Finite Group Databasebeta

← #214817

Subgroups — #214817

Order 512.

1022 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35700C211yesmaximal, characteristic
2#20730C211yesmaximal, characteristic, complemented
3#17352C211yesmaximal, characteristic, complemented
4#9486C212yesmaximal, complemented
5#9394C212yesmaximal, complemented
6#9196C212yesmaximal, complemented
7#9036C212yesmaximal
8#8767C212yesmaximal, complemented
9#8388C212yesmaximal, complemented
10C2×D4×Q8—21no—
11C2×D4×Q8C2211yescharacteristic
12C2×#519C2211yescharacteristic
13#2982C2212yes—
14#2921C2212yes—
15#2621C412yes—
16#2621C2211yescharacteristic
17#2546C412yes—
18#2546C2211yescharacteristic
19#2536C412yes—
20#2536C2211yescharacteristic
21#2429—21no—
22#2429—22no—
23#2415—21no—
24#2415—22no—
25#2392C412yes—
26#2392C2211yescharacteristic
27#2351C2211yescharacteristic
28Q8×C22⋊C4—21no—
29#1822—21no—
30#1801C2212yes—
31#1801C2212yes—
32#1784—21no—
33C4×#101—21no—
34C4×#101—21no—
35#1546C2214yes—
36#1486C2214yes—
37#1461—22no—
38#1461C2212yes—
39#1458—22no—
40#1458C2212yes—
41#1303C2214yes—
42#1286C2214yes—
43C23×Q8—21no—
44C23×Q8D411yescharacteristic
45D4×Q8—44no—
46D4×Q8—48no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8C2×C412yes—
50C2×C4⋊Q8C2311yescharacteristic
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—22no—
57#519—44no—
58C2×C4×Q8—21no—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D412yes—
63C2×C4×Q8C2311yescharacteristic
64C2×C4×Q8C2311yescharacteristic
65C2×C4×D4—22no—
66C2×C4×D4—21no—
67C2×C4×D4C2×C412yes—
68C2×C4×D4C2311yescharacteristic
69C2×#117C2×C412yes—
70C2×#117D411yescharacteristic
71C2×#117D412yes—
72C2×#117C2311yescharacteristic
73C8×Q8—44no—
74#433—44no—
75#431—24no—
76#431C2314yes—
77#423C2314yes—
78C2×#101—22no—
79C2×#101—22no—
80#407—22no—
81C4×M16—22no—
82#398—22no—
83#398—21no—
84#398C2×C412yes—
85#398C2×C412yes—
86#398C2×C412yes—
87#398C2×C412yes—
88#398C2×C411yescharacteristic
89#397—22no—
90#397—21no—
91#397C2×C411yescharacteristic
92#397C2×C412yes—
93#396—22no—
94#396—21no—
95#394—22no—
96#394—21no—
97#394C2×C412yes—
98#394C2×C411yescharacteristic
99#392C2×C411yescharacteristic
100#391—22no—
101#391—21no—
102#391—22no—
103#391—21no—
104#389—21no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#382C2×C412yes—
113#382C2311yescharacteristic
114#380—22no—
115#380—21no—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2311yescharacteristic
118C4×C22⋊C4—22no—
119C4×C22⋊C4—21no—
120#343—22no—
121#343—22no—
122#332—44no—
123D4⋊C8—44no—
124#324—24no—
125#324—22no—
126#323—22no—
127C22×Q8—24no—
128C22×Q8—24no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×Q8—42no—
132C22×Q8—44no—
133C22×Q8—44no—
134C22×Q8—44no—
135C22×Q8C22⋊C412yes—
136C22×Q8C2×D411yescharacteristic
137C22×Q8C2×D411yescharacteristic
138C22×D4—21no—
139C22×D4C22×C411yescharacteristic
140C23×C4—22no—
141C23×C4—21no—
142C23×C4C22⋊C412yes—
143C23×C4C2×D411yescharacteristic
144C2×M16—24no—
145C4⋊Q8—24no—
146C4⋊Q8—44no—
147C4⋊Q8—48no—
148C4⋊Q8—44no—
149C22⋊Q8—44no—
150C22⋊Q8—48no—
151C22⋊Q8—48no—
152C22⋊Q8—44no—
153C22⋊Q8—48no—
