Finite Group Databasebeta

← #190252

Subgroups — #190252

Order 512.

1087 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34126C211yesmaximal, characteristic
2#20726C211yesmaximal, characteristic
3C2×#1460C211yesmaximal, characteristic
4#8771C212yesmaximal, complemented
5#8770C212yesmaximal, complemented
6#8765C212yesmaximal, complemented
7#8764C212yesmaximal, complemented
8#8092C212yesmaximal, complemented
9#7958C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11#2920C2212yes—
12C2×#431C2212yes—
13#2357C2211yescharacteristic
14#2357C2211yescharacteristic
15#2353C412yes—
16#2353C412yes—
17#2349C2211yescharacteristic
18#2349C2211yescharacteristic
19#2345—21no—
20#2342—22no—
21D4×C22⋊C4—21no—
22#2293—21no—
23#2290—22no—
24#2283C2211yescharacteristic
25#2280C412yes—
26C42×D4—21no—
27C4×#117C2211yescharacteristic
28C2×C4×C22⋊C4C412yes—
29#1821—21no—
30#1800C2212yes—
31#1800C2212yes—
32#1783—21no—
33C4×#100—21no—
34C4×#100—21no—
35#1460—22no—
36#1460C2212yes—
37#1460C2212yes—
38C2×#323—22no—
39#1303C2214yescomplemented
40#1303C2214yescomplemented
41#1286C2214yescomplemented
42#1286C2214yescomplemented
43C23×D4D411yescharacteristic
44C24×C4C2×C412yes—
45C2×C4×D4—21no—
46C2×C4×D4—21no—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4C2×C412yes—
52C2×C4×D4D411yescharacteristic
53C2×C4×D4D411yescharacteristic
54C2×C4×D4D412yes—
55C2×C4×D4C2311yescharacteristic
56C2×C4×D4C2311yescharacteristic
57C2×#117—21no—
58C2×#117—22no—
59C22×C4⋊C4D411yescharacteristic
60C22×C22⋊C4—22no—
61C22×C22⋊C4D412yes—
62C22×C42—22no—
63C22×C42C2311yescharacteristic
64#432—22no—
65#432—22no—
66#431—22no—
67#431—22no—
68#431—22no—
69#431—22no—
70#431—22no—
71#431C2314yes—
72#423C2314yes—
73C2×C4⋊C8—22no—
74C2×#100—22no—
75C2×#100—22no—
76#407—22no—
77C2×C22⋊C8—22no—
78C4×M16—22no—
79C2×C4×C8—22no—
80#390—21no—
81#390—21no—
82#389—21no—
83#389—21no—
84#388—22no—
85#388—22no—
86#388—22no—
87#388—22no—
88#387—22no—
89#387—22no—
90#386—22no—
91#386—22no—
92#385—22no—
93#385—22no—
94#383C2×C412yes—
95#382C2311yescharacteristic
96#382C2311yescharacteristic
97#381—21no—
98#381C2×C412yes—
99#381C2×C412yes—
100#380—22no—
101#380—21no—
102#380C2×C412yes—
103#380C2×C412yes—
104#379—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4C2311yescharacteristic
110C4×C4⋊C4C2311yescharacteristic
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—21no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4C2×C412yes—
118C4×C22⋊C4C2×C412yes—
119C4×C22⋊C4C2×C411yescharacteristic
120C4×C22⋊C4C2×C412yes—
121C4×C22⋊C4C2×C411yescharacteristic
122#376—21no—
123#376C2×C411yescharacteristic
124#376C2×C411yescharacteristic
125C2×#95C2×C412yes—
126C2×#95C2×C412yes—
127C43—21no—
128#343—22no—
129#343—22no—
130#323—24no—
131#323—24no—
132C25C22⋊C412yes—
133C22×D4—24no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—42no—
137C22×D4C22⋊C412yes—
138C22×D4C2×D411yescharacteristic
139C22×D4C2×D411yescharacteristic
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—84no—
147C23×C4C22⋊C412yes—
148C23×C4C22⋊C412yes—
149C23×C4C22⋊C412yes—
150C23×C4C22×C412yes—
151C23×C4C22×C412yes—
152C23×C4C22×C411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C2×M16—24no—
156C22×C8—42no—
157C4×D4—24no—
158C4×D4—24no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—44no—
170C4×D4—44no—
171#117—44no—
172#117—84no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C22⋊C4—21no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—48no—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C2×D412yes—
199C2×C22⋊C4C2×D412yes—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—84no—
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42#4112yes—
