Finite Group Databasebeta

← #482102

Subgroups — #482102

Order 512.

1171 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36029C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic, complemented
3#17351C211yesmaximal, characteristic, complemented
4#11877C214yesmaximal, complemented
5#11875C214yesmaximal, complemented
6#11693C212yesmaximal, complemented
7#11691C212yesmaximal, complemented
8C2×#550—21no—
9C2×#550C411yescharacteristic, complemented
10C2×#520C2211yescharacteristic
11#2972—21no—
12C2×#410—21no—
13#2619C412yescomplemented
14#2619C2211yescharacteristic
15#2541C411yescharacteristic, complemented
16#2541C2212yes—
17#2535C411yescharacteristic, complemented
18#2535C2212yes—
19#2428—22no—
20#2428—21no—
21#2416—21no—
22#2416—22no—
23#2392C412yescomplemented
24#2392C2211yescharacteristic
25#2351C411yescharacteristic, complemented
26#2341—21no—
27#2316—21no—
28#1799—22no—
29#1799C2214yes—
30#1740C2212yes—
31#1461—21no—
32#1459C2214yes—
33#1458C2212yes—
34C2×#324—22no—
35#1344C2214yes—
36#1343C2218yes—
37#1343C2214yes—
38C22×#41—21no—
39C22×#41D411yescharacteristic
40#550—44no—
41#550—44no—
42#550—44no—
43C2×#127—21no—
44C2×#127—22no—
45C2×#127C2×C412yes—
46C2×#127C2×C411yescharacteristic
47C2×#121—21no—
48C2×#121—22no—
49C2×#121—22no—
50C2×#121—21no—
51C2×#121—22no—
52C2×#121—22no—
53C2×#121—22no—
54C2×#121—22no—
55C2×#121—21no—
56#520—44no—
57C2×C4×Q8—21no—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×D4—22no—
61C2×C4×D4—21no—
62C2×C4×D4C2×C412yes—
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4D412yes—
66C2×C4×D4D411yescharacteristic
67C2×C4×D4C2312yes—
68C2×#117—22no—
69C2×#117C2×C412yes—
70C2×#117C2311yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D412yes—
73C8×D4—44no—
74#431—22no—
75#410—44no—
76C2×#99—24no—
77C2×C22⋊C8—24no—
78#398C2×C411yescharacteristic
79#397C2×C412yes—
80#397C2311yescharacteristic
81#396—22no—
82#396—21no—
83#394—22no—
84#394—21no—
85#394—22no—
86#394—21no—
87#394C2×C411yescharacteristic
88#394C2×C412yes—
89#392—22no—
90#392—21no—
91#392C2×C412yes—
92#392C2×C412yes—
93#392C2×C412yes—
94#392C2312yes—
95#392C2311yescharacteristic
96#390—21no—
97#390—22no—
98#390—22no—
99#390—21no—
100#389—22no—
101#389—21no—
102#389—44no—
103#386—22no—
104#386—22no—
105#386—22no—
106#382C2×C412yes—
107#382C2×C411yescharacteristic
108#380—21no—
109#380—22no—
110#380—24no—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C411yescharacteristic
113C4×C22⋊C4—22no—
114C4×C22⋊C4—21no—
115#342—22no—
116#342C2314yes—
117#324—22no—
118#324—24no—
119#323C2314yes—
120C2×#41—42no—
121C2×#41—42no—
122C2×#41—44no—
123C2×#41—44no—
124C2×#41—44no—
125C2×#41D814yes—
126C2×#41SD1614yes—
127C22×Q8C22⋊C411yescharacteristic
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4C22⋊C411yescharacteristic
138C22×D4C22×C412yes—
139C22×D4C22×C411yescharacteristic
140C22×D4C2×D412yes—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4C22⋊C412yes—
144C23×C4C22⋊C412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C22×C8—44no—
149#127—44no—
150#127—42no—
151#127—44no—
152#127—44no—
153#127—44no—
154#121—48no—
155#121—44no—
156#121—44no—
157#121—44no—
158#121—48no—
159#121—42no—
160#121—42no—
161#121—44no—
162#121—84no—
163#121—84no—
164#121—84no—
165C4×Q8—24no—
166C4×Q8—42no—
167C4×Q8—44no—
168C4×D4—24no—
169C4×D4—42no—
170C4×D4—44no—
171C4×D4—42no—
172C4×D4—44no—
173C4×D4—44no—
174C4×D4—48no—
175C4×D4—42no—
176C4×D4—42no—
177#117—42no—
178#117—44no—
179#117—44no—
180#117—44no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D412yes—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4C22⋊C412yes—
213C2×C22⋊C4C22×C412yes—
214C2×C22⋊C4C2×D411yescharacteristic
215C2×C22⋊C4C2411yescharacteristic
216C2×C42—22no—
217C2×C42—21no—
218C2×C42—42no—
219C2×C42C22⋊C412yes—
220C2×C42C22×C412yes—
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C4⋊C8—42no—
224#99—48no—
225C22⋊C8—24no—
226C22⋊C8—44no—
227C4×C8—42no—
228#95—21no—
229#95—22no—
230#95—21no—
231#95—22no—
232#95—21no—
