Finite Group Databasebeta

← #222995

Subgroups — #222995

Order 512.

940 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34945C211yesmaximal, characteristic
2D4×C4⋊C8C211yesmaximal, characteristic
3C4×#436C211yesmaximal, characteristic
4#9870C211yesmaximal, characteristic, complemented
5#9865C211yesmaximal, characteristic, complemented
6#9077C212yesmaximal, complemented
7#8937C212yesmaximal, complemented
8#8711C212yesmaximal, complemented
9#8564C211yesmaximal, characteristic, complemented
10C4×Q8⋊C8C211yesmaximal, characteristic, complemented
11#7895C212yesmaximal, complemented
12C2×#436—21no—
13C2×C8×D4—21no—
14C22×C4⋊C8C2212yes—
15C2×C4×M16C2212yes—
16#2573—22no—
17#2571C411yescharacteristic
18#2568C411yescharacteristic
19#2547C411yescharacteristic
20#2538C411yescharacteristic
21#2514—21no—
22#2494C412yes—
23#2447—21no—
24#2421C412yes—
25#2401—22no—
26C4×C4⋊Q8C2211yescharacteristic
27C4×#124—21no—
28C4×C22⋊Q8C2212yes—
29#2295—21no—
30#2283C2212yes—
31C42×Q8C2211yescharacteristic
32C42×D4C2211yescharacteristic
33#1939—21no—
34#1850—22no—
35#1842C2211yescharacteristic
36#1797C2212yes—
37#1770—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40#1753—22no—
41C4×C22⋊C8C2212yes—
42C42×C8C2211yescharacteristic
43C2×Q8⋊C8C2211yescharacteristic
44C2×Q8⋊C8C2211yescharacteristic
45C2×Q8⋊C8C2212yes—
46#1279C2211yescharacteristic
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2211yescharacteristic
53C22×M16—22no—
54C23×C8—22no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#124—21no—
57C2×C22⋊Q8C2×C412yes—
58C2×C4×Q8—21no—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C22×C42C2312yes—
66#436—28no—
67C8×D4—44no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—42no—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2311yescharacteristic
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C4×M16—24no—
80C4×M16—42no—
81C2×#97—22no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8C2312yes—
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88#401—22no—
89#400—22no—
90#400—22no—
91#400—22no—
92#397—22no—
93#397—22no—
94#397—22no—
95#396—22no—
96#391—21no—
97#391—21no—
98#391—21no—
99#391—21no—
100#391C2×C412yes—
101#390—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389C2×C412yes—
106#389C2×C412yes—
107#388—22no—
108#388—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#385—22no—
113#384C2×C411yescharacteristic
114#383—22no—
115#382—21no—
116#382C2×C412yes—
117#382C2×C412yes—
118#381—22no—
119#380C2×C412yes—
120#380C2×C412yes—
121#380C2×C412yes—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4C2312yes—
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4C2×C412yes—
137C4×C22⋊C4D412yes—
138C4×C22⋊C4D412yes—
139#376—22no—
140C43—21no—
141C43C2311yescharacteristic
142#336—22no—
143#336—21no—
144#336—22no—
145#336—21no—
146Q8⋊C8—22no—
147Q8⋊C8—22no—
148Q8⋊C8—42no—
149C22×Q8—21no—
150C22×Q8C22×C411yescharacteristic
151C22×D4C22⋊C411yescharacteristic
152C23×C4C22×C412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C2×M16—28no—
156C2×M16—44no—
157C2×M16—44no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—44no—
165C22×C8—44no—
166C22×C8—82no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169#124—44no—
170C22⋊Q8—24no—
171C22⋊Q8—42no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—41no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×D4—24no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C22⋊C412yes—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C412yes—
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2411yescharacteristic
232C4⋊C8—24no—
233C4⋊C8—21no—
234C4⋊C8—22no—
235C4⋊C8—21no—
236C4⋊C8—22no—
237C4⋊C8—21no—
238C4⋊C8—44no—
239C4⋊C8—42no—
240C4⋊C8—81no—
241C4⋊C8#4112yes—
242C22⋊C8—28no—
243C22⋊C8—44no—
244#97—44no—