214C2×C42#4112yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217C2×C42C2411yescharacteristic
218C4⋊C8—24no—
219C4⋊C8—24no—
220C4⋊C8—24no—
221C4⋊C8—42no—
222#100—44no—
223#100—44no—
224C22⋊C8—24no—
225C22⋊C8—42no—
226C22⋊C8—42no—
227C22⋊C8—42no—
228#97—42no—
229C4×C8—24no—
230C4×C8—42no—
231C4×C8—42no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—21no—
236#95—21no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—21no—
241#95—21no—
242#95C22×C411yescharacteristic
243#95C22×C411yescharacteristic
244#95C22×C412yes—
245#95C22×C412yes—
246#95C22×C411yescharacteristic
247#95C22×C411yescharacteristic
248C24—22no—
249C24—22no—
250C24—24no—
251C24—42no—
252C24—44no—
253C24—82no—
254C24C2×C22⋊C411yescharacteristic
255C24C2×C22⋊C412yes—
256C24C2×C22⋊C412yes—
257C2×D4—24no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—24no—
264C2×D4—48no—
265C2×D4—48no—
266C2×D4—42no—
267C2×D4#13514yes—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—84no—
295C22×C4—84no—
296C22×C4—84no—
297C22×C4—84no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4#11711yescharacteristic
309C22×C4#11711yescharacteristic
310C22×C4#11711yescharacteristic
311C22×C4#11711yescharacteristic
312C22×C4#12312yes—
313C22×C4#12312yes—
314C22×C4#12312yes—
315C22×C4#12312yes—
316C22×C4C23×C411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319M16—44no—
320M16—44no—
321C2×C8—44no—
322C2×C8—44no—
323C2×C8—84no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—42no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C42—42no—
347C42—44no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—82no—
354C42—82no—
355C42#11712yes—
356C42#11712yes—
357C42#12312yes—
358C42#12312yes—
359C42#12312yes—
360C42#12312yes—
361C42C2×#4112yes—
362C42C2×#4112yes—
363C23—22no—
364C23—21no—
365C23—22no—
366C23—21no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—41no—
372C23—44no—
373C23—44no—
374C23—42no—
375C23—42no—
376C23—44no—
377C23—42no—
378C23—42no—
379C23—44no—
380C23—82no—
381C23—82no—
382C23—88no—
383C23—88no—
384C23#38111yescharacteristic
385C23#38111yescharacteristic
386C23#41614yes—
387C23C22×C22⋊C411yescharacteristic
388D4—28no—
389D4—48no—
390D4—44no—
391C2×C4—24no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—88no—
438C2×C4—88no—
439C2×C4#38112yes—
440C2×C4#38112yes—
441C2×C4#38111yescharacteristic
442C2×C4#38111yescharacteristic
443C2×C4C2×#11711yescharacteristic
444C2×C4C2×#11711yescharacteristic
445C2×C4C2×#12312yes—
446C2×C4C2×#12312yes—
447C2×C4C2×#13512yes—
448C2×C4#57512yes—
449C8—84no—
450C8—84no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—24no—
457C22—21no—
458C22—21no—
459C22—42no—
460C22—42no—
461C22—44no—
462C22—44no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—88no—
469C22—82no—
470C22—82no—
471C22—88no—
472C22#182712yes—
473C22C2×#38111yescharacteristic
474C22C2×#41611yescharacteristic
475C22#289611yescharacteristic
476C4—22no—
477C4—22no—
478C4—22no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—81no—
487C4—81no—
488C4—82no—
489C4—82no—
490C4#182912yes—
491C4#309114yes—
492C2—21no—
493C2—21no—
494C2—22no—
495C2—41no—
496C2—44no—
497C2—82no—
498C2#2069911yescharacteristic
499C2C2×#182711yescharacteristic
500C2C2×#182911yescharacteristic
subgroupG/H
centerC2×C4#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#381
prime radicalC23×D4D4
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
4. #8771C2semidirect
5. #8770C2semidirect
6. #8765C2semidirect
7. #8764C2semidirect
8. #8092C2semidirect
9. #7958C2semidirect
39. #1303C22semidirect
40. #1303C22semidirect
41. #1286C22semidirect
42. #1286C22semidirect

Maximal subgroups

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

subgroupindex
#341262
#207262
C2×#14602
#87712
#87702
#87652
#87642
#80922
#79582