233#95—44no—
234#95C22×C411yescharacteristic
235#95C22×C411yescharacteristic
236#95C22×C412yes—
237C24—22no—
238C24—21no—
239C24C2×C22⋊C412yes—
240C24C22×D411yescharacteristic
241#41—48no—
242#41—48no—
243#41—816no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8#10214yes—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—41no—
252C2×D4—42no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—42no—
256C2×D4—44no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—44no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—42no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—82no—
271C2×D4—81no—
272C2×D4—82no—
273C2×D4—81no—
274C2×D4—82no—
275C2×D4—82no—
276C2×D4—82no—
277C2×D4—82no—
278C2×D4—82no—
279C2×D4—82no—
280C2×D4—82no—
281C2×D4—82no—
282C2×D4—82no—
283C2×D4#10214yes—
284C2×D4C2×D814yes—
285C2×D4C2×SD1614yes—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4#10218yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×D812yes—
316C22×C4C2×SD1612yes—
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C2×C8—44no—
320C2×C8—164no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—24no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—44no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—44no—
332C4⋊C4—42no—
333C4⋊C4—84no—
334C4⋊C4—164no—
335C4⋊C4#9914yes—
336C22⋊C4—24no—
337C22⋊C4—42no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—42no—
341C22⋊C4—44no—
342C22⋊C4—42no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—42no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—82no—
352C22⋊C4—81no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C42—22no—
357C42—24no—
358C42—22no—
359C42—42no—
360C42—44no—
361C42—42no—
362C42—84no—
363C23—22no—
364C23—21no—
365C23—21no—
366C23—22no—
367C23—42no—
368C23—41no—
369C23—42no—
370C23—41no—
371C23—41no—
372C23—42no—
373C23—41no—
374C23—42no—
375C23—42no—
376C23—41no—
377C23—41no—
378C23—42no—
379C23—41no—
380C23—42no—
381C23—41no—
382C23—42no—
383C23—42no—
384C23—41no—
385C23C2×#10214yes—
386C23C22×C22⋊C411yescharacteristic
387C23C22×D811yescharacteristic
388C23C22×SD1611yescharacteristic
389Q8—28no—
390D4—28no—
391D4—44no—
392D4—44no—
393D4—88no—
394D4—82no—
395D4—82no—
396D4—88no—
397D4—82no—
398D4—84no—
399D4—84no—
400D4—164no—
401D4—162no—
402D4—162no—
403D4—162no—
404D4—164no—
405D4—164no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—42no—
419C2×C4—48no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4—88no—
439C2×C4—84no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—162no—
444C2×C4C2×#9914yes—
445C2×C4C2×#9912yes—
446C2×C4C2×#10214yes—
447C2×C4C2×#10212yes—
448C2×C4C2×#10212yes—
449C2×C4#42112yes—
450C8—84no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—24no—
456C22—41no—
457C22—41no—
458C22—44no—
459C22—41no—
460C22—42no—
461C22—41no—
462C22—42no—
463C22—81no—
464C22—82no—
465C22—81no—
466C22—82no—
467C22—82no—
468C22—81no—
469C22—82no—
470C22—81no—
471C22—82no—
472C22—81no—
473C22—81no—
474C22—82no—
475C22—81no—
476C22—82no—
477C22—82no—
478C22—81no—
479C22C22×#9911yescharacteristic
480C22C22×#10211yescharacteristic
481C22C2×#42111yescharacteristic
482C4—24no—
483C4—22no—
484C4—24no—
485C4—42no—
486C4—41no—
487C4—41no—
488C4—41no—
489C4—42no—
490C4—41no—
491C4—81no—
492C4—82no—
493C4—81no—
494C4—164no—
495C4#151914yes—
496C2—21no—
497C2—21no—
498C2—41no—
499C2—41no—
500C2—44no—
501C2—161no—
502C2—162no—
503C2C2×#151911yescharacteristic
504C2#3335111yescharacteristic
505C2#3335911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#36029C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20634C2semidirect
3. #17351C2semidirect
4. #11877C2semidirect
5. #11875C2semidirect
6. #11693C2semidirect
7. #11691C2semidirect
9. C2×#550C4semidirect
13. #2619C4semidirect
15. #2541C4semidirect
17. #2535C4semidirect
23. #2392C4semidirect
25. #2351C4semidirect

Maximal subgroups

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

subgroupindex
#360292
#206342
#173512
#118772
#118752
#116932
#116912