Finite Group Databasebeta

← #374146

Subgroups — #374146

Order 512.

2105 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic, complemented
2#36626C211yesmaximal, characteristic
3C2×#1896C212yesmaximal
4C2×C4×#95C211yesmaximal, characteristic, complemented
5#11209C212yesmaximal, complemented
6#11053C212yesmaximal, complemented
7#10933C212yesmaximal, complemented
8#10886C212yesmaximal, complemented
9#10719C212yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C23×C42C2212yes—
12C23×C42C2211yescharacteristic
13#2920—28no—
14C2×#432—22no—
15C2×#431—22no—
16C2×#423C2212yes—
17C22×#104C2214yes—
18C2×#407C2212yes—
19C2×C4×M16C2212yes—
20#2855C412yescomplemented
21#2855C412yescomplemented
22#2854C412yescomplemented
23#2854C412yescomplemented
24#2853—22no—
25#2849—22no—
26C4×#123—22no—
27C4×#123—22no—
28#2297C2211yescharacteristic
29#2297C2211yescharacteristic
30#2296C2211yescharacteristic
31#2296C2211yescharacteristic
32#2280C2211yescharacteristic
33C22×#95—22no—
34#1896—28no—
35C2×#339C2214yes—
36C2×#339C2214yes—
37C2×#339C2214yes—
38C4×#95—22no—
39C4×#95C2212yes—
40C4×#95C2212yes—
41C22×#41C2311yescharacteristic
42C24×C4—22no—
43C24×C4D412yes—
44C24×C4C2311yescharacteristic
45C22×M16C2314yes—
46#525—24no—
47#525—24no—
48C2×#123—22no—
49C2×#123—22no—
50C2×#123—21no—
51C2×C22⋊Q8C2×C412yes—
52C2×#121C2×C412yes—
53C2×#120—21no—
54C2×C4×D4—22no—
55C2×#117C2311yescharacteristic
56C2×#117C2311yescharacteristic
57C2×#117C2311yescharacteristic
58C22×C42—24no—
59C22×C42—24no—
60C22×C42—22no—
61C22×C42—28no—
62C22×C42—24no—
63C22×C42—24no—
64C22×C42—44no—
65C22×C42C2×C412yes—
66C22×C42C2×C412yes—
67C22×C42D412yes—
68C22×C42D412yes—
69C22×C42D412yes—
70C22×C42C2314yes—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73#432—216no—
74#431—216no—
75#423—28no—
76C2×C4⋊C8—24no—
77C2×#104—28no—
78C2×#104—28no—
79C2×#104—44no—
80#407—24no—
81#407D418yes—
82C2×C22⋊C8—24no—
83C4×M16—28no—
84C2×#97—22no—
85C2×C4×C8—22no—
86#391C2×C411yescharacteristic, complemented
87#391C2×C411yescharacteristic, complemented
88#391C2×C412yescomplemented
89#390C2×C411yescharacteristic, complemented
90#390C2×C411yescharacteristic, complemented
91#390C2×C412yescomplemented
92#389C2×C412yescomplemented
93#389C2×C412yescomplemented
94#389C2×C412yescomplemented
95#389C2×C412yescomplemented
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#386—22no—
105#386—22no—
106#386—22no—
107#386—22no—
108#385—22no—
109#385—22no—
110#385—22no—
111#385—22no—
112#381—22no—
113#381—22no—
114#381—21no—
115#380C2×C412yes—
116#380C2×C412yes—
117#379—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—22no—
125C2×#95—216no—
126C2×#95—24no—
127C2×#95—24no—
128#339—28no—
129#339—28no—
130#339—44no—
131C25C2×D411yescharacteristic
132C2×#41—22no—
133C2×#41—22no—
134C2×#41—24no—
135C2×#41—22no—
136C2×#41—22no—
137C22×Q8C22×C411yescharacteristic
138C22×D4—22no—
139C22×D4—21no—
140C22×D4C22×C411yescharacteristic
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—216no—
145C23×C4—28no—
146C23×C4—28no—
147C23×C4—24no—
148C23×C4—28no—
149C23×C4—21no—
150C23×C4—44no—
151C23×C4—48no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4C22⋊C412yes—
155C23×C4C22⋊C412yes—
156C23×C4C22⋊C412yes—
157C23×C4C22⋊C412yes—
158C23×C4C22×C412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D414yes—
164C23×C4#4114yes—
165C23×C4C2411yescharacteristic
166C2×M16—24no—
167C2×M16—24no—
168C2×M16—24no—
169C2×M16—24no—
170C2×M16—24no—
171C2×M16C2×D418yes—
172C22×C8—24no—
173#123—48no—
174#123—48no—
175#123—44no—
176C22⋊Q8—28no—
177#121—28no—
178#120—44no—
179C4×D4—44no—
180C4×D4—44no—
181#117—22no—
182#117—22no—
183#117—24no—
184#117—22no—
185#117—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C412yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C42—28no—
207C2×C42—28no—
208C2×C42—24no—
209C2×C42—24no—
210C2×C42—28no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—28no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—24no—
226C2×C42—416no—
227C2×C42—416no—
228C2×C42—44no—
229C2×C42—44no—
230C2×C42C22⋊C412yes—
231C2×C42C22⋊C412yes—
232C2×C42C22⋊C412yes—
233C2×C42C22⋊C412yes—
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C412yes—
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42#4118yes—
241C4⋊C8—216no—
242#104—416no—
