Finite Group Databasebeta

← #214830

Subgroups — #214830

Order 512.

904 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35846C211yesmaximal, characteristic
2#20640C211yesmaximal, characteristic
3#17353C211yesmaximal, characteristic
4#9484C212yesmaximal, complemented
5#9394C212yesmaximal, complemented
6#9195C212yesmaximal, complemented
7#9033C212yesmaximal, complemented
8#8770C212yesmaximal, complemented
9#8389C212yesmaximal, complemented
10C2×#556C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2982C2212yes—
13#2905C2212yes—
14C2×#413—21no—
15#2638C412yes—
16#2621C2211yescharacteristic
17#2583C412yes—
18#2581C2211yescharacteristic
19#2546C2211yescharacteristic
20#2484C412yes—
21#2478—22no—
22#2466—22no—
23#2447—21no—
24#2432—21no—
25#2396C2211yescharacteristic
26#2395C412yes—
27#2365—21no—
28#2360—21no—
29#2357C2211yescharacteristic
30#1801—21no—
31#1801C2212yes—
32#1800—21no—
33#1800C2212yes—
34C2×#343—21no—
35#1546C2214yes—
36#1487C2214yes—
37#1460—22no—
38#1460C2212yes—
39#1458—22no—
40#1458C2212yes—
41#1303C2214yes—
42#1286C2214yes—
43C22×#41D411yescharacteristic
44C22×M16—22no—
45#556—44no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8C2311yescharacteristic
48C2×#126—22no—
49C2×#124—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8C2×C412yes—
52#520—44no—
53C2×C4×Q8—21no—
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8C2311yescharacteristic
56C2×C4×Q8C2311yescharacteristic
57C2×C4×D4C2×C412yes—
58C2×C4×D4D411yescharacteristic
59C2×C4×D4D412yes—
60C2×C4×D4C2311yescharacteristic
61C2×#117C2×C412yes—
62C2×#117D412yes—
63C2×#117C2311yescharacteristic
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4D411yescharacteristic
66C8×Q8—44no—
67#431—24no—
68#431C2314yes—
69#424—44no—
70#423—22no—
71#423C2314yes—
72#413—44no—
73C2×#101—22no—
74C2×#100—22no—
75#400C2×C412yes—
76#400C2×C411yescharacteristic
77#399C2×C411yescharacteristic
78#398—21no—
79#398—21no—
80#398C2×C412yes—
81#398C2×C412yes—
82#398C2×C412yes—
83#397—22no—
84#397—22no—
85#397—21no—
86#397C2×C412yes—
87#396—21no—
88#395C2×C412yes—
89#395C2×C411yescharacteristic
90#394—22no—
91#394—22no—
92#393C2×C411yescharacteristic
93#391—21no—
94#391—21no—
95#391—21no—
96#390—21no—
97#389—22no—
98#389—22no—
99#389—21no—
100#389—22no—
101#388—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#385—22no—
107#385—22no—
108#385—22no—
109#385—22no—
110#382C2311yescharacteristic
111#381—22no—
112#381C2×C412yes—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2311yescharacteristic
117C4×C22⋊C4—22no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4C2×C412yes—
120#343—22no—
121#343—22no—
122#332—44no—
123Q8⋊C8—44no—
124#324—22no—
125#323—24no—
126#323—22no—
127C2×#41—22no—
128C2×#41—22no—
129C2×#41—24no—
130C2×#41—42no—
131C22×Q8—21no—
132C22×Q8C22×C411yescharacteristic
133C22×Q8C2×D411yescharacteristic
134C22×D4C22⋊C412yes—
135C22×D4C2×D411yescharacteristic
136C23×C4C22⋊C412yes—
137C23×C4C22⋊C412yes—
138C23×C4C22⋊C412yes—
139C23×C4C22×C412yes—
140C23×C4C2×D411yescharacteristic
141C2×M16—24no—
142C2×M16—42no—
143C2×M16—44no—
144C2×M16—44no—
145C22×C8—42no—
146C4⋊Q8—24no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149#126—48no—
150#124—44no—
151C22⋊Q8—44no—
152C22⋊Q8—44no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—42no—
