Finite Group Databasebeta

← #224910

Subgroups — #224910

Order 512.

908 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36597C211yesmaximal, characteristic
2#21346C212yesmaximal, complemented
3#9689C212yesmaximal, complemented
4#9689C212yesmaximal, complemented
5#9685C212yesmaximal, complemented
6#8583C211yesmaximal, characteristic, complemented
7#8583C211yesmaximal, characteristic, complemented
8#8579C211yesmaximal, characteristic, complemented
9#8224C211yesmaximal, characteristic, complemented
10#8224C211yesmaximal, characteristic, complemented
11C4×#329C211yesmaximal, characteristic, complemented
12C2×#559C411yescharacteristic
13C2×#556C411yescharacteristic
14#2858C2211yescharacteristic
15#2858C2211yescharacteristic
16#2858C2211yescharacteristic
17#2554—22no—
18#2554—22no—
19#2546C411yescharacteristic
20#2545C411yescharacteristic
21#2545C411yescharacteristic
22#2544C411yescharacteristic
23#2544C411yescharacteristic
24#2541C411yescharacteristic
25#2426—22no—
26#2363—22no—
27C4×#124C2211yescharacteristic
28C4×#124C2211yescharacteristic
29C4×#124C2211yescharacteristic
30C42×Q8C2211yescharacteristic
31#1985—22no—
32#1940—22no—
33#1916—22no—
34#1916—22no—
35#1767C2212yes—
36#1767C2212yes—
37#1767C2212yes—
38#1745C2212yes—
39#1745C2212yes—
40C4×#97C2212yes—
41C2×#329C2212yes—
42C2×#329C2212yes—
43C2×#329C2212yes—
44C2×#329C2212yes—
45C2×#329C2212yes—
46C2×#329C2212yes—
47#1295C2211yescharacteristic
48#1295C2211yescharacteristic
49#1295C2211yescharacteristic
50#1295C2211yescharacteristic
51#559—44no—
52#556—44no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#127C2×C411yescharacteristic
57C2×#127C2×C411yescharacteristic
58C2×#127C2×C411yescharacteristic
59C2×#126—21no—
60C2×#126—21no—
61C2×#124—21no—
62C2×#124C2×C411yescharacteristic
63C2×#124C2×C411yescharacteristic
64C2×#124C2×C411yescharacteristic
65C2×#124C2×C411yescharacteristic
66C2×#124C2×C411yescharacteristic
67C2×#124C2×C411yescharacteristic
68C2×#124C2311yescharacteristic
69C2×#124C2311yescharacteristic
70C2×#124C2311yescharacteristic
71C2×#123—21no—
72C2×#123—21no—
73C2×C22⋊Q8—21no—
74C2×#121—22no—
75C2×C4×Q8C2×C411yescharacteristic
76C2×C4×Q8C2×C411yescharacteristic
77C2×C4×Q8C2×C411yescharacteristic
78C2×C4×Q8C2×C411yescharacteristic
79C2×C4×Q8D412yes—
80C2×C4×D4—21no—
81C2×C4×D4C2×C411yescharacteristic
82C2×C4×D4C2×C411yescharacteristic
83C2×C4×D4C2×C411yescharacteristic
84C2×#117—21no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×#97—22no—
89C2×#97C2312yes—
90C2×#97C2312yes—
91C2×#97C2312yes—
92C2×#97C2312yes—
93C2×C4×C8—22no—
94#398—22no—
95#396—22no—
96#396—22no—
97#394—22no—
98#394—22no—
99#392—22no—
100#391C2×C411yescharacteristic
101#391C2×C411yescharacteristic
102#391C2×C411yescharacteristic
103#390—22no—
104#390—22no—
105#390—22no—
106#390—22no—
107#390—22no—
108#390—22no—
109#390C2×C411yescharacteristic
110#390C2×C411yescharacteristic
111#390C2×C411yescharacteristic
112#389—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#388—22no—
117#388—22no—
118#386—22no—
119#386—22no—
120#382C2311yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D412yes—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C43C2311yescharacteristic
131C43C2311yescharacteristic
132C43C2311yescharacteristic
133#336—22no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#329—24no—
141#329—24no—
142#329—24no—
143#329—42no—
144#329—42no—
145#329—42no—
146C22×Q8C22⋊C411yescharacteristic
147C22×Q8C22×C411yescharacteristic
148C22×Q8C22×C411yescharacteristic
149C22×Q8C22×C411yescharacteristic
150C22×D4—21no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—21no—
154C22×D4C22×C411yescharacteristic
155C22×D4C22×C411yescharacteristic
156C22×D4C22×C411yescharacteristic
