Finite Group Databasebeta

← #162466

Subgroups — #162466

Order 512.

757 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35184C211yesmaximal, characteristic
2#22734C211yesmaximal, characteristic, complemented
3#20570C211yesmaximal, characteristic, complemented
4#7709C212yesmaximal, complemented
5#7696C212yesmaximal
6#7689C212yesmaximal, complemented
7#7676C212yesmaximal
8#7590C211yesmaximal, characteristic, complemented
9#7574C211yesmaximal, characteristic
10#7571C211yesmaximal, characteristic, complemented
11#7560C211yesmaximal, characteristic
12C2×#446—21no—
13C2×C8×Q8—21no—
14#2863C2212yes—
15#2861C2212yes—
16#2654—22no—
17#2568C411yescharacteristic
18#2552C411yescharacteristic
19#2547C411yescharacteristic
20#2538C411yescharacteristic
21#2525—22no—
22#2495—21no—
23#2453C412yes—
24#2448C412yes—
25#2423—21no—
26#2323—21no—
27#2314—21no—
28C4×C4⋊Q8C2211yescharacteristic
29#2293C2211yescharacteristic
30C42×Q8C2211yescharacteristic
31#1939—21no—
32#1939—22no—
33#1831C2212yes—
34#1831C2211yescharacteristic
35#1771—22no—
36#1770—21no—
37#1767C2212yes—
38C4×C4⋊C8C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1745C2212yes—
42#1744C2211yescharacteristic
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125C2×C412yes—
55C2×#124C2×C412yes—
56C2×C22⋊Q8—21no—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×#117—21no—
63#446—28no—
64C8×Q8—44no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—21no—
71C2×#97—22no—
72C2×#97—22no—
73C2×C4×C8—21no—
74C2×C4×C8—22no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77#400—22no—
78#400—22no—
79#399—22no—
80#398—22no—
81#398—22no—
82#397—22no—
83#397—22no—
84#396—22no—
85#391—21no—
86#391C2×C412yes—
87#391C2×C412yes—
88#391C2×C412yes—
89#391C2312yes—
90#390C2×C412yes—
91#390C2×C411yescharacteristic
92#390C2311yescharacteristic
93#389—22no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389C2×C412yes—
98#389C2×C412yes—
99#389C2×C411yescharacteristic
100#389C2311yescharacteristic
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—21no—
107#388—21no—
108#388—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#384—21no—
114#384C2×C411yescharacteristic
115#383—22no—
116#382—21no—
117#380—21no—
118#380—21no—
119#380—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D412yes—
129C4×C22⋊C4—21no—
130C43C2311yescharacteristic
131C43C2312yes—
132#336—22no—
133#336—21no—
134#336—21no—
135#336—22no—
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142C22×Q8C22⋊C411yescharacteristic
143C22×Q8C22×C412yes—
144C22×D4C22×C411yescharacteristic
145C23×C4—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—21no—
152C4⋊Q8—24no—
153#125—82no—
154#124—82no—
155C22⋊Q8—44no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—82no—
161C4×D4—24no—
162#117—42no—
163#117—42no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42C22×C412yes—
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C412yes—
199C2×C42C22×C411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2411yescharacteristic
209C4⋊C8—24no—
210C4⋊C8—28no—
211C4⋊C8—44no—
212C4⋊C8—42no—
213C4⋊C8—44no—
214C4⋊C8—42no—
215#97—28no—
216#97—82no—
217C4×C8—24no—
218C4×C8—44no—
219C4×C8—42no—
220C4×C8—81no—
221#95—21no—
222#95—21no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—21no—
232#95—22no—
233#95—22no—
234#95C22⋊C412yes—
235#95C22⋊C412yes—
236#95C22⋊C412yes—
237C24—21no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—24no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8#10414yes—
244C2×D4—22no—
245C2×D4—41no—
246C2×D4—41no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C23×C411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—24no—
269C2×C8—22no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—42no—
279C2×C8—41no—
280C2×C8—42no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—41no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—24no—
287C4⋊C4—22no—
288C4⋊C4—24no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—81no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—44no—
313C22⋊C4—81no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C42—24no—
318C42—22no—
319C42—22no—
320C42—21no—
321C42—22no—
322C42—21no—
323C42—22no—
324C42—22no—
325C42—24no—
326C42—41no—
327C42—42no—
328C42—41no—
329C42—42no—
330C42—41no—
331C42—42no—
332C42—82no—
333C42#13114yes—
334C42AGL(1,ℤ/8)12yes—
335C42#13712yes—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23C22×C22⋊C411yescharacteristic
344Q8—28no—
345Q8—84no—
346D4—82no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—84no—
387C2×C4#40812yes—
388C2×C4#40812yes—
389C2×C4#41712yes—
390C2×C4#41712yes—
391C2×C4C2×#10414yes—
392C2×C4C2×#10412yes—
393C2×C4C2×#13112yes—
394C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
395C2×C4C2×#13711yescharacteristic
396C2×C4#57711yescharacteristic
397C2×C4#57811yescharacteristic
398C8—42no—
399C8—44no—
400C8—42no—
401C8—42no—
402C8—82no—
403C8—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22C2×#40811yescharacteristic
412C22#288411yescharacteristic
413C22#288811yescharacteristic
414C22C2×#41711yescharacteristic
415C22#289811yescharacteristic
416C22C22×#10411yescharacteristic
417C22#290111yescharacteristic
418C4—24no—
419C4—22no—
420C4—22no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—22no—
426C4—42no—
427C4—42no—
428C4—42no—
429C4—41no—
430C4—81no—
431C4—81no—
432C4—81no—
433C4—82no—
434C4—82no—
435C4#148314yes—
436C2—161no—
437C2#1739011yescharacteristic
438C2C2×#148311yescharacteristic
439C2#1748411yescharacteristic
440C2#1754911yescharacteristic
441C2#1762811yescharacteristic
442C2#1765511yescharacteristic
443C2#1776411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22734C2semidirect
3. #20570C2semidirect
4. #7709C2semidirect
6. #7689C2semidirect
8. #7590C2semidirect
10. #7571C2semidirect

Maximal subgroups

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

subgroupindex
#351842
#227342
#205702
#77092
#76962
#76892
#76762
#75902
#75742
#75712
#75602