Finite Group Databasebeta

← #214833

Subgroups — #214833

Order 512.

1059 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35737C211yesmaximal, characteristic
2#20731C211yesmaximal, characteristic, complemented
3#17351C211yesmaximal, characteristic, complemented
4#9485C212yesmaximal, complemented
5#9393C212yesmaximal, complemented
6#9196C212yesmaximal, complemented
7#9036C212yesmaximal, complemented
8#8767C212yesmaximal, complemented
9#8388C212yesmaximal, complemented
10C2×D4×Q8C2211yescharacteristic
11C2×#548—21no—
12C2×#520C2211yescharacteristic
13#2972C2212yes—
14#2921C2212yes—
15#2621C412yes—
16#2621C2211yescharacteristic
17#2547C412yes—
18#2547C2211yescharacteristic
19#2535C412yes—
20#2535C2211yescharacteristic
21#2427—22no—
22#2427—21no—
23#2417—21no—
24#2417—22no—
25#2392C412yes—
26#2392C2211yescharacteristic
27#2351C2211yescharacteristic
28Q8×C22⋊C4—21no—
29#1821—21no—
30#1801C2212yes—
31#1800C2212yes—
32#1785—21no—
33C4×#101—21no—
34C4×#100—21no—
35#1546C2214yes—
36#1486C2214yes—
37#1461C2212yes—
38#1459—22no—
39#1458C2212yes—
40C2×#324—22no—
41#1303C2214yes—
42#1286C2214yes—
43C22×#41D411yescharacteristic
44C23×Q8—21no—
45D4×Q8—44no—
46#548—48no—
47C2×C4⋊Q8C2×C412yes—
48C2×C4⋊Q8C2311yescharacteristic
49C2×#124—22no—
50C2×#124—21no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—22no—
55C2×#121—22no—
56C2×#121—21no—
57#520—44no—
58C2×C4×Q8—21no—
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8C2311yescharacteristic
61C2×C4×D4—22no—
62C2×C4×D4—21no—
63C2×C4×D4C2×C412yes—
64C2×C4×D4C2×C412yes—
65C2×C4×D4D412yes—
66C2×C4×D4D411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117C2×C412yes—
70C2×#117C2311yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D412yes—
73C8×D4—44no—
74#433—44no—
75#431C2314yes—
76#423C2314yes—
77C2×#101—22no—
78C2×#100—22no—
79#407—22no—
80C2×C22⋊C8—24no—
81C4×M16—22no—
82#398—22no—
83#398—21no—
84#398C2×C411yescharacteristic
85#398C2×C412yes—
86#397—22no—
87#397—21no—
88#397C2×C412yes—
89#397C2×C412yes—
90#397C2×C412yes—
91#397C2×C412yes—
92#397C2×C411yescharacteristic
93#396—22no—
94#396—21no—
95#394—22no—
96#394—21no—
97#394C2×C411yescharacteristic
98#392C2×C412yes—
99#392C2×C411yescharacteristic
100#391—22no—
101#391—21no—
102#391—22no—
103#391—21no—
104#389—21no—
105#389—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#382C2×C412yes—
110#382C2311yescharacteristic
111#380—22no—
112#380—22no—
113#380—22no—
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—22no—
125#324—24no—
126#323—22no—
127C2×#41—24no—
128C2×#41—24no—
129C2×#41—42no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×Q8—44no—
133C22×Q8—44no—
134C22×Q8—44no—
135C22×Q8C2×D411yescharacteristic
136C22×D4—21no—
137C22×D4—22no—
138C22×D4C22⋊C412yes—
139C22×D4C22×C411yescharacteristic
140C22×D4C2×D411yescharacteristic
141C23×C4—22no—
142C23×C4—21no—
143C23×C4C22⋊C412yes—
144C23×C4C22⋊C412yes—
145C23×C4C22⋊C412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C2×M16—24no—
149C22×C8—44no—
150C4⋊Q8—24no—
151C4⋊Q8—44no—
152#124—48no—
153#124—44no—
154C22⋊Q8—44no—
155C22⋊Q8—48no—
156C22⋊Q8—44no—
157C22⋊Q8—48no—
158#121—48no—
159#121—44no—
160C4×Q8—24no—
161C4×Q8—42no—
162C4×Q8—44no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—44no—