245C4×C8—24no—
246C4×C8—21no—
247C4×C8—22no—
248C4×C8—21no—
249C4×C8—22no—
250C4×C8—21no—
251C4×C8—42no—
252C4×C8—42no—
253C4×C8#4112yes—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—21no—
263#95—21no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—21no—
268#95—22no—
269#95C22⋊C412yes—
270#95C22⋊C412yes—
271#95C22⋊C412yes—
272C24C2×C22⋊C412yes—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8#13112yes—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4#10414yes—
285C22×C4—21no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—42no—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C23×C411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317M16—416no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—24no—
323C2×C8—22no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—41no—
336C2×C8—88no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—22no—
350C4⋊C4—42no—
351C4⋊C4—44no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—44no—
358C4⋊C4—81no—
359C4⋊C4—82no—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—44no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—82no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—22no—
378C42—41no—
379C42—42no—
380C42—41no—
381C42—41no—
382C42—42no—
383C42—42no—
384C42—41no—
385C42—41no—
386C42—42no—
387C42—42no—
388C42—81no—
389C42#13112yes—
390C42#13112yes—
391C42AGL(1,ℤ/8)12yes—
392C42#13712yes—
393C42C2×#4111yescharacteristic
394C42C2×#4111yescharacteristic
395C42H(5,𝔽2)11yescharacteristic
396C422−1+411yescharacteristic
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23C2×#10414yes—
404C23C22×C22⋊C411yescharacteristic
405Q8—22no—
406Q8—22no—
407Q8—41no—
408Q8—41no—
409Q8—84no—
410D4—28no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4#40812yes—
458C2×C4#40812yes—
459C2×C4#41712yes—
460C2×C4#41712yes—
461C2×C4C2×#10412yes—
462C2×C4C4×#4111yescharacteristic
463C2×C4#52011yescharacteristic
464C2×C4#54611yescharacteristic
465C2×C4#54611yescharacteristic
466C2×C4#54711yescharacteristic
467C2×C4#54711yescharacteristic
468C2×C4C2×#13111yescharacteristic
469C2×C4C2×#13111yescharacteristic
470C2×C4C2×#13111yescharacteristic
471C2×C4#56811yescharacteristic
472C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
473C2×C4C2×#13711yescharacteristic
474C2×C4#57711yescharacteristic
475C2×C4#57811yescharacteristic
476C8—24no—
477C8—42no—
478C8—44no—
479C8—42no—
480C8—81no—
481C8—82no—
482C22—24no—
483C22—24no—
484C22—24no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22#233911yescharacteristic
490C22C2×#40811yescharacteristic
491C22#288011yescharacteristic
492C22C2×#41711yescharacteristic
493C22#289811yescharacteristic
494C22C22×#10411yescharacteristic
495C22#290111yescharacteristic
496C4—22no—
497C4—22no—
498C4—21no—
499C4—21no—
500C4—41no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—82no—
506C4—81no—
507C4—82no—
508C4—81no—
509C4—81no—
510C4#148314yes—
511C4#296611yescharacteristic
512C4#296911yescharacteristic
513C4#299311yescharacteristic
514C4#299411yescharacteristic
515C4#332411yescharacteristic
516C4#332811yescharacteristic
517C4#335311yescharacteristic
518C4#335611yescharacteristic
519C2—44no—
520C2C2×#148311yescharacteristic
521C2#1746911yescharacteristic
522C2#1751511yescharacteristic
523C2#1751911yescharacteristic
524C2#2247411yescharacteristic
525C2#2250411yescharacteristic
526C2#2250511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9870C2semidirect
5. #9865C2semidirect
6. #9077C2semidirect
7. #8937C2semidirect
8. #8711C2semidirect
9. #8564C2semidirect
10. C4×Q8⋊C8C2semidirect
11. #7895C2semidirect

Maximal subgroups

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

subgroupindex
#349452
D4×C4⋊C82
C4×#4362
#98702
#98652
#90772
#89372
#87112
#85642
C4×Q8⋊C82
#78952