Finite Group Databasebeta

← #164207

Subgroups — #164207

Order 512.

901 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36272C211yesmaximal, characteristic
2#22643C211yesmaximal, characteristic, complemented
3#20368C211yesmaximal, characteristic, complemented
4#7700C212yesmaximal, complemented
5#7685C211yesmaximal, characteristic, complemented
6#7683C212yesmaximal, complemented
7#7668C211yesmaximal, characteristic, complemented
8#7628C212yesmaximal, complemented
9#7616C211yesmaximal, characteristic
10#7580C212yesmaximal, complemented
11#7565C211yesmaximal, characteristic
12C2×#552—21no—
13C2×#405—21no—
14#2859C2212yes—
15#2857C2211yescharacteristic
16#2611—22no—
17#2553—22no—
18#2552C411yescharacteristic
19#2542C411yescharacteristic
20#2496C412yes—
21#2489—21no—
22#2422C412yes—
23#2415—21no—
24#2365C411yescharacteristic
25#2360—21no—
26D4×C4⋊C4C411yescharacteristic
27C4×#121C2211yescharacteristic
28#2293C2212yes—
29C4×#117C2211yescharacteristic
30#1939—21no—
31#1924—22no—
32#1839C2212yes—
33#1797C2211yescharacteristic
34#1770—21no—
35#1767C2212yes—
36C4×C22⋊C8C2211yescharacteristic
37#1746—22no—
38#1745C2212yes—
39#1744C2212yes—
40#1729—21no—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
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
75C2×C4⋊C8—21no—
76C2×C4⋊C8—22no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79#405—28no—
80C2×#97—22no—
81C2×#97—22no—
82C2×C4×C8—22no—
83C2×C4×C8—21no—
84#400—22no—
85#399—22no—
86#398—22no—
87#397—22no—
88#397—22no—
89#394—22no—
90#391—21no—
91#391—21no—
92#390C2×C412yes—
93#390C2311yescharacteristic
94#389—21no—
95#389—21no—
96#389—22no—
97#389—22no—
98#389C2×C412yes—
99#389C2×C412yes—
100#389C2×C412yes—
101#389C2312yes—
102#389C2311yescharacteristic
103#388—21no—
104#388—22no—
105#388—22no—
106#387—22no—
107#387—22no—
108#386—21no—
109#385—22no—
110#385—22no—
111#384C2×C411yescharacteristic
112#384C2×C412yes—
113#384C2×C412yes—
114#382—22no—
115#380—21no—
116#380—21no—
117#380C2×C412yes—
118#380C2×C411yescharacteristic
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C22⋊C4—21no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4C2×C412yes—
128C4×C22⋊C4D412yes—
129#376D412yes—
130C43C2312yes—
131#336—22no—
132#336—22no—
133#336—21no—
134#336—21no—
135#336—22no—
136#336—22no—
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2312yes—
142#336C2311yescharacteristic
143C22×Q8—21no—
144C22×D4—21no—
145C22×D4C22×C411yescharacteristic
146C23×C4—21no—
147C23×C4—22no—
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C2×M16—24no—
152C2×M16—42no—
153C2×M16—42no—
154C22×C8—24no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—42no—
162C22×C8—42no—
163C4⋊Q8—42no—
164#126—48no—
165#125—44no—
166#125—44no—
167#125—44no—
168#123—48no—
169C22⋊Q8—44no—
170C22⋊Q8—42no—
171C22⋊Q8—42no—
172C22⋊Q8—44no—
173#121—22no—
174#121—41no—
175C4×Q8—44no—
176C4×D4—24no—
177C4×D4—42no—
178C4×D4—42no—
179C4×D4—42no—
180C4×D4—42no—
181#117—22no—
182#117—22no—
183#117—24no—
184#117—44no—
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—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4C22⋊C412yes—
218C2×C22⋊C4C22⋊C412yes—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—42no—
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C412yes—
224C2×C42C22×C412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D412yes—
231C2×C42C2411yescharacteristic
232C4⋊C8—42no—
233C4⋊C8—44no—
234C22⋊C8—42no—
235C22⋊C8—81no—
236#97—28no—
237#97—82no—
238C4×C8—28no—
239C4×C8—42no—
240#95—21no—
241#95—22no—
242#95—21no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—21no—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253C24—21no—
254C24C2×C22⋊C411yescharacteristic
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×D4—21no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—81no—
264C2×D4#13112yes—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—82no—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4AGL(1,ℤ/8)12yes—
294C22×C4AGL(1,ℤ/8)12yes—
295C22×C4C23×C411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298M16—48no—
299C2×C8—22no—
300C2×C8—24no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—48no—
307C2×C8—42no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—42no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—41no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—21no—
323C4⋊C4—22no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—42no—
339C4⋊C4—44no—
340C4⋊C4—82no—
341C4⋊C4—81no—
342C4⋊C4#10414yes—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—44no—
346C22⋊C4—42no—
347C22⋊C4—44no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—44no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—44no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—24no—
365C42—42no—
366C42—41no—
367C42—41no—
368C42—42no—
369C42—44no—
370C42—42no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—22no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23#41812yes—
385C23C22×C22⋊C411yescharacteristic
386Q8—84no—
387D4—42no—
388D4—42no—
389D4—84no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—88no—
435C2×C4—84no—
436C2×C4#41012yes—
437C2×C4#41812yes—
438C2×C4C2×#10414yes—
439C2×C4C2×#10412yes—
440C2×C4C2×#13111yescharacteristic
441C2×C4#56811yescharacteristic
442C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
443C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
444C2×C4#57711yescharacteristic
445C2×C4#57811yescharacteristic
446C8—42no—
447C8—42no—
448C8—42no—
449C8—44no—
450C8—82no—
451C8—81no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22#288011yescharacteristic
467C22C2×#41011yescharacteristic
468C22#288911yescharacteristic
469C22C2×#41811yescharacteristic
470C22#289811yescharacteristic
471C22C22×#10411yescharacteristic
472C22#290111yescharacteristic
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—24no—
480C4—41no—
481C4—42no—
482C4—42no—
483C4—42no—
484C4—41no—
485C4—81no—
486C4—82no—
487C4—81no—
488C4—81no—
489C4—82no—
490C4#150514yes—
491C2—42no—
492C2—82no—
493C2#1741311yescharacteristic
494C2#1751111yescharacteristic
495C2#1751811yescharacteristic
496C2C2×#150511yescharacteristic
497C2#1761211yescharacteristic
498C2#1765311yescharacteristic
499C2#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. #20368C2semidirect
4. #7700C2semidirect
5. #7685C2semidirect
6. #7683C2semidirect
7. #7668C2semidirect
8. #7628C2semidirect
10. #7580C2semidirect

Maximal subgroups

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

subgroupindex
#362722
#226432
#203682
#77002
#76852
#76832
#76682
#76282
#76162
#75802
#75652