Finite Group Databasebeta

← #381089

Subgroups — #381089

Order 512.

1617 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3C42×C22⋊C4C211yesmaximal, characteristic, complemented
4#11209C212yesmaximal, complemented
5#11201C212yesmaximal, complemented
6#11053C212yesmaximal, complemented
7#10971C212yesmaximal
8#10923C212yesmaximal, complemented
9#10878C212yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#423C2212yes—
13C22×#104C2214yes—
14C2×#407C2212yes—
15#2855C412yescomplemented
16#2855C412yescomplemented
17#2854C412yescomplemented
18#2854C412yescomplemented
19#2853—22no—
20#2849—22no—
21C4×#123—22no—
22C4×#123—22no—
23#2297C2211yescharacteristic
24#2297C2211yescharacteristic
25#2296C2211yescharacteristic
26#2296C2211yescharacteristic
27#2280C2211yescharacteristic
28C2×C4×C22⋊C4—22no—
29C2×C4×C22⋊C4C2212yes—
30C2×C43C2212yes—
31#1797—22no—
32#1797—22no—
33C2×#339C2214yes—
34C2×#339C2214yes—
35C2×#339C2214yes—
36#1728C2212yes—
37C4×#95—22no—
38C4×#95C2212yes—
39C22×#41C2311yescharacteristic
40C24×C4D412yes—
41C24×C4C2311yescharacteristic
42C22×M16C2314yes—
43#525—44no—
44C2×#123—22no—
45C2×#123—22no—
46C2×#123—21no—
47C2×C22⋊Q8C2×C412yes—
48C2×#121C2×C412yes—
49C2×#120—21no—
50C2×C4×D4—22no—
51C2×#117C2311yescharacteristic
52C2×#117C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C22×C22⋊C4—22no—
55C22×C42—24no—
56C22×C42—44no—
57C22×C42C2×C412yes—
58C22×C42C2×C412yes—
59C22×C42D412yes—
60C22×C42D412yes—
61C22×C42D412yes—
62C22×C42C2314yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65#423—44no—
66C2×C4⋊C8—24no—
67C2×#104—28no—
68C2×#104—44no—
69C2×#104—44no—
70#407—44no—
71C2×C22⋊C8—24no—
72#391C2×C411yescharacteristic
73#391C2×C411yescharacteristic
74#391C2×C412yes—
75#390C2×C411yescharacteristic
76#390C2×C411yescharacteristic
77#390C2×C412yes—
78#389C2×C412yes—
79#389C2×C412yes—
80#389C2×C412yes—
81#389C2×C412yes—
82#388—22no—
83#388—22no—
84#388—22no—
85#388—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#386—22no—
91#386—22no—
92#386—22no—
93#386—22no—
94#385—22no—
95#385—22no—
96#385—22no—
97#385—22no—
98#381—22no—
99#381—22no—
100#381—21no—
101#380C2×C412yes—
102#380C2×C412yes—
103#379—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—22no—
107C4×C22⋊C4—216no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—28no—
113C2×#95—22no—
114C2×#95—24no—
115C43—44no—
116#339—28no—
117#339—44no—
118#339—44no—
119#336—22no—
120#336—22no—
121C25C2×D411yescharacteristic
122C2×#41—24no—
123C2×#41—24no—
124C2×#41—22no—
125C2×#41—22no—
126C22×Q8C22×C411yescharacteristic
127C22×D4—22no—
128C22×D4—21no—
129C22×D4C22×C411yescharacteristic
130C23×C4—24no—
131C23×C4—28no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—44no—
138C23×C4—44no—
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22⋊C412yes—
142C23×C4C22⋊C412yes—
143C23×C4C22×C412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2411yescharacteristic
149C2×M16—24no—
150C2×M16—24no—
151C2×M16—24no—
152C2×M16—24no—
153C2×M16—24no—
154C2×M16—24no—
155C22×C8—24no—
156#123—48no—
157#123—44no—
158#123—48no—
159C22⋊Q8—44no—
160#121—44no—
161#120—44no—
162C4×D4—48no—
163#117—24no—
164#117—24no—
165#117—22no—
166#117—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C4⋊C4C22×C412yes—
176C2×C22⋊C4—216no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—216no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—24no—
200C2×C42—44no—
201C2×C42—44no—
202C2×C42—44no—
203C2×C42—44no—
204C2×C42—44no—
205C2×C42—48no—
206C2×C42—48no—
207C2×C42C22⋊C412yes—
208C2×C42C22⋊C412yes—
209C2×C42C22⋊C412yes—
210C2×C42C22⋊C412yes—
