Finite Group Databasebeta

← #195039

Subgroups — #195039

Order 512.

1506 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2905C211yesmaximal, characteristic, complemented
2#34633C211yesmaximal, characteristic
3#17485C212yesmaximal, complemented
4C2×#1458C211yesmaximal, characteristic, complemented
5#9035C218yesmaximal, complemented
6#8095C212yesmaximal, complemented
7C23×M16—21no—
8C2×#532C2211yescharacteristic
9C2×#525—21no—
10C22×#117C2211yescharacteristic
11C2×#431C2212yes—
12#2905—24no—
13#2905—22no—
14#2905C2214yes—
15C2×#424—24no—
16C2×#423—21no—
17C2×#423C2212yes—
18#2621C412yes—
19#2620—22no—
20#2619C412yes—
21#2618—21no—
22#2618C412yes—
23#2617—21no—
24#2617C412yes—
25#2392—22no—
26#2392C2214yes—
27#2297—21no—
28#2285C2211yescharacteristic
29#1492—24no—
30#1486C2218yescomplemented
31#1480—24no—
32#1458—22no—
33#1458C2214yes—
34C2×#324—21no—
35C2×#323—21no—
36#1303C2218yescomplemented
37C22×#41—21no—
38C24×C4D411yescharacteristic
39C22×M16—22no—
40C22×M16—24no—
41C22×M16—22no—
42C22×M16—24no—
43C22×M16—44no—
44C22×M16—44no—
45C23×C8—22no—
46#532—22no—
47#532—22no—
48#532—22no—
49#532—22no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#127C2×C411yescharacteristic
52C2×#124—21no—
53#525—44no—
54#525—44no—
55C2×#123—22no—
56C2×C22⋊Q8—22no—
57C2×#121—22no—
58C2×#121—21no—
59C2×#121C2×C412yes—
60C2×#120—22no—
61C2×C4×D4—24no—
62C2×C4×D4—21no—
63C2×C4×D4C2312yes—
64C2×#117—21no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117C2×C412yes—
70C2×#117D412yes—
71C2×#117D412yes—
72C2×#117C2312yes—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C22⋊C4C2×C411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42C2311yescharacteristic
79#431—22no—
80#431—22no—
81#431—22no—
82#431C2314yes—
83#424—216no—
84#424—216no—
85#423—24no—
86#423—24no—
87#423—22no—
88#423—22no—
89#423D414yescomplemented
90#423C2314yes—
91C2×C4⋊C8—24no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8D414yescomplemented
95C2×C22⋊C8—22no—
96C2×C4×C8—22no—
97#398—22no—
98#398C2×C414yes—
99#397—22no—
100#397C2×C414yes—
101#396—24no—
102#396—24no—
103#394—22no—
104#394—24no—
105#394—24no—
106#394C2×C414yes—
107#392—22no—
108#392C2×C414yes—
109#391—22no—
110#390—22no—
111#386—22no—
112#386—22no—
113#384—21no—
114#382C2×C412yes—
115#380—21no—
116#380C2×C412yes—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2312yes—
119C4×C22⋊C4—24no—
120D4⋊C8—216no—
121#324—24no—
122#323—24no—
123C25C22⋊C411yescharacteristic
124C2×#41—41no—
125C2×#41—41no—
126C2×#41—41no—
127C2×#41—41no—
128C2×#41—42no—
129C2×#41—42no—
130C2×#41—42no—
131C2×#41—42no—
132C22×Q8—21no—
133C22×D4—22no—
134C22×D4—24no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4C22×C412yes—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C412yes—
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4D814yes—
150C23×C4SD1614yes—
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C2×M16—28no—
156C2×M16—24no—
157C2×M16—24no—
158C2×M16—48no—
159C2×M16—416no—
160C2×M16—48no—
161C2×M16—48no—
162C2×M16—44no—
163C2×M16—44no—
164C22×C8—28no—
165C22×C8—24no—
166C22×C8—48no—
167C22×C8—42no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170#127—22no—
171#127—22no—
172#124—42no—
173#124—42no—
174#123—48no—
175C22⋊Q8—48no—
176#121—24no—
177#121—24no—
178#121—48no—
179#121—42no—
180#121—42no—
181#120—48no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—48no—
185C4×D4—48no—
186C4×D4—42no—
187C4×D4—42no—
188#117—24no—
189#117—24no—
190#117—24no—
191#117—24no—
192#117—24no—
193#117—24no—
194#117—24no—
195#117—24no—
196#117—42no—
197#117—42no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4C22⋊C412yes—
224C2×C22⋊C4C22⋊C412yes—
225C2×C22⋊C4C22×C412yes—
226C2×C22⋊C4C2×D412yes—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42C22⋊C412yes—
233C2×C42C22×C411yescharacteristic
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42#4112yes—
240C2×C42#4112yes—
241C2×C42C2411yescharacteristic
242C4⋊C8—216no—
243C4⋊C8—24no—
244C4⋊C8—24no—
245C4⋊C8—24no—
246C4⋊C8—24no—
247C4⋊C8—42no—
248C4⋊C8C2×D418yes—
249C22⋊C8—24no—
250C22⋊C8—42no—
251C4×C8—24no—
252C4×C8—42no—
253#95—24no—
254#95—24no—
255#95—22no—
256#95C22×C414yes—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—22no—
