Finite Group Databasebeta

← #190406

Subgroups — #190406

Order 512.

1372 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic
3C2×#1797C211yesmaximal, characteristic
4#8709C214yesmaximal, complemented
5#8706C214yesmaximal, complemented
6#7893C214yesmaximal, complemented
7C22×#117—21no—
8C23×C42C2211yescharacteristic
9C2×#431C2212yes—
10C22×C4⋊C8C2212yes—
11C22×C22⋊C8—21no—
12#2285—21no—
13#2285—21no—
14#2285—21no—
15#2285—21no—
16#2285C412yes—
17#2285C412yes—
18#2285C2212yes—
19#2285C2212yes—
20C2×#384C411yescharacteristic
21C2×#384C411yescharacteristic
22C2×#382—21no—
23C2×#380—21no—
24C2×#380C412yes—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2212yes—
27#1817—21no—
28#1797—21no—
29#1797—21no—
30#1797—21no—
31#1797—21no—
32#1797C2212yes—
33#1797C2212yes—
34#1779—21no—
35C4×C22⋊C8C2212yes—
36C4×C22⋊C8C2212yes—
37C2×#336—21no—
38#1279C22116yescomplemented
39C24×C4C2×C411yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4D411yescharacteristic
42C23×C8—22no—
43C2×#117—22no—
44C2×#117—22no—
45C2×#117—22no—
46C2×#117—22no—
47C2×#117—44no—
48C2×#117—44no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4C2×C412yes—
55C22×C22⋊C4—22no—
56C22×C42—28no—
57C22×C42—22no—
58C22×C42C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70#431—28no—
71C2×C4⋊C8—28no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8C2314yes—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—44no—
81C2×C4×C8C2314yes—
82#384—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384C2×C412yes—
87#384C2×C412yes—
88#384C2×C412yes—
89#384C2×C412yes—
90#382—22no—
91#382—22no—
92#382—22no—
93#382—22no—
94#380—22no—
95#380—22no—
96#380—22no—
97#380—22no—
98#380—22no—
99#380—22no—
100#380C2×C414yes—
101#380C2×C414yes—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4C2×C414yes—
109C4×C4⋊C4C2314yes—
110C2×#95—22no—
111C2×#95—22no—
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116C25C22⋊C411yescharacteristic
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—24no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4D814yes—
137C23×C4SD1614yes—
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C22×C8—24no—
148C22×C8—24no—
149C22×C8—44no—
150C22×C8—44no—
151C22×C8—44no—
152#117—48no—
153#117—48no—
154#117—44no—
155#117—44no—
156#117—44no—
157#117—44no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4C22×C414yes—
175C2×C4⋊C4C22×C414yes—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—48no—
179C2×C42—24no—
180C2×C42—24no—
181C2×C42—24no—
182C2×C42—24no—
183C2×C42—24no—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—44no—
193C2×C42—44no—
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C411yescharacteristic
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42#4118yes—
210C2×C42C2411yescharacteristic
211C4⋊C8—28no—
212C4⋊C8—44no—
213C4⋊C8—44no—
214C22⋊C8—44no—
215C22⋊C8—44no—
216C22⋊C8—44no—
217C22⋊C8—44no—
218C22⋊C8—44no—
219C4×C8—28no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224C24—22no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24—41no—
229C24—41no—
230C24#10214yes—
231C24C2×C22⋊C411yescharacteristic
232C24C2×C22⋊C411yescharacteristic
233C24C2×C22⋊C411yescharacteristic
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—216no—
252C22×C4—216no—
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—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—416no—
282C22×C4—416no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4#10214yes—
290C22×C4#10214yes—
291C22×C4#10214yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4#11714yes—
302C22×C4#12314yes—
303C22×C4#12314yes—
304C22×C4#13114yes—
305C22×C4C2×D812yes—
306C22×C4C2×D812yes—
307C22×C4C2×D812yes—
308C22×C4C2×SD1612yes—
309C22×C4C2×SD1612yes—
310C22×C4C2×SD1612yes—
311C22×C4#13714yes—
312C22×C4C23×C411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—44no—
318C2×C8—44no—
319C2×C8—44no—
320C2×C8—44no—
321C2×C8—88no—
322C4⋊C4—28no—
323C4⋊C4—28no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C22⋊C4—48no—
335C22⋊C4—48no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—44no—
349C42—44no—
350C42—48no—
351C42#11714yes—
352C42#12314yes—
353C42#12314yes—
354C42C2×#4114yes—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—28no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—28no—
370C23—21no—
371C23—44no—
372C23—44no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23#38112yes—
377C23#40812yes—
378C23C2×#10212yes—
379C23C2×#10212yes—
380C23C2×#10212yes—
381C23#41912yes—
382C23#42112yes—
383C23C22×C22⋊C411yescharacteristic
384C2×C4—28no—
385C2×C4—28no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—28no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—88no—
430C2×C4—88no—
431C2×C4#38112yes—
432C2×C4#38112yes—
433C2×C4#38112yes—
434C2×C4#40812yes—
435C2×C4C2×#10212yes—
436C2×C4C2×#10212yes—
437C2×C4C2×#10212yes—
438C2×C4#41912yes—
439C2×C4#42112yes—
440C2×C4#48018yes—
441C2×C4#48318yes—
442C2×C4C2×#11712yes—
443C2×C4C2×#12312yes—
444C2×C4C2×#12312yes—
445C2×C4C2×#13112yes—
446C2×C4#56812yes—
447C2×C4C22×D811yescharacteristic
448C2×C4C22×SD1611yescharacteristic
449C2×C4C2×#13712yes—
450C8—84no—
451C8—84no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—416no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22#148014yes—
473C22#182314yes—
474C22C2×#38111yescharacteristic
475C22C2×#40811yescharacteristic
476C22#287911yescharacteristic
477C22C22×#10211yescharacteristic
478C22C2×#41911yescharacteristic
479C22C2×#42111yescharacteristic
480C22C2×#42111yescharacteristic
481C4—22no—
482C4—22no—
483C4—22no—
484C4—22no—
485C4—22no—
486C4—22no—
487C4—22no—
488C4—41no—
489C4—41no—
490C4—84no—
491C4—84no—
492C4#182314yes—
493C4#292414yes—
494C4C2×#48012yes—
495C4C2×#48312yes—
496C2—24no—
497C2—24no—
498C2—42no—
499C2C2×#148011yescharacteristic
500C2#1746411yescharacteristic
501C2#1748511yescharacteristic
502C2#1749611yescharacteristic
503C2#2068511yescharacteristic
504C2C2×#182311yescharacteristic
505C2#2070511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8709C2semidirect
5. #8706C2semidirect
6. #7893C2semidirect
38. #1279C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22852
#207142
C2×#17972
#87092
#87062
#78932