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D412yes—
217C2×C42#4118yes—
218C4⋊C8—48no—
219#104—416no—
220C22⋊C8—48no—
221#95—216no—
222#95—28no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233C24—22no—
234C24—24no—
235C24—22no—
236C24—22no—
237C24—21no—
238C24—42no—
239C24C2×C22⋊C412yes—
240C24C22×D411yescharacteristic
241#41—44no—
242#41—44no—
243#41—44no—
244#41—44no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—44no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—216no—
261C22×C4—22no—
262C22×C4—28no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—432no—
292C22×C4—48no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4#10418yescomplemented
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C23×C411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C2×#4114yes—
312C22×C4H(5,𝔽2)14yes—
313C22×C4H(5,𝔽2)12yes—
314C22×C42−1+412yes—
315M16—44no—
316M16—44no—
317M16—44no—
318M16—44no—
319C2×C8—44no—
320C2×C8—44no—
321C2×C8—44no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C22⋊C4—232no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C42—24no—
344C42—22no—
345C42—22no—
346C42—24no—
347C42—22no—
348C42—22no—
349C42—28no—
350C42—28no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—42no—
355C42—432no—
356C42—42no—
357C42—42no—
358C42—41no—
359C42—41no—
360C42—44no—
361C42—44no—
362C23—22no—
363C23—22no—
364C23—28no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—48no—
372C23—48no—
373C23—44no—
374C23—44no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—42no—
379C23—41no—
380C23C2×#10414yes—
381C23C22×C22⋊C411yescharacteristic
382C23#52512yes—
383C23#53412yes—
384C23#53712yes—
385Q8—44no—
386D4—44no—
387D4—88no—
388D4—84no—
389C2×C4—216no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—216no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—48no—
404C2×C4—48no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—48no—
408C2×C4—416no—
409C2×C4—416no—
410C2×C4—48no—
411C2×C4—48no—
412C2×C4—44no—
413C2×C4—48no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4C2×#10414yes—
432C2×C4C2×#10414yes—
433C2×C4#42114yes—
434C2×C4C4×#4112yes—
435C2×C4#51812yes—
436C2×C4#52012yes—
437C2×C4#53212yes—
438C2×C4#53412yes—
439C2×C4#53511yescharacteristic
440C2×C4#53611yescharacteristic
441C2×C4#53814yes—
442C2×C4#53914yes—
443C2×C4#54014yes—
444C2×C4#54214yes—
445C8—84no—
446C22—24no—
447C22—24no—
448C22—22no—
449C22—22no—
450C22—24no—
451C22—24no—
452C22—24no—
453C22—416no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—82no—
459C22—82no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22#229711yescharacteristic
464C22#230811yescharacteristic
465C22#231111yescharacteristic
466C22#263014yes—
467C22C22×#10412yes—
468C22C2×#42112yes—
469C4—24no—
470C4—24no—
471C4—24no—
472C4—416no—
473C4—42no—
474C4—42no—
475C4—82no—
476C4—82no—
477C4—81no—
478C4—81no—
479C4—81no—
480C4#233514yes—
481C4#266112yes—
482C4#266312yes—
483C2—24no—
484C2—44no—
485C2—161no—
486C2#1428711yescharacteristic
487C2#2141511yescharacteristic
488C2#2141611yescharacteristic
489C2#2177312yes—
490C2#2185612yes—
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
2. #20626C2semidirect
3. C42×C22⋊C4C2semidirect
4. #11209C2semidirect
5. #11201C2semidirect
6. #11053C2semidirect
8. #10923C2semidirect
9. #10878C2semidirect
15. #2855C4semidirect
16. #2855C4semidirect
17. #2854C4semidirect
18. #2854C4semidirect
303. C22×C4#104semidirect

Maximal subgroups

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

subgroupindex
#366262
#206262
C42×C22⋊C42
#112092
#112012
#110532
#109712
#109232
#108782