243C22⋊C8—216no—
244#97—28no—
245C4×C8—28no—
246#95—216no—
247#95—216no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258C24—22no—
259C24—21no—
260C24—21no—
261C24—24no—
262C24—24no—
263C24—21no—
264C24C2×C22⋊C412yes—
265C24#12114yes—
266C24C22×D411yescharacteristic
267#41—44no—
268#41—44no—
269#41—44no—
270#41—44no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—44no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—28no—
294C22×C4—28no—
295C22×C4—28no—
296C22×C4—28no—
297C22×C4—216no—
298C22×C4—232no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—28no—
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—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—432no—
321C22×C4—416no—
322C22×C4—432no—
323C22×C4—416no—
324C22×C4—432no—
325C22×C4—48no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4#10418yescomplemented
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C412yes—
338C22×C4C4×D418yes—
339C22×C4#12014yes—
340C22×C4#12114yes—
341C22×C4#12314yes—
342C22×C4C23×C411yescharacteristic
343C22×C4C22×D412yes—
344C22×C4C22×D411yescharacteristic
345C22×C4C2×#4112yes—
346C22×C4C2×#4114yes—
347C22×C4H(5,𝔽2)12yes—
348C22×C42−1+412yes—
349M16—28no—
350M16—28no—
351M16—44no—
352M16—44no—
353C2×C8—28no—
354C2×C8—44no—
355C2×C8—44no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C42—22no—
378C42—24no—
379C42—22no—
380C42—21no—
381C42—21no—
382C42—22no—
383C42—21no—
384C42—22no—
385C42—21no—
386C42—24no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—28no—
391C42—28no—
392C42—416no—
393C42—41no—
394C42—41no—
395C42—42no—
396C42—42no—
397C42—44no—
398C42—44no—
399C23—28no—
400C23—28no—
401C23—28no—
402C23—28no—
403C23—22no—
404C23—216no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—24no—
410C23—24no—
411C23—22no—
412C23—41no—
413C23—42no—
414C23—42no—
415C23—41no—
416C23—41no—
417C23#38614yes—
418C23C2×#10414yes—
419C23#42114yes—
420C23C22×C22⋊C411yescharacteristic
421C23C2×#12112yes—
422C23#52512yes—
423C23#53511yescharacteristic
424C23#53611yescharacteristic
425Q8—44no—
426D4—44no—
427D4—88no—
428D4—82no—
429D4—82no—
430C2×C4—22no—
431C2×C4—216no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—216no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—216no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—48no—
450C2×C4—48no—
451C2×C4—48no—
452C2×C4—48no—
453C2×C4—464no—
454C2×C4—416no—
455C2×C4—416no—
456C2×C4—48no—
457C2×C4—48no—
458C2×C4—48no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4#38614yes—
475C2×C4C2×#10414yes—
476C2×C4C2×#10414yes—
477C2×C4C2×C4×D412yes—
478C2×C4C4×#4112yes—
479C2×C4#52012yes—
480C2×C4C2×#12011yescharacteristic
481C2×C4C2×#12311yescharacteristic
482C2×C4#53212yes—
483C2×C4#53712yes—
484C2×C4#54614yes—
485C2×C4#54714yes—
486C2×C4#55114yes—
487C2×C4#55214yes—
488C8—48no—
489C22—216no—
490C22—216no—
491C22—24no—
492C22—22no—
493C22—22no—
494C22—24no—
495C22—24no—
496C22—24no—
497C22—24no—
498C22—22no—
499C22—24no—
500C22—22no—
501C22—24no—
502C22—81no—
503C22—82no—
504C22—82no—
505C22—81no—
506C22—81no—
507C22#187118yes—
508C22C2×#38611yescharacteristic
509C22#229711yescharacteristic
510C22#231111yescharacteristic
511C22#241112yes—
512C22#241312yes—
513C22C22×#10412yes—
514C22C2×#42112yes—
515C4—24no—
516C4—24no—
517C4—24no—
518C4—416no—
519C4—42no—
520C4—42no—
521C4—81no—
522C4—82no—
523C4—82no—
524C4—81no—
525C4—81no—
526C4#235914yes—
527C4#243012yes—
528C4#249812yes—
529C2—22no—
530C2—28no—
531C2—22no—
532C2—161no—
533C2#1413511yescharacteristic
534C2C2×#187112yes—
535C2#2141511yescharacteristic
536C2#2141711yescharacteristic
537C2#2189912yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2920C2semidirect
4. C2×C4×#95C2semidirect
5. #11209C2semidirect
6. #11053C2semidirect
7. #10933C2semidirect
8. #10886C2semidirect
9. #10719C2semidirect
20. #2855C4semidirect
21. #2855C4semidirect
22. #2854C4semidirect
23. #2854C4semidirect
86. #391C2×C4semidirect
87. #391C2×C4semidirect
88. #391C2×C4semidirect
89. #390C2×C4semidirect
90. #390C2×C4semidirect
91. #390C2×C4semidirect
92. #389C2×C4semidirect
93. #389C2×C4semidirect
94. #389C2×C4semidirect
95. #389C2×C4semidirect
332. C22×C4#104semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
#366262
C2×#18962
C2×C4×#952
#112092
#110532
#109332
#108862
#107192