Finite Group Databasebeta

← #223840

Subgroups — #223840

Order 512.

930 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36185C211yesmaximal, characteristic
2#21108C212yesmaximal, complemented
3#9895C212yesmaximal, complemented
4#9123C212yesmaximal, complemented
5#9120C212yesmaximal, complemented
6#8721C211yesmaximal, characteristic, complemented
7#8721C211yesmaximal, characteristic, complemented
8#8583C211yesmaximal, characteristic, complemented
9#8224C211yesmaximal, characteristic, complemented
10#8031C211yesmaximal, characteristic, complemented
11#8028C211yesmaximal, characteristic, complemented
12C2×#556C411yescharacteristic
13C2×#555C411yescharacteristic
14C2×#432C2212yes—
15C2×C4×M16C2212yes—
16#2862C2211yescharacteristic
17#2850C2211yescharacteristic
18#2850C2211yescharacteristic
19#2567C411yescharacteristic
20#2542C411yescharacteristic
21#2521—22no—
22#2521—22no—
23#2500C411yescharacteristic
24#2500C411yescharacteristic
25#2499C411yescharacteristic
26#2499C411yescharacteristic
27#2427—22no—
28#2362—22no—
29C4×#124C2211yescharacteristic
30#2285C2211yescharacteristic
31#2285C2211yescharacteristic
32C42×D4C2211yescharacteristic
33#1923—22no—
34#1916—22no—
35#1870—22no—
36#1851—22no—
37#1767C2212yes—
38#1749C2212yes—
39#1749C2212yes—
40#1745C2212yes—
41C2×#329C2212yes—
42C2×#329C2212yes—
43#1295C2211yescharacteristic
44#1295C2211yescharacteristic
45#1295C2211yescharacteristic
46#1295C2212yes—
47#1295C2212yes—
48#1295C2211yescharacteristic
49#1295C2212yes—
50#1295C2212yes—
51C22×M16—22no—
52#556—44no—
53#555—44no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#126—21no—
57C2×#126—21no—
58C2×#124—21no—
59C2×#124—21no—
60C2×#124—21no—
61C2×#124C2311yescharacteristic
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8C2×C411yescharacteristic
64C2×C22⋊Q8C2×C411yescharacteristic
65C2×#121—21no—
66C2×#121C2×C411yescharacteristic
67C2×#121C2×C411yescharacteristic
68C2×C4×Q8—21no—
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4D412yes—
73C2×#117—21no—
74C2×#117C2×C411yescharacteristic
75C2×#117C2×C411yescharacteristic
76C2×#117C2×C411yescharacteristic
77C2×#117C2×C411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#432—24no—
81#432—42no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C4×M16—24no—
88C4×M16—42no—
89C2×#97C2312yes—
90C2×#97C2312yes—
91C2×#97C2312yes—
92C2×#97C2312yes—
93C2×C4×C8—22no—
94#399—22no—
95#397—22no—
96#397—22no—
97#395—22no—
98#394—22no—
99#394—22no—
100#391—22no—
101#391—22no—
102#391C2×C411yescharacteristic
103#391C2×C411yescharacteristic
104#391C2×C411yescharacteristic
105#390C2×C411yescharacteristic
106#390C2×C411yescharacteristic
107#390C2×C411yescharacteristic
108#389—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#387—22no—
115#387—22no—
116#386—22no—
117#386—22no—
118#384C2×C411yescharacteristic
119#384C2×C411yescharacteristic
120#384C2×C411yescharacteristic
121#384C2×C411yescharacteristic
122#382C2×C411yescharacteristic
123#382C2×C411yescharacteristic
124#382C2311yescharacteristic
125#382C2311yescharacteristic
126#382C2311yescharacteristic
127#380—22no—
128#380—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4D412yes—
133C4×C22⋊C4—22no—
134C4×C22⋊C4C2×C411yescharacteristic
135C4×C22⋊C4C2×C411yescharacteristic
136C4×C22⋊C4D412yes—
137C4×C22⋊C4D412yes—
138C43C2311yescharacteristic
139#336—22no—
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144#329—24no—
145#329—42no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8C22×C411yescharacteristic
149C22×D4—21no—
150C22×D4—21no—
151C22×D4C22⋊C411yescharacteristic
152C22×D4C22×C411yescharacteristic
153C23×C4—21no—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C2×M16—44no—
159C2×M16—44no—
160C2×M16—44no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167#126—44no—
168#126—44no—
169#124—22no—
170#124—21no—
171#124—21no—
172#124—44no—