166C4×D4—44no—
167C4×D4—42no—
168C4×D4—44no—
169C4×D4—48no—
170C4×D4—44no—
171#117—24no—
172#117—44no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22×C412yes—
205C2×C22⋊C4C22×C411yescharacteristic
206C2×C22⋊C4C2×D411yescharacteristic
207C2×C42—22no—
208C2×C42—21no—
209C2×C42C22⋊C412yes—
210C2×C42C22×C412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2411yescharacteristic
213C4⋊C8—24no—
214C4⋊C8—24no—
215C4⋊C8—24no—
216#101—44no—
217#100—44no—
218C22⋊C8—24no—
219C22⋊C8—44no—
220C22⋊C8—44no—
221#97—42no—
222C4×C8—24no—
223C4×C8—42no—
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
238C24C2×C22⋊C412yes—
239#41—416no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—44no—
248C2×Q8—816no—
249C2×Q8#13514yes—
250C2×D4—24no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—24no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—44no—
259C2×D4—42no—
260C2×D4—44no—
261C2×D4—82no—
262C2×D4—81no—
263C2×D4AGL(1,ℤ/8)14yes—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C412yes—
293C22×C4C23×C411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296M16—44no—
297M16—44no—
298C2×C8—44no—
299C2×C8—44no—
300C2×C8—164no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—24no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—81no—
317C4⋊C4—82no—
318C22⋊C4—22no—
319C22⋊C4—24no—
320C22⋊C4—22no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—44no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—42no—
331C22⋊C4—82no—
332C22⋊C4—81no—
333C42—24no—
334C42—22no—
335C42—42no—
336C42—44no—
337C42—42no—
338C42H(5,𝔽2)14yes—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—42no—
346C23—41no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23#41814yes—
356C23C22×C22⋊C411yescharacteristic
357Q8—28no—
358Q8—88no—
359Q8—84no—
360D4—24no—
361D4—28no—
362D4—48no—
363D4—42no—
364D4—88no—
365D4—84no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—48no—
378C2×C4—44no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—84no—
399C2×C4—88no—
400C2×C4—84no—
401C2×C4—162no—
402C2×C4—162no—
403C2×C4#41614yes—
404C2×C4#51812yes—
405C2×C4#53412yes—
406C2×C4#53412yes—
407C2×C4C2×#13512yes—
408C2×C4C2×AGL(1,ℤ/8)12yes—
409C8—84no—
410C8—84no—
411C22—21no—
412C22—21no—
413C22—24no—
414C22—44no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—41no—
421C22—42no—
422C22—41no—
423C22—41no—
424C22—82no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22#230811yescharacteristic
431C22C2×#41611yescharacteristic
432C22C2×#41811yescharacteristic
433C4—24no—
434C4—22no—
435C4—22no—
436C4—41no—
437C4—41no—
438C4—42no—
439C4—41no—
440C4—42no—
441C4—81no—
442C4—82no—
443C4—81no—
444C4—82no—
445C4—81no—
446C4#318814yes—
447C2—21no—
448C2—21no—
449C2—44no—
450C2—81no—
451C2—82no—
452C2#2176111yescharacteristic
453C2#2178011yescharacteristic
454C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#35737C2
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. #20731C2semidirect
3. #17351C2semidirect
4. #9485C2semidirect
5. #9393C2semidirect
6. #9196C2semidirect
7. #9036C2semidirect
8. #8767C2semidirect
9. #8388C2semidirect

Maximal subgroups

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

subgroupindex
#357372
#207312
#173512
#94852
#93932
#91962
#90362
#87672
#83882