261C24—81no—
262C24—81no—
263C24#10214yes—
264C24C2×C22⋊C412yes—
265C24C2×C22⋊C411yescharacteristic
266#41—82no—
267#41—82no—
268#41—82no—
269#41—82no—
270#41—82no—
271#41—82no—
272#41—82no—
273#41—82no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×D4—24no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—48no—
282C2×D4—44no—
283C2×D4—44no—
284C2×D4—42no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—82no—
290C2×D4—82no—
291C2×D4—84no—
292C2×D4—84no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4—81no—
326C22×C4—81no—
327C22×C4#10218yes—
328C22×C4#10214yes—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C4×D412yes—
336C22×C4C4×D412yes—
337C22×C4#12011yescharacteristic
338C22×C4#12011yescharacteristic
339C22×C4#12112yes—
340C22×C4#12112yes—
341C22×C4#12311yescharacteristic
342C22×C4#12311yescharacteristic
343C22×C4C2×D814yes—
344C22×C4C2×D812yes—
345C22×C4C2×SD1614yes—
346C22×C4C2×SD1612yes—
347C22×C4AGL(1,ℤ/8)12yes—
348C22×C4#13712yes—
349C22×C4C23×C411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352M16—416no—
353M16—44no—
354M16—44no—
355M16—48no—
356C2×C8—28no—
357C2×C8—416no—
358C2×C8—48no—
359C2×C8—44no—
360C2×C8—84no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4—48no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—44no—
376C22⋊C4—48no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—48no—
380C22⋊C4—84no—
381C22⋊C4—84no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—42no—
387C42—42no—
388C42C4×D412yes—
389C42C4×D412yes—
390C42#12012yes—
391C42#12112yes—
392C42#12112yes—
393C42#12312yes—
394C42C22×D412yes—
395C42C2×#4112yes—
396C23—24no—
397C23—22no—
398C23—22no—
399C23—24no—
400C23—41no—
401C23—41no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—44no—
407C23—44no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—81no—
412C23—81no—
413C23—84no—
414C23—84no—
415C23#38611yescharacteristic
416C23#38611yescharacteristic
417C23C2×#10214yes—
418C23C2×#10212yes—
419C23#41712yes—
420C23C22×C22⋊C411yescharacteristic
421Q8—82no—
422Q8—82no—
423D4—48no—
424D4—84no—
425D4—816no—
426D4—84no—
427D4—84no—
428D4—82no—
429D4—82no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—28no—
437C2×C4—28no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
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—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4#38612yes—
467C2×C4#38612yes—
468C2×C4#38611yescharacteristic
469C2×C4#38611yescharacteristic
470C2×C4C2×#10214yes—
471C2×C4C2×#10212yes—
472C2×C4#41712yes—
473C2×C4#44712yes—
474C2×C4#44712yes—
475C2×C4#44812yes—
476C2×C4#44812yes—
477C2×C4#48012yes—
478C2×C4#48012yes—
479C2×C4#48112yes—
480C2×C4#48112yes—
481C2×C4C2×C4×D412yes—
482C2×C4C2×#12011yescharacteristic
483C2×C4C2×#12112yes—
484C2×C4C2×#12311yescharacteristic
485C2×C4C22×D811yescharacteristic
486C2×C4C22×SD1611yescharacteristic
487C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
488C2×C4C2×#13711yescharacteristic
489C8—48no—
490C8—84no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—24no—
498C22—42no—
499C22—42no—
500C22—44no—
501C22—48no—
502C22—42no—
503C22—42no—
504C22—82no—
505C22—84no—
506C22—84no—
507C22—84no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22#187412yes—
512C22#187412yes—
513C22C2×#38611yescharacteristic
514C22C22×#10211yescharacteristic
515C22C2×#41711yescharacteristic
516C4—24no—
517C4—22no—
518C4—22no—
519C4—22no—
520C4—22no—
521C4—42no—
522C4—42no—
523C4—84no—
524C4—82no—
525C4#187412yes—
526C4#187412yes—
527C4C2×#44711yescharacteristic
528C4C2×#44811yescharacteristic
529C4C2×#48011yescharacteristic
530C4C2×#48111yescharacteristic
531C2—21no—
532C2—22no—
533C2—21no—
534C2—22no—
535C2—44no—
536C2—162no—
537C2C2×#187411yescharacteristic
538C2#2094711yescharacteristic
539C2#2096411yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radical#34633C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2905C2semidirect
3. #17485C2semidirect
4. C2×#1458C2semidirect
5. #9035C2semidirect
6. #8095C2semidirect
30. #1486C22semidirect
36. #1303C22semidirect
89. #423D4semidirect
94. C2×C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29052
#346332
#174852
C2×#14582
#90352
#80952