Finite Group Databasebeta

← #165431

Subgroups — #165431

Order 512.

919 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36406C211yesmaximal, characteristic
2#22000C212yesmaximal, complemented
3#7722C212yesmaximal, complemented
4#7707C212yesmaximal, complemented
5#7667C212yesmaximal, complemented
6#7649C211yesmaximal, characteristic, complemented
7#7612C212yesmaximal, complemented
8#7601C211yesmaximal, characteristic, complemented
9#7578C212yesmaximal, complemented
10C2×#556C411yescharacteristic
11C2×#547C411yescharacteristic
12#2862C2211yescharacteristic
13#2857C2212yes—
14#2553—22no—
15#2553—22no—
16#2546C411yescharacteristic
17#2542C411yescharacteristic
18#2499C412yes—
19#2432C412yes—
20#2427—22no—
21#2360—22no—
22C4×#124C2211yescharacteristic
23#2297C2212yes—
24C42×D4C2211yescharacteristic
25#1924—22no—
26#1917—22no—
27#1869—22no—
28#1850—22no—
29#1839C2212yes—
30#1831C2212yes—
31#1817C2212yes—
32#1797C2212yes—
33#1749C2212yes—
34C2×#336C2212yes—
35#1272C2211yescharacteristic
36#1272C2212yes—
37#1272C2212yes—
38#1272C2212yes—
39#1272C2212yes—
40#1272C2212yes—
41#1272C2211yescharacteristic
42#1270C2211yescharacteristic
43#1270C2212yes—
44#1270C2211yescharacteristic
45C22×#41C2×C412yes—
46C23×C8—22no—
47#556—44no—
48#547—44no—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×#126—22no—
53C2×#124—21no—
54C2×#124C2×C411yescharacteristic
55C2×#123—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8C2×C412yes—
58C2×C22⋊Q8C2×C412yes—
59C2×#121—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D412yes—
65C2×#117C2×C412yes—
66C22×C4⋊C4—21no—
67C22×C42C2312yes—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×#97—22no—
74#399—21no—
75#399—21no—
76#398—21no—
77#398—21no—
78#397—22no—
79#397—22no—
80#394—22no—
81#394—22no—
82#391—22no—
83#391C2×C412yes—
84#391C2×C412yes—
85#391C2311yescharacteristic
86#391C2312yes—
87#390C2×C412yes—
88#390C2311yescharacteristic
89#389—22no—
90#389—22no—
91#389—22no—
92#388—22no—
93#387—22no—
94#387—22no—
95#386—22no—
96#386—22no—
97#385—22no—
98#385—22no—
99#384C2×C412yes—
100#382—22no—
101#382C2×C411yescharacteristic
102#380—22no—
103#380—22no—
104C4×C4⋊C4C2×C411yescharacteristic
105C4×C4⋊C4D412yes—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4C2×C412yes—
110C4×C22⋊C4D412yes—
111C4×C22⋊C4D412yes—
112C43C2311yescharacteristic
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336C2312yes—
119#336C2312yes—
120#336C2312yes—
121#336C2312yes—
122C2×#41—24no—
123C2×#41—42no—
124C2×#41—44no—
125C2×#41—82no—
126C22×Q8—21no—
127C22×Q8C22×C411yescharacteristic
128C22×Q8C22×C412yes—
129C22×D4—22no—
130C22×D4C22⋊C411yescharacteristic
131C22×D4C22×C411yescharacteristic
132C23×C4—21no—
133C23×C4—22no—
134C23×C4C22×C412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—44no—
142C22×C8—82no—
143C22×C8—82no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146C4⋊Q8—82no—
147#126—48no—
148#124—22no—
149#124—41no—
150#124—44no—
151#123—48no—
152C22⋊Q8—48no—
153C22⋊Q8—44no—
154C22⋊Q8—44no—
155#121—44no—
156C4×Q8—42no—
157C4×Q8—44no—
158C4×D4—24no—
159C4×D4—42no—
160C4×D4—44no—
161#117—44no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C411yescharacteristic
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4C22⋊C412yes—
187C2×C22⋊C4C22⋊C412yes—
188C2×C22⋊C4C22⋊C412yes—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—82no—
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2411yescharacteristic
200C4⋊C8—82no—
201C4⋊C8—82no—
202C22⋊C8—44no—
203C22⋊C8—82no—
204C22⋊C8—82no—
205#97—82no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95C22⋊C412yes—
215#95C22⋊C412yes—
216#95C22⋊C411yescharacteristic
217#95C22⋊C411yescharacteristic
218C24—21no—
219C24C2×C22⋊C412yes—
220#41—88no—
221#41—88no—
222C2×Q8—21no—
223C2×Q8—22no—
224C2×Q8—21no—
225C2×Q8—22no—
226C2×Q8—44no—
227C2×Q8—41no—
228C2×Q8—42no—
229C2×Q8—41no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×D4—21no—
233C2×D4—42no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—44no—
237C2×D4—44no—
238C2×D4—82no—
239C2×D4#10014yes—
240C2×D4#13112yes—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—82no—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C412yes—
269C22×C4C23×C411yescharacteristic
270C22×C4C22×D412yes—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—88no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—41no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—42no—
302C22⋊C4—44no—
303C22⋊C4—42no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—44no—
311C42—21no—
312C42—21no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—42no—
317C42—42no—
318C42—41no—
319C42—42no—
320C42—42no—
321C42—44no—
322C42—42no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—41no—
331C23—41no—
332C23—42no—
333C23—42no—
334C23C2×#10014yes—
335C23C22×C22⋊C411yescharacteristic
336Q8—81no—
337Q8—81no—
338Q8—82no—
339Q8—82no—
340Q8—84no—
341D4—44no—
342D4—42no—
343D4—42no—
344D4—88no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—84no—
383C2×C4—88no—
384C2×C4C2×#10012yes—
385C2×C4#42114yes—
386C2×C4C2×#13111yescharacteristic
387C2×C4#56811yescharacteristic
388C2×C4#57712yes—
389C2×C4#57812yes—
390C8—82no—
391C8—82no—
392C8—82no—
393C8—82no—
394C22—24no—
395C22—24no—
396C22—24no—
397C22—42no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—44no—
403C22—81no—
404C22—81no—
405C22—82no—
406C22#288011yescharacteristic
407C22#288311yescharacteristic
408C22C22×#10011yescharacteristic
409C22#289812yes—
410C22C2×#42112yes—
411C4—22no—
412C4—21no—
413C4—21no—
414C4—22no—
415C4—22no—
416C4—24no—
417C4—42no—
418C4—42no—
419C4—41no—
420C4—42no—
421C4—81no—
422C4—82no—
423C4—82no—
424C4—81no—
425C4—81no—
426C2—44no—
427C2—82no—
428C2#1742111yescharacteristic
429C2#1752312yes—
430C2#1759812yes—
431C2#1767511yescharacteristic
432C2#1777911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#547C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22000C2semidirect
3. #7722C2semidirect
4. #7707C2semidirect
5. #7667C2semidirect
6. #7649C2semidirect
7. #7612C2semidirect
8. #7601C2semidirect
9. #7578C2semidirect

Maximal subgroups

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

subgroupindex
#364062
#220002
#77222
#77072
#76672
#76492
#76122
#76012
#75782