Finite Group Databasebeta

← #162436

Subgroups — #162436

Order 512.

765 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35207C211yesmaximal, characteristic
2#22835C211yesmaximal, characteristic, complemented
3#22008C211yesmaximal, characteristic, complemented
4#7731C212yesmaximal, complemented
5#7718C212yesmaximal, complemented
6#7671C212yesmaximal, complemented
7#7658C212yesmaximal, complemented
8#7591C211yesmaximal, characteristic, complemented
9#7574C211yesmaximal, characteristic, complemented
10#7570C211yesmaximal, characteristic, complemented
11#7560C211yesmaximal, characteristic, complemented
12#2854C2212yes—
13#2851C2212yes—
14#2653—22no—
15#2563C411yescharacteristic
16#2563C411yescharacteristic
17#2552C411yescharacteristic
18#2537C411yescharacteristic
19#2494—21no—
20#2494C412yes—
21#2490C412yes—
22#2458—22no—
23#2422—21no—
24#2324—21no—
25#2314—21no—
26C4×C4⋊Q8C2211yescharacteristic
27#2293C2211yescharacteristic
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1954—21no—
31#1939—21no—
32#1923—22no—
33#1923—22no—
34#1831C2211yescharacteristic
35#1817C2212yes—
36#1817C2212yes—
37#1797C2212yes—
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C22⋊C8C2212yes—
42#1744C2211yescharacteristic
43#1271C2212yes—
44#1271C2212yes—
45#1271C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125C2×C412yes—
55C2×#123—22no—
56C2×#121—21no—
57C2×#121C2×C412yes—
58C2×C4×Q8—21no—
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×#117—21no—
64C22×C42C2312yes—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76C2×C4×C8—22no—
77#401—22no—
78#398—22no—
79#397—22no—
80#397—22no—
81#394—22no—
82#394—22no—
83#393—22no—
84#393—22no—
85#391—21no—
86#390—21no—
87#390—22no—
88#390C2×C412yes—
89#390C2×C411yescharacteristic
90#390C2311yescharacteristic
91#389—21no—
92#389—21no—
93#389C2×C412yes—
94#389C2×C411yescharacteristic
95#389C2312yes—
96#389C2311yescharacteristic
97#388—22no—
98#388—22no—
99#388—22no—
100#388—21no—
101#388—21no—
102#388—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#386—22no—
107#385—22no—
108#385—22no—
109#384—21no—
110#384C2×C411yescharacteristic
111#382—21no—
112#382C2×C412yes—
113#380—21no—
114#380—21no—
115#380—21no—
116#380C2×C412yes—
117#380C2×C412yes—
118#380C2×C412yes—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C22⋊C4—21no—
126C4×C22⋊C4C2×C412yes—
127C4×C22⋊C4D412yes—
128C4×C22⋊C4D412yes—
129C43C2311yescharacteristic
130#336—22no—
131#336—22no—
132#336—21no—
133#336—21no—
134#336C2311yescharacteristic
135#336C2312yes—
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140C22×Q8—21no—
141C22×D4—22no—
142C22×D4C22⋊C411yescharacteristic
143C22×D4C22×C411yescharacteristic
144C23×C4—21no—
145C23×C4C22×C412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C4⋊Q8—24no—
155#125—82no—
156#123—82no—
157#123—82no—
158#121—44no—
159#121—82no—
160C4×Q8—81no—
161C4×Q8—81no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—42no—
167#117—42no—
168#117—42no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22⋊C412yes—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—82no—
203C2×C42C22×C412yes—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2411yescharacteristic
213C4⋊C8—42no—
214C4⋊C8—42no—
215C4⋊C8—42no—
216C4⋊C8—82no—
217C22⋊C8—44no—
218C22⋊C8—82no—
219C22⋊C8—82no—
220C22⋊C8—82no—
221C4×C8—42no—
222C4×C8—42no—
223C4×C8—81no—
224C4×C8—82no—
225#95—21no—
226#95—21no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—21no—
235#95—22no—
236#95C22⋊C412yes—
237#95C22⋊C412yes—
238#95C22⋊C412yes—
239C24—21no—
240C24C2×C22⋊C412yes—
241C2×Q8—42no—
242C2×Q8—81no—
243C2×Q8—81no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—44no—
251C2×D4—82no—
252C2×D4—82no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C2×C8—22no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—41no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—81no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—42no—
322C22⋊C4—44no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—81no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C42—21no—
336C42—21no—
337C42—22no—
338C42—42no—
339C42—41no—
340C42—42no—
341C42—41no—
342C42—42no—
343C42—41no—
344C42—42no—
345C42—82no—
346C42#13114yes—
347C42#13512yes—
348C42#13512yes—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23C22×C22⋊C411yescharacteristic
364Q8—84no—
365D4—44no—
366D4—82no—
367D4—164no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4#40812yes—
412C2×C4#40812yes—
413C2×C4#41612yes—
414C2×C4#41612yes—
415C2×C4#42112yes—
416C2×C4C2×#13112yes—
417C2×C4C2×#13511yescharacteristic
418C2×C4C2×#13511yescharacteristic
419C2×C4#57611yescharacteristic
420C2×C4#57711yescharacteristic
421C8—42no—
422C8—81no—
423C8—82no—
424C8—81no—
425C8—81no—
426C8—82no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22C2×#40811yescharacteristic
442C22#288311yescharacteristic
443C22#288911yescharacteristic
444C22C2×#41611yescharacteristic
445C22#289711yescharacteristic
446C22C2×#42111yescharacteristic
447C22#290111yescharacteristic
448C4—22no—
449C4—22no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—22no—
455C4—41no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—81no—
460C4—81no—
461C4—82no—
462C4—82no—
463C4—81no—
464C4#148214yes—
465C2—82no—
466C2—161no—
467C2#1738811yescharacteristic
468C2C2×#148211yescharacteristic
469C2#1748311yescharacteristic
470C2#1759911yescharacteristic
471C2#1761711yescharacteristic
472C2#1768211yescharacteristic
473C2#1772711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2537C4
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. #22835C2semidirect
3. #22008C2semidirect
4. #7731C2semidirect
5. #7718C2semidirect
6. #7671C2semidirect
7. #7658C2semidirect
8. #7591C2semidirect
9. #7574C2semidirect
10. #7570C2semidirect
11. #7560C2semidirect

Maximal subgroups

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

subgroupindex
#352072
#228352
#220082
#77312
#77182
#76712
#76582
#75912
#75742
#75702
#75602