173#124—44no—
174#124—44no—
175C22⋊Q8—48no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178#121—44no—
179#121—42no—
180#121—42no—
181C4×Q8—42no—
182C4×Q8—44no—
183C4×D4—24no—
184C4×D4—42no—
185C4×D4—44no—
186#117—44no—
187#117—42no—
188#117—42no—
189#117—42no—
190#117—42no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4C22⋊C411yescharacteristic
222C2×C22⋊C4C22⋊C411yescharacteristic
223C2×C22⋊C4C22⋊C412yes—
224C2×C22⋊C4C22⋊C412yes—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42C22×C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2411yescharacteristic
245C4⋊C8—44no—
246C4⋊C8—44no—
247C22⋊C8—44no—
248C22⋊C8—44no—
249C22⋊C8—44no—
250#97—22no—
251#97—22no—
252#97—22no—
253#97—22no—
254#97—22no—
255#97—22no—
256#97—22no—
257#97#4114yes—
258C4×C8—44no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95C22⋊C412yes—
268#95C22⋊C412yes—
269#95C22⋊C412yes—
270C24—21no—
271C24C2×C22⋊C411yescharacteristic
272C24C2×C22⋊C411yescharacteristic
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—44no—
276C2×Q8—42no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—42no—
280C2×Q8—82no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—22no—
284C2×D4—42no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—82no—
290C2×D4—82no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4—42no—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C412yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D412yes—
325M16—416no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C4⋊C4—24no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—42no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—44no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4#10014yes—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—24no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—44no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—44no—
372C22⋊C4—82no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—42no—
383C42—41no—
384C42—42no—
385C42—42no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42—42no—
390C42C2×#4112yes—
391C42H(5,𝔽2)11yescharacteristic
392C422−1+411yescharacteristic
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—42no—
407C23—42no—
408C23C22×C22⋊C411yescharacteristic
409Q8—81no—
410Q8—81no—
411Q8—84no—
412Q8—162no—
413D4—44no—
414D4—81no—
415D4—81no—
416D4—162no—
417D4—162no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—84no—
461C2×C4C2×#10012yes—
462C2×C4C2×#10012yes—
463C2×C4C2×#10012yes—
464C2×C4#42114yes—
465C2×C4C4×#4111yescharacteristic
466C2×C4#52011yescharacteristic
467C2×C4#54612yes—
468C2×C4#54712yes—
469C8—44no—
470C8—44no—
471C8—82no—
472C8—82no—
473C22—42no—
474C22—42no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—42no—
480C22—42no—
481C22—41no—
482C22—41no—
483C22—82no—
484C22—81no—
485C22—81no—
486C22—82no—
487C22—81no—
488C22#233911yescharacteristic
489C22C22×#10011yescharacteristic
490C22#288911yescharacteristic
491C22C2×#42112yes—
492C22#290112yes—
493C4—21no—
494C4—21no—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—24no—
500C4—22no—
501C4—42no—
502C4—41no—
503C4—42no—
504C4—81no—
505C4—81no—
506C4—81no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4—81no—
511C2—81no—
512C2—81no—
513C2—161no—
514C2#1766311yescharacteristic
515C2#1768312yes—
516C2#1768811yescharacteristic
517C2#2247911yescharacteristic
518C2#2250612yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#555C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21108C2semidirect
3. #9895C2semidirect
4. #9123C2semidirect
5. #9120C2semidirect
6. #8721C2semidirect
7. #8721C2semidirect
8. #8583C2semidirect
9. #8224C2semidirect
10. #8031C2semidirect
11. #8028C2semidirect

Maximal subgroups

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

subgroupindex
#361852
#211082
#98952
#91232
#91202
#87212
#85832
#82242
#80312
#80282