154C22⋊Q8—44no—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—44no—
158C4×Q8—44no—
159C4×Q8—42no—
160C4×Q8—42no—
161C4×Q8—44no—
162C4×D4—24no—
163C4×D4—44no—
164C4×D4—48no—
165C4×D4—44no—
166#117—24no—
167#117—44no—
168#117—44no—
169#117—42no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C22×C411yescharacteristic
202C2×C22⋊C4C2×D411yescharacteristic
203C2×C42—22no—
204C2×C42—21no—
205C2×C42C22⋊C412yes—
206C2×C42C22⋊C412yes—
207C2×C42C22⋊C412yes—
208C2×C42C22×C412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2411yescharacteristic
212C4⋊C8—24no—
213C4⋊C8—24no—
214C4⋊C8—24no—
215C4⋊C8—44no—
216#101—44no—
217#101—44no—
218C22⋊C8—24no—
219C22⋊C8—44no—
220#97—42no—
221C4×C8—24no—
222C4×C8—42no—
223C4×C8—44no—
224#95—22no—
225#95—21no—
226#95—22no—
227#95—21no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95C22×C411yescharacteristic
234#95C22×C412yes—
235#95C22×C411yescharacteristic
236C24—21no—
237C24C2×C22⋊C411yescharacteristic
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—42no—
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—416no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—44no—
252C2×Q8—816no—
253C2×Q8#13514yes—
254C2×Q8AGL(1,ℤ/8)14yes—
255C2×D4—22no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C412yes—
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287M16—44no—
288M16—44no—
289C2×C8—44no—
290C2×C8—44no—
291C4⋊C4—22no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—24no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—44no—
317C22⋊C4—42no—
318C22⋊C4—44no—
319C22⋊C4—42no—
320C22⋊C4—44no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—44no—
324C42—24no—
325C42—22no—
326C42—24no—
327C42—42no—
328C42—42no—
329C42—42no—
330C42—44no—
331C42—42no—
332C42H(5,𝔽2)14yes—
333C23—22no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23C22×C22⋊C411yescharacteristic
346Q8—28no—
347Q8—28no—
348Q8—48no—
349Q8—88no—
350Q8—84no—
351D4—24no—
352D4—42no—
353D4—48no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—48no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—88no—
389C2×C4—84no—
390C2×C4#41614yes—
391C2×C4#41814yes—
392C2×C4#51812yes—
393C2×C4#53412yes—
394C2×C4#53412yes—
395C2×C4C2×#13512yes—
396C2×C4C2×AGL(1,ℤ/8)12yes—
397C8—84no—
398C8—84no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—41no—
405C22—42no—
406C22—41no—
407C22—41no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22#230811yescharacteristic
414C22C2×#41611yescharacteristic
415C22C2×#41811yescharacteristic
416C4—24no—
417C4—24no—
418C4—22no—
419C4—22no—
420C4—42no—
421C4—41no—
422C4—41no—
423C4—42no—
424C4—41no—
425C4—42no—
426C4—82no—
427C4—81no—
428C4—82no—
429C4—81no—
430C4#318814yes—
431C2—21no—
432C2—21no—
433C2—81no—
434C2—82no—
435C2#2176111yescharacteristic
436C2#2178011yescharacteristic
437C2#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. #20730C2semidirect
3. #17352C2semidirect
4. #9486C2semidirect
5. #9394C2semidirect
6. #9196C2semidirect
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
#207302
#173522
#94862
#93942
#91962
#90362
#87672
#83882