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C4⋊Q8—42no—
165C4⋊Q8—81no—
166C4⋊Q8—81no—
167#127—42no—
168#127—81no—
169#127—81no—
170#126—44no—
171#126—44no—
172#124—22no—
173#124—22no—
174#124—22no—
175#124—21no—
176#124—21no—
177#124—21no—
178#124—21no—
179#124—21no—
180#124—21no—
181#124—42no—
182#124—42no—
183#124—42no—
184#124—42no—
185#124—44no—
186#124—81no—
187#124—81no—
188#123—44no—
189#123—44no—
190C22⋊Q8—44no—
191#121—48no—
192C4×Q8—24no—
193C4×Q8—44no—
194C4×Q8—42no—
195C4×Q8—42no—
196C4×Q8—42no—
197C4×D4—42no—
198C4×D4—42no—
199C4×D4—42no—
200C4×D4—44no—
201#117—44no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C22⋊C411yescharacteristic
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C412yes—
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C22×C411yescharacteristic
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2411yescharacteristic
252C4⋊C8—44no—
253C4⋊C8—44no—
254C4⋊C8—44no—
255#97—22no—
256#97—22no—
257#97—22no—
258#97—22no—
259#97—22no—
260#97—22no—
261#97—22no—
262#97—44no—
263#97#4114yes—
264C4×C8—44no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95C22⋊C412yes—
270#95C22⋊C412yes—
271#95C22⋊C412yes—
272C24—21no—
273C24—21no—
274C24—21no—
275C2×Q8—22no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—42no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—44no—
306C2×D4—82no—
307C2×D4—82no—
308C2×D4—82no—
309C2×D4—82no—
310C2×D4—82no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—44no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C412yes—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D412yes—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4#10014yes—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—44no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—44no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—24no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42—42no—
393C42—42no—
394C42—42no—
395C42—42no—
396C42—42no—
397C42C2×#4112yes—
398C42H(5,𝔽2)11yescharacteristic
399C422−1+411yescharacteristic
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23C22×C22⋊C411yescharacteristic
416Q8—44no—
417Q8—81no—
418Q8—81no—
419Q8—81no—
420Q8—81no—
421Q8—81no—
422Q8—81no—
423D4—81no—
424D4—81no—
425D4—81no—
426D4—81no—
427D4—81no—
428D4—81no—
429D4—84no—
430D4—164no—
431D4—164no—
432D4—162no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—84no—
468C2×C4—162no—
469C2×C4—162no—
470C2×C4C2×#10012yes—
471C2×C4C2×#10012yes—
472C2×C4C2×#10012yes—
473C2×C4#42114yes—
474C2×C4C4×#4111yescharacteristic
475C2×C4#52011yescharacteristic
476C2×C4#54812yes—
477C2×C4#55012yes—
478C8—44no—
479C8—44no—
480C8—82no—
481C8—82no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22#234111yescharacteristic
498C22C22×#10011yescharacteristic
499C22#288811yescharacteristic
500C22C2×#42112yes—
501C22#290112yes—
502C4—21no—
503C4—21no—
504C4—24no—
505C4—22no—
506C4—22no—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—42no—
511C4—42no—
512C4—41no—
513C4—41no—
514C4—41no—
515C4—81no—
516C4—81no—
517C4—81no—
518C4—81no—
519C4—81no—
520C2—161no—
521C2—161no—
522C2—161no—
523C2#1766311yescharacteristic
524C2#1767912yes—
525C2#1768511yescharacteristic
526C2#2253611yescharacteristic
527C2#2255112yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2541C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21346C2semidirect
3. #9689C2semidirect
4. #9689C2semidirect
5. #9685C2semidirect
6. #8583C2semidirect
7. #8583C2semidirect
8. #8579C2semidirect
9. #8224C2semidirect
10. #8224C2semidirect
11. C4×#329C2semidirect

Maximal subgroups

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

subgroupindex
#365972
#213462
#96892
#96852
#85832
#85792
#82242
C4×#3292