Finite Group Databasebeta

← #164136

Subgroups — #164136

Order 512.

937 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36272C211yesmaximal, characteristic
2#22632C211yesmaximal, characteristic, complemented
3#20430C211yesmaximal, characteristic, complemented
4#7685C211yesmaximal, characteristic, complemented
5#7680C212yesmaximal, complemented
6#7668C211yesmaximal, characteristic, complemented
7#7663C212yesmaximal, complemented
8#7616C211yesmaximal, characteristic
9#7614C212yesmaximal, complemented
10#7565C211yesmaximal, characteristic
11#7560C212yesmaximal, complemented
12C2×#552—21no—
13C2×#424—21no—
14C2×#405—21no—
15#2859C2212yes—
16#2857C2211yescharacteristic
17#2611—22no—
18#2553—22no—
19#2552C411yescharacteristic
20#2542C411yescharacteristic
21#2496C412yes—
22#2489—21no—
23#2422C412yes—
24#2415—21no—
25#2365C411yescharacteristic
26#2360—21no—
27D4×C4⋊C4C411yescharacteristic
28C4×#121C2211yescharacteristic
29#2293C2212yes—
30C4×#117C2211yescharacteristic
31#1939—21no—
32#1923—22no—
33#1831C2212yes—
34#1797C2211yescharacteristic
35C8×C4⋊C4—21no—
36#1768—22no—
37#1767C2212yes—
38C4×C4⋊C8C2212yes—
39C4×C22⋊C8C2211yescharacteristic
40#1744C2212yes—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C22×M16—21no—
54C23×C8—21no—
55#552—44no—
56#552—44no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#126—22no—
59C2×#125—21no—
60C2×#125—22no—
61C2×#123—22no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×#121C2×C411yescharacteristic
66C2×C4×Q8—21no—
67C2×C4×D4—21no—
68C2×C4×D4C2×C412yes—
69C2×C4×D4C2×C411yescharacteristic
70C2×#117—22no—
71C2×#117D411yescharacteristic
72C2×#117D411yescharacteristic
73C22×C4⋊C4—21no—
74C22×C42C2311yescharacteristic
75#424—28no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—22no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82#405—44no—
83C2×#97—22no—
84C2×C4×C8—22no—
85C2×C4×C8—22no—
86C2×C4×C8—21no—
87#400—22no—
88#399—22no—
89#398—22no—
90#397—22no—
91#397—22no—
92#394—22no—
93#391—21no—
94#391—21no—
95#390C2×C412yes—
96#390C2311yescharacteristic
97#389—21no—
98#389—21no—
99#389—22no—
100#389—22no—
101#389C2×C412yes—
102#389C2×C412yes—
103#389C2×C412yes—
104#389C2312yes—
105#389C2311yescharacteristic
106#388—21no—
107#388—22no—
108#388—22no—
109#387—22no—
110#387—22no—
111#386—21no—
112#385—22no—
113#385—22no—
114#384C2×C411yescharacteristic
115#384C2×C412yes—
116#384C2×C412yes—
117#382—22no—
118#380—21no—
119#380—21no—
120#380C2×C412yes—
121#380C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C22⋊C4—21no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4C2×C412yes—
131C4×C22⋊C4D412yes—
132#376D412yes—
133C43C2312yes—
134#336—22no—
135#336—22no—
136#336—21no—
137#336—21no—
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2312yes—
143#336C2311yescharacteristic
144C22×Q8—21no—
145C22×D4—21no—
146C22×D4C22×C411yescharacteristic
147C23×C4—21no—
148C23×C4—22no—
149C23×C4C22×C411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C2×M16—24no—
153C2×M16—42no—
154C2×M16—42no—
155C22×C8—24no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—42no—
163C22×C8—42no—
164C4⋊Q8—42no—
165#126—48no—
166#125—44no—
167#125—44no—
168#125—44no—
169#123—48no—
170C22⋊Q8—42no—
171C22⋊Q8—42no—
172C22⋊Q8—44no—
173C22⋊Q8—44no—
174#121—22no—
175#121—41no—
176C4×Q8—44no—
177C4×D4—24no—
178C4×D4—42no—
179C4×D4—42no—
180C4×D4—42no—
181C4×D4—42no—
182#117—24no—
183#117—44no—
184#117—44no—
185#117D814yes—
186#117SD1614yes—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C412yes—
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4C22⋊C412yes—
219C2×C22⋊C4C22⋊C412yes—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—42no—
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C412yes—
225C2×C42C22×C412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2411yescharacteristic
233C4⋊C8—28no—
234C4⋊C8—28no—
235C4⋊C8—42no—
236C4⋊C8—82no—
237C22⋊C8—42no—
238C22⋊C8—81no—
239#97—44no—
240C4×C8—42no—
241C4×C8—44no—
242C4×C8—44no—
243#95—21no—
244#95—22no—
245#95—21no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—21no—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256C24—21no—
257C24C2×C22⋊C411yescharacteristic
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×D4—21no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—81no—
267C2×D4#13112yes—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—82no—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×D812yes—
297C22×C4C2×SD1612yes—
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301M16—48no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—24no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—48no—
310C2×C8—42no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—41no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—82no—
342C4⋊C4—81no—
343C4⋊C4#10214yes—
344C4⋊C4#10214yes—
345C4⋊C4#10414yes—
346C22⋊C4—24no—
347C22⋊C4—44no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—44no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4#10218yes—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—24no—
366C42—42no—
367C42—41no—
368C42—41no—
369C42—42no—
370C42—44no—
371C42—42no—
372C42C2×D814yes—
373C42C2×SD1614yes—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—22no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23C2×#10212yes—
388C23C22×C22⋊C411yescharacteristic
389Q8—84no—
390D4—42no—
391D4—42no—
392D4—84no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—88no—
436C2×C4—84no—
437C2×C4#41012yes—
438C2×C4C2×#10214yes—
439C2×C4C2×#10214yes—
440C2×C4C2×#10212yes—
441C2×C4C2×#10414yes—
442C2×C4C2×#10412yes—
443C2×C4C2×#13111yescharacteristic
444C2×C4#56811yescharacteristic
445C2×C4C22×D811yescharacteristic
446C2×C4C22×SD1611yescharacteristic
447C2×C4#57711yescharacteristic
448C2×C4#57811yescharacteristic
449C8—42no—
450C8—42no—
451C8—42no—
452C8—44no—
453C8—81no—
454C8—82no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22#288011yescharacteristic
470C22C2×#41011yescharacteristic
471C22#288911yescharacteristic
472C22C22×#10211yescharacteristic
473C22#289811yescharacteristic
474C22C22×#10411yescharacteristic
475C22#290111yescharacteristic
476C4—21no—
477C4—21no—
478C4—24no—
479C4—24no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—24no—
485C4—41no—
486C4—42no—
487C4—41no—
488C4—81no—
489C4—82no—
490C4—81no—
491C4—81no—
492C4—82no—
493C4#150214yes—
494C2—42no—
495C2—82no—
496C2#1741311yescharacteristic
497C2#1750711yescharacteristic
498C2#1751811yescharacteristic
499C2C2×#150211yescharacteristic
500C2#1760511yescharacteristic
501C2#1765311yescharacteristic
502C2#1772111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121C2×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
2. #22632C2semidirect
3. #20430C2semidirect
4. #7685C2semidirect
5. #7680C2semidirect
6. #7668C2semidirect
7. #7663C2semidirect
9. #7614C2semidirect
11. #7560C2semidirect

Maximal subgroups

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

subgroupindex
#362722
#226322
#204302
#76852
#76802
#76682
#76632
#76162
#76142
#75652
#75602