Finite Group Databasebeta

← #190266

Subgroups — #190266

Order 512.

1030 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34190C211yesmaximal, characteristic
2#20726C211yesmaximal, characteristic
3C2×#1460C211yesmaximal, characteristic
4#8771C212yesmaximal, complemented
5#8770C212yesmaximal, complemented
6#8765C212yesmaximal, complemented
7#8764C212yesmaximal, complemented
8#8096C212yesmaximal, complemented
9#7956C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11#2920C2212yes—
12C2×#431C2212yes—
13#2357C2211yescharacteristic
14#2357C2211yescharacteristic
15#2353C412yes—
16#2353C412yes—
17#2349C2211yescharacteristic
18#2349C2211yescharacteristic
19#2344—21no—
20#2342—22no—
21D4×C22⋊C4—21no—
22#2319—21no—
23#2318—22no—
24#2283C2211yescharacteristic
25#2283C2211yescharacteristic
26#2280C412yes—
27C2×#381C412yes—
28#2275—21no—
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—22no—
58C22×C4⋊C4D411yescharacteristic
59C22×C22⋊C4—22no—
60C22×C22⋊C4D412yes—
61C22×C42C2311yescharacteristic
62#432—22no—
63#432—22no—
64#431—22no—
65#431—22no—
66#431—22no—
67#431—22no—
68#431—22no—
69#431C2314yes—
70#423C2314yes—
71C2×C4⋊C8—22no—
72C2×#100—22no—
73C2×#100—22no—
74#407—22no—
75C2×C22⋊C8—22no—
76C4×M16—22no—
77C2×C4×C8—22no—
78#390—21no—
79#390—21no—
80#389—21no—
81#389—21no—
82#388—22no—
83#388—22no—
84#388—22no—
85#388—22no—
86#387—22no—
87#387—22no—
88#386—22no—
89#386—22no—
90#385—22no—
91#385—22no—
92#383—21no—
93#383C2×C411yescharacteristic
94#383C2×C411yescharacteristic
95#382—21no—
96#382C2311yescharacteristic
97#382C2311yescharacteristic
98#381—22no—
99#381—22no—
100#381C2×C412yes—
101#381C2×C412yes—
102#381C2×C411yescharacteristic
103#381C2×C412yes—
104#381C2×C411yescharacteristic
105#380—22no—
106#380—21no—
107#380—21no—
108#380C2×C412yes—
109#380C2×C412yes—
110#379—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2311yescharacteristic
115C4×C4⋊C4C2311yescharacteristic
116C4×C22⋊C4—22no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4C2×C412yes—
122C4×C22⋊C4C2×C412yes—
123#376—21no—
124#376C2×C412yes—
125C2×#95—22no—
126C2×#95C2×C412yes—
127C2×#95C2×C412yes—
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—24no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—24no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—44no—
170C4×D4—44no—
171#117—84no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—84no—
195C2×C22⋊C4C22⋊C412yes—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C2×D412yes—
198C2×C22⋊C4C2×D412yes—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42#4112yes—
208C2×C42#4112yes—
209C2×C42#4112yes—
210C2×C42#4112yes—
211C2×C42C2411yescharacteristic
212C4⋊C8—24no—
213C4⋊C8—24no—
214C4⋊C8—24no—
215C4⋊C8—42no—
216#100—44no—
217#100—44no—
218C22⋊C8—24no—
219C22⋊C8—42no—
220C22⋊C8—42no—
221C22⋊C8—42no—
222#97—42no—
223C4×C8—24no—
224C4×C8—42no—
225C4×C8—42no—
226#95—21no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—22no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95C22×C411yescharacteristic
241#95C22×C411yescharacteristic
242#95C22×C412yes—
243#95C22×C412yes—
244#95C22×C411yescharacteristic
245#95C22×C411yescharacteristic
246C24—22no—
247C24—22no—
248C24—24no—
249C24—42no—
250C24—44no—
251C24—82no—
252C24C2×C22⋊C411yescharacteristic
253C24C2×C22⋊C412yes—
254C24C2×C22⋊C412yes—
255C2×D4—24no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—24no—
262C2×D4—48no—
263C2×D4—48no—
264C2×D4—42no—
265C2×D4#13514yes—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—84no—
289C22×C4—84no—
290C22×C4—81no—
291C22×C4—84no—
292C22×C4—84no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4#11711yescharacteristic
307C22×C4#11711yescharacteristic
308C22×C4#11711yescharacteristic
309C22×C4#11711yescharacteristic
310C22×C4#12312yes—
311C22×C4#12312yes—
312C22×C4#12312yes—
313C22×C4#12312yes—
314C22×C4C23×C411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317M16—44no—
318M16—44no—
319C2×C8—44no—
320C2×C8—44no—
321C2×C8—84no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—81no—
328C4⋊C4—81no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—42no—
336C22⋊C4—44no—
337C22⋊C4—42no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C42—42no—
345C42—42no—
346C42—82no—
347C42—82no—
348C42—82no—
349C42#11712yes—
350C42#11712yes—
351C42#12312yes—
352C42#12312yes—
353C42#12312yes—
354C42#12312yes—
355C42C2×#4112yes—
356C42C2×#4112yes—
357C23—22no—
358C23—21no—
359C23—22no—
360C23—21no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—41no—
366C23—44no—
367C23—44no—
368C23—42no—
369C23—42no—
370C23—44no—
371C23—42no—
372C23—42no—
373C23—44no—
374C23—82no—
375C23—82no—
376C23—88no—
377C23—88no—
378C23#38111yescharacteristic
379C23#38111yescharacteristic
380C23#41614yes—
381C23C22×C22⋊C411yescharacteristic
382D4—28no—
383D4—48no—
384D4—44no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—164no—
432C2×C4—164no—
433C2×C4#38112yes—
434C2×C4#38112yes—
435C2×C4#38111yescharacteristic
436C2×C4#38111yescharacteristic
437C2×C4C2×#11711yescharacteristic
438C2×C4C2×#11711yescharacteristic
439C2×C4C2×#12312yes—
440C2×C4C2×#12312yes—
441C2×C4C2×#13512yes—
442C2×C4#57512yes—
443C8—84no—
444C8—84no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—24no—
451C22—21no—
452C22—21no—
453C22—42no—
454C22—42no—
455C22—44no—
456C22—44no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—88no—
463C22—82no—
464C22—82no—
465C22—88no—
466C22#182912yes—
467C22C2×#38111yescharacteristic
468C22C2×#41611yescharacteristic
469C22#289611yescharacteristic
470C4—22no—
471C4—22no—
472C4—22no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—42no—
478C4—42no—
479C4—81no—
480C4—81no—
481C4—82no—
482C4—82no—
483C4—81no—
484C4—81no—
485C4#309114yes—
486C2—21no—
487C2—21no—
488C2—22no—
489C2—41no—
490C2—44no—
491C2—82no—
492C2#2069911yescharacteristic
493C2#2070811yescharacteristic
494C2C2×#182911yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ 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. #8096C2semidirect
9. #7956C2semidirect
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
#341902
#207262
C2×#14602
#87712
#87702
#87652
#87642
#80962
#79562