Finite Group Databasebeta

← #216128

Subgroups — #216128

Order 512.

922 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2C2×#1842C211yesmaximal, characteristic
3#20777C211yesmaximal, characteristic
4#9874C212yesmaximal, complemented
5#9643C212yesmaximal, complemented
6#8937C214yesmaximal
7#8568C214yesmaximal, complemented
8C22×C4⋊C8—21no—
9C22×C4⋊C8—21no—
10C22×C4⋊C8C2212yes—
11#2864—21no—
12#2863—21no—
13#2863C412yes—
14#2862C411yescharacteristic
15#2859C411yescharacteristic
16#2857C411yescharacteristic
17#2857C411yescharacteristic
18#2855—21no—
19#2855C412yes—
20#2851—21no—
21C4×C4⋊Q8C2211yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C4×#125—21no—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8C2212yes—
26#2295—21no—
27#2295C2212yes—
28#2283—21no—
29C2×C43C2211yescharacteristic
30#1842—21no—
31#1842—21no—
32#1842C2212yes—
33#1842C2212yes—
34#1839—21no—
35#1831—21no—
36#1817—21no—
37#1797—21no—
38C4×C4⋊C8C2212yes—
39C4×C22⋊C8C2212yes—
40#1744C2212yes—
41C2×Q8⋊C8C2218yes—
42#1279C2218yes—
43C23×C8—22no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8C2×C412yes—
49C2×C4×Q8—21no—
50C2×C4×Q8C2312yes—
51C22×C42C2×C411yescharacteristic
52C22×C42C2×C411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42C2311yescharacteristic
58C2×C4⋊C8—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—44no—
65C2×C4⋊C8—44no—
66C2×C4⋊C8—44no—
67C2×C4⋊C8C2314yes—
68C2×C22⋊C8—22no—
69C2×C4×C8C2314yes—
70#391—21no—
71#391—21no—
72#391—21no—
73#391—21no—
74#391C2×C412yes—
75#391C2×C412yes—
76#391C2×C412yes—
77#391C2×C412yes—
78#389—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389—21no—
86#389C2×C412yes—
87#389C2×C412yes—
88#389C2×C412yes—
89#389C2×C412yes—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#385—22no—
99#385—22no—
100#384C2×C411yescharacteristic
101#384C2×C411yescharacteristic
102#383—22no—
103#382—21no—
104#382—21no—
105#382—21no—
106#381—22no—
107#380—21no—
108#380—21no—
109#380—21no—
110#380C2×C412yes—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2312yes—
117C4×C22⋊C4—22no—
118#376—22no—
119C43C2×C411yescharacteristic
120C43C2×C411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125C43C2311yescharacteristic
126C43C2311yescharacteristic
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131Q8⋊C8—48no—
132C22×Q8—21no—
133C22×Q8C22×C412yes—
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C22×C8—24no—
142C22×C8—24no—
143C22×C8—44no—
144C22×C8—44no—
145C22×C8—44no—
146C4⋊Q8—42no—
147C4⋊Q8—42no—
148#125—44no—
149C22⋊Q8—28no—
150C22⋊Q8—44no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—42no—
154C4×Q8—42no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C42—24no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42—42no—
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42SD1614yes—
196C2×C42Q1614yes—
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2411yescharacteristic
213C4⋊C8—28no—
214C4⋊C8—44no—
215C4⋊C8—44no—
216C4⋊C8—44no—
217C4⋊C8—44no—
218C4⋊C8—44no—
219C4⋊C8—44no—
220C22⋊C8—44no—
221C4×C8—24no—
222C4×C8—24no—
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—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237C24C2×C22⋊C411yescharacteristic
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4#10314yes—
268C22×C4#10314yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4#13114yes—
281C22×C4C2×SD1612yes—
282C22×C4C2×Q1612yes—
283C22×C4C23×C411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C2×C8—44no—
287C2×C8—44no—
288C2×C8—44no—
289C2×C8—44no—
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—88no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C42—22no—
312C42—21no—
313C42—21no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—21no—
320C42—21no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42—42no—
336C42—42no—
337C42—42no—
338C42—42no—
339C42#10314yes—
340C42#10314yes—
341C42C2×SD1612yes—
342C42C2×SD1612yes—
343C42C2×Q1612yes—
344C42C2×Q1612yes—
345C42AGL(1,ℤ/8)12yes—
346C42#13712yes—
347C42H(5,𝔽2)12yes—
348C422−1+412yes—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23#40812yes—
354C23C2×#10312yes—
355C23#42112yes—
356C23C22×C22⋊C411yescharacteristic
357Q8—48no—
358Q8—84no—
359C2×C4—28no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—28no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—28no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4#40812yes—
408C2×C4C2×#10312yes—
409C2×C4C2×#10312yes—
410C2×C4C2×#10312yes—
411C2×C4C2×#10312yes—
412C2×C4C2×#10312yes—
413C2×C4#41712yes—
414C2×C4#41712yes—
415C2×C4#42112yes—
416C2×C4#52012yes—
417C2×C4#53511yescharacteristic
418C2×C4#53611yescharacteristic
419C2×C4#53712yes—
420C2×C4C2×#13112yes—
421C2×C4#56812yes—
422C2×C4C22×SD1611yescharacteristic
423C2×C4C22×Q1611yescharacteristic
424C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
425C2×C4C2×#13711yescharacteristic
426C8—84no—
427C8—84no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22#148114yes—
435C22#231111yescharacteristic
436C22C2×#40811yescharacteristic
437C22#287911yescharacteristic
438C22C22×#10311yescharacteristic
439C22C2×#41711yescharacteristic
440C22C2×#42111yescharacteristic
441C22C2×#42111yescharacteristic
442C4—22no—
443C4—22no—
444C4—22no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—22no—
450C4—22no—
451C4—22no—
452C4—41no—
453C4—41no—
454C4—82no—
455C4—82no—
456C4—82no—
457C4—82no—
458C4#299214yes—
459C4#320811yescharacteristic
460C4#320911yescharacteristic
461C4#322911yescharacteristic
462C4#323011yescharacteristic
463C2—24no—
464C2C2×#148111yescharacteristic
465C2#1746611yescharacteristic
466C2#1748611yescharacteristic
467C2#1749211yescharacteristic
468C2#2185211yescharacteristic
469C2#2185711yescharacteristic
470C2#2186011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9874C2semidirect
5. #9643C2semidirect
7. #8568C2semidirect

Maximal subgroups

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

subgroupindex
#366352
C2×#18422
#207772
#98742
#96432
#89372
#85682