Finite Group Databasebeta

← #214865

Subgroups — #214865

Order 512.

1043 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#35700C211yesmaximal, characteristic
2#20639C211yesmaximal, characteristic, complemented
3#17371C211yesmaximal, characteristic, complemented
4#9490C212yesmaximal, complemented
5#9402C212yesmaximal, complemented
6#9195C212yesmaximal, complemented
7#9035C212yesmaximal
8#8767C212yesmaximal, complemented
9#8388C212yesmaximal, complemented
10C2×D4×Q8—21no—
11C2×D4×Q8C2211yescharacteristic
12C2×#519C2211yescharacteristic
13#2983C2212yes—
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#1467—22no—
38#1464—22no—
39#1461C2212yes—
40#1458C2212yes—
41#1303C2214yes—
42#1286C2214yes—
43C23×Q8—21no—
44C23×Q8D411yescharacteristic
45C22×M16—22no—
46D4×Q8—44no—
47D4×Q8—48no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2311yescharacteristic
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—22no—
58#519—44no—
59C2×C4×Q8—21no—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D412yes—
64C2×C4×Q8C2311yescharacteristic
65C2×C4×Q8C2311yescharacteristic
66C2×C4×D4—22no—
67C2×C4×D4—21no—
68C2×C4×D4C2×C412yes—
69C2×C4×D4C2311yescharacteristic
70C2×#117C2×C412yes—
71C2×#117D411yescharacteristic
72C2×#117D412yes—
73C2×#117C2311yescharacteristic
74#446—44no—
75#432—24no—
76#431C2314yes—
77#424—44no—
78#423—22no—
79#423C2314yes—
80C2×#101—22no—
81C2×#101—22no—
82C2×#101—44no—
83#398—22no—
84#398—21no—
85#398C2×C412yes—
86#398C2×C412yes—
87#398C2×C412yes—
88#398C2×C412yes—
89#398C2×C411yescharacteristic
90#397—22no—
91#397—21no—
92#397C2×C411yescharacteristic
93#397C2×C412yes—
94#396—22no—
95#396—21no—
96#394—22no—
97#394—21no—
98#394C2×C412yes—
99#394C2×C411yescharacteristic
100#392C2×C411yescharacteristic
101#391—22no—
102#391—21no—
103#391—22no—
104#391—21no—
105#389—21no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#382C2×C412yes—
114#382C2311yescharacteristic
115#380—22no—
116#380—21no—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2311yescharacteristic
119C4×C22⋊C4—22no—
120C4×C22⋊C4—21no—
121#343—22no—
122#343—22no—
123#332—44no—
124D4⋊C8—44no—
125#324—22no—
126#324—24no—
127#323—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—24no—
132C22×Q8—21no—
133C22×Q8—42no—
134C22×Q8—44no—
135C22×Q8—44no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8C22⋊C412yes—
139C22×Q8C2×D411yescharacteristic
140C22×Q8C2×D411yescharacteristic
141C22×D4—21no—
142C22×D4C22×C411yescharacteristic
143C23×C4—22no—
144C23×C4—21no—
145C23×C4C22⋊C412yes—
146C23×C4C2×D411yescharacteristic
147C2×M16—24no—
148C2×M16—42no—
149C2×M16—44no—
150C2×M16—44no—
151C22×C8—42no—
152C4⋊Q8—24no—
153C4⋊Q8—44no—
154C4⋊Q8—48no—
155C4⋊Q8—44no—
156C22⋊Q8—44no—
157C22⋊Q8—48no—
158C22⋊Q8—48no—
159C22⋊Q8—44no—
160C22⋊Q8—48no—
161C22⋊Q8—44no—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8—44no—
165C4×Q8—44no—
166C4×Q8—42no—
167C4×Q8—42no—
168C4×Q8—44no—
169C4×D4—24no—
170C4×D4—44no—
171C4×D4—48no—
172C4×D4—44no—
173#117—24no—
174#117—44no—
175#117—44no—
176#117—42no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22×C411yescharacteristic
209C2×C22⋊C4C2×D411yescharacteristic
210C2×C42—22no—
211C2×C42—21no—
212C2×C42C22⋊C412yes—
213C2×C42C22⋊C412yes—
214C2×C42C22⋊C412yes—
215C2×C42C22×C412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2411yescharacteristic
219C4⋊C8—24no—
220C4⋊C8—24no—
221C4⋊C8—24no—
222C4⋊C8—44no—
223C4⋊C8—44no—
224#101—44no—
225#101—44no—
226C22⋊C8—24no—
227#97—44no—
228C4×C8—24no—
229#95—22no—
230#95—21no—
231#95—22no—
232#95—21no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—21no—
238#95C22×C411yescharacteristic
239#95C22×C412yes—
240#95C22×C411yescharacteristic
241C24—21no—
242C24C2×C22⋊C411yescharacteristic
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—21no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—42no—
252C2×Q8—48no—
253C2×Q8—48no—
254C2×Q8—44no—
255C2×Q8—44no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—44no—
261C2×Q8—88no—
262C2×Q8—88no—
263C2×Q8AGL(1,ℤ/8)12yes—
264C2×Q8#13712yes—
265C2×D4—22no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—44no—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C412yes—
295C22×C4AGL(1,ℤ/8)14yes—
296C22×C4C23×C411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299M16—44no—
300M16—44no—
301M16—84no—
302C2×C8—44no—
303C2×C8—44no—
304C2×C8—84no—
305C4⋊C4—22no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—44no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—42no—
334C22⋊C4—44no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C42—24no—
339C42—24no—
340C42—22no—
341C42—42no—
342C42—42no—
343C42—42no—
344C42—44no—
345C42—42no—
346C42H(5,𝔽2)14yes—
347C23—21no—
348C23—21no—
349C23—22no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23#41812yes—
359C23C22×C22⋊C411yescharacteristic
360Q8—24no—
361Q8—24no—
362Q8—48no—
363Q8—42no—
364Q8—42no—
365Q8—88no—
366Q8—82no—
367Q8—82no—
368D4—24no—
369D4—42no—
370D4—48no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—48no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—88no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4#41714yes—
410C2×C4#41812yes—
411C2×C4#51812yes—
412C2×C4#53412yes—
413C2×C4#53412yes—
414C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
415C2×C4C2×AGL(1,ℤ/8)12yes—
416C2×C4C2×#13711yescharacteristic
417C8—84no—
418C8—84no—
419C22—21no—
420C22—21no—
421C22—22no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—41no—
426C22—41no—
427C22—42no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22#230811yescharacteristic
434C22C2×#41811yescharacteristic
435C22C2×#41711yescharacteristic
436C4—22no—
437C4—24no—
438C4—22no—
439C4—22no—
440C4—42no—
441C4—42no—
442C4—41no—
443C4—42no—
444C4—41no—
445C4—42no—
446C4—82no—
447C4—81no—
448C4—82no—
449C4—81no—
450C4#319212yes—
451C4#319312yes—
452C2—21no—
453C2—21no—
454C2—42no—
455C2—82no—
456C2#2176411yescharacteristic
457C2#2178011yescharacteristic
458C2#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. #17371C2semidirect
4. #9490C2semidirect
5. #9402C2semidirect
6. #9195C2semidirect
8. #8767C2semidirect
9. #8388C2semidirect

Maximal subgroups

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

subgroupindex
#357002
#206392
#173712
#94902
#94022
#91952
#90352
#87672
#83882