Finite Group Databasebeta

← #164650

Subgroups — #164650

Order 512.

840 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36473C211yesmaximal, characteristic
2#22643C211yesmaximal, characteristic, complemented
3#20774C211yesmaximal, characteristic, complemented
4#7726C212yesmaximal, complemented
5#7712C212yesmaximal, complemented
6#7685C211yesmaximal, characteristic, complemented
7#7668C211yesmaximal, characteristic, complemented
8#7622C212yesmaximal, complemented
9#7616C211yesmaximal, characteristic
10#7604C212yesmaximal, complemented
11#7565C211yesmaximal, characteristic
12C2×#556—21no—
13#2860C2212yes—
14#2857C2211yescharacteristic
15#2604—22no—
16#2551C411yescharacteristic
17#2542C411yescharacteristic
18#2521—22no—
19#2496C412yes—
20#2492—21no—
21#2491C412yes—
22#2415—21no—
23#2365C411yescharacteristic
24#2362—21no—
25#2354C411yescharacteristic
26C4×C4⋊Q8C2212yes—
27C4×#121C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1939—21no—
30#1924—22no—
31#1840—22no—
32#1839C2212yes—
33#1839C2212yes—
34#1839C2212yes—
35#1797C2211yescharacteristic
36#1770—21no—
37#1767C2212yes—
38C4×C22⋊C8C2211yescharacteristic
39#1729—21no—
40#1729—21no—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1271C2212yes—
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2211yescharacteristic
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C22×M16—21no—
54C23×C8—21no—
55#556—44no—
56#556—44no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C412yes—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×#126—22no—
61C2×#126—22no—
62C2×#124—22no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×#121C2×C411yescharacteristic
67C2×C4×Q8—21no—
68C2×C4×Q8—22no—
69C2×C4×D4—21no—
70C2×C4×D4C2×C411yescharacteristic
71C2×#117—21no—
72C2×#117D411yescharacteristic
73C2×#117D411yescharacteristic
74C22×C42C2311yescharacteristic
75C2×C4⋊C8—21no—
76C2×C4⋊C8—22no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×#97—22no—
80C2×C4×C8—21no—
81#400—22no—
82#399—22no—
83#397—22no—
84#395—22no—
85#394—22no—
86#393—22no—
87#391—21no—
88#391—22no—
89#391—21no—
90#390C2×C412yes—
91#390C2×C412yes—
92#390C2311yescharacteristic
93#389—21no—
94#389—22no—
95#389—21no—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2311yescharacteristic
99#389C2312yes—
100#388—21no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#387—22no—
107#387—22no—
108#386—21no—
109#384C2×C412yes—
110#384C2×C411yescharacteristic
111#380—21no—
112#380—21no—
113#380C2×C412yes—
114#380C2×C411yescharacteristic
115C4×C4⋊C4—22no—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C22⋊C4—21no—
124C4×C22⋊C4C2×C412yes—
125C4×C22⋊C4D412yes—
126#376D412yes—
127C43C2312yes—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—21no—
135#336—21no—
136#336C2312yes—
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142C22×Q8—21no—
143C22×Q8—22no—
144C22×D4—21no—
145C22×D4C22×C411yescharacteristic
146C23×C4—21no—
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C2×M16—42no—
151C2×M16—42no—
152C2×M16—42no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—42no—
160C22×C8—42no—
161C22×C8—42no—
162C4⋊Q8—24no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4⋊Q8—44no—
166#126—44no—
167#126—44no—
168#126—48no—
169#124—48no—
170C22⋊Q8—44no—
171C22⋊Q8—44no—
172C22⋊Q8—42no—
173C22⋊Q8—42no—
174#121—22no—
175#121—41no—
176C4×Q8—44no—
177C4×Q8—44no—
178C4×Q8—44no—
179C4×D4—42no—
180C4×D4—42no—
181C4×D4—42no—
182#117—22no—
183#117—22no—
184#117—42no—
185#117—44no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4C22⋊C412yes—
216C2×C22⋊C4C22⋊C412yes—
217C2×C42—22no—
218C2×C42—21no—
219C2×C42—22no—
220C2×C42—42no—
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C412yes—
223C2×C42C22×C412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2411yescharacteristic
231C4⋊C8—42no—
232C4⋊C8—44no—
233C22⋊C8—42no—
234C22⋊C8—81no—
235#97—82no—
236C4×C8—42no—
237#95—21no—
238#95—22no—
239#95—21no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—21no—
245#95—22no—
246#95C22⋊C412yes—
247#95C22⋊C412yes—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250C24—21no—
251C24C2×C22⋊C411yescharacteristic
252C2×Q8—44no—
253C2×Q8—44no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—82no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—42no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—42no—
264C2×D4—81no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4AGL(1,ℤ/8)12yes—
291C22×C4AGL(1,ℤ/8)12yes—
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295M16—84no—
296C2×C8—22no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—41no—
314C2×C8—84no—
315C4⋊C4—21no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—44no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—81no—
337C4⋊C4#13112yes—
338C22⋊C4—24no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—44no—
342C22⋊C4—42no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—42no—
346C22⋊C4—44no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—82no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—42no—
361C42—41no—
362C42—41no—
363C42—42no—
364C42—42no—
365C42—44no—
366C42—44no—
367C42—42no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23#41812yes—
382C23C22×C22⋊C411yescharacteristic
383Q8—88no—
384Q8—84no—
385D4—81no—
386D4—81no—
387D4—84no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—84no—
434C2×C4#41012yes—
435C2×C4#41812yes—
436C2×C4#42112yes—
437C2×C4C2×#13111yescharacteristic
438C2×C4#56811yescharacteristic
439C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
440C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
441C2×C4#57711yescharacteristic
442C2×C4#57811yescharacteristic
443C8—42no—
444C8—82no—
445C8—81no—
446C8—81no—
447C8—81no—
448C8—82no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22#288011yescharacteristic
464C22C2×#41011yescharacteristic
465C22#288911yescharacteristic
466C22C2×#41811yescharacteristic
467C22#289811yescharacteristic
468C22C2×#42111yescharacteristic
469C22#290111yescharacteristic
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—41no—
477C4—42no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4—41no—
483C4—81no—
484C4—82no—
485C4—81no—
486C4—82no—
487C4#151414yes—
488C2—42no—
489C2—161no—
490C2#1742011yescharacteristic
491C2#1751111yescharacteristic
492C2#1752511yescharacteristic
493C2C2×#151411yescharacteristic
494C2#1761211yescharacteristic
495C2#1768311yescharacteristic
496C2#1775911yescharacteristic
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. #22643C2semidirect
3. #20774C2semidirect
4. #7726C2semidirect
5. #7712C2semidirect
6. #7685C2semidirect
7. #7668C2semidirect
8. #7622C2semidirect
10. #7604C2semidirect

Maximal subgroups

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

subgroupindex
#364732
#226432
#207742
#77262
#77122
#76852
#76682
#76222
#76162
#76042
#75652