156C4×Q8—44no—
157C4×D4—24no—
158C4×D4—44no—
159C4×D4—42no—
160C4×D4—44no—
161#117—24no—
162#117—44no—
163#117—44no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4C22⋊C412yes—
194C2×C22⋊C4C22×C411yescharacteristic
195C2×C22⋊C4C2×D412yes—
196C2×C22⋊C4C2×D411yescharacteristic
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42C22⋊C412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2411yescharacteristic
205C4⋊C8—24no—
206C4⋊C8—24no—
207C4⋊C8—24no—
208C4⋊C8—44no—
209C4⋊C8—44no—
210#101—44no—
211#100—44no—
212C22⋊C8—24no—
213C4×C8—24no—
214C4×C8—44no—
215#95—22no—
216#95—21no—
217#95—21no—
218#95—22no—
219#95—21no—
220#95—22no—
221#95—22no—
222#95—21no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—21no—
228#95C22×C411yescharacteristic
229#95C22×C412yes—
230#95C22×C411yescharacteristic
231C24C2×C22⋊C412yes—
232C24C2×C22⋊C411yescharacteristic
233#41—48no—
234#41—48no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—81no—
243C2×Q8AGL(1,ℤ/8)12yes—
244C2×Q8#13712yes—
245C2×D4—24no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—24no—
251C2×D4—42no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C412yes—
276C22×C4#13514yes—
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280M16—44no—
281M16—44no—
282M16—84no—
283C2×C8—44no—
284C2×C8—44no—
285C2×C8—84no—
286C4⋊C4—22no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—24no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—81no—
305C4⋊C4—82no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4—24no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—82no—
316C22⋊C4—81no—
317C42—24no—
318C42—22no—
319C42—42no—
320C42—44no—
321C42—42no—
322C42—41no—
323C42—41no—
324C42—82no—
325C42H(5,𝔽2)14yes—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—41no—
332C23—41no—
333C23—42no—
334C23—42no—
335C23#41612yes—
336C23C22×C22⋊C411yescharacteristic
337Q8—24no—
338Q8—24no—
339Q8—44no—
340Q8—82no—
341Q8—82no—
342D4—48no—
343D4—42no—
344D4—42no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—48no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—84no—
381C2×C4—162no—
382C2×C4—162no—
383C2×C4#41612yes—
384C2×C4#41714yes—
385C2×C4#51812yes—
386C2×C4#53412yes—
387C2×C4#53412yes—
388C2×C4C2×#13512yes—
389C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
390C2×C4C2×#13711yescharacteristic
391C8—84no—
392C8—84no—
393C22—21no—
394C22—21no—
395C22—22no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—81no—
403C22—82no—
404C22—82no—
405C22#230811yescharacteristic
406C22C2×#41611yescharacteristic
407C22C2×#41711yescharacteristic
408C4—22no—
409C4—24no—
410C4—22no—
411C4—22no—
412C4—42no—
413C4—42no—
414C4—41no—
415C4—42no—
416C4—41no—
417C4—81no—
418C4—82no—
419C4—82no—
420C4—81no—
421C4—81no—
422C4#318812yes—
423C4#319012yes—
424C2—21no—
425C2—21no—
426C2—42no—
427C2—82no—
428C2#2176311yescharacteristic
429C2#2177911yescharacteristic
430C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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. #9484C2semidirect
5. #9394C2semidirect
6. #9195C2semidirect
7. #9033C2semidirect
8. #8770C2semidirect
9. #8389C2semidirect

Maximal subgroups

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

subgroupindex
#358462
#206402
#173532
#94842
#93942
#91952
#90332
#87702
#83892