83C2×#101—44no—
84#398—22no—
85#398—21no—
86#398C2×C412yes—
87#398C2×C412yes—
88#398C2×C412yes—
89#398C2×C412yes—
90#398C2×C411yescharacteristic
91#397—22no—
92#397—21no—
93#397C2×C411yescharacteristic
94#397C2×C412yes—
95#396—22no—
96#396—21no—
97#394—22no—
98#394—21no—
99#394C2×C412yes—
100#394C2×C411yescharacteristic
101#392C2×C411yescharacteristic
102#391—22no—
103#391—21no—
104#391—22no—
105#391—21no—
106#389—21no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#382C2×C412yes—
115#382C2311yescharacteristic
116#380—22no—
117#380—21no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2311yescharacteristic
120C4×C22⋊C4—22no—
121C4×C22⋊C4—21no—
122#343—22no—
123#343—22no—
124#332—44no—
125D4⋊C8—44no—
126#324—24no—
127#324—22no—
128#323—22no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—44no—
134C22×Q8—44no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8C22⋊C412yes—
138C22×Q8D814yes—
139C22×Q8SD1614yes—
140C22×Q8C2×D411yescharacteristic
141C22×Q8C2×D411yescharacteristic
142C22×D4—21no—
143C22×D4C22×C411yescharacteristic
144C23×C4—22no—
145C23×C4—21no—
146C23×C4C22⋊C412yes—
147C23×C4C2×D411yescharacteristic
148C2×M16—24no—
149C2×M16—42no—
150C2×M16—44no—
151C2×M16—44no—
152C22×C8—42no—
153C4⋊Q8—24no—
154C4⋊Q8—44no—
155C4⋊Q8—48no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C22⋊Q8—44no—
159C22⋊Q8—48no—
160C22⋊Q8—48no—
161C22⋊Q8—42no—
162C22⋊Q8—42no—
163C22⋊Q8—48no—
164C22⋊Q8—42no—
165C22⋊Q8—42no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—44no—
169C4×Q8—44no—
170C4×Q8—42no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×Q8—42no—
174C4×D4—24no—
175C4×D4—44no—
176C4×D4—48no—
177C4×D4—42no—
178C4×D4—42no—
179#117—24no—
180#117—44no—
181#117—44no—
182#117—42no—
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—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4C22⋊C412yes—
214C2×C22⋊C4C22×C411yescharacteristic
215C2×C22⋊C4C2×D411yescharacteristic
216C2×C42—22no—
217C2×C42—21no—
218C2×C42C22⋊C412yes—
219C2×C42C22⋊C412yes—
220C2×C42C22⋊C412yes—
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#101—44no—
232C22⋊C8—24no—
233C4×C8—24no—
234C4×C8—44no—
235#95—22no—
236#95—21no—
237#95—22no—
238#95—21no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—21no—
244#95C22×C411yescharacteristic
245#95C22×C412yes—
246#95C22×C411yescharacteristic
247C24—21no—
248C24C2×C22⋊C411yescharacteristic
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—48no—
256C2×Q8—48no—
257C2×Q8—44no—
258C2×Q8—44no—
259C2×Q8—42no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—42no—
263C2×Q8—44no—
264C2×Q8—88no—
265C2×Q8—88no—
266C2×Q8#10218yes—
267C2×Q8C2×D812yes—
268C2×Q8C2×D812yes—
269C2×Q8C2×SD1612yes—
270C2×Q8C2×SD1612yes—
271C2×D4—22no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4#10218yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×D812yes—
301C22×C4C2×SD1612yes—
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305M16—44no—
306M16—44no—
307M16—84no—
308C2×C8—44no—
309C2×C8—44no—
310C2×C8—84no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—24no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—42no—
340C22⋊C4—44no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—44no—
344C22⋊C4—42no—
345C42—24no—
346C42—24no—
347C42—22no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42—41no—
352C42—41no—
353C42—44no—
354C42H(5,𝔽2)14yes—
355C23—21no—
356C23—21no—
357C23—22no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23C2×#10212yes—
367C23C22×C22⋊C411yescharacteristic
368Q8—24no—
369Q8—24no—
370Q8—24no—
371Q8—24no—
372Q8—48no—
373Q8—88no—
374Q8—82no—
375Q8—82no—
376D4—24no—
377D4—42no—
378D4—48no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—41no—
393C2×C4—48no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—88no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4C2×#10214yes—
417C2×C4C2×#10214yes—
418C2×C4C2×#10212yes—
419C2×C4#51812yes—
420C2×C4#53412yes—
421C2×C4#53412yes—
422C2×C4C22×D811yescharacteristic
423C2×C4C22×SD1611yescharacteristic
424C2×C4#57512yes—
425C8—84no—
426C8—84no—
427C22—21no—
428C22—21no—
429C22—22no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—41no—
434C22—41no—
435C22—42no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22#230811yescharacteristic
442C22C22×#10211yescharacteristic
443C22#289611yescharacteristic
444C4—22no—
445C4—22no—
446C4—22no—
447C4—24no—
448C4—24no—
449C4—42no—
450C4—41no—
451C4—42no—
452C4—41no—
453C4—42no—
454C4—81no—
455C4—82no—
456C4—82no—
457C4—81no—
458C4#318612yes—
459C4#318712yes—
460C2—21no—
461C2—21no—
462C2—42no—
463C2—82no—
464C2#2175811yescharacteristic
465C2#2178011yescharacteristic
466C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#392C2×C4
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. #20639C2semidirect
3. #17352C2semidirect
4. #9486C2semidirect
5. #9394C2semidirect
6. #9195C2semidirect
8. #8766C2semidirect
9. #8388C2semidirect

Maximal subgroups

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

subgroupindex
#357002
#206392
#173522
#94862
#93942
#91952
#90352
#87662
#83882