Finite Group Databasebeta

← #162472

Subgroups — #162472

Order 512.

798 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35186C211yesmaximal, characteristic
2#22734C211yesmaximal, characteristic, complemented
3#20570C211yesmaximal, characteristic, complemented
4#7706C212yesmaximal, complemented
5#7696C212yesmaximal, complemented
6#7686C212yesmaximal, complemented
7#7676C212yesmaximal, complemented
8#7587C211yesmaximal, characteristic
9#7577C211yesmaximal, characteristic, complemented
10#7569C211yesmaximal, characteristic
11#7562C211yesmaximal, characteristic, complemented
12C2×#446—21no—
13C2×C8×Q8—21no—
14#2861C2212yes—
15#2860C2212yes—
16#2651—22no—
17#2582C411yescharacteristic
18#2552C411yescharacteristic
19#2548C411yescharacteristic
20#2548C411yescharacteristic
21#2512—21no—
22#2468—22no—
23#2454C412yes—
24#2448—21no—
25#2447C412yes—
26#2326—21no—
27#2319—21no—
28C4×#127C2211yescharacteristic
29#2295C2211yescharacteristic
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#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C2×#127C2×C411yescharacteristic
54C2×#126—22no—
55C2×#125C2×C412yes—
56C2×#124—21no—
57C2×#124C2×C412yes—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4C2×C412yes—
65#446—28no—
66C8×Q8—44no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—22no—
73C2×#97—22no—
74C2×#97—22no—
75C2×C4×C8—21no—
76C2×C4×C8—22no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79#399—22no—
80#398—22no—
81#398—22no—
82#397—22no—
83#397—22no—
84#396—22no—
85#394—22no—
86#394—22no—
87#391—21no—
88#391C2×C412yes—
89#391C2×C411yescharacteristic
90#391C2311yescharacteristic
91#390—21no—
92#390—21no—
93#390—21no—
94#390—21no—
95#390—21no—
96#390C2×C412yes—
97#390C2×C412yes—
98#390C2×C412yes—
99#390C2312yes—
100#389—21no—
101#389—21no—
102#389C2×C412yes—
103#389C2×C412yes—
104#389C2×C411yescharacteristic
105#389C2311yescharacteristic
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#387—21no—
114#387—21no—
115#385—22no—
116#385—22no—
117#384—21no—
118#384C2×C411yescharacteristic
119#381—22no—
120#380—22no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D411yescharacteristic
129#376—21no—
130C43C2311yescharacteristic
131C43C2312yes—
132#336—21no—
133#336—22no—
134#336—21no—
135#336—22no—
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142C22×Q8C22⋊C411yescharacteristic
143C22×Q8C22×C411yescharacteristic
144C22×D4—21no—
145C22×D4—21no—
146C22×D4C22×C412yes—
147C23×C4—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—21no—
154#127—24no—
155#126—82no—
156#126—82no—
157#125—82no—
158#124—44no—
159#124—82no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—24no—
165C4×D4—42no—
166C4×D4—42no—
167C4×D4—81no—
168C4×D4—81no—
169C4×D4—82no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42C22×C412yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C412yes—
205C2×C42C22×C411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2411yescharacteristic
215C4⋊C8—24no—
216C4⋊C8—28no—
217C4⋊C8—44no—
218C4⋊C8—42no—
219C4⋊C8—44no—
220C4⋊C8—42no—
221#97—28no—
222#97—82no—
223C4×C8—24no—
224C4×C8—44no—
225C4×C8—42no—
226C4×C8—81no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241C24—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8#10414yes—
248C2×D4—24no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—81no—
256C2×D4—81no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—44no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—24no—
279C2×C8—22no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—41no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—81no—
309C4⋊C4—81no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4—82no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C42—24no—
328C42—22no—
329C42—22no—
330C42—21no—
331C42—22no—
332C42—21no—
333C42—22no—
334C42—22no—
335C42—24no—
336C42—41no—
337C42—42no—
338C42—41no—
339C42—42no—
340C42—41no—
341C42—42no—
342C42—42no—
343C42#13114yes—
344C42AGL(1,ℤ/8)12yes—
345C42#13712yes—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23C22×C22⋊C411yescharacteristic
354Q8—28no—
355Q8—82no—
356D4—84no—
357D4—84no—
358D4—84no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—164no—
399C2×C4#40812yes—
400C2×C4#40812yes—
401C2×C4#41712yes—
402C2×C4#41712yes—
403C2×C4C2×#10414yes—
404C2×C4C2×#10412yes—
405C2×C4C2×#13112yes—
406C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
407C2×C4C2×#13711yescharacteristic
408C2×C4#57611yescharacteristic
409C2×C4#57711yescharacteristic
410C8—42no—
411C8—44no—
412C8—42no—
413C8—42no—
414C8—82no—
415C8—81no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22C2×#40811yescharacteristic
424C22#288411yescharacteristic
425C22#288811yescharacteristic
426C22C2×#41711yescharacteristic
427C22#289711yescharacteristic
428C22C22×#10411yescharacteristic
429C22#290111yescharacteristic
430C4—24no—
431C4—22no—
432C4—22no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—22no—
438C4—42no—
439C4—42no—
440C4—41no—
441C4—42no—
442C4—81no—
443C4—82no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4#148314yes—
448C2—162no—
449C2#1739011yescharacteristic
450C2C2×#148311yescharacteristic
451C2#1748311yescharacteristic
452C2#1755011yescharacteristic
453C2#1762811yescharacteristic
454C2#1765511yescharacteristic
455C2#1776211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C2×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. #7706C2semidirect
5. #7696C2semidirect
6. #7686C2semidirect
7. #7676C2semidirect
9. #7577C2semidirect
11. #7562C2semidirect

Maximal subgroups

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

subgroupindex
#351862
#227342
#205702
#77062
#76962
#76862
#76762
#75872
#75772
#75692
#75622