Finite Group Databasebeta

← #160639

Subgroups — #160639

Order 512.

906 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34332C211yesmaximal, characteristic
2#22835C211yesmaximal, characteristic
3#22008C211yesmaximal, characteristic
4#7600C212yesmaximal, complemented
5#7593C212yesmaximal, complemented
6#7591C211yesmaximal, characteristic, complemented
7#7583C212yesmaximal, complemented
8#7577C211yesmaximal, characteristic, complemented
9#7573C212yesmaximal, complemented
10#7570C211yesmaximal, characteristic, complemented
11#7562C211yesmaximal, characteristic, complemented
12C2×#554C411yescharacteristic
13#2370—21no—
14#2369C411yescharacteristic
15#2360—22no—
16#2360C412yes—
17Q8×C4⋊C4C411yescharacteristic
18D4×C4⋊C4—21no—
19D4×C4⋊C4C411yescharacteristic
20#2345C412yes—
21#2330—21no—
22#2329—22no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×C22⋊Q8C2212yes—
25#2295C2211yescharacteristic
26C2×#389C2212yes—
27#2275—21no—
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1954—21no—
31#1939—21no—
32#1923—22no—
33#1923—22no—
34#1831C2211yescharacteristic
35#1817C2212yes—
36#1817C2212yes—
37#1797C2212yes—
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C22⋊C8C2212yes—
42#1744C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2212yes—
50#1271C2212yes—
51#1271C2212yes—
52#1271C2212yes—
53#554—24no—
54#554C818yescomplemented
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125—21no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8C2×C412yes—
59C2×C4×Q8—21no—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2×C412yes—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C22×C4⋊C4—22no—
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4C2×C412yes—
70C22×C42C2312yes—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—22no—
83#391—21no—
84#391—22no—
85#391C2×C411yescharacteristic
86#391C2311yescharacteristic
87#389—21no—
88#389—21no—
89#389—21no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389C2×C411yescharacteristic
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2312yes—
98#389C2311yescharacteristic
99#387—21no—
100#387—22no—
101#387—21no—
102#387—22no—
103#387—22no—
104#387—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#384—21no—
109#384C2×C411yescharacteristic
110#380—22no—
111#380C2×C412yes—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4C2×C412yes—
127C4×C22⋊C4D412yes—
128C4×C22⋊C4D412yes—
129#376—21no—
130C2×#95—22no—
131C43C2311yescharacteristic
132#336—22no—
133#336—22no—
134#336—21no—
135#336—21no—
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142C22×Q8—21no—
143C22×Q8C22×C411yescharacteristic
144C22×D4C22⋊C411yescharacteristic
145C23×C4—22no—
146C23×C4—22no—
147C23×C4C22×C412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C4⋊Q8—22no—
157C4⋊Q8C2×C814yes—
158#125—28no—
159C22⋊Q8—24no—
160C22⋊Q8—216no—
161C22⋊Q8C2×C818yes—
162C4×Q8—22no—
163C4×Q8—81no—
164C4×Q8—81no—
165C4×Q8—82no—
166C4×Q8C2×C814yes—
167C4×D4—28no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—42no—
171C4×D4—44no—
172C4×D4C2×C814yes—
173C4×D4M1614yes—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C2×C818yes—
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22⋊C412yes—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—82no—
211C2×C42C22×C412yes—
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2411yescharacteristic
221C4⋊C8—42no—
222C4⋊C8—42no—
223C4⋊C8—42no—
224C4⋊C8—82no—
225C22⋊C8—44no—
226C22⋊C8—82no—
227C22⋊C8—82no—
228C22⋊C8—82no—
229C4×C8—42no—
230C4×C8—42no—
231C4×C8—81no—
232C4×C8—82no—
233#95—22no—
234#95—22no—
235#95—21no—
236#95—21no—
237#95—22no—
238#95—21no—
239#95—22no—
240#95—22no—
241#95—21no—
242#95—21no—
243#95—22no—
244#95—21no—
245#95—21no—
246#95C22⋊C412yes—
247#95C22⋊C412yes—
248#95C22⋊C412yes—
249C24C2×C22⋊C412yes—
250C2×Q8—21no—
251C2×Q8—24no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—81no—
255C2×Q8—81no—
256C2×Q8C22×C812yes—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4C22⋊C814yes—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—28no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C22×C814yes—
290C22×C4C2×M1614yes—
291C22×C4C23×C411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C2×C8—22no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—41no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—28no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—28no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—81no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—81no—
334C4⋊C4C22⋊C814yes—
335C4⋊C4C22×C812yes—
336C4⋊C4C22×C814yes—
337C22⋊C4—28no—
338C22⋊C4—24no—
339C22⋊C4—42no—
340C22⋊C4—44no—
341C22⋊C4C22⋊C818yes—
342C42—21no—
343C42—21no—
344C42—24no—
345C42—22no—
346C42—42no—
347C42—41no—
348C42—42no—
349C42—41no—
350C42—42no—
351C42—41no—
352C42—42no—
353C42—82no—
354C42—82no—
355C42—82no—
356C42C22×C812yes—
357C42C2×M1612yes—
358C42#13712yes—
359C42#13712yes—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23C2×C22⋊C814yes—
367C23C22×C22⋊C411yescharacteristic
368Q8—42no—
369Q8—42no—
370Q8—84no—
371D4—28no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—88no—
418C2×C4—164no—
419C2×C4C2×C22⋊C812yes—
420C2×C4C2×C22⋊C814yes—
421C2×C4C2×C22⋊C812yes—
422C2×C4#41912yes—
423C2×C4#41912yes—
424C2×C4#42112yes—
425C2×C4C23×C811yescharacteristic
426C2×C4C22×M1611yescharacteristic
427C2×C4C2×#13711yescharacteristic
428C2×C4C2×#13711yescharacteristic
429C2×C4#57611yescharacteristic
430C2×C4#57711yescharacteristic
431C8—42no—
432C8—81no—
433C8—82no—
434C8—81no—
435C8—81no—
436C8—82no—
437C22—24no—
438C22—24no—
439C22—24no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22C22×C22⋊C811yescharacteristic
445C22#288311yescharacteristic
446C22#288911yescharacteristic
447C22C2×#41911yescharacteristic
448C22#289711yescharacteristic
449C22C2×#42111yescharacteristic
450C22#290111yescharacteristic
451C4—22no—
452C4—22no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—24no—
458C4—21no—
459C4—21no—
460C4—41no—
461C4—42no—
462C4—44no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—81no—
467C4—82no—
468C4—81no—
469C4#147914yes—
470C2—44no—
471C2#1735111yescharacteristic
472C2C2×#147911yescharacteristic
473C2#1743811yescharacteristic
474C2#1759911yescharacteristic
475C2#1762711yescharacteristic
476C2#1768211yescharacteristic
477C2#1776111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7600C2semidirect
5. #7593C2semidirect
6. #7591C2semidirect
7. #7583C2semidirect
8. #7577C2semidirect
9. #7573C2semidirect
10. #7570C2semidirect
11. #7562C2semidirect
54. #554C8semidirect

Maximal subgroups

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

subgroupindex
#343322
#228352
#220082
#76002
#75932
#75912
#75832
#75772
#75732
#75702
#75622