Finite Group Databasebeta

← #164758

Subgroups — #164758

Order 512.

753 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36481C211yesmaximal, characteristic
2#22632C211yesmaximal, characteristic
3#20754C211yesmaximal, characteristic
4#7733C212yesmaximal, complemented
5#7687C211yesmaximal, characteristic
6#7669C212yesmaximal, complemented
7#7667C211yesmaximal, characteristic
8#7653C212yesmaximal, complemented
9#7617C211yesmaximal, characteristic
10#7590C212yesmaximal, complemented
11#7566C211yesmaximal, characteristic
12C2×#424—21no—
13#2863C2212yes—
14#2857C2211yescharacteristic
15#2606—22no—
16#2552C411yescharacteristic
17#2546C411yescharacteristic
18#2522—22no—
19#2507—21no—
20#2441—21no—
21#2429C412yes—
22#2423C412yes—
23#2370—21no—
24#2365C411yescharacteristic
25Q8×C4⋊C4C411yescharacteristic
26#2347—21no—
27C4×C4⋊Q8C2212yes—
28C4×C22⋊Q8C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1939—21no—
31#1923—22no—
32#1832—22no—
33#1831C2212yes—
34#1831C2212yes—
35#1831C2212yes—
36#1797C2211yescharacteristic
37C8×C4⋊C4—21no—
38C4×C22⋊C8C2211yescharacteristic
39#1744C2212yes—
40#1729—21no—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2211yescharacteristic
47#1272C2211yescharacteristic
48#1272C2212yes—
49#1272C2212yes—
50#1271C2211yescharacteristic
51#1271C2212yes—
52#1271C2211yescharacteristic
53C22×M16—21no—
54C23×C8—21no—
55C2×C4⋊Q8C2×C412yes—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#125—21no—
58C2×C22⋊Q8C2×C411yescharacteristic
59C2×C22⋊Q8C2×C412yes—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8C2×C412yes—
63C2×#117D411yescharacteristic
64C2×#117D411yescharacteristic
65C22×C42C2311yescharacteristic
66#424—44no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C4×C8—22no—
74C2×C4×C8—21no—
75#399—22no—
76#398—22no—
77#398—22no—
78#398—22no—
79#398—22no—
80#396—22no—
81#391C2×C412yes—
82#391C2×C412yes—
83#391C2311yescharacteristic
84#391C2312yes—
85#389—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—22no—
90#389—22no—
91#389—21no—
92#389—21no—
93#389—22no—
94#389—22no—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2311yescharacteristic
98#388—22no—
99#388—22no—
100#387—21no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#385—21no—
107#384—21no—
108#384C2×C412yes—
109#384C2×C411yescharacteristic
110#383—22no—
111#380—21no—
112#380—21no—
113#380C2×C411yescharacteristic
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C22⋊C4—21no—
125C4×C22⋊C4C2×C412yes—
126C4×C22⋊C4D412yes—
127#376D412yes—
128C43C2312yes—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—21no—
133#336—21no—
134#336—22no—
135#336C2312yes—
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141C22×Q8C22×C412yes—
142C22×Q8C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×M16—42no—
147C2×M16—42no—
148C2×M16—42no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—42no—
156C22×C8—42no—
157C22×C8—42no—
158C4⋊Q8—24no—
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161#125—44no—
162C22⋊Q8—22no—
163C22⋊Q8—41no—
164C22⋊Q8—82no—
165C4×Q8—42no—
166C4×Q8—82no—
167C4×Q8—82no—
168#117—42no—
169#117SD1614yes—
170#117Q1614yes—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22⋊C412yes—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—42no—
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C412yes—
207C2×C42C22×C412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2411yescharacteristic
215C4⋊C8—42no—
216C4⋊C8—44no—
217C4⋊C8—44no—
218C4⋊C8—82no—
219C22⋊C8—42no—
220C22⋊C8—81no—
221C4×C8—42no—
222C4×C8—44no—
223#95—21no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—22no—
235#95—22no—
236#95C22⋊C412yes—
237#95C22⋊C412yes—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240C24C2×C22⋊C411yescharacteristic
241C2×Q8—21no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8#13112yes—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×SD1612yes—
273C22×C4C2×Q1612yes—
274C22×C4C23×C411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277M16—84no—
278C2×C8—22no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—84no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—81no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4#10314yes—
322C4⋊C4#10314yes—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—81no—
326C22⋊C4—82no—
327C22⋊C4#10318yes—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—42no—
335C42—41no—
336C42—41no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42—42no—
341C42—82no—
342C42—82no—
343C42C2×SD1614yes—
344C42C2×Q1614yes—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23C2×#10312yes—
352C23C22×C22⋊C411yescharacteristic
353Q8—42no—
354Q8—42no—
355Q8—82no—
356Q8—82no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4#41012yes—
404C2×C4C2×#10314yes—
405C2×C4C2×#10314yes—
406C2×C4C2×#10312yes—
407C2×C4#42112yes—
408C2×C4C2×#13111yescharacteristic
409C2×C4#56811yescharacteristic
410C2×C4C22×SD1611yescharacteristic
411C2×C4C22×Q1611yescharacteristic
412C2×C4#57711yescharacteristic
413C2×C4#57811yescharacteristic
414C8—42no—
415C8—81no—
416C8—82no—
417C8—81no—
418C8—81no—
419C8—82no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22#288011yescharacteristic
428C22C2×#41011yescharacteristic
429C22#288811yescharacteristic
430C22C22×#10311yescharacteristic
431C22#289811yescharacteristic
432C22C2×#42111yescharacteristic
433C22#290111yescharacteristic
434C4—21no—
435C4—21no—
436C4—24no—
437C4—24no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—41no—
443C4—42no—
444C4—42no—
445C4—42no—
446C4—41no—
447C4—82no—
448C4—81no—
449C4—82no—
450C4—81no—
451C4—81no—
452C4#151814yes—
453C2—42no—
454C2#1741511yescharacteristic
455C2#1750911yescharacteristic
456C2#1752511yescharacteristic
457C2C2×#151811yescharacteristic
458C2#1760511yescharacteristic
459C2#1768411yescharacteristic
460C2#1772211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7733C2semidirect
6. #7669C2semidirect
8. #7653C2semidirect
10. #7590C2semidirect

Maximal subgroups

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

subgroupindex
#364812
#226322
#207542
#77332
#76872
#76692
#76672
#76532
#76172
#75902
#75662